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abstract: |
Preliminary first-principles calculations on the magnetic behavior of ultra-thin epitaxial multilayers between the BiFeO$_3$ magnetoelectric compound and various types of spacers are presented. As spacer we have considered i) InP semiconductor, ii) Fe which is a ferromagnet, and iii) metallic V. In all cases under study the growth axis of the multilayer was the \[001\]. Our results indicate that the magnetic properties are seriously downgraded for the ultrathin BiFeO$_3$ multilayers independent of the nature of the spacer and in some cases under study magnetism even vanishes. More extensive calculations are needed to establish a more clear view of the physical properties of the interfaces involving the BiFeO$_3$ compound. The present manuscript completes the study presented in two recent research articles \[K. Koumpouras and I. Galanakis, *J. Magn. Magn. Mater.* 323, 2328 (2011); *ibid*, *J. Spintron. Magn. Nanomater.* 1, in press (2012)\].\
**Keywords:** Electronic Structure Calculations, Magnetism, Multiferroics, DFT, Ferrites
author:
- 'K. Koumpouras'
- 'I. Galanakis'
title: 'Suppression of magnetism in BiFeO$_3$ ultrathin epitaxial multilayers'
---
Introduction
============
Among the most interesting new classes of materials under intense investigation in Spintronics[@Zutic; @Felser; @Zabel] are the so-called multiferroics which combine several ferroic orders like ferromagnetism, ferroelectricity, ferroelasticity etc.[@Review; @Review5; @Review6] The compounds combining the electric and magnetic order[@Review] have several potential applications like magnetic-field sensors and electric-write magnetic-read random-access memories.[@Hetero; @General1; @Zhang; @Review7; @Review8] Magnetic order and ferroelectricity have different origins[@Picozzi; @Picozzi2; @General2] and thus the materials exhibiting the magnetoelectric effect are few and the coupling between the magnetic and electric properties is weak. An alternative route to achieve a strong coupling could be the growth of thin film heterostructures and several advances have been made towards the magnetic control of ferroelectricity[@General3; @General4; @Review2] and the electric control of thin film magnetism.[@Films3; @Review3; @Review4]
Bismuth ferrite is probably the most studied representative of magnetoelectric materials. Bulk BiFeO$_3$ crystallizes in a perovskite-like pseudocubic structure instead of a ferrite one and is classified as a ferroelectric G-type antiferromagnet.[@Kiselev] Several first-principles calculations have been carried out to study the properties of bulk BiFeO$_3$[@BiFeO3-Calc] and we refer readers to Ref. for an overview of the literature on this compounds. Since the single-component crystals like BiFeO$_3$ present only a weak magnetoelectric effect, an alternative route to achieve a more strong effect has been proposed to be the growth of heterostructures where epitaxial strain can enhance the phenomenon.[@Films1; @Trilayers; @Kovachev; @Yamauchi] Thus first-principles calculations of such heterostructures involving alternating layers of BiFeO$_3$ and various spacers can serve as a test-ground to study the behavior of electric and magnetic properties of films. The study of the latter in the case of ultrathin epitaxial films is the aim of the present manuscript.
In a recent publication (Ref. ) we have presented extended first-principles calculations, employing the Quantum-ESPRESSO[@QE] ab-initio electronic structure method in conjunction with the Generalized-Gradient Approximation (GGA) in the Perdew-Burke-Erzenhof formulation,[@GGA] on the electronic and magnetic properties of BiFeO$_3$ alloy as a function of the lattice constant in the case of the cubic perovskite structure (see figure 1 in Ref. ). In Ref. we have extended this study to cover also the case of Mn substation for Fe. All compounds under study in these two references exhibited significant magnetic properties with spin magnetic moments at the Fe and Mn sites of the order of 3 $\mu_B$. In the present contribution we expand these two studies using the same ab-initio method with grids of the same density in the **k**-space to cover also the case of ultrathin epitaxial films. Although these films are ideal cases and cannot be, with few exceptions, be realized experimentally, they can serve as starting point to understand the interplay between the interface structure and the magnetic properties of BiFeO$_3$ films with other spacers. For completeness we took into account three different spacers to cover different cases of electronic properties at the interface: InP semiconductor, metallic V and ferromagnetic Fe. In Section II we present the structure of the multilayers and the results for the case of the InP spacer, in section III the cases of Fe and V metallic spacers and finally in Section IV we present the main conclusions of our work.
IP spacer
=========
Structure of the multilayer
---------------------------
Along the \[001\] direction BiFeO$_3$ can be considered to consist of alternating layers of BiO and FeO$_2$ as shown in Fig. \[fig1\]. The in-plane unit cell is a square of the same lattice constant as the cubic unit cell of the bulk BiFeO$_3$. InP crystallizes in the zincblende structure, similar to GaAs, as most of the III-V semiconductors, and has an experimental lattice constants of 5.87 Å. Along the \[001\] direction, the zincblende structure can be viewed as consisting of alternating pure In and pure P atomic layers as shown in Fig. \[fig2\]. To fully describe the zincblende structure we have to consider, except the In and P atoms, also the occurrence of vacant sites (voids); there are exactly two no-equivalent voids within the zincblende unit cell. As shown in Fig. \[fig2\] along the \[001\] direction there are not two but four alternating non-equivalent atomic layers. The difference between the A(B) and C(D) layers is that the In(P) atoms have exchanged sites with the voids. This exchange of atomic positions is important since it lead to four non-equivalent interfaces between InP and BiFeO$_3$ in our study. Finally we have to mention that the lattice constant of InP coincides with the BiFeO$_3$ lattice constant multiplied by the square root of two: $a_{InP}=\sqrt{2} \times a_{BiFeO_3}$. Thus as shown in Fig. \[fig1\] the diagonal of the BiFeO$_3$ two-dimensional square unit cell can be assumed to be the side of the corresponding unit cell of the InP (denoted with red color in the figure) and thus epitaxial growth between InP and BiFeO$_3$ can be assumed.
![Schematic representation of the structure of the BiO (a) and FeO$_2$ (b) possible terminations of the BiFeO$_3$ alloy along the \[001\] direction. With the red lines we denote the limits of the two-dimensional unit cell of InP (see Fig. \[fig2\]) which has a lattice constant: $a_{InP}=\sqrt{2}\times a_{BiFeO_3}$. []{data-label="fig1"}](fig1.eps){width="\columnwidth"}
For the InP/BiFeO$_3$ multilayers we studied four different cases with respect to the relative position of the atoms at the interface between the two spacers. Along the growth axis we took into account eight atomics layers which are repeated in the \[001\] direction. Initially we studied the structure ...FeO$_2$//In/P/In/P/BiO/FeO$_2$/BiO/FeO$_2$//In..., and thus in the first case we have two non-equivalent interface in our structure: a P/Bi and an In/Fe ones. A close examination of Figs. \[fig1\] and \[fig2\] reveals that, in the case just mentioned :(a) with respect to the P/Bi interface P atoms are situated in the diagonal connecting Bi atoms (compare (a) in Fig. \[fig1\] with the D layer in Fig. \[fig2\]), and (2) with respect to the In/Fe interface In atoms are situated in the midpoint between Fe nearest neighbors (take into account A layer in Fig. \[fig2\] and combine it with the (b) in Fig. \[fig1\]).
![Structure of the atomic layers of InP along the \[001\] direction taking into account also the vacant sites. The structure is built using four alternating pure atomic layers A, B, C and D. The difference between A(B) and C(D) is that the In(P) atoms and the voids have exchanged sites.[]{data-label="fig2"}](fig2.eps){width="\columnwidth"}
In the second case under study In and P atoms have exchanged sites with respect to the first case under study and thus now we have along the \[001\] direction a ...FeO$_2$//P/In/P/In/BiO/FeO$_2$/BiO/FeO$_2$//P... structure, *i.e.* the inequivalent interfaces are now In/Bi and Fe/P. Notice that as in the Fe/In interface in case 1, also in the P/Fe interface, which occurs in case 2, the P atoms at the interface are located in the midpoints between the Fe atoms (if we examine only the in-plane projection of the multilayer) just above the oxygen atoms and this results to vanishing magnetism in case 2 under study as will be discussed later in the next subsection.
As we mentioned above the structure of InP along the \[001\] direction should be viewed as consisting of four atomic layers (see Fig. \[fig2\]) since voids play a crucial role in interfaces. If in the two previous cases 1 and 2 we exchange layer A(B) with layer C(D) in Fig. \[fig2\] we get two new cases 3 and 4 with different arrangement of the atoms at the interface. In case 3(4), the succession of the atomic layers is similar to case 1(2). When we compare case 1(2) with case 3(4), we conclude that the P(In)/Bi interfaces are similar since the P(In) atoms are now situated at the other diagonal connecting the Bi atoms. Critical is the other In(P)/FeO$_2$ interface since now in case 3(4) the In(P) atoms are located just above the Fe atoms while in case 1(2) they were located above the oxygen atoms at the midpoints between neighboring Fe atoms.
Results and discussion
----------------------
In all cases under study our results converged to the ferromagnetic solution independently of the initial conditions and initial arrangement of the spin moments. As in-plane lattice constant we have chosen the $\sqrt(2) \times 4.153$ Å $=5.874$ Å (11.1 a.u.) which is the experimental lattice constant of InP and moreover for the lattice constant of 4.153 Å BiFeO$_3$ exhibits very pronounced magnetic properties.[@Koumpouras] As a result of the epitaxial growth along the \[001\] axis the lattice constant was 14.182 Å (26.8 a.u.).
![Density of states (DOS) projected on the Fe d-orbitals for the Fe atoms in the BiFeO$_3$ spacer (Fe$^{bulk}$ at the middle of the spacer and Fe$^{int}$ at the FeO$_2$ interface for all four cases under study (see text for explanation). Positive DOS corresponds to the majority spin (spin-up) electrons and negative DOS to the minority spin (spin-down) electrons. The Fermi level has been assigned to the zero energy. []{data-label="fig3"}](fig3.eps){width="\columnwidth"}
In Fig. \[fig3\] we present the density of states (DOS) of the Fe d-orbitals for all four cases under study. We denote as Fe$^{bulk}$ the Fe atoms within the BiFeO$_3$ spacer and with Fe$^{int}$ the Fe atoms located at the interface. We have also to note here that for the multilayer structures under study, we have a complete lift of the degeneracy of the Fe d-orbitals and we cannot refer anymore to the double-degenerated $e_g$ and triple degenerated $t_{2g}$ orbitals as in bulk BiFeO$_3$. In case 2 where we have a P/Bi interface our calculations have converged to a non-magnetic solution. The loss of magnetism in this case should be attributed to the reduced hybridization between the neighboring Fe and P atoms at the interface. The latter have as valence 3p electrons which are less extended in space with respect to the In 5p valence states and do not hybridize with the Fe d-orbitals at the interface. In case I, the d-band is shifted to higher energy values for the spin-down electrons with respect to the spin-up electrons. This is more easy to visualize for the Fe$^{int}$ atoms since their hybridization with the p-orbitals of In leads to more pronounced electronic properties (smaller bandwidth with respect to the Fe$^{bulk}$ atoms and thus more intense picks). The latter is also reflected on the spin magnetic moments presented in Table \[table1\] where the spin magnetic moments are considerable larger for Fe$^{int}$ with respect to Fe$^{bulk}$.
-------- ------------- ------------ -- -- -- -- -- -- -- -- -- --
Fe$^{bulk}$ Fe$^{int}$
Case 1 0.357 1.158
Case 2 0 0
Case 3 0.247 0.746
Case 4 0.296 0.919
-------- ------------- ------------ -- -- -- -- -- -- -- -- -- --
: Fe spin magnetic moments in $\mu_B$ within the BiFeO$_3$ spacer for the InP/BiFeO$_3$ epitaxial ultrathin multilayers for all four cases under study (see text for explanation of different cases). The BiFeO$_3$ spacer contains two inequivalent atoms: Fe$^{bulk}$ at the middle of the BiFeO$_3$ spacer and Fe$^{int}$ at the interface.[]{data-label="table1"}
In Fig. \[fig3\] we also present the DOS of the Fe atoms for the cases 3 and 4 where we have changed the positions of the In(P) atoms at the interface. More precisely, as we discussed in the previous section, the In(P) atoms at the interface with FeO are not any more situated above the Fe atoms instead of the O atoms in cases 1 and 2. This leads to increased hybridization also in case 4, where we have a P/Fe interface and now the multilayer converges to a magnetic solution contrary to case 2 where we had a non-magnetic configuration. If we compare cases 1 and 3, where the Fe/In contact appears, we can notice that in case 3 the splitting of the Fe$^{int}$ d-bands is smaller due to the alteration in the Fe 3d-In 5p orbitals hybridization resulting also in smaller spin magnetic moments as shown in Table \[table1\]. But overall the obtained DOS are similar to the ones obtained for case 1 and presented also in Fig. \[fig3\].
V and F spacers
===============
We continue our study using two metallic spacers, ferromagnetic Fe and normal metallic V, instead of InP. Both Fe and V crystallize in the bcc structure. The zincblende structure of InP, if we take into account the vacant sites and ignore the different chemical species, is in reality a bcc one. Thus the structure of the Fe/BiFeO$_3$ and V/BiFeO$_3$ multilayers is similar to the structure of the InP/BiFeO$_3$ where all sites are occupied exclusively by Fe or V atoms. We have studied 5 cases for both multilayers where we have just varied the lattice constant. Case 1 corresponds to the same lattice constant as for InP/BiFeO$_3$ studied in the previous section. Cases 2 and 3 correspond to a uniform compression of the lattice parameter in all directions of 10 and 5 % , respectively, and cases 4 and 5 of a uniform expansion of the lattice parameter of 5 and 10 % , respectively. Our aim is to study the behavior of the magnetic properties upon hyrdrostatic pressure. In cases 1, 4 and 5 we converged to a ferromagnetic solution irrespectively of the initial arrangement of the spin magnetic moments. In contrast in the compressed cases 2 and 3 for Fe/BiFeO$_3$ we could not get convergence while for the case of the V spacer we converged to a non magnetic solution. This behavior stems from the reduced volume around the Fe atoms which leaded to a large compression of the Fe d-orbitals and suppression of magnetism as expected by the magnetovolume effect which has been extensively studied for transition metal atoms. In the multilayers under study we have also two inequivalent interfaces: a V(Fe)/BiO and a V(Fe)/FeO$_2$ contact. In the first interface the four V(Fe) atoms are located within the two diagonals connecting the Bi atoms as shown in Fig. \[fig4\]a while in the second interface they are located just above the Fe atoms of FeO$_2$ as shown in Fig. \[fig4\]b.
![Schematic representations of the V/BiO (a) and V/FeO$_2$ interfaces in the case of the V/BiFeO$_3$ multilayer. For the Fe/BiFeO$_3$ multilayer under study Fe atoms simply substitute the V atoms since both V and Fe have a bcc structure. []{data-label="fig4"}](fig4.eps){width="\columnwidth"}
![DOS projected on the Fe d-orbitals for the Fe atoms within the BiFeO$_3$ spacer for both V/BiFeO$_3$ and Fe/BiFeO$_3$ multilayers. Results are for the two expanded lattice parameters denoted as cases 4 and 5 (see text for details). Details as in Fig. \[fig3\]. []{data-label="fig5"}](fig5.eps){width="\columnwidth"}
We will start our discussion concentrating on the Fe atoms within the BiFeO$_3$ spacer in the case of the Fe/BiFeO$_3$ multilayer and in Table \[table2\] we have gathered the Fe spin magnetic moments for both Fe atoms in the inside of the BiFeO$_3$ spacer (Fe$^{bulk}$) and at the interface (Fe$^{int}$). As mentioned above for the compressed cases 2 and 3 we were not able to converge to a solution. For case 1 we converged to a non-magnetic solution and for the more expanded lattice parameters (cases 4 and 5) we converged to a magnetic solution. The spin magnetic moments are larger for the more expanded case 5. This behavior is expected by the well studied magnetovolume effect. For the late transition metal atoms like Fe, the spin-splitting of the d-states increases with the atomic volume. Thus as we expand the lattice the tendency to magnetism increases leading to larger spin moments while compression of the lattice eventually leads to loss of magnetism. A similar situation occurs also for the case of V/BiFeO$_3$ but now the tendency to magnetism is stronger for the same lattice parameter. For cases 2 and 3 we converged to a non-magnetic solution while we got a ferromagnetic configuration for case 1 contrary to the Fe/BiFeO$_3$ multilayer. Fe spin magnetic moments are larger for the V spacer but the largest calculated value, which we got for Fe$^{int}$ in case 5 as shown in Table \[table2\], is 1.3 $\mu_B$ almost half the value in pure bulk Fe. Although magnetism is present for the cases with the more expanded lattice constant the magnetic properties are seriously downgraded with respect to pure BiFeO$_3$ bulk crystals.
-------- ------------- ------------ ------------- ------------ -- -- -- -- -- -- -- --
Fe$^{Bulk}$ Fe$^{int}$ Fe$^{Bulk}$ Fe$^{int}$
Case 1 0.146 0.777 0 0
Case 2 0 0 – –
Case 3 0 0 – –
Case 4 0.269 1.252 0.224 0.620
Case 5 0.325 1.301 0.339 0.815
-------- ------------- ------------ ------------- ------------ -- -- -- -- -- -- -- --
: Fe spin magnetic moments in $\mu_B$ within the BiFeO$_3$ spacer for the V(Fe)/BiFeO$_3$ epitaxial ultrathin multilayers for all cases under study. Case 1 corresponds to a BiFeO$_3$ in-plane lattice constant of 4.153 Å, cases 2 and 3 to uniform compression by 10 and 5 % respectively, and cases 4 and 5 uniform expansion by 5 and 10 % respectively. Zero values means that we have converged to a non-magnetic solution and “–” that we were not able to achieve convergence. Details as in Table \[table1\].[]{data-label="table2"}
In the case 1 of Fe/BiFeO$_3$, presented in the lower panel of Fig. \[fig7\], both Fe atoms within the BiFeO$_3$ spacer are non magnetic and the DOS is the same for both spin directions, while in the case of V/BiFeO$_3$ a small splitting of the d-bands appears in accordance to the spin magnetic moments presented in Table \[table2\]. This splitting increases as we move to case 4 and 5 (presented in Fig. \[fig5\]) following the increase of the spin magnetic moments. We can make two remarks with respect to the presented DOS. First, for the Fe atoms at the interface (Fe$^{int}$) the weight of the d-states is shifted close to the Fermi level with respect to the Fe$^{bulk}$ atoms as a consequence of the increased hybridization at the interface. Second, if we compare the behavior of the same Fe atom for the same lattice constant in the two multilayers under study, the exchange splitting of the d-bands is larger in the case of the V spacer (spin-up states are deeper in energy and spin-down states higher in energy) in accordance with the larger spin magnetic moments in this case.
Effect of the change of V(Fe) positions at the interface
--------------------------------------------------------
![Same as Fig. \[fig4\] but now we have exchanged the V atomic layers. In the left (a) panel the V/BiO interface and in the right panel (b) the V/FeO$_2$ interface. Notice that with respect to Fig. \[fig4\], the V atoms are located now exactly at the top of the O and Bi atoms. We do not present the V layers and the BiO(FeO$_2$) layers in the same square as in Fig. \[fig4\] to make the atomic positions more clear.[]{data-label="fig6"}](fig6.eps){width="\columnwidth"}
In this subsection we present results again for the V/BiFeO$_3$ and Fe/BiFeO$_3$ multilayers for the lattice constant of case 5 for which magnetism is more pronounced. And thus the in-plane lattice constant is 6.46 Å and the out of plane 15.61 Å . The difference with the multilayers studied just above is that we have changed the position of the V(Fe) atoms at the interface. As shown in Fig. \[fig6\] the V(Fe) atom at the interface with FeO are now located just above the oxygen atoms of the BiFeO$_3$ spacer and not the Fe ones. The change of the local environment leads to reduced hybridization of d-orbitals of the transition-metal atoms at the interface due to the larger Fe-V(Fe) distance and thus to smaller induced spin magnetic moments in the BiFeO$_3$ layer. For both V/ and Fe/BiFeO$_3$ multilayers the Fe$^{int}$ spin magnetic moment is 0.26 $\mu_B$, while the Fe$^{bulk}$ moment is 0.11 $\mu_B$ for the case of the V spacer and only 0.002 $mu_B$ for the case of the Fe spacer. Our discussion on the spin magnetic moments is reflected also on the DOS presented in the upper panel of Fig. \[fig7\] where the imbalance between the spin-up and spin-down states is very small. The only noticeable effect is the shift of the Fe$^{int}$ d-states between the two multilayers; for the V spacer the Fe$^{int}$ states are more concentrated around the Fermi level. The Fe$^{bulk}$ atoms are shielded from the interface due to the surrounding Bi and O atoms and the DOS is more similar for both type of V and Fe spacers as reflected also on the spin magnetic moments.
![DOS projected on the Fe d-orbitals for the Fe atoms within the BiFeO$_3$ spacer for both V/BiFeO$_3$ and Fe/BiFeO$_3$ multilayers. Results are for the lattice constant of 4.153 Å. In the bottom panel we present the results for the interface structure of Fig. \[fig4\] and in the upper panel for the interface structure in Fig. \[fig6\]. Details as in Fig. \[fig3\]. []{data-label="fig7"}](fig7.eps){width="\columnwidth"}
Behavior of V and F atoms within the V(F)spacer
-----------------------------------------------
In this section we will shortly refer to the magnetic properties of the V and Fe atoms within the transition metal spacers. There are four transition metal atoms within each atomic layer and we have 4 atomic layers in our spacer which we count starting from the interface with the FeO$_2$ (we denote it as layer 1) and end with the atomic layer at the interface with BiO. We will start our discussion for the V spacer in the case of the V/BiFeO$_3$ multilayer. V atoms present vanishing spin magnetic moments as shown in Table \[table3\] in all cases under study and only in the case 5 (largest lattice constant under study), do the V atoms in the layer close to the FeO$_2$ layer show a small spin magnetic moment of about 0.13 $\mu_B$.
--------- -------- -------- -------- ----------- -- -- -- -- -- -- -- -- -- --
case 1 case 4 case 5 case 5-II
Layer 1 0.016 -0.010 -0.003 -0.003
0.062 0.130 0.095 0.039
Layer 2 -0.005 0.015 0.016 0.007
Layer 3 -0.015 -0.003 -0.004 0.015
0.002 -0.006 -0.009 -0.002
Layer 4 -0.005 0.005 0.010 0.003
case 1 case 4 case 5 case 5-II
Layer 1 0 -0.026 -0.129 -0.110
-0.040 -0.174 0.479
Layer 2 0 0.017 0.209 -0.023
Layer 3 0 -0.005 -0.021 -0.022
-0.012 -0.202 0.058
Layer 4 0 0.015 0.116 -0.010
--------- -------- -------- -------- ----------- -- -- -- -- -- -- -- -- -- --
: V(Fe) spin magnetic moments in $\mu_B$ within the V(Fe) spacer for the V(Fe)/BiFeO$_3$ epitaxial ultrathin multilayers. We do not present results for the compressed cases 2 and 3 since the V/BiFeO$_3$ was found non-magnetic while for Fe/BiFeO$_3$ we could not get convergence. We have 4 atomic layers of V(Fe) spacer and each one contains four V(Fe) atoms as shown in Fig. \[fig4\]. We count the layers starting from the one at the V(Fe)/FeO$_2$ interface and finishing at the V(Fe)/BiO interface. With “case 5-II” we denote the structure presented in Fig. \[fig6\]. Notice that within each layer there are two inequivalent with respect to their magnetic properties Fe(V) atoms, *e.g* at the V/FeO$_2$ interface presented in Fig. \[fig4\] there are the V atoms at the diagonals located just above at the Fe atoms and the V atoms at the corners and the middle of the square unit cell. Near the BiO interface all V(Fe) atoms as shown in Fig. \[fig4\] have the same nearest neighbors environment and for all atoms in layer 4 close to the V(Fe)/BiO interface we got the same value of the spin magnetic moment for all V(Fe) atoms within the layer.[]{data-label="table3"}
Similar is the situation for the Fe spacer where for the two smaller lattice constants we could not even converge our calculations. As shown in Table \[table3\] only for the largest value of the lattice constant we got significant values of the Fe spin magnetic moments, which even for this case, are considerable smaller than the spin magnetic moments of Fe atoms of the BiFeO$_3$ spacer ($\sim$0.2 $\mu_B$ for Fe spacer compared to $\sim$ 0.8 $\mu_B$ for BiFeO$_3$ spacer) and are about one order of magnitude smaller than in bulk Fe.
We have also included in Table \[table3\] the results for the interface structure of Fig. \[fig6\], denoted as “case 5-II” where we have changed the positions of the V(Fe) atoms at the interface. Overall also in this case the magnetic properties of the spacer are not significant. Only in the case of the Fe/BiFeO$_3$ multilayer half the Fe atoms at layer 1 (located at the interface with FeO$_2$) present a significant spin magnetic moment of about 0.5 $\mu_B$ which is still much lower than the bulk Fe value.
Conclusions
===========
We expand our study on the magnetic properties BiFeO$_3$ presented in Refs. and to the case of ultrathin epitaxial multilayers using the Quantum-ESPRESSO first-principles electronic structure method.[@QE] We have studied several cases of these ultrathin epitaxial BiFeO$_3$ multilayers using different types of spacers covering a wide range of electronic materials: InP semiconductor, ferromagnetic Fe and metallic V. Irrespectively of the spacer the magnetic properties of the multilayers were found to seriously degrade due to the low-symmetry of the film and in some cases magnetism vanished completely. Thus ultrathin multilayers of BiFeO$_3$ are not suitable for spintronic applications. More extended calculations covering more thick multilayers and including quantum molecular dynamics are needed in the future to determine in more detail the properties of the interfaces of BiFeO$_3$ with other alloys.
Financial support from the University of Patras (K. Karatehodori 2008 program Nr. C588) is acknowledged.
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abstract: 'The influence of the electron-vibron coupling on the transport properties of a strongly interacting quantum dot built in a suspended carbon nanotube is analyzed. The latter is probed by a charged AFM tip scanned along [the]{} axis [of the CNT]{} which induces oscillations of the chemical potential and of the linear conductance. These oscillations are due to the competition between finite-size effects and the formation of a Wigner molecule for strong interactions. Such oscillations are shown to be suppressed by the electron-vibron coupling. The suppression is more pronounced in the regime of [weak]{} Coulomb interactions, which ensures that probing Wigner correlations in such a system is in principle possible.'
address:
- 'Dipartimento di Fisica, Università di Genova, Via Dodecaneso 33, 16146, Genova, Italy.'
- 'CNR-SPIN, Via Dodecaneso 33, 16146, Genova, Italy.'
author:
- 'N. Traverso [Ziani]{}$^{1,2}$, F. [Cavaliere]{}$^{1,2}$, M. [Sassetti]{}$^{1,2}$'
title: Probing Wigner correlations in a suspended carbon nanotube
---
When Coulomb interaction dominates over the kinetic energy of electronic systems, Wigner molecules of electrons can emerge [@vignale]. After more than 75 years [@wigner], however the details of the appearance of such a strongly correlated state of matter are only partially known: the theoretical modeling and the experimental realization of Wigner molecules are still challenging problems [@wigmol1; @wigmol2; @1Dwig], even though quantum dots [@koudots], allowed a breakthrough in the problem. As long as the theoretical modeling is concerned, the description of 2-dimensional (2D) Wigner molecules has relied mainly on numerical techniques [@2Dnum2; @Egger99; @2Dnum3; @2Dnum5; @2Dnum6; @2Dnum7; @serra; @2Dnum8; @2Dnum10], while in 1 dimension (1D), aside from numerical techniques [@kramer; @xia; @bortz; @pederiva; @bedu; @szafran; @wire3; @polini; @shulenburger; @secchi1; @sgm2; @astrak; @burke; @polini2; @silva], analytical methods can be employed [@giamarchi; @schulz; @mantelli; @safi; @sablikov; @fiete1; @075], mainly thanks to the Luttinger liquid theory [@giamarchi; @haldanefluid; @voit; @delft]. On the experimental side, in 2D optical spectroscopy has been employed [@wigmol1; @wigmol2]; in 1D transport techniques based on momentum resolved tunneling between quantum wires [@fili2; @wire1; @wire2], and on the magnetic properties of the Wigner molecule [@nature] have been used. Recently other experimental set ups for the detection of 1D Wigner molecules have been proposed [@sgm2; @AFM; @secchi2; @sgm1; @linear; @epl; @njp]: the transport properties of 1D quantum dots perturbed by local probes such as AFM and STM tips have been demonstrated to be effective in the detection of the Wigner molecule. [In]{} many of the experimental realizations[,]{} quantum dots are however surrounded by metallic gates which unavoidably screen the Coulomb interactions, demoting the formation of the molecule itself. A possible candidate in which screening effects can be dramatically reduced is a suspended carbon nanotube (CNT) [@nature; @secchiarxiv], a system in which rather strong Coulomb interactions can be attained. In such a system, however, the electronic degrees of freedom are strongly coupled to the vibrational ones as it occurs in nano-electro-mechanical systems (NEMS) [@braig; @nems1; @nems2; @4; @10; @15; @6]: is has indeed been demonstrated that suspended CNTs can behave as mechanical resonators [@resonator]. Such a source of fluctuations can potentially be detrimental to the formation of the Wigner molecule.\
In this work we investigate the transport properties of a one dimensional quantum dot, built in a suspended and interacting CNT, scanned by a negatively charged AFM tip free to move along the axis of the CNT. Employing the Luttinger liquid theory, we address both the renormalization induced on the chemical potential and on the linear conductance peak by the presence of the AFM tip. In the absence of electron vibron coupling these quantities have been demonstrated to be effective in the detection of the Wigner molecule. [We]{} show that care must be taken, since the coupling to a vibron indeed suppresses the oscillations induced by the tip. However, since electron-electron interaction reduces the effect of the electron-vibron coupling, we can conclude that a strongly interacting, suspended CNT may be an excellent candidate to investigate the physics of Wigner molecules.\
As a first step we briefly remind the low energy model for finite size interacting CNTs. It is well established [@egger] that the low energy properties of any finite interacting metallic (or semiconducting, if the Fermi point is not close to the charge neutrality point [@noi]) CNT can be described in terms of a four-channel Luttinger liquid. The Hamiltonian ($\hbar=1$) ${H}$ is ${H}=\sum_{j=1}^4{H}_{j}$, with [@yo; @grifoni] $$\label{eq:hambos}
{H}_{j}=\frac{1}{2}E_{j}\left({N}_{j}-\delta_{1,j} N_g\right)^{2}+\sum_{\mu\geq 1}\mu\omega_{j}{b}_{j,\mu}^{\dagger}{b}_{j,\mu}\, ,$$ where $E_j=\pi v_F/(4Lg_j^2)$, $v_F$ is the Fermi velocity and $g_j$ is the Luttinger liquid parameters of each channel, with $0<g_1\equiv g\leq 1$ and $g_2=g_3=g_4=1$. For a metallic CNT $v_{F}=8\cdot 10^5\ \mathrm{m/s}$ [@saito], while considerably lower velocities can be found in semiconducting CNTs [@noi]. Here, ${N}_j$ is the number of excess electrons in the channel $j$ with $N_1$ the total number of excess electrons, and $N_g\propto V_g$ is due to the inclusion of a gate contact capacitively coupled to the dot. Finally, the second term to the r.h.s. of Eq. (\[eq:hambos\]) describes collective excitations with ${b}_{j,\mu}$ bosonic operators and $\omega_j=\pi v_F/Lg_j$. The electron field operator reads $$\label{eq:fieldopop}
{\Psi}_{s}(\mathbf{r})=\sum_{r=\pm 1}\sum_{\alpha=\pm 1}f_{r,\alpha}(\mathbf{r})e^{irq_{\mathrm{F}}x}{\psi}_{+1,r\alpha,s}(rx),$$ where $$\begin{aligned}
{\psi}_{+1,\alpha,s}(x)&=&\frac{{\eta}_{\alpha,s}}{\sqrt{2\pi\tilde{a}}}e^{-i\theta_{\alpha,s}}e^{i\frac{\pi
x}{4L}\left({N}_{1}+\alpha
{N}_{2}+s{N}_{3}+\alpha s
{N}_{4}\right)}\cdot\nonumber\\ &&e^{\frac{i}{2}\left[{\phi}_{1}(x)+\alpha{\phi}_{2}(x)+s{\phi}_{3}(x)+\alpha
s {\phi}_{4}(x)\right]}\, ,\label{eq:fieldopdiag}\end{aligned}$$ and $$\begin{aligned}
\phi_{j}(x)&=&\sum_{\mu\geq 1}\left\{\frac{\cos{(q_{\mu}x)}}{\sqrt{\mu g_{j}}}\left[{b}_{j,\mu}+{b}_{j,\mu}^{\dagger}\right]\nonumber\right.\\
&+&\left.i\sqrt{\frac{g_{j}}{\mu}}\sin(q_{\mu}x)\left[{b}_{j,\mu}-{b}_{j,\mu}^{\dagger}\right]\right\}\, .\end{aligned}$$ One has $q_{\mu}=\pi\mu/L$, $[{\theta}_{\alpha,s},{N}_{\alpha',s'}]=i\delta_{s,s'}\delta_{\alpha,\alpha'}$, $N_{\alpha,s}={N}_{1}+\alpha
{N}_{2}+s{N}_{3}+\alpha s
{N}_{4}$ and $\tilde a\ll L$ a cutoff length. The functions $f_{r,\alpha}(\mathbf{r})$ consist of a superposition of wavefunctions for $p_{z}$ orbitals, peaked around the positions of atoms in the CNT [@saito]. Finally, $q_F$ is the distance in the reciprocal lattice between a Fermi point and the closest Dirac point, considering an effective n doping for the CNT [@noi]. Throughout this work, we will assume a reference state with $N_{0}=4\kappa$ (with $\kappa>0$ an integer) electrons, and $q_{F}=\pi N_{0}/4L$.\
When the CNT is suspended the interplay between the electronic and the mechanical degrees of freedom must be addressed. Among the possible quantized oscillation modes (vibrons) [@suzu; @mart; @penn; @mahan; @eros; @flens; @alves; @izu], the most relevant for the electronic properties considered in this work are the lowest stretching ones [@sapmaz]. We assume that the vibron extends over the whole length of the CNT, clamped at the CNT ends. The vibron Hamiltonian is $$\label{eq:vibh0}
{H}_{\mathrm v}=\frac{{P_{0}}^2}{2M}+\frac{M\omega_{0}^2}{2}{X}_{0}^2\, ,$$ where $\omega_{0}=\pi v_{\mathrm s}/{L}$ with $v_{\mathrm s}\approx 2.4\cdot 10^{4}$ m/s and $M=2\pi \mathcal{W}{L}\rho_{0}$ is the CNT mass, with $\mathcal{W}$ the radius and $\rho_{0}\approx6.7\cdot10^{-7}$ Kg/$\mathrm{m}^2$. Here, $\sqrt{2M}X_{0}=\left(b_{0}+b_{0}^{\dagger}\right)$ is the amplitude operator of the strain field in the fundamental mode ${u}(\mathbf{r})=\sqrt{2}{X_{0}}\sin\left(\frac{\pi x}{{L}}\right)$. In the elastic limit, the form of the Hamiltonian ${H}_{\mathrm{d-v}}$ coupling electronic and vibronic degrees of freedom is that of a deformation potential $H_{\mathrm{d-v}}=c\int\ \mathrm{d}x\ R(\mathbf{r})\partial_{x}u(x)$, where $R(\mathbf{r})=\sum_{s}\Psi^{\dagger}_{s}(\mathbf{r})\Psi_{s}(\mathbf{r})$ is the electron density and $c\approx 30\ \mathrm{eV}$.\
Neglecting rapidly oscillating terms with a wavelength $\approx a$ (with $a$ the CNT lattice constant), the electron vibron coupling simplifies as [@egger; @noi; @proc] $H_{\mathrm{d-v}}\sim c\int_{0}^{L}\ \mathrm{d}x\ \rho(x)\partial_{x}u(x)$. Keeping only the lowest relevant harmonics in the density bosonic expansion [@sablikov; @haldanefluid] one obtains ${\rho}(x)=\rho^{LW}(x)+F\rho^F(x)+(1-F)\rho^W(x)$ where $\rho^{LW}(x)$ is the long wave contribution, $\rho^F(x)$ are the Friedel oscillations [@yo], due to finite size effects, and $\rho^W(x)$ is the Wigner contribution, due to interaction effects. The parameter $0\leq F\leq 1$ cannot be calculated within the Luttinger liquid theory: numerically one finds $F=1$ in the non interacting case, while $F\ll 1$ in the limit of very strong interactions [@bortz]. In the bosonization language one has $$\begin{aligned}
\rho^{LW}(x)\!&=&\!\frac{N_1}{L}+\frac{4q_F}{\pi}-\frac{\partial_x}{\pi}\sum_{\alpha,s}\varphi_{\alpha,s}(x)\nonumber\\
\rho^F(x)\!&=&\!-\sum_{\alpha,s}\frac{N_{\alpha,s}}{L}\cos\left[\!\mathcal{L}_{F}(x)\!+\!\frac{2\pi x N_{\alpha,s}}{L}\!-\!2\varphi_{\alpha,s}(x)\!\right]\!,\nonumber\\
\rho^W(x)\!&=&\!-\frac{N_1}{L}\cos\left[\mathcal{L}_{W}(x)+\frac{2\pi x N_{1}}{L}-4\varphi_{1}(x)\right],\nonumber\end{aligned}$$ with $\mathcal{L}_{F}(x)=2q_{F}(x)-2g^2h(x)$, $\mathcal{L}_{W}(x)=4\mathcal{L}_{F}(x)$, $h(x)=[\phi_1(x),\phi_1(-x)]/(4i)$, and $\varphi_{\alpha,s}(x)=[\phi_{\alpha,s}(-x)-\phi_{\alpha,s}(x)]/2$, $\varphi_{1}(x)=[\phi_{1}(-x)-\phi_{1}(x)]/2$.\
The long-wave part has a typical length scale $\approx L$, while the Friedel and Wigner contributions oscillate with typical wavelengths $(2q_{F})^{-1}$ and $(8q_{F})^{-1}$ respectively. In this work, we will consider a CNT dot with with a not too small number of electrons $N\gg 1$ so that the Friedel and Wigner terms of the density oscillate much faster than the strain field. As a result, the only relevant contribution to $H_{\mathrm{d-v}}$ stems from the long-wave part of the density and reads $$\label{eq:ephint00}
{H}_{\mathrm{d-v}}=\frac{c}{2\pi}\int_{0}^{L}\mathrm{d}x\ \partial_{x}\left[\phi_{1}(x)-\phi_{1}(-x)\right] \partial_{x}{u}_{p}(x)\, .$$ Introducing $B_{\mu}=ib_{1,\mu}$ and $\sqrt{2\mu\omega_{1}}X_{\mu}$ (with a conjugated $P_{\mu}$ satisfying $[X_{\mu},P_{\nu}]=\delta_{\mu,\nu}$) the electron-vibron coupling casts into $$\label{eq:ephint00}
{H}_{\mathrm{d-v}}=\sqrt{M}X_{0}\sum_{\mu\geq 1}C_{\mu}X_{\mu}\, .$$ where $\quad C_{\mu\geq 1}=2\lambda_{\mathrm{m}}\omega_{0}^{3/2}\sqrt{\omega_{1}}L^{-1}\int_{0}^{L}{\mathrm d}x\ \cos\left(\frac{\mu\pi x}{L}\right)\cos\left(\frac{\pi}{{L}}x\right)$, with $c\lambda_{\mathrm m}^{-1}=\sqrt{\rho_{0}\pi\mathcal{W}v_{\mathrm s}}$. For a typical CNT, $\lambda_{\mathrm{m}}\approx 2$ [@noi]. We will assume this value in the following.\
It can be readily seen that the operator ${h}_1={H}_1+{H}_{\mathrm{d-v}}+H_{V}$ is manifestly quadratic in the operators $X_{0}$, $P_{0}$, $X_{\mu}$, and $P_{\mu}$ and thus can be diagonalized exactly [@ullersma]. The diagonal form of $h_{1}$ is $${h}_1=\frac{1}{2}E_{1}\left({N}_{1}-N_{g}\right)^{2}+\sum_{\mu\geq0}\left(\frac{{\bar{P}}_{\mu}^{2}}{2}+\Omega_{\mu}^{2}\frac{{\bar{X}}_{\mu}^{2}}{2}\right)\, .$$ For the case of a vibron in its fundamental mode, spanning the entire dot, one finds $C_{\mu}=\lambda_{m}\omega_{0}^{3/2}\sqrt{\omega_{1}}\delta_{\mu,1}$, i.e. the vibron couples to the lowest-lying plasmon mode only. As a result, $\Omega_{\mu\geq 2}\equiv\mu\omega_{1}$ with the corresponding plasmonic modes remaining completely unaffected. The energy of the two lowest-lying collective modes are the positive roots of $\left(\epsilon^2-1\right)\left(\epsilon^2-r^2\right)=\lambda_{\mathrm{m}}^{2}r$ where $\epsilon=\Omega_{\mu}/\omega_{0}$ ($\mu=0,1$) and $r=\omega_{1}/\omega_{0}=v_{F}/(g v_{s})$. Clearly, the parameter $r$ governs the nature of the solutions. Even in semiconducting CNTs, one finds $r>1$ since $v_{s}$ does not exceed $v_{F}$. In this regime, it can be readily seen that the mode with $\mu=0$ has energy $\Omega_{0}<\omega_{0}$ and represents a vibron dressed by the plasmonic mode, while $\Omega_{1}>\omega_{1}$ with $\Omega_{1}\approx\omega_{1}$ is the energy of a collective mode almost unaffected by the coupling. As a consequence, we can further simplify $h_{1}$ neglecting the slight renormalization of the first plasmonic mode, to obtain $$\label{eq:hamcolfin}
{h}_1=H_{1}+\frac{{\bar{P}}_{0}^{2}}{2}+\Omega_{0}^{2}\frac{{\bar{X}}_{0}^{2}}{2}\, .$$ Within the above approximation, the operator $\phi_{1}(x)$, entering the field operators $\Psi_{s}(x)$ transforms upon the diagonalization into $\bar{\phi}_{1}(x)=\phi_{1}(x)+\phi_{0}(x)$, with the contribution $$\label{eq:phi0}
\phi_{0}(x)=\alpha_{0}(x)\bar{X}_{0}+\beta_{0}(x)\bar{P}_{0}\, ,$$ stemming from the coupled vibronic mode. We find $$\label{eq:alpha0}
\alpha_{0}(x)\approx-\sqrt{2g\omega_{1}}\lambda_{\mathrm{m}}r^{-3/2}k_{0}\sin{\left(\frac{\pi x}{L}\right)}$$ and $\beta_{0}(x)=\left[L/(\pi g\omega_{1})\right]\partial_{x}\alpha_{0}(x)$ with $k_{0}^{-2}=1-\lambda_{m}^{2}/r^{2}$.\
The total Hamiltonian $H_d$ of the vibrating CNT $H_d=h_1+H_2+H_3+H_4$ is separated in two additive contributions: the one of zero modes $H_N$, which only depends on $N_{i}$, $i=1,..,4$ and the bosonic one $H_b$. The eigenstates are $|\{N_i\},\{n^{(\mu)}_{i}\}\rangle$, where $n^{(\mu)}_i$ is the number of bosonic excitations in the $i$-th channel, with momentum $\pi\mu/L$. For a given number of particles $N$, the ground state $|N\rangle$ is obtained minimizing the zero mode energy with the constraint $N_1$ and setting $n^{(\mu)}_{i}=0$ $\forall\mu,i$.\
We now turn to the transport properties of the suspended CNT, tunnel coupled to two lateral contacts and capacitively coupled to a charged AFM tip, located at $0\leq x_0\leq L$ and kept at a potential $V_{tip}<0$. A scheme of the set up is shown in Fig.1.
![Scheme of the setup. See text for further details.[]{data-label="fig:fig1"}](Traverso1.pdf){width="6.8cm"}
The source and drain ($\lambda=S,D$) contacts are non-interacting Fermi gases at potential $-V/2$ and $V/2$ respectively, with Hamiltonians $H_\lambda$. The tunneling Hamiltonians connecting the dot and the lead $\lambda$ read [@noi; @proc; @bercioux; @giacomo] $${H}_{\lambda}^{t}=t_{0}\sum_{\lambda=S,D}\sum_{\alpha,s,q}{\psi}_{+1,\alpha,s}(x_{\lambda}){c}_{\lambda,s}(q)+\mathrm{h.c.}\,,$$ where we assumed symmetric barriers with $t_{0}$ the tunneling amplitude. Here, ${c}_{\lambda,s}(q)$ are the operators for an electron with momentum $q$ and spin $s$ in the non-interacting lead $\lambda$ and ${x}_{1}=0$, ${x}_{2}=L$ are the positions of the tunneling contacts.\
The capacitive coupling between the dot and the tip (assumed non magnetic [@biggio]) is given by $H_{AFM}=H_F+H_W$ with, $$\begin{aligned}
H_F&=&V^{(F)}\sum_{\alpha,s}\cos\left[\mathcal{L}_{F}(x)+\frac{2\pi N_{\alpha,s} x}{L}-2{\varphi}_{\alpha,s}(x)\right],\nonumber\\
H_W&=&V^{(W)}\cos\left[\mathcal{L}_{W}(x)+\frac{2\pi N_1 x}{L}-4{\varphi}_{1}(x)\right].\nonumber \end{aligned}$$ Here, $V^{(\xi)}\propto|V_{tip}|$ ($\xi=F,W$) parameterizes the strength of coupling between the tip and the Friedel or Wigner contributions to the density. We have assumed in this work a sufficiently sharp tip, whose width $\delta$ is larger than the CNT lattice constant but smaller than the wavelength of the Wigner oscillations. The coupled tip induces a renormalization of both the chemical potential of the dot and of its linear conductance.\
To the lowest perturbative order in $V^{\xi}$ is given by $\mu=\mu_{0}+\delta\mu(x_{0})$ where $\mu_{0}=\langle N+1|H_{d}|N+1\rangle-\langle N|H_{d}|N\rangle$ is the chemical potential in the absence of the tip and $\delta\mu(x_0)=\langle N+1|H_{AFM}|N+1\rangle-\langle N|H_{AFM}|N\rangle$ the first order tip correction. Here, and in the rest of the paper, for definiteness we assume $N_1=4n$ (with $n\geq 0$ an integer). One finds $$\delta\mu(x_0)=\sum_{\xi=F,W}V^{(\xi)}[\zeta^{(\xi)}(N+1,x_0)-\zeta^{(\xi)}(N,x_0)]\, ,$$ where $$\begin{aligned}
\zeta^{(F)}(n,x)\!&=&\!K(x)^{\frac{3+g}{4}}V(x)^{\frac{1}{4}}\!\sum_{\alpha,s}\!\cos\left[\mathcal{L}_F(x)\!+\!\frac{2\pi x N_{\alpha,s}}{L}\right],\nonumber\\
\zeta^{(W)}(n,x)\!&=&\!K(x)^{4g}V(x)^{4}\cos\left[\mathcal{L}_W(x)\!+\!\frac{2\pi x n}{L}\right],\label{eq:basta}\\
K(x)\!&=&\!\frac{\sinh\left(\frac{\pi\tilde{a}}{2L}\right)}{\sqrt{\sinh^2\left(\!\frac{\pi\tilde{a}}{2L}\right)\!+\!\sin^2\left(\frac{\pi x}{L}\!\right)}}\ \ ;\,\,V(x)\!=\!e^{-\frac{\alpha^2_0(x)}{\Omega_0}}.\nonumber\end{aligned}$$ In Eq. (\[eq:basta\]) we have $N_{\alpha,s}=N_1/4$ if $n=N_1$, while $N_{\alpha^*,s^*}=N_1/4+1$ for a given $\alpha=\alpha^*$, $s=s^*$ and $N_{\alpha\neq\alpha^*,s\neq s^*}=N_1/4$ if $n=N_1+1$. This is due to the fourfold degeneracy of the ground state with $N+1$ electrons. The chemical potential corrections consist of an oscillating pattern, oscillating in accordance to the Friedel or the Wigner length scale, modulated by an envelope function slowly varying on the scale of $L$. The envelope functions are composed by a term $K(x)$, stemming from the collective modes of the CNT at energies $\mu\omega_{1}$ and present also in the absence of the electron-vibron coupling [@AFM], and by $V(x)$ which originates from the electron-vibron coupling. Both $K(x)\leq 1$ and $V(x)\leq 1$ contribute to suppress the oscillatory pattern: the coupling of mechanical and electrical degrees of freedom leads therefore to an [*additional*]{} suppression of both Friedel and Wigner oscillations, induced by increased fluctuations of the charge degree of freedom. The term $K(x)$ exhibits different power laws as a function of the Luttinger parameter $g$ in the Friedel and in the Wigner channels: the suppression of the Wigner fluctuations is most severe than that of the Friedel channel when $g\to 1$, while in the strong interactions regime one has no suppression of the Wigner oscillations due to the high energy collective modes of the CNT, in contrast with the Friedel oscillations which are still damped even for $g=0$.\
In the sequential tunneling regime, the linear conductance can be evaluated setting up a rate equation for the occupation probability of the dot states. Tunneling rates between dot ground states with $N=N_{0}+N_{1}$ and $N+1$ electrons are evaluated to the second order in $t_{0}$ by means of the Keldysh technique [@AFM; @blum; @haupt; @master]. The effects of the AFM tip are evaluated as a perturbation to first order in $V^{(\xi)}$. In the linear regime and for low temperature $k_{B}T<\Omega_{0}$ tunneling rates attain the form $\Gamma^{(\lambda)}_{N\to N+1}=\gamma^{(\lambda)}(x_{0})f(\mu)$ and $\Gamma^{(\lambda)}_{N_1+1\to N_1}=\gamma^{(\lambda)}(x_{0})f(-\mu)$ where $f(E)=\left[1+e^{\beta E}\right]^{-1}$ is the Fermi function with $\beta^{-1}=k_{B}T$. The tunneling rates $\gamma^{(\lambda)}(x_{0})$ can be evaluated explicitly following a procedure analogous to that outlined in Ref. [@AFM]. Here we just quote the final result $\gamma^{(\lambda)}(x_{0})=\gamma_{0}^{(\lambda)}\left[1+\delta\gamma^{(F)}(x_0)+\delta\gamma^{(W)}(x_{0})\right]$ with $\gamma_{0}^{(\lambda)}=\nu_{0}|t_{0}|^{2}(\pi\tilde{a})^{-1}\left(1-e^{-\pi \tilde{a}/L}\right)^{-(3+g)/4}$ where $\nu_{0}$ is the leads density of states, and $$\delta\gamma^{(\xi)}= 2V^{(\xi)}N^{(\xi)}(x_{0})\sum_{\mathbf{m}\neq\mathbf{0},\delta={i,f}}\frac{1}{\Lambda}
B^{(\xi,\delta)}_{\mathbf{m}}C_{\mathbf{m}}^{(\delta,\xi)}(x_0)\, .\nonumber$$ Several quantities have been introduced: $\mathbf{m}=(m_1,m_2,m_3,m_4)$ is a vector of four integers,$\Lambda=\epsilon_\rho(m_1+m_2)+\epsilon_\sigma(m_3+m_4)$, the coefficients are given by $B^{F,i}_{\mathbf{m}}=b_{m_{1}}^{+,1/4}b_{m_{2}}^{-,1/4}b_{m_{3}}^{+,3/4}b_{m_{4}}^{-,3/4}$, and, $B^{W,i}_{\mathbf{n},\mathbf{m}}=b_{m_{1}}^{+,1}b_{m_{2}}^{-,1}\delta_{m_{3},0}\delta_{m_{4},0}$, while ${B}_{\mathbf{n},\mathbf{m}}^{\xi,f}$ is expressed in terms of $B_{\mathbf{m}}^{\xi}$ as ${B}_{m_{1},m_{2},m_{3},m_{4}}^{\xi,f}=B_{m_{2},m_{1},m_{4},m_{3}}^{\xi,i}$ with $$\begin{aligned}
b_{l}^{+,\kappa}&=&\left(-e^{-\frac{\pi\alpha}{L}}\right)^{l}\left(1-e^{-\frac{\alpha\pi}{L}}\right)^{\kappa}\frac{\Gamma(1+\kappa)\theta(l)}{l!\Gamma(1+\kappa-l)}\\
b_{l}^{-,\kappa}&=&\left(e^{-\frac{\pi\alpha}{L}}\right)^{l}\left(1-e^{-\frac{\alpha\pi}{L}}\right)^{-\kappa}\frac{\Gamma(l+\kappa)}{l!\Gamma(\kappa)}\theta(l).\end{aligned}$$ The oscillations of the tunneling rate as a function of the tip position are encoded in the functions $C_{\mathbf{n},\mathbf{m}}^{(\delta,\xi)}(x_0)$, given by $$C_{\mathbf{m}}^{(\delta,\xi)}(x_0)\!=\!\cos\left[\!\mathcal{L}^{(\delta)}_{F/W}(x_0)\!+\!\frac{\pi(m_{1}\!+\!m_{3}\!-\!m_{2}\!-\!m_{4})x_0}{L}\!\right],$$ with $\mathcal{L}_{F}^{(i)}(x_0)=\mathcal{L}_{F}(x_{0})+\pi N_1 x_{0}/2L$, $\mathcal{L}_{W}^{(i)}(x_0)=\mathcal{L}_{F}(x_{0})+2\pi N_1 x_{0}/L$, $\mathcal{L}^{(f)}_{F}(x_0)=\mathcal{L}^{(i)}_{F}(x_0)+\frac{\pi x_0}{2L}$, $\mathcal{L}^{(f)}_{W}(x_0)=\mathcal{L}^{(i)}_{W}(x_0)+\frac{2\pi x_0}{L}$, and finally the pre-factors $N^{(\xi)}(x_{0})$ are given by $N^{(F)}(x_0)=K(x_0)^{\frac{3+g}{4}}V(x_0)^{\frac{1}{4}}$ and $N^{(W)}(x_0)=K(x_{0})^{4g}V(x_0)^{4}$.\
In terms of the above quantities, the differential conductance $G$ is given by [@AFM] $$\begin{aligned}
G&=&\frac{4\beta e^2 \Delta(x_{0})f(-\mu)}{4+e^{\beta\mu}}\label{eq:cond}\\
\Delta(x_{0})&=&\frac{\gamma^{(S)}(x_{0})\gamma^{(D)}(x_{0})}{\gamma^{(S)}(x_{0})+\gamma^{(D)}(x_{0})}\, .\end{aligned}$$ The factor 4 in Eq. (\[eq:cond\]) stems from the fourfold degeneracy of the ground state with $N_1+1$ electrons. We now turn to a discussion of the results.
![Transport properties of a suspended CNT dot near the transition between 16 and 17 electrons for mild interactions, $g=0.7$, when the AMF tip is placed at $x_{0}$. (a) Corrections to the chemical potential $\delta\mu(x_0)$ (units $V^{(F)}$) as a function of $x_0$ (units $L$); (b) Density plot of the linear conductance $G$ (units $e^2\gamma_{0}^{(S)}/\omega_{2}$) as a function of $x_{0}$ (units $L$) and $\bar{N}_g=N_{g}-3\omega_{2}/8E_{1}$; (c) Maximum of the linear conductance as a function of $x_{0}$ (units $L$). In panels (a) and (c), blue (red) curves have been calculated neglecting (including) electron-vibron coupling. Other parameters: $V^{(F)}=V^{(W)}=0.05\omega_{2}$, $k_{B}T=0.1\Omega_{0}$, $\lambda_{m}=2$, $v_{F}/v_{s}=4.6$ appropriate for a semiconducting CNT and $\tilde{a}=L/50$.[]{data-label="fig:fig2"}](Traverso2.pdf){width="7cm"}
Figure \[fig:fig2\] shows the calculated linear conductance and chemical potential shift for a CNT tuned about the transition between $N=16$ electrons ($N_0=16$, $N_1=0$) and $N=17$ electrons ($N_0=16$, $N_1=1$) for the case of mild interactions, $g=0.7$. Panel (a) shows the oscillations of the chemical potential shift $\delta\mu(x_{0})$ as a function of the tip position $x_{0}$. In the weak interaction regime, Friedel oscillations dominate over the Wigner ones: as a result a typical shape with $N_0/4+1=5$ maxima and $N_0/4=4$ minima is shown. This corresponds to the wavelength of the Friedel oscillations which is not determined by the total number of electrons but by the number of electrons in the sector $\alpha,\sigma$ involved in transport, in analogy with the behaviour of a two-channel quantum wire [@sgm2; @AFM]. The blue curve represents the results in the absence of electron-vibron coupling: only minor ripples near the dot edges signal the weak influence of the Wigner molecule. The red curve shows the results obtained including the electron-vibron coupling: a suppression of the oscillations is evident, which also washes away the small features near the border. The overall shape of $\delta\mu(x_{0})$ is, however, completely analogous to the static case. Here and in the following, we have chosen a semiconducting CNT with a fairly small Fermi velocity $v_{F}/v_{s}=4.6$, in order to enhance the electron-vibron coupling. Panel (b) shows the linear conductance as a color map, as a function of the tip position and the number of charges induced by the gate. The latter quantity is responsible for tuning $\mu_{0}$ and hence for bringing the dot into the resonance, which occurs when $\mu=\mu_{0}+\delta\mu(x_0)=(k_{B}T/2)\ln(4)$, see Eq. (\[eq:cond\]). Since $\delta\mu(x_0)$ oscillates, the value of $N_{g}$ needed to obtain the resonance fluctuates, allowing to map the shift in chemical potential. Also the intensity of the conductance oscillates as the tip is scanned along the CNT. Panel (c) shows the height of the maximum $G_{m}=\mathrm{max}_{N_{g}}\{G\}$ as a function of $x_{0}$. The overall qualitative features are analogous to those of the chemical potential with the presence of 5 peaks and 4 valleys. The static CNT displays small ripples at the border of the dot, similarly to the behaviour of the chemical potential. However, such features basically disappear as a vibrating dot is considered, along with a suppression of the conductance.
![Same as in Fig. \[fig:fig2\] but for a strongly interacting CNT, with $g=0.2$.[]{data-label="fig:fig3"}](Traverso3.pdf){width="7cm"}
Figure \[fig:fig3\] shows the case of a strongly interacting CNT, most favorable to observe the emergence of Wigner correlations. The chemical potential shift now distinctly shows 17 maxima and 16 minima, with an oscillation characterized by the Wigner wavelength. In the case of a vibrating CNT, a suppression of the oscillations is observed. Still, the qualitative shape of the chemical potential shift is unaffected. Indeed, as $g\to 0$, the parameter $r$ increases, which leads to $\alpha_{0}(x)\to 0$, thus making the contribution due to vibrons small. Also in this case the oscillations of the intensity of the conductance show oscillations with a period shorter than the Friedel one, However, the regime of a Wigner molecule is not fully developed in this quantity. The small suppression of the differential conductance maximum in the case of a vibrating dot does not alter qualitatively the results.\
In conclusion we have calculated the correction to the chemical potential and the linear conductance of a 1D quantum dot built in an interacting and suspended CNT, perturbed by a charged AFM tip capacitively coupled to the dot. We demonstrated that both quantities depend on the electron-vibron coupling, whose effect is to suppress the oscillations of the chemical potential and of the conductance, induced by the Wigner and Friedel oscillations of the density. We have also shown that for realistic devices with a fairly strong electron-vibron coupling this suppression is not enough to produce qualitative modifications of the oscillating patterns. Indeed, in the case of strongly interacting nanotubes, the large mismatch of velocities between the plasma modes and the strain implies a strong reduction of the effects induced by vibrons, producing an even more favorable situation to observe the effects of Wigner molecules in a transport experiment.\
[*Acknowledgements.*]{} Financial support by the EU- FP7 via ITN-2008-234970 NANOCTM is gratefully acknowledged.
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abstract: 'In this paper we will study the statistics of the unit geodesic flow normal to the boundary of a hyperbolic manifold with non-empty totally geodesic boundary. Viewing the time it takes this flow to hit the boundary as a random variable, we derive a formula for its moments in terms of the orthospectrum. The first moment gives the average time for the normal flow acting on the boundary to again reach the boundary and the zeroth moment recovers Basmajian’s identity. Furthermore, we are able to give explicit formulae for the first moment in the surface case as well as for manifolds of odd dimension. In dimension two, the summation terms are dilogarithms. In dimension three, we are able to find the moment generating function for this length function.'
address: |
Department of Mathematics\
Boston College\
140 Commonwealth Ave.\
Chestnut Hill, MA 02467\
nicholas.vlamis@bc.edu
author:
- 'Nicholas G. Vlamis'
bibliography:
- 'average-length.bib'
nocite: '[@*]'
title: Moments of a Length Function on the Boundary of a Hyperbolic Manfiold
---
Introduction
============
Let $M$ be a compact hyperbolic manifold with non-empty totally geodesic boundary. An [*orthogeodesic*]{} for $M$ is a geodesic arc with endpoints normal to $\partial M$ (see [@basmajian]). We will denote the collection of orthogeodesics by $O_M = \{{\alpha}_i\}$. Let $\ell_i$ denote the length of ${\alpha}_i$, then the collection $L_M = \{\ell_i\}$ (with multiplicities) is known as the [*orthospectrum*]{}. As we will be summing over the orthospectrum, it is import to note that $O_M$ is a countable collection: this can be seen by doubling the manifold and observing that the orthogeodesics correspond to a subset of the closed geodesics in the double.
Given $x\in \partial M$, let ${\alpha}_x$ be the geodesic emanating from $x$ normal to $\partial M$. Then, as the limit set is measure zero, for almost every $x\in \partial M$ we have that ${\alpha}_x$ terminates in $\partial M$; hence, the length of ${\alpha}_x$ is finite. This allows us to define the measurable function $L: \partial M \to {\mathbb R}$ given by $L(x) = length(\alpha_x)$. Let $dV$ denote the hyperbolic volume measure on $\partial M$, then $V(\partial M)$ is finite allowing us to define the probability measure $dm = dV/V(\partial M)$ on $\partial M$, so that $(\partial M, dm)$ is a probability space. This lets us view $L:\partial M \to {\mathbb R}$ as a random variable. Given a random variable $X$ on a probability space with measure $p$, the [*$k^{th}$-moment of $X$*]{} is defined to be $E[X^k] = \int X^k dp$, where $E[X]$ denotes the expected value. Let $A_k(M)$ be the $k^{th}$ moment of $L$. In particular, $A_1(M)$ is the expected value of $L$. In this paper we will show that the positive moments of $L$ are finite and encoded in the orthospectrum:
\[finiteness\] Let $M= M^{n}$ be an $n$-dimensional compact hyperbolic manifold with nonempty totally geodesic boundary, then $A_k(M)$ is finite for all $k\in {\mathbb Z}^{\geq 0}$.
\[ortho\] Let $M= M^{n}$ be an $n$-dimensional compact hyperbolic manifold with nonempty totally geodesic boundary, then for all $k\in {\mathbb Z}^{\geq 0}$ $$A_k(M) = \frac{1}{V(\partial M)} \sum_{\ell \in L_M} F_{n,k}(\ell),$$ where $$F_{n,k}(x) =\Omega_{n-2}\int_0^{\log\coth(x/2)}\left[ \log\left(\frac{\coth x +\cosh r }{\coth x-\cosh r}\right)\right]^k \sinh^{n-2}(r)\ dr$$ and $\Omega_n$ is the volume of the standard $n$-sphere.
As corollaries we can write the function $F_{n,1}(x)$ in dimension 2 and all odd dimensions without integrals. In the following corollary ${\mathrm{Li}_2}(x)$ is the standard dilogarithm (see [@lewin]). We will also write $\ell(\partial S)$ for sum of the lengths of each boundary component of a surface $S$.
\[surface\] Let $S$ be a compact hyperbolic surface with nonempty totally geodesic boundary. Then $$A_1(S) = \frac2{\ell(\partial M)} \sum_{\ell\in L_M}\left[{\mathrm{Li}_2}\left(-\tanh^2\frac {\ell}2\right)-{\mathrm{Li}_2}\left(\tanh^2\frac {\ell}2\right)+\frac {\pi^2}4\right]$$
\[odd\] Let $M$ be an $n$-dimensional compact hyperbolic manifold with nonempty totally geodesic boundary where $n$ is odd. Then $$\resizebox{1\hsize}{!} {$A_1(M) = \frac{2\Omega_{n-2}}{Vol(\partial M)}\sum_{\ell\in L_M}\sum_{j=0}^{\frac{n-3}2}\frac{(-1)^{\frac{n-3}2-j}\binom{\frac{n-3}2}{j}}{2j+1}\coth^{2j+1}(\ell)\left[\log(2\cosh\ell)-\ell_i\tanh^{2j+1}(\ell)+\sum_{k=1}^j\frac{1-\tanh^{2k}(\ell)}{2k}\right].$}$$
The rest of the paper is dedicated to understanding the asymptotics of the $F_{n,k}$’s and finding the moment generating function in dimension 3. The motivation of this paper comes from recent work of Bridgeman and Tan in [@bt], where the authors study the moments of the hitting function associated to the unit tangent bundle of a manifold (i.e. the time it takes the geodesic flow of a vector to reach the boundary). In the paper they are able to show the moments are finite and give an explicit formula for the expected value in the surface case as well as relate the orthospectrum identities of Basmajian (see below) and Bridgeman (see [@b] and [@bk]) as different moments of the hitting function. Observe that $F_{n,0}(x) = V_{n-1}(\log\coth(x/2))$, where $V_n(r)$ is the volume of the hyperbolic $n$-ball of radius $r$. As $A_0(M) = 1$, Theorem \[ortho\] gives the volume of the boundary in terms of the orthospectrum. This identity is known as Basmajian’s identity:
If $M$ is a compact hyperbolic $n$-manifold with totally geodesic boundary $S$, and $\ell_i$ denotes the (ordered) orthospectrum of $M$, with multiplicity, then $$Area(S) = \sum_i V_{n-1}(\log\coth\frac {\ell_i}2),$$ where $V_n(r)$ is the volume of the hyperbolic $n$-ball of radius $r$.
Note that by combining Theorem \[ortho\] and Basmajian’s identity we see that $A_k(M)$ depends solely on the orthospectrum.
Acknowledgements {#acknowledgements .unnumbered}
----------------
I would like to thank my adviser, Martin Bridgeman, for his guidance.
Kleinian Groups
===============
For $n\geq 2$, let $\operatorname{Isom}^+({\mathbb H}^n)$ be the space of orientation preserving isometries of hyperbolic $n$-space. With the topology of uniform convergence on the space of isometries, we define a [*Kleinian group*]{} to be a discrete torsion-free subgroup of $\operatorname{Isom}^+({\mathbb H}^n)$. If $\Gamma<\operatorname{Isom}^+({\mathbb H}^n)$ is a Kleinian group, then ${\mathbb H}^n/\Gamma$ is a hyperbolic manifold, i.e. a Riemannian manifold of constant curvature $-1$.
In the Poincaré model of hyperbolic space we can identify the boundary of ${\mathbb H}^n$ with the $(n-1)$-sphere called the [*sphere at infinity*]{} and denoted $S^{n-1}_\infty$. Pick $x\in {\mathbb H}^n$ and define the [*limit set of $\Gamma$*]{} to be the set $\Lambda_\Gamma = \overline{\Gamma x} \cap S^{n-1}_\infty$. Note that this definition is independent of the choice of $x$. Define the [*convex hull*]{} $\mathrm{CH}(\Lambda_\Gamma)$ of the limit set $\Lambda_\Gamma$ to be the smallest convex subset of ${\mathbb H}^n$ containing all the geodesics in ${\mathbb H}^n$ with endpoints in $\Lambda_\Gamma$. As $\Lambda_\Gamma$ is $\Gamma$-invariant, so is $\mathrm{CH}(\Lambda_\Gamma)$ and so we can take the quotient of $\mathrm{CH}(\Lambda_\Gamma)$ by $\Gamma$, which we call the [*convex core*]{} and denote $\mathrm{C}(\Gamma)$. A Kleinian group is [*convex cocompact*]{} if its associated convex core is compact (see [@thurston]).
With these definitions at hand, we recall the following two theorems from Patterson-Sullivan theory (see [@nicholls]):
Let $\Gamma<\mathrm{Isom}^+({\mathbb H}^n)$ be a convex cocompact Kleinian group and let $\delta = \delta(\Gamma)$ be the Hausdorff dimension of the limit set of $\Gamma$. Then for $r\geq r_0$, $$N_x(r) = \left|\{\gamma\in \Gamma\colon d(\gamma(x),x)<r\}\right| \leq ae^{\delta r},$$ for some constant $a$ depending on $\Gamma$ and $x$.
Let $\Gamma<\mathrm{Isom}^+({\mathbb H}^n)$ be a convex cocompact Kleinian group and let $\delta = \delta(\Gamma)$ be the Hausdorff dimension of the limit set of $\Gamma$. Then $\delta = n-1$ if and only if ${\mathbb H}^n/\Gamma$ is finite volume.
Finite Moments
==============
Let $M = M^n$ be a compact $n$-dimensional hyperbolic manifold with totally geodesic boundary. As above, let $L$ denote the time to the boundary of the unit normal flow on the boundary. We let $dV$ be the induced hyperbolic volume measure on the boundary and define $dm = dV/V(\partial M)$, so that $(\partial M, m)$ is a probability space and $L:\partial M \to {\mathbb R}$ is a random variable on this space. We let $A_k(\partial M) = E[L^k]=\int_{\partial M} L^k\, dm$ be the $k^{th}$ moment of $L$. In this section we will show that $A_k(M)$ is finite for all nonnegative integers $k$. We first state a basic fact from hyperbolic geometry.
Let $U$ be a hyperplane in ${\mathbb H}^n$ and $B_R$ a hyperbolic $n$-ball of radius $R$ a distance $s$ from $U$. The orthogonal projection of $B_R$ to $U$ has radius $r<\log\coth(s/2)$.
Let $p\in \partial B_R$ be the point closest to $U$, so that $d(p,U) = s$ and let $V$ be the hyperplane containing $p$ such that $d(U,V) = s$. Then the orthogonal projection of $B_R$ is contained in the orthogonal projection of $V$. The orthogonal projection of $V$ to $U$ has radius $\log\coth(s/2)$ (see [@basmajian]), implying that $r < \log\coth(s/2)$ as desired.
We can now show that $A_k(M)$ is finite:
[**Theorem \[finiteness\]**]{} [*Let $M= M^{n}$ be an $n$-dimensional compact hyperbolic manifold with nonempty totally geodesic boundary, then $A_k(M)$ is finite for all $k\in {\mathbb Z}^{\geq 0}$.*]{}
We want to work in hyperbolic space: identify the universal cover ${\widetilde}M$ of $M$ with a convex subset of ${\mathbb H}^n$, so that $\pi_1M=\Gamma < \mathrm{Isom}^+({\mathbb H}^n)$ is a convex cocompact Kleinian group. As $M$ has a finite number of disjoint boundary components and we are investigating the integral over the boundary, it is enough to prove finiteness for a single component. Fix $S\subset \partial M$ a component and a lift ${\widetilde}S\subset {\widetilde}M$ of $S$ (note: ${\widetilde}S$ is a copy of ${\mathbb H}^{n-1}$ sitting in ${\mathbb H}^n$). Let $U$ be a convex fundamental domain for the action of $\Gamma$ on ${\widetilde}M$. Pick $p\in U$ and let $B_R(p)$ be a ball centered at $p$ of radius $R$ such that $U\subset B_R(p)$. Set $V= U\cap {\widetilde}S$ be a fundamental domain for the action of $\mathrm{Stab}({\widetilde}S)<\Gamma$ on ${\widetilde}S$. Define $n_t: V \to {\mathbb H}^n$ to be the unit geodesic flow normal to ${\widetilde}S$ into ${\widetilde}M$ for a time $t$ and set $X_t = \{x\in V\colon n_t(x) \in {\widetilde}M\}.$ Define $\pi: {\mathbb H}^n \to {\widetilde}S$ to be orthogonal projection.
We will now bound $V(X_t)$ for $t\geq r_0$. If $x\in X_t$, then $n_t(x)\in \gamma U$ for some $\gamma\in \Gamma$. If $n_t(x)\in n_t(X_t)\cap \gamma U$, then $d(p,\gamma(p)) < t+2R$. Let $\Gamma_t =\{\gamma\in \Gamma\colon n_t( X_t)\cap \gamma U \neq \emptyset\}$, then from the above theorem, we know that $|\Gamma_t| \leq N_p(t+2R) \leq ae^{\delta(t+2R)}$, where $\delta$ is the Hausdorff dimension of the limit set of $\Gamma$. As $n_t(X_t)\subset \bigcup_{\gamma\in\Gamma_t} \gamma U$ and $\pi(n_t(X_t))=X_t$, we have $$V(X_t) \leq \sum_{\gamma\in\Gamma_t}V(\pi(\gamma U)).$$ Now, fix $\gamma\in \Gamma_t$, then $\gamma\cdot U \subset B_R(\gamma\cdot p).$ Suppose that $B_R(\gamma\cdot p)$ is a distance $s$ from $V$ and let $r$ be the radius of its projection, we then have that $t < r+s+2R$ by the triangle inequality; in particular, $s>t-r-2R$. Furthermore, as orthogonal projection is always distance decreasing in hyperbolic space, $r< R$, so that $s>t-3R$. From the above lemma, we can conclude that $$r \leq \log\coth(s/2)\leq \log\coth\left(\frac{t-3R}2\right)\equiv f(t).$$ As the above bound for the radius does not depend on $\gamma$, we have $$V(X_t)\leq |\Gamma_t|V_{n-1}\left(f(t)\right)\leq N_p(t+2R) V_{n-1}\left(f(t))\right)\leq ae^{\delta(t+2r)}V_{n-1}\left(f(t)\right),$$ where $V_{n}(r)$ is the volume of a $n$-dimensional hyperbolic ball of radius $r$. We observe two asymptotics: 1) $\lim_{x\to \infty} e^{-x}\log\coth(x/2) = 2$ and 2) $\lim_{r\to0} V_n(r)/r^n = C_n$ for some constant $C_n>0$. From these facts and the above inequality, we see that $$\lim_{t\to\infty} e^{(n-1-\delta)t}\cdot V(X_t)\leq A,$$ for some constant $A$. From the theorem stated above, we know that $n-1-\delta>0$.
We now move to the moments. We have setup the following situation: $$\begin{aligned}
\int_S L^k dV &= \sum_{t=0}^\infty \int_{L^{-1}(t,t+1)} L^k dV \leq \sum_{t=0}^{\infty} (t+1)^k\int_{L^{-1}(t,t+1)}dV
\leq \sum_{t=0}^\infty (t+1)^k V(X_t).\end{aligned}$$ But, we saw that the asymptotics of $V(X_t)$ are less than a multiple of $e^{-bt}$ with $b>0$, which implies the above sum converges since $\sum(t+1)^ke^{-bt}$ converges.
The Moments as a Sum over the Orthospectrum
===========================================
Basmajian’s Ball Decomposition of the Boundary
----------------------------------------------
In this section we introduce a decomposition of $\partial M$ into a disjoint union of $n-1$ balls (affectionately known as “leopard spots"). We will recall Danny Calegari’s method of accomplishing this in [@calegari].
Let $X$ and $Y$ be totally geodesic copies of ${\mathbb H}^{n-1}$ sitting inside of ${\mathbb H}^n$ with disjoint closure in ${\mathbb H}^n\cup S^{n-1}_\infty$. A [*chimney*]{} is the closure of the union of the geodesic arcs from $X$ to $Y$ that are perpendicular to $X$.
The distance between the hyperplanes $X$ and $Y$ defining the chimney is realized by a unique geodesic perpendicular to both planes, called the [*core*]{}, the length of which is the [*height*]{} of the chimney. The chimney cuts out a disk in $X$, which is called the [*base*]{}. Let ${\alpha}$ be the geodesic containing the core and ${\beta}$ a geodesic containing a diameter of the base. Then ${\alpha}$ and ${\beta}$ span a copy of ${\mathbb H}^2$ in ${\mathbb H}^n$. Furthermore, the intersection of this plane with the chimney cuts out half an ideal quadrilateral with orthospectrum $\{2\ell, 2r\}$, where $\ell$ is the length of the core and $r$ the radius of the base. We then have $\sinh(r)\sinh(\ell) = 1$, which yields $r = \log\coth\frac{\ell}2$ (see [@beardon]).
Let $M$ be a compact hyperbolic $n$-manifold with totally geodesic boundary $S$. Let $M_S$ be the covering space of $M$ associated to $S$. Then $M_S$ has a canonical decomposition into a piece of zero measure, together with two chimneys of height $\ell_i$ for each $\ell_i$ in the (unordered) orthospectrum.
If we take the bases of the chimney’s in the decomposition of the above theorem, then we get a decomposition of $\partial M$ into $(n-1)$-balls. With this decomposition, we can give the quick proof of Basmajian’s identity in [@calegari]:
Recall that we are working with the ordered orthospectrum. $S$ in $M_S$ is decomposed into a set of measure zero together with the union of the bases of the chimneys. Thus $$Area(S) = \sum_i V_{n-1}\left(\log\coth\frac{\ell_i}2\right),$$ where $V_n(r)$ is the volume of a hyperbolic $n$-ball of radius $r$.
Deriving the Length Function
----------------------------
Let $U_i$ be the interior of the ball associated to $\ell_i \in L(M)$. By above, the union of the $U_i$’s is a full measure set in $S = \partial M$. The measurable function $L: S \to {\mathbb R}$ assigning to each $x\in S$ the length of the arc emanating perpendicularly from $S$ at $x$ can be written as $L = \sum_{\ell_i\in L(M)} L_i$, where $L_i = L|U_i : U_i \to{\mathbb R}$ since the $U_i$’s are pairwise disjoint. As a chimney has rotational symmetry about its core, we see that $L(x)$ for $x\in S$ depends only on the distance between $x$ and the core, ie $L_i$ is a function of the radius; hence, deriving a formula for $L_i$ is a problem in the hyperbolic plane. Associated to each $U_i$ are two components of the boundary, $R_i$ and $T_i$, and two lifts of these components to hyperplanes in ${\mathbb H}^n$, ${\widetilde}R_i$ and ${\widetilde}T_i$. If $x\in R_i$, then we are interested in the chimney with its base in ${\widetilde}R_i$ and the lift of $x$ sitting in ${\widetilde}R_i$, call it ${\widetilde}x$. There is a unique copy of ${\mathbb H}^2\subset {\mathbb H}^n$ containing the core of the chimney, ${\widetilde}x$, and the geodesic connecting the two. The geodesic contained in this plane going through ${\widetilde}x$ and intersecting ${\widetilde}R_i$ perpendicularly intersects ${\widetilde}T_i$; furthermore, the length of this arc is $L_i(x)$. The diagram showing this situation in ${\mathbb H}^2$ is shown in Figure \[fig:figure\].
![A Lambert quadrilateral showing the setup for $L(r)$.[]{data-label="fig:figure"}](lambert2.png)
We see that $L_i(x)$ is the length of a side in a Lambert quadrilateral (a hyperbolic quadrilateral with three right angles). Let $r$ be the distance from $x$ to the core, then as we noted $L_i$ is solely a function of the radius, we will write $L_i(x) = L_i(r)$. From hyperbolic trigonometry we find $\coth L_i(r) = \operatorname{sech}(r)\coth(\ell_i)$ (see [@beardon]) or $$\label{eq: length}
L_i(r) = \operatorname{arccoth}(\operatorname{sech}(r)\coth(\ell_i)) = \frac12 \log \left(\frac{\coth \ell_i + \cosh r}{\coth \ell_i - \cosh r}\right),$$ where the second equality holds as $\operatorname{sech}(r)\coth(\ell_i) > 1$ on the domain of interest $r\in [0, \log\coth(\ell_i/2))$.
Proof of Theorem 1.2
--------------------
For completeness, we restate the result:
[**Theorem \[ortho\].**]{} [*Let $M= M^{n}$ be an $n$-dimensional compact hyperbolic manifold with nonempty totally geodesic boundary, then for all $k\in {\mathbb Z}^{\geq 0}$ $$A_k(M) = \frac{1}{V(\partial M)} \sum_{\ell \in L_M} F_{n,k}(\ell),$$ where $$F_{n,k}(x) =\Omega_{n-2}\int_0^{\log\coth(x/2)}\left[ \log\left(\frac{\coth x +\cosh r }{\coth x-\cosh r}\right)\right]^k \sinh^{n-2}(r)\ dr$$ and $\Omega_n$ is the volume of the standard $n$-sphere.*]{}
From the additivity property of measures we have $\int L^k\,\, dm = \sum \int_{U_i} L_i^k \, dm$. As $dm = dV/V(\partial M)$ and $dV$ is the $(n-1)$-dimensional hyperbolic volume form we can write it in spherical coordinates as $$dm = \frac{1}{V(\partial M)}\sinh^{n-2}(r)\, dr\, d\Omega_{n-2},$$ where $d\Omega_{n-2}$ is the volume form on the standard unit sphere. Above we saw that $L_i$ is a function solely of the radius and $U_i$ is a $n-1$ dimensional hyperbolic ball with radius $\log(\coth(\ell_i/2))$, so that $$\begin{aligned}
\int_{U_i} L_i^k \, dm &= \frac1{V(\partial M)} \int_0^{\log(\coth(\ell_i/2))} L_i^k(r) \sinh^{n-2}(r)\, dr\, d\Omega_{n-2} \\
&= \frac{\Omega_{n-2}}{V(\partial M)} \int_0^{\log(\coth(\ell_i/2))} L_i^k(r) \sinh^{n-2}(r)\, dr,\end{aligned}$$ where we write $\Omega_{n-2}$ for the volume of the standard $(n-2)$-dimensional unit sphere. Define $F_{n,k}(x)$ as stated in the theorem, so that the equality holds for $A_k(M)$ by .
Surface Case
============
Dilogarithms
------------
For $|z|<1$ in ${\mathbb C}$ the dilogarithm is defined as $${\mathrm{Li}_2}(z) = \sum_{n=1}^\infty \frac{z^n}{n^2}.$$ Using the Taylor series for $\log(1-z)$ about $z=0$, we can write $${\mathrm{Li}_2}(z) = \int_z^0 \frac{\log(1-z)}{z}dz.$$ One can then take a branch of $\log(z)$ in order to analytically continue ${\mathrm{Li}_2}(z)$ to the complex plane minus a branch cut. The standard definition of the dilogarithm assumes the branch cut for $\log(z)$ is along $(-\infty,0]$; however, for our purposes we will be interested in a different branch cut. Define the function ${\mathscr{D}}(z)$ to be the resulting dilogarithm by using the branch cut along $(-i\infty,0]$ for $\log(z)$ such that $\log(-1) = i\pi$. We note that ${\mathrm{Li}_2}(z) = {\mathscr{D}}(z)$ for $z\in (-\infty, 1)$.
The dilogarithm ${\mathrm{Li}_2}(z)$ has the following well-known identity (see [@lewin]): $${\mathrm{Li}_2}(z) + {\mathrm{Li}_2}\left(\frac 1z\right) = -\frac12 \log^2(-z) - \frac{\pi^2}6.$$ This identity is verified by differentiating both sides. As ${\mathrm{Li}_2}' = {\mathscr{D}}'$ and ${\mathrm{Li}_2}(z) = {\mathscr{D}}(z)$ on the negative real axis, the identity holds for ${\mathscr{D}}$; hence, $$\label{eq: identity}
{\mathscr{D}}(z) + {\mathscr{D}}\left(\frac1z\right) = -\frac12\log^2(-z) - \frac{\pi^2}6.$$ (The branch of logarithm being used should be clear from context.)
Deriving the formula for $F_{2,1}(x)$
-------------------------------------
For a fixed value of $x$, we define the map $H_x: [0, \log\coth(x/2)] \to {\mathbb C}$ as follows: $$H_x(r) = {\mathscr{D}}(-e^{-r}\coth(x/2))-{\mathscr{D}}(e^{-r}\coth(x/2))+{\mathscr{D}}(-e^{-r}\tanh(x/2))-{\mathscr{D}}(e^{-r}\tanh(x/2)).$$
$$\frac{d(\Re H_x)}{dr} = \log\frac{\coth x + \cosh r}{\coth x - \cosh r}$$
We first calculate $H_x'$ and then take real parts. Given the definition of the dilogarithm and the fact that $\coth(x/2)+\tanh(x/2) = 2\coth x$, we have that $$\begin{aligned}
H_x'(r) &= \log(1+e^{-r}\coth(x/2))-\log(1-e^{-r}\coth(x/2))+\\
&+\log(1+e^{-r}\tanh(x/2))-\log(1-e^{-r}\tanh(x/2))\\
& = \resizebox{.9\hsize}{!}{$\log[(1+e^{-r}\coth(x/2))(1+e^{-r}\tanh(x/2))]-\log[(1-e^{-r}\coth(x/2))(1-e^{-r}\tanh(x/2))]$}\\
& = \log[2e^{-r}(\cosh r + \coth x)] - \log[2e^{-r}(\cosh r - \coth x)]\\
&= \log(\coth x + \cosh r) - \log(\cosh r - \coth x)\\
& = \log \frac{\coth x + \cosh r}{\coth x - \cosh r} - i\pi.\end{aligned}$$ Given the domain for $H_x$, the argument of the logarithm above is always a positive real number.
We therefore see that $F_{2,1}(x) = 2\cdot\Re[H_x(\log\coth(x/2)) - H_x(0)]$ as $\Omega_0= 2$. For a surface $S$ with boundary, let $\ell(\partial S)$ be the sum of the lengths of the boundary components. Given the above we can now prove the following:
[**Corollary \[surface\].**]{} [*Let $S$ be a compact hyperbolic surface with nonempty totally geodesic boundary. Then $$A_1(S) = \frac2{\ell(\partial S)} \sum_{\ell\in L_M}\left[{\mathrm{Li}_2}\left(-\tanh^2\frac {\ell}2\right)-{\mathrm{Li}_2}\left(\tanh^2\frac {\ell}2\right)+\frac {\pi^2}4\right].$$*]{}
From the above formulation of $F_{2,1}(x)$, we get the following: $$\begin{aligned}
F_{2,1}(x) = 2\cdot\Re&\left[{\mathscr{D}}(a) + {\mathscr{D}}\left(\frac1a\right) - {\mathscr{D}}\left(-a\right)-{\mathscr{D}}\left(-\frac1a\right)+ {\mathscr{D}}\left(-\frac1{a^2}\right)-{\mathscr{D}}\left(\frac1{a^2}\right) - \frac {\pi^2}{4}\right],\end{aligned}$$ where $a = \coth\left(\frac x2\right)$. From applying $\eqref{eq: identity}$ twice we see that: $${\mathscr{D}}(a) + {\mathscr{D}}\left(\frac1a\right) - {\mathscr{D}}\left(-a\right)-{\mathscr{D}}\left(-\frac1a\right) = \frac12(\log^2(a) - \log^2(-a)).$$ Recalling that $\log(-1) = i\pi$, for $a>0$ we have $\log(a) - \log(-a) = -\log(-1)$, so that simplifying the above $${\mathscr{D}}(a) + {\mathscr{D}}\left(\frac1a\right) - {\mathscr{D}}\left(-a\right)-{\mathscr{D}}\left(-\frac1a\right) = \frac{\pi^2}2 - i\pi\log a \text{ for } a>0.$$ In particular, for positive values of $a$, the real part is always $\pi^2/2$. As $\ell_i$ is always positive this identity holds in the case of interest. Furthermore, ${\mathrm{Li}_2}\left(\pm\tanh(\ell_i/2)\right) = {\mathscr{D}}\left(\pm\tanh(\ell_i/2)\right)$ as $\pm\tanh(\ell_i/2) \in (-1,1)$; hence, the result follows.
Asymptotics for $F_{2,1}(x)$
----------------------------
We will use the following notation throughout the rest of the paper: For $f,g:{\mathbb R}\to {\mathbb R}$ we will write $f\sim g$ if $\lim_{x\to\infty} [f(x)/g(x)] = 1$. This is clearly an equivalence relation on real-valued functions. Below we find the asymptotic behavior of $F_{2,1}(x)$ from our above result; we note that we will also come to the same result later in the paper when we study the asymptotics of $F_{n,k}$ from the integral definition.
Let $F_{2,1}(x)$ be defined as above, then $F_{2,1}(x)\sim 8xe^{-x}.$
We start with the following observation: $$\lim_{x\to1} \frac{{\mathrm{Li}_2}(-x)-{\mathrm{Li}_2}(x)+\pi^2/4}{(x-1)\log(1-x)} = 1,$$ which is a direct application of L’Hôpital’s rule and the definition of the dilogarithm. From this, we can gather the following: $$\begin{aligned}
F_{2,1}(x) &\sim 2(\tanh^2(x/2)-1)\log(1-\tanh^2(x/2)) =4 \operatorname{sech}^2(x/2)\log\cosh(x/2)\\
&=4\left(\frac{2}{e^\frac x2+e^{-\frac x2}}\right)^2\log\left[e^\frac x2\left(\frac{1+e^{-x}}2\right)\right] \sim 8xe^{-x}\end{aligned}$$
Odd Dimensions
==============
In this section we will write an explicit formula for $A_1(M^n)$ with $n$ odd. For $n$ odd, we can simplify the integral in the definition of $F_{n,k}$ by using the substitution $u=\cosh r$ to get: $$F_{n,1}(x) = \Omega_{n-2}\int_1^{\coth x} (u^2-1)^{\frac{n-3}2}\log\frac{\coth x + u}{\coth x - u} \, du.$$ An elementary calculation gives the following integrals (up to a constant) when $m$ is even: $$\resizebox{\hsize}{!}{$F^+_m(u,y) = \int u^m \log(y + u) \, du = \frac{1}{m+1}\left[\left(u^{m+1}+y^{m+1}(x)\right)\log(y+u) + \sum_{k=1}^{m+1}(-1)^{m-k}\frac{y^{m-k+1}u^k}{k}\right]$}$$ $$F^-_m(u,y) = \int u^m\log(y- u)\, du = \frac{1}{m+1}\left[\left(u^{m+1}-y^{m+1}(x)\right)\log(y-u) - \sum_{k=1}^{m+1}\frac{y^{m-k+1}u^k}{k}\right].$$
Now set $f_m(x) = F^+_m(\coth x, \coth x)-F^+_m(1,\coth x)+F^-_m(1,\coth x) - F^-_m(\coth x,\coth x)$. After some routine manipulation and simplification, we find: $$f_m(x) = \frac{2\coth^{m+1}(x)}{m+1}\left[\log(2\cosh x)-x\tanh^{m+1}(x)+\sum_{k=1}^{\frac m2} \frac{1-\tanh^{2k}(x)}{2k}\right].$$ If we expand out the binomial in $F_{n,1}(x)$, we find $$F_{n,1}(x) = \Omega_{n-2}\sum_{j=0}^{\frac{n-3}2}(-1)^{\frac{n-3}2-j}\binom{\frac{n-3}2}{j}f_{2j}(x).$$ We then immediately have:
[**Corollary \[odd\].**]{} [*Let $M$ be an $n$-dimensional compact hyperbolic manifold with nonempty totally geodesic boundary where $n$ is odd. Then $$\begin{aligned}
\resizebox{\hsize}{!}{$A_1(M) = \frac{2\Omega_{n-2}}{Vol(\partial M)}\sum_{\ell_i\in L_M}\sum_{j=0}^{\frac{n-3}2}\frac{(-1)^{\frac{n-3}2-j}\binom{\frac{n-3}2}{j}}{2j+1}\coth^{2j+1}(\ell_i)\left[\log(2\cosh\ell_i)-\ell_i\tanh^{2j+1}(\ell_i)+\sum_{k=1}^j\frac{1-\tanh^{2k}(\ell_i)}{2k}\right].$}\end{aligned}$$*]{}
The Asymptotics of $F_{n,k}$
============================
In this section, we explore the asymptotic behavior of the $F_{n,k}$’s.
For all $n,k\in {\mathbb Z}^+$ $$\lim_{x\to\infty} \frac{e^{-(n-1)x}F_{n,k}(x)}{x^k} = \frac{2^{n+k-1}\Omega_{n-2}}{n-1},$$ Furthermore, for $n$ odd we have $$\lim_{x\to0}{x^{n-2}}F_{n,1}(x) = \frac{2}{n-2}[\log 2 + \frac12 H_{(n-1)/2}],$$ where $H_n$ is the $n^{th}$ harmonic number.
Recall that $F_{n,k}(x) = \Omega_{n-2} \int_0^{\log\coth(x/2)} L^k_x(r) \sinh^{n-2}(r)\, dr$. Using the substitution $u=\cosh r$, we have $$F_{n,k}(x) = \Omega_{n-2} \int_1^{\coth x} (u^2-1)^{\frac{n-3}2} \left[\log\left(\frac{\coth x+u}{\coth x- u}\right)\right]^k du.$$ For the moment, let $n\geq 3$, so that $(n-3)/2 \geq 0$, then $$\begin{aligned}
F_{n,k}(x)& \geq 2^{\frac{n-3}2}\Omega_{n-2}\int_1^{\coth x} (u-1)^{\frac{n-3}2}\left[\log\left(\frac{\coth x + u}{\coth x-u}\right)\right]^k du\\
F_{n,k}(x) & \leq (\coth x +1)^{\frac{n-3}2}\Omega_{n-2}\int_1^{\coth x} (u-1)^{\frac{n-3}2}\left[\log\left(\frac{\coth x + u}{\coth x-u}\right)\right]^k du.\end{aligned}$$ As $\coth x+1$ approaches 2 in the limit as $x$ goes to infinity, we see from the above two inequalities that $$F_{n,k}(x) \sim 2^{\frac{n-3}2}\Omega_{n-1}\int_1^{\coth x} (u-1)^{\frac{n-3}2}\left[\log\left(\frac{\coth x + u}{\coth x-u}\right)\right]^k du.$$ In the case $n=2$, the inequalities above are reversed, but yield the same result; hence, what follows will hold for all $n$. We now look at the following two inequalities: $$\resizebox{\hsize}{!}{$
\int_1^{\coth x} (u-1)^{\frac{n-3}2}\left[\log\left(\frac{\coth x + u}{\coth x-u}\right)\right]^k du \geq \int_1^{\coth x} (u-1)^{\frac{n-3}2}\left[\log(\coth x+1) - \log(\coth x -u)\right]^k du$}$$ $$\resizebox{\hsize}{!}{$
\int_1^{\coth x} (u-1)^{\frac{n-3}2}\left[\log\left(\frac{\coth x + u}{\coth x-u}\right)\right]^k du \leq \int_1^{\coth x} (u-1)^{\frac{n-3}2}\left[\log(2\coth x) - \log(\coth x -u)\right]^k du$}$$
Note for large $x$ that $\coth x - u<1$ for all $u\in [1,\coth x]$, so that $\log(\coth x - u) < 0$. As both $\log(2\coth x)$ and $\log(\coth x+1)$ limit to $\log 2$, we see that both the integrals in the inequalities are asymptotic to $\int_1^{\coth x} (u-1)^{(n-3)/2} [\log(\coth x- u)]^k du$. Let us write $a(x) = \coth x -1$ and $v = \frac{u-1}a$, so that we now have $$\resizebox{\hsize}{!}{$
F_{n,k}(x) \sim 2^{\frac{n-3}2}\Omega_{n-2}a^{\frac {n-1}2}\int_0^1 v^{\frac{n-3}2} [\log(a-av)]^k dv = 2^{\frac{n-3}2}\Omega_{n-2}a^{\frac {n-1}2}\int_0^1 v^{\frac{n-2}2} \left[\log a + \log\left(1-v\right)\right]^k dv.$}$$ As $\int_0^1 v^{(n-3)/2} [\log(1-v)]^m dv$ is finite for all $m$, we find that $$F_{n,k}(x) \sim (-1)^k 2^{\frac{n-3}2} \Omega_{n-2}(\log a)^{k}a^{\frac {n-1}2} \int_0^1 v^{\frac{n-3}2} dv =\frac {2^\frac {n-1}2 \Omega_{n-2}}{n-1} \left(\log\frac1a\right)^k a^{\frac{n-1}2}.$$ Since, $a(x) = \coth x -1 \sim 2e^{-2x}$, we get the stated result.
When $n$ is odd, we have the following when $x$ approaches 0: As $x$ tends to 0, it is easy to see that $\tanh^{m+1}(x)f_m(x)$ is finite. As $\lim_{x\to 0}[x \coth x]$ is finite, we see that $\lim_{x\to 0} {x^{m+1}}f_m(x)<\infty$. Again, as $F_{n,1}(x)$ is a sum of the $f_m$’s, the largest exponent dominates, which gives the result.
The Moment Generating Function in Dimension 3
=============================================
Let $M$ be a hyperbolic 3-manifold with totally geodesic boundary and let $S = \partial M$. We define the [*moment-generating function*]{} $M_L(t) = E[e^{tL}]$, where $E[X]$ denotes the expected value of a random variable $X$ with respect to our probability measure $dm = dV/V(\partial M)$. The moment-generating function encodes all the moments of $L$ in its derivatives: $A_k(M)=E[L^k] = M_L^{(k)}(0)$. In particular, by calculating $M_L(0)$ we will recover Basmajian’s identity and $A_1(M)$ by calculating $M_L'(0)$. The goal of this section is to prove that following theorem:
\[generating\] If $M$ is a compact hyperbolic 3-manifold with totally geodesic boundary $S$, then $$M_L(t) = \frac{4\pi}{V(S)}\sum_{\ell_i\in L_M} \coth(\ell_i)\cdot B\left(\frac{1-\tanh \ell_i}{2},1-t,1+t\right),$$ where $B$ is the incomplete beta function.
Hypergeometric Function and Incomplete Beta Function
----------------------------------------------------
The [*hypergeometric functions*]{} $_2F_1(a,b,c,z)$ for $z\in{\mathbb C}$ with $|z|<1$ are given by the power series:
$$_2F_1(a,b,c,z) = \sum_{n=0}^\infty \frac{(a)_n(b)_n}{(c)_n} \frac{z^n}{n!},$$ provided $c\notin {\mathbb Z}^{\leq 0}$ and where $$(a)_n = \left\{
\begin{array}{lr}
1 & \text{for }n = 0 \\
a(a+1)\cdots(a+n-1) & \text{for }n >0
\end{array} \right. .$$ We will use the following identity below: $$(1-z)^{-a} = {_2F_1}(a,1,1,z).$$
The [*incomplete beta functions*]{} $B(x,a,b)$ are defined as $$B(x,a,b) = \int_0^x s^{a-1}(1-s)^{b-1}ds.$$ We can also write an incomplete beta function in terms of a hypergeometric function as follows (see [@luke]): $$\label{eqn: beta}
B(x,a,b) = \frac{x^a}a {_2F_1}(a,1-b,a+1,x).$$
We present two calculations as technical lemmas that will allow us to derive the moment generating function.
$$\frac{\partial}{\partial x} {_2F_1}(1+t,t,2+t,x) = \frac{1+t}x \left[(1-x)^{-t} - {_2F_1}(1+t,t,2+t,x)\right]$$
We calculate: $$\begin{aligned}
\frac{\partial}{\partial x} {_2F_1}(1+t,t,2+t,x) &= \sum_{n=1}^\infty \frac{(t)_n(1+t)_n}{(2+t)_n} \frac{x^{n-1}}{(n-1)!} \\
&= \frac{1+t}x \sum_{n=1}^\infty(t)_{n}\frac{n}{t+n+1}\frac{x^n}{n!}\\
&= \frac{1+t}x \sum_{n=1}^\infty \left[(t)_n-\frac{(t)_n(1+t)}{t+n+1}\right]\frac{x^n}{n!} \\
&= \frac{1+t}x\left[ \sum_{n=0}^\infty (t)_n\frac{x^n}{n!}- \sum_{n=0}^\infty \frac{(t)_n(1+t)_n}{(t+2)_n} \frac{x^n}{n!}\right] \\
& = \frac{1+t}{x}[{_2F_1}(t,1,1,x)-{_2F_1}(1+t,t,2+t,x)]\\
&=\frac{1+t}x \left[(1-x)^{-t} - {_2F_1}(1+t,t,2+t,x)\right]\end{aligned}$$
Let $g(u,a,t) = (1+t)^{-1}(a+u)^{t+1}(2a)^{-t}{_2F_1}\left(1+t,t,2+t, \frac{a+u}{2a}\right),$ then $$\frac{\partial g}{\partial u} = \left(\frac{a+u}{a-u}\right)^t.$$
This is an immediate consequence of the previous lemma.
Proof of Theorem \[generating\]
-------------------------------
We can now find the moment generating function of $L$.
Let $S = \partial M$ and recall that $\Omega_1 = 2\pi$. By definition, $$\begin{aligned}
M_L(t) &= E[e^{tL}] = \int_S e^{tL}dm = \sum_i \int_{U_i} e^{tL_i}dm \\
&= \frac{2\pi}{V(S)}\sum_i\int_{0}^{\log(\coth(\ell_i/2))} \left(\frac{\coth(\ell_i)+\cosh r}{\coth(\ell_i)-\cosh r}\right)^t \sinh r\, dr.\\
&= \frac{2\pi}{V(S)}\sum_i\int_1^{\coth(\ell_i)}\left(\frac{\coth(\ell_i)+u}{\coth(\ell_i)-u}\right)^t du,\end{aligned}$$ where $u = \cosh r$. From the above lemma, we then have that $$M_L(t) = (2\pi/V(S))\sum_{\ell_i\in L_M}[g(\coth(\ell_i),\coth(\ell_i),t) - g(0,\coth(\ell_i),t)].$$ After expanding the above terms using the definition of $g$, some simplifications get us to: $$\resizebox{\hsize}{!}{$
M_L(t) =\frac{2\pi}{V(S)} \sum_{\ell_i\in L_m} \frac{\coth(\ell_i)}{1+t}\left[2{_2F_1}(1+t,t,2+t,1)-\frac12\left(\frac{1+\tanh(\ell_i)}{2}\right)^{t+1}{_2F_1}\left(1+t,t,2+t,\frac{1+\tanh(\ell_i)}2\right)\right].$}$$ By this becomes $$M_L(t) =\frac{4\pi}{V(S)} \sum_{\ell_i\in L_M} \coth(\ell_i)\left[B(1,1+t,1-t)-B\left(\frac{1+\tanh(\ell_i)}2,1+t,1-t\right)\right].$$ It is left to investigate $B(1,1+t,1-t) - B(a, 1+t,1-t)$: $$\resizebox{\hsize}{!}{$
B(1,1+t,1-t)-B(a,1+t,1-t) = \int_a^1s^t(1-s)^{-t}ds = -\int_{1-a}^0(1-u)^tu^{-t} du = B(1-a,1-t,1+t),$}$$ where $u = 1-a$. Therefore, we can conclude $$M_L(t) = \frac{4\pi}{V(S)}\sum_{\ell_i\in L_M}\coth(\ell_i) \cdot B\left(\frac{1-\tanh(\ell_i)}2,1-t,1+t\right).$$
Recovering Basmajian’s Identity in Dimension 3
----------------------------------------------
As $M_L(0) = 1$ we have $$1 = \frac{4\pi}{V(S)} \sum_{\ell_i\in L_M} \coth(\ell_i)\cdot B\left(\frac{1-\tanh(\ell_i)}2,1,1\right)$$ and as $B(a,1,1) = a$, we have $$V(S) = \sum_{\ell_i\in L_M} 2\pi(\coth(\ell_i) - 1) = \sum_{\ell_i\in L_M}\frac{2\pi e^{-\ell_i}}{\sinh(\ell_i)} = \sum_{\ell_i\in L_M} V_2(\log(\coth(\ell_i/2))),$$ where $V_2(r)$ is the area of a hyperbolic circle of radius $r$.
|
---
author:
- 'Mark Bun [^1]'
- 'Roi Livni [^2]'
- 'Shay Moran [^3]'
bibliography:
- 'biblio.bib'
title: |
An Equivalence Between Private Classification\
and Online Prediction
---
Introduction
============
This paper continues the study of the close relationship between differentially-private learning and online learning.
#### Differentially-Private Learning.
Statistical analyses and computer algorithms play significant roles in the decisions which shape modern society. The collection and analysis of individuals’ data drives computer programs which determine many critical outcomes, including the allocation of community resources, decisions to give loans, and school admissions.
While data-driven and automated approaches have obvious benefits in terms of efficiency, they also raise the possibility of unintended negative impacts, especially against marginalized groups. This possibility highlights the need for [*responsible*]{} algorithms that obey relevant ethical requirements (see e.g. [@Oneil2016weapons]).
[*Differential Privacy*]{} (DP) [@DworkMNS06] plays a key role in this context. Its initial (and primary) purpose was to provide a formal framework for ensuring individuals’ privacy in the statistical analysis of large datasets. But it has also found use in addressing other ethical issues such as [*algorithmic fairness*]{} (see, e.g. [@DworkHPRZ12; @cummings19fairness]).
Many tasks which involve sensitive data arise in machine learning (e.g. in medical applications and in social networks). Consequently, a large body of practical and theoretical work has been dedicated to understand which learning tasks can be performed by DP learning algorithms. The simplest and most extensively studied model of learning is the private PAC model [@Valiant84; @KasiviswanathanLNRS11], which captures binary classification tasks under differential privacy. A partial list of works on this topic includes [@KasiviswanathanLNRS11; @BeimelBKN14; @BunNSV15; @FeldmanX15; @BeimelNS16; @BunDRS18; @Beimel19Pure; @AlonLMM19; @kaplan2019privately].
In this manuscript we make progress towards characterizing what tasks are DP PAC-learnable by demonstrating a qualitative equivalence with online-learnable tasks.
#### Online Learning.
Online learning is a well-studied branch of machine learning which addresses algorithms making real-time predictions on sequentially arriving data. Such tasks arise in contexts including recommendation systems and advertisement placement. The literature on this subject is vast and includes several books, e.g. [@cesabianchi06prediction; @Shalev-Shwartz12book; @Hazan16oco].
[*Online Prediction*]{}, or [*Prediction with Expert Advice*]{} is a basic setting within online learning. Let $\mathcal{H} = \{h:X\to \{\pm1\} \}$ be a class of predictors (also called experts) over a domain $X$. Consider an algorithm which observes examples $(x_1,y_1)\ldots (x_T,y_T)\in X\times\{\pm 1\}$ in a sequential manner. More specifically, in each time step $t$, the algorithm first observes the instance $x_t$, then predicts a label $\hat{y}_t\in\{\pm 1\}$, and finally learns whether its prediction was correct. The goal is to minimize the [*regret*]{}, namely the number of mistakes compared to the best expert in $\mathcal{H}$: $$\sum_{t=1}^T 1[y_t\neq \hat{y}_t] - \min_{h^*\in\mathcal{H}} \sum_{t=1}^T 1[y_t\neq h^*(x_t)].$$ In this context, a class $\mathcal H$ is said to be online learnable if for every $T$, there is an algorithm that achieves sublinear regret $o(T)$ against any sequence of $T$ examples. The [*Littlestone dimension*]{} is a combinatorial parameter associated to the class ${\mathcal{H}}$ which characterizes its online learnability [@Littlestone87online; @Bendavid09agnostic]: $\mathcal H$ is online learnable if and only if it has a finite Littlestone dimension $d<\infty$. Moreover, the best possible regret $R(T)$ for online learning of ${\mathcal{H}}$ satisfies $$\Omega (\sqrt{dT}) \leq R(T) \leq O(\sqrt{dT\log T}).$$ Furthermore, if it is known that if one of the experts never errs (i.e. in the realizable mistake-bound model), then the optimal regret is exactly $d$.
#### Stability.
While at a first glance it may seem that online learning and differentially-private learning have little to do with one another, a line of recent works has revealed a tight connection between the two [@Agarwal17dponline; @Abernathy17onlilnedp; @AlonLMM19; @bousquet2019passing; @NeelRW19; @Joseph2019TheRO; @Gonen19privateonline].
At a high-level, this connection appears to boil down to the notion of stability, which plays a key role in both topics. On one hand, the definition of differential privacy is itself a form of stability; it requires robustness of the output distribution of an algorithm when its input undergoes small changes. On the other hand, stability also arises as a central motif in online learning paradigms such as [*Follow the Perturbed Leader*]{} [@Kalai02geometricalgorithms; @kalai05efficient] and [*Follow the Regularized Leader*]{} [@abernethy08competing; @Shalev07ftrl; @Hazan16oco].
In their monograph [@DworkR14], Dwork and Roth identified stability as a common factor of learning and differential privacy: [*“Differential privacy is enabled by stability and ensures stability… we observe a tantalizing moral equivalence between learnability, differential privacy, and stability.”*]{} This insight has found formal manifestations in several works. For example, Abernethy et al. used DP inspired stability methodology to derive a unified framework for proving state of the art bounds in online learning [@Abernathy17onlilnedp]. In the opposite direction, Agarwal and Singh showed that certain standard stabilization techniques in online learning imply differential privacy [@Agarwal17dponline].
Stability plays a key role in this work as well. Our main result, which shows that any class with a finite Littlestone dimension can be privately learned, hinges on the following form of stability: for $\eta > 0$ and $n\in\mathbb{N}$, a learning algorithm ${\mathcal{A}}$ is [*$(n,\eta)$-globally stable*]{}[^4] with respect to a distribution ${\mathcal{D}}$ over examples if there exists an hypothesis $h$ whose frequency as an output is at least $\eta$. Namely, $$\Pr_{S\sim {\mathcal{D}}^n}[{\mathcal{A}}(S) = h] \geq \eta.$$ Our argument follows by showing that every ${\mathcal{H}}$ can be learned by a globally-stable algorithm with parameters $\eta = \exp(-d), n=\exp(d)$, where $d$ is the Littlestone dimension of ${\mathcal{H}}$. As a corollary, we get an equivalence between global stability and differential privacy (which can be viewed as a form of local stability). That is, the existence of a globally stable learner for ${\mathcal{H}}$ is equivalent to the existence of a differentially-private learner for it (and both are equivalent to having a finite Littlestone dimension).
#### Littlestone Classes.
It is natural to ask which classes have finite Littlestone dimension. First, note that every finite class ${\mathcal{H}}$ has Littlestone dimension $d \leq \log\lvert {\mathcal{H}}\rvert$. There are also many natural and interesting infinite classes with finite Littlestone dimension. For example, let $X=\mathbb{F}^n$ be an $n$-dimensional vector space over a field $\mathbb{F}$ and let ${\mathcal{H}}\subseteq\{\pm 1\}^X$ consist of all (indicators of) affine subspaces of dimension $\leq d$. The Littlestone dimension of ${\mathcal{H}}$ is $d$. More generally, any class of hypotheses that can be described by polynomial *equalities* of constant degree has finite Littlestone dimension.[^5] This can be generalized even further to classes that are definable in [*stable theories*]{}. This (different, still) notion of stability is deep and well-explored in model theory. We refer the reader to [@Chase19modelmachine], Section 5.1 for more examples of stable theories and the Littlestone classes they correspond to.
#### Organization.
The rest of this manuscript is organized as follows. In Section \[sec:results\] we formally state our main results and discuss some implications. Section \[sec:overview\] overviews some of the main ideas in the proofs. Sections \[sec:preliminaries\] - \[sec:wrapping\] contain complete proofs, and the last section (Section \[sec:conc\]) concludes the paper with some suggestions for future work.
Main Results {#sec:results}
------------
We next present our main results. We begin with the statements concerning the relationship between online learning and differentially-private learning. In Section \[sec:stability\] we present and discuss the notion of global stability, and finally in Section \[sec:boosting\] we discuss an implication for private boosting. Throughout this section some standard technical terms are used. For definitions of these terms we refer the reader to Section \[sec:preliminaries\].
\[thm:main\] Let ${\mathcal{H}}\subseteq\{\pm 1\}^X$ be a class with Littlestone dimension $d$, let ${\varepsilon},\delta \in (0, 1)$ be privacy parameters, and let $\alpha,\beta \in (0, 1/2)$ be accuracy parameters. For $$n = O\left(\frac{16^d \cdot d^2 \cdot (d + \log(1/\beta\delta))}{\alpha{\varepsilon}}\right) = O_d\left(\frac{\log(1/\beta\delta)}{\alpha{\varepsilon}}\right)$$ there exists an $({\varepsilon},\delta)$-DP learning algorithm such that for every realizable distribution ${\mathcal{D}}$, given an input sample $S\sim {\mathcal{D}}^n$, the output hypothesis $f={\mathcal{A}}(S)$ satisfies ${\operatorname{loss}}_{{\mathcal{D}}}(f)\leq \alpha$ with probability at least $1-\beta$, where the probability is taken over $S\sim {\mathcal{D}}^n$ as well as the internal randomness of ${\mathcal{A}}$.
A similar result holds in the agnostic setting:
\[thm:agnostic\] Let ${\mathcal{H}}\subseteq\{\pm 1\}^X$ be a class with Littlestone dimension $d$, let ${\varepsilon}$, and $\delta \in (0, 1)$ be privacy parameters, and let $\alpha,\beta \in (0, 1/2)$ be accuracy parameters. For $$n = O\left(\frac{16^d \cdot d^2 \cdot (d + \log(1/\beta\delta))}{\alpha\epsilon} +\frac{\textrm{VC}({\mathcal{H}})+\log 1/\beta}{\alpha^2\epsilon} \right)$$ there exists an $({\varepsilon},\delta)$-DP learning algorithm such that for every distribution ${\mathcal{D}}$, given an input sample $S\sim {\mathcal{D}}^n$, the output hypothesis $f={\mathcal{A}}(S)$ satisfies $${\operatorname{loss}}_{{\mathcal{D}}}(f)\leq \min_{h\in {\mathcal{H}}} {\operatorname{loss}}_{{\mathcal{D}}}(h)+ \alpha$$ with probability at least $1-\beta$, where the probability is taken over $S\sim {\mathcal{D}}^n$ as well as the internal randomness of ${\mathcal{A}}$.
follows from by Theorem 2.3 in [@alon2020closure] which provides a general mechanism to transform a learner in the realizable setting to a learner in the agnostic setting[^6]. We note that formally the transformation in [@alon2020closure] is stated for a constant ${\varepsilon}=O(1)$. Taking ${\varepsilon}=O(1)$ is without loss of generality as a standard “secrecy-of-the-sample” argument can be used to convert this learner into one which is $({\varepsilon}, \delta)$-differentially private by increasing the sample size by a factor of roughly $1/{\varepsilon}$ and running the algorithm on a random subsample. See [@KasiviswanathanLNRS11; @Vadhan17] for further details.
\[thm:equivalence\] The following statements are equivalent for a class ${\mathcal{H}}\subseteq \{\pm 1\}^X$:
1. ${\mathcal{H}}$ is online learnable.
2. ${\mathcal{H}}$ is approximate differentially-privately PAC learnable.
Theorem \[thm:equivalence\] is a corollary of Theorem \[thm:main\] (which gives $1\to 2$) and the result by Alon et al. [@AlonLMM19] (which gives $2\to 1$). We comment that a quantitative relation between the learning and regret rates is also implied: it is known that the optimal regret bound for ${\mathcal{H}}$ is $\tilde \Theta_d(\sqrt{T})$, where the $\tilde \Theta_d$ conceals a constant which depends on the Littlestone dimension of ${\mathcal{H}}$ [@Bendavid09agnostic]. Similarly, we get that the optimal sample complexity of agnostically privately learning ${\mathcal{H}}$ is $\Theta_d(\frac{\log({1}/(\beta\delta))}{\alpha^2{\varepsilon}})$.
We remark however that the above equivalence is mostly interesting from a theoretical perspective, and should not be regarded as an efficient transformation between online and private learning. Indeed, the Littlestone dimension dependencies concealed by the $\tilde \Theta_d(\cdot)$ in the above bounds on the regret and sample complexities may be very different from one another. For example, there are classes for which the $\Theta_d(\frac{\log({1}/(\beta\delta))}{\alpha{\varepsilon}})$ bound hides a $\mathrm{poly}(\log^*(d))$ dependence, and the $\tilde \Theta_d(\sqrt{T})$ bound hides a $\Theta(d)$ dependence. One example which attains both of these dependencies is the class of thresholds over a linearly ordered domain of size $2^d$ [@AlonLMM19; @kaplan2019privately].
### Global Stability {#sec:stability}
Our proof of Theorem \[thm:main\], which establishes that every Littlestone class can be learned privately, hinges on an intermediate property which we call [*global stability*]{}:
Let $n\in\mathbb{N}$ be a sample size and $\eta > 0$ be a global stability parameter. An algorithm ${\mathcal{A}}$ is $(n,\eta)$-globally-stable with respect to a distribution ${\mathcal{D}}$ if there exists an hypothesis $h$ such that $$\Pr_{S\sim{\mathcal{D}}^n}[A(S) = h] \geq \eta.$$
While global stability is a rather strong property, it holds automatically for learning algorithms using a finite hypothesis class. By an averaging argument, every learner using $n$ samples which produces a hypothesis in a finite hypothesis class ${\mathcal{H}}$ is $(n, 1/|{\mathcal{H}}|)$-globally-stable. The following proposition generalizes “Occam’s Razor" for finite hypothesis classes to show that global stability is enough to imply similar generalization bounds in the realizable setting.
\[prop:gs\] Let ${\mathcal{H}}\subseteq\{\pm 1\}^X$ be a class, and assume that ${\mathcal{A}}$ is a learner for ${\mathcal{H}}$ (i.e. ${\operatorname{loss}}_S({\mathcal{A}}(S))=0$ for every realizable sample $S$). Let ${\mathcal{D}}$ be a realizable distribution such that ${\mathcal{A}}$ is $(n,\eta)$-globally-stable with respect to ${\mathcal{D}}$, and let $h$ be a hypothesis such that $\Pr_{S\sim{\mathcal{D}}^n}[A(S) = h] \geq \eta$, as guaranteed by the definition of global stability. Then, $${\operatorname{loss}}_{\mathcal{D}}(h) \leq \frac{\ln(1/\eta)}{n}.$$
Let $\alpha$ denote the loss of $h$, i.e. ${\operatorname{loss}}_{\mathcal{D}}(h) = \alpha$, and let $E_1$ denote the event that $h$ is consistent with the input sample $S$. Thus, $\Pr[E_1] = (1-\alpha)^n$. Let $E_2$ denote the event that ${\mathcal{A}}(S)=h$. By assumption, $\Pr[E_2]\geq \eta$. Now, since ${\mathcal{A}}$ is consistent we get that $E_2\subseteq E_1$, and hence that $\eta\leq(1-\alpha)^n$. This finishes the proof (using the fact that $1-\alpha \leq e^{-\alpha}$ and taking the logarithm of both sides).
Another way to view global stability is in the context of *pseudo-deterministic algorithms* [@Gat11pseudo]. A pseudo-deterministic algorithm is a randomized algorithm which yields some fixed output with high probability. Thinking of a realizable distribution ${\mathcal{D}}$ as an instance on which PAC learning algorithm has oracle access, a globally-stable learner is one which is “weakly” pseudo-deterministic in that it produces some fixed output with probability bounded away from zero. A different model of pseudo-deterministic learning, in the context of learning from membership queries, was defined and studied by Oliveira and Santhanam [@OliveiraS18].
We prove Theorem \[thm:main\] by constructing, for a given Littlestone class ${\mathcal{H}}$, an algorithm ${\mathcal{A}}$ which is globally stable with respect to realizable distribution.
### Boosting for Approximate Differential Privacy {#sec:boosting}
Our characterization of private learnability in terms of the Littlestone dimension has new consequences for boosting the privacy and accuracy guarantees of differentially-private learners. Specifically, it shows that the existence of a learner with weak (but non-trivial) privacy and accuracy guarantees implies the existence of a learner with any desired privacy and accuracy parameters — in particular, one with $\delta(n) = \exp(-\Omega(n))$.
\[thm:adp-boost\] There exists a constant $c > 0$ for which the following holds. Suppose that for some sample size $n_0$ there is an $({\varepsilon}_0, \delta_0)$-differentially private learner $\cal W$ for a class ${\mathcal{H}}$ satisfying the guarantee $$\Pr_{S\sim {\mathcal{D}}^{n_0}}[{\operatorname{loss}}_{{\mathcal{D}}}({\cal W}(S)) > \alpha_0 ] < \beta_0$$ for ${\varepsilon}_0 = 0.1, \alpha_0 = \beta_0 = 1/16$, and $\delta_0 \le c/n_0^2\log n_0$.
Then there exists a constant $C_{\mathcal{H}}$ such that for every $\alpha, \beta, {\varepsilon}, \delta \in (0, 1)$ there exists an $({\varepsilon}, \delta)$-differentially private learner for ${\mathcal{H}}$ with $$\Pr_{S\sim {\mathcal{D}}^{n}}[{\operatorname{loss}}_{{\mathcal{D}}}({{\mathcal{A}}}(S)) > \alpha] < \beta$$ whenever $n \ge C_{\mathcal{H}}\cdot \log(1/\beta\delta)/\alpha{\varepsilon}$.
Given a weak learner $\cal W$ as in the statement of Theorem \[thm:adp-boost\], the results of [@AlonLMM19] imply that ${\operatorname{Ldim}}({\mathcal{H}})$ is finite. Hence Theorem \[thm:main\] allows us to construct a learner for ${\mathcal{H}}$ with arbitrarily small privacy and accuracy, yielding Theorem \[thm:adp-boost\]. The constant $C_{{\mathcal{H}}}$ in the last line of the theorem statement suppresses a factor depending on ${\operatorname{Ldim}}({\mathcal{H}})$.
Prior to our work, it was open whether arbitrary learning algorithms satisfying approximate differential privacy could be boosted in this strong a manner. We remark, however, that in the case of *pure* differential privacy, such boosting can be done algorithmically and efficiently. Specifically, given an $({\varepsilon}_0, 0)$-differentially private weak learner as in the statement of Theorem \[thm:adp-boost\], one can first apply random sampling to improve the privacy guarantee to $(p{\varepsilon}_0, 0)$-differential privacy at the expense of increasing its sample complexity to roughly $n_0 /p$ for any $p \in (0, 1)$. The Boosting-for-People construction of Dwork, Rothblum, and Vadhan [@DworkRV10] (see also [@BunCS20]) then produces a strong learner by making roughly $T \approx \log(1/\beta)/\alpha^2$ calls to the weak learner. By composition of differential privacy, this gives an $({\varepsilon}, 0)$-differentially private strong learner with sample complexity roughly $n_0 \cdot \log(1/\beta)/\alpha^2{\varepsilon}$.
What goes wrong if we try to apply this argument using an $({\varepsilon}_0, \delta_0)$-differentially private weak learner? Random sampling still gives a $(p{\varepsilon}_0, p\delta_0)$-differentially private weak learner with sample complexity $n_0 / p$. However, this is not sufficient to improve the $\delta$ parameter of the learner *as a function of the number of samples $n$*. Thus the strong learner one obtains using Boosting-for-People still at best guarantees $\delta(n) = \tilde{O}(1/n^2)$. Meanwhile, Theorem \[thm:adp-boost\] shows that the existence of a $(0.1, \tilde{O}(1/n^2))$-differentially private learner for a given class implies the existence of a $(0.1, \exp(-\Omega(n))$-differentially private learner for that class.
We leave it as an interesting open question to determine whether this kind of boosting for approximate differential privacy can be done algorithmically.
Proof Overview {#sec:overview}
==============
We next give an overview of the main arguments used in the proof of Theorem \[thm:main\]. The proof consist of two parts: (i) we first show that every class with a finite Littlestone dimension can be learned by a globally-stable algorithm, and (ii) we then show how to generically obtain a differentially-private learner from any globally-stable learner.
Step 1: Finite Littlestone Dimension $\implies$ Globally-Stable Learning
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Let ${\mathcal{H}}$ be a concept class with Littlestone dimension $d$. Our goal is to design a globally-stable learning algorithm for ${\mathcal{H}}$ with stability parameter $\eta = \exp(-d)$ and sample complexity $n=\exp(d)$. We will sketch here a weaker variant of our construction which uses the same ideas but is simpler to describe.
The property of ${\mathcal{H}}$ that we will use is that it can be online learned in the realizable setting with at most $d$ mistakes (see Section \[sec:online\] for a brief overview of this setting). Let ${\mathcal{D}}$ denote a realizable distribution with respect to which we wish to learn in a globally-stable manner. That is, ${\mathcal{D}}$ is a distribution over examples $(x,c(x))$ where $c\in{\mathcal{H}}$ is an unknown target concept. Let $\mathcal{A}$ be a learning algorithm that makes at most $d$ mistakes while learning an unknown concept from ${\mathcal{H}}$ in the online model. Consider applying $\mathcal{A}$ on a sequence $S=((x_1,c(x_1))\ldots (x_n,c(x_n)))\sim{\mathcal{D}}^n$, and denote by $M$ the random variable counting the number of mistakes $\mathcal{A}$ makes in this process. The mistake-bound guarantee on ${\mathcal{A}}$ guarantees that $M\leq d$ always. Consequently, there is $0\leq i \leq d$ such that $$\Pr[M=i] \geq \frac{1}{d+1}.$$ Note that we can identify, with high probability, an $i$ such that $\Pr[M=i] \geq 1/2d$ by running ${\mathcal{A}}$ on $O(d)$ samples from ${\mathcal{D}}^n$. We next describe how to handle each of the $d+1$ possibilities for $i$. Let us first assume that $i=d$, namely that $$\Pr[M=d] \geq \frac{1}{2d}.$$ We claim that in this case we are done: indeed, after making $d$ mistakes it must be the case that ${\mathcal{A}}$ has completely identified the target concept $c$ (or else ${\mathcal{A}}$ could be presented with another example which forces it to make $d+1$ mistakes). Thus, in this case it holds with probability at least $1/2d$ that ${\mathcal{A}}(S)=c$ and we are done. Let us next assume that $i=d-1$, namely that $$\Pr[M=d-1] \geq \frac{1}{2d}.$$ The issue with applying the previous argument here is that before making the $d$’th mistake, ${\mathcal{A}}$ can output many different hypotheses (depending on the input sample $S$). We use the following idea: draw two samples $S_1,S_2 \sim {\mathcal{D}}^n$ independently, and set $f_1 = {\mathcal{A}}(S_1)$ and $f_2={\mathcal{A}}(S_2)$. Condition on the event that the number of mistakes made by ${\mathcal{A}}$ on each of $S_1,S_2$ is exactly $d-1$ (by assumption, this event occurs with probability at least $(1/2d)^2$) and consider the following two possibilities:
- $\Pr[f_1=f_2]\geq\frac{1}{4}$,
- $\Pr[f_1=f_2] < \frac{1}{4}$.
If (i) holds then using a simple calculation one can show that there is $h$ such that $\Pr[A(S) = h] \geq \frac{1}{(2d)^2}\cdot \frac{1}{4}$ and we are done. If (ii) holds then we apply the following [*“random contest”*]{} between $S_1,S_2$:
1. Pick $x$ such that $f_1(x)\neq f_2(x)$ and draw $y\sim\{\pm 1\}$ uniformly at random.
2. If $f_1(x)\neq y$ then the output is ${\mathcal{A}}(S_1 \circ (x,y))$, where $S_1\circ (x,y)$ denotes the sample obtained by appending $(x,y)$ to the end of $S$. In this case we say that $S_1$ “won the contest”.
3. Else, $f_2(x)\neq y$ then the output is ${\mathcal{A}}(S_2 \circ (x,y))$. In this case we that $S_2$ “won the contest”.
Note that adding the auxiliary example $(x,y)$ forces ${\mathcal{A}}$ to make exactly $d$ mistakes on $S_i\circ (x,y)$. Now, if $y\sim\{\pm 1\}$ satisfies $y = c(x) $ then by the mistake-bound argument it holds that ${\mathcal{A}}(S_i\circ (x,y))=c$. Therefore, since $\Pr_{y\sim\{\pm 1\}}[c(x)=y] = 1/2$, it follows that $$\Pr_{S_1,S_2, y}[{\mathcal{A}}(S_i\circ (x,y))=c] \geq \frac{1}{(2d)^2}\cdot \frac{3}{4}\cdot\frac{1}{2} =\Omega(1/d^2),$$ and we are done.
Similar reasoning can be used by induction to handle the remaining cases (the next one would be that $\Pr[M=d-2] \geq \frac{1}{2d}$, and so on). The proof we present in Section \[sec:LSstable\] is based on a similar idea of performing “random contests,” although the construction becomes more complex to handle other issues, such as generalization, which were not addressed here. For more details we refer the reader to the complete argument in Section \[sec:LSstable\].
Step 2: Globally-Stable Learning $\implies$ Differentially-Private Learning
---------------------------------------------------------------------------
Given a globally-stable learner ${\mathcal{A}}$ for a concept class ${\mathcal{H}}$, we can obtain a differentially-private learner using standard techniques in the literature on private learning and query release. If ${\mathcal{A}}$ is a $(\eta, m)$-globally stable learner with respect to a distribution ${\mathcal{D}}$, we obtain a differentially-private learner using roughly $m/\eta$ samples from that distribution as follows. We first run ${\mathcal{A}}$ on $k \approx 1/\eta$ independent samples, non-privately producing a list of $k$ hypotheses. We then apply a differentially-private “Stable Histograms” algorithm [@KorolovaKMN09; @BunNS16] to this list which allows us to privately publish a short list of hypotheses that appear with frequency $\Omega(\eta)$. Global stability of the learner ${\mathcal{A}}$ guarantees that with high probability, this list contains *some* hypothesis $h$ with small population loss. We can then apply a generic differentially-private learner (based on the exponential mechanism) on a fresh set of examples to identify such an accurate hypothesis from the short list.
Preliminaries {#sec:preliminaries}
=============
PAC Learning
------------
We use standard notation from statistical learning; see, e.g., [@Shalev14book]. Let $X$ be any “domain” set and consider the “label” set $Y=\{\pm 1\}$. A [*hypothesis*]{} is a function $h : X\to Y$, which we alternatively write as an element of $Y^X$. An [*example*]{} is a pair $(x, y) \in X\times Y$. A [*sample*]{} $S$ is a finite sequence of examples.
Let ${\mathcal{D}}$ be a distribution over $X \times \{\pm 1\}$. The population loss of a hypothesis $h : X \to \{\pm 1\}$ is defined by $${\operatorname{loss}}_{{\mathcal{D}}}(h) = \Pr_{(x, y) \sim {\mathcal{D}}}[h(x) \ne y].$$ Let $S=\bigl((x_i,y_i)\bigr)_{i=1}^n$ be a sample. The empirical loss of $h$ with respect to $S$ is defined by $${\operatorname{loss}}_{S}(h) = \frac{1}{n}\sum_{i=1}^n1[h(x_i)\neq y_i].$$
Let ${\mathcal{H}}\subseteq Y^X$ be a [*hypothesis class*]{}. A sample $S$ is said to be [*realizable by ${\mathcal{H}}$*]{} if there is $h\in H$ such that ${\operatorname{loss}}_S(h)=0$. A distribution ${\mathcal{D}}$ is said to be [*realizable by ${\mathcal{H}}$*]{} if there is $h\in H$ such that ${\operatorname{loss}}_{\mathcal{D}}(h)=0$. A [*learning algorithm*]{} $A$ is a (possibly randomized) mapping taking input samples to output hypotheses. We also use the following notation: for samples $S,T$, let $S\circ T$ denote the combined sample obtained by appending $T$ to the end of $S$.
Online Learning {#sec:online}
---------------
#### Littlestone Dimension.
The Littlestone dimension is a combinatorial parameter that captures mistake and regret bounds in online learning [@Littlestone87online; @Bendavid09agnostic].[^7] Its definition uses the notion of [*mistake trees*]{}. A mistake tree is a binary decision tree whose internal nodes are labeled by elements of $X$. Any root-to-leaf path in a mistake tree can be described as a sequence of examples $(x_1,y_1),...,(x_d,y_d)$, where $x_i$ is the label of the $i$’th internal node in the path, and $y_i=+1$ if the $(i+1)$’th node in the path is the right child of the $i$’th node and $y_i = -1$ otherwise. We say that a mistake tree $T$ is [*shattered* ]{}by ${\mathcal{H}}$ if for any root-to-leaf path $(x_1,y_1),...,(x_d,y_d)$ in $T$ there is an $h\in {\mathcal{H}}$ such that $h(x_i)=y_i$ for all $i\leq d$ (see Figure \[fig:shatteredtree\]). The Littlestone dimension of ${\mathcal{H}}$, denoted ${\operatorname{Ldim}}({\mathcal{H}})$, is the depth of largest complete tree that is shattered by ${\mathcal{H}}$. We say that ${\mathcal{H}}$ is a Littlestone class if it has finite Littlestone dimension.
![[]{data-label="fig:shatteredtree"}](shatteredtree.pdf)
#### Mistake Bound and the Standard Optimal Algorithm (${\mathsf{SOA}}$).
The simplest setting in which learnability is captured by the Littlestone dimension is called the [*mistake-bound model*]{} [@Littlestone87online]. Let ${\mathcal{H}}\subseteq \{\pm 1\}^X$ be a fixed hypothesis class known to the learner. The learning process takes place in a sequence of trials, where the order of events in each trial $t$ is as follows:
- the learner receives an instance $x_t\in X$,
- the learner responses with a prediction $\hat y_t\in \{\pm 1\}$, and
- the learner is told whether or not the response was correct.
We assume that the examples given to the learner are realizable in the following sense: For the entire sequence of trials, there is a hypothesis $h\in {\mathcal{H}}$ such that $y_t = h(x_t)$ for every instance $x_t$ and correct response $y_t$. An algorithm in this model learns ${\mathcal{H}}$ with mistake bound $M$ if for every realizable sequence of examples presented to the learner, it makes a total of at most $M$ incorrect predictions.
Littlestone showed that the minimum mistake bound achievable by any online learner is exactly ${\operatorname{Ldim}}({\mathcal{H}})$ [@Littlestone87online]. Furthermore, he described an explicit algorithm, called the [*Standard Optimal Algorithm*]{} (${\mathsf{SOA}}$), which achieves this optimal mistake bound.
[**Standard Optimal Algorithm (${\mathsf{SOA}}$)**]{}\
1. Initialize ${\mathcal{H}}_1 = {\mathcal{H}}$.
2. For trials $t = 1, 2, \dots$:
- For each $b \in \{\pm 1\}$ and $x \in X$, let ${\mathcal{H}}_t^b(x) = \{h \in {\mathcal{H}}_t : h(x) = b\}$. Define $h : X \to \{\pm 1\}$ by $h_t(x) = {\operatorname{argmax}}_b {\operatorname{Ldim}}({\mathcal{H}}_t^{b}(x))$.
- Receive instance $x_t$.
- Predict $\hat{y}_t = h_t(x_t)$.
- Receive correct response $y_t$.
- Update ${\mathcal{H}}_{t+1} = {\mathcal{H}}_t^{y_t}(x_t)$.
#### Extending the ${\mathsf{SOA}}$ to non-realizable sequences.
Our globally-stable learner for Littlestone classes will make use of an optimal online learner in the mistake bound model. For concreteness, we pick the ${\mathsf{SOA}}$ (any other optimal algorithm will also work). It will be convenient to extend the ${\mathsf{SOA}}$ to sequences which are not necessarily realizable by a hypothesis in ${\mathcal{H}}$. We will use the following simple extension of the ${\mathsf{SOA}}$ to non-realizable samples:
\[def:soaext\] Consider a run of the ${\mathsf{SOA}}$ on examples $(x_1,y_1),\ldots, (x_m,y_m)$, and let $h_t$ denote the predictor used by the ${\mathsf{SOA}}$ after seeing the first $t$ examples (i.e., $h_t$ is the rule used by the ${\mathsf{SOA}}$ to predict in the $(t+1)$’st trial). Then, after observing both $x_{t+1},y_{t+1}$ do the following:
- If the sequence $(x_1,y_1),\ldots, (x_{t+1},y_{t+1})$ is realizable by some $h\in{\mathcal{H}}$ then apply the usual update rule of the ${\mathsf{SOA}}$ to obtain $h_{t+1}$.
- Else, set $h_{t+1}$ as follows: $h_{t+1}(x_{t+1}) = y_{t+1}$, and $h_{t+1}(x)=h_t(x)$ for every $x\neq x_{t+1}$.
Thus, upon observing a non-realizable sequence, this update rule locally updates the maintained predictor $h_t$ to agree with the last example.
Differential Privacy
--------------------
We use standard definitions and notation from the differential privacy literature. For more background see, e.g., the surveys [@DworkR14; @Vadhan17]. For $a, b, {\varepsilon}, \delta \in [0, 1]$ let $a\approx_{{\varepsilon},\delta} b$ denote the statement $$a\leq e^{{\varepsilon}}b + \delta ~\text{ and }~ b\leq e^{\varepsilon}a + \delta.$$ We say that two probability distributions $p,q$ are [*$({\varepsilon},\delta)$-indistinguishable*]{} if $p(E) \approx_{{\varepsilon},\delta} q(E)$ for every event $E$.
\[def:private\] A randomized algorithm $$A: (X\times \{\pm 1\})^m \to \{\pm 1\}^X$$ is $({\varepsilon},\delta)$-differentially-private if for every two samples $S,S'\in (X\times \{\pm 1\})^n$ that disagree on a single example, the output distributions $A(S)$ and $A(S')$ are $({\varepsilon},\delta)$-indistinguishable.
We emphasize that $({\varepsilon}, \delta)$-indistinguishability must hold for every such pair of samples, even if they are not generated according to a (realizable) distribution.
The parameters ${\varepsilon},\delta$ are usually treated as follows: ${\varepsilon}$ is a small constant (say $0.1$), and $\delta$ is negligible, $\delta = n^{-\omega(1)}$, where $n$ is the input sample size. The case of $\delta=0$ is also referred to as [*pure differential privacy*]{}. Thus, a class ${\mathcal{H}}$ is privately learnable if it is PAC learnable by an algorithm $A$ that is $({\varepsilon}(n),\delta(n))$-differentially private with ${\varepsilon}(n) \leq 0.1$, and $\delta(n) \leq n^{-\omega(1)} $.
Globally-Stable Learning of Littlestone Classes {#sec:LSstable}
===============================================
Theorem Statement
-----------------
The following statement any class ${\mathcal{H}}$ with a bounded Littlestone dimension can be learned by a globally-stable algorithm.
\[thm:littlestone-frequent\] Let ${\mathcal{H}}$ be a hypothesis class with Littlestone dimension $d\geq 1$, let $\alpha>0$, and set $$m = (8^{d+1}+1)\cdot\Bigl\lceil\frac{d}{\alpha}\Bigr\rceil.$$ Then there exists a randomized algorithm $G : (X \times \{\pm 1\})^m \to \{\pm 1\}^X$ with the following properties. Let ${\mathcal{D}}$ be a realizable distribution and let $S\sim {\mathcal{D}}^m$ be an input sample. Then there exists a hypothesis $f$ such $$\Pr[G(S) = f] \geq \frac{1}{(d+1)2^{d+1}} \text{ and } {\operatorname{loss}}_{{\mathcal{D}}}(f) \leq \alpha.$$
The distributions ${\mathcal{D}}_k$
-----------------------------------
The Algorithm $G$ is obtained by running the ${\mathsf{SOA}}$ on a sample drawn from a carefully tailored distribution. This distribution belongs to a family of distributions which we define next. Each of these distributions can be sampled from using black-box access to i.i.d. samples from ${\mathcal{D}}$. Recall that for a pair of samples $S,T$, we denote by $S\circ T$ the sample obtained by appending $T$ to the end of $S$. Define a sequence of distributions ${\mathcal{D}}_k$ for $k\geq 0$ as follows:
[**Distributions ${\mathcal{D}}_k$**]{}\
Let $n$ denote an “auxiliary sample” size (to be fixed later) and let ${\mathcal{D}}$ denote the target realizable distribution over examples. The distributions ${\mathcal{D}}_k = {\mathcal{D}}_k({\mathcal{D}},n)$ are defined by induction on $k$ as follows:
1. ${\mathcal{D}}_0$: output the empty sample $\emptyset$ with probability 1.
2. Let $ k\ge 1 $. If there exists a $f$ such that $$\Pr_{S \sim {\mathcal{D}}_{k-1}, T\sim{\mathcal{D}}^n}[{\mathsf{SOA}}(S\circ T) = f] \geq 2^{-d},$$ or if ${\mathcal{D}}_{k-1}$ is undefined then ${\mathcal{D}}_k$ is undefined.
3. Else, ${\mathcal{D}}_k$ is defined recursively by the following process:
- Draw $S_0,S_1\sim {\mathcal{D}}_{k-1}$ and $T_0,T_1\sim{\mathcal{D}}^n$ independently.
- Let $f_0={\mathsf{SOA}}(S_0\circ T_0)$, $f_1={\mathsf{SOA}}(S_1\circ T_1)$.
- If $f_0=f_1$ then go back to step (i).
- Else, pick $x\in \{x: f_0(x)\neq f_1(x)\}$ and sample $y\sim\{\pm 1\}$ uniformly.
- If $f_0(x)\neq y$ then output $S_0 \circ T_0\circ \bigl((x,y)\bigr)$ and else output $S_1\circ T_1\circ \bigl((x,y)\bigr)$.
Please see Figure \[fig:Dk\] for an illustration of sampling $S\sim {\mathcal{D}}_k$ for $k=3$.
We next observe some basic facts regarding these distributions. First, note that whenever ${\mathcal{D}}_k$ is well-defined, the process in Item 3 terminates with probability 1.
Let $k$ be such that ${\mathcal{D}}_k$ is well-defined and consider a sample $S$ drawn from ${\mathcal{D}}_k$. The size of $S$ is $\lvert S\rvert = k\cdot(n + 1)$. Among these $k\cdot(n+1)$ examples there are $k\cdot n$ examples drawn from ${\mathcal{D}}$ and $k$ examples which are generated in Item 3(iv). We will refer to these $k$ examples as . Note that during the generation of $S\sim {\mathcal{D}}_k$ there are examples drawn from ${\mathcal{D}}$ which do not actually appear in $S$. In fact, the number of such examples may be unbounded, depending on how many times Items 3(i)-3(iii) were repeated. In Section \[sec:monte\] we will define a “Monte-Carlo” variant of ${\mathcal{D}}_k$ in which the number of examples drawn from ${\mathcal{D}}$ is always bounded. This Monte-Carlo variant is what we actually use to define our globally-stable learning algorithm, but we introduce the simpler distributions ${\mathcal{D}}_k$ to clarify our analysis.
The $k$ tournament examples satisfy the following important properties.
\[obs:mistakebound\] Let $k$ be such that ${\mathcal{D}}_k$ is well-defined and consider running the ${\mathsf{SOA}}$ on the concatenated sample $S\circ T$, where $S\sim {\mathcal{D}}_k$ and $T\sim {\mathcal{D}}^n$. Then
1. Each tournament example forces a mistake on the ${\mathsf{SOA}}$. Consequently, the number of mistakes made by the ${\mathsf{SOA}}$ when run on $S\circ T$ is at least $k$.
2. ${\mathsf{SOA}}(S\circ T)$ is consistent with $T$.
The first item follows directly from the definition of $x$ in Item 3(iv) and the definition of $S$ in Item 3(v). The second item clearly holds when $S\circ T$ is realizable by ${\mathcal{H}}$ (because the ${\mathsf{SOA}}$ is consistent). For non-realizable $S\circ T$, Item 2 holds by our extension of the ${\mathsf{SOA}}$ in Definition \[def:soaext\].
![[]{data-label="fig:Dk"}](DrawingfromDk.pdf)
### The Existence of Frequent Hypotheses
The following lemma is the main step in establishing global stability.
\[lem:frequentdist\] There exists $k\leq d$ and an hypothesis $f:X\to\{\pm 1\}$ such that $$\Pr_{S\sim {\mathcal{D}}_k, T\sim{\mathcal{D}}^n}[{\mathsf{SOA}}(S\circ T) = f] \geq 2^{-d}.$$
Suppose for the sake of contradiction that this is not the case. In particular, this means that ${\mathcal{D}}_d$ is well-defined and that for every $f$:
$$\label{eq:3}
\Pr_{S\sim {\mathcal{D}}_d, T\sim{\mathcal{D}}^n}[{\mathsf{SOA}}(S\circ T) = f] < 2^{-d}.$$
We show that this cannot be the case when $f=c$ is the target concept (i.e., for $c\in{\mathcal{H}}$ which satisfies ${\operatorname{loss}}_{\mathcal{D}}(c)=0$). Towards this end, note that with probability $2^{-d}$ over $S\sim {\mathcal{D}}_d$ we have that all $d$ tournament examples are consistent with $c$. Indeed, this follows since in each tournament example $(x_i,y_i)$, the label $y_i$ is drawn independently of $x_i$ and of the sample constructed thus far. So, $y_i=c(x_i)$ with probability $1/2$ independently for each tournament example.
Therefore, with probability $2^{-d}$ we have that $S\circ T$ is consistent with $c$ (because all examples in $S\circ T$ which are drawn from ${\mathcal{D}}$ are also consistent with $c$). Now, since each tournament example forces a mistake on the ${\mathsf{SOA}}$ (Observation \[obs:mistakebound\]), and since the ${\mathsf{SOA}}$ does not make more than $d$ mistakes on realizable samples, it follows that if all tournament examples in $S\sim {\mathcal{D}}_d$ are consistent with $c$ then ${\mathsf{SOA}}(S)={\mathsf{SOA}}(S\circ T)=c$. Thus,
$$\Pr_{S\sim {\mathcal{D}}_d, T\sim{\mathcal{D}}^n}[{\mathsf{SOA}}(S\circ T) = c] \geq 2^{-d},$$ which contradicts Equation \[eq:3\] and finishes the proof.
### Generalization
The next lemma shows that only hypotheses $f$ that generalize well satisfy the conclusion of Lemma \[lem:frequentdist\] (note the similarity of this proof with the proof of Proposition \[prop:gs\]):
\[lem:gen\] Let $k$ be such that ${\mathcal{D}}_k$ is well-defined. Then every $f$ such that $$\Pr_{S\sim {\mathcal{D}}_k, T\sim{\mathcal{D}}^n}[{\mathsf{SOA}}(S\circ T) = f] \geq 2^{-d}$$ satisfies ${\operatorname{loss}}_{{\mathcal{D}}}(f) \le \frac{d}{ n}$.
Let $f$ be a hypothesis such that $\Pr_{S\sim {\mathcal{D}}_k, T \sim {\mathcal{D}}^n}[{\mathsf{SOA}}(S \circ T) = f] \geq 2^{-d}$ and let $\alpha={\operatorname{loss}}_{{\mathcal{D}}}(h)$. We will argue that $$\label{eq:4}
2^{-d} \leq (1-\alpha)^{n}.$$ Define the events $A,B$ as follows.
1. $A$ is the event that ${\mathsf{SOA}}(S\circ T) = f$. By assumption, $\Pr[A] \geq 2^{-d}$.
2. $B$ is the event that $f$ is consistent with $T$. Since $\lvert T\rvert = n$, we have that $\Pr[B] = (1-\alpha)^{n}$.
Note that $A \subseteq B$: Indeed, ${\mathsf{SOA}}(S\circ T)$ is consistent with $T$ by the second item of Observation \[obs:mistakebound\]. Thus, whenever ${\mathsf{SOA}}(S\circ T)=f$, it must be the case that $f$ is consistent with $T$. Hence, $\Pr[A]\leq \Pr[B]$, which implies Inequality \[eq:4\] and finishes the proof (using the fact that $1-\alpha \leq 2^{-\alpha}$ and taking logarithms on both sides).
The Algorithm $G$
-----------------
### A Monte-Carlo Variant of ${\mathcal{D}}_k$ {#sec:monte}
Consider the following first attempt of defining a globally-stable learner $G$: (i) draw $i\in\{0\ldots d\}$ uniformly at random, (ii) sample $S\sim{\mathcal{D}}_i$, and (iii) output ${\mathsf{SOA}}(S\circ T)$, where $T\sim {\mathcal{D}}^n$. The idea is that with probability $1/(d+1)$ the sampled $i$ will be equal to a number $k$ satisfying the conditions of Lemma \[lem:frequentdist\], and so the desired hypothesis $f$ guaranteed by this lemma (which also has low population loss by Lemma \[lem:gen\]) will be outputted with probability at least $2^{-d}/(d+1)$.
The issue here is that sampling $f\sim {\mathcal{D}}_i$ may require an unbounded number of samples from the target distribution ${\mathcal{D}}$ (in fact, ${\mathcal{D}}_i$ may even be undefined). To circumvent this possibility, we define a Monte-Carlo variant of ${\mathcal{D}}_k$ in which the number of examples drawn from ${\mathcal{D}}$ is always bounded.
[**The Distributions $\tilde {\mathcal{D}}_k$ (a Monte-Carlo variant of ${\mathcal{D}}_k$)**]{}\
1. Let $n$ be the auxiliary sample size and $N$ be an upper bound on the number of examples drawn from ${\mathcal{D}}$.
2. $\tilde {\mathcal{D}}_0$: output the empty sample $\emptyset$ with probability 1.
3. For $k> 0$, define $\tilde {\mathcal{D}}_k$ recursively by the following process:
- [**Throughout the process, if more than $N$ examples from ${\mathcal{D}}$ are drawn (including examples drawn in the recursive calls), then output “Fail”.**]{}
- Draw $S_0,S_1\sim \tilde {\mathcal{D}}_{k-1}$ and $T_0,T_1\sim{\mathcal{D}}^n$ independently.
- Let $f_0={\mathsf{SOA}}(S_0\circ T_0)$, $f_1={\mathsf{SOA}}(S_1\circ T_1)$.
- If $f_0=f_1$ then go back to step (i).
- Else, pick $x\in \{x: f_0(x)\neq f_1(x)\}$ and sample $y\sim\{\pm 1\}$ uniformly.
- If $f_0(x)\neq y$ then output $S_0 \circ T_0\circ \bigl((x,y)\bigr)$ and else output $S_1\circ T_1\circ \bigl((x,y)\bigr)$.
Note that $\tilde {\mathcal{D}}_k$ is well-defined for every $k$, even for $k$ such that ${\mathcal{D}}_k$ is undefined (however, for such $k$’s the probability of outputting “Fail” may be large).
It remains to specify the upper bound $N$ on the number of examples drawn from ${\mathcal{D}}$ in $\tilde {\mathcal{D}}_k$. Towards this end, we prove the following bound on the expected number of examples from ${\mathcal{D}}$ that are drawn during generating $S\sim{\mathcal{D}}_k$:
\[lem:avgsample\] Let $k$ be such that ${\mathcal{D}}_k$ is well-defined, and let $M_k$ denote the number of examples from ${\mathcal{D}}$ that are drawn in the process of generating $S\sim {\mathcal{D}}_k$. Then, $$\mathbb{E}[M_k] \leq 4^{k+1}\cdot n.$$
Note that $\mathbb{E}[M_0]=0$ as ${\mathcal{D}}_0$ deterministically produces the empty sample. We first show that for all $0 < i < k$, $$\label{eq:1}
\mathbb{E}[M_{i+1}] \leq 4\mathbb{E}[M_{i}] + 4n,$$ and then conclude the desired inequality by induction.
To see why Inequality \[eq:1\] holds, let the random variable $R$ denote the number of times Item 3(i) was executed during the generation of $S\sim {\mathcal{D}}_{i+1}$. That is, $R$ is the number of times a pair $S_0,S_1\sim {\mathcal{D}}_i$ and a pair $T_0,T_1\sim {\mathcal{D}}^n$ were drawn. Observe that $R$ is distributed geometrically with success probability $\theta$, where: $$\begin{aligned}
\theta &= 1 - \Pr_{S_0,S_1, T_0,T_1}\bigl[{\mathsf{SOA}}(S_0\circ T_0) = {\mathsf{SOA}}(S_1\circ T_1)\bigr] \\
&= 1 - \sum_{f}\Pr_{S, T}\bigl[{\mathsf{SOA}}(S\circ T) = f\bigr]^2\\
&\geq 1-2^{-d},\end{aligned}$$ where the last inequality follows because $i< k$ and hence ${\mathcal{D}}_i$ is well-defined, which implies that $\Pr_{S, T}\bigl[{\mathsf{SOA}}(S\circ T) = f\bigr]\leq 2^{-d}$ for all $h$.
Now, the random variable $M_{i+1}$ can be expressed as follows: $$M_{i+1} = \sum_{j=1}^\infty M_{i+1}^{(j)},$$ where $$M_{i+1}^{(j)} =
\begin{cases}
0 &\text{if } R < j,\\
\text{$\#$ of examples drawn from ${\mathcal{D}}$ in the $j$'th execution of Item 3(i)} &\text{if } R\geq j.
\end{cases}$$ Thus, $\mathbb{E}[M_{i+1}] = \sum_{j=1}^\infty\mathbb{E}[M_{i+1}^{(j)}]$. We claim that $$\mathbb{E}[M_{i+1}^{(j)}] = (1-\theta)^{j-1}\cdot (2\mathbb{E}[M_i] + 2n).$$ Indeed, the probability that $R\geq j$ is $(1-\theta)^{j-1}$ and conditioned on $R\geq j$, in the $j$’th execution of Item 3(i) two samples from ${\mathcal{D}}_{i}$ are drawn and two samples from ${\mathcal{D}}^n$ are drawn. Thus $$\mathbb{E}[M_{i+1}] = \sum_{j=1}^\infty(1-\theta)^{j-1}\cdot (2\mathbb{E}[M_i] + 2n)= \frac{1}{\theta}\cdot (2\mathbb{E}[M_i] + 2n) \leq 4\mathbb{E}[M_i] + 4n,$$ where the last inequality is because $\theta \geq 1- 2^{-d}\geq 1/2$. This gives Inequality \[eq:1\]. Next, using that $\mathbb{E}[M_0]=0$, a simple induction gives $$\mathbb{E}[M_{i+1}]\leq (4+4^2+\ldots+ 4^{i+1})n \leq 4^{i+2}n,$$ and the lemma follows by taking $i+1=k$.
### Completing the Proof of Theorem \[thm:littlestone-frequent\]
Our globally-stable learning algorithm $G$ is defined as follows.
[**Algorithm $G$**]{}\
1. Consider the distribution $\tilde {\mathcal{D}}_k$, where the auxiliary sample size is set to $n=\lceil \frac{d}{\alpha}\rceil$ and the sample complexity upper bound is set to $N=8^{d+1}\cdot n$.
2. Draw $k\in \{0,1,\ldots, d\}$ uniformly at random.
3. Output $h={\mathsf{SOA}}(S\circ T)$, where $T\sim {\mathcal{D}}^n$ and $S\sim \tilde {\mathcal{D}}_k$.
First note that the sample complexity of $G$ is $\lvert S\rvert + \lvert T\rvert \leq N+n = (8^{d+1}+1)\cdot\lceil\frac{d}{\alpha}\rceil$, as required. It remains to show that there exists a hypothesis $f$ such that: $$\Pr[G(S) = f] \geq \frac{2^{-(d+1)}}{d+1} \text{ and } {\operatorname{loss}}_{{\mathcal{D}}}(f) \leq \alpha.$$ By Lemma \[lem:frequentdist\], there exists $k^*\leq d$ and $f^*$ such that $$\Pr_{S\sim {\mathcal{D}}_{k^*}, T\sim{\mathcal{D}}^n}[{\mathsf{SOA}}(S\circ T) = f^*] \geq 2^{-d}.$$ By Lemma \[lem:gen\], $${\operatorname{loss}}_{{\mathcal{D}}}(f^*) \leq \frac{d}{n} \leq \alpha.$$ We claim that $G$ outputs $f^*$ with probability at least $2^{-(d+1)}$. To see this, let $M_{k^*}$ denote the number of examples drawn from ${\mathcal{D}}$ during the generation of $S\sim {\mathcal{D}}_{k^*}$. Lemma \[lem:avgsample\] and an application of Markov’s inequality yield $$\begin{aligned}
\Pr\bigl[M_{k^*} > 8^{d+1}\cdot n\bigr] &\leq \Pr\bigl[M_{k^*} > 2^{d+1}\cdot 4^{k^*+1}\cdot n\bigr] \tag{because $k^*\leq d$}\\
&\leq 2^{-(d+1)}. \tag{by Markov's inequality, since $\mathbb{E}[M_{k^*}]\leq 4^{k^*+1}\cdot n$}\end{aligned}$$ Therefore, $$\begin{aligned}
\Pr_{S\sim \tilde {\mathcal{D}}_{k^*}, T\sim {\mathcal{D}}^n}[{\mathsf{SOA}}(S\circ T) = f^*] &= \Pr_{S\sim {\mathcal{D}}_{k^*}, T\sim {\mathcal{D}}^n}[{\mathsf{SOA}}(S\circ T) = f^* \text{ and } M_{k^*} \leq 8^{d+1}\cdot n] \\
&\geq 2^{-d} - 2^{-(d+1)} = 2^{-(d+1)}.\end{aligned}$$ Thus, since $k=k^*$ with probability $1/(d+1)$, it follows that $G$ outputs $f^*$ with probability at least $\frac{2^{-(d+1)}}{d+1}$ as required.
Globally-Stable Learning Implies Private Learning {#sec:stableprivate}
=================================================
In this section we prove that any globally-stable learning algorithm yields a differentially-private learning algorithm with finite sample complexity.
Tools from Differential Privacy
-------------------------------
We begin by stating a few standard tools from the differential privacy literature which underlie our construction of a learning algorithm.
Let $X$ be a data domain and let $S \in X^n$. For an element $x \in X$, define ${\operatorname{freq}}_S(x) = \frac{1}{n} \cdot \#\{i \in [n] : x_i = x\}$, i.e., the fraction of the elements in $S$ which are equal to $x$.
\[lem:histograms\] Let $X$ be any data domain. For $$n \ge O\left(\frac{\log(1/\eta\beta\delta)}{\eta {\varepsilon}}\right)$$ there exists an $({\varepsilon}, \delta)$-differentially private algorithm ${\mathsf{Hist}}$ which, with probability at least $1-\beta$, on input $S = (x_1, \dots, x_n)$ outputs a list $L \subseteq X$ and a sequence of estimates $a \in [0, 1]^{|L|}$ such that
- Every $x$ with ${\operatorname{freq}}_S(x) \ge \eta$ appears in $L$ and
- For every $x \in L$, the estimate $a_x$ satisfies $|a_x - {\operatorname{freq}}_S(x)| \le \eta$.
Using the Exponential Mechanism of McSherry and Talwar [@McSherryT07], Kasiviswanathan et al. [@KasiviswanathanLNRS11] described a generic differentially-private learner based on approximate empirical risk minimization.
\[lem:generic\] Let $H \subseteq \{\pm 1\}^X$ be a collection of hypotheses. For $$n = O\left(\frac{\log|H| +\log(1/\beta)}{\alpha {\varepsilon}}\right)$$ there exists an ${\varepsilon}$-differentially private algorithm ${\mathsf{GenericLearner}}: (X \times \{\pm 1\})^n \to H$ such that the following holds. Let ${\mathcal{D}}$ be a distribution over $(X \times \{\pm 1\})$ such that there exists $h^* \in H$ with $${\operatorname{loss}}_{{\mathcal{D}}}(h^*) \le \alpha.$$ Then on input $S \sim {\mathcal{D}}^n$, algorithm ${\mathsf{GenericLearner}}$ outputs, with probability at least $1-\beta$, a hypothesis $\hat{h} \in H$ such that $${\operatorname{loss}}_{\mathcal{D}}(\hat{h}) \le 2\alpha.$$
Our formulation of the guarantees of this algorithm differ slightly from those of [@KasiviswanathanLNRS11], so we give its standard proof for completeness.
The algorithm ${\mathsf{GenericLearner}}(S)$ samples a hypothesis $h \in H$ with probability proportional to $\exp(-{\varepsilon}n {\operatorname{loss}}_S(h) / 2)$. This algorithm can be seen as an instantiation of the Exponential Mechanism [@McSherryT07]; the fact that changing one sample changes the value of ${\operatorname{loss}}_S(h)$ by at most $1$ implies that ${\mathsf{GenericLearner}}$ is ${\varepsilon}$-differentially private.
We now argue that ${\mathsf{GenericLearner}}$ is an accurate learner. Let $E$ denote the event that the sample $S$ satisfies the following conditions:
1. For every $h \in H$ such that ${\operatorname{loss}}_{{\mathcal{D}}}(h) > 2\alpha$, it also holds that ${\operatorname{loss}}_{S}(h) > 5\alpha/3$, and
2. For the hypothesis $h^* \in H$ satisfying ${\operatorname{loss}}_{{\mathcal{D}}}(h^*) \le \alpha$, it also holds that ${\operatorname{loss}}_{S}(h^*) \le 4\alpha / 3$.
We claim that $\Pr[E] \ge 1-\beta/2$ as long as $n \ge O(\log(|H|/\beta) / \alpha)$. To see this, let $h \in H$ be an arbitrary hypothesis with ${\operatorname{loss}}_D(h) > 2\alpha$. By a multiplicative Chernoff bound[^8] we have ${\operatorname{loss}}_S(h) > 7\alpha / 4$ with probability at least $1 - \beta/(4|H|)$ as long as $n \ge O(\log(|H|/\beta) / \alpha)$. Taking a union bound over all $h \in H$ shows that condition 1. holds with probability at least $1 - \beta/4$. Similarly, a multiplicative Chernoff bound ensures that condition 2 holds with probability at least $1 - \beta/4$, so $E$ holds with probability at least $1-\beta/2$.
Now we show that conditioned on $E$, the algorithm ${\mathsf{GenericLearner}}(S)$ indeed produces a hypothesis $h$ with ${\operatorname{loss}}_D(\hat{h}) \le 2\alpha$. This follows the standard analysis of the accuracy guarantees of the Exponential Mechanism. Condition 2 of the definition of event $E$ guarantees that ${\operatorname{loss}}_S(h^*) \le 4\alpha / 3$. This ensures that the normalization factor in the definition of the Exponential Mechanism is at least $\exp(-2{\varepsilon}\alpha n /3)$. Hence by a union bound, $$\Pr[{\operatorname{loss}}_S(\hat{h}) > 5\alpha/3] \le |H| \cdot \frac{\exp(-5{\varepsilon}\alpha n / 6)}{\exp(-2{\varepsilon}\alpha n / 3)} = |H| e^{-{\varepsilon}\alpha n / 6}.$$ Taking $n \ge O(\log(|H|/\beta) / \alpha{\varepsilon})$ ensures that this probability is at most $\beta / 2$. Given that ${\operatorname{loss}}(\hat{h}) \le 5\alpha / 3$, Condition 1 of the definition of event $E$ ensures that ${\operatorname{loss}}_{{\mathcal{D}}}(\hat{h}) \le 2\alpha$. Thus, for $n$ sufficiently large as described, we have overall that ${\operatorname{loss}}_{{\mathcal{D}}}(\hat{h}) \le 2\alpha$ with probability at least $1- \beta$.
Construction of a Private Learner
---------------------------------
We now describe how to combine the Stable Histograms algorithm with the Generic Private Learner to convert any globally-stable learning algorithm into a differentially-private one.
\[thm:selection\] Let ${\mathcal{H}}$ be a concept class over data domain $X$. Let $G : (X \times \{\pm 1\})^m \to \{\pm 1\}^X$ be a randomized algorithm such that, for ${\mathcal{D}}$ a realizable distribution and $S \sim {\mathcal{D}}^m$, there exists a hypothesis $h$ such that $\Pr[G(S) = h] \ge \eta$ and ${\operatorname{loss}}_{{\mathcal{D}}}(h) \le \alpha / 2$.
Then for some $$n = O\left(\frac{m \cdot \log(1/\eta\beta\delta)}{\eta{\varepsilon}} + \frac{\log(1/\eta\beta)}{\alpha{\varepsilon}}\right)$$ there exists an $({\varepsilon}, \delta)$-differentially private algorithm $M: (X \times \{\pm 1\})^n \to \{\pm 1\}^X$ which, given $n$ i.i.d. samples from $\mathcal{D}$, produces a hypothesis $\hat{h}$ such that ${\operatorname{loss}}_{{\mathcal{D}}}(\hat{h}) \le \alpha$ with probability at least $1-\beta$.
Theorem \[thm:selection\] is realized the learning algorithm $M$ described below. Here, the parameter $$k = O\left(\frac{\log(1/\eta\beta\delta)}{\eta{\varepsilon}}\right)$$ is chosen so that Lemma \[lem:histograms\] guarantees Algorithm ${\mathsf{Hist}}$ succeeds with the stated accuracy parameters. The parameter $$n' = O\left(\frac{\log(1/\eta\beta)}{\alpha{\varepsilon}}\right)$$ is chosen so that Lemma \[lem:generic\] guarantees that ${\mathsf{GenericLearner}}$ succeeds on a list $L$ of size $|L| \le 2/\eta$ with the given accuracy and confidence parameters.
[**Differentially-Private Learner $M$**]{}\
1. Let $S_1, \dots, S_k$ each consist of $m$ i.i.d. samples from ${\mathcal{D}}$. Run $G$ on each batch of samples producing $h_1 = G(S_1), \dots, h_k = G(S_k)$.
2. Run the Stable Histogram algorithm ${\mathsf{Hist}}$ on input $H = (h_1, \dots, h_k)$ using privacy parameters $({\varepsilon}/2, \delta)$ and accuracy parameters $(\eta/8, \beta/3)$, producing a list $L$ of frequent hypotheses.
3. Let $S'$ consist of $n'$ i.i.d. samples from ${\mathcal{D}}$. Run ${\mathsf{GenericLearner}}(S')$ using the collection of hypotheses $L$ with privacy parameter ${\varepsilon}/ 2$ and accuracy parameters $(\alpha / 2, \beta/3)$ to output a hypothesis $\hat{h}$.
We first argue that the algorithm $M$ is differentially private. The outcome $L$ of step 2 is generated in a $({\varepsilon}/2, \delta)$-differentially-private manner as it inherits its privacy guarantee from ${\mathsf{Hist}}$. For every fixed choice of the coin tosses of $G$ during the executions $G(S_1), \dots, G(S_k)$, a change to one entry of some $S_i$ changes at most one outcome $h_i \in H$. Differential privacy for step 2 follows by taking expectations over the coin tosses of all the executions of $G$, and for the algorithm as a whole by simple composition.
We now argue that the algorithm is accurate. Using standard generalization arguments, we have that with probability at least $1-\beta/3$, $$\left|{\operatorname{freq}}_H(h) - \Pr_{S \sim {\mathcal{D}}^m}[G(S) = h]\right| \le \frac{\eta}{8}$$ for every $h \in \{\pm 1\}^X$ as long as $k \ge O(\log(1/\beta)/\eta)$. Let us condition on this event. Then by the accuracy of the algorithm ${\mathsf{Hist}}$, with probability at least $1-\beta/2$ it produces a list $L$ containing $h^*$ together with a sequence of estimates that are accurate to within additive error $\eta / 8$. In particular, $h^*$ appears in $L$ with an estimate $a_{h^*} \ge \eta - \eta/8 - \eta/8 \ge 3\eta / 4$.
Now remove from $L$ every item $h$ with estimate $a_h < 3\eta / 4$. Since every estimate is accurate to within $\eta / 8$, this leaves a list with $|L| \le 2/\eta$ that contains $h^*$ with ${\operatorname{loss}}_{{\mathcal{D}}}(h^*) \le \alpha$. Hence, with probability at least $1-\beta/3$, step 3 succeeds in identifying $h^*$ with ${\operatorname{loss}}_{{\mathcal{D}}}(h^*) \le \alpha/2$.
The total sample complexity of the algorithm is $k\cdot m + n'$ which matches the asserted bound.
Wrapping up (Proof of Theorem \[thm:main\]) {#sec:wrapping}
===========================================
We now combine Theorem \[thm:littlestone-frequent\] (finite Littlestone dimension $\implies$ global stability) with Theorem \[thm:selection\] (global stability $\implies$ private learnability) to prove Theorem \[thm:main\].
Let ${\mathcal{H}}$ be a hypothesis class with Littlestone dimension $d$ and let ${\mathcal{D}}$ be any realizable distribution. Then Theorem \[thm:littlestone-frequent\] guarantees, for $m = O(8^d \cdot d / \alpha)$, the existence of a randomized algorithm $G : (X \times \{\pm 1\})^m \to \{\pm 1\}^X$ and a hypothesis $f$ such that $$\Pr[G(S) = f] \ge \frac{1}{(d+1)2^{d+1}} \text{ and } {\operatorname{loss}}_{{\mathcal{D}}}(f) \le \alpha / 2.$$ Taking $\eta = 1/(d+1)2^{d+1}$, Theorem \[thm:selection\] gives an $({\varepsilon}, \delta)$-differentially private learner with sample complexity $$n = O\left(\frac{m \cdot \log(1/\eta\beta\delta)}{\eta{\varepsilon}} + \frac{\log(1/\eta\beta)}{\alpha{\varepsilon}}\right) = O\left(\frac{16^d \cdot d^2 \cdot (d + \log(1/\beta\delta))}{\alpha{\varepsilon}}\right).$$
Conclusion {#sec:conc}
==========
We conclude this paper with a few suggestions for future work.
1. [**Sharper Quantitative Bounds.**]{} Our upper bound on the differentially-private sample complexity of a class ${\mathcal{H}}$ has an exponential dependence on the Littlestone dimension ${\operatorname{Ldim}}({\mathcal{H}})$, while the lower bound by [@AlonLMM19] depends on $\log^*({\operatorname{Ldim}}({\mathcal{H}}))$. The recent work by [@kaplan2019privately] shows that for some classes the lower bound is nearly tight (up to a polynomial factor). It would be interesting to determine whether the upper bound can be improved in general.
2. [**Characterizing Private Query Release.**]{} Another fundamental problem in differentially-private data analysis is the query release, or equivalently, data sanitization problem: Given a class ${\mathcal{H}}$ and a sensitive dataset $S$, output a synthetic dataset $\hat{S}$ such that $h(S) \approx h(\hat{S})$ for every $h \in {\mathcal{H}}$. Does finite Littlestone dimension characterize when this task is possible? Such a statement would follow if one could make our private learner for Littlestone classes *proper* [@bousquet2019passing].
3. [**Oracle-Efficient Learning.**]{} Neel, Roth, and Wu [@NeelRW19] recently began a systematic study of oracle-efficient learning algorithms: Differentially-private algorithms which are computationally efficient when given oracle access to their non-private counterparts. The main open question left by their work is whether *every* privately learnable concept class can be learned in an oracle-efficient manner. Our characterization shows that this is possible if and only if Littlestone classes admit oracle-efficient learners.
4. [**General Loss Functions.**]{} It is natural to explore whether the equivalence between online and private learning extends beyond binary classification (which corresponds to the 0-1 loss) to regression and other real-valued losses.
5. [**Global Stability.**]{} It would be interesting to perform a thorough investigation of global stability and to explore potential connections to other forms of stability in learning theory, including uniform hypothesis stability [@Bousquet02stability], PAC-Bayes [@McAllester99PACB], local statistical stability [@Ligett19adaptive], and others.
6. [**Differentially-Private Boosting.**]{} Can the type of private boosting presented in Section \[sec:boosting\] be done algorithmically, and ideally, efficiently?
Acknowledgements {#acknowledgements .unnumbered}
================
We would like to thank Amos Beimel and Uri Stemmer for pointing out, and helping to fix, a mistake in the derivation of \[thm:agnostic\] in a previous version. We also thank Yuval Dagan for providing useful comments and for insightful conversations.
[^1]: Department of Computer Science, Boston University *mbun@bu.edu*
[^2]: Department of Electrical Engineering, Tel Aviv University *rlivni@tauex.tau.ac.il*
[^3]: Google AI Princeton *shaymoran1@gmail.com*
[^4]: The word [*global*]{} highlights a difference with other forms of algorithmic stability. Indeed, previous forms of stability such as DP and [*uniform hypothesis stability*]{} [@Bousquet02stability] are local in the sense that they require output robustness subject to [*local*]{} changes in the input. However, the property required by global stability captures stability with respect to resampling the entire input.
[^5]: Note that if one replaces “equalities” with “inequalities” then the Littlestone dimension may become unbounded while the VC dimension remains bounded. This is demonstrated, e.g., by halfspaces which are captured by polynomial inequalities of degree $1$.
[^6]: Theorem 2.3 in [@alon2020closure] is based on a previous realizable-to-agnostic transformation from [@beimel2015learning] which applies to [*proper*]{} learners. Here we require the more general transformation from [@alon2020closure] as the learner implied by may be improper.
[^7]: It appears that the name “Littlestone dimension” was coined in [@Bendavid09agnostic].
[^8]: I.e., for independent random variables $Z_1, \dots, Z_n$ whose sum $Z$ satisfies $\operatorname*{\mathbb{E}}[Z] = \mu$, we have for every $\delta \in (0, 1)$ that $\Pr[Z \le (1-\delta)\mu] \le \exp(-\delta^2\mu / 2)$ and $\Pr[Z \ge (1 + \delta)\mu] \le \exp(-\delta^2\mu / 3)$.
|
---
author:
- 'Kazuhiro [Kuboki]{}[^1]'
title: Effect of Band Structure on the Symmetry of Superconducting States
---
(\#1)
The superconducting (SC) state of Sr$_2$RuO$_4$ attracts much attention, since it is likely to have a spin-triplet pairing symmetry.[@Maeno; @Rice; @Ishida; @Luke; @Mack] The triplet Cooper pairs in $^3$He are formed due to the ferromagnetic spin fluctuation, so that it may be natural to assume that a SC state in Sr$_2$RuO$_4$ is also realized by the same mechanism. However, recent neutron scattering experiments showed that the ferromagnetic (${\vec q} = 0$) spin fluctuation is not large, but the peak is located near ${\vec q} = (\pm 2\pi/3, \pm 2\pi/3)$,[@neutron] consistent with band structure calculations.[@Mazin] In view of this fact it is proposed that the antiferromagnetic spin fluctuation may lead to the spin-triplet SC state.[@Sato; @Ogata] Similar results have also been obtained in different contexts.[@Kagan; @Chubukov]
In this article we study the effect of the band structure on the symmetry of SC states. First we treat a single-band tight-binding model on a square lattice using a mean-field approximation (MFA). We find that the spin-triplet and the spin-singlet SC states, together with states with their coexistence can occur for the same type of interaction simply by changing the shape of the Fermi surface.[@tsuchi] Micnas [*et al*]{}.[@Micnas] studied the stability of various SC states in a model similar to ours, but they have determined only the bare $T_{\rm c}$. Namely they solved only the linearized self-consistency equations. We will solve the self-consistency equations without linearization to determine the phase diagram, and clarify the reason for the change of the symmetry of the SC state as the band structure is changed.
Experimental results on Sr$_2$RuO$_4$ seem to indicate that the SC state in this system has a line (or lines) of nodes.[@Nishi; @Ishida2; @Bonalde; @Matsuda; @Lupien] In order to explain these results theoretically,[@Miyake; @Graf; @Maki; @Kuroki] Hasegawa [*et al*]{}.[@Hase] proposed an interlayer-pairing state which has horizontal lines of nodes based on a symmetry argument. We will examine the stability of this type of interlayer-pairing state when the in-plane and the interlayer interactions compete.
First we consider a tight-binding model on a square lattice whose Hamiltonian is given by $$\begin{array}{rl}
H = & \displaystyle -\sum_{i,j,\sigma} t_{ij}
c^\dagger_{i\sigma}c_{j\sigma}
- \mu \sum_{i\sigma} c^\dagger_{i\sigma} c_{i\sigma} \\
& \\
- & \displaystyle \sum_{i,j} V_{ij}
n_{i\uparrow}n_{j\downarrow}
\end{array}$$ where $\mu$ is the chemical potential, and $t_{ij}$ is defined as $$t_{ij} = \displaystyle
t \sum_{\delta=\pm {\hat x},\pm {\hat y}} \delta_{i,j+\delta}
+ t' \sum_{\delta=\pm {\hat x} \pm {\hat y}} \delta_{i,j+\delta}.$$ Namely, $t$ ($t'$) is the transfer integral for the (next) nearest-neighbor sites and ${\hat x}$ (${\hat y}$) is the unit vector in the $x$ ($y$) direction (lattice constant is taken to be unity). Similarly the nearest-neighbor attractive interaction $V_{ij}$ is defined as $V_{ij} = V \sum_{\delta=\pm {\hat x}, \pm {\hat y}}
\delta_{i,j+\delta}$ ($V > 0$). This Hamiltonian is decoupled by a standard mean-field procedure $$\begin{array}{rl}
n_{i\uparrow}n_{j\downarrow} = &
c^\dagger_{i\uparrow}c_{i\uparrow}
c^\dagger_{j\downarrow}c_{j\downarrow} \\
& \\
\to & \displaystyle
\Delta_{ij} c^\dagger_{j\downarrow}c^\dagger_{i\uparrow}
+ \Delta_{ij}^{*} c_{i\uparrow}c_{j\downarrow}
- |\Delta_{ij}|^2
\end{array}$$ with $\Delta_{ij} \equiv \langle c_{i\uparrow}c_{j\downarrow} \rangle$ being the SC order parameter (OP). On the square lattice the $d_{x^2-y^2}$- ($\Delta_d$), extended $s$- ($\Delta_s$), $p_x$- ($\Delta_{p_x}$) and $p_y$-wave ($\Delta_{p_y}$) symmetries are possible for the nearest-neighbor interaction, and the corresponding OP’s are defined as $$\begin{array}{rl}
\Delta_d(i) = & \displaystyle (\Delta_{i,i+x} + \Delta _{i,i-x}
- \Delta_{i,i+y} - \Delta_{i,i-y})/4 \\
%& \\
\Delta_s(i) = & \displaystyle (\Delta_{i,i+x} + \Delta_{i,i-x}
+ \Delta_{i,i+y} + \Delta_{i,i-y})/4 \\
%& \\
\Delta_{p_{x(y)}}(i) = & \displaystyle
i (\Delta_{i,i+x(y)} - \Delta_{i,i-x(y)})/2 .
\end{array}$$ Assuming that these OP’s are uniform (i.e., independent of $i$) we obtain the following self-consistency equations: $$\begin{array}{rl}
\Delta_{d(s)} = & \displaystyle \frac{V}{4N}\sum_k
\omega_{d(s)}(k)\frac{\Delta_k}{E_k}\tanh \Bigl(\frac{E_k}{2T}\Bigr) \\
& \\
\Delta_{p_{x(y)}} = & \displaystyle \frac{V}{2N}\sum_k
\omega_{p_{x(y)}}(k)\frac{\Delta_k}{E_k}\tanh \Bigl(\frac{E_k}{2T}\Bigr)
\end{array}$$ where $N$ and $T$ are the total number of lattice sites and the temperature, respectively, and $$\begin{array}{rl}
E_k = & \displaystyle \sqrt{\xi_k^2 + |\Delta_k|^2} \\
& \\
\xi_k = & \displaystyle -2t(\cos k_x + \cos k_y) -4t'\cos k_x \cos k_y -\mu \\
& \\
\Delta_k = & \displaystyle 2 \sum_{j=d,s,p_x,p_y} \omega_j(k) \Delta_j
\end{array}$$ with $$\begin{array}{rl}
\omega_d(k) = & \displaystyle \cos k_x - \cos k_y, \\
%& \\
\omega_s(k) = & \displaystyle \cos k_x + \cos k_y, \\
%& \\
\omega_{p_{x(y)}} = & \displaystyle \sin k_{x(y)}.
\end{array}$$ In the following we will solve the self-consistency equations to determine the phase giagram in the plane of $T$ and $\mu$. These equations are solved by the iteration method, starting from various sets of initial values. When several solutions are obtained for the same set of parameters ($\mu$, $T$), the state with the lowest free energy is adopted as the true one. Here we note that the SC long-range order cannot exist at finite temperature in a purely two dimensional (2D) system. However, $T_c$ obtained within the MFA in purely 2D systems can give a reasonable estimate of $T_c$ in the presence of small three dimensionality.
In Fig.1 we show the phase diagram in the plane of $T$ and $\mu$ (or, the electron density), for $t =1$, $t' = 0$ and $V = 1.5$. (Due to the particle-hole symmetry, the result for $-\mu$ is the same as that for $\mu$, and so it is not shown.) It is seen that a $d_{x^2-y^2}$-wave SC state is stabilized near half-filling ($\mu \sim 0$), while an extended $s$-wave state occurs at high (and low) densities ($\mu \sim \pm 4t$). In the region between $d$- and $s$-wave states spin-triplet $(p_x \pm ip_y)$-wave states appear. The $(p_x + ip_y)$- and the $(p_x - ip_y)$-states are degenerate but different states, and they transform each other under parity ${\cal P}$ and time-reversal ${\cal T}$ transformation. Then the system breaks ${\cal P}$ and ${\cal T}$ symmetries spontaneously, and these states are usually denoted as the chiral $p$-wave SC states. Near the boundary between triplet $(p_x \pm ip_y)$ and singlet ($d$ or $s$) states we find states where the spin-singlet and the spin-triplet OP’s coexist. These coexisting states, $(d \pm ip_y)$- and $(s \pm ip_y)$-states, are degenerate with $(d \pm ip_x)$- and $(s \pm ip_x)$- states, respectively.[@mixture] The $(d \pm ip_x \pm ip_y)$ and $(s \pm ip_x \pm ip_y)$ states are slightly higher in energy so that they are only local minima of the free energy. There is no reason (regarding symmetry) which precludes the coexistence of spin-triplet and spin-singlet SCOP’s, and it is the energy that decides which state should appear. Actually the coexistence of $d$- and $p$-wave OP’s has been found in superconductor/(anti)ferromagnet bilayer systems, where the proximity effect induces the imbalance of spin-up and spin-down electron densities.[@prox]
The above results show that both triplet and singlet SC states can occur with the same type of interaction. In order to understand this we use the Ginzburg-Landau (GL) theory by expanding the free energy with respect to $\Delta$’s[@SigUe] $$\begin{array}{rl}
{\cal F}_\Delta = & \displaystyle \frac{1}{S} \int d^2r \Bigl(\sum_{j=d,s,p_x,p_y}
\big[\alpha_j|\Delta_j|^2 + \beta |\Delta_j|^4
\big] \\
& \\
+ & \displaystyle \sum_{i \not= j} \big[\gamma_{ij}^{(1)} |\Delta_i|^2
|\Delta_j|^2 + \gamma_{ij}^{(2)} \big(\Delta_i^2\Delta_j^{*2} + c.c.
\big)\big]\Bigr)
\end{array}$$ where the gradient and higher order terms are discarded, and $S$ is the area of the system. As $T$ is decreased the OP with the highest (bare) transition temperature $T_c^{(0)}$ ($\alpha(T_c^{(0)}) = 0$) appears. The explicit forms of $\alpha$’s are given as $$\begin{array}{rl}
\alpha_{d(s)} = & \displaystyle 4V \Bigl(1 - \frac{V}{N}\sum_k
\omega_{d(s)}^2(k)\frac{\tanh(\xi_k/2T)}{2\xi_k}\Bigr) \\
& \\
\alpha_{p_{x(y)}} = & \displaystyle 2V \Bigl(1 - \frac{V}{N}\sum_k
\omega_{p_{x(y)}}^2(k) \frac{\tanh(\xi_k/2T)}{\xi_k}\Bigr).
\end{array}$$ The Fermi surface (FS) near the band edge ($\mu \sim \pm 4t$) is close to the $\Gamma$ point or ${\mib k} = (\pm \pi, \pm \pi)$, so that $|\omega_s(k)|$ is large on the FS and thus $\Delta_s$ is favored. On the other hand the FS at half-filling is the square connecting four points $(\pm \pi, 0), (0,\pm \pi)$, and $\omega_s(k)$ vanishes there. Then $\Delta_s$ is suppressed near half-filling and $\Delta_d$ is favored. For intermediate $\mu$ ($\mu \sim \pm 2t$), the FS comes close to the points $k_x =\pm \pi/2$ or $k_y = \pm \pi/2$ so that $|\omega_{p_{x(y)}}|$ can be large. Then the $p$-wave states have the highest $T_c$ in the region between $d$- and $s$-wave states ($p_x$ and $p_y$ states are degenerate).
When more than one $\alpha$ become negative there may be a coexistence of several OP’s. In this case $\gamma$ terms will play important roles. We can explicitly show that $\beta_i > 0$, $\gamma_{ij}^{(1)} > 0$ and $\gamma_{ij}^{(2)} > 0$ ($i,j = d, s, p_x, p_y$). The fact $\gamma_{ij}^{(2)} > 0$ indicates that the OP’s would form complex rather than real combinations (if they coexist), and in this case the nodes are removed and the system gains more condensation energy. This is the reason why the chiral ($p_x \pm ip_y$)-state (rather than ($p_x \pm p_y$)-state) appears. The coexisting states also have the complex combinations of OP’s due the same reason. In Fig.1 the region of $(d + ip_y)$-state is much wider than that of $(s + ip_y)$-state. This can be understood as follows. The nodes in the $d$-wave state can be removed by the introduction of the $ip_x$ component, while in the $s$-state there is already a full gap so that the lowering of the energy due to the second OP is much smaller.
Whether or not the above argument is correct can be tested by considering the case of $t = 0$, $t' \not =1$, i.e., with only the next-nearest-neighbor hopping terms. The Fermi surface at half-filling consists of the lines $k_x = \pm \pi/2$ and $k_y = \pm \pi/2$, so that the $(p_x \pm ip_y)$-state should be most favored near half-filling. This is actually the case as shown in Fig.2. Here the chiral SC state appears at and near half-filling, and $d$- or extended $s$-wave state occurs away from half-filling.
Next we examine the stability of interlayer-pairing SC states in a tetragonal system with a weak interlayer interaction. The Hamiltonian in this case is $$\begin{array}{rl}
H = & \displaystyle H_{2D} + H_\perp \\
& \\
H_\perp = & \displaystyle
- t_\perp \sum_i \sum_{\delta=\pm {\hat z}} c^\dagger_i c_{i+\delta}
- V_\perp \sum_i \sum_\delta n_{i\uparrow} n_{i+\delta,\downarrow}
\end{array}$$ where the summation on $\delta$ in the second term is taken over $\delta=\pm {\hat z} \pm {\hat x}, \pm {\hat z} \pm {\hat y}$ and $H$ in eq.(1) is redefined as $H_{2D}$. Here ${\hat z}$ denotes the unit vector in the $z$-direction with a lattice constant $c$. For the $t_\perp$-term we simply take the nearest-neighbor hopping, but we do not consider the nearest-neighbor interaction because of the following reason. The interaction terms with $\delta = \pm {\hat z}$ (with coupling constant $V_\perp^{(0)}$) may lead to the OP of the form $\Delta_{i,i \pm z}$. This OP is invariant under the rotation around the $z$-axis so that it is decoupled from the in-plane OP (denoted as $\Delta_\parallel$) if $\Delta_\parallel$ has a $d$- or $p$-wave symmetry. Then $V_\perp^{(0)}$ of the order of $V$ (in-plane coupling constant) is necessary to stabilize $\Delta_{i,i \pm z}$. When $\Delta_\parallel$ has an $s$-wave symmetry, it couples to $\Delta_{i,i\pm z}$ and then the latter becomes finite even for an infinitesimal $V_\perp^{(0)}$. Since we are not interested in the $s$-wave case, we do not consider $V_\perp^{(0)}$ in the following.
We decouple $H_\perp$ using the same procedure as in 2D case. The possible symmetries of the OP’s are $$\begin{array}{rl}
(\alpha) & \sin k_x \cos k_zc, \ \ \sin k_y \cos k_zc \\
(\beta) & \cos k_x \sin k_zc, \ \ \cos k_y \sin k_zc \\
(\gamma) & \sin k_x \sin k_zc, \ \ \sin k_y \sin k_zc \\
(\delta) & \cos k_x \cos k_zc, \ \ \cos k_y \cos k_zc.
\end{array}$$ Here ($\alpha$) and ($\beta$) (($\gamma$) and ($\delta$)) are spin-triplet (spin-singlet) states. (Note the states with $x$ and $y$ interchanged are degenerate.) Among these eight OP’s we are interested in the states in ($\alpha$), since their complex combinations, i.e., ($\sin k_x \pm i\sin k_y)\cos k_zc$ are proposed to describe the SC state of Sr$_2$RuO$_4$.[@Hase] This state has horizontal lines of nodes, and this behavior is consistent with the experimental results. For $t_\perp = 0$ all interlayer pairing OP’s ($\Delta_\perp$) do not couple to $\Delta_\parallel$, while for $t_\perp \not= 0$ some of $\Delta_\perp$ may have a bilinear coupling (in the sense of GL theory) to $\Delta_\parallel$ if both OP’s have the same symmetry. In the latter case $\Delta_\perp$ can be finite once $V_\perp \not= 0$. In view of this we take $t=1, t'=0, V_1=1.5$ and $\mu=-2$, since $\Delta_\parallel$ has the $(p_x \pm ip_y)$-symmetry for these values of parameters, and OP’s of $(\alpha)$ may couple to $\Delta_\parallel$. In the following we consider $(\sin k_x + i\sin k_y)\cos k_zc$- and $(\sin k_x + i\sin k_y)\sin k_zc$-wave OP’s, and denote them as $\Delta_\perp^{(c)}$ and $\Delta_\perp^{(s)}$, respectively. Then we calculate $\Delta_\parallel$, $\Delta_\perp^{(c)}$ and $\Delta_\perp^{(s)}$ self-consistently as functions of $V_\perp$.
In Fig.3 the results for $t_\perp = 0$ at $T = 0$ is shown. In this case $\Delta_\parallel$ and $\Delta_\perp$ are not coupled. Since the FS has no warping along the $z$-axis, it is not energetically favorable to introduce the second component of OP ($\Delta_\perp$) because all parts of the Fermi surface are already gapped by $\Delta_\parallel$. Thus $\Delta_\perp$ is absent for $V_\perp < V_{\rm c}$ ($V_{\rm c} \sim V$). When $V_\perp > V_{\rm c}$, $\Delta_\perp^{(c)}$ and $\Delta_\perp^{(s)}$ appear simultaneously, since these two states are degenerate when $t_\perp$. Then the state for large $V_\perp$ has ($\Delta_\perp^{(c)} + i\Delta_\perp^{(s)}$)-symmetry and is fully gapped. The in-plane OP is excluded due to the same reason which suppresses $\Delta_\perp$ when $V_\perp < V_{\rm c}$. Thus we conclude that the state does not have nodes irrespective of the value of $V_\perp$ if $t_\perp = 0$.
Next we consider the case $t_\perp \not= 0$. In this case $\Delta_\parallel$ couples to $\Delta_\perp^{(c)}$. Then the latter can be finite once $V_\perp$ becomes finite. Now the gap function is $$\Delta_k =
(\sin k_x + i\sin k_y)(\Delta_\parallel + \Delta_\perp^{(c)}\cos k_zc
+ i\Delta_\perp^{(s)}\sin k_zc).$$ Since $\Delta_\perp^{(c)}$ is induced by the bilinear coupling to $\Delta_\parallel$, their relative phase is either $0$ or $\pi$, while $\Delta_\perp^{(s)}$ favors a phase $\pm \pi/2$ relative to them due to $\gamma$ terms. In Fig.4 the results are shown for $t_\perp = 0.4$. Here $|\Delta_\parallel| > |\Delta_\perp| $ (and $\Delta_\perp^{(s)} = 0$) for small $V_\perp$ so that there is a full gap. For large values of $V_\perp$, $|\Delta_\perp^{(c)}| > |\Delta_\parallel|$. However, $i\Delta_\perp^{(s)}$ component appears before $|\Delta_\perp^{(c)}|$ exceeds $|\Delta_\parallel|$. Then the state is again fully gapped except an accidental case where $|\Delta_\parallel^{(c)}| = |\Delta_\perp|$. We have also examined other values of $t_\perp$, and the SC state (with $(p_x \pm ip_y)$-symmetry for the in-plane OP) always has a full gap except an accidental case, unless $t_\perp$ becomes comparable to $t$.
In summary we have studied the symmetry of the SC states in a single-band tight-binding model with an attractive interaction between nearest-neighbor sites. It is shown that the spin-triplet and the spin-singlet SC states, and even their coexistence can occur as the band structure is changed. These results can be understood by considering the change of the shape of the Fermi surface. The present result implies that the band structure is an important factor to determine the symmetry of the SC state. We have also examined the stability of interlayer-pairing states with line nodes. These states are difficult to be stabilized in a model with such a simple band structure (Fermi surface) as that used in the present work. Experimental results of Sr$_2$RuO$_4$ seem to indicate that the SC state has a spin-triplet symmetry, and that there is a line (or lines) of nodes in the excitation gap. In order to give consistent interpretations of these facts, it would be necessary to consider the model which takes into account the more realistic crystal (and band) structure of Sr$_2$RuO$_4$.[@Hase2] This problem will be examined elsewhere.
The author is grateful to M. Sigrist, Y. Hasegawa, Y. Tanaka and H. Fukuyama for useful discussions. He also thanks Ken Yokoyama for letting him know Ref.10, 11. This work was supported in part by a Grant-in-Aid for Scientific Research from the Ministry of Education, Science, Sports and Culture of Japan.
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K. Miyake and O. Narikiyo: Phys. Rev. Lett. [**83**]{} (1999) 1423.
M.J. Graf and A.V. Balatsky: Phys. Rev. B [**62**]{} (2000) 9697.
T. Dahm, H. Won and K. Maki: cond-mat/0006301.
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A tiny amount of $s$-wave ($d$-wave) component is mixed in the $d + ip_y$ ($s + ip_y$) state. This is because the $x$- and the $y$- directions are not equivalent in this state, and so the absolute values of OP’s for both directions are not the same.
K. Kuboki: J. Phys. Soc. Jpn. [**68**]{} (1999) 3150.
For a review see M. Sigrist and K. Ueda, Rev. Mod. Phys., [**63**]{}, 239 (1991).
Y. Hasegawa; private communications.
[**Fig. 1**]{} Phase diagram in the plane of $T$ and $\mu$. Parameters used are $t = 1, t' =0$ and $V = 1.5$.
[**Fig. 2**]{} Phase diagram in the plane of $T$ and $\mu$. Parameters used are $t = 0, t' =1$ and $V = 1.5$. A narrow region between $s$ ($d$) and $p_x +ip_y$ states is an $s + ip_y$ ($d + ip_y$) state.
[**Fig. 3**]{} The $V_\perp$ dependence of SC order parameters for $t=1, t'=0, V=1.5, \mu=-2, T = 0$ and $t_\perp=0$. Note that all order parameters are non-dimensional.
[**Fig. 4**]{} The $V_\perp$ dependence of SC order parameters for $t=1, t'=0, V=1.5, \mu=-2, T = 0$ and $t_\perp=0.4$. Note that all order parameters are non-dimensional.
[^1]: E-mail: kuboki@phys.sci.kobe-u.ac.jp
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---
abstract: 'This note studies (practical) asymptotic stability of nonlinear networked control systems whose protocols are not necessarily uniformly globally exponentially stable. In particular, we propose a Lyapunov-based approach to establish (practical) asymptotic stability of the networked control systems. Considering so-called modified Round Robin and Try-Once-Discard protocols, which are only uniformly globally asymptotically stable, we explicitly construct Lyapunov functions for these two protocols, which fit our proposed setting. In order to optimize the usage of communication resource, we exploit the following transmission policy: wait for a certain minimum amount of time after the last sampling instant and then check a state-dependent criterion. When the latter condition is violated, a transmission occurs. In that way, the existence of the minimum amount of time between two consecutive transmission is established and so-called Zeno phenomenon, therefore, is avoided. Finally, illustrative examples are given to verify the effectiveness of our results.'
author:
- 'Seyed Hossein Mousavi, Navid Noroozi, Anton H. J. de Ruiter, and Roman Geiselhart [^1] [^2] [^3]'
title: Stability analysis of networked control systems with not necessarily UGES protocols
---
Hybrid systems, networked control systems, scheduling protocols, Laypunov methods
Introduction
============
Networked Control Systems (NCSs) are classes of systems where plant and controller are connected through a communication network. The importance to consider the underlying digital communication layer when designing and analyzing control systems has been widely recognized.
The so-called [*emulation approach*]{} is the most popular method in analysis and design of *nonlinear* NCS [@Walsh.2001; @Nesic.2004; @Carnevale.2007; @Heemels.2010]. In this approach, first, the communication network constraints are ignored and a continuous-time controller is designed for the continuous-time plant. Then, it is shown that the stability and performance for network-based implementation of the system is maintained in an appropriate sense, if the data transmission frequency is sufficiently high [and the scheduling protocol meet some desired stability]{}. Emulation-based approaches are favorable because of their simplicity and also from the point that various standard tools in the continuous-time domain can still be applied through the controller design process. Basically, knowing the maximum allowable transmission interval (MATI) is a key issue in the emulation design procedure of [*time-triggered*]{} NCSs, since it quantifies the stability margin in terms of the transmission period. As a seminal contribution, in [@Nesic.2004] the authors develop a *hybrid systems* framework for modeling and ${\cal L}_p$-stability analysis of nonlinear NCSs, where transmission and scheduling effects induced by a communication channel are taken into account. In particular, an *explicit* formula for computing an upper bound for the MATI is given in [@Nesic.2004]. In [@Carnevale.2007], the results in [@Nesic.2004] have been improved by providing a *less* conservative explicit upper bound of the MATI. Over the last decade, the works [@Nesic.2004; @Carnevale.2007] have received great attention and further extensions toward handling other issues due to a digital network such as communication time-delays [@Heemels.2010], quantization effects [@Nesic.2009b], and reference tracking problems [@Postoyan.2014]. However, in all the mentioned works, only a class of scheduling protocols, including [*classic*]{} Round Robin (RR) and try-once-discard (TOD) protocols, is considered in which the Lyapunov function of the discrete-time system induced by the protocol has to *exponentially* decay.
A central idea for optimizing the usage of communication resources is to lower data transmission over the communication network. One way towards this end, is modifying existing standard protocols such that less often data is transmitted through the network. The authors in [@Nesic.2004b] study stability of NCSs with protocols with a lower data transmission rate. In particular, two [*modified*]{} versions of RR and TOD protocols are introduced in [@Nesic.2004b], which are shown *not* to be uniformly globally exponentially stable (UGES) but uniformly globally asymptotically stable (UGAS). However, no explicit formula for the computation of the MATI is provided in [@Nesic.2004b].
As the first contribution of this paper, we introduce Lyapunov-based conditions for stability analysis of NCSs, which not only relax the need for exponential decay-[rate]{} of the Lyapunov function associated with the discrete-time system induced by the protocol, but also provides an explicit upper bound for the MATI. Note that the MATI formula proposed in [@Carnevale.2007], is developed based on the Lyapunov-based conditions that [demand an *exponential* decay-rate of the Lyapunov function associated with the protocol; hence, they are particularly applicable to NCSs with UGES protocols. It should be noted that, in the context of discrete-time systems, any Lyapunov function associated with a globally asymptotically stable discrete-time system can also be scaled to satisfy the required exponential decay-rate assumption [@Grune.2014 Theorem 2.3]. However, in general, it may not be an easy task to find such a scaling function. Also, in case the proper scaling function is found, the resulting Lyapunov function does not necessarily satisfy all the other conditions required in [@Carnevale.2007].]{} [In this regard and]{} compared with [@Carnevale.2007], we only consider an *asymptotic* decay-rate for the Lyapunov function associated with the protocol. Consequently a wider class of protocols are incorporated, while an explicit formula for MATI is still provided. [However, the reusling upper bound in this case could become very small. In order to relax conservatism, the upper-bound for MATI is computed online at each transmission instant (see Theorem \[thm:ncs-sgc\] for more details).]{}
[As the second contribution of this work, we explicitly construct Lyapunov functions for the modified RR and TOD protocols. The resulting Lyapunov functions and the associated gain functions fit into the Lyapunov-based conditions given to establish the (practical) asymptotic stability of the NCSs.]{} [We note that although the UGAS property of the modified TOD and RR are shown in [@Nesic.2004b], the results are not constructive, meaning that *no* explicit Lyapunov function and the associated gains are provided.]{}
Another idea toward reduction of the rate of data transmission is to use [*event-triggered*]{} control mechanisms. Generally speaking, in an event-based control system, data communication between plant and controller is scheduled using a pre-designed triggering condition. In other words, in such systems data is not transmitted unless a specific triggering condition is violated. It has been widely recognized an event-triggering mechanism is effectively able to optimize the usage of the network, while the desired stability and performance are preserved; see, *e.g.*, [@Astrom.1999; @Tabuada.2007].
[Similar to [@Dolk.2017; @Abdelrahim.2016], to further reduce the rate of data communication, we combine an event-triggering condition called deadband control ([@Otanez.2002; @Antsaklis.2013]) and the time-regularization subject to scheduling. To be more precise, we exploit the following transmission policy: $i$) we first wait for $T$ units of times after the last sampling instant, where $T$ corresponds to the MATI already discussed earlier as the first contribution of this work; $ii$) we then check the deadband condition. When the latter condition is violated, a transmission occurs. In a deadband control mechanism, no new information is broadcast over the network, if the norm of the network-induced error ([*i.e.*]{} the difference between the last transmitted and current outputs values) lies within a certain deadband. We finally note that from item $i$), the existence of the minimum amount of time between two consecutive transmission is established, which implies that so-called Zeno phenomenon [@Goebel.2012] is avoided.]{}
The rest of this paper is organized as follows. Section \[sec:Pre\] and \[sec:problem-statement\], respectively, contain preliminaries and the problem statement. In Section \[sec:protocols\], stability of the modified TOD and RR protocols is investigated. Section \[sec:main-results\] proposes new conditions for stability analysis of NCSs with UGAS protocols as well as an event-triggering policy for further reduction of the data transmission rate. Simulation results are given in Section \[sec:example\] and finally, concluding remarks are provided in Section \[sec:conclusions\].
Preliminaries {#sec:Pre}
=============
In this note, ${\R_{\geq 0}}({\R_{> 0}})$ and ${\Z_{\geq 0}}({\field{N}})$ are the nonnegative (positive) real and nonnegative (positive) integer numbers, respectively. [For a set $\mathcal{S} \subset {\R^{n}}$, $\operatorname{cl}(\mathcal{S})$ denotes the closure of $\mathcal{S}$.]{} The standard Euclidean norm is denoted by ${\left|\cdot\right|}$. [We denote the floor function by $\lfloor \cdot \rfloor$.]{} We write $(x,y)$ to represent $[x^\top,y^\top]^\top$ for any pair $(x,y) \in {\R^{n}}\times {\R^{m}}$. The identity $n$ by $n$ matrix is denoted by $I_n$. A function $\rho \colon {\R_{\geq 0}}\to {\R_{\geq 0}}$ is positive definite if it is continuous, zero at zero and positive elsewhere. A positive definite function $\alpha$ is of class $\K$ ($\alpha \in \K$) if it is strictly increasing. It is of class $\Kinf$ ($\alpha \in \Kinf$) if $\alpha \in \K$ and also $\alpha(s) \to \infty$ if $s \to \infty$. A continuous function $\gamma \colon {\R_{\geq 0}}\to {\R_{\geq 0}}$ is of class $\mathcal{L}$ ($\gamma \in \mathcal{L}$) if it is decreasing and $\lim_{s \to \infty} \gamma (s) \to 0$. A function $\beta \colon {\R_{\geq 0}}\times {\R_{\geq 0}}\to {\R_{\geq 0}}$ is of class $\KL$ ($\beta \in \KL$), if for each $s \geq 0$, $\beta(\cdot,s) \in \K$, and for each $r \geq 0$, $\beta (r,\cdot) \in \mathcal{L}$. A function $\beta \colon {\R_{\geq 0}}\times {\R_{\geq 0}}\times {\R_{\geq 0}}\to {\R_{\geq 0}}$ is of class $\mathcal{KLL}$ ($\beta \in \mathcal{KLL}$), if for each $s \geq 0$, $\beta (\cdot,s,\cdot) \in \KL$ and $\beta (\cdot,\cdot,s) \in \mathcal{KL}$. For a locally Lipschitz function $U: {\R^{n}}\to {\R_{\geq 0}}$ and a vector $y \in {\R^{n}}$, $U^{\circ}(x;y):= {\limsup}_{h \to 0^+, z\to x}(U(z+hy)-U(z))/h$. For a continuously differentiable $U$, $U^{\circ}$ reduces to the standard directional derivative $\left\langle \nabla U(x),y \right\rangle$, where $\nabla U$ denotes gradient.
Due to lack of space, the basics of hybrid dynamical systems are not presented here. The reader is referred to [@Goebel.2012] for detailed information.
Problem Statement {#sec:problem-statement}
=================
Consider the nonlinear plant model $$\begin{aligned}
\label{eq:e01}
\dot{x}_p= f_p (x_p,u), \ \ y = g_p (x_p),\end{aligned}$$ where $x_p \in {\R^{n_p}}$ is the plant state, $u \in {\R^{n_u}}$ is the control input, and $y \in {\R^{n_y}}$ is the plant output.
It is assumed that the plant is output feedback stabilizable and we know a continuous-time controller which globally asymptotically stabilizes the origin of system in the absence of a packet-based communication network. We focus on dynamic controllers of the form $$\begin{aligned}
\label{eq:e06n}
\dot{x}_c=f_c (x_c,y), \ \ u = g_c (x_c),\end{aligned}$$ where $x_c \in \R^{n_c}$ is the controller state.
We consider the scenario where the plant and controller are connected via a packet-based communication network that is composed of $\ell \in {\field{N}}$ nodes. Each node corresponds to a collection of sensors and/or actuators of the plant and the controller.
The network imposes different constraints on the communication of both $u$ and $y$. In this paper, we concentrate on the effect arisen due to sampling and scheduling. Data transmissions only happen at some time instants $t_j, j \in {\field{N}}$, satisfying $0\leq t_0 < t_1 < t_2 < \!\dots$. The structure of the network is in a way that at each transmission instant, a single node is granted access to the network. This scheduling is carried out by the transmission policy.
The overall system can be modelled as the following impulsive system [@Nesic.2004] $$\begin{aligned}
\label{eq:e09n}
&\begin{array}{rcll}
\dot{x}_p &=& f_p (x_p,\hat{u}) & t \in [t_{j-1},t_j] \\
y &=& g_p (x_p) \\
\dot{x}_c &=& f_c (x_c,\hat{y}) & t \in [t_{j-1},t_j] \\
u &=& g_c (x_c) \\
{\dot{\hat{y}}} &=& \hat{f}_p (x_c,\hat y, \hat u) & t \in [t_{j-1},t_j]\\
{\dot{\hat{u}}} &=& \hat{f}_c (x_p,\hat y, \hat u) & t \in [t_{j-1},t_j] \\
\hat{y} (t^+_j) &=& y (t_j) + h_y (j,e(t_j)) \\
\hat{u} (t^+_j) &=& u (t_j) + h_u (j,e(t_j)),
\end{array}\end{aligned}$$ where $\hat{y} \in \R^{n_y}$ and $\hat{u} \in \R^{n_u}$ are, respectively, the estimate of measurements at the controller side and the currently available estimates of the true controller output at the plant side. These two variables are generated by the holding functions $\hat f_p$ and $\hat f_c$ between two successive transmission instants. For the case that zero-order-hold devices are exploited, we would have $\hat f_p = 0$ and $\hat f_c = 0$. The functions $h_y$ and $h_u$, respectively, accommodate the effect of the transmission protocol on the updates of $\hat y (t_j)$ and $\hat u (t_j)$ at the transmission time $t_j$. Moreover, $e := (e_y,e_u) \in \R^{n_e}$ represents the network-induced errors in which $e_y := \hat{y} - y\in{\R^{n_y}}$ and $e_u := \hat{u} - u\in{\R^{n_u}}$. Given $x := (x_p,x_c) \in \R^{n_x}, h := (h_y,h_u) \in {\R^{n_u + n_y}}$, we rewrite as
\[eq:e05\] $$\begin{aligned}
\dot{x} &=& f (x,e), \label{eq:e05a} \\
\dot{e} &=& g (x,e), \label{eq:e05b} \\
e (t^+_j) &=& h (j,e(t_j)), \label{eq:e05c}\end{aligned}$$
where $f \colon \R^{n_x} \times \R^{n_e} \to \R^{n_x}$ and $g \colon \R^{n_x} \times \R^{n_e} \to \R^{n_e} $ are defined by $f (x,e) := \big( f_p (x_p , g_c (x_c) + e_u),
f_c (x_c, g_p(x_p) + e_y)\big)$, $g (x,e) := \big( g_1(x,e), g_2(x,e)\big)$ with $g_1(x,e) := \hat{f}_p (x_p,x_c,g_p (x_p) + e_y, g_c (x_c) + e_u) - \frac{\partial g_p}{\partial x_p} (x_p) f_p (x_p , g_c(x_p)+e_u)$ and $g_2(x,e) := \hat{f}_c (x_p,x_c,g_p(x_p) + e_y, g_c(x_c) + e_u) -\frac{\partial g_c}{\partial x_c} (x_c) f_c (x_c , g_p(x_p)+e_y).$ The function $h$ in is called the scheduling protocol.
In order to study the protocol stability properties, the [*protocol-induced*]{} discrete-time system is defined by $$\label{eq:discrete_prot}
e(i+1) = h(i,e(i)).$$
\[D:UGAS\] The discrete-time system is uniformly globally asymptotically stable (UGAS) with a Lyapunov function $W \colon {\Z_{\geq 0}}\times {\R^{n_e}} \to {\R_{\geq 0}}$ if there exist $\ul\alpha_e,\ol\alpha_e \in \Kinf$ and $\sigma \in \K$ with [$\sigma (s) < s$ for all $s \in (0,\infty)$]{} such that for all $j \in {\Z_{\geq 0}}, e \in \R^{n_e}$ the following holds $$\begin{aligned}
\ul\alpha_e ({\left|e\right|}) & \leq & W(j,e) \leq \ol\alpha_e ({\left|e\right|}), \label{eq:de18} \\
W(j+1,h(j,e)) & \leq & \sigma(W(j,e)) . \label{eq:de19}
\end{aligned}$$
In [@Carnevale.2007; @Nesic.2004], Lyapunov-based conditions are imposed on protocol-induced system , [where the function $\sigma$ in needs to be linear.]{} Hence, the conditions are specifically applicable to uniformly globally exponentially stable (UGES) protocols. As shown in [@Nesic.2004b], not every protocol is UGES. The modified TOD and RR are two examples of such protocols, which are described in more details in Section \[sec:protocols\].
In this note, we propose a transmission policy which has two different phases depending on the value of the network-induced error $e$ and a predefined deadband $d \in {\R_{\geq 0}}$. Define $$\label{eq:Ed}
{E_d := \{ e\in \R^{n_e}: |e_i|\leq \frac{d}{\ell}, \ \text{for} \ i=1,\cdots,\ell \}}.$$ [The proposed transmission paradigm is given as follows: For $T_j > 0, j \in {\field{N}}$ units of times after the last transmission instant (*i.e.* $t_j$) *no* data is transmitted over the network, then]{}
- if $e \in E_d$, no data is still transmitted over the network. Basically, this condition prevents data transmission when there is no remarkable change (quantified by $d$) in the nodes’ output information.
- if $e \notin E_d$, this conditions holds, data transmission is scheduled by a UGAS protocol.
[We note that Zeno phenomenon is avoided by the use of the proposed transmission paradigm.]{} We aim to study the stability properties of the NCS under the above-explained network policy as well as provide an explicit formula for computing [$\{T_j\}_{j \in {\Z_{\geq 0}}}$]{} such that the desired stability property is achieved.
UGAS Protocols {#sec:protocols}
==============
Here we examine the modified versions of TOD and RR originally introduced in [@Nesic.2004b]. Propositions \[P:TOD\] and \[P:RR\] below *explicitly* construct Lyapunov functions satisfying (\[eq:de18\]) and (\[eq:de19\]) for the modified TOD and RR protocols, respectively. The proofs of Propositions \[P:TOD\] and \[P:RR\] can be found in Appendix.
Modified TOD
------------
This protocol behaves for large $e$ in the same way as TOD protocol, but for small $e$ the error jumps are smaller. In this case the transmission protocol is defined by $$\begin{aligned}
\label{eq:tod}
h (i,e) = (I - \Psi(e) ) e ,\end{aligned}$$ where $\Psi(e) := \mathrm{diag}(\psi_1(e) I_{n_1},\psi_2(e) I_{n_2},\dots,\psi_\ell(e) I_{n_\ell})$, and for each $j \in \{1,2,\dots,\ell \}$ $$\begin{aligned}
\label{eq:tod-psi}
&\!\!\! \psi_j (e) := \left\{\begin{array}{lcl}
\!\mathrm{sat} ({\left|e_j\right|}) & \! \mathrm{if}\! & \!j\! =\! \min (\mathrm{arg} \max_j {\left|e_j\right|}), \\
\!0 & & \!\!\!\!\!\!\!\!\mathrm{otherwise},
\end{array}\right.\end{aligned}$$ with $\mathrm{sat} (s) := \min \{s,1\}$ for all $s \geq 0$. As discussed in [@Nesic.2004b], this protocol is *not* UGES and transmits less data compared to its original version.
The original TOD protocol is also described by , but $\psi_j (e)$ is defined as $$\begin{aligned}
\label{eq:tod-psi-trad}
\psi_j (e) := \left\{\begin{array}{lcl}
1 & \mathrm{if} & j = \min (\mathrm{arg} \max_j {\left|e_j\right|}), \\
0 & & \mathrm{otherwise}.
\end{array}\right.
\end{aligned}$$
\[P:TOD\] The modified TOD protocol is UGAS with the Lyapunov function $|e|$ with the associated gain functions $\ul \alpha_e(s) = \ol \alpha_e(s) = s$, and $$\begin{aligned}
\label{eq:sigmaTOD}
\sigma(s) = \left\{ \begin{array}{lcl}
s \sqrt{1 - {\ell^{-3/2}} s } & \mathrm{if} & 0\leq s \leq \sqrt{\ell}, \\
\sqrt{({\ell -1})/{\ell}} s & \mathrm{if} & s > \sqrt{\ell}.
\end{array} \right. \end{aligned}$$
Modified RR
-----------
This protocol behaves exactly in the same way as RR for large error $e$, but it transmits less frequently for small $e$. In this case the transmission policy is defined as $$\label{MRR}
h(i,e) = (I - \Delta(i,e))e,$$ where $\Delta(i,e) = \text{diag} \{ \delta_1(i,e) I_{n_1},\cdots,\delta_{\ell} I_{n_{\ell}} \}$, and $$\label{RR-delta}
\delta_k(i,e)\! :=\!\left\{\begin{array}{lcl}
1 & \mathrm{if} &\! \!\!{\left|e\right|}>0, i = \floor{\frac{1}{\mathrm{sat} ({\left|e\right|})}}(k+j \ell),j \in \mathbb{N}, \\
0 & & \!\!\!\mathrm{otherwise}.
\end{array}\right.$$ [Basically, $\delta_i$ switches on and off the action corresponding to node $i$ in the policy equation ]{}.
The original RR protocol is also described by , but $\delta_k(i,e)$ is defined as $$\label{RR-delta-trad}
\delta_k(i,e) :=\left\{\begin{array}{lcl}
1 & \mathrm{if} & i = k+j \ell, j \in \mathbb{N}, \\
0 & & \mathrm{otherwise}.
\end{array}\right.$$
\[P:RR\] The modified RR protocol is UGAS with the Lyapunov function $W(i,e) = \sqrt{\sum_{k=i}^{+\infty}{|\phi(k,i,e)|^2}}$, where $\phi(k,i,e)$ is a solution to with $h$ given by , at sample $k$, starting at initial time $i$ with initial condition $e$. Moreover the associated gain functions in and are $\ul\alpha_e(s)=s$, $\ol\alpha_e(s)=\max \left\{ \ell \sqrt{2s}, \sqrt{\ell} s \right\}$, and $$\begin{aligned}
\label{eq:sigmaRR}
\sigma(s) = \left\{ \begin{array}{lcl}
s \sqrt{1 - {s^2}/({4\ell^4}}) & \mathrm{if} & 0\leq s < 2 \sqrt{\ell^3}, \\
\sqrt{{(\ell -1)}/{\ell}} s & \mathrm{if} & s \geq 2 \sqrt{\ell^3}.
\end{array} \right.\end{aligned}$$
We note that and are nonlinear functions, which are not upper bounded by a linear function less than the identity function. In particular, the modified TOD and RR are *not* UGES but proved to be UGAS. Although these two protocols are studied in [@Nesic.2004b], no *explicit* Lyapunov and associated gain functions are provided therein.
Stability Properties of the NCS with UGAS Protocols {#sec:main-results}
===================================================
In this paper, we provide an explicit state-dependent bound for MATI, which guarantees global (practical) asymptotic stability of the proposed NCS. Toward this end, we first transform the NCS model into a hybrid system such that the analytical tools of [@Goebel.2012] can be exploited to infer the stability properties of the system. In particular, we introduce the auxiliary clock variable $\tau \in {\R_{\geq 0}}$ representing the time elapsed since the last transmission instant. We also introduce $\kappa \in {\Z_{\geq 0}}$ to count the number of transmissions. Denote $\xi := (x,e,\tau,\kappa)$, $F(\xi) :=(f(x,e),g(x,e),1,0)$ and $G(\xi) := (x,h(\kappa,e),0,\kappa+1)$. The following hybrid system representation of the NCS is obtained $$\begin{aligned}
\label{ncs-hybrid}
& \mathcal{H}_\mathrm{NCS} := \left\{ \begin{array}{l}
\dot{\xi} = F (\xi) \;\; \quad \xi \in \mathcal{C}, \\
\xi^+ = G (\xi) \quad \xi \in \mathcal{D},
\end{array} \right. &\end{aligned}$$ $\!\!$where $\C$ and $\D$ denote the flow and jump sets respectively. According to the network data transmission paradigm described in Section \[sec:problem-statement\], these sets are defined by $\C$ $=$ $\{ (x,e,\tau,\kappa) \colon$ $\tau \in [0,T_j] \ \mathrm{or} \ {e \in E_d} \}$ and $\D = \{ (x,e,\tau,\kappa) \colon$ $ { \tau \geq T_j} \ \mathrm{and} \ {e \in \tilde E_d} \}$, [where $\tilde E_d := \operatorname{cl} \big(\R^{n_e} \backslash E_d\big)$]{}.
In the sequel, a Lyapunov-based approach for (practical) asymptotic stability of the NCS is presented. To this end, [ and similar to [@Carnevale.2007] and [@Postoyan.2014]]{}, we make the following assumptions.
\[A:01\] Consider the NCS . There exist a locally Lipschitz function $V \colon \R^{n_x} \to {\R_{\geq 0}}$, a function $W \colon {\Z_{\geq 0}}\times \R^{n_e} \to {\R_{\geq 0}}$ [that is locally Lipschitz in its second argument]{}, a continuous function $H \colon \R^{n_x} \to {\R_{\geq 0}}$, $\ul\alpha_x,\ol\alpha_x,\ul\alpha_e,\ol\alpha_e \in \Kinf, \sigma \in \K$ with $\sigma(s) < s$ for $s > \ul\alpha_e(d)$, and real numbers $L,\gamma > 0$, and $\eta >0$ such that for all $x \in \R^{n_x}$ it holds $$\begin{aligned}
& \underline\alpha_x ({\left|x\right|}) \leq V (x) \leq \overline\alpha_x ({\left|x\right|}), & \label{eq:e17}
\end{aligned}$$ and for almost all $x \in \R^{n_x}$, for all $e \in \R^{n_e}$ and all $\kappa
\in {\Z_{\geq 0}}$ it holds that $$\begin{aligned}
\langle \nabla V (x) , f(x,e) \rangle \leq & - \eta V(x) - \eta W^2(\kappa,e) - H^2(x) + \gamma^2 W^2 (\kappa,e) . \label{eq:e18}
\end{aligned}$$ Moreover, we have
& \_e ([|e|]{}) W (,e) \_e ([|e|]{}), [\_[0]{}]{}, e \^[n\_e]{}, \[eq:e19\]\
& W (+1,h(,e)) ( W (,e) ), [\_[0]{}]{}, e E\_d; \[eq:e20\]
for almost all $e \in \R^{n_e}$, all $x \in \R^{n_x}$ and all $\kappa \in {\Z_{\geq 0}}$ $$\left\langle {\partial W (\kappa,e)}/{\partial e} , g(x,e) \right\rangle \leq \!L W (\kappa,e) \!+\! H(x) . \label{eq:e21}$$
From equations and , the emulated controller assures an ISS-like property for the system $\dot x = f(x,e)$ with $W(\kappa,e)$ as input. Different classes of linear and nonlinear systems satisfy and (see [@Postoyan.2014]). [From conditions and (\[eq:e20\]) the protocol only needs to be uniformly globally *practically* stable.]{}
Let the parameters $\gamma, L$ come from Assumption \[A:01\], [$\lambda \in (0,1)$]{} and define the notation $$\label{eq:MATIbound}
\small
\mathcal{T}(\gamma,L,\lambda) := \!\!
\left\{ \begin{array}{ll}
\!\!\frac{1}{L r}\tan^{-1}\left(\frac{r(1-\lambda)}{2\frac{\lambda}{\lambda+1} (\frac{\gamma}{L}-1) +1+\lambda}\right) & \gamma>L, \\
\!\!\frac{1}{L} \left( \frac{1-\lambda}{1+\lambda} \right) & L = \gamma, \\
\!\!\frac{1}{L r}\tanh^{-1}\left(\frac{r(1-\lambda)}{2\frac{\lambda}{\lambda+1} (\frac{\gamma}{L}-1) +1+\lambda}\right) & \gamma<L,
\end{array}\right.$$ where $r := \sqrt{{\left|(\gamma / L)^2-1\right|}}$. Theorem \[thm:ncs-sgc\] contains the main result of this note. The proof is provided in Appendix.
\[thm:ncs-sgc\] Consider system (\[ncs-hybrid\]) and let Assumptions \[A:01\] hold. Generate a sequence $\{\lambda_{j}\}_{j \in {\Z_{\geq 0}}}$ with $\lambda_{j} \in (0,1)$ for all $j \in {\Z_{\geq 0}}$ as follows: $\lambda_{0} \geq \sigma(W(0,e_0))/{W(0,e_0)}$, $e_0 := e(0,0)$ and at all the other transmission instants $(t_{j},j-1) \in E$ with $E$ as the hybrid time domain, $\lambda_{j} \geq \sigma(W(\kappa(t_{j},j-1),e(t_{j},j-1)))/W(\kappa(t_{j},j-1),e(t_{j},j-1))$. Accordingly, generate $\{T_{j}\}_{j \in {\Z_{\geq 0}}}$, where [$T_j \in (0,\mathcal{T}(\gamma,L,\lambda_j)]$]{}. There exist $\beta \in \mathcal{KLL}$ and $\delta > 0$ such that for all $x_0 := x(0,0) \in \R^{n_x}$, all $e_0 \in {\R^{n_e}}$, all $\kappa (0,0) \in {\Z_{\geq 0}}$, all $\tau (0,0) \in {\R_{\geq 0}}$ and all $(t,j) \in E$ a solution of satisfies $$\label{eq:GApS}
{\left|x(t,j)\right|} \leq \max \big\{ \beta \left({\left|(x_0,e_0)\right|},t, j \right) , \delta \big\},$$ with $\delta \leq \frac{ 2\lambda_{\max} (\gamma^2 - \eta)\overline \alpha_e^2(d)}{\lambda_{\min} \eta (1-e^{-\eta \epsilon})}$.
[From condition (\[eq:e20\]), we can always generate a sequence $\{\lambda_{j}\}_{j \in {\Z_{\geq 0}}}$ satisfying the respective statement of Theorem \[thm:ncs-sgc\].]{}
Theorem \[thm:ncs-sgc\] provides an explicit upper bound for [$T_j$]{}, assuring practical *asymptotic* stability property in the sense of . Moreover, an explicit relation between the *quality-of-control* (QoC) parameter $\delta$ (which quantifies the [the ultimate bound of states convergence]{}) and the *quality-of-service* (QoS) deadband parameter $d$ (affecting the data transmission rate) is provided. Hence, there is an explicit tradeoff between the system performance of the channel resources usage. In particular, the deadband parameter $d$ can be used for resource-aware control design in NCSs. For more details on tradeoffs between QoC and QoS, see [@Borgers2017].
Although in [@Nesic.2004b] the stability property of [*time-triggered*]{} NCSs with not necessarily UGES protocols is studied, [*no*]{} explicit formula for the computation of MATI is provided. On the contrary, Theorem \[thm:ncs-sgc\] proposes [an explicit upper bound for the minimum time between two consecutive transmission instants (*i.e.* $T_j$), which can be viewed as MATI when only time-regularization subject to scheduling is exploited (*i.e.* the deadband event-triggering mechanism is bypassed).]{}
[We note that $T_j$ is computed at each transmission instant and it determines the minimum amount of time elapsed until the next transmission occurs. More specifically, at each transmission instant $t_{j}$, $\mathcal{T}(\gamma,L,\lambda_{j})$ is calculated, where $\lambda_{j}$ is generated by Theorem \[thm:ncs-sgc\]. Then we take $T_j \in (0, \mathcal{T}(\gamma,L,\lambda_{j})]$. Finally the next transmission instant $t_{j+1}$ is scheduled according to the transmission policy: after $T_j$ unit of times the triggering condition [ ${\left|e_i\right|}\leq d/\ell$]{} for all $i \in \{1,\dots,\ell \}$ is continuously evaluated until it is violated. ]{}
\[rrrr\] Theorem \[thm:ncs-sgc\] is also valid if $T_j$ for all $j\in{\Z_{\geq 0}}$ is computed for a fixed value of $\lambda_\mathrm{max}$, where $\lambda_\mathrm{max}$ is defined as $\lambda_\mathrm{max} := \max\limits_j \lambda_{j}$. However, the state-dependent strategy provides less conservative values for $T_j$, compared to the fixed one. See Section \[sec:example\] below for an illustrative example.
[Basically, at each transmission instant, $T_j$ is updated based on the Lyapunov function decay rate ($\lambda_{j}^{-1}$)]{}. From the value computed for $\lambda_{j}$ (generated by Theorem \[thm:ncs-sgc\]) in the *modified* TOD for $W(\kappa,e) > \sqrt{\ell}$ is $\sqrt{\frac{\ell-1}{\ell}}$, which is the same as the one for the *classic* TOD (see [@Nesic.2004]). [This results in the same upper bound for $T_j$ for both protocols in this region]{}. However, for $W(\kappa,e) \leq \sqrt{\ell}$, as $W(\kappa,e)$ decreases, $\lambda_{j}$ approaches 1. [Consequently, the Lyapunov function decay rate decreases, which in the view of leads to a tighter upper bound for $T_j$ for the modified TOD.]{} Similar arguments hold for the modified RR.
Illustrative Example {#sec:example}
====================
The proposed approach is verified by applying to a batch reactor system which is a benchmark example in the NCS literature (see [@Nesic.2004; @Carnevale.2007; @Heemels.2010]). The reactor dynamics are modeled by the linear equations $\dot x_p = A_p x + B_p u$ and $y = C_p x$, in which $n_p=4$, $n_u=2$, $n_y=2$. The dynamic controller is of the form $\dot x_c = A_c x_c + B_c y$ and $u=C_c x_c$ with $n_c=2$. We assume that only the plant outputs are sent to the controller over the network and so $\ell =2$. Using these parameters, functions $f$ and $g$ in are formulated as $f(x,e) = A_{11} x + A_{12} e$ and $g(x,e) = A_{21} x + A_{22} e$, where the values for $A_{11}, A_{12}, A_{21}$ and $A_{22}$ are given in [@Heemels.2010]. Two different transmission protocols are considered and examined for this setup.
Modified TOD
------------
In this case, the transmission policy $h$ in is described by and . The data transmission deadband $d$ is varied in the interval $[0.1,0.8]$ to illustrate its effectiveness on data communication reduction. To verify Assumption \[A:01\], define $V(x) = x^T P x$, where $P$ is a positive definite matrix, and $W(i,e)=|e|$. Taking $H(x) = M|A_{12}x|$ and $L = M|A_{22}|$ (with $|\frac{\partial W}{\partial e}| \leq M = 1$), is clearly satisfied and $L$ is obtained as $L=15.73$. Computing the derivative of $V$, one can show that holds if one can solve the following linear matrix inequalities (LMIs) for some $\epsilon > 0$ $$\label{eq:LMIs}
\begin{array}{l}
\small
\! \! \! \! \! \! \! \begin{bmatrix}
A_{11}^T P + P A_{11} + M^2 A_{21}^T A_{21} + \epsilon I & P A_{12} \\
A_{12}^T P & (\epsilon-\gamma^2) I
\end{bmatrix} \preceq 0 .
\end{array}$$ [From , the smaller $\gamma$ is, the larger $T_j$ is obtained.]{} Let $\epsilon = 0.001$ and solve LMIs with the objective of minimizing $\gamma^2$. This gives $\gamma =16.92$. For the simulation, $T_j$ is updated as $T_j =\Gamma(\gamma,L,\lambda_{j})$, where $\lambda_{j}$ is generated by Theorem \[thm:ncs-sgc\] and . Simulations are carried out for different values of $d \in [0.1,0.8]$ and initial conditions (randomly) satisfying ${\left|x_0\right|}<2$ and ${\left|e_0\right|}<1.5$. Fig. \[fig:tau\_vs\_d\] shows that the average values of transmission intervals, denoted by $\ol T$, versus the deadband $d \in [0.1, 0.8]$ follows a trend, ascending from $20$ ms to $230$ ms. However, as expected, this happens at the expense of a [larger ultimate bound of states convergence]{}. This is shown by Fig. \[fig:statesTOD\]. From Remark \[rrrr\], we may use a fixed $T_j$ for all $j\in{\Z_{\geq 0}}$, i.e. $\underline T := \mathcal{T}(\gamma,L,\lambda_\mathrm{max})$. Table \[tab:MATIs\] reports $\underline T$’s for different values of $d$ varying from $0.1$ to $0.8$. In this case, values for $\overline{T}$ are also obtained, as depicted by Fig. \[fig:tau\_vs\_d\_cons\]. Obviously, smaller $\overline{T}$ in this case are considerably smaller than those in Fig.\[fig:tau\_vs\_d\].
![Average of transmission intervals as a function of the deadband $d$, while using modified TOD protocol, [with varying $\lambda_j$]{}.[]{data-label="fig:tau_vs_d"}](tau_vs_d.eps){width=".6\textwidth" height="5cm"}
{height="8cm" width=".8\textwidth"}
\[fig:statesTOD\]
![Average of transmission intervals as a function of the deadband $d$ , while using the modified TOD protocol, [with a fixed $\lambda_j = \lambda_\mathrm{max}$]{} for all $j\in {\Z_{\geq 0}}$.[]{data-label="fig:tau_vs_d_cons"}](M_TOD_Conservative.eps){width=".6\textwidth" height="5cm"}
Modified RR
-----------
In this case the transmission protocol $h$ is modeled by and . Using the formulation for $L$ given in the previous part and setting $M$ to $\sqrt{\ell}$ (which is obtained using a similar approach as [@Nesic.2004]), $L$ is calculated as $L=22.24$. Having solved the LMIs with the objective of minimizing $\gamma$, we obtain $\gamma=23.93$. The rest of parameters are set similar to the previous part and simulations are done for different values of $d$. The results for the average values of $T_j$ versus $d$ are depicted in Fig \[fig:tau\_vs\_d\_RR\]. Moreover, the trajectories for the first and second states of the plant are shown for two different deadband values in Fig. \[fig:statesRR\].
![ Average of transmission intervals as a function of the deadband $d$, while using modified RR protocol.[]{data-label="fig:tau_vs_d_RR"}](tau_vs_d_RR.eps){width=".6\textwidth" height="5cm"}
![ Trajectories for the first and second states of the plant, for the cases $d=0.1$ (solid) and $d=0.6$ (dashed), while using modified RR protocol.[]{data-label="fig:statesRR"}](x1x2RR.eps){width=".8\textwidth" height="8cm"}
Conclusions {#sec:conclusions}
===========
The stability of a class of NCSs with not necessarily UGES protocols was investigated. A set of UGAS protocols Lyapunov conditions was provided, and particularly, modified versions of RR and TOD protocols were proved to satisfy the conditions. Moreover, an explicit formula for computation of the upper bound for the minimum amount of time between two consecutive transmission was provided. Simulation results were provided to show the effectiveness of the approach.
[10]{} \[1\][\#1]{} url@samestyle \[2\][\#2]{} \[2\][[l@\#1=l@\#1\#2]{}]{}
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[**Proof of Proposition \[P:TOD\]**]{}. Consider arbitrary $e$ and suppose without loss of generality that ${\left|e_1\right|} = \max_j {\left|e_j\right|}$. Then we would have ${\left|e_1\right|}^2 \geq \frac{1}{\ell} {\left|e\right|}^2$. Now consider two cases: i) If ${\left|e_1\right|} \geq 1$, we have with the same arguments as those in [@Nesic.2004 Section IV] that $W (h(e)) \leq \sqrt{{(\ell -1)}/{\ell}} \ W(e)$. ii) Assume that ${\left|e_1\right|} < 1$. It follows from that ${\left|h(e)\right|}^2 = \sum_{j=2}^\ell {\left|e_j\right|}^2 + (1-{\left|e_1\right|})^2 {\left|e_1\right|}^2 = {\left|e\right|}^2 + {\left|e_1\right|}^3 ({\left|e_1\right|}-2)$. Using the last equality and the facts ${\left|e_1\right|} < 1$ and ${\left|e_1\right|}^2 \geq \frac{1}{\ell} {\left|e\right|}^2$, we have $W (h(e)) \leq \sqrt{{\left|e\right|}^2 - {\left|e_1\right|}^3 } \leq W (e) \sqrt{1 - \frac{1}{\ell^{3/2}} W(e)}$.
It follows from the results of cases i) and ii) that $W (h(e)) \leq \sigma (W(e))$, where $\sigma : [0,+\infty) \to {\R_{\geq 0}}$ is defined by . Moreover, it is not hard to see that $\sigma \in \Kinf$ and therefore condition is satisfied. Trivially, is also satisfied with $\ul\alpha_e(s)= \ol\alpha_e(s)= s$. $\Box$
[**Proof of Proposition \[P:RR\]**]{}. Without loss of generality it is assumed that $|e_j|>0$ for all $j \in \{1,2,\dots,\ell \}$. Define vectors $e^{(i)} \in {\R^{n}}$ as follows: $e^{(i)}=0$ if $j \leq i$, and $e^{(i)}=0$ otherwise, where $i \in \{1,2,\dots,\ell \}$ and $e^{(0)} := e$. [Denote $m$ ($0 \leq m \leq \ell$) as the number of transmissions after which the value of $|\phi(k,i,e)|$ drops below 1/2.]{} In the other words $|\phi(k,i,e)| \geq 1/2$ if $i \leq k \leq i+m-1$, and $|\phi(k,i,e)| < 1/2$ otherwise. [(Note that this always happens since if $|e_j|>1/2$ for all $j=1,\dots, \ell$, then, based on the definition of the protocol, the modified RR turns into the classic RR and the error vector becomes zero after $\ell$ steps; see [@Nesic.2004].)]{} Using this fact and based on the protocol definition given in (\[MRR\]) and (\[RR-delta\]), it is derived that $\sum_{k=i}^{i+m}{|\phi(k,i,e)|^2} \! \leq \! m |e|^2$, and $\sum_{k=i+m+1}^{+\infty}{|\phi(k,i,e)|^2} \! \leq \! (\ell + m+1) \floor{\frac{1}{{\left|e^{(m+1)}\right|}}}|e^{(m+1)}|^2 + \dots + (\ell + \ell - 1)\floor{\frac{1}{{\left|e^{(\ell-1)}\right|}}}|e^{(\ell-1)}|^2 \leq (\ell - m) (\ell + \frac{\ell+m}{2})|e| \leq (\ell - m) (2\ell)|e|$. Using these two inequalities we have $W(i,e) = \sqrt{\sum_{k=i}^{+\infty}{|\phi(k,i,e)|^2}} \leq \sqrt{\frac{m}{\ell}(\ell |e|^2) + \frac{\ell-m}{\ell}(2 \ell^2|e|)}. $ As $\frac{m}{\ell} + \frac{\ell-m}{\ell} =1$ and $0 \leq m \leq \ell$, we get $W(i,e) \leq \max_{0\leq m \leq \ell} \sqrt{\frac{m}{\ell}(\ell |e|^2) + \frac{\ell-m}{\ell}(2 \ell^2|e|)}$. To find the max term in the last inequality, we consider two different cases: i) If $\ell |e|^2 \leq 2 \ell^2 |e|$, the maximum happens at $m=0$, and so $W(i,e) \leq \ell \sqrt{2|e|}$. On the other hand, since $W(i,e) = \sqrt{\sum_{k=i}^{+\infty}{|\phi(k,i,e)|^2}}$, we have $W(i+1,h(i,e)) = \sqrt{W^2(i,e) - |e|^2}$. It is obtained that $W(i+1,h(i,e)) \leq W(i,e)\sqrt{1 - \frac{W^2(i,e)}{4 \ell^4}}$. ii) If $\ell |e|^2 \geq 2 \ell^2 |e|$, the maximum happens at $m=\ell$, and so $W(i,e) \leq \sqrt{\ell}|e|$. Using this inequality alongside with the fact $W(i+1,h(i,e)) = \sqrt{W^2(i,e) - |e|^2}$ results in $W(i+1,h(i,e)) \leq \sqrt{\frac{\ell-1}{\ell}} W(i,e)$.
It follows from the results of cases i) and ii) that $W(i+1,h(i,e)) \leq \sigma(W(i,e))$, where $\sigma : [0,+\infty) \to {\R_{\geq 0}}$ is defined by . We note that $\sigma \in \Kinf$. Thus condition is satisfied. We also have $W(i,e) \leq \max \left\{ \ell \sqrt{2{\left|e\right|}}, \sqrt{\ell} {\left|e\right|} \right\}$. On other hand, from the definition of $W(i,e)$ we have that $W(i,e) \geq \sqrt{|\phi(i,i,e)|^2} = {\left|e\right|}$. Thus condition is satisfied with the functions $\ul\alpha_e(s)=s$ and $\ol\alpha_e(s)=\max \left\{ \ell \sqrt{2s}, \sqrt{\ell} s \right\}$. $\,$$\Box$
[**Proof of Theorem. \[thm:ncs-sgc\]**]{} Let $T_j \leq \mathcal{T}(\gamma,L,\lambda_{j})$ be given. Also, let the triple $(\gamma,L,\lambda_{j})$ generate $\phi_{j} \colon [0,T_j] \to \R$ via $$\begin{aligned}
\label{eq:e25}
\dot{\phi}_{j} = -2 L \phi_{j} - \gamma (\phi_{j}^2+1), \qquad \phi_{j}(0) = \lambda_{j}^{-1}, \end{aligned}$$ with $\lambda_{j} \in (0,1)$. As [@Carnevale.2007], it can be easily shown that $\phi_{j}(\tau) \in [\lambda_{j},\lambda_{j}^{-1}]$ for all $\tau \in [0,T_j]$.
[Similar to [@Abdelrahim.2016]]{}, define the hybrid Lyapunov function $U (\xi) := V(x) + \max \{ \gamma \phi_{j}(\tau)W^2(\kappa,e),0 \}$, for all $\xi \in \C \cup \D$. It is follows from the definition of $U$, and that $\underline\alpha_x(|x|) \leq U(\xi) \leq \overline\alpha_x(|x|) + \gamma \lambda_{\min}^{-1} \overline\alpha_e^2(|e|)$, where $\lambda_{\min}:= \min\limits_j \lambda_j$.
Let $\xi \in \C$ and consider two different cases: i) If $\phi_{j}(\tau) \geq 0$, it follows from , and (after some calculation) that $U^{\circ} (\xi;F(\xi)) \leq - \eta V(x) - \eta W^2(\kappa,e) \leq \! - \ol\eta U(\xi),$ where $\ol\eta := \eta \min\{1,\lambda_{\min}/\gamma\}$. ii) If $\phi_{j}(\tau)<0$, according to , it is derived that $U^{\circ} (\xi;F(\xi)) \leq - \eta V(x) - (\eta - \gamma^2) W^2(\kappa,e) - H^2(x).$ Since $\phi_{j}(\tau) \in [\lambda_{j},\lambda_{j}^{-1}]$ for all $\tau \in [0,T_j]$, it is concluded from $\phi_{j}(\tau)<0$ that $\tau > T_j$. From this fact and based on the definition of $\C$, it is inferred that $|e| \leq d$. Using this and also in the the last inequality we get $U^{\circ} (\xi;F(\xi)) \leq - \eta U(\xi) + (\gamma^2 - \eta)\overline \alpha_e^2(d)$. By the results of cases i) and ii) the inequality $$\label{eq:Udott}
U^{\circ} (\xi;F(\xi)) \leq - \eta U(\xi) + \hat d,$$ holds for all $\xi \in \C$, with $\hat d = (\gamma^2 - \eta)\overline \alpha_e^2(d)$.
Moreover, for any $\xi \in \mathcal{D}$ we have $U (\xi^+) = V(x^+) + \gamma \phi_{j}(\tau^+) [W(\kappa^+,e^+)]^2$. It follows from , and the fact $\phi(\tau^+) = \lambda^{-1}_{j+1}$ that $U (\xi^+) \leq V(x) + \gamma \lambda^{-1}_{j+1} [\sigma (W(\kappa,e)) ]^2$. From the generation of $\{\lambda_{j}\}_{j \in {\Z_{\geq 0}}}$, we have that $ U (\xi^+) \leq V(x) + ({\lambda_{j+1}}/{\lambda_{j}}) \gamma \lambda_{j} [W (\kappa,e)]^2. $ Since $\phi_{j}(\tau) > \lambda_{j}$, it is concluded that $ U (\xi^+) \leq V(x) + ({\lambda_{j+1}}/{\lambda_{j}}) \gamma \phi_{j}(\tau) [W (\kappa,e)]^2. $ If $\lambda_{j+1}/ \lambda_{j} < 1$ it is trivially concluded that $U(\xi^+) \leq U(\xi)$. Now assume that $\lambda_{j+1}/ \lambda_{j} \geq 1$. Then, from the last inequality the following holds $ U (\xi^+) \leq {\lambda_{j+1}}/{\lambda_{j}} U(\xi)$. Partition $E := \bigcup\limits_{k = 0}^{J} {\left( {\left[ {{t_k},{t_{k + 1}}} \right],k} \right)}$ with $t_0 = 0$, $t \in [t_{J},t_{J+1}]$ and $U_0 := U(0,0)$. Concatenating flows and jumps yields $$\begin{aligned}
\label{eq:e86}
& U (t,j) \leq \bar U_0 e^{- \eta (t/2 + \epsilon j/2)} + \tilde d\end{aligned}$$ for all $(t,j) \in E$, where $\epsilon$ is the minimum inter-jump time and $\bar U_0 = \frac{\lambda_{\max}}{\lambda_{\min}}U(0,0)$, $\tilde d = \frac{ \lambda_{\max}\hat d}{\lambda_{\min} \eta (1-e^{-\eta \epsilon})}$, $\lambda_\mathrm{max}:=\max_j\lambda_j$.
By the lower and upper bounds on $U$, and , we conclude with $\beta(s,t,j) = \underline \alpha_x^{-1}(2 \overline \alpha(s) e^{-\eta(t/2+\epsilon j/2)})$, and $\delta = 2 \tilde d$, where $\overline\alpha(\cdot) := \lambda_{\max}/\lambda_{\min}[\overline\alpha_x(\cdot) + \gamma \lambda_{\min}^{-1} \overline\alpha_e^2(\cdot)]$. $\Box$
[^1]: S. H. Mousavi and A. H. J. de Ruiter are with the department of Aerospace Engineering, Ryrson University, Toronto, Ontario, Canada, emails: `{seyedhossein.mousavi,aderuiter}@ryerson.ca`.
[^2]: Navid Noroozi is with the University of Passau, Faculty of Computer Science and Mathematics, Innstraße 33, 94032 Passau, Germany, email: `navid.noroozi@uni-passau.de`. The work of N. Noroozi was supported by the Alexander von Humboldt Foundation.
[^3]: Roman Geiselhart is with the University of Ulm, Institute of Measurement, Control and Microtechnology, Albert-Einstein-Allee 41, 89081 Ulm, Germany, email: `roman.geiselhart@uni-ulm.de`.
|
---
abstract: 'The current status of $\bar{B} \to X_s \gamma$ decay rate calculations is summarized. Missing ingredients at the NNLO level are listed. The global normalization factor and non-perturbative effects are discussed. Arguments are presented that results for the cutoff-enhanced perturbative corrections have been misused in Ref. [@Becher:2006pu] by applying them in the region $E_\gamma \in [1.0, 1.6]\,$GeV, which means that the corresponding numerical effect on ${\cal B}\left( \bar{B} \to X_s \gamma,~ E_{\gamma} > 1.6\,{\rm GeV}\right)$ is unreliable.'
author:
- 'M. Misiak'
title: QCD Calculations of Radiative Decays
---
Introduction \[sec:intro\]
==========================
The motivation for precision studies of radiative $B$ decays is well known. First, they are sensitive to new physics loop effects that often arise at the same order in the electroweak couplings as the leading Standard Model (SM) contributions. Moreover, the inclusive $\bar{B} \to X_s \gamma$ rate is well approximated by the perturbatively calculable radiative decay rate of the b-quark. The CLEO [@Chen:2001fj], BELLE [@Abe:2001hk] and BABAR [@Aubert:2005cu] measurements have been combined by HFAG [@Barberio:2008fa] to get \[bexp\] [B]{}( |[B]{} X\_s )\_[exp]{} = ( 3.52 0.23 0.09 ) 10\^[-4]{} for $E_\gamma > 1.6\,$GeV. The corresponding SM prediction that was published two years ago[^1] reads [@Misiak:2006zs] \[bth\] [B]{}( |[B]{} X\_s )\_[SM]{} = ( 3.15 0.23 ) 10\^[-4]{}. Its consistency with Eq. (\[bexp\]) provides strong constrains on many extensions of the SM (see, e.g., Ref. [@Olive:2008vv]).
Resummation of large logarithms $\left(\alpha_s \ln M_W^2/m_b^2\right)^n$ in the calculation of the decay rate is most conveniently performed after decoupling the electroweak bosons and the top quark. In the resulting effective theory, the relevant flavour-changing weak interactions are given by a linear combination of dimension-five and -six operators[^2] O\_[1,2]{} &=& (|[s]{} \_i c)(|[c]{} ’\_i b),
[l]{}\
\
O\_[3,4,5,6]{} &=& (|[s]{} \_i b) [\_q]{} (|[q]{} ’\_i q),
[l]{}\
\
O\_7 &=& [e m\_b]{}[16 \^2]{} |[s]{}\_L \^ b\_R F\_,
[l]{}\
\
O\_8 &=& [g m\_b]{}[16 \^2]{} |[s]{}\_L \^ T\^a b\_R G\^a\_.
[l]{}\
\[ops\] One begins with perturbatively calculating their Wilson coefficients $C_i$ at the renormalization scale $\mu_0 \sim (M_W, m_t)$. Next, the Renormalization Group Equations (RGE) are used for the evolution of $C_i$ down to the scale $\mu_b \sim m_b/2$. Finally, the operator on-shell matrix elements are calculated at $\mu_b$.
The Wilson coefficient RGE are governed by Anomalous Dimension Matrices (ADM’s) that are derived from ultraviolet divergences in the Feynman diagrams with operator insertions. Around 20000 four-loop diagrams with $O_{1,2}$ insertions have been calculated in Ref. [@Czakon:2006ss] to make the large logarithm resummation complete up to ${\cal O}\left[ \alpha_s^2 \left(\alpha_s \ln M_W^2/m_b^2\right)^n \right]$, i.e. at the Next-to-Next-to-Leading-Order (NNLO) in QCD. Including such corrections is necessary to suppress the theoretical uncertainty in Eq. (\[bth\]) down to the level of the experimental error in Eq. (\[bexp\]). The numerical effect of the four-loop ADM’s on the branching ratio amounts to around $-4\%$ for $\mu_b = 2.5\,$GeV. At present, all the relevant Wilson coefficients $C_i(\mu_b)$ are known at the NNLO [@Czakon:2006ss; @Bobeth:1999mk]. However, evaluation of the matrix elements at this order is still in progress — see Sec. \[sec:melem\].
The global normalization factor \[sec:normal\]
==============================================
In order to reduce parametric uncertainties stemming from the CKM angles as well as from the $c$- and $b$-quark masses, one writes the branching ratio as follows [@Gambino:2001ew] \[main\] [B]{}(|[B]{} X\_s )\_[E\_ > E\_0]{} &=& [B]{}(|[B]{} X\_c e |)\_[exp]{} | [ V\^\*\_[ts]{} V\_[tb]{}]{}[V\_[cb]{}]{} |\^2\
&& [6 \_[em]{}]{}[C]{} , where $\alpha_{\rm em} = \alpha_{\rm em}^{\rm on~shell}$, and $N(E_0)$ denotes the non-perturbative correction (see Sec. \[sec:nonpert\]). The $m_c$-dependence of $\bar{B} \to X_c e \bar\nu$ is accounted for by \[phase1\] C = | [V\_[ub]{}]{}[V\_[cb]{}]{} |\^2 [(|[B]{} X\_c e |)]{}[ (|[B]{} X\_u e |)]{}, while $P(E_0)$ is defined by the perturbative ratio \[pert.ratio\] [( b X\_s )\_[E\_ > E\_0]{}]{}[ |V\_[cb]{}/V\_[ub]{}|\^2 ( b X\_u e |)]{} = | [ V\^\*\_[ts]{} V\_[tb]{}]{}[V\_[cb]{}]{} |\^2 [6 \_[em]{}]{} P(E\_0). The NNLO expression for the phase-space factor$C$ (\[phase1\]) is a known function of $m_c/m_b$ as well as non-perturbative parameters that affect $N(E_0)$, too. All these quantities are determined in a single fit from the measured spectrum of the inclusive semileptonic decay $\bar{B} \to X_c e
\bar{\nu}$. The fits are performed using either the 1S or the kinetic renormalization schemes. The corresponding results for $C$ and $m_c$ read (
[c]{} C\
m\_c(m\_c)
) = {
[ll]{} (
[l]{} 0.582 0.016\
1.224 0.057
), & ,\
(
[l]{} 0.546\^[+0.023]{}\_[-0.033]{}\
1.267
), & .\
. The above $\overline{\rm MS}$-scheme values of $m_c$ have been obtained from the 1S- and kinetic-scheme ones using the three-loop and two-loop relations, respectively. The three-loop relation for the kinetic scheme is not yet known. The ratio $C$ is scheme-independent, but it is affected by the so-called weak annihilation contribution $B_{\rm WA}$ that remains unknown. Since $B_{\rm
WA}$ cancels out in Eq. (\[main\]), fixing its value is a matter of convention in the present context. Here, we follow the convention of Ref. [@Gambino:2008fj], namely $B_{\rm WA}(\mu = m_b/2)=0$.
The difference between the two determinations of $C$ amounts to $1.6\sigma$ when counted in terms of the upper error of the very recent kinetic-scheme result [@Gambino:2008fj]. It is a consequence of using different experimental data sets, methodology and renormalization schemes. Fortunately, the effects of changing $C$ and $m_c$ partially compensate each other in Eq. (\[main\]) because $\partial/\partial m_c\; P(E_0) < 0$. For $E_0=1.6\,$GeV, I find \[bbth\] [B]{}( |[B]{} X\_s ) = {
[ll]{} ( 3.15 0.23 ) 10\^[-4]{}, & ,\
( 3.25 0.24 ) 10\^[-4]{}, & ,
. where the first result is just that of Ref. [@Misiak:2006zs], while the second one has been obtained using the same code but with the input parameters from Refs. [@Gambino:2008fj; @Buchmuller:2005zv].
The actual value of ${\cal B}\left( \bar{B} \to X_s \gamma \right)$ in Ref. [@Gambino:2008fj] is somewhat larger than $3.25 \times 10^{-4}$ because $P(E_0)$ was calculated there using the one-loop rather than two-loop determination of $m_c(m_c)$ from $m_c^{\rm kin}$. In principle, using the one-loop relation is allowed at ${\cal O}(\alpha_s^2)$ because $P(E_0)$ becomes $m_c$-dependent only at ${\cal O}(\alpha_s)$.
Cutoff-enhanced corrections \[sec:cutoff\]
==========================================
The perturbative ratio $P(E_0)$ in Eq. (\[pert.ratio\]) depends on the cutoff energy $E_0$ via the dimensionless parameter = 1 - [2 E\_0]{}[m\_b]{}. For very small $\delta$, i.e. close to the kinematical endpoint $E_0=m_b/2$, the usual (“fixed-order”) perturbative expansion breaks down because the corrections behave like powers of $\ln\delta$. In that region, one needs to resum large logarithms of $\delta$. Such a resummation of the cutoff-enhanced corrections (i.e. corrections enhanced by powers of $\ln\delta$) has been performed up to the NNLO in Refs. [@Becher:2005pd; @Becher:2006pu]. These results constitute a valuable contribution to our knowledge of the photon energy spectrum in the endpoint region. However, they need to be treated with extreme care further from the endpoint, where logarithms of $\delta$ no longer dominate. Naively, one might expect that resummation of small logarithms does not hurt, even if it is not an improvement. Unfortunately, this is not the case because the logarithmic and non-logarithmic terms undergo a strong cancellation away from the endpoint.
![Approximate cancellation of the logarithmic ($\phi^{(1)}_L$) and non-logarithmic ($\phi^{(1)}_N$) terms in $\phi^{(1)}$ away from the endpoint. The exact expressions from Eq. (\[split\]) are represented by the solid lines. The Taylor expansions of $\phi^{(1)}_L$ and $\phi^{(1)}_N$ around $E_0=0$ up to ${\cal O}(E_0^3)$ are shown by the dashed lines. \[fig:phi1\]](figure1.ps){width="80mm"}
In order to illustrate this issue in a simple manner, let us consider only the dominant photonic dipole operator $O_7$ in Eq. (\[ops\]). When all the other operators are neglected, the fixed-order expression for the cutoff-dependence of $P(E_0)$ is given by \[ppratio\] [P(E\_0)]{}[P(0)]{} = 1 + [\_s]{} \^[(1)]{}() + [\_s\^2]{}[\^2]{} \^[(2)]{}() + …Each of the functions $\phi^{(k)}$ can be split into two parts: $\phi^{(k)}_L$ that is polynomial in $\ln \delta$, and $\phi^{(k)}_N$ that vanishes at the endpoint. The explicit expressions for $k=1$ read [@Ali:1990tj] \^[(1)]{} &=& \^[(1)]{}\_L + \^[(1)]{}\_N,\
\^[(1)]{}\_L() &=& -[2]{}[3]{} \^2 -[7]{}[3]{} - [31]{}[9]{},\
\^[(1)]{}\_N() &=& [10]{}[3]{} + [1]{}[3]{} \^2 - [2]{}[9]{} \^3 + [1]{}[3]{} (-4) . \[split\] In Fig. \[fig:phi1\], $\phi^{(1)}_L$, $\phi^{(1)}_N$ and their sum are plotted as functions of $E_0$. The endpoint is located at $m_b/2 \simeq
2.34\,$GeV. One can see that $\phi^{(1)}_L$ begins to dominate around $E_0 =
2\,$GeV. On the other hand, already in the vicinity of $E_0 = 1.6\,$GeV, the cancellation of the two components of $\phi^{(1)}$ is very strong.
One may wonder whether similar cancellations occur at higher orders, too. A positive answer concerning $\phi^{(2)}$ is immediate because this function can easily be derived from the results of Ref. [@Melnikov:2005bx]. The corresponding plot is presented in Fig. \[fig:phi2\] for the case when $\alpha_s = \alpha_s^{n_f=3}(m_b)$ in Eq. (\[ppratio\]).
![Same as Fig. \[fig:phi1\] but for $\phi^{(2)}$ in the case when $\alpha_s = \alpha_s^{n_f=3}(m_b)$ in Eq. (\[ppratio\]). \[fig:phi2\]](figure2.ps){width="80mm"}
No explicit results at order ${\cal O}(\alpha_s^3)$ are available. However, we know on general grounds that all the $\phi^{(k)}$ behave like $\left( 2 E_0/m_b
\right)^4$ at small $E_0$. Two powers of $E_0$ originate from $F_{\mu\nu}$ in the vertex $O_7$ in Eq. (\[ops\]), and two additional powers come from the phase-space measure $E_\gamma dE_\gamma$. Consequently, the Taylor expansions of $\phi^{(k)}_L$ and $\phi^{(k)}_N$ at small $E_0$ up to ${\cal O}(E_0^3)$ must exactly cancel each other. These Taylor expansions are shown by the dashed lines in Figs. \[fig:phi1\] and \[fig:phi2\]. One can see that both $\phi^{(1)}_L$ and $\phi^{(2)}_L$ are well approximated by the dashed lines in the region below $1.6\,$GeV. It must also be the case at higher orders because $\phi^{(k)}_L$ are polynomial in $\ln\delta$ that is well approximated by the same expansion in the considered region (see Fig. \[fig:log\]). Thus, cancellations like those shown in Figs. \[fig:phi1\] and \[fig:phi2\] are expected to occur at any order in $\alpha_s$ (see “Note Added”).
In the approach of Refs. [@Becher:2005pd; @Becher:2006pu], logarithms of $\delta$ have been resummed at the NNLO in $\phi^{(k)}_L$, while $\phi^{(k)}_N$ have been retained in the fixed order. More precisely, Eq. (\[ppratio\]) has been effectively re-expressed as \[ppratioN\] [P(E\_0)]{}[P(0)]{} = X + [\_s(\_b) ]{} \^[(1)]{}\_N + [\_s\^2(\_b)]{}[\^2]{} \^[(2)]{}\_N, and $X$ has been calculated up to ${\cal O}\left( \alpha_s^2\times \alpha_s^n \ln^m \delta \right)$, with $n=0,1,2,\ldots$, and $m=n,n+1,\ldots,m_{\rm max}(n)$. This is a reasonable approximation only in the region very close to the endpoint where no cancellation between the two components takes place. Elsewhere, it leads to overestimating the numerical effect of the ${\cal O}(\alpha_s^3)$ terms in Eq. (\[ppratio\]) by a factor of order $\left| \phi^{(3)}_L/\phi^{(3)} \right| \sim \left( m_b/(2 E_0) \right)^4$ that amounts to around 4.6 for $E_0 = 1.6\,$GeV, and around 30 for $E_0 = 1.0\,$GeV.
Unfortunately, it was precisely the range$E_\gamma \in [1.0, 1.6]\,$GeV where the authors of Ref. [@Becher:2006pu] applied their results to calculate the effect on ${\cal B}\!\left( \bar{B} \to X_s \gamma,~ E_{\gamma} > 1.6\,{\rm GeV}\right)$. They adopted the fixed-order result at $E_0 = 1.0\;$GeV from Ref. [@Misiak:2006zs] ( |[B]{} X\_s , E\_ > 1.0[GeV]{}) = 3.27 10\^[-4]{}, and supplemented it with their own numerical value of $\left[ P(1.0)-P(1.6) \right]/P(0)$ that follows from Eq. (\[ppratioN\]). That value is almost twice larger than in the fixed-order NNLO calculation. In the end, their result for the branching ratio with a cutoff at $E_0 = 1.6\,$GeV was considerably lower than the one in Eq. (\[bth\]). In view of the above remarks, their prediction should be considered unreliable.
![$\ln \delta$ (solid) and its expansion (dashed) up to ${\cal O}(E_0^3)$. \[fig:log\]](figure3.ps){width="70mm"}
Status of the NNLO QCD calculations of the matrix elements \[sec:melem\]
========================================================================
On the l.h.s. of Eq. (\[pert.ratio\]) that defines $P(E_0)$, thedenominator is already known at the NNLO [@vanRitbergen:1999gs; @Pak:2008qt]. In the expression for the numerator ( b X\_s )\_[E\_ > E\_0]{} &=& |V\_[tb]{} V\_[ts]{}\^|\^2\
&& \_[i,j=1]{}\^8 C\_i\^[eff]{} C\_j\^[eff]{} G\_[ij]{}(E\_0), \[rate\] the quantities $G_{ij}$ are determined by the matrix elements of $O_1$, …, $O_8$. So far, only $G_{77}$ has been evaluated up to ${\cal O}(\alpha_s^2)$ in a complete manner [@Melnikov:2005bx; @Asatrian:2006ph; @Asatrian:2006rq]. The remaining $G_{ij}$ are fully known at the Next-to-Leading Order (NLO), i.e. up to ${\cal O}(\alpha_s)$ (see Ref. [@Buras:2002er] for a complete list of references). At the NNLO, it is practically sufficient to restrict considerations to $G_{ij}$ with $i,j \in
\{1,2,7,8\}$ because the four-quark penguin operators have small Wilson coefficients. It is often convenient to apply the optical theorem, and calculate $G_{ij}$ by summing imaginary parts of the $b$-quark propagator Feynman diagrams. One can see in Fig. \[fig:Gij\] that imaginary parts of three-, four- and five-loop diagrams occur at the NNLO in $G_{77}$, $G_{27}$ and $G_{22}$, respectively.
![Examples of Feynman diagrams that contribute to $G_{77}$, $G_{27}$ and $G_{22}$ at the NNLO. The dashed vertical lines mark the unitarity cuts. Black squares represent the operators $O_2$ and $O_7$ in Eq. (\[ops\]). \[fig:Gij\]](figure4a.eps "fig:"){width="23mm"} ![Examples of Feynman diagrams that contribute to $G_{77}$, $G_{27}$ and $G_{22}$ at the NNLO. The dashed vertical lines mark the unitarity cuts. Black squares represent the operators $O_2$ and $O_7$ in Eq. (\[ops\]). \[fig:Gij\]](figure4b.eps "fig:"){width="23mm"}\
![Examples of Feynman diagrams that contribute to $G_{77}$, $G_{27}$ and $G_{22}$ at the NNLO. The dashed vertical lines mark the unitarity cuts. Black squares represent the operators $O_2$ and $O_7$ in Eq. (\[ops\]). \[fig:Gij\]](figure4c.eps "fig:"){width="35mm"} ![Examples of Feynman diagrams that contribute to $G_{77}$, $G_{27}$ and $G_{22}$ at the NNLO. The dashed vertical lines mark the unitarity cuts. Black squares represent the operators $O_2$ and $O_7$ in Eq. (\[ops\]). \[fig:Gij\]](figure4d.eps "fig:"){width="25mm"}\
![Examples of Feynman diagrams that contribute to $G_{77}$, $G_{27}$ and $G_{22}$ at the NNLO. The dashed vertical lines mark the unitarity cuts. Black squares represent the operators $O_2$ and $O_7$ in Eq. (\[ops\]). \[fig:Gij\]](figure4e.eps "fig:"){width="35mm"} ![Examples of Feynman diagrams that contribute to $G_{77}$, $G_{27}$ and $G_{22}$ at the NNLO. The dashed vertical lines mark the unitarity cuts. Black squares represent the operators $O_2$ and $O_7$ in Eq. (\[ops\]). \[fig:Gij\]](figure4f.eps "fig:"){width="25mm"}
A relatively simple set of the NNLO contributions is given by diagrams with quark loops on the gluon lines. The quark in the loop is either massive (charm and bottom) or treated as massless (up, down and strange). Such contributions are already known [@Asatrian:2006rq; @Bieri:2003ue; @Ligeti:1999ea; @Boughezal:2007ny] for all the $G_{ij}$ with $i,j \in \{1,2,7,8\}$, except for the massless case in $G_{18}$ and $G_{28}$. The BLM [@Brodsky:1982gc] (or large-$\beta_0$) approximation for the complete NNLO correction is derived from the massless quark results [@Bieri:2003ue; @Ligeti:1999ea].
In Ref. [@Misiak:2006ab], the asymptotic behaviour for $m_c \gg m_b/2$ was calculated for all the non-BLM NNLO corrections to $G_{ij}$ with $i,j \in
\{1,2,7,8\}$, except for $G_{78}$ and $G_{88}$. Next, an interpolation in $m_c$ of these corrections was performed assuming that the interpolated quantities vanish at $m_c=0$. The result of that procedure was an essential input for the NNLO estimate in Eq. (\[bth\]). The overall error of around 7% in the branching ratio was obtained by combining in quadrature four types of uncertainties: 5% non-perturbative, 3% parametric, 3% higher-order, and 3% due to the $m_c$-interpolation ambiguity.
The results in Eqs. (\[bth\]) and (\[bbth\]) do not include several contributions to the branching ratio that are known at present. These additional effects are summarized in Tab. \[tab:add\]. They sum up to around $+1.6\%$, which is small when compared to the overall uncertainty of around $7\%$. Therefore, Eq. (\[bbth\]) can still be treated as an up-to-date SM prediction.
------------------------------------------------------------------------------------- ----------
The BLM terms from Ref. [@Ligeti:1999ea] $+2.0\%$
$O_8$ in the 4-loop ADM’s [@Czakon:2006ss] $-0.3\%$
$b$ and $c$ loops on gluon lines [@Pak:2008qt; @Asatrian:2006rq; @Boughezal:2007ny] $+1.6\%$
Non-perturbative ${\cal O}(\alpha_s \Lambda/m_b$) effects [@Lee:2006wn] $-1.5\%$
Non-perturbative collinear effects [@Kapustin:1995fk] $-0.2\%$
Total $+1.6\%$
------------------------------------------------------------------------------------- ----------
: Additional known effects not included in Eq. (\[bbth\]).
\[tab:add\]
As the reader has already noticed, even in the $m_c$-interpolation approach of Ref. [@Misiak:2006ab], there are still some missing NNLO ingredients, namely:
- the BLM contributions to $G_{18}$ and $G_{28}$,
- the large-$m_c$ results for $G_{78}$ and $G_{88}$.
Their numerical effect on the branching ratio is expected to remain within the estimated higher-order uncertainty of around 3%. The calculation of the most interesting $G_{78}$ is very advanced [@calc.78.yyy], and the results should become available soon (for any value of $m_c$). As far as $G_{88}$ is concerned, its calculation in the large-$m_c$ limit will automatically give the result for any value of $m_c$.
For the full NNLO calculation, the currently missing ingredients (apart from the ones listed above) are the non-BLM corrections to
- $G_{17}$ and $G_{27}$,
- $G_{11}$, $G_{12}$ and $G_{22}$,
- $G_{18}$ and $G_{28}$.
The calculation of [*(i)*]{} in the $m_c=0$ limit is quite advanced, but also extremely difficult and time-consuming — see Ref. [@Boughezal:2007km] for the status reports. Once it is finished, the main challenge will amount to finding [*(ii)*]{}, even for $m_c=0$. The corrections [*(iii)*]{} are expected to be numerically less important.
When the non-BLM corrections are known at$m_c=0$ sometime in the future, the interpolation in $m_c$ is still going to be necessary. However, our error estimates should become more solid once the BLM approximation is no longer used at the boundary.
Finding all the non-BLM corrections for the actual value of $m_c \simeq m_b/4$ is even more difficult, but it must be considered at some point, too. A calculation of the IR-divergent two-particle-cut contributions to [*(i)*]{} is being currently performed for arbitrary $m_c$ [@Boughezal:2007km]. The IR-convergent two-particle-cut contributions to [*(ii)*]{} for arbitrary $m_c$ are already known because they are given by products of the NLO corrections. The $(n\geq3)$-particle-cut contributions to [*(ii)*]{} vanish at the endpoint, so their numerical relevance should be diminished by the high photon energy cutoff.
Apart from the NNLO corrections, there are other perturbative contributions to $\Gamma ( b \rightarrow X_s \gamma )_{E_{\gamma} > E_0}$ that have been neglected so far, namely tree-level diagrams with the $u$-quark analogues of $O_{1,2}$ and the four-quark penguin operators $O_{3,4,5,6}$. Such contributions are suppressed with respect to the leading term by $|(C_{1,2}^u,
C_{3,4,5,6})/C_7|^2 \leq 0.2$, where $C_{1,2}^u = (V_{us}^* V_{ub})/(V_{ts}^*
V_{tb}) C_{1,2}$, as well as by the high photon energy cutoff. A quantitative verification of how small they really are should become available soon [@Kaminski:2008yyy].
Non-perturbative effects \[sec:nonpert\]
========================================
Let us begin with considering a simplified world where $m_c=m_b$. There, in the decay of the $\bar{B}$ meson, a high-energy photon ($E_\gamma \sim m_b/2$) can be produced in four different ways:
- Hard: The photon is emitted directly from the hard process of the $b$-quark decay.
- Conversion: The $b$-quark decays in a hard way into quarks and gluons only. Next, one of the decay products scatters in a non-soft radiative manner with the remnants of the $\bar B$ meson. This can be viewed as a parton-to-photon conversion in the QCD medium.
- Collinear: In the process of hadronization, a collinear photon is emitted.
- Annihilation: An energetic $q\bar q$ state produced in the $\bar{B}$ meson decay disintegrates radiatively.
The hard way would be the only one if no other operators but $O_7$ were present in the effective theory. Non-perturbative effects in such a case were first analyzed in Ref. [@Falk:1993dh]. They arise as corrections of order $\Lambda^2/m_b^2$ to $\Gamma ( b \rightarrow X_s \gamma )_{E_{\gamma} > E_0}$ when $(m_b-2E_0) \sim m_b$. Moreover, these ${\cal O}(\Lambda^2/m_b^2)$ terms cancel out in $N(E_0)$ in Eq. (\[main\]) with the analogous non-perturbative corrections to the charmless semileptonic rate. Thus, we are left with the small ${\cal O}(\Lambda^3/m_b^3)$ effects [@Bauer:1997fe]. The ${\cal O}\left(\Lambda^2/(m_b-2E_0)^2\right)$ corrections are also small [@Neubert:2004dd] for $E_0 \leq 1.6\,$GeV. All these terms are included in Eq. (\[bbth\]), and affect the branching ratio by around $-0.7\%$.
The photon production via conversion is suppressed both by $\alpha_s$ (due to the non-soft scattering) and by $\Lambda/m_b$ (due to dilution of the target). The analysis in Ref. [@Lee:2006wn] shows that no other suppression factors occur. An effect on the branching ratio of roughly $-1.5 \pm 1.5\%$ was found in that paper (see Tab. \[tab:add\]).
In our simplified case ($m_c=m_b$), the tree-level decay $b \to ({\rm quarks\;\&\;gluons})_s$ is possible only via the operators $O_{3,4,5,6,8}$, and by the $u$-quark analogues of $O_{1,2}$. Consequently, the collinear photon emission is suppressed either by $\alpha_s |C_8/C_7|^2
\leq 0.08$, or by $|(C_{1,2}^u, C_{3,4,5,6})/C_7|^2 \leq 0.2$. Moreover, there is an additional suppression by products of the quark electric charges and, most importantly, by the high photon energy cutoff. The results of Ref. [@Kapustin:1995fk] lead to an estimate that the non-perturbative collinear effects due to $O_8$ amount to around $-0.2\%$ in the branching ratio for $E_0 = 1.6\,$GeV (see Tab. \[tab:add\]).
As far as annihilation is concerned, photons originating from decays of $\pi^0$, $\eta$, $\eta'$ and $\omega$ are removed on the experimental side as (huge) background. Contributions from the other established $q\bar q$ mesons are negligible. The corresponding perturbative diagrams are responsible for only around 0.1% of the total rate for $E_0 = 1.6\,$GeV. Once the assumption $m_c\!=\!m_b$ is relaxed, the ${\cal O}(\Lambda^2/m_b^2)$ hard effects from $O_{1,2}$ get replaced by a series of the form [@Buchalla:1997ky] [\^2]{}[m\_c\^2]{} \_[n=0]{}\^ b\_n ( [m\_b]{}[m\_c\^2]{} )\^n, with quickly decreasing coefficients $b_n$. The calculable leading term has been included in Eq. (\[bbth\]). It affects the branching ratio by around $+3.1\%$. All the quantitatively estimated non-perturbative effects that have been mentioned so far sum up to (-0.7-0.2-1.5+3.1)% = +0.7%. However, since their evaluation is often very uncertain, and the knowledge of ${\cal O}(\alpha_s \Lambda/m_b)$ contributions is by no means complete, a non-perturbative error of $\pm 5\%$ has been assumed in Eq. (\[bbth\]), as already mentioned Sec. \[sec:melem\]. Probably the most interesting of all the unknown ${\cal O}(\alpha_s \Lambda/m_b)$ effects originate from charm annihilation in the massive $(\bar cs)(\bar qc)$ intermediate states ($q=u$ or $d$), for the actual value of $m_c \simeq m_b/4$. One should remember that the error in Eq. (\[bexp\]) is affected by a non-perturbative theoretical uncertainty, too. It follows from the fact that the actual measurements are not performed with $E_0 \simeq 1.6\,$GeV. The most precise experimental results correspond to significantly higher photon energy cutoffs for which the ${\cal O}\left(\Lambda^n/(m_b-2E_0)^n\right)$ effects are no longer small. These effects are described by a non-perturbative shape function [@Neubert:1993ch] that is constrained by the semileptonic data. The very recent analysis [@Ligeti:2008ac] of this function and its effects on the $\bar{B} \to X_s \gamma$ photon spectrum can hopefully provide input for the future HFAG averages.
Conclusions \[sec:concl\]
=========================
Thanks to the efforts of the past years, the uncertainties in ${\cal
B}\!\left( \bar{B} \to X_s \gamma \right)$ have reached the level of around $\pm 7\%$ on both the experimental and theoretical sides. A significant progress in the perturbative calculations is expected in the near future. However, understanding the ${\cal O}(\alpha_s \Lambda/m_b)$ non-perturbative effects remains the key issue.
Note Added
==========
After the first version of this article was submitted to the arXiv, Einan Gardi pointed out to me that the approximate cancellation of logarithmic and non-logarithmic terms (Sec. \[sec:cutoff\]) has already been discussed in Ref. [@Andersen:2006hr].
I would like to thank the organizers of theHQL 2008 conference for hospitality. I am grateful to P. Gambino, C. Greub and T. Ewerth for helpful remarks. This work has been supported in part by the Polish Ministry of Science and Higher Education as a research project N N202 006334 (in years 2008-11), and by the EU-RTN Programme, contract no. MRTN–CT-2006-035482, “Flavianet”.
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[^1]: More recent contributions are discussed in Secs. \[sec:normal\]–\[sec:nonpert\].
[^2]: The specific matrices $\Gamma_i$ and $\Gamma'_i$ can be found in Ref. [@Chetyrkin:1996vx].
|
---
abstract: 'In this paper we continue our study of the moduli space of stable bundles of rank two and degree $1$ on a very general quintic surface. The goal in this paper is to understand the irreducible components of the moduli space in the first case in the “good” range, which is $c_2=10$. We show that there is a single irreducible component of bundles which have seminatural cohomology, and conjecture that this is the only component for all stable bundles.'
address:
- |
CNRS, Laboratoire J. A. Dieudonné, UMR 7351\
Université de Nice-Sophia Antipolis\
06108 Nice, Cedex 2, France
- |
CNRS, Laboratoire J. A. Dieudonné, UMR 7351\
Université de Nice-Sophia Antipolis\
06108 Nice, Cedex 2, France
author:
- Nicole Mestrano
- Carlos Simpson
title: 'Seminatural bundles of rank two, degree one and $c_2=10$ on a quintic surface'
---
[^1]
This paper is the next in a series, starting with [@MestranoSimpson], in which we study the moduli spaces of rank two bundles of odd degree on a very general quintic hypersurface $X\subset {{\mathbb P}}^3$. This series is dedicated to Professor Maruyama, who brought us together in the study of moduli spaces, a subject in which he was one of the first pioneers.
In the first paper, we showed that the moduli space $M_X(2,1,c_2)$, of stable bundles of rank $2$, degree $1$ and given $c_2$, is empty for $c_2\leq 3$, irreducible for $4\leq c_2\leq 9$, and good (i.e. generically smooth of the expected dimension) for $c_2\geq 10$. On the other hand, Nijsse has shown that the moduli space is irreducible for $c_2\geq 15$ [@Nijsse] using the techniques of O’Grady [@OGradyIrred] [@OGradyBasic]. This leaves open the question of irreducibility for $10\leq c_2\leq 14$.
\[all\] The moduli space $M_X(2,1,10)$ is irreducible.
We haven’t yet formulated an opinion about the cases $11\leq c_2\leq 14$.
In the present paper, due to lack of time and for length reasons, we treat a special case of the conjecture: the case of bundles with [*seminatural cohomology*]{}, meaning that only at most one of $h^0(E(n))$, $h^1(E(n))$ or $h^2(E(n))$ can be nonzero for each $n$. Let $M^{sn}_X(2,1,10)$ denote the open subvariety of the moduli space consisting of bundles with seminatural cohomology. In Section \[seminatural\] we show that the seminatural condition is a consequence of assuming just $h^0(E(1))=5$. The main result of this paper is:
\[main\] The moduli space $M^{sn}_X(2,1,10)$ is irreducible.
Recall from [@MestranoSimpson] that our inspiration to look at this question came from the recent results of Yoshioka, for the case of Calabi-Yau surfaces originating in [@Mukai]. Yoshioka shows that the moduli spaces are irreducible for all positive values of $c_2$, when $X$ is an abelian or K3 surface [@YoshiokaAbelian] [@YoshiokaK3]. His results apply for example when $X$ is a general quartic hypersurface. We thought it was a natural question to look at the case of a quintic hypersurface, which is one of the first cases where $X$ has general type, with $K_X={{\mathcal O}}_X(1)$ being as small as possible.
[*Remark on the difficulty of this project:*]{} We were somewhat surprised by the diversity of techniques needed to treat this question. Much of the difficulty stems from the possibilities of overdetermined intersections which need to be ruled out at various places in the argument. This question is inherently very delicate, because there is not, to our knowledge, any general principle which would say whether the moduli space is “supposed to be” irreducible or not. On the one hand, the present case is close to the abelian or K3 case, so it isn’t too surprising if the moduli space remains irreducible; however on the other hand, at some point new irreducible components will be appearing as has been shown by the first author in [@Mestrano]. So, we are led to analyse a number of cases for various aspects of the argument. If any case is mistakenly ignored, it might hide a new irreducible component which would then be missed.
A natural question to wonder about is whether “derived algebraic geometry” could help here, but it would seem that those techniques need to be further developed in order to apply to some basic geometric situations such as we see here. Furthermore, each place in the argument where some case is ruled out, constitutes a possible reason for there to be additional irreducible components in more complicated situations (such as on a sextic hypersurface). So, in addition to the theorem itself which only goes a little way into the range that remains to be treated, much of the interest lies in the geometric situations which are encountered along the way.
Notations
=========
Throughout the paper, $X\subset {{\mathbb P}}^3$ denotes a very general quintic hypersurface, and $E$ is a stable rank two vector bundle of degree one[^2] with determinant $\bigwedge ^2E\cong {{\mathcal O}}_X(1)$ such that $c_2(E)=10$. The moduli space of stable bundles in general has been the subject of much work [@Gieseker; @GiesekerCons; @GiesekerLi; @HuybrechtsLehn; @Maruyama; @MaruyamaET; @MaruyamaTransform; @Mukai; @OGradyIrred; @OGradyBasic; @YoshiokaAbelian; @YoshiokaK3], but the special case $M_X(2,1,10)$ considered here goes into a somewhat new and uncharted territory.
Note that $Pic(X)={{\mathbb Z}}$ is generated by ${{\mathcal O}}_X(1)$. The canonical bundle is $K_X={{\mathcal O}}_X(1)$. For any $n$ we have $H^1({{\mathcal O}}_X(n))=0$. For $n\leq 4$ the map $H^0({{\mathcal O}}_{{{\mathbb P}}^3}(n))\rightarrow H^0({{\mathcal O}}_X(n))$ is an isomorphism.
The Hilbert polynomial of $E$ is $\chi (E(n))= 5n^2$. In particular $\chi (E)=0$. We will be assuming that $E$ is general in some irreducible component of the moduli space. From the previous paper [@MestranoSimpson] using some techniques for bounding the singular locus which had also been introduced in [@Langer] and [@Zuo], it follows that $E$ is unobstructed, so if $End^0(E)$ denotes the trace-free part of $End(E)$ then $H^2(End^0(E))=0$. Note however that $H^2({{\mathcal O}}_X)={{\mathbb C}}^4$, indeed it is dual to $H^0({{\mathcal O}}_X(1))=H^0({{\mathcal O}}_{{{\mathbb P}}^3}(1))$. Thus $$H^2(E\otimes E^{\ast})\cong {{\mathbb C}}^4.$$ The dual bundle is given by $E^{\ast}=E(-1)$, so duality says that $H^i(E(n))\cong H^{2-i}(E(-n))$.
The dimension of any irreducible component of the moduli space is the expected one, $20$. The subspace of bundles $E$ with $H^0(E)\neq 0$ has dimension $<20$, see our previous paper [@MestranoSimpson], so a general $E$ has $H^0(E)=0$. It follows from duality that $H^2(E)=0$ and by $\chi (E)=0$ we get $H^1(E)=0$. Throughout the paper (except at one place in Section \[onquintic\]), we consider only bundles with $H^0(E)=0$.
Duality says that $H^2(E(1))$ is dual to $H^0(E(-1))=0$. Since $\chi (E(1))=5$, if we set $f:= h^1(E(1))$ then $h^0(E(1))=5+f$. In particular there are at least $5$ linearly independent sections of $E(1)$ which may be viewed as maps $s:{{\mathcal O}}_X(-1)\rightarrow E$ or equivalently $s:{{\mathcal O}}_X\rightarrow E(1)$. Note that the zero set of $s$ has to be of codimension $2$, as any codimension-one component would be a divisor integer multiple of the hyperplane class but $h^0(E)=0$ so this can’t happen. If we choose one such map $s$, then we get the [*standard exact sequence*]{} $$\label{standardexact}
0\rightarrow {{\mathcal O}}_X (-1)\rightarrow E \rightarrow J_P(2)\rightarrow 0$$ and its twisted versions such as $$\label{standardexact1}
0\rightarrow {{\mathcal O}}_X \rightarrow E(1) \rightarrow J_P(3)\rightarrow 0.$$ As a notational matter, $J_P$ denotes the ideal of $P$ in $X$ or in ${{\mathbb P}}^3$. Which one it is should be clear from context, for example it is the ideal of $P\subset X$ in the above sequences, and we choose not to weigh down the notation with an extra subscript.
Calculation of the Chern class $c_2(E)=10$ shows that $P\subset X$ is a subscheme of length $20$. It is a union of possibly nonreduced points, which are locally complete intersections i.e. defined by two equations. Furthermore, as is classically well-known [@EisenbudGreenHarris] [@Iarrobino], $P$ satisfies the Cayley-Bacharach condition for ${{\mathcal O}}_X(4)$ which we denote by $CB(4)$, saying that any subscheme $P'\subset P$ of colength $1$ imposes the same number of conditions as $P$ on sections of ${{\mathcal O}}_X(4)$. The extension class is governed by an element of $H^1(J_P(4))^{\ast}$, and we have the exact sequence $$0\rightarrow H^0(J_P(4))\rightarrow H^0({{\mathcal O}}_X(4))\rightarrow {{\mathcal O}}_P(4)\rightarrow H^1(J_P(4))\rightarrow 0.$$ To get a locally free $E$, the extension class should be nonzero on each vector coming from a point in $P$, the existence of such being exactly $CB(4)$. Note that $h^0({{\mathcal O}}_X(4)) = 35$, and define $e:= h^1(J_P(4))-1$. Then $e\geq 0$ (the extension can’t be split, indeed this is part of the $CB(4)$ condition), and $h^0(J_P(4))=16+e$.
The “well-determined” case is when $e=0$. Then the extension class is well-defined up to a scalar multiple which doesn’t affect the isomorphism class of $E$, and the existence of the nonzero class in $H^1(J_P(4))$ is expected to impose $16$ conditions on the $20$ points, giving the expected dimension of the Hilbert scheme of such subschemes $P\subset {{\mathbb P}}^3$ to be $44$. In Section \[seminatural\], we will show $f=0\Rightarrow e=0$ and in that case the bundle $E$ has seminatural cohomology.
Here are a few techniques which will often be useful throughout the paper.
If a zero-dimensional subscheme $P\subset X$ satisfies $CB(n)$ then it satisfies $CB(m)$ for any $m\leq n$.
Suppose $P'\subset P$ has colength $1$. Choose a section $g\in H^0({{\mathcal O}}_X(n-m))$ nonvanishing at all points of $P$, then if $f\in H^0(J_{P'}(m))$ we have $fg\in H^0(J_{P'}(n))$. By $CB(n)$, $fg$ vanishes on $P$, but $g$ is a unit near any point of $P$ so $f$ vanishes on $P$, proving $CB(m)$.
The results of [@BGS] allow us to estimate the dimension of the Hilbert scheme of $0$-dimensional subschemes of a curve, as was used in some detail in [@MestranoSimpson]. Mainly, as soon as the curve is locally planar, the space of subschemes of length $\ell$ has dimension $\leq \ell$.
If $W\subset X$ is a divisor and $P$ is a zero-dimensional subscheme, we obtain the [*residual subscheme*]{} $P^{\perp}$ of $P$ with respect to $W$, such that $\ell (P^{\perp}) +\ell (P\cap W)=\ell (P)$. It is characterized by the property that sections of ${{\mathcal O}}_X(n)(-W)$ which vanish on $P^{\perp}$, map to sections of ${{\mathcal O}}_X(n)$ vanishing on $P$. If $P$ is reduced then $P^{\perp}$ is just the union of those point of $P$ which are not in $W$; if $P$ contains some nonreduced schematic points then the structure of $P^{\perp}$ may be more complicated.
\[inquadric\] If $P$ satisfies $CB(3)$ and $P''\subset P$ is a subscheme of colength $2$, suppose $P''$ is contained in a quadric. Then $P$ is contained in the same quadric.
The residual subscheme $P^{\perp}$ for the quadric has length $\leq 2$. If it is nonempty, we can choose a linear form containing a subscheme of colength $1$ of $P^{\perp}$, corresponding to a subscheme $P^1\subset P$ of colength $1$. Applying $CB(3)$ to the product of the quadric and the linear form is a contradiction, so $P^{\perp}=\emptyset$ and we’re done.
Restriction to a plane section {#planesection}
==============================
Suppose $H\subset {{\mathbb P}}^3$ is a hyperplane, and let $Y:=H\cap X$. By the genericity assumption on $X$ in particular $Pic(X)$ generated by ${{\mathcal O}}_X(1)$, we get that $Y$ has to be reduced and irreducible. Its canonical sheaf is ${{\mathcal O}}_Y(2)$. When $Y$ is smooth, then, it is a plane curve of degree $5$ and genus $6$. We have an exact sequence $$0\rightarrow E \rightarrow E(1)\rightarrow E_Y(1)\rightarrow 0.$$ From the vanishing of $H^i(E)$ it follows that $H^2(E(1))=0$ (but this is also clear from duality), and $$H^0(E(1))\stackrel{\cong}{\longrightarrow} H^0(E_Y(1)),$$ $$H^1(E(1))\stackrel{\cong}{\longrightarrow} H^1(E_Y(1)).$$
Suppose $L\subset {{\mathbb P}}^3$ is a line. A generic $X$ doesn’t contain any lines, so $L\cap X$ is a finite subscheme of length $\ell (L\cap X)=5$. We claim that for a general plane $H$ containing $L$, the intersection $Y=H\cap X$ is smooth. This holds by Bertini’s theorem away from the base locus of the linear system of planes passing through $L$, so we just have to see that it also holds at a point $x\in L\cap X$. Note that $T_xL\subset T_xX$ is a one-dimensional subspace. A general $H$ will have tangent space which is a general plane in $T_x{{\mathbb P}}^3$ containing $T_xL$. Thus, a general plane $H$ containing $L$ has tangent space $T_xH$ which doesn’t contain $T_xX$; in particular the intersection $T_xH\cap T_xX=T_xL$ is transverse. This implies that $H\cap X$ is smooth at $x$. This works for all the finitely many points $x\in L\cap X$, so the general section $Y=H\cap X$ is smooth. It is therefore a smooth plane curve of degree $5$ and genus $6$. Notice that $L\subset H$ so $L\cap X \subset Y$.
Pick $Y$ as in the previous paragraph, suppose $Q\subset L\cap X$ is a finite subscheme of length $4$, and suppose that $x\in H^0(E(1))$ is a section vanishing on $Q$. Then $s|_Y$ is a section of $H^0(E(1))$ vanishing on $Q\subset Y$. As $Y$ is smooth, the finite subscheme $Q$ is a Cartier divisor. The section $s|_Y$ corresponds to a map ${{\mathcal O}}_Y\rightarrow E(1)$ which, since it vanishes on $Q$, gives a map $${{\mathcal O}}_Y(Q)\rightarrow E(1).$$ Let $Q'\subset Y$ be the divisor of zeros of $s$, in particular $Q\subset Q'$, and $s$ extends to a strict map, i.e. an inclusion of a sub-vector bundle $${{\mathcal O}}_Y(Q')\hookrightarrow E(1).$$ The quotient line bundle is ${{\mathcal O}}_Y(3-Q')$ where the notation here combines ${{\mathcal O}}_Y(3)$ which is three times the hyperplane divisor (which has degree $5$ on $Y$), with the divisor $Q'$. In particular ${{\mathcal O}}_Y(3-Q')$ is a line bundle of degree $15-\ell (Q')$. We obtain an exact sequence $$0\rightarrow {{\mathcal O}}_Y(Q')\rightarrow E_Y(1)\rightarrow {{\mathcal O}}_Y(3-Q')\rightarrow 0,$$ leading to the long exact sequence of cohomology. This construction will be used many times in Section \[baseloci\].
Another useful construction is the following. Write $L\cap Y=x+y+u+v+w$, possibly with some of the points being the same. Take a linear form containing $w$ as an isolated zero, and divide by the equation of $L$. This gives a meromorphic function whose polar divisor is $x+y+u+v$. Equivalently, ${{\mathcal O}}_Y(x+y+u+v)$ has a section nonvanishing at the points $x,y,u,v$. This will be used often without too much further notice below.
The seminatural condition {#seminatural}
=========================
\[snhyp\] Assume that $h^0(E(1))=5$, and that $H^i(E)=0$ for $i=0,1,2$.
Recall that the second part is true for any $E$ general in its irreducible component as discussed above.
The goal of this section is to show that \[snhyp\] implies $f=0$ and $E$ has seminatural cohomology, which in this case means $H^0(E(n))=0$ for $n\leq 0$, $H^2(E(n))=0$ for $n\geq 0$, and $H^1(E(n))=0$ for all $n$. Our main Theorem \[main\] is the statement that there is only a single irreducible component corresponding to such bundles, so Hypothesis \[snhyp\] will be in effect throughout the rest of the paper.
\[sncase\] If $h^0(E(1))=5$ then $f=0$, in other words $H^1(E(1))=0$. If $s:{{\mathcal O}}(-1)\rightarrow E$ has scheme of zeros $P$, then saying $h^0(E(1))=5$ is equivalent to requiring that $h^0(J_P(3))=4$, and saying that all $h^i(E)=0$ is equivalent to requiring that $h^0(J_P(2))=0$.
As discussed above, $h^2(E(1))=0$ so the fact that $\chi (E(1))=5$ gives the first statement. For the second statement, use the fact that $H^1({{\mathcal O}}_X(n))=0$ for all $n$, and the long exact sequences of cohomology for the extension $E(1)$ of $J_P(3)$ by ${{\mathcal O}}_X$ and similarly $E$ of $J_P(2)$ by ${{\mathcal O}}_X(-1)$.
A first part of the seminatural condition is easy to see.
\[seminat02\] Under our hypothesis \[snhyp\], $H^0(E(n))=0$ for $n\leq 0$, and $H^2(E(n))=0$ for $n\geq 0$.
Since $H^0(E)=0$ it follows that $H^0(E(n))=0$ for all $n\leq 0$, and for $n\geq 0$, $H^2(E(n))$ is dual to $H^0(E(-n))=0$.
The main step towards the seminatural condition is the next twist:
We also have $H^1(E(2))=0$.
If $Y=H\cap X$ is a smooth plane section, we claim $H^0(E_Y(-1))=0$. If not, then we would get an inclusion ${{\mathcal O}}_Y(1)\hookrightarrow E_Y$, hence ${{\mathcal O}}_Y(2)\hookrightarrow E_Y(1)$. However, $Y$ is a curve of genus $6$ and $K_Y={{\mathcal O}}_Y(2)$ so $H^0({{\mathcal O}}_Y(2))$ has dimension $6$. This gives $h^0(E_Y(1))\geq 6$. Consider the exact sequence $$0\rightarrow E\rightarrow E(1)\rightarrow E_Y(1)\rightarrow 0.$$ The fact that $H^1(E)=0$ implies that $H^0(E(1))$ surjects onto $H^0(E_Y(1))$, so $h^0(E(1))\geq 6$. This is a contradiction to $h^0(E(1))=5$, showing that $H^0(E_Y(-1))=0$.
To show that $H^1(E(2))=0$, it suffices by duality to show that $H^1(E(-2))=0$. Consider the exact sequence $$0\rightarrow E(-2)\rightarrow E(-1)\rightarrow E_Y(-1)\rightarrow 0.$$ Again by duality from Lemma \[sncase\], $H^1(E(-1))=0$, so the long exact sequence gives an isomorphism between $H^0(E_Y(-1))$ and $H^1(E(-2))$. From the previous paragraph we obtain $H^1(E(-2))=0$. This proves the proposition.
\[seminatcor\] Under Hypothesis \[snhyp\], $E$ has seminatural cohomology: $H^1(E(n))=0$ for all $n$.
By duality it suffices to consider $n\geq 0$ and we have already done $n=0,1,2$. Consider the case $n=3$. This could be done by continuing as in the previous proposition but here is another argument. Choose an inclusion $s:{{\mathcal O}}(-1)\rightarrow E$, and let $P$ be the subscheme of zeros of $s$. Choose a general plane section $Y=H\cap X$ such that $H$ passes through one point $z\in P$ in a general direction. Then $s|_Y$ has a zero at $z$, of multiplicity $m$ with $1\leq m\leq 5$. Indeed, $P$ cannot contain a $6$-fold fat point whose length is $21$, because $P$ has length $20$. Thus the multiplicity of a general plane section of $P$ at any point $z$ is $\leq 5$. The section $s$ restricted to $Y$ therefore induces a strict inclusion of vector bundles from ${{\mathcal O}}_Y(m\cdot z)$ to $E(1)$, hence an exact sequence, of the form $$0\rightarrow {{\mathcal O}}_Y(2+m\cdot z) \rightarrow E(3) \rightarrow {{\mathcal O}}_Y(5 - m\cdot z) \rightarrow 0.$$ Note that the sub-line bundle has degree $10+m$ and the quotient line bundle has degree $25-m$, so both of these have vanishing $H^1$ by duality. It follows that $H^1(E(3))=0$. For any $n\geq 4$ a similar argument (but $Y$ doesn’t even need to pass through a point of $P$) shows that $H^1(E(n))=0$.
It follows that $e=0$, which is to say that for any inclusion $s:{{\mathcal O}}(-1)\rightarrow E$, if $P$ is the subscheme of zeros of $s$ then $h^0(J_P(4))=16$.
Choose an inclusion $s$ and consider the exact sequence $$0\rightarrow {{\mathcal O}}_X(1)\rightarrow E(2)\rightarrow J_P(4)\rightarrow 0.$$ Notice that $H^2({{\mathcal O}}(1)) = H^2(K_X)={{\mathbb C}}$. The long exact sequence of cohomology then reads $$0\rightarrow H^1(E(2))\rightarrow H^1(J_P(4))\rightarrow {{\mathbb C}}\rightarrow 0,$$ since $H^2(E(2))=0$ and $H^1({{\mathcal O}}_X(n))=0$ for all $n$. The previous conclusion says the term on the left $H^1(E(2))$ vanishes, so $H^1(J_P(4))={{\mathbb C}}$. It is generated by the nonzero extension class governing the exact sequence corresponding to $s$. On the other hand we have $$0\rightarrow J_P(4)\rightarrow {{\mathcal O}}_X(4)\rightarrow {{\mathcal O}}_P(4)\rightarrow 0$$ so the map $H^0({{\mathcal O}}_X(4)) = {{\mathbb C}}^{35}\rightarrow {{\mathcal O}}_P(4) = {{\mathbb C}}^{20}$ has cokernel $H^1(J_P(4))$ of dimension $1$. It follows that the kernel $H^0(J_P(4))$ has dimension $16$.
\[snunique\] Pick a section $s\in H^0(E(1))$ and let $P$ be its subscheme of zeros. The extension class defining $E$ as an extension of $J_P(2)$ by ${{\mathcal O}}_X(-1)$ is unique up to a scalar.
Recall from where $e$ was defined that the space of extensions, $H^1(J_P(4))$, has dimension $e+1$. Thus, the condition $e=0$ means that this is a line: the extension is unique up to scalars, and for a given subscheme $P$ there is a unique bundle extension $E$ up to isomorphism.
\[H0EYzero\] If $Y=H\cap X$ is a plane section, then $H^0(E_Y)=0$. Also, $H^0(E(1))\stackrel{\cong}{\rightarrow} H^0(E_Y(1))$ and $H^1(E_Y(1))=0$.
Consider the exact sequence $$0\rightarrow E(-1) \rightarrow E\rightarrow E_Y \rightarrow 0.$$ From $H^0(E)=0$ and $H^1(E(-1))=0$ we get $H^0(E_Y)=0$. Similarly, the exact sequence $$0\rightarrow E \rightarrow E(1)\rightarrow E_Y(1) \rightarrow 0$$ together with $H^i(E)=0$ gives $H^i(E(1))\stackrel{\cong}{\rightarrow} H^i(E_Y(1))$.
The structure of the base loci {#baseloci}
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Let $B_2\subset X$ be the subset of points where all sections of $H^0(E(1))$ vanish. Let $B_1\subset X$ be the subset of points $x$ such that the image of $H^0(E(1))\rightarrow E(1)_x$ has dimension $\leq 1$ (in particular $B_2\subset B_1$). These are the [*base loci*]{} of sections of $E(1)$. In this section, we obtain some information about these base loci, which will allow us to to deduce, in Section \[Pstructure\], that the zero-scheme of a general section $s$ has some fairly strong general position properties.
There is at most one point in $B_2$ {#atmostone}
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\[B2onepoint\] The subset $B_2$ has at most one point and if it exists, then the sections of $H^0(E(1))$ define this reduced point as a subscheme.
Suppose $p\neq q$ are two points of $B_2$. Then all sections of $E(1)$ vanish at $p$ and $q$. Consider a plane section $Y=H\cap X$ such that $p,q \in Y$, but $Y$ general for this property, in particular $Y$ is smooth (Section \[planesection\]). The map $$E(1)_p \oplus E(1)_q \rightarrow H^1(E_Y(1-p-q))$$ is injective. Furthermore it is surjective since $H^1(E_Y(1))=0$.
Let $L$ denote the line through $p$ and $q$. It intersects $Y$ in a divisor denoted $p+q+u+v+w$. Some of the points $u,v,w$ may be equal or equal to $p$ or $q$.
We have an exact sequence $$0\rightarrow E_Y \rightarrow E_Y(1)\rightarrow E_{L\cap Y}(1)\rightarrow 0,$$ and on the other hand, the exact sequence $$0\rightarrow E(-1)\rightarrow E\rightarrow E_Y\rightarrow 0$$ gives $H^1(E_Y)\stackrel{\cong}{\rightarrow} H^2(E(-1))\cong H^0(E(1))^{\ast}\cong {{\mathbb C}}^5$. Hence the image of $H^0(E_Y(1))\rightarrow E_{L\cap Y}(1)$ has codimension $5$, and since $L\cap Y$ is a finite subscheme of length $5$, $E_{L\cap Y}(1)\cong {{\mathbb C}}^{10}$ so the image has dimension $5$ too.
We may impose the condition of vanishing at two points $u,v$ and obtain a nonzero section $s\in H^0(E_Y(1)(-p-q-u-v))$. This has the required meaning when some of the points coincide, using the previous paragraph. However, the section $s$ then doesn’t vanish at the third point $w$, otherwise we would get a section in $H^0(E_Y(1)(-L\cap Y))=H^0(E_Y)$ contradicting Corollary \[H0EYzero\].
This section generates a sub-line bundle $M\subset E_Y(1)$, with $M={{\mathcal O}}_Y(p+q+u+v+D)$ for an effective divisor $D$ not passing through $w$. Note that ${{\mathcal O}}_Y(p+q+u+v)$ has a nonzero section, corresponding to the quotient of a linear form (on the plane $H$) vanishing at $w$ but not along $L$, divided by a linear form vanishing along $L$. If $D$ doesn’t contain both $p$ and $q$ then we would get a section of $E(1)$ nonvanishing at one of those points, contradicting our assumption $p,q\in B_2$. Therefore $D\geq p+q$. It follows that $w\neq p,q$. The same reasoning works for $u$ and $v$ too, so $u,v,w$ are three points distinct from $p$ or $q$.
Our section $s$ comes from a section in $H^0(E(1))$ corresponding to ${{\mathcal O}}(-1)\rightarrow E$, and the subscheme of zeros $P$ contains $p,q,u,v$. These are four points on the line $L$, so any cubic form vanishing at $P$ has to vanish along $L$. In particular, elements of $H^0(J_P(3))$ vanish at $w$. This implies that elements of $H^0(E(1))$ evaluate at $w$ to elements in the line $M_w\subset E(1)_w$. Thus $w\in B_1$.
This reasoning holds even if $w$ coincides say with $v$; it means that all elements of $J^0(J_P(3))$ have to vanish in the tangent direction corresponding to the additional point $w$ glued onto $v$, which still gives a rank one condition on the values of sections of $E(1)$ at the point $w$.
The same reasoning holds for $u$ and $v$. If at least two of the points $u,v,w$ are distinct, then we obtain this way at least two points of $B_1$ along the line $L$. Then, vanishing at these two points consists of two conditions, so we can impose further vanishing at the third point (even if it is a tangential point at one of the other two) and obtain a nonzero section which vanishes at all five points. As before this yields a nonzero section of $H^0(E_Y)$ contradicting Corollary \[H0EYzero\].
It remains to consider the case when all three points are the same, that is to say $L\cap Y=p+q+3u$ with $u\neq p,q$, and choosing a section vanishing at $p,q$ and two times at $u$ generates a subbundle $M={{\mathcal O}}_Y(ap+bq+2u +D)\hookrightarrow E(1)$ with $D$ an effective divisor distinct from $p,q,u$, and $a,b\geq 2$. Recall that if either $a=1$ or $b=1$ then this would give a section of $E(1)$ nonvanishing at $p$ or $q$ contradicting our assumption $p,q\in B_2$.
As above, we have $u\in B_1$. Therefore, choosing a section in $H^0(E_Y(1)(-2u))$ represents only $3$ conditions rather than $4$, hence there are two linearly independent such sections $s_1,s_2$. We claim that the values of these two sections, in $E_Y(1)(-2u)_u$, are linearly independent. Indeed, otherwise a combination of the two would vanish again at $u$ and this would give a section of $E_Y(1)(-3u)$ which also vanishes at $p, q\in B_2$. This would give a nonzero element of $H^0(E_Y)$ which can’t happen.
Let $a$ and $b$ be the smallest possible orders of vanishing of $s_i$ at $p$ and $q$ respectively, and by linear combinations we can assume that both of them vanish to those orders. They give maps $$M_1={{\mathcal O}}_Y(ap+bq+2u +D_1)\stackrel{s_1}{\rightarrow} E_Y(1),$$ $$M_2={{\mathcal O}}_Y(ap+bq+2u +D_2)\stackrel{s_2}{\rightarrow} E_Y(1),$$ and the resulting map $$M_1\oplus M_2 \rightarrow E_Y(1)$$ has image of rank $2$ at the point $u$ by the previous paragraph. Therefore it is injective. It follows that ${\rm deg}(M_1\oplus M_2)\leq {\rm deg}(E_Y(1))= 15$. Suppose ${\rm deg}(D_1)$ is the smaller of the two, then we get $a+b+2+{\rm deg}(D_1)\leq 7$. We may also by symmetry assume $a\geq b$. Write $M=M_1$ and $D=D_1$. There are three possibilities: $$M = {{\mathcal O}}_Y (2p+2q+2u + d), \;\;\; D=(d),\;\;\; {\rm deg}(M)= 7$$ $$M = {{\mathcal O}}_Y (2p+2q+2u), \;\;\; D=0,\;\;\; {\rm deg}(M)= 6$$ or $$M = {{\mathcal O}}_Y (3p+2q+2u), \;\;\; D=0,\;\;\; {\rm deg}(M)= 7.$$
In each case, let $N:= E_Y(1)/M = {{\mathcal O}}_Y(3)\otimes M^{-1}$ be the quotient bundle. Recall that ${{\mathcal O}}_Y(3)={{\mathcal O}}_Y(3p+3q + 9u)$ and $K_Y={{\mathcal O}}_Y(2)={{\mathcal O}}_Y(2p+2u+6v)$. We have an exact sequence $$H^0(N)\rightarrow N_p\oplus N_q \rightarrow H^1(N(-p-q))\rightarrow H^1(N) .$$ The rightmost map is dual to $$H^0(K_Y \otimes N^{-1}) \rightarrow H^0(K_Y \otimes N^{-1} (p+q)).$$ Notice however that $N^{-1}= {{\mathcal O}}_Y(-3)\otimes M$ so $K_Y\otimes N^{-1}= M(-1)=M(-p-q-3u)$. Hence our rightmost map is dual to $$H^0(M(-p-q-3u)) \rightarrow H^0(M(-3u)).$$ This map is surjective; indeed, the condition $p,q\in B_2$ means that all sections of $M$ must vanish at $p$ and $q$, and sections of $M(-3u)$ are in particular sections of $M$, so every element of $H^0(M(-3u))$ must come from an element of $H^0(M(-p-q-3u))$. This surjectivity translates by duality to the statement that the rightmost map in the above exact sequence, is injective. It follows that $H^0(N)\rightarrow N_p\oplus N_q$ is surjective.
In other words, the values of global sections of $N$ at $p$ and $q$ span a two-dimensional space. Since on the other hand the values of sections of $E_Y(1)$ must vanish at $p$ and $q$, this implies from the exact sequence $$H^0(E_Y(1))\rightarrow H^0(N) \rightarrow H^1(M)$$ that we have $h^1(M)\geq 2$.
Consider now the three cases, the first case being $M = {{\mathcal O}}_Y (2p+2q+2u + d)$, with $\chi (M)= 2$, so $h^1(M)\geq 2$ implies that $h^0(M)\geq 4$. Vanishing at $2u$ imposes two conditions which leaves $h^0(M(-2u))= h^0({{\mathcal O}}_Y(2p+2q+d) \geq 2$. These sections must vanish at $p$ and $q$, so we get $h^0({{\mathcal O}}_Y(p+q+d) \geq 2$. Now, our two independent sections of ${{\mathcal O}}_Y(p+q+d)$ cannot vanish at both $p$ and $q$ because $Y$ is not ${{\mathbb P}}^1$ so there are no functions with a single nontrivial pole at $d$. We get a section of ${{\mathcal O}}_Y(p+q+d)$ whose value at one of $p$ or $q$ is nonzero. Multiplying this by the section of ${{\mathcal O}}_Y(p+q+2u)$ nonvanishing at $p$ and $q$, gives a section of $M$ nonvanishing at $p$ or $q$, a contradiction which treats the first case.
In the next case, $M={{\mathcal O}}_Y(2p+2q+2u)$ with $\chi (M) = 1$ so $h^1(M)\geq 2$ gives $h^0(M)\geq 3$. As usual, sections of $M$ have to vanish at $p$ and $q$ so $h^0(M(-p-q))=h^0({{\mathcal O}}_Y(p+q+2u))\geq 3$. But notice that ${{\mathcal O}}_Y(p+q+2u)={{\mathcal O}}_Y(1)(-u)$. The map ${{\mathbb C}}^3=H^0({{\mathcal O}}_H(1))\rightarrow H^0({{\mathcal O}}_Y(1))$ is an isomorphism; and ${{\mathcal O}}_Y(1)$ is generated by its global sections. Hence, vanishing of a section at $u$ imposes a nontrivial condition, giving $h^0({{\mathcal O}}_Y(1)(-u))=2$. This contradicts the previous estimation of $\geq 3$. This contradiction completes this case.
In the last case, $M={{\mathcal O}}_Y(3p+2q+2u)$ with $\chi (M) = 2$ so $h^1(M)\geq 2$ gives $h^0(M)\geq 4$. This is similar to the first case. Vanishing at $2u$ imposes two conditions, and then the sections must further vanish at $p$ and $q$, which leaves $h^0(M(-2u))= h^0(M(-p-q-2u))= h^0({{\mathcal O}}_Y(2p+q+d) \geq 2$. If we have a section here which is nonzero at either $p$ or $q$, then multiplying it by the section of ${{\mathcal O}}_Y(p+q+2u)$ nonvanishing at $p$, gives a section of $M$ nonvanishing at $p$ or $q$, a contradiction. Therefore, both sections in $H^0({{\mathcal O}}_Y(2p+q+d)$ have to vanish further at $p$ and $q$. This would give $h^0({{\mathcal O}}_Y(p+d))\geq 2$. That can happen only if $Y$ is a hyperelliptic curve.
But a smooth plane curve of degree $5$ is never hyperelliptic. If $Y$ were hyperelliptic, for a general $y\in Y$ let $y'$ be the conjugate by the involution; there is a meromorphic function $f$ with polar divisor $y+y'$. The line $L$ through $y$ and $y'$ meets $Y$ in $5$ different points, otherwise the map sending $L$ to the point of higher multiplicity would be a ${{\mathbb P}}^1\rightarrow Y$. Write $L\cap Y = y+y'+u+v+w$. A linear form vanishing at one of the other points, say $w$, divided by the equation of $L$, gives a meromorphic function $g$ whose polar locus is $y+y'+u+v$. Then $[1:f:g]$ provides a degree $4$ map to ${{\mathbb P}}^2$. By looking at the genus, it can’t be injective, also it spans the plane so it isn’t a degree $4$ map to a line. The only other case would be a degree $2$ map to a conic. But in that case, a linear combination of $f$ and $g$ would have polar divisor $u+v$. Doing the same for the other possibilities gives a $3$ dimensional space of functions with poles $\leq u+v+w$, but $Y$ can’t have a degree $3$ spanning map to ${{\mathbb P}}^2$. This shows that $Y$ cannot be hyperelliptic, so this case is also ruled out.
We have now finished showing that it is impossible to have two distinct points $p,q\in B_2$. The same proof works equally well if $q$ is infinitesimally near $p$; this double point defines a tangent direction, and $L$ should be chosen as the tangent line in this direction. The main case as before is when $L\cap Y = 2p+3u$, and as before we get three cases: either $M={{\mathcal O}}_Y(4p+2u+d)$, $M= {{\mathcal O}}_Y(4p + 2u)$, or $M= {{\mathcal O}}_Y(5p+2u)$. The main principle here is that sections of $M$ have to vanish on both $p$ and the nearby point $q$, that is to say they have to vanish to order $2$ at $p$. With this, the same proofs as above hold, so this shows that if $B_2$ is nonempty, then it is a single reduced point. This completes the proof of the proposition.
Local structure of $B_1$ at a point of $B_2$ {#localstrucB2}
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For the next discussion, we assume that there is a point $p'\in B_2$, unique by above. Consider the schematic structure of $B_1$ around this point $p'$. An argument similar to the above, allows us to show that $B_1$ can’t contain the third infinitesimal neighborhood of $p'$; however, we haven’t been able to rule out the possibility that it might contain the second neighborhood. We will formulate this statement precisely in the form of the following lemma, even though we haven’t really defined the schematic structure of $B_1$. Recall that $H^0(E(1))\stackrel{\cong}{\rightarrow} H^0(E_Y(1))$.
Suppose $Y\subset X$ is a general plane section passing through $p'$. Choose $s\in H^0(E_Y(1))$, vanishing to order $1$ at $p'$. Let $M\subset E_Y(1))$ be the sub-line bundle generated by $s$. Then there exists a section $t\in H^0(E_Y(1))$ such that the projection of $t$ as a section of $N:= E_Y(1)/M$ vanishes to order at most $2$ at $p'$.
Suppose on the contrary that all sections vanish to order $\geq 3$ in $N$. As this is true on a general $Y$, we may also specialize $Y$ and it remains true. In particular, choose a tangent line $L$ to $X$ at $p'$ such that the second fundamental form vanishes. Choose a smooth plane section $Y$ corresponding to a plane containing $L$. Then $Y\cap L$ is a divisor of class ${{\mathcal O}}_Y(1)$, and we can write $$Y\cap L = 3p' + u+v,$$ where, as far as we know for now, $u$ and $v$ might be the same, and one or both might be equal to $p'$.
Choose a nonzero section $s\in H^0(E_Y(1))$ which has only a simple zero at $p'$. Recall that this is possible, by the result that $B_2$ is reduced in the previous proposition. Let $M\subset E_Y(1)$ be the subline bundle generated by $s$, and let $N:=E_Y(1)/M$ be the quotient. The contrary hypothesis says that all sections of $E_Y(1)$ vanish to order $3$ at $p'$, when projected into $N$. This means that the condition of a section vanishing to order $3$ at $p'$ imposes only two additional conditions. Indeed, etale-locally we can choose a basis for $E_Y(1)$ compatible with the subbundle $M$, and impose two conditions stating that the first coordinate (corresponding to $M$) vanishes to order $3$ (it automatically vanishes to order $1$ already). This implies that the section vanishes, since the second coordinate vanishes to order $3$ by hypothesis.
Now since $h^0(E_Y(1))=5$, we can impose two further conditions and obtain a section $t$ vanishing at $u$. The divisor of vanishing of $t$ is therefore $ap'+u+D$ where $a\geq 3$. If $a=3$ then we would get a morphism $${{\mathcal O}}_Y(3p'+u)\rightarrow E_Y(1)$$ nonzero at $p'$, but the line bundle ${{\mathcal O}}_Y(3p'+u)$ has a section nonvanishing at $p'$, and this would contradict $p'\in B_2$. Therefore we can conclude that $a\geq 4$.
We now note that $u$ and $v$ must be distinct from $p'$. For example, if $Y\cap L=4p'+u$, choose a section $s$ vanishing at $u$, and as described above, we can assume vanishing to order $3$ at $p'$ which imposes two additional conditions. If $s$ vanishes to order $\geq 4$ at $p'$ this would give a section in $H^0(E_Y)$ which can’t happen, so we can assume that $M={{\mathcal O}}_Y(3p'+u)$ and again this has a section nonvanishing at $p'$, contradicting $p'\in B_2$. So, this case can’t happen.
Similarly if $Y\cap L=5p'$, vanishing to order $3$ imposes two conditions, and vanishing to order $4$ imposes two more conditions so again there is a section $s$ which generates $M={{\mathcal O}}_Y(4p')\subset E_Y(1)$, but this $M$ has a section nonvanishing at $p'$ contradicting $p'\in B_2$. From these arguments we conclude that $u,v$ are different from $p'$.
Next, use the fact that a cubic polynomial on $L$ vanishing at $4$ points in $L\cap X$, must also vanish on the fifth point. Suppose first that $u\neq v$. Our section $t$ viewed as a section of $E(1)$ defines a zero-scheme $P$, which contains its zeros on $Y$. In particular, $P$ contains the scheme $3p'$ on $L$ as well as the scheme $u$. Note on the other hand that $v\not \in P$ otherwise we would get a section in $H^0(E_Y)$. We conclude that any element of $H^0(J_P(3))$ has to vanish at $v$. It follows that $v\in B_1$. By symmetry, we get also $u\in B_1$. Now, vanishing of sections at $u$ and $v$ imposes $2$ conditions, and vanishing at $3p'\subset Y$ imposes $2$ conditions as discussed above. This gives a section in $H^0(E_Y(1))$ vanishing at all of $Y\cap L$, hence a nonzero section of $H^0(E_Y)$. We get a contradiction in this case.
To finish the proof of the lemma, we have to treat the case where $u=v$, that is $Y\cap L = 3p'+2u$. As described previously, $u\neq p'$. Basically the same argument as before gives $u\in B_1$. Indeed, we can consider a section $t$ which vanishes at $3p'+u$. It can’t have a zero of order $2$ at $u$ otherwise we would get $H^0(E_Y)\neq 0$. Let $M\subset E_Y(1)$ be the subline bundle generated by $t$, and let $N$ be the quotient. Write $M={{\mathcal O}}_Y(ap'+u+D)$ with $a\geq 4$ and $D$ disjoint from $p',u$. We may also consider $t$ as a section defined over $X$, inducing a quotient morphism $E(1)\rightarrow {{\mathcal O}}_X(3)$. When restricted to $Y$ this provides a morphism $E_Y(1)\rightarrow {{\mathcal O}}_Y(3)$ which is the same as the map to $N$ generically. Hence it must factor through $E_Y(1)\rightarrow N \rightarrow {{\mathcal O}}_Y(3)$. This is more precisely given by $N={{\mathcal O}}_Y(3)(-ap'-u-D)$. Sections of $E(1)$ map to sections of ${{\mathcal O}}_X(3)$ vanishing on the zero locus $P$ of $t$, which contains $3p'+u\subset L$ (a subscheme of length $4$). These sections must vanish on all of $L$, hence they vanish on $2u$. Thus the image of any section in $N$ has to be a section of ${{\mathcal O}}_Y(3)(-ap'-2u-D)=N(-u)$. This means that the sections of $E(1)$, evaluated at $u$, must lie in $M_u$. In other words, $u\in B_1$ as claimed.
Vanishing of a section at $u$ therefore imposes a single condition. So there are two linearly independent sections $t_1,t_2$ which vanish at $3p'+u$. No nonzero linear combination of these can have a zero of order $2$ at $u$. It follows that the derivatives of $t_1$ and $t_2$ at $u$ are linearly independent. Let $M_1$ and $M_2$ denote the sub-line bundles of $E_Y(1)$ generated by the $t_i$. We have $$M_i = {{\mathcal O}}_Y(a_ip'+u + D_i)$$ with $a_i\geq 4$ and $D_i\cap u=\emptyset$. But the line bundle ${{\mathcal O}}_Y(3p'+u)$ has a section nonvanishing at $u$, so $M_i$ has a section nonvanishing at $u$. But the $M_i(u)\subset E(1)_u$ are generated by the derivatives of $t_i$, which are linearly independent. Thus the $M_i(u)$ generate $E(1)_u$. But as there are sections of $M_i$ nonvanishing at $u$, this contradicts $u\in B_1$. This completes the proof of the lemma.
\[lengthtwoatB2\] Suppose $p'\in B_2$. Then for a general section $s\in H^0(E(1))$, the scheme of zeros of $s$ locally at $p'$ is either the reduced point $p'$, or a length $2$ subscheme (infinitesimal tangent vector) at $p'$.
From the proposition before, the sections of $E(1)$ define $p'$ as a reduced subscheme. This means that for any tangent direction, there is at least one section whose derivative in that direction doesn’t vanish. So, if $Y\subset X$ is a generic curve through $p'$, then the zero scheme $P$ of a general section $s$ has $P\cap Y=\{ p'\}$ being a reduced subscheme locally at $p'$. It follows that $P$ is curvilinear at $p'$.
Assume that the general $P$ has length $\geq 3$ locally at $p'$. Consider the two sections $s,t$ given by the previous lemma. Their zero sets are therefore curvilinear subschemes of length $\geq 3$ at $p'$. Given $s$, we may choose $t$ general, then the zero set of $t$ is transverse to that of $s$ at $p'$. For otherwise this would mean that the tangent directions of the zero sets are always the same, but that would give an infinitesimal tangent vector in $B_2$ contradicting the above proposition. So these curvilinear subschemes are transversal. We may choose local coordinates at $p'$ so that they go along the coordinate axes, up to order $3$ at least. If $x,y$ are these coordinates with $p'=(0,0)$, we may write $$s=xa, \;\;\; t=yb \mbox{ modulo terms of order }3$$ where $a$ and $b$ are sections of $E(1)$ nonvanishing at $p'$. Furthermore, $Y$ is transverse to $(x=0)$. We may assume that the sub-line bundle of $E_Y(1)$ generated by $s$ is generated by $a|_Y$.
Notice that if $b(0,0)$ is linearly independent from $a(0,0)$ then $s+t=xa+yb$ would be a section whose zero scheme is the reduced point $p'$ so we would be done. Therefore we may assume, after possibly multiplying by a scalar, that $b(0,0)=a(0,0)$.
The conclusion of the lemma says that $t$ is not a section of this subline bundle to order $3$, which means that $b|_Y$ is not a multiple of $a$ to order $2$ i.e. modulo quadratic terms. We may therefore write $$b= a + xb_x + yb_y + \ldots$$ with one of $b_x,b_y$ nonzero modulo $a(0,0)$. Look at the section $s+\lambda t$ for variable $\lambda \in {{\mathbb C}}$; its leading term is $(x+\lambda y)a(0,0)$. By our contrary hypothesis, we suppose that the zero set of this section is curvilinear to order $3$ for all $\lambda$, which means that there is a factorization of $s+\lambda t$ as a multiple of a single section of $E(1)$, up to terms of order $3$. The first term has to be $(x+\lambda y)a(0,0)$, so we can write $$s+\lambda t = (x+\lambda y + q(x,y))(a + xf_x + yf_y).$$ This expands to $$xa + \lambda y (a + x b_x + yb_y) = (x+\lambda y + q(x,y))(a + xf_x + yf_y)$$ or simplifying (always modulo terms of order $3$), $$\lambda y (x b_x + yb_y) = q(x,y)a + (x+\lambda y)(xf_x + yf_y).$$ Now compare terms modulo the section generated by $a$; we get $$f_x = 0, \,\, f_y = b_y \mbox{ modulo } a(0,0)$$ and from the $xy$ term we get $\lambda b_x = f_y$ again modulo $a(0,0)$. Putting these together gives that $\lambda b_x=b_y$ modulo $a(0,0)$, for all $\lambda$. This is possible only if $b_x$ and $b_y$ are multiples of $a(0,0)$; but the conclusion of the previous lemma said that this wasn’t the case. This contradiction completes the proof of the corollary.
The above discussion may seem somewhat complicated: let us explain the geometric picture, in terms of a schematic notion of the base locus $B_1$. The problem is that $B_1$ could have some “layers” surrounding the point $p'\in B_2$. Locally, this would mean that the subsheaf of $E(1)$ generated by global sections, looks like a rank $1$ subsheaf over $B_1$, the layers of which would give a certain infinitesimal neighborhood of $p'$. In the lemma, we say that if we cut by a general plane section $Y$ going through $p'$, then the intersection with $B_1$ has length at most $2$. Intuitively this means that while $B_1$ might have a single layer around $p'$, it can’t have two layers. Notice that in some directions $B_1$ might be bigger, but in a general direction it has length $2$. Then, in the corollary, we say that if the general section has a curvilinear zero set of length $\geq 3$, that would mean that $B_1$ had to have at least two layers around $p'$.
Dimension of the CB-Hilbert scheme {#sec-dimensions}
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Let ${\bf H}_X$ denote the Hilbert scheme of subschemes $P\subset X$ which satisfy $CB(4)$. Let ${\bf H}_{{{\mathbb P}}^3}$ denote the Hilbert scheme of subschemes $P\subset {{\mathbb P}}^3$ which satisfy $CB(4)$. We call these the [*CB-Hilbert schemes*]{}.
Let ${\bf H}_X^{sn}$ and ${\bf H}^{sn}_{{{\mathbb P}}^3}$ denote the subschemes parametrizing $P$ such that $h^0(J_P(3))= 4$ and $h^0(J_P(2))=0$, and (in the second case) such that $P$ is contained in at least one smooth quintic surface. In that case, as we have seen in Lemma \[sncase\], any bundle $E$ extending $J_P(2)$ by ${{\mathcal O}}_X(-1)$ has seminatural cohomology, so we call them the [*seminatural CB-Hilbert schemes*]{}. Furthermore, as in Corollary \[snunique\], the isomorphism class of $E$ is uniquely determined by $P$. Since $E$ is stable, it doesn’t have any nontrivial automorphisms.
\[dimensions\] The seminatural CB-Hilbert scheme ${\bf H}_X^{sn}$ has pure dimension $24$; the seminatural CB-Hilbert scheme ${\bf H}^{sn}_{{{\mathbb P}}^3}$ has pure dimension $44$. Denote by ${\bf H}_X^{sn}[2]$ and ${\bf H}^{sn}_{{{\mathbb P}}^3}[2]$ the fiber bundles over these, parametrizing pairs $(P,U)$ where $P$ is a seminatural CB Hilbert point, and $U\subset H^0(J_P(3))$ is a $2$-dimensional subspace. These have pure dimensions $28$ and $48$ respectively.
A point in ${\bf H}_X^{sn}$ corresponds to a choice of bundle $E$ in $M^{sn}_X(2,1,10)$ plus a section $s\in H^0(E(1))$ up to scalar. As the moduli space has dimension $20$ and, for the seminatural case, ${{\mathbb P}}H^0(E(1))$ has dimension $4$, the total dimension of ${\bf H}_X^{sn}$ is $24$. The Hilbert scheme of pairs $(P,X)$ with $P\in {\bf H}_X^{sn}$ fibers over the $55$ dimensional space of quintics $X$ (note that $h^0({{\mathcal O}}_{{{\mathbb P}}^3}(5))=56$) with $24$-dimensional fibers, so it has dimension $79$. On the other hand, for a fixed $P\in {\bf H}^{sn}_{{{\mathbb P}}^3}$, the space of quintics $X$ containing $P$ is ${{\mathbb P}}H^0(J_P(5))$. Notice that if $P$ is contained in at least one $X$ then the discussion of Section \[seminatural\] implies that $h^1(J_P(5))=0$ so $h^0(J_P(5))= 36$ and the space of quintics containing $P$ is an open subset of ${{\mathbb P}}^{35}$. So, the dimension of the Hilbert scheme ${\bf H}^{sn}_{{{\mathbb P}}^3}$ is $79-35=44$. The fiber bundles parametrizing choices of $U\subset H^0(J_P(3))$ are bundles of Grassmanians of dimension $4$, so they have dimensions $28$ and $48$ respectively. All irreducible components have the same dimension because the same discussion works for all of them.
The base locus $B_1$ has dimension zero
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\[B1dimzero\] Suppose $E$ is a general point of its irreducible component. The subset $B_1$ of points at which $E(1)$ is not generated by global sections, has dimension $0$. Equivalently, if $s$ is a general section of $E(1)$ and $P$ its subscheme of zeros, then the base locus in $X$ of the linear system of cubics $H^0(J_P(3))$, has dimension $0$ (it remains possible that the base locus in ${{\mathbb P}}^3$ could have dimension $1$, indeed that will be a major case treated in Section \[sec-ccc\] below).
The proof takes up the rest of this section, using three further lemmas. Note first the equivalence of the two formulations. The section $s$ generates a rank one subsheaf of $E(1)$ at all points outside $P$. Thus, if $B_1$ had positive dimension, this would mean that all sections restrict to multiples of $s$ on $B_1$, so all sections of $H^0(J_P(3))$ would factor as a function vanishing on $B_1$ times some other function. So the second statement implies the first. In the other direction, suppose all elements of the linear system factored as $fg$ where $g$ is a fixed form, either linear or quadratic. Then the zero set of $g$ would provide a positive dimensional component of $B_1$.
To be proven, is that the elements of the linear system $H^0(J_P(3))$ cannot all share a common factor $g$. Suppose to the contrary that they did, and let $W\subset X$ be the zero-set of $g$. It is a divisor either in the linear system ${{\mathcal O}}_X(1)$ or ${{\mathcal O}}_X(2)$, which is to say that it is either a plane section or a conic section of $X$.
Let $P^{\perp}$ be the residual subscheme of $P$ along $W$ (i.e. roughly speaking $P-P\cap W$). Recall that we are assuming that $P$ is not contained in a conic section, so $P\not\subset W$ and $P^{\perp}$ is nonempty. The statement that elements of $H^0(J_P(3))$ vanish along $W$, means that the map $$H^0(J_{P^{\perp}}(3)(-W))\rightarrow H^0(J_P(3))$$ is an isomorphism. Recall also that the right hand side has dimension $4$ in our situation, so we get $h^0(J_{P^{\perp}}(3)(-W))=4$ too.
It is now easy to rule out the case where $W$ is a conic section. Indeed, in that case we would have $h^0(J_{P^{\perp}}(1))=4$, but $h^0({{\mathcal O}}_X(1))=4$ and the space of sections generates ${{\mathcal O}}_X(1)$ everywhere, so there are at most $3$ sections vanishing on a nonempty subscheme $P^{\perp}$ giving a contradiction.
Therefore, we may now say that $W$ is a plane section of $X$. From above, $h^0(J_{P^{\perp}}(2))=4$.
Next, we claim that $P^{\perp}$ satisfies $CB(3)$, that is Cayley-Bacharach for ${{\mathcal O}}_X(3)$ which is the same as ${{\mathcal O}}_X(4)(-W)$. Indeed, if $f$ is a section of ${{\mathcal O}}_X(3)$ and $g$ is the equation of $W$ then $fg$ is a section of ${{\mathcal O}}_X(4)$. Suppose $P^3\subset P^{\perp}$ is a colength $1$ subscheme. Then it induces a colength $1$ subscheme $P'\subset P$ such that $P^3$ is the residual of $P'$, notice that ${{\mathcal O}}_{P^{\perp}}$ may be viewed as the ideal $(g)$ inside ${{\mathcal O}}_P$ so an ideal of length $1$ in ${{\mathcal O}}_{P^{\perp}}$ gives an ideal of length $1$ in ${{\mathcal O}}_P$. Now if $f$ vanishes on $P^3$ then $fg$ vanishes on $P'$, so by $CB(4)$ for $P$ we get that $fg$ vanishes on $P$ which in turn says that $f$ vanishes on $P^{\perp}$. This proves that $P^{\perp}$ satisfies $CB(3)$ as claimed. It follows that $P^{\perp}$ also satisfies $CB(2)$.
The next remark is that $P^{\perp}$ is not contained in a plane, for if it were then the union of this plane with the one defining $W$ would be a conic containing $P$, contrary to our situation.
We have the following lemma, which is a preliminary version of the structural result of Proposition \[structureofP\] below. Notice that here we haven’t yet shown that $B_1$ has dimension $0$, so we use the specific current situation in the proof instead.
\[Pdecomp\] In the situation of the proof of the present proposition, consider a general section $t\in H^0(E(1))$, and let $P\subset X$ be its subscheme of zeros. Then $P$ decomposes as a disjoint union $P=P'\sqcup P''$ such that $P''$ is reduced, and $P'$ is either empty, consists of a point $p'$, or an infinitesimal tangent vector at $p'$, in the latter two cases $p'$ is the unique point of $B_2$.
Choose first any section $s\in H^0(E(1))$ with zero-scheme $P$, corresponding to a subsheaf ${{\mathcal O}}_X(-1)\subset E(1)$. Let $r$ be another section linearly independent from $s$, and let $F\subset E(1)$ be the subsheaf generated by $r$ and $s$. Let $K:= E(1)/F$ be the quotient. Let $\tilde{r}$ be the image of $r$ considered as a section of $J_P(3)$. Under our hypothesis of the proof of the proposition, the zero scheme $Z(\tilde{r})$ decomposes as $W\cup D$ where $D$ is a conic section. For $r$ sufficiently general, $D$ doesn’t contain $W$ (otherwise the conic section $2W$ would be a common zero of the linear system, and we have ruled that out). Thus, $Z(\tilde{r})$ is smooth on the complement of a finite set. Now, we can choose $t$ so that it is nonvanishing at all isolated points of $B_1$ (except maybe $p'\in B_2$), and at the finite set of singularities. Thus, if the zero scheme $P(t)$ of $t$ meets $W$, it meets it at a point where $Z(\tilde{r})$ is reduced (which we think of heuristically, as points where $B_1$ is reduced even though we haven’t given a scheme structure to $B_1$). Furthermore, since $P(t)$ moves (except maybe at $p'\in B_2$), its intersection with $W$ is reduced. On the other hand, at points located on $W$, $P(t)$ has to be locally contained in $W$, otherwise we could add a small multiple of $r$ to split off the part of the subscheme sticking out of $W$. These together imply that the points of $P(t)$ contained in $W$ are reduced (except possibly at $B_2$). The points outside of $W$ are reduced because they are located at places where $E(1)$ is generated by global sections, again with the possible exception of $p'\in B_2$. From the discussion of the above subsection, the local structure of $P(t)$ near the possible single point $p'\in B_2$ is at most an infinitesimal tangent vector of length $2$. This gives the claim of the lemma.
The length of $P^{\perp}$ is $\leq 10$.
Consider first the case where $B_2$ is empty, or $P$ is reduced at $p'\in B_2$, or else $p'\not \in W$. In this case, $P=P^W\cup P^{\perp}$ with $P^W=P\cap W$ a reduced subscheme. For fixed $W$, the dimension of the space of choices of $P^W$ is $\leq \ell (P^W)$. On the other hand, $P^{\perp}$ is located at the intersection $C_1\cap C_2\cap X$ where $C_1$ and $C_2$ are conics, whose intersection has dimension $1$. Furthermore, no component of $C_1\cap C_2$ is contained in $X$, indeed the former has degree $4$ while curves in $X$ have degre $\geq 5$ because of the condition $Pic(X)=\langle {{\mathcal O}}_X (1)\rangle$. Therefore, $C_1\cap C_2\cap X$ is a finite set. Since $P^{\perp}$ is reduced except for a possible tangent vector at the unique point $p'\in B_2$, we get that the dimension of the set of choices of $P^{\perp}$ for a given $C_1, C_2$, is $\leq 1$. On the other hand, suppose $C_1,C_2,C_3$ are three general conics through $P^{\perp}$. If their intersection is finite, it contains at most $8$ points; but with $\ell (P^{\perp})\geq 11$ this can’t happen and we must have a nontrivial curve in the intersection; this means that a double intersection $C_1\cap C_2$ has to split into two pieces. The dimension of the space of such double intersections is the dimension of the Grassmanian of $2$-planes in $H^0({{\mathcal O}}(2))={{\mathbb C}}^{10}$, this Grassmanian has dimension $16$. However, as may be seen by a calculation of the possible cases of splitting, the subvariety of the Grassmanian corresponding to double intersections which split into at least two components, is $\leq 14$. Together with the possible one dimensional choice of tangent vector at $p'$, we get altogether that the space of choices for $P^{\perp}$ together with the two-dimensional subspace spanned by $C_1,C_2$, is $\leq 15$. Putting in $P^W$, we get that the dimension of the space of choices of $P$ plus a $2$-dimensional subspace of $H^0(J_P(3))$, is less than $15+3+\ell (P^W)$. The $3$ is for the space of choices of plane section $W$. Now if $\ell (P^{\perp})\geq 11$ then $\ell (P^W)\leq 9$ and this dimension is $\leq 27$. The dimension of the corresponding bundle over the seminatural CB-Hilbert scheme ${\bf H}_X^{sn}[2]$ is $28$ (Proposition \[dimensions\]) so such a bundle $E$ cannot be general in its irreducible component.
We are left to treat the case where the unique point $p'\in B_2$ lies on $W$, and $P$ includes a tangent vector here. Let $P^1$ denote the subscheme of $P$ located set-theoretically along $W$, and $P^2$ the complement. Then the dimension of the space of choices of $P^1$ is still $\ell (P^1)$, and the same argument as above gives that the dimension of the space of choices of $P^2$ plus a two-dimensional subspace of conics, is $\leq 15$. We get as before $\ell (P^2)\leq 10$. On the other hand, the tangent vector at $p'$ might go outside of $W$ and contribute to $P^{\perp}$. If the tangent vector stays inside $W$ then $P^{\perp}=P^2$ and we are done. If the tangent vector goes outside of $W$, then the estimate from above says only that $\ell (P^{\perp})\leq 11$; however, we get an additional condition saying that the conics have to vanish at this point $p'\in W$, and this condition (which may be seen, for example, as a condition on the choice of $P^1$ once $P^2$ and the conics are fixed) gets us back to the estimate $\ell (P^{\perp})\leq 10$.
There is a plane section $V\subset X$ such that $V\cap P^{\perp}$ has length $\geq 5$.
Suppose not, that is to say, suppose that any plane section meets $P^{\perp}$ in a subscheme of length $\leq 4$. In order to obtain a contradiction, we show that under this hypothesis, $\ell (P^{\perp})\geq 11$.
Choose a plane section meeting $P^{\perp}$ in a subscheme of length $\geq 3$, call this intersection $P^{\perp}_+$ and let $P^{\perp}_-$ denote the residual subscheme. Then the condition $CB(3)$ for $P^{\perp}$ implies $CB(2)$ for $P^{\perp}_-$. The results of our previous paper [@MestranoSimpson] therefore apply: (a) $\ell (P^{\perp}_-)\geq 4$; (b) if $\ell (P^{\perp} _-)=4$ or $5$ then $P^{\perp}_-$ is contained in a line; (c) if $\ell (P^{\perp}_-)=6$ or $7$ then $P^{\perp}_-$ is contained in a plane.
However, our hypothesis for the proof of the lemma says that no plane contains a subscheme of $P^{\perp}$ of length $\geq 5$, so the cases $\ell (P^{\perp}_-)=5$, $6$, $7$ can’t happen. If $\ell (P^{\perp}_-)=4$ then $P^{\perp}_-$ is contained in a line, and we can choose a plane which meets furthermore a point of $P^{\perp}_+$, again giving a plane with more than $5$ points. This shows that we must have $\ell (P^{\perp}_-)\geq 8$ and since $\ell (P^{\perp}_+)\geq 3$ we get $\ell (P^{\perp})\geq 11$ as claimed (under the hypothesis contrary to the lemma). This contradicts the estimate of the previous lemma, which completes the proof of the present one.
Now choose a plane section $V$ such that $V\cap P^{\perp}$ has maximal length. Write $P^{\perp}_+=P^{\perp}\cap V$ and let $P^{\perp} _-$ be the residual subscheme with respect to $V$. Then $P^{\perp}_-$ satisfies $CB(2)$. If $\ell (P^{\perp}_+)=5$ then $\ell (P^{\perp}_-)\leq 5$ and by [@MestranoSimpson], $P^{\perp}_-$ must consist of $4$ or $5$ points on a line. Choose a new plane section passing through this line but not meeting $P^{\perp}_+$; we conclude that $P^{\perp}_+$ must also consist of $5$ points on a line, but then in fact we could choose a plane section meeting $P^{\perp}$ in $6$ points. Thus the case $\ell (P^{\perp}_+)=5$ doesn’t happen. If $\ell (P^{\perp}_+)\geq 7$ then $P^{\perp}_-$ would consist of $\leq 3$ points, but there are no such subschemes satisfying $CB(2)$, so this can’t happen either. We conclude that $\ell (P^{\perp}_+)=6$, hence $P^{\perp}_-$ must be $4$ points on a line. If $y$ is any point of $P^{\perp}_+$ then there is a plane containing $P^{\perp}_-$ and $y$, so the remaining points of $P^{\perp}_+$ are either on this same plane, or else contained in a line. If the two lines meet at a point, this would give a plane section containing too many points. Hence, we conclude that there are $2$ skew lines containing at least $8$ of the $10$ points in $P^{\perp}$. Because of $CB(3)$ for $P^{\perp}$, in fact all of the points must be on the two skew lines.
Count now the dimension of the space of such configurations: there are $8$ parameters for the two skew lines. Once this configuration is fixed, the subscheme $P^{\perp}$ is specified up to a finite set of choices. The choice of $W$ counts for $3$, and the choice of $10$ points in $W$ counts for $10$. The full dimension of this space of choices is therefore $\leq 21$. As in the proof of the previous lemma, the case where our double point at $p'\in B_2$ lies on $W$ but the tangent direction extends out of $W$, doesn’t add an extra dimension because we get a point participating in $P^{\perp}$ which constrains the choice of points on $W$. In view of the fact that ${\rm dim}({\bf H}^{ns}_X)=24$, this situation cannot happen for a general $E$.
This contradiction completes the proof of the proposition: for a general $E$ in its irreducible component, the base locus $B_1$ has dimension $0$.
The structure of a general zero scheme $P$ {#Pstructure}
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The previous results were as close as we could get to saying that $E(1)$ is generated by global sections, with the techniques we could find. Choose a general section $s\in H^0(E(1))$ and let $P$ denote its scheme of zeros. If $y\in B_1$ (but not in $B_2$) then a general section will not vanish at $y$. Furthermore, if there is a point $p'$ in $B_2$ then the structure of $P$ near $p'$ is at most an infinitesimal tangent vector.
\[structureofP\] Suppose $s\in H^0(E(1))$ is a general element, and let $P$ be the subscheme of zeros. We can write $P=P'\cup P''$ where $P'$ consists of the possible point of $P$ located at $B_2$, and $P''$ is all the rest. With this notation, $P''$ consists of $18$, $19$ or $20$ isolated points, and $P'$ is respectively an infinitesimal tangent vector at $p'\in B_2$; or the isolated point $p'$; or empty. At any point $y\in P''$, the map $$H^0(J_P(3))\rightarrow J_y /J_y^2 (3)$$ is surjective, meaning that $y$ is locally the complete intersection of two general sections of $H^0(J_P(3))$.
By Proposition \[B1dimzero\], the base locus $B_1$ has dimension zero. For any point $z\in B_1$ with $z\not \in B_2$, a sufficiently general section $s\in H^0(E(1))$ is nonzero at $z$. Therefore, for $s$ general the scheme of zeros $P$ doesn’t meet $B_1$ except possibly at $B_2$. Divide $P$ into two pieces, $P'$ at $B_2$ and $P''$ which doesn’t meet either $B_1$ or $B_2$. By Proposition \[B2onepoint\], the base locus $B_2$ consists of at most one point which we shall denote by $p'$ if it exists. Therefore, $P'$ is either empty or has a single point. By Corollary \[lengthtwoatB2\], the zero scheme of a general $s$ at $p'$ has length at most two. So, if $P'$ has a point, then it is scheme-theoretically either this reduced point, or an infinitesimal tangent vector there.
At a point $y\in P''$, since $y$ is not in the base locus $B_1$, it means that $E(1)$ is generated by global sections at $y$. From the standard exact sequence for $s$ we see that $E(1)_y = J_y /J_y^2 (3)$, so the generation of $E(1)_y$ by global sections is exactly the surjectivity of the last claim in the proposition.
The points of $P''$ are “interchangeable”. This can be phrased using Galois theory. Write $H^0(E(1))= {{\mathbb A}}^5_{{{\mathbb C}}}= {\rm Spec}{{\mathbb C}}[t_1,\ldots , t_5]$. Put $K= {{\mathbb C}}(t_1,\ldots , t_5)$ and let ${\bf s}\in {{\mathbb A}}^5_K$ be the tautological point. Think of ${\bf s}\in H^0(X_K, E(1))$. Let $P\subset X_K$ be the subscheme of zeros. The decomposition $P=P'\cup P''$ is canonical, hence defined over $K$. On the other hand, $P''$ consists of $18$ to $20$ points, but the points are only distinguishable over $\overline{K}$, which is to say $P'' _{\overline{K}}\subset X(\overline{K})$ is a set with $18$, $19$ or $20$ points. The Galois group ${\rm Gal}(\overline{K}/K)$ acts.
\[doublytransitive\] The action of ${\rm Gal}(\overline{K}/K)$ on the set $P''_{\overline{K}}$ is doubly transitive: it means that any pair of points can be mapped to any other pair.
For general $s$, the part $P''$ is contained in the open subset $X^g$ where $E(1)$ is generated by global sections. Suppose $x_0,y_0$ and $x_1,y_1$ are two pairs of points in $P''$. Consider a continuous path of pairs $(x(t),y(t))$ contained in $X^g\times X^g$ defined for $t\in [0,1]\subset {{\mathbb R}}$, with $(x(0),y(0))=(x_0,y_0)$ and $(x(t),y(t))=(x_0,y_0)$. Vanishing of a section at $x(t)$ and $y(t)$ imposes $4$ conditions on elements of the $5$-dimensional space $H^0(E(1))$, so we get a family of sections $s(t)$ leading to a family of subschemes $P''(t)$. For general choice of path, the $P''(t)$ will all be reduced with $18$, $19$ or $20$ points. At $t=0$ and $t=1$, the section is the same as $s$ up to a scalar since it is uniquely determined by the vanishing conditions. We obtain an element of the fundamental group of an open subset of the parameter space of sections $s$, whose action on the covering determined by the points in $P''$, sends $(x_0,y_0)$ to $(x_1,y_1)$. This shows that the action is doubly transitive, and it is the same as the Galois action after applying the Grothendieck correspondence between Galois theory and covering spaces.
\[singlecomponent\] Suppose $P$ is the scheme of zeros of a general section $s\in H^0(E(1))$, written $P=P'\cup P''$ as above. Let $Z\subset {{\mathbb P}}^3$ be the intersection of two cubic hypersurfaces corresponding to general elements of $H^0(J_P(3))$. This $Z$ is a complete intersection: ${\rm dim}(Z)=1$. There is a single irreducible component $Z''$ of $Z$ such that $P''$ is contained in the smooth locus of $Z''$. At points of $P''$, $Z''$ is transverse to $X$. The only irreducible components of $Z$ which can be non-reduced are those, other than $Z''$, which are fixed as the cubic hypersurfaces vary.
The points of $P''$ lie in the subset $X^g$ where sections generate $E(1)$. In the standard exact sequence , sections of $E(1)$ map to sections of $J_P(3)$, and the fiber $E(1)_x$ maps to $J_x/J_x^2(3)$ for $x\in P''$. As sections generate the fiber, it implies that sections of $J_P(3)$ generate the ideal $J_x$ (which is the maximal ideal at $x$). Two general sections therefore have linearly independent derivatives at $x\in P''$ when restricted to $X$, so the same is true of the cubics in ${{\mathbb P}}^3$ which means that $Z$ is a transverse complete intersection at any $x\in P''$. The doubly transitive action from the previous proposition implies that for general sections and general choice of $Z$, the points of $P''$ must all lie in the same irreducible component $Z''$ of $Z$. Note, on the other hand, that any other component of $Z$ must also have dimension $1$, otherwise we would get a $1$-dimensional base locus $B_1$ of sections of $E(1)$ on $X$ and this possibility has been ruled out in Proposition \[B1dimzero\].
Note that $Z''$ is reduced since its smooth locus is nonempty. If $Z_i$ is a non-reduced component, then by Sard’s theorem it has to be a fixed part of the family of complete intersections of the form $Z$.
Complete intersections of two cubics {#completeintersections}
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We need to know something about what curves $Z$ can arise as the complete intersection of two cubics in ${{\mathbb P}}^3$. The degree of $Z$ is $9$. If $Z$ is smooth, then $K_Z= {{\mathcal O}}_Z(2)$ is a line bundle of degree $18$, so the genus of $Z$ is $10$.
The choice of $Z$ corresponds to a choice of two-dimensional subspace $U\subset H^0({{\mathcal O}}_{{{\mathbb P}}^3}(3))={{\mathbb C}}^{20}$; furthermore $U=H^0(J_Z(3))$ and $h^0({{\mathcal O}}_Z(3))= 18$ (as may be seen from the exact sequences of restriction to one of the cubics $C$ and then from $C$ to $Z$). The dimension of the Grassmanian of $2$-dimensional subspaces of ${{\mathbb C}}^{20}$ is $2\cdot (20-2)=36$. Denote this Grassmanian by ${\bf G}$; we have a universal family $${\bf Z} \subset {\bf G}\times {{\mathbb P}}^3.$$ Let ${\bf G}_{rci}$ denote the subset of $U$ defining a reduced complete intersection, i.e. such that the fiber $Z_U$ of ${\bf Z}$ over $U$ has dimension $1$ and is reduced.
Suppose $Z=Z'\cup Z''$ is a decomposition with $d':= {\rm deg}(Z')$, $d'':={\rm deg}(Z'')$, so $d"'+d'' = 9$ and we may assume $d'\leq d''$. We aren’t saying necessarily that the pieces $Z'$ and $Z''$ are irreducible, though. This gives $d'\leq 4$.
\[leq5inconic\] Suppose $Z$ is a complete intersection of two cubic hypersurfaces in ${{\mathbb P}}^3$. If $Z_i$ is a reduced irreducible component of $Z$ of degree $\leq 5$, then either $Z_i$ is contained in a quadric, or the normalization of $Z_i$ has genus $g=0$ or $1$ and the space of such curves has dimension $\leq 20$.
If ${\rm deg}(Z_i)\leq 4$ then it is contained in a quadric, so we may assume the degree is $5$. Let $Y\rightarrow Z_i$ be the normalization and let $g$ denote the genus of $Y$. Projecting from a point on $Z_i$ gives a presentation of $Y$ as the normalization of a plane curve of degree $4$, so it has genus $g\leq 3$. The line bundle ${{\mathcal O}}_Y(2)$ has degree $10$ which is therefore in the range $\geq 2g-1$, so $h^0({{\mathcal O}}_Y(2))=11-g$. If $g\geq 2$ then this is $\leq 9$ and the map $H^0({{\mathcal O}}_{{{\mathbb P}}^3}(2))\rightarrow H^0({{\mathcal O}}_Y(2))$ is not injective, giving a quadric containing $Z_i$.
If $g=0$, $Y\cong {{\mathbb P}}^1$, and the embedding to ${{\mathbb P}}^3$ corresponds to a map ${{\mathbb C}}^4\rightarrow H^0({{\mathcal O}}_Y(1))\cong {{\mathbb C}}^6$. This yields $24$ parameters, minus $1$ for scalars, minus $3$ for $Aut({{\mathbb P}}^1)$, so there are $20$ parameters.
If $g=1$, the moduli space of elliptic curves provided with a line bundle ${{\mathcal O}}_Y(1)$ of degree $5$, has dimension $1$ (the line bundles are all equivalent via translations). Here $h^0({{\mathcal O}}_Y(1))={{\mathbb C}}^5$ so the space of parameters for the embedding has dimension $19$, this gives a $20$ dimensional space altogether.
\[63\] Let ${\bf G}_{rci}(6,3)$ denote the locally closed subset of ${\bf G}_{rci}$ parametrizing complete intersections $Z_U$ such that $Z_U = Z'\cup Z''$ with $Z''$ irreducible of degree $6$. The degree $3$ piece $Z'$ is allowed to have other irreducible components. Then ${\bf G}_{rci}(6,3)$ is the union of four irreducible components parametrizing: (a) the case where $Z'$ is a rational normal space cubic; (b) the case where $Z'$ is a plane cubic; (c) the case where $Z'$ is a disjoint union of a plane conic and a line; and (d) the case where $Z'$ is a disjoint union of three lines.
These have dimensions $28$, $30$, $26$ and $24$ respectively. For (a) there is an open set on which $Z_U=Z' \cup Z''$ with $Z'$ and $Z''$ being smooth and meeting transversally in $8$ points.
We divide into cases corresponding to the piece $Z'$ obtained by removing the degree $6$ irreducible component $Z''$. In the first case (a), we include all degree $3$ curves $Z'$ which are connected chains of rational curves with no loops or self-intersections, which then have to span ${{\mathbb P}}^3$. In these cases, $H^0({{\mathcal O}}_{Z'} (1))$ is always $4$-dimensional, and $Z'$ deforms to a smooth rational normal space cubic. For a given $Z'$, the space of choices of $U$ is the Grassmanian of $2$-planes in $H^0(J_{Z'}(3))$ and $H^0({{\mathcal O}}_{Z'} (3))$ has dimension $10$; one can check (case by case) that the restriction map from $H^0({{\mathcal O}}_{{{\mathbb P}}^3}(3))$ is surjective, so $h^0(J_{Z'}(3))=10$ and the Grassmanian has dimension $16$. The space of choices of rational normal space cubic is irreducible, equal to the space of choices of basis for the $4$-dimensional space $H^0({{\mathcal O}}_{Z'}(1))$ ($16d$), modulo scalars ($1d$) and the automorphisms of the rational curve $Z'$ ($3d$). In the case of a chain the dimension of the automorphism group goes up so those pieces are of smaller dimension in the closure of the open set where $Z'$ is smooth. The dimension of this component is therefore $${\rm dim}{\bf G}_{rci}(6,3)^{(a)} = 16 + 16 - 1 - 3 = 28.$$ A general point corresponds to a smooth $Z'$ with general choice of $U$ yielding a smooth curve $Z''$ of degree $6$ meeting $Z'$ at $8$ points.
The remaining possibilities are (b), (c) and (d), which are irreducible and one counts the dimensions as: (b) a plane $3d$ plus a cubic $9d$ plus a subspace of $H^0(J_{Z'}(3))={{\mathbb C}}^{11}$, $18d$ for a total of $30$; (c) a plane $3d$ plus a conic $5d$ plus a disjoint line $4d$ plus a subspace of $H^0(J_{Z'}(3))={{\mathbb C}}^9$, $14d$ for a total of $26$; (d) three lines $12d$ plus a subspace of $H^0(J_{Z'}(3))={{\mathbb C}}^8$, $12d$ for a total of $24$.
\[72\] Let ${\bf G}_{rci}(7,2)$ denote the locally closed subset of ${\bf G}_{rci}$ parametrizing complete intersections $Z_U$ such that $Z_U = Z'\cup Z''$ with $Z''$ irreducible of degree $7$. The degree $2$ piece $Z'$ is allowed to have other irreducible components. Then ${\bf G}_{rci}(7,2)$ is the union of two irreducible components parametrizing: (a) the case where $Z'$ is a plane conic; and (b) the case where $Z'$ is a disjoint union of two lines.
These have dimensions $30$ and $28$ respectively. For (a) there is an open set on which $Z_U=Z' \cup Z''$ with $Z'$ and $Z''$ being smooth and meeting transversally in $6$ points.
The complementary curve $Z'$ has degree $2$. If irreducible, it has to be a plane conic. If reducible, it is the union of two lines. If the lines meet, this still corresponds to (a), if they are disjoint it is case (b). Both cases have irreducible spaces of parameters.
To count the dimensions, in case (a) the choice of plane $H\cong {{\mathbb P}}^2\subset {{\mathbb P}}^3$ is $3d$, the choice of conic in the plane is $5d$, and $h^0({{\mathcal O}}_{Z'}(3))= 7$. One can check that the restriction map is surjective (since $Z'$ is reduced there are only two cases, an irreducible conic or two crossing lines); so $h^0(J_{Z'}(3))= 13$ and the Grassmanian of $2$-planes in here has dimension $22$. The total dimension is therefore $3+5+22=30$.
In case (b) the choice of two disjoint lines is $8$ dimensional, and $h^0(J_{Z'}(3))= 12$; the Grassmanian of $2$-planes has dimension $20$ so the total dimension here is $28$.
\[81\] The subvariety ${\bf G}_{rci}(8,1)$ parametrizing $Z_U = Z'\cup Z''$ with $Z''$ irreducible of degree $8$, is irreducible of dimension $32$. It has an open set on which $Z''$ is smooth and meets the line $Z'$ transversally in $4$ points.
Note that $Z'$ has to be a line. The space of lines has dimension $4$ and $h^0(J_{Z'}(3))=16$. The Grassmanian of $2$-planes $U$ in $H^0(J_{Z'}(3))$ has dimension $28$, so the total dimension is $32$. The general element is contained in a smooth cubic surface, on which the relevant linear system defining $Z''$ has no base points so a general $Z''$ is smooth; the intersection $Z'\cap Z''$ has $4$ points by the adjunction formula.
The common curve case {#sec-ccc}
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Consider a general bundle $E$ in its irreducible component, a general section $s\in H^0(E(1))$, and a general two-dimensional subspace $U\subset W:= H^0(J_P(3))$. Let $Z\subset {{\mathbb P}}^3$ be the intersection defined by $U$, which is a complete intersection by Corollary \[singlecomponent\]. Write $P=P'\cup P''$ as usual and let $Z''\subset Z$ be the irreducible component containing $P''$.
Let $Q\subset {{\mathbb P}}^3$ be the intersection of the four independent cubics spanning $W=H^0(J_P(3))$. It is contained in $Z$, indeed it is the intersection of $Z$ with the other two cubics spanning the complement of $U\subset W$. Hence ${\rm dim}(Q)\leq 1$, and also of course $P\subset Q$. Write $Q=Q_0\cup Q_1$ where $Q_1$ is the union of $1$-dimensional pieces of $Q$ and $Q_0$ is the remaining $0$-dimensional part. Notice that $Q_1\cap P$ and $Q_0 \cap P$ correspond to Galois invariant pieces in the situation where $s$ is a generic geometric point, so by Proposition \[doublytransitive\] it follows that if $P''\cap Q_1$ is nonempty then $P''\subset Q_1$ and similarly for $Q_0$.
Our situation therefore breaks down into two distinct cases: –the [*common curve case*]{} when the $1$-dimensional part $Q_1$ contains the big variable part $P''$; or —the [*variable curve case*]{} when $P''\subset Q_0$.
In this section, we would like to rule out the first possibility; reasoning by contradiction suppose on the contrary that we are in the common curve case. Since $Q_1\subset Z$, and by Corollary \[singlecomponent\] there is a single irreducible component $Z''$ of $Z$ containing $P''$, it follows that $Z''\subset Q_1$. The common curve case is therefore equivalent to the following hypothesis, which will be in vigour throughout the section until it is ruled out.
\[ccc\] All of the sections in $W=H^0(J_P(3))$ vanish along $Z''$.
\[containedquartic\] Suppose that ${\rm deg}(Z'')\neq 6$. Then $Z''$ is contained in a quadric, from which it follows that $P$ is contained in a quadric.
If we can show that $Z''$ is contained in a quadric, then it follows that $P$ is contained in the same quadric by Lemma \[inquadric\]. Suppose ${\rm deg}(Z'')\leq 5$. Then by Lemma \[leq5inconic\], either $Z''$ is contained in a quadric or it runs in a space of dimension $\leq 20$. In the latter case, for each choice of $Z''$ we have a space of possible choices of $P$ of dimension $20$, $19+3=22$, or $18+5=23$ depending on whether $P'$ is empty, a single point, or an infinitesimal tangent vector. In all cases, this results in a space of possible subschemes $P$ of dimension $\leq 43<44$, so by Proposition \[dimensions\] it can’t contribute to a general point in the irreducible component.
Suppose ${\rm deg}(Z'')=d\geq 7$ (and of course $Z''\subset Z$ so $d\leq 9$). Choose a hyperplane $H\cong {{\mathbb P}}^2\subset {{\mathbb P}}^3$ not passing through a point of $P''$, and let $A:= H\cap Z''\subset {{\mathbb P}}^2$ be the intersection. It is finite of length $d$. Using Hypothesis \[ccc\], we get a $4$-dimensional space $W$ of sections of $H^0(J_{Z''}(3))$ (one can note that, by the same argument as Lemma \[inquadric\], sections of ${{\mathcal O}}(3)$ vanishing on $Z''$ vanish also on $P$ so $H^0(J_{Z''}(3))=H^0(J_{P}(3))$). Consider the restriction map $$r:W\rightarrow H^0(J_{{{\mathbb P}}^2, A}(3)).$$ If $w$ lies in the kernel, it means that $w$ factors as the linear form defining $H$ times a quadric. Then this quadric contains $Z''$, so it contains $P$ by Lemma \[inquadric\].
We now show that $r$ is not injective.
\[cubicconditions\] Suppose $A\subset {{\mathbb P}}^2$ is a subscheme of length $7$. If $A$ doesn’t contain $5$ points on a line (i.e. the intersection with any line has length $\leq 4$) then $A$ imposes $7$ independent conditions on $H^0({{\mathcal O}}_{{{\mathbb P}}^2}(3))$.
Choose a line $L$ with maximal value of the length $\ell$ of $L\cap A$. Then $2\leq \ell \leq 4$. On the $10$ dimensional space of cubics we can try to impose $3$ further conditions and should prove that this makes sections vanish. Imposing $0$,$1$ or $2$ additional conditions on cubics restricted to $L$ makes them vanish. The residual subscheme $A'$ of $A$ with respect to $L$ has length $7-\ell$, and we have to show that it imposes this number of conditions on conics. Again choosing a line $L'$ with maximal contact (of order $2\leq \ell ' \leq 3$) with $A'$, imposing $0$ or $1$ additional conditions we get vanishing of the conics on $L'$; we are left with a further residual subscheme $A''$ of length $7-\ell -\ell '$, which is between $0$ and $3$; however if $A''$ consisted of $3$ colinear points that would imply $\ell \geq 3$ so $A''$ would have length $\leq 2$ and this is ruled out. Hence $A''$ imposes independent conditions on linear sections. We conclude that $A$ imposed $7$ independent conditions on cubics.
To finish the proof of Lemma \[containedquartic\], consider the subscheme $A$ from that proof. It has length $7$, $8$ or $9$. Since $H$ was general, no $5$ points of $A$ lie on a line. Applying the previous lemma to a subscheme of length $7$, we see that $A$ has to impose at least $7$ conditions on cubics. But $r(W)$ is a subspace of the $10$-dimensional $H^0({{\mathcal O}}_{{{\mathbb P}}^2}(3))$, vanishing on $A$. Thus ${\rm dim}(r(W))\leq 3$ showing that $r$ can’t be injective. This completes the proof.
To finish this section, we just have to consider the case when ${\rm deg}(Z'')=6$. Consider first the case where a general $Z$ is reduced, and apply Lemma \[63\]. Notice that for each choice of $Z$ a reduced complete intersection $Z=Z_U$, $U\in {\bf G}_{rci}(6,3)$, the space of possible choices of $P$ has dimension $\leq 20$. From $P\subset Z$ this is clear when $P$ is reduced. The other possibility is that $P'$ is an infinitesimal tangent vector. In that case, $P''$ is to be chosen in the smooth subset of $Z''$, giving an $18$ dimensional space of choices. When $P'$ is in the smooth part of $Z'$ it really only corresponds to a $1$ dimensional space of choices, giving $19$ in all; when $P'$ is in the singular set of $Z$, the choice of $p'$ is $0$-dimensional and the choice of tangent vector $\leq 2$-dimensional, so we get a space of choices of dimension $\leq 20$ in all.
From Lemma \[63\], the space of choices of pairs $(P,Z)$ in case (a) has dimension $\leq 28+20=48$. This is the same as the dimension of the component of the Hilbert scheme we are looking at. However, a general pair $(Z,P)$ with $Z=Z_U$ in the $28$ dimensional piece ${\bf G}_{rci}(6,3)^{(a)}$ and $P\subset Z$ general, doesn’t occur. Indeed, the degree $6$ piece $Z''$ is a smooth curve of genus $3$ so there are $22$ sections of ${{\mathcal O}}(4)$ on $Z''$, and imposing up to $20$ conditions can’t make the sections vanish there; on the other hand, we could start by imposing up to $2$ independent conditions from $P'$ on the degree $3$ piece $Z'$. Thus, a general choice of $P\subset Z$ imposes $20$ conditions on the $27$-dimensional space $H^0({{\mathcal O}}_Z(4))$, leaving only $7$ sections to add to $h^0(J_Z(4))=8$ giving $15$. So, for a general choice of $P\subset Z$ we have $h^0(J_P(4))=15$ and $P$ can’t satisfy $CB(4)$. Hence the space of $(P,Z)$ such that $Z$ decomposes with a degree $6$ piece $Z''$, is a proper subspace of our irreducible component so for general bundles $E$ this case doesn’t occur.
In case (b) of Lemma \[63\], consider the plane $H$ containing $Z'$. The subspace $U$ of cubics vanishing on $Z$ has dimension $2$, whereas $H^0(J_{H,Z'}(3))$ has dimension $1$ (the plane cubic $Z'$ determines its equation uniquely up to a scalar). Therefore the restriction map from $U$ to $H^0({{\mathcal O}}_H(3))$ is not injective; but an element $u\in U$ mapping to zero on $H$ must be a product of a quadric and the linear equation of $H$. This gives a quadric containing $Z''$ and hence $P$. So, for bundles with $h^0(E)=0$, this case doesn’t occur.
In cases (c) and (d) of Lemma \[63\], the total dimension is $\leq 26+20$ which is too small, so these don’t contribute for general bundles $E$.
This completes the analysis of the case where a general $Z$ is reduced. If a general $Z$ is non-reduced, the non-reduced components $Z_i$ must be fixed, but different from $Z''$. As $Z''$ is also fixed (when we vary the two-dimensional subspace $U$ of cubics), there must be at least one variable component $Z_j$. The degree of the complementary piece to $Z''$ is $3$ so the only possibility is a fixed line of multiplicity two, and a variable line of multiplicity $1$. But then we have a $4$ dimensional space of cubics passing through the degree $8$ curve $Z'' \cup Z_i$, so as in the proof of Lemma \[containedquartic\], this would give a quartic containing $Z''\cup Z_i$.
We have finished the proof of the following theorem ruling out the common curve case.
\[variablecurve\] Hypothesis \[ccc\] leads to a contradiction. Therefore, for a general seminatural bundle $E$ in its irreducible component and a general section $s\in H^0(E(1))$ defining a scheme of zeros $P$, if the intersection of the four cubics passing through $P$ has a $1$-dimensional piece $Q_1$, then the big interchangeable collection of points $P''\subset P$ doesn’t meet $Q_1$. In other words, we are in the variable curve case.
The reducible variable curve case {#sec-rvcc}
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The common curve case is ruled out by the previous section. Hence we are in the variable curve case, when $P'' \subset Q_0$. It means that the choice of $W$, which determines $Q$, then determines $P''$ and hence almost $P'$ (note however that $P'$ could still be in a $1$-dimensional piece of $Q_1$). Let $U\subset W$ be a general $2$-dimensional subspace determining a complete intersection $Z=Z_U$. In this section, we consider the case when $Z$ is not irreducible, a possibility which we would like to rule out.
As was argued before, the points of $P''$ are indistinguishable under the Galois group; the subspace $U$ may be chosen defined over the same field as $P$, so $P''$ must be contained in the smooth points of a single irreducible component $Z''$ of $Z$. Write $Z=Z'\cup Z''$ where the remaining piece $Z'$ is allowed to be reducible.
Applying Lemmas \[leq5inconic\] (as in the first paragraph of the proof of \[containedquartic\]) and \[inquadric\] as well as the hypothesis $h^0(E)=0$ so $P$ is not contained in a quadric, gives that ${\rm deg}(Z'')\geq 6$.
The idea is to use a dimension count. The dimensions of the cases go all the way up to ${\rm dim}{\bf G}_{rci}(8,1) = 32$. However, the subspace $W$ determines $P''$, and in turn $W$ is determined by a smaller subset of points than $P$, so the dimension count can still work.
Choose a subscheme $P_{16}\subset P$ of length $16$ as follows: start with $P_2$ of length $2$ containing $P'$. Note that $P_2$ imposes $2$ independent conditions on $H^0({{\mathcal O}}_{{{\mathbb P}}^3}(3))$. Then for $3\leq i\leq 16$ let $P_i:=P_{i-1}\cup \{ p_i\}$ with $p_i$ chosen in $P''$ such that it imposes a nontrivial condition on $H^0(J_{P_{i-1}}(3))$. This exists because $$h^0(J_P(3))=4< 20-(i-1)=h^0(J_{P_{i-1}}(3)).$$ For $i=16$ we get $P_{16}$ imposing $16$ independent conditions, and $P'\subset P_{16}$. It follows that $$W = H^0(J_P(3))= H^0(J_{P_{16}}(3)).$$ In particular, $W$ is determined by $P_{16}$. However, the remaining four points of $P-P_{16}$ are all in $P''$, in particular they are reduced points. Because of the “variable curve case” Theorem \[variablecurve\], the intersection $Q$ of the cubics in $W$ has dimension $0$ at the points of $P''$; therefore, the locations of the remaining four points are determined (up to a finite choice) by $W$. We get that $P$ is determined by $P_{16}$.
We may now count the dimension of the space of choices of pair $(P,Z)$ where $Z=Z_U$ for a general subspace $U\subset W$. The space of choices of $Z$ containing a degree $6$ or degree $7$ piece, is $\leq 30$. The dimension of the space of choices of $P_{16}$ inside $Z$ is $\leq 16$ if we assume $Z$ reduced, or $\leq 17$ in any case, so the total dimension there is $\leq 47$ which is too small. For the case of $Z$ containing a piece $Z''$ of degree $8$, we get a dimension of $32+16=48$ so this looks possible. However, the general element $Z$ of the parameter space corresponds to the union of a smooth degree $8$ curve $Z''$ meeting a line $Z'$ in $4$ points. In order to get to dimension $48$, we must have $P$ general, in particular $P''$ is a general collection of $18$ points in $Z''$. Now $Z''$ has genus $7$. The line bundle ${{\mathcal O}}_{Z'}(3)(-P'')$ is a general one of degree $6$, which on a curve of genus $7$ will not have any sections. Hence, all cubics containing $P$ must vanish on $Z''$, which would put us back into the “constant curve case”. So, this case doesn’t occur.
We have finished ruling out the possibility that $Z$ would be reducible, resulting in the following theorem.
\[Zirred\] For a general seminatural bundle $E$ in its irreducible component and a general section $s\in H^0(E(1))$ defining a scheme of zeros $P$, choose a general $2$-dimensional subspace $U\subset W=H^0(J_P(3))$ defining a complete intersection $Z_U$. Then $Z_U$ is irreducible.
Subschemes of an irreducible degree $9$ curve {#deg9curve}
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In this section we complete the proof that the Hilbert scheme ${\bf H}^{sn}_{{{\mathbb P}}^3}$ is irreducible, by treating the case $P\subset Z_U$ where $Z_U$ is an irreducible complete intersection of degree $9$.
We first indicate how to construct an open set of the Hilbert scheme. Consider a smooth complete intersection curve $Z_U$ for a general $2$-dimensional subspace $U\subset H^0({{\mathcal O}}_{{{\mathbb P}}^3}(3))$. The Grassmanian of choices of $U$ has dimension $36$ and there is a dense open set where $Z=Z_U$ is smooth of genus $10$.
Now $P\subset Z$ will be a subscheme of length $20$, which is a Cartier divisor since $Z$ is smooth. By varying any collection of $10$ points, we obtain a family which surjects to the Jacobian ${\rm Jac}^{20}(Z)$. The line bundle $L={{\mathcal O}}_Z(4)(-P)$ has degree $36-20=16$. Note that the map $$H^0({{\mathcal O}}_{{{\mathbb P}}^3}(4))\rightarrow H^0({{\mathcal O}}_Z(4))$$ is surjective, with kernel of dimension $8$. Hence, in order to obtain $h^0(J_P(4))=16$ one should ask for $h^0({{\mathcal O}}_Z(4)(-P))= 8$ that is, $h^0(L)=8$. As $g=10$ we get $\chi (L)=16+1-10= 7$. The condition $h^0(L)=8$ is therefore equivalent to $h^1(L)=1$ or by duality, $h^0(K_Z\otimes L^{-1})= 1$. Now, $K_Z={{\mathcal O}}_Z(2)$ has degree $18$, so $M:= K_Z\otimes L^{-1}$ is a line bundle of degree $2$. Asking for it to have a section is equivalent to asking that $M\cong {{\mathcal O}}_Z(x+y)$ for a degree $2$ effective divisor $(x)+(y)\in Z^{(2)}\subset {\rm Jac}^2(X)$. The dimension of choices of $M$ is $2$ and the space of choices is irreducible. For each choice of $M$, we have $L:= K_Z\otimes M^{-1}$, and the space of choices of divisor $P$ such that ${{\mathcal O}}_Z(4)(-P)=L$ is a projective space of dimension $$\# (P) - {\rm dim} ({\rm Jac}(Z))= 10.$$ Putting these together, we get an irreducible $12$ dimensional space of choices of $P\subset Z$ such that $h^0(J_P(4))=16$. Including the variation of $Z$ in a $36$ dimensional space, these fit together to form an irreducible $48$ dimensional variety.
If we replace $P$ by a subscheme $P_1\subset P$ of colength $1$ in the above argument, then $M$ changes to $M_1=M(z)={{\mathcal O}}_Z(x+y+z)$ where $(z)=P-P_1$. As this is a general point of $Z$, we still have $h^0(M_1)=1$ giving the Cayley-Bacharach condition $CB(4)$ for $P$. Hence, there is a dense open subset of the $48$ dimensional variety parametrizing pairs $(Z,P)$ where $P$ satisfies $CB(4)$. This is our irreducible component of ${\bf H}^{sn}_{{{\mathbb P}}^3}[2]$. Abstracting out the choice of $Z$ gives an irreducible $44$-dimensional component of the Hilbert scheme ${\bf H}^{sn}_{{{\mathbb P}}^3}$.
\[only\] The irreducible component constructed above is the only one in ${\bf H}^{sn}_{{{\mathbb P}}^3}$.
The argument above shows the basic idea. However, we need to do some more work to treat the case when $Z$ is singular and specially the possibility of a point or infinitesimal tangent vector in $P'$. The first step is to rule out this last possibility.
A general $P$ in its irreducible component is reduced.
Given $P$ we can choose a quintic surface $X$ containing it, and write $P=P'\cup P''$. We have $P''$ reduced and if $P'$ is non-reduced, it consists of a single infinitesimal tangent vector. Furthermore we may assume that $P$ is at a smooth point of its Hilbert scheme. Choose a local smoothing infinitesimal deformation of $P'$; we would like to extend that to a deformation of $P$ preserving the $CB(4)$ condition. As the Cayley-Bacharach property is open, it is equivalent to preserving the property $h^0(J_P(4))=16$. One can check that the obstruction to finding a deformation of $P''$ which, when added to the given deformation of $P'$, preserves $h^0(J_P(4))$, would be the existence of a section $t\in H^0(J_P(4))$ such that $t$ vanishes to order $2$ at all the points of $P''$ in ${{\mathbb P}}^3$.
Consider a complete intersection of cubics $Z$ containing $P$, and we may assume that $P''$ lies on the smooth locus of $Z$. From the results of the previous sections, we may assume that $Z$ is an irreducible curve of degree $9$. Hence ${{\mathcal O}}_Z(4)$ is a line bundle of degree $36$. Our section $t$ vanishes at $2P''\subset Z$, but also at the points of $P'$. Together these are at least $38$ points, so it follows that $t$ vanishes on $Z$. Let $C\subset {{\mathbb P}}^3$ be one of the cubics defining $Z$. The residual of the scheme $2P''$ of multiplicity $2$ at $P''$, intersected with $C$, consists of all the points of $P''$. The restriction $t|_C$, divided by the other equation of $Z$, corresponds to a linear section vanishing at these points; but the points of $P''$ are not all contained in a plane (indeed they are not even contained in a quadric), so $t|_C=0$. Then $t$ divided by the equation of $C$ is a linear form again vanishing on $P''$, so it is zero. Thus, $t=0$.
This proves that the obstruction to lifting our smoothing deformation of $P'$ to a deformation of $P$, vanishes. Therefore, for a general point $P$ the piece $P'$ has to consist of at most a single reduced point. This proves the lemma.
Suppose next that $P$ is a Cartier divisor on $Z$. This will always be the case at points of $P''$ which are smooth points of $Z$, but it remains a possibility that $P'$ is a non-movable point at a singularity of $Z$. We will deal with this problem below, but for now in the interest of better explaining the argument, assume that $L:= {{\mathcal O}}_Z(4)(-P)$ is a line bundle which we may think of as being a restriction from a small analytic neighborhood of $Z$.
Now $Z$ is a complete intersection, so duality still applies. This can be seen, for example, by using Serre duality on ${{\mathbb P}}^3$ and the equations for $Z$ which provide resolutions for ${{\mathcal O}}_Z$; the local $Ext$ sheaves may be tensored with $L$ which exists on a neighborhood of $Z$. We get $$H^i(Z,L|_Z) \cong H^{1-i}(Z, L^{-1}\otimes{{\mathcal O}}_Z (2))^{\ast}.$$ Applying this to $L= {{\mathcal O}}_Z(4)(-P)$, we get $$h^1(L|_Z)= h^0({{\mathcal O}}_Z(-2)(P)).$$ On the other hand, $\chi (L) = 7$ and as before, $h^0(J_P(4))= h^0(L|_Z) + 8$, so the condition $h^0(J_P(4))=16$ is equivalent to $h^1(L|_Z)=1$, which in turn is equivalent to asking that the degree $2$ line bundle ${{\mathcal O}}_Z(-2)(P)$ be effective.
The Picard scheme ${\rm Pic}^0(Z)$ is still a group scheme, hence smooth; and its tangent space at the origin is $H^1({{\mathcal O}}_Z)$. The exact sequences for $Z\subset C\subset {{\mathbb P}}^3$ (where $C$ is one of the cubics cutting out $Z$) give $H^1({{\mathcal O}}_Z)\cong H^3({{\mathcal O}}_{{{\mathbb P}}^3}(-6)) = H^0({{\mathcal O}}_{{{\mathbb P}}^3}(2))^{\ast}$ which is $10$-dimensional. So the group scheme, as well as its torsors ${\rm Pic}^d(Z)$ are $10$ dimensional. An infinitesimal argument with exact sequences also shows that for $10$ general points in $Z$, the map from the product of their tangent spaces to the Picard scheme is surjective. As $P$ consists of $20$ points, and the Picard scheme has dimension $10$, at least $10$ points can move generally, keeping the same divisor $P$.
The effective divisors form a two dimensional subscheme of ${\rm Pic}^2(Z)$. Thus, at a general $P\subset Z$ satisfying $h^0(J_P(4))=16$, the Hilbert scheme of such $P$ has dimension $12$. The locus of singular $Z$ has dimension $\leq 35$, so the pairs $(Z,P)$ with $Z$ singular lie on a subscheme of dimension $\leq 47$, and cannot therefore correspond to a general bundle $E$ in its irreducible component. This finishes the proof of Theorem \[only\] in the case where $P$ corresponds to a Cartier divisor.
Some further argument is needed for the general case. The reader may calculate directly that the dimension of the space of $(Z,P)$ such that $Z$ is a nodal curve and $P$ contains a point $p'$ located at the node, is $<48$ and doesn’t contribute. This indicates that we don’t get a new irreducible component in this way.
To give a more complete argument, consider $(Z,P)$ with $Z$ singular (but still reduced and irreducible) and $P$ including a point $p'\in P'$ located at a singular point of $Z$. Consider general hyperplanes $H\subset {{\mathbb P}}^3$ passing through $p'$, let $K:= (H\cap Z)_{p'}$ (meaning the local piece of $H\cap Z$ at $p'$) and let $P^+=P''\cup K$. This is now a Cartier divisor on $Z$ so the previous considerations apply. Let $\ell$ denote the length of $K$. The condition that $Z$ is not contained in a plane means that the general intersection $H\cap Z$ can’t be concentrated at a single point, on the other hand $p'$ is singular in $Z$, so $2\leq \ell \leq 8$. The exact sequences for complete intersections imply that $K$ imposes $\ell$ independent conditions on cubics.
Our point $p'$ is in the base locus $B_2$ for the bundle $E$, meaning that sections in $H^0(J_{X,P}(3))$ vanish to order $\geq 2$ at $p'$ in $X$. This is true for any general quintic $X$ passing through $P$, so sections of $H^0(J_{{{\mathbb P}}^3,P}(3))$ vanish to order $\geq 2$ at $p'$ in ${{\mathbb P}}^3$. In particular, $Z$ contains the multiplicity two fat point at $p'$. In turn, this implies that $K$ contains the multiplicity two fat point at $p'$ in $H$.
We have $h^0(J_{P^+}(3))\geq 17-\ell$, which translates, using duality and calculating the Euler characteristic, into $h^0({{\mathcal O}}_{Z}(-2)(P^+))\geq 1$. That is to say, ${{\mathcal O}}_X(-2)(P^+)$ is an effective line bundle of degree $\ell +1$ (the case $\ell = 1$ would correspond to the case treated previously). The dimension of the space of choices of $P^+$ satisfying this effectivity condition, at general $P''$ in its linear system, is $\leq 11+\ell$. Note that since $P$ is reduced, $P^+$ determines $P$.
For a given $K\subset H$, the space of choices of $Z$ passing through $K$ is the Grassmanian of $2$-planes in ${{\mathbb C}}^{20-\ell}$, so it has dimension $2(18-\ell ) = 36-2\ell$. We consider the space of choices of $(p',H,Z,P)$. The choices of $p'\in H$ form a $5$ dimensional space. Let $k$ denote the dimension of the space of choices of $K\subset H$ located at a given point $p'$. Then altogether, the space of choices of $(p',H,Z,P)$ has dimension $$\leq 5 + (11+\ell ) + (36-2\ell ) + k = 52 +k-\ell .$$ This should be compared with the dimension of the Hilbert scheme, plus the number of choices of $H$ ($2$-dimensional) for each $P$, which is to say $50$.
The dimension count is now taken care of by noting that $K\subset H$ contains the fat point of multiplicity $2$ at $p'$ and this part is fixed without parameters. The remaining parameters for the choice of $K$ therefore correspond to the length of the remaining subscheme, which is to say $k\leq \ell -3$. This gives a count of $\leq 49$ for the space of $(p',H,Z,P)$ corresponding to the singular situation, which is $<50$ so it doesn’t contribute to the general points of the Hilbert scheme of $(Z,P)$. This completes the proof of Theorem \[only\].
Bundles on the quintic {#onquintic}
======================
To complete the proof of Theorem \[main\], we should go back from the Hilbert scheme of $CB(4)$ subschemes in ${{\mathbb P}}^3$, to the Hilbert scheme of $CB(4)$ subschemes of a general quintic $X$. Note first of all that we have looked above at the Hilbert scheme ${\bf H}^{sn}_{{{\mathbb P}}^3}[2]$ of pairs $(Z,P)$. However, for a given $P$ the space of choices of $Z$ is just a Grassmanian of $2$-planes $U\subset W\cong {{\mathbb C}}^4$. So, irreducibility of the $48$-dimensional Hilbert scheme $\{ (Z,P)\}$ implies irreducibility of the $44$-dimensional Hilbert scheme ${\bf H}^{sn}_{{{\mathbb P}}^3}$ which for brevity we denote just by $\{ P\}$ and so forth. Consider now the incidence variety of pairs $(P,X)$ such that $X$ is a smooth quintic hypersurface containing $P$. The map $\{ (P,X)\} \rightarrow \{ P\}$ is a fibration in projective spaces of dimension $35$, indeed by the seminatural condition $P$ imposes $20$ conditions on the $56$ dimensional space $H^0({{\mathcal O}}_{{{\mathbb P}}^3}(5))$ and we should also divide out by scalars. Thus, the incidence variety $\{ (P,X)\} $ is irreducible of dimension $79$. The space of quintics denoted $\{ X\}$ is an open subset of ${{\mathbb P}}^{55}$, and the Hilbert scheme of choices of $P$ for a given general $X$, is the $24$-dimensional fiber of the map $$\{ (P,X)\}\rightarrow \{ X\} .$$ Up to now, we have shown that the source of this map is irreducible. An additional argument is needed to show that the fibers are irreducible. We will use the same argument as was used in [@MestranoSimpson], which was pointed out to us by A. Hirschowitz.
The idea is to say that there is a specially determined irreducible component of each fiber; then this component is invariant under the Galois action of the Galois group of the function field of the base, on the collection of irreducible components of the fiber. On the other hand, irreducibility of the total space means that the Galois group acts transitively on the set of irreducible components of the fiber, and together these imply that the fiber is irreducible.
In order to isolate a special irreducible component, notice that the singular locus of the moduli space of bundles was identified in [@MestranoSimpson]. It has a some explicit irreducible components corresponding to the choice of $CB(2)$ subschemes of length $10$ in $X$, yielding the case of bundles with $H^0 (E)\neq 0$ (this is the case we have been explicitly avoiding throughout the bulk of the argument above). We consider the $19$-dimensional component of the singular locus whose general point is a bundle $E$ fitting into an exact sequence $$0\rightarrow {{\mathcal O}}_X\rightarrow E \rightarrow J_R(1)\rightarrow 0$$ where $R\subset Y$ is a general collection of $10$ points on $Y=X\cap C$ for a quadric $C$.
For a general such bundle $E$, there is a unique co-obstruction, which is to say a unique exact sequence as above, and the Zariski normal space to the singular locus may naturally be identified with $H^1(E)$ which has dimension $2$. The second order obstruction map is the same as the quadratic form associated to the symmetric bilinear form obtained from duality $H^1(E)\cong H^1(E^{\ast}(1))^{\ast}=H^1(E^{\ast})$. This quadratic form defines a pair of lines inside $H^1(E)$. These are the two actual normal directions of the moduli space of bundles along the singular locus at $E$. In order to show that this component of the singular locus meets a canonically defined irreducible component of the moduli space, it suffices to show that these two lines are interchanged as $R$ moves about in the Hilbert scheme of $10$-tuples of points in $Y$.
The $2$-dimensional space $H^1(E)$ together with its quadratic form, depend only on the arrangement $R\subset {{\mathbb P}}^3$ of $10$ points on a quadric $C\cong {{\mathbb P}}^1\times {{\mathbb P}}^1$, in a way we now explain. The homogeneous coordinates of the $10$ points give a map ${{\mathbb C}}^4\rightarrow {{\mathbb C}}^{10}$. We get a map ${\rm Sym}^2({{\mathbb C}}^4)\rightarrow {{\mathbb C}}^{10}$, and the equation of the quadric $C$ is an element of the kernel; as ${\rm Sym}^2({{\mathbb C}}^4)$ has dimension $10$ itself, there is an element $\xi =(\xi _1,\ldots , \xi _{10})$ in the cokernel, unique up to scalars. The $CB(2)$ condition, which holds for general $R$, corresponds to asking that $\xi _i\neq 0$ for all $1\leq i\leq 10$. Therefore $\xi$ defines a nondegenerate symmetric bilinear form on ${{\mathbb C}}^{10}$ denoted $$\langle X,Y\rangle := X \Delta (\xi )Y^t = \sum_{i=1}^{10} \xi _ix_iy_i$$ The condition that $\xi$ vanish on the image of ${\rm Sym}^2({{\mathbb C}}^4)$ says that ${{\mathbb C}}^4\subset {{\mathbb C}}^{10}$ is an isotropic subspace. In other words, it is contained in its orthogonal subspace ${{\mathbb C}}^4\subset ({{\mathbb C}}^4)^{\perp} \cong {{\mathbb C}}^6$. The quotient ${{\mathbb C}}^6 / {{\mathbb C}}^4$ is our two-dimensional space $H^1(E)$ and $\Delta (\xi )$ induces a quadratic form on here. We are interested in the two isotropic lines. Fix $9$ of the points in a general way; then our two dimensional subspace with quadratic form, depends on a single choice of $r_{10}\in C$. A calculation shows that the discriminant divisor of the quadratic form contains reduced components in $C$. So if one has a curve of points $r_{10}\in Y$ which intersect this divisor transversally, the two lines are interchanged when we go around the intersection point on the curve. Now, one can choose $X$ to pass through the given $r_1,\ldots , r_9$ as well as transversally through a general reduced point on the discriminant divisor. For such $X$, the tangent directions are interchanged as $R$ moves around in $Y=X\cap C$, so the same is also true for any general $X$.
This completes the construction of a specified irreducible component of the moduli space of bundles. Notice that for the singular points $E$ constructed above, we still have $H^1(E(1))=0$, so a soon as we move off the singular locus to get $H^1(E)=0$, this gives a bundle with seminatural cohomology. Thus, our specified irreducible component corresponds to bundles with seminatural cohomology. Now, Hirschowitz’s argument plus Theorem \[only\] saying that the Hilbert scheme of choices $\{ P\}$ is irreducible, combine to show that there is only one irreducible component in the moduli space of stable bundles on $X$ of degree $1$ and $c_2=10$ having seminatural cohomology. This completes the proof of Theorem \[main\].
Some ideas for the non-seminatural case {#someideas}
=======================================
We indicate here how one should be able to treat Conjecture \[all\]. Notice that we made the hypothesis that $H^1(E(1))=0$, and this implied seminatural cohomology. So, in the non-seminatural case we have $h^1(E(1))\geq 1$, and $h^0(E(1))\geq 6$. If $s:{{\mathcal O}}\rightarrow E(1)$ with subscheme of zeros $P$ then $h^0(J_P(3))\geq 5$.
The first step will be to show that sections of $E(1)$ have a base locus consisting of at most one point $p'$, and that a general $P$ has to be reduced at $P'$, with $19$ points making up $P''$ with doubly-transitive Galois action. This should be similar to our arguments of Sections \[baseloci\] and \[Pstructure\].
One can also point out, right away, that this allows us to rule out the “common curve case” as in Section \[sec-ccc\], indeed even in the case when $Q_1$ has degree $6$, the same argument as we used for degrees $7$ and $8$ works to show that $Q_1$ would have to be contained in a quadric.
So, we are in the variable curve case. If $Z$ is a complete intersection of two cubics passing through $P$, then $Z=Z'\cup Z''$ with $Z''$ irreducible, containing $P''$ in its smooth locus. Part of the argument consisted of ruling out ${\rm deg}(Z'')<9$ by a dimension count. Here we can’t just transpose the arguments, indeed the dimension of the Hilbert scheme of possible collections $P$ might be strictly smaller than $44$, because each $P$ can contribute a positive dimensional space of extension classes.
So we should divide the argument into two cases. If $h^1(J_P(4))=1$, i.e. $e=0$ in the notations of Section \[notations\], then the dimension of ${\rm Ext}^1(J_P(2),{{\mathcal O}}_X(-1))$ is $1$ and the extension class is unique up to scalars. In this case, the dimension of the Hilbert scheme $\{ P\}$ remains $44$ (and including the complete intersection curve $Z$ gives $\{ (Z,P)\}$ of dimension $48$). The dimension count may then proceed as we have done and this should allow us to treat this case.
In the case when $h^1(J_P(4))\geq 2$, each choice of $P$ corresponds to a positive dimensional space of choices of extension class up to scalars. However, in this case we can degenerate the extension class to one which no longer satisfies the Cayley-Bachrach condition—meaning that, viewed as a dual element to ${{\mathcal O}}_P(4)$, it vanishes on one or more points.
The doubly transitive Galois action on $P$ implies that the images of the points $P$ in the projective space of extension classes, cannot generically bunch up in groups of more than one. Therefore, it is possible to degenerate the extension class towards one which vanishes at exactly one point of $P$. This means an extension which corresponds to a torsion-free sheaf $E$ with a singularity at a single point. It therefore corresponds to a point in the boundary of the moduli space, at the boundary component coming from $M_X(2,1,9)$. This boundary piece has codimension $1$ and we should be able to analyze the nearby bundles and conclude that we remain in the principal irreducible component (indeed it suffices to say that nearby bundles have seminatural cohomology).
The technique of localizing our picture on the boundary of the moduli space is obviously a necessary and important one which needs to be further developed in order to treat this type of question. This will be left for a future work.
Another interesting direction will be to look at Reider’s theory of nonabelian Jacobians [@Reider1] [@Reider2] for bundles on a general quintic surface. The structures we have encountered in an [*ad hoc*]{} way in the course of our proof, are actually pieces of Reider’s theory.
[A]{}
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[^1]: This research project was initiated on our visit to Japan supported by JSPS Grant-in-Aid for Scientific Research (S-19104002)
[^2]: This represents a change in notation from [@MestranoSimpson], where we considered bundles of degree $-1$. For the present considerations, bundles of degree $1$ are more practical in terms of Hilbert polynomial. We apologize for this inconvenience, but luckily the indexation by second Chern class stays the same. Indeed, if $E$ has degree $1$ then $c_2(E)=c_2(E(-1))$ as can be seen for example on the bundle $E={{\mathcal O}}_X\oplus {{\mathcal O}}_X(1)$ with $c_2(E)=c_2(E(-1))=0$. Thus, the moduli space of stable bundles $M_X(2,1,c_2)$ we look at here is isomorphic to $M_X(2,-1,c_2)$ considered in [@MestranoSimpson].
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abstract: |
[In this paper we employ some operator techniques to establish some refinements and reverses of the Callebaut inequality involving the geometric mean and Hadamard product under some mild conditions. In particular, we show $$\begin{aligned}
K&\left(\frac{M^{2t-1}}{m^{2t-1}},2\right)^{r'}
\sum_{j=1}^n(A_j\sharp_{s}B_j)\circ \sum_{j=1}^n(A_j\sharp_{1-s}B_j)
\nonumber\\&\,\,+\left(\frac{t-s}{t-1/2}\right)\left(\sum_{j=1}^n(A_j\sharp_{t}B_j)\circ \sum_{j=1}^n(A_j\sharp_{1-t}B_j)
-\sum_{j=1}^n(A_j\sharp B_j)\circ \sum_{j=1}^n(A_j\sharp B_j)\right)\nonumber
\\&\leq \sum_{j=1}^n(A_j\sharp_{t}B_j)\circ \sum_{j=1}^n(A_j\sharp_{1-t} B_j)\,,\end{aligned}$$ where $A_j, B_j\in{\mathbb B}({\mathscr H})\,\,(1\leq j\leq n)$ are positive operators such that $0<m' \leq B_j\leq m <M \leq A_j\leq M'\,\,(1\leq j\leq n)$, either $1\geq t\geq s>{\frac{1}{2}}$ or $0\leq t\leq s<\frac{1}{2}$, $r'=\min\left\{\frac{t-s}{t-1/2},\frac{s-1/2}{t-1/2}\right\}$ and $K(t,2)=\frac{(t+1)^2}{4t}\,\,(t>0)$.]{}
address: 'Department of Mathematics, Faculty of Mathematics, University of Sistan and Baluchestan, Zahedan, Iran.'
author:
- Mojtaba Bakherad
title: Some reversed and refined Callebaut inequalities via Kontorovich constant
---
Introduction and preliminaries
==============================
Let ${\mathbb B}({\mathscr H})$ denote the $C^*$-algebra of all bounded linear operators on a complex Hilbert space ${\mathscr H}$ with the identity $I$. An operator $A\in{\mathbb B}({\mathscr H})$ is called positive if $\langle Ax,x\rangle\geq0$ for all $x\in{\mathscr H }$, and we then write $A\geq0$. We write $A>0$ if $A$ is a positive invertible operator. The set of all positive invertible operators is denoted by ${\mathbb B}({\mathscr H})_+$. For self-adjoint operators $A, B\in{\mathbb B}({\mathscr H})$, we say $B\geq A$ if $B-A\geq0$. The Gelfand map $f\mapsto f(A)$ is an isometric $*$-isomorphism between the $C^*$-algebra $C(\rm{sp}(A))$ of a complex-valued continuous functions on the spectrum $\rm{sp}(A)$ of a self-adjoint operator $A$ and the $C^*$-algebra generated by $I$ and $A$. If $f, g\in C({\rm sp}(A))$, then $f(t)\geq g(t)\,\,(t\in{\rm sp}(A))$ implies that $f(A)\geq g(A)$.
It is known that the Hadamard product can be presented by filtering the tensor product $A \otimes B$ through a positive linear map. In fact, $A\circ B=U^*(A\otimes B)U$, where $U:{\mathscr H}\to {\mathscr H}\otimes{\mathscr H}$ is the isometry defined by $Ue_j=e_j\otimes e_j$, where $(e_j)$ is an orthonormal basis of the Hilbert space ${\mathscr H}$; see [@paul].
For $A, B\in{\mathbb B}({\mathscr H})_+$, the operator geometric mean $A\sharp B$ is defined by $A\sharp B=A^{\frac{1}{2}}\left(A^{\frac{-1}{2}}BA^{\frac{-1}{2}}\right)^{\frac{1}{2}}A^{\frac{1}{2}}.$ For $\alpha\in(0,1)$, the operator weighted geometric mean is defined by $$\begin{aligned}
A\sharp_\alpha
B=A^{\frac{1}{2}}\left(A^{\frac{-1}{2}}BA^{\frac{-1}{2}}\right)^{\alpha}A^{\frac{1}{2}}.\end{aligned}$$ Callebaut [@CAL] showed the following refinement of the Cauchy–Schwarz inequality $$\begin{aligned}
\label{choshi}
\left(\sum_{j=1}^n x_j^{\frac{1}{2}}y_j^{\frac{1}{2}}\right)^2&\leq
\sum_{j=1}^n x_j^{\frac{1+s}{2}}y_j^{\frac{1-s}{2}}\sum_{j=1}^n x_j^{\frac{1-s}{2}}y_j^{\frac{1+s}{2}}\nonumber
\\&\leq\sum_{j=1}^n x_j^{\frac{1+t}{2}}y_j^{\frac{1-t}{2}}\sum_{j=1}^n x_j^{\frac{1-t}{2}}y_j^{\frac{1+t}{2}}\nonumber
\\&\leq \left(\sum_{j=1}^n x_j\right)\left(\sum_{j=1}^ny_j\right),
\end{aligned}$$ where $ x_j, y_j \,\,(1\leq j\leq n)$ are positive real numbers and either $1\geq t\geq s>{\frac{1}{2}}$ or $0\leq t\leq s<\frac{1}{2}$. This is indeed an extension of the Cauchy–Schwarz inequality.\
Wada [@wada] gave an operator version of the Callebaut inequality by showing that if $A, B\in{\mathbb B}({\mathscr H})_+$, then [$$\begin{aligned}
(A\sharp B)\otimes(A\sharp B)&\leq\frac{1}{2}\left\{(A\sharp_\alpha B)\otimes (A\sharp_{1-\alpha} B)+(A\sharp_{1-\alpha} B)\otimes(A\sharp_{\alpha} B)\right\}\\&\leq \frac{1}{2}\left\{(A\otimes B)+(B\otimes A)\right\},\end{aligned}$$ ]{} where $\alpha\in[0,1]$. In [@caleba] the authors showed another operator version of the Callebaut inequality $$\begin{aligned}
\label{34rf}
\sum_{j=1}^n(A_j\sharp B_j)\circ \sum_{j=1}^n(A_j\sharp B_j)\nonumber
&\leq\sum_{j=1}^n(A_j\sharp_{s}B_j)\circ \sum_{j=1}^n(A_j\sharp_{1-s}B_j)\nonumber
\\&\leq \sum_{j=1}^n(A_j\sharp_tB_j)\circ \sum_{j=1}^n(A_j\sharp_{1-t} B_j)\nonumber\\&\leq\left(\sum_{j=1}^nA_j\right)\circ \left(\sum_{j=1}^nB_j\right)\,,\end{aligned}$$ where $A_j, B_j\in{\mathbb B}({\mathscr H})_+\,\,(1\leq j\leq n)$ and either $1\geq t\geq s>{\frac{1}{2}}$ or $0\leq t\leq s<\frac{1}{2}$.\
In [@mo-moj] the authors presented the following refinement of inequality as follows [$$\begin{aligned}
\label{moj-mo}
\sum_{j=1}^n(A_j\sharp_{s}B_j)&\circ \sum_{j=1}^n(A_j\sharp_{1-s}B_j)\nonumber
\\&\leq\sum_{j=1}^n(A_j\sharp_{s}B_j)\circ \sum_{j=1}^n(A_j\sharp_{1-s}B_j)\nonumber
\\&\,\,+\left(\frac{t-s}{s-1/2}\right)\left(\sum_{j=1}^n(A_j\sharp_{s}B_j)\circ \sum_{j=1}^n(A_j\sharp_{1-s}B_j)
-\sum_{j=1}^n(A_j\sharp B_j)\circ \sum_{j=1}^n(A_j\sharp B_j)\right)\nonumber
\\&\leq \sum_{j=1}^n(A_j\sharp_{t}B_j)\circ \sum_{j=1}^n(A_j\sharp_{1-t} B_j)\,,
\end{aligned}$$]{} in which $A_j, B_j\in{\mathbb B}({\mathscr H})_+\,\,(1\leq j\leq n)$ and either $1\geq t\geq s>{\frac{1}{2}}$ or $0\leq t\leq s<\frac{1}{2}$.\
There have been obtained several Cauchy–Schwarz type inequalities for Hilbert space operators and matrices; see [@aldaz; @ABM; @I-V; @salemi] and references therein.
In this paper, we present some refinements and reverses of the Callebaut inequality involving the weighted geometric mean and Hadamard product of Hilbert space operators.
Further refienements of the Callebaut inequality involving Hadamard product
============================================================================
The Kontorovich constant is $$\begin{aligned}
K(t,2)=\frac{(t+1)^2}{4t}\qquad(t>0).\end{aligned}$$ The classical Young inequality states that $$\begin{aligned}
a^\nu b^{1-\nu}\leq \nu a+(1-\nu)b,\end{aligned}$$ where $a,b\geq0$ and $\nu\in[0,1]$. Recently, Zuo et. al. [@zho] showed an improvement of the Young inequality as follows: $$\begin{aligned}
K\left(\sqrt{\frac{a}{b}},2\right)^{r}a^\nu b^{1-\nu}\leq \nu a+(1-\nu)b,\end{aligned}$$ where $a,b>0$, $\nu\in[0,1]$, $r=\min\left\{\nu,1-\nu\right\}$. Applying this inequality, J. Wu and J. Zhao [@W-Z] showed the following refiniment of the Young inequality $$\begin{aligned}
\label{Wu-Zhao}
K\left(\sqrt{\frac{a}{b}},2\right)^{r'}a^\nu b^{1-\nu}+r\left(
\sqrt{a}-\sqrt{b}\right)^2\leq \nu a+(1-\nu)b,\end{aligned}$$ where $a,b>0$, $\nu\in[0,1]-\left\{\frac{1}{2}\right\}$, $r=\min\left\{\nu,1-\nu\right\}$ and $r'=\min\left\{2r,1-2r\right\}$. Using we get the following lemmas.
Let $a,b>0$ and $\nu\in[0,1]-\left\{\frac{1}{2}\right\}$. Then $$\begin{aligned}
\label{kokol}
K\left(\sqrt{\frac{a}{b}},2\right)^{r'}\left(a^\nu b^{1-\nu}+a^{1-\nu}b^{\nu}\right)+2r\left(
\sqrt{a}-\sqrt{b}\right)^2\leq a+b,\end{aligned}$$ where $r=\min\left\{\nu,1-\nu\right\}$ and $r'=\min\left\{2r,1-2r\right\}.$
\[rar32\] Let $0<m' \leq B\leq m <M \leq A\leq M'$ and either $1\geq t\geq s>{\frac{1}{2}}$ or $0\leq t\leq s<\frac{1}{2}$. Then $$\begin{aligned}
\label{tool}
K\left(\frac{M^{2t-1}}{m^{2t-1}},2\right)^{r'}
&(A^{s}\otimes B^{1-s}+A^{1-s}\otimes B^{s}) \nonumber\\&\,\,\,\,+\left(\frac{t-s}{t-1/2}\right)\left(A^{t}\otimes B^{1-t}+A^{1-t}\otimes B^{t}-2(A^{\frac{1}{2}}\otimes B^{\frac{1}{2}})\right) \nonumber\\&\leq A^{t}\otimes B^{1-t}+A^{1-t}\otimes B^{t}\,,
\end{aligned}$$ where $r'=\min\left\{\frac{t-s}{t-1/2},\frac{s-1/2}{t-1/2}\right\}$.
Let $a>0$. If we replace $b$ by $a^{-1}$ and take $\nu=\frac{1-\mu}{2}$ , then we get $$\begin{aligned}
\label{ttt1}
K(a,2)^{r'}\left(a^\mu+a^{-\mu}\right)+(1-\mu)\left(
a+a^{-1}-2\right)\leq a+a^{-1},\end{aligned}$$ in which $\mu\in(0,1]$ and $r'=\min\left\{1-\mu,\mu\right\}$. Let us fix positive real numbers $\alpha,\beta$ such that $\beta<\alpha$. It follows from $0<m' \leq B\leq m <M \leq A\leq M'$ that $I\leq h=\left (\frac{M}{m}\right)^\alpha\leq A^\alpha\otimes B^{-\alpha}\leq h'=\left(\frac{M'}{m'}\right)^\alpha$ and $\rm{sp}(A^\alpha\otimes B^{-\alpha})\subseteq[h,h']\subseteq(1,+\infty)$. Since the Kontorovich constant $\frac{(a+1)^2}{4a}$ is an increasing function on $(1,+\infty)$, by we have $$\begin{aligned}
K\left(\frac{M^\alpha}{m^\alpha},2\right)^{r'}\left(a^\mu+a^{-\mu}\right)+(1-\mu)\left(
a+a^{-1}-2\right)\leq a+a^{-1},\end{aligned}$$ where $\mu\in(0,1]$ and $r'=\min\left\{1-\mu,\mu\right\}$. Using the functional calculus, if we replace $a$ by the operator $A^\alpha\otimes B^{-\alpha}$ and $\mu$ by $\frac{\beta}{\alpha}$ we have $$\begin{aligned}
\label{theo345}
K\left(\frac{M^\alpha}{m^\alpha},2\right)^{r'}&\left(A^\beta\otimes B^{-\beta}+A^{-\beta}\otimes B^{\beta}\right)\nonumber\\&\,\,\,+\left(1-\frac{\beta}{\alpha}\right)\left(
A^\alpha\otimes B^{-\alpha}+A^{-\alpha}\otimes B^{\alpha}-2I\right)\nonumber\\&\leq A^\alpha\otimes B^{-\alpha}+A^{-\alpha}\otimes B^{\alpha},\end{aligned}$$ where $r'=\min\left\{1-\frac{\beta}{\alpha},\frac{\beta}{\alpha}\right\}$. Multiplying both sides of by $A^\frac{1}{2}\otimes B^\frac{1}{2}$ we reach $$\begin{aligned}
\label{theo3456}
K\left(\frac{M^{\alpha}}{m^{\alpha}},2\right)^{r'}&\left(A^{1+\beta}\otimes B^{1-\beta}+A^{1-\beta}\otimes B^{1+\beta}\right)\nonumber\\&\,\,\,+\left(1-\frac{\beta}{\alpha}\right)\left(
A^{1+\alpha}\otimes B^{1-\alpha}+A^{1-\alpha}\otimes B^{1+\alpha}-2(A\otimes B)\right)\nonumber\\&\leq A^{1+\alpha}\otimes B^{1-\alpha}+A^{1-\alpha}\otimes B^{1+\alpha}.\end{aligned}$$ Now, if we replace $\alpha, \beta, A, B$ by $2t-1, 2s-1, A^\frac{1}{2}, B^\frac{1}{2}$ respectively, in , we obtain $$\begin{aligned}
K\left(\frac{M^{2t-1}}{m^{2t-1}},2\right)^{r'
}&(A^{s}\otimes B^{1-s}+A^{1-s}\otimes B^{s}) \\&\,\,\,\,+\left(\frac{t-s}{t-1/2}\right)\left(A^{t}\otimes B^{1-t}+A^{1-t}\otimes B^{t}-2(A^{\frac{1}{2}}\otimes B^{\frac{1}{2}})\right) \\&\leq A^{t}\otimes B^{1-t}+A^{1-t}\otimes B^{t}\,\end{aligned}$$ for either $1\geq t\geq s>{\frac{1}{2}}$ or $0\leq t\leq s<\frac{1}{2}$ and $r'=\min\left\{\frac{t-s}{t-1/2},\frac{s-1/2}{t-1/2}\right\}$.
We are ready to prove the first result of this section.
\[vow13\] Let $0<m' \leq B_j\leq m <M \leq A_j\leq M'\,\,(1\leq j\leq n)$ and either $1\geq t\geq s>{\frac{1}{2}}$ or $0\leq t\leq s<\frac{1}{2}$. Then $$\begin{aligned}
\label{maman}
K&\left(\frac{M^{2t-1}}{m^{2t-1}},2\right)^{r'}
\sum_{j=1}^n(A_j\sharp_{s}B_j)\circ \sum_{j=1}^n(A_j\sharp_{1-s}B_j)
\nonumber\\&\,\,+\left(\frac{t-s}{t-1/2}\right)\left(\sum_{j=1}^n(A_j\sharp_{t}B_j)\circ \sum_{j=1}^n(A_j\sharp_{1-t}B_j)
-\sum_{j=1}^n(A_j\sharp B_j)\circ \sum_{j=1}^n(A_j\sharp B_j)\right)\nonumber
\\&\leq \sum_{j=1}^n(A_j\sharp_{t}B_j)\circ \sum_{j=1}^n(A_j\sharp_{1-t} B_j)\,,\end{aligned}$$ where $r'=\min\left\{\frac{t-s}{t-1/2},\frac{s-1/2}{t-1/2}\right\}$.
Put $C_j=A_j^{-{1\over2}}B_jA_j^{-{1\over2}}\,\,(1\leq j\leq n)$. By inequality we get $$\begin{aligned}
\label{evs}
K\left(\frac{M^{2t-1}}{m^{2t-1}},2\right)^{r'}&\left(C_j^{s}\otimes C_i^{1-s}+C_j^{1-s}\otimes C_i^{s}\right)\nonumber\\&\,\,\,\,+\left(\frac{t-s}{t-1/2}\right)
\left(C_j^{t}\otimes C_i^{1-t}+C_j^{1-t}\otimes C_i^{t}-2\big(C_j^{\frac{1}{2}}\otimes C_i^{\frac{1}{2}}\big)\right)\nonumber\\&\leq C_j^{t}\otimes C_i^{1-t}+C_j^{1-t}\otimes C_i^{t}\qquad(1\leq i,j\leq n).
\end{aligned}$$ Multiplying both sides of by $A_j^{\frac{1}{2}}\otimes A_i^{\frac{1}{2}}$ we get
$$\begin{aligned}
\label{9090}
K\left(\frac{M^{2t-1}}{m^{2t-1}},2\right)^{r'}&\left((A_j\sharp_{s} B_j)\otimes (A_i\sharp_{1-s} B_i)+(A_j\sharp_{1-s} B_j)\otimes (A_i\sharp_{s}B_i)\right)\nonumber\\&
\,\,\,\,+\left(\frac{t-s}{t-1/2}\right)
\Big((A_j\sharp_{t} B_j)\otimes (A_i\sharp_{1-t} B_i)+(A_j\sharp_{1-t} B_j)\nonumber\\&\,\,\,\,\otimes (A_i\sharp_{t} B_i)-2(A_j\sharp B_j)\otimes (A_i\sharp B_i)\Big)\nonumber\\&\leq
(A_j\sharp_{t}B_j)\otimes (A_i\sharp_{1-t} B_i)+(A_j\sharp_{1-t}B_j)\otimes (A_i\sharp_{t}B_i)\,
\end{aligned}$$
for all $1\leq i,j\leq n$. Therefore [$$\begin{aligned}
&K\left(\frac{M^{2t-1}}{m^{2t-1}},2\right)^{r'}\left(\sum_{j=1}^n(A_j\sharp_{s}B_j)\circ \sum_{j=1}^n(A_j\sharp_{1-s}B_j)\right)
\\ & \,+\left(\frac{t-s}{t-1/2}\right)\left(\sum_{j=1}^n(A_j\sharp_{t}B_j)\circ \sum_{j=1}^n(A_j\sharp_{1-t}B_j)
-\Big(\sum_{j=1}^nA_j\sharp B_j\Big)\circ\Big(\sum_{j=1}^n A_j\sharp B_j\Big)\right)\\&=
\frac{1}{2}\sum_{i,j=1}^n\Big[K\left(\frac{M^{2t-1}}{m^{2t-1}},2\right)^{r'}\Big((A_j\sharp_{s} B_j)\circ (A_i\sharp_{1-s} B_i)+(A_j\sharp_{1-s} B_j)\circ (A_i\sharp_{s}B_i)\Big)\nonumber\\&
\,+\left(\frac{t-s}{t-1/2}\right)
\left((A_j\sharp_{t} B_j)\circ (A_i\sharp_{1-t} B_i)+(A_j\sharp_{1-t} B_j)\circ (A_i\sharp_{t} B_i)-2(A_j\sharp B_j)\circ (A_i\sharp B_i)\right)\Big]\\&\leq \frac{1}{2}\sum_{i,j=1}^n\left((A_j\sharp_{t}B_j)\circ (A_i\sharp_{1-t} B_i)+(A_j\sharp_{1-t}B_j)\circ (A_i\sharp_{t}B_i)\right)\qquad(\textrm{by inequality\,\eqref{9090}})\\&
=\sum_{j=1}^n(A_j\sharp_{t}B_j)\circ \sum_{j=1}^n(A_j\sharp_{1-t} B_j)\,.\end{aligned}$$]{}
It follows from $$\left(\frac{t-s}{t-1/2}\right)\left(\sum_{j=1}^n(A_j\sharp_{t}B_j)\circ \sum_{j=1}^n(A_j\sharp_{1-t}B_j)
-\sum_{j=1}^n(A_j\sharp B_j)\circ \sum_{j=1}^n(A_j\sharp B_j)\right)\geq0\,,$$ where $A_j, B_j\in {\mathbb B}({\mathscr H})_+\,\,(1\leq j\leq n)$, either $1\geq t\geq s>{\frac{1}{2}}$ or $0\leq t\leq s<\frac{1}{2}$ and $K(t,2)=\frac{(t+1)^2}{4t}\geq 1\,\,(t>0)$ that inequality is a refinement of the second inequality of inequalities and .
We conclude an application of Theorem \[vow13\] for numerical cases which is a refinement of inequality .
Let $0<m' \leq y_j\leq m <M \leq x_j\leq M'\,\,(1\leq j\leq n)$ and either $-1\leq t\leq s<0$ or $1\geq t\geq s>0$. Then $$\begin{aligned}
&\sum_{j=1}^nx_j^{{1+s}\over2}y_j^{{1-s\over2}} \sum_{j=1}^nx_j^{{1-s}\over2}y_j^{{1+s\over2}}
\\&\leq K\left(\frac{M^{t}}{m^{t}},2\right)^{r'}
\sum_{j=1}^nx_j^{{1+s}\over2}y_j^{{1-s\over2}} \sum_{j=1}^nx_j^{{1-s}\over2}y_j^{{1+s\over2}}
\\&\,\,+\left(\frac{t-s}{t}\right)\left(\sum_{j=1}^nx_j^{{1+t}\over2}y_j^{{1-t\over2}} \sum_{j=1}^nx_j^{{1-t}\over2}y_j^{{1+t\over2}}
-\left(\sum_{j=1}^nx_j^{1\over2}y_j^{1\over2}\right)^2 \right)
\\&\leq \sum_{j=1}^nx_j^{{1+t}\over2}y_j^{{1-t\over2}} \sum_{j=1}^nx_j^{{1-t}\over2}y_j^{{1+t\over2}}\,,\end{aligned}$$ where $r'=\min\left\{\frac{t-s}{t},\frac{s}{t}\right\}$.
[If we put $A_j=x_j, B_j=y_j\,\,(1\leq j\leq n)$ in Theorem \[vow13\] and inequality , then we get $$\begin{aligned}
&\sum_{j=1}^nx_j^{{1-s}}y_j^{{s}} \sum_{j=1}^nx_j^{{s}}y_j^{{1-s}}
\\&\leq K\left(\frac{M^{2t-1}}{m^{2t-1}},2\right)^{r'}
\sum_{j=1}^nx_j^{{1-s}}y_j^{{s}} \sum_{j=1}^nx_j^{{s}}y_j^{{1-s}}
\\&\,\,+\left(\frac{t-s}{t-1/2}\right)\left(\sum_{j=1}^nx_j^{{1-t}}y_j^{{t}} \sum_{j=1}^nx_j^{{t}}y_j^{{1-t}}
-\left(\sum_{j=1}^nx_j^{1\over2}y_j^{1\over2}\right)^2 \right)
\\&\leq \sum_{j=1}^nx_j^{{1-t}}y_j^{{t}} \sum_{j=1}^nx_j^{{t}}y_j^{{1-t}}\,,\end{aligned}$$ where $r'=\min\left\{\frac{t-s}{t-1/2},\frac{s-1/2}{t-1/2}\right\}$ and either $0\leq t\leq s<\frac{1}{2}$ or $1\geq t\geq s>{\frac{1}{2}}$. Now if we replace $s$ by ${s+1\over2}$ and $t$ by ${t+1\over2}$, respectively, where either $-1\leq t\leq s<0$ or $1\geq t\geq s>0$, then we reach the desired inequalities.]{}
Let $a,b>0$ and $\nu\in(0,1)$. Then $$\begin{aligned}
a^\nu b^{1-\nu}+a^{1-\nu} b^\nu +2r(\sqrt{a}-\sqrt{b})^2+r'\left(2\sqrt{ab}+a+b-2a^{\frac{1}{4}}b^{\frac{3}{4}}-2a^{\frac{3}{4}}b^{\frac{1}{4}}\right) \leq a+b\,,\end{aligned}$$ where $r=\min\{\nu,1-\nu\}$ and $r'=\min\{2r,1-2r\}$.
Let $a,b>0$, $\nu\in(0,1)$, $r=\min\{\nu,1-\nu\}$ and $r'=\min\{2r,1-2r\}$. Applying [@zha Lemma 1], we have the inequalities $$a^{1-\nu} b^\nu+\nu(\sqrt{a}-\sqrt{b})^2+r'(\sqrt[4]{ab}-\sqrt{a})^2 \leq(1-\nu) a+\nu b\,,$$ where $0<\nu\leq\frac{1}{2}$ and $$a^{1-\nu} b^\nu+(1-\nu)(\sqrt{a}-\sqrt{b})^2+r'(\sqrt[4]{ab}-\sqrt{b})^2 \leq(1-\nu) a+\nu b\,,$$ where $\frac{1}{2}<\nu<1$. Summing these inequalities we get the desired result.
\[rar3234\] Let $A_j, B_j\in{\mathbb B}({\mathscr H})_+\,\,(1\leq j\leq n)$ and either $1\geq t\geq s>{\frac{1}{2}}$ or $0\leq t\leq s<\frac{1}{2}$. Then $$\begin{aligned}
&(A^{s}\otimes B^{1-s}+A^{1-s}\otimes B^{s}) \\&\,\,\,\,+\left(\frac{t-s}{t-1/2}\right)\left(A^{s}\otimes B^{1-s}+A^{1-s}\otimes B^{s}-2(A^{\frac{1}{2}}\otimes B^{\frac{1}{2}})\right)\\&\,\,\,\,+r'\left(A^{t}\otimes B^{1-s}+A^{1-s}\otimes B^{s}+2(A^{\frac{1}{2}}\otimes B^{\frac{1}{2}})-2A^{\frac{1+2s}{4}}\otimes B^{\frac{3-2s}{4}}-2A^{\frac{3-2s}{4}}\otimes B^{\frac{1+2s}{4}}\right) \\&\leq A^{t}\otimes B^{1-t}+A^{1-t}\otimes B^{t}\,,
\end{aligned}$$ where $r'=\min\left\{\frac{t-s}{t-1/2},\frac{s-1/2}{t-1/2}\right\}$.
Using Lemma \[rar3234\] and the same argument in the proof of Lemma \[rar32\], we get another refinement of inequality .
\[okmn345\] Let $A_j, B_j\in{\mathbb B}({\mathscr H})_+\,\,(1\leq j\leq n)$ and either $1\geq t\geq s>{\frac{1}{2}}$ or $0\leq t\leq s<\frac{1}{2}$. Then $$\begin{aligned}
\label{maman2}
&\sum_{j=1}^n(A_j\sharp_{s}B_j)\circ \sum_{j=1}^n(A_j\sharp_{1-s}B_j)\nonumber
\\&\,\,+\left(\frac{t-s}{t-1/2}\right)\left(\sum_{j=1}^n(A_j\sharp_{s}B_j)\circ \sum_{j=1}^n(A_j\sharp_{1-s}B_j)
-\sum_{j=1}^n(A_j\sharp B_j)\circ \sum_{j=1}^n(A_j\sharp B_j)\right)\nonumber
\\&\,\,+r'\Big(\sum_{j=1}^n(A_j\sharp_{s}B_j)\circ \sum_{j=1}^n(A_j\sharp_{1-s}B_j)
+\sum_{j=1}^n(A_j\sharp B_j)\circ \sum_{j=1}^n(A_j\sharp B_j)\nonumber\\&\,\,\,\,-2\sum_{j=1}^n(A_j\sharp_{\frac{3-2s}{4}}B_j)\circ \sum_{j=1}^n(A_j\sharp_{\frac{1+2s}{4}}B_j)\Big)\nonumber
\\&\leq \sum_{j=1}^n(A_j\sharp_{t}B_j)\circ \sum_{j=1}^n(A_j\sharp_{1-t} B_j)\,,
\end{aligned}$$ where $r'=\min\left\{\frac{t-s}{t-1/2},\frac{s-1/2}{t-1/2}\right\}$.
If $A_j, B_j\in{\mathbb B}({\mathscr H})_+\,\,(1\leq j\leq n)$ and either $1\geq t\geq s>{\frac{1}{2}}$ or $0\leq t\leq s<\frac{1}{2}$, then [$$\begin{aligned}
\sum_{j=1}^n(A_j\sharp_{s}B_j)\circ \sum_{j=1}^n(A_j\sharp_{1-s}B_j)
+\sum_{j=1}^n(A_j\sharp B_j)\circ \sum_{j=1}^n(A_j\sharp B_j)-2\sum_{j=1}^n(A_j\sharp_{\frac{3-2s}{4}}B_j)\circ \sum_{j=1}^n(A_j\sharp_{\frac{1+2s}{4}}B_j)\end{aligned}$$]{} and [$$\begin{aligned}
\left(\frac{t-s}{t-1/2}\right)\left(\sum_{j=1}^n(A_j\sharp_{s}B_j)\circ \sum_{j=1}^n(A_j\sharp_{1-s}B_j)
-\sum_{j=1}^n(A_j\sharp B_j)\circ \sum_{j=1}^n(A_j\sharp B_j)\right)\end{aligned}$$]{} are positive operators. So inequality is another refinement of the second inequality of inequalities and .
If we put $B_j=I\,\,(1\leq j\leq n)$ in Theorem \[okmn345\], then we get the next result.
Let $A_j\in{\mathbb B}({\mathscr H})_+\,\,(1\leq j\leq n)$ and either $1\geq t\geq s>{\frac{1}{2}}$ or $0\leq t\leq s<\frac{1}{2}$. Then $$\begin{aligned}
&\sum_{j=1}^nA_j^{1-s}\circ \sum_{j=1}^nA_j^s
\\&\,\,+\left(\frac{t-s}{t-1/2}\right)\left(\sum_{j=1}^nA_j^{1-s}\circ \sum_{j=1}^nA_j^s
-\sum_{j=1}^nA_j^{\frac{1}{2}}\circ \sum_{j=1}^nA_j^{\frac{1}{2}}\right)\nonumber
\\&\,\,+r'\Big(\sum_{j=1}^nA_j^{1-s}\circ \sum_{j=1}^nA_j^s
+\sum_{j=1}^nA_j^{\frac{1}{2}}\circ \sum_{j=1}^nA_j^{\frac{1}{2}}\\&\,\,\,\,-2\sum_{j=1}^nA_j^{\frac{1+2s}{4}}\circ \sum_{j=1}^nA_j^{\frac{3-2s}{4}}\Big)
\\&\leq \sum_{j=1}^nA_j^{1-t}\circ \sum_{j=1}^nA_j^t\,,
\end{aligned}$$ where $r'=\min\left\{\frac{t-s}{t-1/2},\frac{s-1/2}{t-1/2}\right\}$.
If in Theorem \[okmn345\] we replace $A_j, B_j, s, t$, by $x_j, y_j, {s+1\over2}, {t+1\over2}\,\,(1\leq j\leq n)$, respectively, then we reach another refinement of inequality .
Let $x_j,y_j\,\,(1\leq j\leq n)$ be positive numbers and either $-1\leq t\leq s<0$ or $1\geq t\geq s>0$. Then $$\begin{aligned}
&\sum_{j=1}^nx_j^{1+s\over2}y_j^{1-s\over2} \sum_{j=1}^nx_j^{1-s\over2}y_j^{1+s\over2}
\\&\,\,+\left(\frac{t-s}{t}\right)\left(\sum_{j=1}^nx_j^{1+s\over2}y_j^{1-s\over2} \sum_{j=1}^nx_j^{1-s\over2}y_j^{1+s\over2}
-\left(\sum_{j=1}^nx_j^{1\over2}y_j^{1\over2}\right)^2\right)
\\&\,\,+r'\Big(\sum_{j=1}^nx_j^{1+s\over2}y_j^{1-s\over2} \sum_{j=1}^nx_j^{1-s\over2}y_j^{1+s\over2}
+\left(\sum_{j=1}^nx_j^{1\over2}y_j^{1\over2}\right)^2\\&\,\,\,\,-2\sum_{j=1}^nx_j^{2+s\over4}y_j^{2-s\over4} \sum_{j=1}^nx_j^{2-s\over4}y_j^{2+s\over4}\Big)
\\&\leq \sum_{j=1}^nx_j^{1+t\over2}y_j^{1-t\over2} \sum_{j=1}^nx_j^{1-t\over2}y_j^{1+t\over2}\,,
\end{aligned}$$ where $r'=\min\left\{\frac{t-s}{t},\frac{s-1}{t-1}\right\}$.
Some reverses of the Callebaut type inequality
===============================================
In [@W-Z], the authors showed a reverse of the young inequality as follows: $$\begin{aligned}
\label{laenana}
\nu a+(1-\nu)b\leq K\left(\sqrt{\frac{a}{b}},2\right)^{-r'}a^\nu b^{1-\nu}+s\left(
\sqrt{a}-\sqrt{b}\right)^2,\end{aligned}$$ in which $a,b>0$, $\nu\in[0,1]-\left\{\frac{1}{2}\right\}$, $r=\min\left\{\nu,1-\nu\right\}$, $r'=\min\left\{2r,1-2r\right\}$ and $s=\max\left\{\nu,1-\nu\right\}$. Applying we have the next result.
Let $a,b>0$ and $\nu\in[0,1]-\left\{\frac{1}{2}\right\}$. Then $$\begin{aligned}
a+b\leq K\left(\sqrt{\frac{a}{b}},2\right)^{-r'}\left(a^\nu b^{1-\nu}+a^{1-\nu}b^{\nu}\right)+2s\left(
\sqrt{a}-\sqrt{b}\right)^2,\end{aligned}$$ where $r=\min\left\{\nu,1-\nu\right\}$, $r'=\min\left\{2r,1-2r\right\}$ and $s=\max\left\{\nu,1-\nu\right\}$.\
In particular, If $\nu\in[0,\frac{1}{2})$, then $$\begin{aligned}
\label{now12}
a+a^{-1}\leq K(a,2)^{-r'}\left(a^{1-2\nu}+a^{-(1-2\nu)}\right)+2(1-\nu)\left(
a^\frac{1}{2}-a^\frac{-1}{2}\right)^2.\end{aligned}$$
Now, utilizing inequality and the same argument in the proof of Lemma \[rar32\] we can accomplish the corresponding result.
\[dear\] Let $0<m' \leq B\leq m <M \leq A\leq M'$ and either $1\geq t\geq s>{\frac{1}{2}}$ or $0\leq t\leq s<\frac{1}{2}$. Then $$\begin{aligned}
A^{t}\otimes B^{1-t}+A^{1-t}\otimes B^{t}&\leq K\left(\frac{M^{2t-1}}{m^{2t-1}},2\right)^{-r'}
(A^{s}\otimes B^{1-s}+A^{1-s}\otimes B^{s}) \nonumber\\&+\left(\frac{s-1/2}{t-1/2}\right)\left(A^{t}\otimes B^{1-t}+A^{1-t}\otimes B^{t}-2(A^{\frac{1}{2}}\otimes B^{\frac{1}{2}})\right)\,,
\end{aligned}$$ where $r'=\min\left\{\frac{t-s}{t-1/2},\frac{s-1/2}{t-1/2}\right\}$.
As a consequence of Lemma \[dear\] we have the following result.
\[mainth\] Let $0<m' \leq B_j\leq m <M \leq A_j\leq M'\,\,(1\leq j\leq n)$. Then $$\begin{aligned}
\sum_{j=1}^n(A_j&\sharp_{t}B_j)\circ \sum_{j=1}^n(A_j\sharp_{1-t} B_j)\leq K\left(\frac{M^{2t-1}}{m^{2t-1}},2\right)^{-r'}
\sum_{j=1}^n(A_j\sharp_{s}B_j)\circ \sum_{j=1}^n(A_j\sharp_{1-s}B_j)
\\&\,\,+\left(\frac{s-1/2}{t-1/2}\right)\left(\sum_{j=1}^n(A_j\sharp_{t}B_j)\circ \sum_{j=1}^n(A_j\sharp_{1-t}B_j)
-\sum_{j=1}^n(A_j\sharp B_j)\circ \sum_{j=1}^n(A_j\sharp B_j)\right)\end{aligned}$$ for either $1\geq t\geq s>{\frac{1}{2}}$ or $0\leq t\leq s<\frac{1}{2}$ and $r'=\min\left\{\frac{t-s}{t-1/2},\frac{s-1/2}{t-1/2}\right\}$.
If we put $t=1$ and $1\geq s>{\frac{1}{2}}$ in Theorem \[mainth\], then we get a reverse of the third inequality of $$\begin{aligned}
\left(\sum_{j=1}^nA_j \right)\circ& \left(\sum_{j=1}^n B_j\right)\leq K\left(\frac{M}{m},2\right)^{-r'}
\sum_{j=1}^n(A_j\sharp_{s}B_j)\circ \sum_{j=1}^n(A_j\sharp_{1-s}B_j)
\\&\,\,+\left({2s-1}\right)\left(\left(\sum_{j=1}^nA_j \right)\circ \left(\sum_{j=1}^n B_j\right)
-\sum_{j=1}^n(A_j\sharp B_j)\circ \sum_{j=1}^n(A_j\sharp B_j)\right),\end{aligned}$$ where $r'=\min\left\{{2-2s},{2s-1}\right\}$.
If we replace $A_j, B_j, s, t$, by $x_j, y_j, {s+1\over2}, {t+1\over2}\,\,(1\leq j\leq n)$ in Theorem \[mainth\], respectively, then we reach a reverse of the second inequality of
Let $0<m' \leq y_j\leq m <M \leq x_j\leq M'\,\,(1\leq j\leq n)$. Then $$\begin{aligned}
\sum_{j=1}^nx_j^{1+t\over2}y_j^{1-t\over2}& \sum_{j=1}^nx_j^{1-t\over2}y_j^{1+t\over2}\leq K\left(\frac{M^{2t-1}}{m^{2t-1}},2\right)^{-r'}
\sum_{j=1}^nx_j^{1+s\over2}y_j^{1-s\over2} \sum_{j=1}^nx_j^{1-s\over2}y_j^{1+s\over2}
\\&\,\,+\left(\frac{s-1/2}{t-1/2}\right)\left(\sum_{j=1}^nx_j^{1+t\over2}y_j^{1-t\over2} \sum_{j=1}^nx_j^{1-t\over2}y_j^{1+t\over2}
-\left(\sum_{j=1}^nx_j^{1\over2}y_j^{1\over2}\right)^2\right)\end{aligned}$$ for either $1\geq t\geq s>{\frac{1}{2}}$ or $0\leq t\leq s<\frac{1}{2}$ and $r'=\min\left\{\frac{t-s}{t-1/2},\frac{s-1/2}{t-1/2}\right\}$.
Let $0<m' \leq B_j\leq m <M \leq A_j\leq M'\,\,(1\leq j\leq n)$ and either $1\geq t\geq s>{\frac{1}{2}}$ or $0\leq t\leq s<\frac{1}{2}$. Then $$\begin{aligned}
K\left({h^{2t-1}},2\right)^{r'}&
\sum_{j=1}^n(A_j\sharp_{s}B_j)\circ \sum_{j=1}^n(A_j\sharp_{1-s}B_j)
+\left(\frac{t-s}{t-1/2}\right)\left(\sqrt{h}-
\sqrt{\frac{1}{h}}\right)^2\\&\leq\sum_{j=1}^n(A_j\sharp_{t}B_j)\circ \sum_{j=1}^n(A_j\sharp_{1-t} B_j)\\&\leq K\left({h^{2t-1}},2\right)^{-r'}
\sum_{j=1}^n(A_j\sharp_{s}B_j)\circ \sum_{j=1}^n(A_j\sharp_{1-s}B_j)
\\&\qquad+\left(\frac{s-1/2}{t-1/2}\right)\left(\sqrt{h'}-\sqrt{\frac{1}{h'}}\right)^2,\end{aligned}$$ where $h=\frac{M}{m}$, $h'=\frac{M'}{m'}$ and $r'=\min\left\{\frac{t-s}{t-1/2},\frac{s-1/2}{t-1/2}\right\}$.
Since the function $f(a)=a-\frac{1}{a}$ is increasing on $(0,\infty)$, we have $$\begin{aligned}
\left(\sqrt{h}-\sqrt{\frac{1}{h}}\right)^2 \leq \left(\sqrt{ a}-\sqrt{\frac{1}{a}}\right)^2\leq\left(\sqrt{h'}-\sqrt{\frac{1}{h'}}\right)^2\,\,\,(h\leq a\leq h').\end{aligned}$$ Applying inequalities , and the same argument in the proof of Theorem \[vow13\] we get the desired result.
**Acknowledgement.** The author would like to sincerely thank the anonymous referee for some useful comments and suggestions. The author also would like to thank the Tusi Mathematical Research Group (TMRG).
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|
---
abstract: 'Starting from the mean-field solution of a spin-orbital model of LiNiO$_2$, we derive an effective quantum dimer model (QDM) that lives on the triangular lattice and contains kinetic terms acting on 4-site plaquettes and 6-site loops. Using exact diagonalizations and Green’s function Monte Carlo simulations, we show that the competition between these kinetic terms leads to a resonating valence bond (RVB) state for a finite range of parameters. We also show that this RVB phase is connected to the RVB phase identified in the Rokhsar-Kivelson model on the same lattice in the context of a generalized model that contains both the 6-site loops and a nearest-neighbor dimer repulsion. These results suggest that the occurrence of an RVB phase is a generic feature of QDM with competing interactions.'
author:
- 'François Vernay$^1$, Arnaud Ralko$^1$, Federico Becca$^2$, and Frédéric Mila$^1$'
date: 'June, 8th 2006'
title: Identification of an RVB liquid phase in a quantum dimer model with competing kinetic terms
---
Introduction
============
After their first derivation by Rokhsar and Kivelson in 1988 in the context of cuprates, [@rokhsar] the hard-core quantum dimer models (QDM) have attracted significant attention. The phase diagrams of the QDM on the square and triangular lattices have been investigated in great details, [@leung; @syljuasen] and, following the pioneering work of Moessner and Sondhi on the triangular lattice, [@moessner] the very existence of a stable resonating valence bond (RVB) phase has been unambiguously demonstrated. [@ralko] The presence of a liquid phase with deconfined vison excitations [@balents] has also been established for a toy model living on the Kagome lattice. [@misguich]
However, the relationship between QDMs and Mott insulators, the physical systems for which they were proposed in the first place, is not straightforward. It is well established by now that the ground state of the S=1/2 Heisenberg model on the square and triangular lattices exhibits long-range magnetic order of Néel and 120 degree type respectively, and this type of order cannot be reached within the variational basis of Rokhsar and Kivelson, which consists of short-range singlet dimers. For a QDM to be a good effective model, one should thus identify models for which the subspace of short-range dimer coverings on a certain lattice is a good variational basis.
The first example of such a case was provided by the S=1/2 Heisenberg model on the trimerized Kagome lattice. [@mila] Indeed, an effective spin-chirality model living on a triangular lattice can be derived, and, at the level of a mean-field decoupling between spin and chirality, the ground state manifold consists of all dimer coverings on the triangular lattice. Going beyond mean-field is thus expected to lead to a relevant effective QDM. Using Rokhsar and Kivelson’s prescription, which consists in truncating the Hamiltonian and inverting the overlap matrix within the basis of dimer coverings, Zhitomirsky has derived such an effective Hamiltonian [@zhitomirsky] and shown that the main competition is between kinetic terms involving loops of length 4 and 6 respectively, and [*not*]{} a competition between a kinetic and a potential term, as for the QDM derived by Rokhsar and Kivelson. The next logical step would be to study the properties of this effective QDM. This is far from easy however. We know from the experience with the standard QDM model on the triangular lattice that the clusters reachable with exact diagonalizations are much too small to allow any significant conclusion regarding the presence of an RVB phase, and since there is no convention leading only to negative off-diagonal matrix elements, it is impossible to perform quantum Monte Carlo simulations.
Recently, two of us came across another model, for which the low energy sector consists of almost degenerate singlet coverings on the triangular lattice. This model is a Kugel-Khomskii [@kugel] model that was derived in the context of LiNiO$_2$, and the mean-field equations that describe the decoupling of the spin and orbital degrees of freedom possess an infinite number of locally stable solutions. These solutions are almost degenerate and correspond to spin singlet (and orbital triplet) dimers on the triangular lattice. [@vernay] Following Rokhsar and Kivelson’s prescription, an effective QDM can also be derived (see Appendix \[derivation\]). As for the S=1/2 Heisenberg model on the trimerized Kagome lattice, it consists of a competition between kinetic terms, with two important differences however. The main term of length 6 lives on loops that have a shape of large triangles, a term absent in the other case. But more importantly, the off-diagonal matrix elements are [*all*]{} negative.
Since the competition between kinetic processes was never investigated before, we have decided to concentrate on the minimal model obtained by keeping only the dominant term of length 6 for clarity. We have checked that the properties of the complete effective model are similar. This minimal model is described by the Hamiltonian:
$$\label{hamilt}
\begin{array}{rcl}
{ H}&=& \ -t \sum
\left(
|\unitlength=1mm
\begin{picture}(6.2,5)
\linethickness{2mm}
\put(0.9,-.7){\line(1,2){1.8}}
\put(3.8,-.7){\line(1,2){1.8}}
\end{picture}
\rangle
\langle
\unitlength=1mm
\begin{picture}(6.5,5)
\linethickness{0.3mm}
\put(3.2,2.6){\line(1,0){3.2}}
\put(0.9,-.7){\line(1,0){3.2}}
\end{picture}
|
+h.c.\right)
- \ t^\prime \sum
\left(
\left|\unitlength=1mm
\begin{picture}(7,6)
\linethickness{2mm}
\put(0.8,-1.7){\line(1,2){1.8}}
\put(6.4,1.6){\line(-1,2){1.8}}
\linethickness{0.2mm}
\put(3.8,-1.7){\line(1,0){3.6}}
\end{picture}
\right\rangle
\left\langle
\unitlength=1mm
\begin{picture}(7,6)
\linethickness{2mm}
\put(1.8,1.6){\line(1,2){1.8}}
\put(7,-1.7){\line(-1,2){1.8}}
\linethickness{0.2mm}
\put(-0.6,-1.7){\line(1,0){3.6}}
\end{picture}
\right|
+h.c.\right)
+ \ V \sum \left(
|\unitlength=1mm
\begin{picture}(6.2,5)
\linethickness{2mm}
\put(0.9,-.7){\line(1,2){1.8}}
\put(3.8,-.7){\line(1,2){1.8}}
\end{picture}
\rangle
\langle
\unitlength=1mm
\begin{picture}(6.2,5)
\linethickness{2mm}
\put(0.9,-.7){\line(1,2){1.8}}
\put(3.8,-.7){\line(1,2){1.8}}
\end{picture}|+
|
\unitlength=1mm
\begin{picture}(6.5,5)
\linethickness{0.3mm}
\put(3.2,2.6){\line(1,0){3.2}}
\put(0.9,-.7){\line(1,0){3.2}}
\end{picture}\rangle
\langle
\begin{picture}(6.5,5)
\linethickness{0.3mm}
\put(3.2,2.6){\line(1,0){3.2}}
\put(0.9,-.7){\line(1,0){3.2}}
\end{picture}
|
\right),
\end{array}$$
where the sums run over the 4-site and 6-site loops with all possible orientations. Although the repulsion is a higher order process, we have included a repulsion term in the Hamiltonian, and we will treat its amplitude $V$ as a free parameter to be able to make contact with the Rokhsar-Kivelson model on the triangular lattice. The hopping amplitudes $t$ and $t^\prime$ are negative, and although the ratio $t^\prime/t$ is in principle fixed by their expression in the perturbative expansion, we will also treat it as a free parameter.
Our central goal in this paper is to determine the nature of the ground state as a function of $t^\prime/t$. With respect to what we already know about QDMs, the main question is whether a competition between kinetic terms can also lead to a liquid phase. As we shall see, the answer to that question is positive, a liquid phase being present in a finite region of the phase diagram in the $t^\prime{-}V$ plane.
The paper is organized as follows. In Section \[method\], we briefly review the basic preliminaries used in the rest of the paper. The results obtained with exact diagonalizations are presented in Section \[resed\], those obtained with quantum Monte Carlo in Section \[resqmc\], and the conclusions in Section \[concl\]. The perturbation calculation which has motivated the investigation of this QDM is finally presented in Appendix \[derivation\].
The method {#method}
==========
In this section, we present a brief introduction to the numerical methods, to the clusters used in the analysis, and to the physical concepts underlying the determination of the phase diagram. More details can be found in Ref. .
Let us first discuss the shape of the finite-size clusters. In general, a finite cluster is defined by two vectors ${\bf T}_1$ and ${\bf T}_2$ and, in order to have the symmetries for rotations by $2\pi/3$, they have to satisfy: [@bernu] $$\begin{aligned}
{\bf{T}}_{1} &=& l {\bf{u}}_{1} + m {\bf{u}}_{2} \nonumber \\
{\bf{T}}_{2} &=& -m {\bf{u}}_{1} + (l+m) {\bf{u}}_{2} \nonumber ,\end{aligned}$$ where $l$ and $m$ are integers and ${\bf u}_1=(1,0)$ and ${\bf u}_2=(1/2,\sqrt{3}/2)$ are the unitary vectors defining the triangular lattice. The number of sites in the cluster is $N= l^2+m^2+lm$. In order to have also the axial symmetry, and therefore all the symmetries of the infinite lattice, we must take either $lm=0$ or $l=m$. The first possibility corresponds to type-A clusters (with the notation of Ref. ), with $N=l^2$ sites; the second one gives rise to type-B clusters, with $N=3 \times l \times l$ (for examples of both cases, see Fig. \[fig:cluster\]). Since for $t^\prime/t=0$ and $V/t=0$ the ground state belongs to a crystalline phase with a 12-site unit cell, [@moessner; @ralko] we will restrict in this work to clusters with a number of sites multiple of 12, in order not to frustrate this order. To limit the finite size effects related to the geometry of the clusters, we will concentrate on type-B clusters, which are always compatible with this order. Note that the 6-dimer loop kinetic term does not introduce further restrictions since all it requires is to be able to accommodate 6-site unit cells.
![\[fig:cluster\] Example of type-A (left) and type-B (right) clusters with 16 and 12 sites, respectively.](fig1.eps){width="45.00000%"}
A very important concept in the QDM is the existence of topological sectors. Indeed, in the triangular lattice, the Hilbert space is split into four disconnected topological sectors on a torus defined by the parity of the number of dimers cutting two lines that go around the two axes of the torus and denoted by $(p,q)$ with $p,q=0$ (respectively $1$) if this number is even (respectively odd). One can convince oneself, by a direct inspection of the effect of the 4-site and 6-site terms, that these numbers are conserved quantities under the action of the Hamiltonian of Eq. (\[hamilt\]). More generally, this is a consequence of the fact that the topological sectors are not coupled by any [*local*]{} perturbation. These topological sectors are extremely useful to distinguish between valence bond solids and spin liquids. Indeed, valence bond solids are only consistent with some topological sectors, whereas RVB spin-liquid phases are characterized by topological degeneracy. Therefore, the main goal will be to investigate whether, in the thermodynamic limit, the topological sectors are degenerate or not. In that respect, it is useful to remember that the $(0,1)$ and $(1,0)$ sectors are always degenerate with either $(0,0)$ or $(1,1)$ (depending on the cluster geometry) since they contain the same configurations rotated by an angle $\pi/3$. [@ralko] One can thus, without any loss of generality, restrict oneself to the analysis of the $(0,0)$ and $(1,1)$ sectors. Therefore, we define the absolute value of the topological gap as: $$\label{topogap}
\Delta E = |E_{00} - E_{11}|,$$ where $E_{00}$ and $E_{11}$ are the total ground-state energies for the topological sectors with $p=q=0$ and $p=q=1$, respectively. This gap is expected to scale to zero with the cluster size in the RVB phase.
Finally, in order to detect a possible dimer order, we also consider the static dimer-dimer correlations $$\label{dimercorr}
D^{i,j}(r-r^\prime) = \langle D^i(r) D^j(r^\prime) \rangle,$$ where $D^i(r)$ is the dimer operator defined as follows: It is a diagonal operator in the configuration space that equals $1$ if there is a dimer from the site $r$ to the site $r+a_i$, with $a_1=(1,0)$, $a_2=(1/2,\sqrt{3}/2)$, or $a_3=(-1/2,\sqrt{3}/2)$ and vanishes otherwise.
The method used is the same one as that used by Ralko and collaborators [@ralko] to determine the phase diagram of the Rokhsar-Kivelson QDM on the triangular lattice and our investigations are Lanczos diagonalizations and Green’s function Monte Carlo (GFMC) simulations. In particular, the GFMC is a zero-temperature stochastic technique based on the power method: Starting from a given wave function and by applying powers of the Hamiltonian, the ground state is statistically sampled to extract its energy and equal-time correlation functions. In principle, as in other Monte Carlo algorithms, in order to reduce the statistical fluctuations, it is very useful to consider the importance sampling, through the definition of a suitable [*guiding function*]{}. Unfortunately, when dealing with dimer models, it is very hard to implement an accurate and, at the same time, efficient guiding function for the crystalline phases. This problem is particularly relevant when the 6-site term becomes dominant. In this case, our simulations suffer from wild statistical fluctuations, deteriorating the convergence of the GFMC. As a consequence, we are not able to reach the largest available size, $432$-site cluster, for all parameters $t^\prime/t$ and $V/t$. By contrast, given the simple form of the spin-liquid ground state (that reduces to a superposition of all the configurations with the same weight at the Rokshar-Kivelson point), in the disordered region we can use the guiding function with all equal weights and obtain very small fluctuations (no fluctuation at the Rokshar-Kivelson point) and, therefore, excellent results with zero computational effort. Combining these facts, the loss of the GFMC convergence can be interpreted as a signal for the appearance of a crystalline phase. Of course, this is not a quantitative criterion, but, as it will be shown in the following, it gives reasonable insight into the emergence of a dimer order.
![\[fig:ED\_12\_36\] (Color online) Difference between $E_{00}$ and $E_{11}$, the total ground-state energies of the topological sectors with $p=q=0$ and $p=q=1$, respectively. The results are found by exact diagonalizations for clusters with 12 and 36 sites.](fig2.eps){width="40.00000%"}
![\[fig:correl\] (Color online) Dimer-dimer correlations on the 36-site cluster. The dimer of reference is the thickest one in the up-right corner. The thicker the line, the farther the value of the correlation from the uniform distribution equal value $1/36$. Solid lines are used when the correlation is higher than $1/36$, and dashed lines when it is lower.](fig3.eps){width="50.00000%"}
Finally it should be mentioned that, since the different topological sectors are completely decoupled (each dimer configuration belonging to one and only one of them) and cannot be connected by the terms contained in the Hamiltonian, within the GFMC it is possible to work in a given topological sector, making it possible to extract the ground-state properties of each of them.
Exact Diagonalizations {#resed}
======================
To get a first idea of the properties of the model, we start with the results we have obtained with exact diagonalizations of finite clusters for the model of Eq. (\[hamilt\]) with $V/t=0$. Let us first begin with the ground-state energy for the 12- and 36-site clusters for both the topological sectors $(0,0)$ and $(1,1)$ (see Fig. \[fig:ED\_12\_36\]). Note that the 12-site cluster is of type B, whereas the 36-site one is of type A. We have that, for both sizes, a level crossing occurs for $t^\prime/t \sim 2$. Below that value, the ground state is in $(1,1)$ topological sector, in agreement with the earlier results of Ref. for $V/t=0$. The main difference between the 12-site and the 36-site clusters is that, for the 36-site cluster, the topological ground-state energies stay very close in a large parameter range for $t^\prime/t \sim 2$. This could suggest that, upon increasing the size, this level crossing might evolve into a phase where these energies are rigorously degenerate, giving rise to a liquid phase without any crystalline order.
In order to give an idea of the various phases, we report in Fig. \[fig:correl\] the dimer-dimer correlations of Eq. (\[dimercorr\]) for the 36-site cluster below, at, and above the level crossing, which takes place at $t^\prime/t=2.2$ for this size. For small $t^\prime/t$, the correlations show a pattern similar to that of the intermediate $\sqrt{12} \times \sqrt{12}$ phase of the standard QDM model, already shown in Ref. . It should be stressed that, since in these calculations the Hamiltonian has the translational symmetry, the 12-site unit cell is not directly visible from Fig. \[fig:correl\], and in order to have a clearer evidence one should break the symmetry by hand. Nonetheless, as it has been shown in Ref. , these results are in perfect agreement with the existence of a $\sqrt{12} \times \sqrt{12}$ phase with a crystalline ground state in the thermodynamic limit. When $t^\prime/t$ is large, another pattern arises, which has never been observed in the standard QDM, and which presents a kind of 6-site triangle ordering. In this case, the dominant kinetic term involving 6 sites \[see the second term of the Hamiltonian (\[hamilt\])\] induces a dimer pattern with the same symmetry, possibly inducing a ground state with a 6-site unit cell in the thermodynamic limit. Also in this case the translational symmetry of the ground state partially masks the existence of a regular dimer pattern. Unfortunately, we will not be able to confirm this prediction since, as stated before, the GFMC algorithm has serious problems of convergence inside this phase and for large clusters.
In any case, for intermediate values of $t^\prime/t$, the correlations decay very rapidly with the distance and are close to those obtained in the liquid phase of the standard QDM with $V/t \lesssim 1$ and $t^\prime/t=0$. [@ralko] This fact gives another evidence of the possible existence of an RVB phase between two ordered phases in the model with competing kinetic terms and without the dimer repulsion. Of course by considering the exact diagonalization results only it is impossible to give definite statements on the stabilization of this liquid phase and, therefore, in the following section, we will consider a more systematic study of the topological gap, in order to unveil the existence of a wide disordered region that develops from the Rokshar-Kivelson point $V/t=1$ of the standard QDM and survives up to $V/t=0$ and finite $t^\prime/t$.
Green’s Function Monte Carlo {#resqmc}
============================
In this section, we use the GFMC method to extend the results of the previous section to larger clusters, with up to $432$ sites, and to map out the phase diagram in the $t^\prime{-}V$ plane.
The case of $V/t=0$
-------------------
Let us first describe the results we have obtained for $V/t=0$ and consider the behavior of the topological gap given in Eq. (\[topogap\]). In Fig. \[fig:topo\_gap1\], for clarity, we divided the results in two sets for $0 \leq t^\prime/t \leq 0.6$ and $t^\prime/t \geq 1$. The first remarkable feature is that, for most parameters, the gap decreases between 12 and 48 sites, regardless of its behavior for larger sizes. Therefore, the possibility to study much larger clusters is crucial for this analysis. Indeed, there is a clear change of behavior for $t^\prime/t \sim 1.6$, a value beyond which our extrapolations give solid evidences in favor of a vanishing gap in the thermodynamic limit. On the other hand, for smaller $t^\prime/t$ ratios, we have a clear evidence that $\Delta E$ increases for large sizes. Therefore, we come to the important conclusion that, the crystalline $\sqrt{12} \times \sqrt{12}$ phase is destroyed by increasing the amplitude of the 6-site term and the system is eventually driven into a liquid RVB phase.
![\[fig:topo\_gap1\] (Color online) Topological gaps for $V/t=0$ as a function of $1/\sqrt{N}$, where $N$ is the number of sites and for different values of $t^\prime/t$. Upper panel: Small values of $t^\prime/t$, where the gap opens for large clusters. Lower panel: Larger values of $t^\prime/t$. For $t^\prime/t \gtrsim 1.6 $, the finite-size gap closes upon increasing the cluster size, signaling a liquid phase.](fig4.eps){width="45.00000%"}
![\[fig:parall\_cs\] (Color online) Dimer-dimer correlation function for a 108-site cluster along a horizontal line as a function of the distance for $t^\prime/t=2$ and $V/t=0$. The dashed line corresponds to $1/36$, the value in the absence of correlations.](fig5.eps){width="50.00000%"}
To have a further confirmation of the existence of this disordered phase, we have calculated the dimer-dimer correlations. The results obtained on the 108-site cluster for $t^\prime/t=2$ are reported in Fig. \[fig:parall\_cs\], where the correlation functions for parallel dimers along the same row are plotted as a function of the distance. Given the small number of clusters available, a precise size scaling of the order parameter is not possible and also a meaningful estimation of the correlation length is very hard. Nevertheless, the behavior is definitely consistent with an exponential decay and the uncorrelated value of $1/36$ is approached very rapidly, as expected in a liquid phase without any crystalline order.
Unfortunately, the larger $t^\prime/t$ region is numerically far more difficult to access. Indeed, as stated above, although the GFMC is in principle numerically exact, we have not been able to find an efficient guiding function to perform the importance sampling and wild statistical fluctuations prevent us to reach a safe convergence for large clusters. In practice, we have access to clusters up to 108 sites, that are still too small to predict the thermodynamic behavior. For instance, for $V/t=0$, the convergence stops before one can observe any increase of the topological gap, and the criterion used for the phase transition on the other side of the RVB phase cannot be used any more. However, the lack of convergence is a clear sign that one enters a new (crystalline) phase. So, if the change of behavior of the topological gap with the size cannot be observed, we take as a definition of the boundary for the phase transition the parameters for which the convergence is not good any more. This is expected to be semi-quantitative, and indeed, as we shall see in the next section when studying the full diagram, the points obtained with this criterion agree reasonably well with those obtained with the reopening of the topological gap.
In summary, although the region where the 6-site term dominates over the usual 4-site dimer flip is not accessible by using GFMC, based on our numerical results for small $t^\prime/t$, we can safely argue that the crystalline $\sqrt{12} \times \sqrt{12}$ phase is destabilized by increasing the 6-site kinetic term, leading to a true disordered ground state with topological degeneracy.
Phase diagram in the $t^\prime{-}V$ plane
-----------------------------------------
In this section we prove that the RVB phase found in the previous paragraph for $V/t=0$ is connected to the one obtained for the standard QDM, i.e., close to the Rokshar-Kivelson point and $t^\prime/t=0$. In order to do that, we have investigated a generalization of the previous model that also includes a repulsion $V$ between dimers facing each other, see Eq. (\[hamilt\]). For $t^\prime/t=0$, this model reduces to the standard QDM, which has an RVB phase for $0.75 \lesssim V/t \lesssim 1$. [@moessner; @ralko] To map out the complete phase diagram of this model, we have done the same analysis of the previous paragraph for different values of $V/t$ between 0 and 1. As an example, we show in Fig. \[fig:topo\_gap2\] the finite-size scaling of the topological gap for $t^\prime/t=1$ and several values of $V/t$. The global behavior is the same as before and we have clear evidence that the topological gap present at $V/t=0$ persists up to $V/t \sim 0.25$, and that it opens again for $V/t \sim 0.8$. In that case, as in many other cases, it turned out to be possible to actually observe the opening of the gap upon leaving again the RVB phase before the convergence problems were too strong.
![\[fig:topo\_gap2\] (Color online) Topological gap for $t^\prime/t=1$ and for various $V/t$ as a function of $1/\sqrt{N}$, where $N$ is the number of sites. Small and large values of $V/t$ have been shown separately in the upper and lower panels for clarity.](fig6.eps){width="45.00000%"}
![\[fig:phasediag\] (Color online) Phase diagram in the $t^\prime{-}V$ plane. A wide disordered region extends all the way from the standard QDM ($t^\prime/t=0$ axis) to the purely kinetic QDM ($V/t=0$ axis). The description of the symbols is given in the text.](fig7.eps){width="40.00000%"}
The resulting phase diagram is depicted in Fig. \[fig:phasediag\], where different symbols have been used depending on whether the boundary was determined from the increase of the correlation length with the size or from the loss of convergence: Solid circles when the closing of the topological gap was observable and solid diamonds when the convergence of the GFMC was lost for large systems. Interestingly enough, these different criteria build a relatively smooth line, a good indication that it can be interpreted as a phase boundary. Note that we did not perform simulations for $V/t>1$ where, for a vanishing $t^\prime/t$, a crystalline phase with staggered dimer order is stabilized. So we cannot exclude that the boundary extends beyond $V/t>1$ for small $t^\prime/t$. Remarkably, the RVB phase we found for $V/t=0$ is connected to the RVB phase reported before for the standard QDM, i.e., $t^\prime/t=0$, the total RVB phase building up a large stripe that encompasses a significant portion of the phase diagram. We have also calculated static correlation function for several values of the parameter, but they merely confirm the identification of the phases and are not reported for brevity.
Conclusions {#concl}
===========
Coming back to our long-term motivation, namely to find an RVB phase in a realistic model of Mott insulators, this paper contains significant results of two sorts. First of all, we have shown (see Appendix \[derivation\]) that, starting from the quasi-degenerate mean-field ground state of a Kugel-Khomskii spin-orbital model, one can construct a QDM with two remarkable properties: It describes a competition between two kinetic terms of comparable magnitude, and all off-diagonal matrix elements in the dimer basis are negative. This has allowed us to implement the GFMC and to investigate the results of the competition between these terms. It turns out that the competition between these kinetic terms leads to the disappearance of the $\sqrt{12}\times \sqrt{12}$ crystalline order when $t^\prime/t \sim 1.6$. This transition is similar to the transition into the RVB phase that happens in the standard QDM upon approaching the Rokhsar-Kivelson point. Indeed, the two phases can be connected into a single RVB phase in the context of a generalized QDM. As far as the numerical investigation of the model is concerned, the main open issue is to pin down the nature of the phase that occurs when the 6-dimer kinetic term dominates. Unfortunately, the GFMC suffers from severe statistical fluctuations whenever the guiding function is not accurate, i.e., for clusters larger that 108 sites and large $t^\prime/t$. Therefore, we cannot make any definite statements on the phase where the 6-site term dominates. Another interesting question is of course the nature of the quantum phase transitions between these phases (continuous or first order). We are currently working on that rather subtle issue in the context of the standard QDM.
The general features of the RVB phase are consistent with the phenomenology of LiNiO$_2$, which exhibits neither orbital nor magnetic long-range order. A number of points deserve further investigation however. The precise form of the QDM does not seem to be an issue: The actual model that can be derived along the Rokhsar-Kivelson lines has more terms (see Appendix \[derivation\]), but preliminary results show that the RVB phase is present in that model as well. The fact that the RVB phase does not contain the point $t^\prime/t=1.34$ derived in the Appendix \[derivation\] is not really an issue either. First of all, this ratio was determined for vanishing Hund’s rule coupling and one vanishing hopping integral, and its precise value in that case should at best be taken as an indication of its order of magnitude in the actual system. Besides, the other 6-site terms pull the RVB region down to smaller values of the relative ratio of the 6-dimer term to the 4-dimer term. What would deserve more attention is the validity of the expansion that leads to the effective QDM. The small parameter of the expansion is not that small ($\alpha=1/\sqrt{2}$, see Appendix \[derivation\]), and it would be very useful to better understand to which extent such an expansion can be controlled. Nevertheless, the present results strongly suggest that the presence of an RVB liquid phase between competing ordered phases is a generic feature of QDM, and that to identify such a phase in a realistic Mott insulator via an effective QDM is very promising.
We acknowledge useful discussions with P. Fazekas, M. Ferrero, and K. Penc. This work was supported by the Swiss National Fund and by MaNEP. F.B. is supported by CNR-INFM and MIUR (COFIN 2005).
Derivation of the model {#derivation}
=======================
LiNiO$_2$ is a layered compound in which the Ni$^{3+}$ ions are in a low spin S=1/2 state with a two-fold orbital degeneracy. A fairly general description of this system is given by a Kugel-Khomskii Hamiltonian defined in terms of two hopping integrals $t_h$ and $t^\prime_h$, the on-site Coulomb repulsion $\tilde U$ and the Hund’s coupling $J$ which, on a given bond, takes the form [@vernay]
$$\begin{aligned}
\mathcal{H}_{ij} &=&
-\frac{2}{\tilde U\!+\!J} \left[
2 t_h t^\prime_h {\bf T}_i {\bf T}_j
- 4 t_h t^\prime_h T^y_i T^y_j
+ (t_h-t^\prime_h)^2 ({\bf n}_{ij}^z{\bf T}_i) ({\bf n}_{ij}^z{\bf T}_j)\right.\nonumber\\
&&\left. + \frac{1}{2} (t_h^2-{t^\prime_h}^2) \left( {\bf n}_{ij}^z{\bf T}_i
+ {\bf n}_{ij}^z{\bf T}_j \right) + \frac{1}{4}(t_h^2+ {t^\prime_h}^2)
\right] \mathcal{P}_{ij}^{S=0}
\nonumber\\
&& -\frac{2}{\tilde U -J} \left[
4 t_h t^\prime_h T^y_i T^y_j
+ \frac{1}{2}(t_h^2+ {t^\prime_h}^2)
+ \frac{1}{2} (t_h^2-{t^\prime_h}^2) \left(
{\bf n}_{ij}^z{\bf T}_i + {\bf n}_{ij}^z {\bf T}_j
\right)
\right] \mathcal{P}_{ij}^{S=0}
\nonumber\\
&& -\frac{2}{\tilde U\!-\!3J} \left[ - 2 t_h t^\prime_h {\bf T}_i {\bf T}_j
-
(t_h-t^\prime_h)^2 ({\bf n}_{ij}^z{\bf T}_i) ({\bf n}_{ij}^z{\bf T}_j)+
\frac{1}{4}(t_h^2+{t^\prime_h}^2)
\right] \mathcal{P}_{ij}^{S=1}
\label{eq:effham3}\end{aligned}$$
with the usual definitions for the projectors on the singlet and triplet states of a pair of spins: $$\mathcal{P}_{ij}^{S=0} = \frac{1}{4} - {\bf S}_i{\bf S}_j
\quad\mbox{and}\quad
\mathcal{P}_{ij}^{S=1} = {\bf S}_i{\bf S}_j+\frac{3}{4},$$ The vector ${\bf n}_{ij}^z$ depends on the type of bond. With the convention of Fig. \[tri\], they are given by: $$\begin{array}{c}
{\bf n}_{12}^z=(0,0,1)\\
{\bf n}_{13}^z=(\frac{\sqrt{3}}{2},0,-\frac{1}{2})\\
{\bf n}_{23}^z=(-\frac{\sqrt{3}}{2},0,-\frac{1}{2}).
\end{array}$$ The operators ${\bf T}_i$ are pseudo-spin operators acting on the orbitals.
On a given bond (see Fig. \[tri\]), a dimer is defined by the following wave function: $$|\Phi_{ij}\rangle=|{\phi}_{ij}^\sigma\rangle\otimes|{\phi}_{ij}^\tau\rangle,$$ where $|\phi_{ij}^\sigma\rangle$ and $|\phi_{ij}^\tau\rangle$ are respectively the spin and orbital components. The spin component is the usual singlet given by: $$|\phi_{ij}^\sigma\rangle=\alpha \left(|\uparrow_i\downarrow_j\rangle-
|\downarrow_i\uparrow_j\rangle\right),$$ regardless of the orientation of the bond. We have denoted by $\alpha$ the normalization coefficient, whose explicit value is of course given by $1/\sqrt{2}$, to be able to keep track of the order in $\alpha$ of various overlaps. The orbital part depends on the bond and is given by: $$\begin{aligned}
|\phi_{12}^\tau\rangle&=&|a_1\rangle|a_2\rangle\\
|\phi_{13}^\tau\rangle&=&|-\frac{1}{2}a_1-\frac{\sqrt{3}}{2}b_1\rangle
|-\frac{1}{2}a_3-\frac{\sqrt{3}}{2}b_3\rangle\\
|\phi_{23}^\tau\rangle&=&|-\frac{1}{2}a_2+\frac{\sqrt{3}}{2}b_2\rangle
|-\frac{1}{2}a_3+\frac{\sqrt{3}}{2}b_3\rangle\end{aligned}$$ with the convention of Fig. \[tri\], and with $|a\rangle=|d_{3z^2-r^2}\rangle$ and $|b\rangle=|d_{x^2-y^2}\rangle$. We also use the convention that the wave function has a plus sign if the two sites are in the order defined by the arrows of Fig. \[tri\], and a minus sign otherwise.
![\[tri\] Sign convention for a triangle](fig8.eps){width="15.00000%"}
![\[b4\] A 4-dimer loop](fig9.eps){width="20.00000%"}
![\[b6\] The 3 types of 6-dimer loops](fig10.eps){width="40.00000%"}
With these definitions, the matrix element of the Hamiltonian on each bond is given by: $$\langle \Phi_{ij}|{\cal H}_{ij}|\Phi_{ij}\rangle=\epsilon\ 2\alpha^2=
\epsilon\ \langle \Phi_{ij}|\Phi_{ij}\rangle,$$ where $\epsilon$ is defined by: $$\epsilon=\frac{-4\tilde Ut_h^2}{\tilde U^2-J^2}<0$$
According to the mean-field analysis of Ref. , the wave functions obtained as tensor products of these wave functions on all dimer coverings on the triangular lattice should constitute a good variational basis if $t^\prime_h\ll t_h$ and $J\ll \tilde U$. In the following, we consider for simplicity the case $t^\prime_h=0$ and $J=0$.
Following Rokhsar and Kivelson, [@rokhsar] the idea is now to perform a unitary transformation to derive the effective QDM. If one defines the overlap matrix by $S_{mn}=\langle\Psi_m|\Psi_n\rangle$, where $|\Psi_m\rangle$ and $|\Psi_n\rangle$ are dimer coverings, the states defined by $$| m \rangle = \sum_n \left(S^{-\frac{1}{2}}\right)_{m,n} | \Psi_n \rangle$$ constitute an orthonormal basis and the matrix elements of the Hamiltonian in this basis are given by: $$\label{hamilteff}
{H}^{\textrm{eff}}_{mn} \equiv \langle m| {\cal H} |n \rangle =\sum_{kl}\left(S^{-\frac{1}{2}}\right)_{mk}
\langle\Psi_k|{\cal H}|\Psi_{l}\rangle
\left(S^{-\frac{1}{2}}\right)_{ln}.$$ The inverse of the square root of the overlap matrix cannot be calculated exactly, but this can be done approximately in the context of an expansion in powers of $\alpha$. Indeed, the overlap matrix can be expanded as: $$\label{overlap}
S= I + {2} A \alpha^4+ {2} B \alpha^6+O[\alpha^8],$$ which leads to: $$\left(S\right)^{-\frac{1}{2}}= I - A\alpha^4 - B\alpha^6+O[\alpha^8].$$ In these expressions, the matrices $A$ and $B$ only have non vanishing matrix elements, equal to $1$, between configurations that are the same except on a 4-dimer loop (see Fig. \[b4\]) or on one of the three types of 6-dimer loop (see Fig. \[b6\]), respectively.
[|\*[5]{}[c|]{}]{} & loop-4 & loop-6 (1)& loop-6 (2) & loop-6 (3)\
$\langle m | {\cal H} | n \rangle$ & $- \frac{66 \epsilon \alpha^4}{16^2}$ & $- \frac{103 \epsilon \alpha^6}{16^2}$ & $- \frac{24 \epsilon \alpha^6}{16^2}$ & $- \frac{177 \epsilon \alpha^6}{16^2}$\
& & & &\
& $\simeq -0.0644 \epsilon$ & $\simeq -0.0503 \epsilon$ & $\simeq -0.0012 \epsilon
$ & $\simeq -0.0864 \epsilon$\
Similarly, the Hamiltonian matrix $\tilde{\cal H}$ defined by $\tilde{\cal H}_{mn}=\langle\Psi_m|{\cal H}|\Psi_n\rangle - \epsilon N_d
\delta_{mn}$, where $N_d$ is the number of dimers, has an expansion in powers of $\alpha$ that reads: $$\label{expansionH}
\tilde{\cal H}= C\alpha^4+ D\alpha^6+O[\alpha^8],$$ where the matrices $C$ and $D$ have non vanishing matrix elements under the same conditions as matrices $A$ and $B$.
Since all these expansions start with $\alpha^4$, it is clear that the first contribution to the diagonal part of ${ H^{\textrm{eff}}}$ will be of order $\alpha^8$. So, to order $\alpha^6$, the effective Hamiltonian will only have off-diagonal matrix elements. Moreover, to this order, these matrix elements are simply given by $\langle m|\tilde{\cal H}|n\rangle=\langle\Psi_m|\tilde{\cal H}|\Psi_n\rangle$. They are tabulated in Table \[tablo\] for configurations that are the same except on a 4-dimer loop (Fig. \[b4\]) or on one of the three types of 6-dimer loop (Fig. \[b6\]). All these matrix elements are negative. Among the 6-dimer loop terms, the matrix element of type (3) is the largest, and its ratio to the 4-dimer loop term is 1.34.
[99]{} D.S. Rokhsar and S.A. Kivelson, , 2376 (1988). P.W. Leung and K.C. Chiu, , 12938 (1996). O.F. Syljuåsen, , 020401(R) (2005). R. Moessner and S.L. Sondhi, , 1881 (2001). A. Ralko, M. Ferrero, F. Becca, D. Ivanov, and F. Mila, , 224109 (2005). L. Balents, M.P.A. Fisher, and S.M. Girvin, , 224412 (2002). G. Misguich, D. Serban, and V. Pasquier, , 137202 (2002). F. Mila, , 2356 (1998). M.E. Zhitomirsky, , 214413 (2005). K.I. Kugel and D.I. Khomskii, Sov. Phys. Usp. [**25**]{}, 232 (1982). F. Vernay, K. Penc, P. Fazekas, and F. Mila, , 014428 (2004). B. Bernu, P. Lecheminant, C. Lhuillier, L. Pierre, , 10048 (1994). P.W. Anderson, Mater. Res. Bull. [**8**]{}, 153 (1973). P. Fazekas and P.W. Anderson: Philos. Mag. [**30**]{}, 423 (1974). A. Ioselevich, D.A. Ivanov, and M.V. Feigel’man, , 174405 (2002). R. Moessner and S.L. Sondhi, , 224401 (2001).
|
SACLAY-T03/053
[**Lepton Electric Dipole Moments\
.2cm from Heavy States Yukawa Couplings** ]{}
0.5cm
**Isabella Masina**
*Service de Physique Théorique [^1] , CEA-Saclay*
*F-91191 Gif-sur-Yvette, France*
**Abstract**
In supersymmetric theories the radiative corrections due to heavy states could leave their footprints in the flavour structure of the supersymmetry breaking masses. We investigate whether present and future searches for the muon and electron EDMs could be sensitive to the CP violation and flavour misalignment induced on slepton masses by the radiative corrections due to the right-handed neutrinos of the seesaw model and to the heavy Higgs triplets of $SU(5)$ GUT. When this is the case, limits on the relevant combination of neutrino Yukawa couplings are obtained. Explicit analytical expressions are provided which accounts for the dependencies on the supersymmetric mass parameters.
.3cm
Introduction
============
In low energy supersymmetric extensions of the Standard Model (SM), unless sparticle masses are considerably increased, present limits on flavour violating (FV) decays and electric dipole moments (EDMs) respectively allow for a quite small amount of fermion-sfermion misalignment in the flavour basis and constrain the phases in the diagonal elements of sfermion masses, involving the parameters $\mu$ and $A$, to be rather small. The bounds on the supersymmetric contribution to lepton (L)FV decays and EDMs have the advantage, as compared to the corresponding squark sector ones, of being not biased by the SM contribution - nor by the non-supersymmetric seesaw [@seesaw] contribution [@pet]. Experimental limits on LFV decays and EDMs are then a direct probe of the flavour and CP pattern of slepton masses - see e.g. Ref. [@ms1] for a recent collection. From a theoretical perspective, understanding why CP phases and deviations from alignment are so strongly suppressed is one of the major problems of low energy supersymmetry, the [*CP and flavour problem*]{}. On the other hand, precisely because FV decays and EDMs provide strong constraints, they can share some light on the features of the (possibly) supersymmetric extension of the SM.
Indeed, it is well known that even if these CP phases and misalignments were absent - or suppressed enough - from the effective broken supersymmetric theory defined at $M_{Pl}$, they would be generated at low energy by the RGE corrections due to other flavour and CP violating sources already present in the theory, in particular from the Yukawa couplings of heavy states like the right-handed neutrinos of the seesaw model [@bormas] and the Higgs triplets of $SU(5)$ grand unified theories (GUT) [@barb]. Effects of radiative origin have the nice features of being naturally small, exactly calculable once specified a certain theory and - most interestingly - if experiments are sensitive to them, they yield limits on some combination of Yukawa couplings [^2].
Recently, it has been stressed that the present (planned) limit on $\meg$ [@megexpp] ($\tmg$ [@tmgexpf]) is sensitive to the lepton-slepton misalignments induced by the radiative corrections in the framework of the supersymmetric seesaw model and that the associated constraints on neutrino Yukawa couplings have a remarkable feedback on neutrino mass model building [@lfv_ss; @imsusy02]. However, LFV decays cannot provide any information on the pattern of CP violation of slepton masses, while EDMs are sensitive to both LF and CP violations. Since the present sensitivities to the electron and muon EDMs, $d_e < 10^{-27}$ e cm [@deexpp] and $d_\mu < 10^{-18}$ e cm [@dmuexpp], could be lowered by planned experiments by up to three to five [@deexpf; @lam] and six to eight [@dmuexpf; @dmuexpff] orders of magnitude respectively, it is natural to wonder [*whether lepton EDMs would explore the range associated to LF and CP violations of radiative origin*]{}.
In this work we analyze the predicted range for $d_e$ and $d_\mu$ when the radiative corrections due to the right-handed neutrinos of the seesaw model and, possibly, the Higgs triplets of $SU(5)$ GUT provide the main source of CP violation and misalignment in slepton masses. This allows to extract many informations. If experimental searches could explore this range, one could obtain limits on the imaginary part of the relevant combination of neutrino Yukawa couplings. Moreover, the eventual discovery of a lepton EDM in this range might be interpreted as indirectly suggesting the existence of such a kind of fundamental particles which are too heavy to be more manifest. On the other hand, finding $d_e$ and/or $d_\mu$ above this predicted range would prove the existence of a source of CP (and likely also LF) violation other than these heavy states.
The pure seesaw case has been considered in Ref. [@ellis-1] - see also the related studies [@ellis-2; @ellis-n] on specific seesaw textures -, where it has been pointed out that threshold effects due to hierarchical right-handed neutrino masses enhance the radiatively-induced ${\cal I}m(A_{ii})$, $i=e,\mu,\tau$. By extensively reappraising this framework, we find that for $\tgb \gtrsim 10$ the amplitude with a LL RR double insertion - proportional to $\tan^3 \beta$ - dominates over the one involving ${\cal I}m(A_{ii})$ - insensitive [^3] to $\tgb$ -, the exact ratio depending on the particular choice of supersymmetric mass parameters. We provide general expressions from which it is easy to recognize the model dependencies and which complete previous analyses. Lepton EDMs are then strongly enhanced in models with large $\tan \beta$. To see how close experiments are getting to the seesaw induced lepton EDMs range, we compare the upper estimates for the radiatively-induced misalignments and CP phases with the corresponding present and planned experimental limits collected in [@ms1].
The range for a seesaw-induced $d_\mu$ turns out to be quite far from present searches and, in particular, its eventual discovery above $10^{-23}$ e cm would prove the existence of a source of CP violation other than the neutrino Yukawa couplings. On the contrary, the present sensitivity to $d_e$ explores the seesaw-induced range in models with large $\tgb$ and small R-slepton masses, say $m_R$ around $100-200$ GeV. The planned improvements for $d_e$ would allow to test also models with $m_R$ up to the TeV region and moderate $\tgb$. When present or planned experimental limits turns out to overlap with these allowed ranges, bounds on the imaginary part of the relevant combinations of neutrino Yukawa couplings are obtained and plotted in the plane $(\ti M_1, m_R)$, respectively the bino and the average R-slepton masses at low energy. These plots allow to check whether any particular seesaw model is consistent with present data and, if so, which level of experimental sensitivity would test it. In any case, an experimentally interesting contribution requires hierarchical right-handed neutrino masses.
This feature is no more necessary when, in addition to the seesaw, a stage of $SU(5)$ grand unification is present above the gauge coupling unification scale. It is well known that the main drawback of minimal $SU(5)$ is that the value of the triplet mass, $M_T$, required by gauge coupling unification is sizeably below the lower bound on $M_T$ derived from proton lifetime (see for instance [@isa_ijmpa] for recent reviews and references). In our analysis we therefore keep the triplet mass as a free parameter. We nevertheless exploit the minimal $SU(5)$ relations between the doublet and triplet Yukawa couplings since in general they are mildly broken in non-minimal versions of $SU(5)$. The simultaneous presence of right-handed neutrinos and heavy triplets turns out to further enhance the amplitude with the LL RR double insertion over the one with ${\cal I}m(A_{ii})$, essentially unaffected by triplets. This cannot be derived by the - otherwise elegant - technique based on the allowed invariants [@romstr].
When right-handed neutrinos and Higgs triplets are simultaneously present, the predicted range for the radiatively-induced $d_e$ is already sizeably excluded by the present experimental limit. This in turn is translated into a strong constraint on the imaginary part of the combination of Yukawa couplings which is relevant for the LL RR amplitude. Such a constraint could be hardly evaded even for large values of the triplet mass and unfavorable supersymmetric mass parameters. The radiatively-induced $d_\mu$ in the presence of triplets should not exceed $10^{-23}$ e cm, as was the case for the pure seesaw. However, at difference of the latter case, in the former one $d_\mu$ only mildly depends on the spectrum of right-handed neutrinos. Notice that planned searches for $d_\mu$ could constrain (depending on the triplet mass mass, of course) the imaginary part of the combination of Yukawa couplings whose absolute value could be independently constrained by the LFV decay $\tmg$.
The paper is organized as follows. In Section 2 we introduce our notations, discuss the framework and make some preliminary considerations. Section 3 considers the pure seesaw case by separately analyzing the flavour conserving (FC) and flavour violating (FV) amplitudes contributing to lepton EDMs. In Section 4 the framework of seesaw and $SU(5)$ is discussed along the same lines. Concluding remarks are drawn in Section 5. Finally, in Appendix A and B we collect the RGE in the case of the seesaw, without and with a minimal $SU(5)$ unification respectively.
Framework and Method {#sec2}
====================
In this section we recall the expression for the lepton EDMs in the mass insertion approximation [@massins; @moroi; @prs; @fms; @ms1] to display the supersymmetric mass parameters that are constrained by the present and projected searches for $d_e$ and $d_\mu$. We then draw some preliminary considerations to introduce our procedure to calculate the radiative 1-loop contribution to these mass parameters. The relevant RGE can be found in the Appendix.
We adopt here the following conventions for the $6 \times 6$ slepton mass matrix in the lepton flavour (LF) basis where the charged lepton mass matrix, $m_{\ell}$, is diagonal: (
[cc]{} \_L\^& \_R\^
) (
[cc]{} m\_[LL]{}\^2 & A\_e\^v\_d - m\_ A\_e v\_d - \^\* m\_ & m\_[RR]{}\^2
) (
[c]{} \_L \_R
) \[sleptonm2\] where $A_e$ is the $3 \times 3$ matrix of the trilinear coupling, the $A-$term. All deviations from alignment in this mass matrix are gathered in the $\d$ matrices, which contain 30 real parameters (including 12 phases) and are defined as: m\^2\_[LL]{}=m\_L\^2 ( + \^[LL]{} ) & &m\^2\_[RR]{}=m\_R\^2 ( + \^[RR]{} )\
A\_e\^v\_d - m\_ &=& (A\_\^\* v\_d - m\_ ) + m\_L m\_R \^[LR]{} where $m_L , m_R$ are average masses for L and R sleptons respectively and $A_{\ell \ell} \sim O(m_{susy} m_\ell/ v_d )$ are the diagonal elements of $A_e$, so that $\delta ^{LR}$ has only non-diagonal, flavour violating, elements.
The supersymmetric contributions to $d_i$, $(i=e,\mu,\tau)$, can be splitted in two parts, involving respectively only flavour conserving (FC) or flavour violating (FV) elements of the slepton mass matrix (\[sleptonm2\]): d\_i = d\_i\^[FC]{} +d\_i\^[FV]{} d\_i\^[FC]{} &= & \[ m\_i [I]{}m() ( I\_B + I\_L - I\_R + I\_2 ) - v\_d [I]{}m(A\_[ii]{}\^\*) I\_B \] \[FC\]\
d\_i\^[FV]{} &= & \[FV\] where $\eta_{\ell} \equiv \i3 - A_{\ell \ell} v_d/ (m_{\ell}\mu \tan \beta) \label{Mapp}$, the functions $I$ are defined as in [@ms1] and terms that are less relevant [^4] or higher order in the $\d$’s matrix elements are omitted. Notice that $\eta_\ell \approx \i3$ for relatively large values of $\tgb$, favored in mSUGRA and for which the LF and CP violations are most likely to be detected. The FC and FV contributions could result from different seeds of CP violation but could also be correlated in many different ways in models. Anyway, the experimental limit can be put on both because, due to the different nature of the many parameters involved, an eventual cancellation between these contributions appears unnatural. The distinction between the FC and FV contributions is also phenomenologically relevant: some of the $|\delta |$’s in the FV terms are already constrained to be smaller than $O(1)$ by LFV decays, while ${\cal I}m(\mu)$ and ${\cal I}m(A_{i i})$ are directly constrained by the EDMs.
Our aim here is to estimate the radiative contribution to the lepton EDMs induced by the seesaw interactions, first alone and subsequently accompanied by a stage of $SU(5)$ GUT. Since the present experimental bounds on LFV decays and EDMs already point towards family blind soft terms ( sparticles with the same quantum numbers must have the same soft terms) with very small CP phases, at the cut-off $\Lambda = M_{Pl}$ corresponding to the decoupling of gravitational interactions we assume real and flavour blind soft terms, namely in eq. (\[sleptonm2\]), m\^2\_[LL]{} = m\^2\_[RR]{} = 02 i3 , A\_e = y\_e a\_0 , with real $a_0$, $m_0$ and $\mu$ term. In the next sections we separately study the FC and FV contributions to $d_i$ and obtain explicit approximate expressions for ${\cal I}m(A_{i i})$ and the products of $\d$’s in (\[FV\]). The effects of a more general family independence assumption are important but not crucial and can be easily included in our analysis. By means of these general approximated expressions we will:\
i) derive the upper prediction for the radiatively induced leptonic EDMs, stressing the model dependences;\
ii) compare it with the experimental limits;\
iii) when allowed by the experiment, obtain an upper bound on the imaginary part of the relevant combination of Yukawa couplings.
A couple of preliminary considerations are in order before presenting the results of the next sections.
On the naive scaling relation
-----------------------------
In the limit that all slepton masses are family independent, the FV contribution vanishes and the FC one is proportional to the mass of the $i-$th lepton (${\cal I}m (A_{ii}) v_d \approx {\cal I}m (a_0) m_{i}$) leading, except an accidental cancellation [^5] with the $\mu$-term amplitude, to the “naive” scaling relation d\_i/d\_j = m\_i/m\_j . \[naive\] Then, due to the present experimental limit on $d_e$, $d_\mu$ could not exceed $d_e m_\mu/m_e \sim 2 \cdot 10^{-25}$ e cm, which roughly corresponds to the planned sensitivity and, if the limit on $d_e$ were still to be lowered, next generation experiments would have no chance of measuring $d_\mu$.
Such considerations could provide interesting informations because (\[naive\]) strictly apply only to the FC $\mu$-contribution to the EDM, while in general both ${\cal I}m (A_{ii})$ and the FV terms may strongly violate it. This is the case for the radiatively induced ${\cal I}m (A_{ii})$. Some of the FV contributions are instead naturally proportional to a different lepton mass, $m_k$, possibly heavier than $m_i$, as discussed in [@fms; @romstr] - and, before, for the quark sector, in [@bs]. In particular, it will turn out in the next sections that the FV contribution can even take over the FC one. Hence, a value of $d_\mu$ above $\sim 2 \cdot 10^{-25}$ e cm is a possibility that deserves experimental tests and, interestingly enough, it would imply the source of lepton EDM being either the FV contribution or a non-universal ${\cal I}m(A_{ii})$, so providing a remarkable hint for our understanding of CP violation.
On the relevant combination of Yukawa couplings
-----------------------------------------------
Once a theory is specified, it is relatively easy to list the combinations of Yukawa couplings appearing in the slepton mass radiative corrections that may contribute to lepton EDMs. However, the actual calculation is more involved as we now turn to discuss.
As an example, let us look for ${\cal I}m(A_{ii})$ by studying the evolution of $A_e$ in the case of degenerate right-handed neutrino masses, $\bar M$, for simplicity. Let us first consider the case of the pure seesaw and define [^6] the hermitian matrices $E \equiv y_e^{\dagger} y_e$, $N \equiv y_{\nu} ^{\dagger} y_{\nu}$. Notice that $E$ is real and diagonal in the LF defining basis and $N$ is diagonalized by a unitary matrix similar to the CKM one with only one phase (even for non-degenerate right-handed neutrinos). By solving the RGE for $A_e$, eq. (\[Ae\_ss\]), linearly in $t_3 \equiv 1/(4 \pi)^2$ $\ln(M_{Pl}/\bar M)$, $A_e$ can only be proportional to $y_e E$ and $y_e N$, whose diagonal elements are real. At $O(t_3^2)$ only $y_e EE, y_e NN, y_e NE$ and $y_e EN$ appear, whose diagonal elements are again real. A potential ${\cal I}m(A_{ii})$ shows up at $O(t_3^4)$, through ${\cal I}m(y_e N [N,E] N)_{ii}$. On the other hand, when also Yukawa interaction of the $SU(5)$ triplets are present, eq. (\[Ae\_su5\]) shows that, at first order in $t_T \equiv 1/(4 \pi)^2$ $\ln(M_{Pl}/ M_T)$, $A_e$ can be proportional to $y_e E, y_e N$ but also to $U^* y_e$, where $U \equiv y_u^\dagger y_u$. The latter have real diagonal elements but allow at $O(t_T^2)$ for a combination with diagonal imaginary part, namely $(U^* y_e N)_{ii}$.
Accordingly, in order to evaluate the actual coefficients in front of the products of Yukawa coupling matrices which are likely to have phases, we solve the RGE for the soft parameters by a Taylor expansion in the small parameters $t_{if}=1/(4 \pi)^2 \ln(Q_i/Q_f)$ associated to the intervals between the successive decoupling thresholds of the various heavy states. For instance, by integrating the RGE for $A_e$ in the case of $SU(5)$ plus seesaw, a non vanishing coefficient is obtained for the combination ${\cal I}m(U^* y_e N )_{ii}$ at $O(t_T^2)$. Now, lepton FV transitions and CP phases are naturally defined in the LF basis where $y_e$ and the Majorana masses $M_R$ are diagonal and real. This basis is not invariant under the RGE evolution and one should diagonalize $y_e$ and $M_R$ again at the lower scale. Therefore, one has to find out the effect of these final rotations.
We adopt the rotating basis method introduced in Ref. [@bs], where the RGE are modified to incorporate the fact that the matrices are defined in the LF basis at each scale, so that $y_e$ and $M_R$ are always diagonal. It is worth to stress that, even if the rotations needed to diagonalize $y_e$ are small, their effect could be crucial for ${\cal I}m(A_{ii})$ and the four products of $\d$’s in eq. (\[FV\]). For instance, in the case of $SU(5)$ plus seesaw it turns out that this correction exactly cancels the term proportional to $t_T^2 {\cal I}m( U^* y_e N )_{ii}$ in $A_e$. On the contrary, these rotations can be safely neglected when deriving approximate expressions for the radiatively-induced LFV decays because the latter are only sensitive to absolute values of $\d$’s.
Lepton EDMs and Seesaw
======================
In this section we consider the predictions for lepton EDMs in the context of the supersymmetric extension of the seesaw model, namely we assume the MSSM supplemented with the seesaw Yukawa interactions and Majorana masses for the right-handed neutrinos as the effective theory valid up to the cut-off $\Lambda = M_{Pl}$ where gravitational interactions decouple. Starting with real and universal boundary conditions at $M_{Pl}$, we solve the RGE displayed in Appendix A by expanding in the small parameters defined by the right-handed neutrino thresholds t\_3= t\_2= t\_1= ,\[thresholds\] with the ordering $M_3 > M_2 >M_1$. We thus obtain approximate analytic expressions for the radiatively induced $\delta$’s and ${\cal I}m(A_{ii})$ which depend on $m_0$, $a_0$ and the Yukawa couplings defined in the LF basis at the scale $\Lambda$. Of course the latter can be immediately translated into the corresponding ones defined at any scale, $M_1$ or $m_{susy}$.
Precisely because of the lepton flavour and CP violating Yukawa couplings, the LF basis is continuously rotated and rephased with the RGE evolution and, as already discussed, we handle this by working in a rotating basis. It turns out that the basis transformation introduces negligible corrections in the FC terms (\[FC\]) but important ones in the FV terms (\[FV\]). Notice also that the seesaw effects stop at the decoupling threshold of the lightest right-handed neutrino, $M_1$, and that the RGE evolution of these effects down to the supersymmetric scales where CP and lepton FV transitions are estimated is generically small and can be neglect for our estimates. We now study separately the FC and the FV contributions.
Flavour conserving contribution
-------------------------------
Starting in the LF basis at $M_{Pl}$ from $A_e = y_{e} a_0$, the seesaw interactions generate a ${\cal I}m(A_{ii})$ in the LF basis at $M_1$. At leading order in the $t$’s and defining $y_\nu^\dagger P_a y_\nu \equiv \a$ ($a=1,2,3$) and the right-handed neutrinos projectors $P_1={\rm diag}(1,0,0)$, $P_2={\rm diag}(1,1,0) $, $P_3 = \i3$, the latter reads [^7]: m (A\_[ii]{}) = 8 a\_0 y\_[e\_i]{} ( t\_2 t\_3 [I]{}m( )\_[ii]{} + t\_1 t\_3 [I]{}m( )\_[ii]{} + t\_1 t\_2 [I]{}m( )\_[ii]{} ) , \[ss\_ae\] where the Yukawas in the r.h.s. are evaluated at $\Lambda$. A similar [^8] formula were previously presented in Refs. [@ellis-1; @ellis-2; @ellis-n], but with the various Yukawas involved defined at different scales. Then, one must be more cautious in drawing general conclusions and the authors validate theirs with specific numerical examples. Our approach offers the advantage that the corresponding results become more transparent and allow for an easier estimate of the effects once a pattern of seesaw parameters is assigned. Anyway, the crucial point [@ellis-1] is that the more right-handed neutrinos are hierarchical, the more the FC contribution increases. Indeed, it vanishes in the limit that right-handed neutrinos are degenerate, in which case a contribution to ${\cal I}m(A_{ii})$ only appears at fourth order, proportional to ${\cal I}m(y_e N[N,E]N)_{ii}$. Notice also that the naive scaling relation is generally violated [@ellis-1] by (\[ss\_ae\]).
Eq. (\[ss\_ae\]) displays a linear dependence on the unknown parameter $a_0$. By defining $A_{ii} \equiv |A_{ii}| e^{i 8 \phi_{A_i}}$, then $ \phi_{A_i} \approx t_2 t_3 {\cal I}m( \2 \3)_{ii} + $ $t_1 t_3 {\cal I}m(\1 \3 )_{ii} + $ $ t_1 t_2 {\cal I}m( \1 \2 )_{ii} $ is completely specified by the seesaw parameters. However, as appears from eq. (\[FC\]) the experimental limit on $d_i$ doesn’t probe directly $\phi_{A_i}$, rather it gives a bound on ${\cal I}m(A_{ii}) v_d \equiv m_{i} {\cal I}m(a_{i})$ once supersymmetric masses are fixed (and up to unnatural conspiracies between the various amplitudes). Figs. \[Fph\_A\], taken from Ref. [@ms1], show the present upper limits on $| {\cal I}m (a_e) |/m_R$ and the planned ones for $| {\cal I}m (a_\mu) |/m_R$ in the plane $(\ti M_1, m_R)$, respectively the bino and R slepton mass at $m_{susy}$. Notice that these limits are quite model independent because, apart from $m_R$ and $\ti M_1$, there is only a mild dependence on $m_L$, which we have fixed as in mSUGRA for definiteness. Indeed, since the $A$-term amplitude in $d_i^{FC}$ arises from pure bino exchange, it does not involve [^9] $\mu$ nor $\tan \beta$. In mSUGRA there is an unphysical region in the plane $(\ti M_1, m_R)$ corresponding to $m_0^2 < 0$ and which has been indicated in light grey in the plots. Anyway, the dark grey region and below is also excluded because $m_R \le \ti M_1$, in contrast with the requirement of neutrality for the LSP.
.5 cm
(-250,250)[Experiment]{} (-190,200)[$ r \ll 1 \downarrow $]{} (-15,118)[$ \swarrow r \approx 1 $]{} (-300,62)[ $ \left( \times \frac{d_e [{\rm e~cm}]}{10^{-27}} \right) $]{} (-86,62)[ $ \left( \times \frac{d_\mu [{\rm e~cm}]}{10^{-24}} \right) $ ]{}
To find an upper estimate for the seesaw induced $d_i^{FC}$, let us evaluate $| {\cal I}m (a_i) |/m_R$ from eq. (\[ss\_ae\]) by considering only its first term. This situation is representative because the terms proportional to $t_1$ are negligible when the lightest right-handed neutrino has smaller Yukawa couplings, as happens in many models and as one would guess from similarity with the charged fermion sectors. Anyway, the following discussion is trivially adapted to other cases. For definiteness, we adopt this set of reference threshold values: $ M_2=10^{12}$ GeV, $M_3=10^{15}$ GeV, $\Lambda=2~10^{18}$ GeV. Since, as demanded by perturbativity, $(\2 \3)_{ii} \le O(1)$, O ( 0.02 ) . \[ae\_ss\] This upper estimate is easily adapted to any given model once the relation between $a_0$ and $m_R$ is made explicit. To make the dependence more manifest, let us introduce the following two mSUGRA situations: a) $a_0^2 = 2 m_0^2$; b) $a_0^2 =\ti M_{1/2}^2$, where $\ti M_{1/2}$ is the universal gaugino mass at the gauge coupling unification scale. In the first case $(a_0/m_R)^2 \approx 2 (1-0.9 r^2)$, with $r \equiv \ti M_1/m_R$. The ratio $a_0/m_R$ thus displays a mild excursion as it decreases from $\sqrt{2}$ down to $0.45$ when moving from the vertical left axes, where $r \ll 1$, to the joining line of the dark grey region, where $r =1$. For case b), the situation is opposite and not so mild: $(a_0/m_R) \approx 2.5~ r$. In the more natural cases in between, the ratio $a_0/m_R$ should thus be rather stable.
Let us now compare the upper estimate (\[ae\_ss\]) with the experimental bound. As appears from eq. (\[FC\]), the latter improves linearly with the experimental sensitivity to $d_i$. Taking $a_0 \sim m_R$, figs. (\[Fph\_A\]) show that values around $0.02$ for $| {\cal I}m (a_{i}) |/m_R$ would require an experimental sensitivity to $d_e$ and $d_\mu$ respectively at the level of $10^{-28}-10^{-29}$ e cm and $2~10^{-26}-2~10^{-27}$ e cm, the exact value depending on the particular point of the $(\ti M_1, m_R)$ plane. However, to avoid charge and color breaking, the constraint $a_0/m_R \le 3$ in general applies. Thus, focusing for instance around $m_R \approx 500$ GeV, the FC contribution cannot exceed $\sim10^{-28}$ e cm for $d_e$ and $\sim 2~ 10^{-26}$ e cm for $d_\mu$, even with highly hierarchical right-handed neutrinos. Notice that the former value could be at hand of future experimental searches for $d_e$, while the latter is at the very limit of the planned experimental sensitivity to $d_\mu$. Allowing for smaller $m_R$ values, $m_R \approx 200$ GeV, the FC seesaw upper estimate comes close to the present limit for $d_e$ while it cannot be more than $\sim 2~10^{-25}$ e cm for $d_\mu$.
The result for $d_\mu$ is thus a kind of “negative” one, since its eventual future discovery above $\sim 2~10^{-25}$ e cm couldn’t be attributed to the radiative FC contribution of the seesaw. Even if the situation for $d_e^{FC}$ appears more optimistic, it is worth to underline that the previous upper estimate actually applies to models with at least four $O(1)$ neutrino Yukawa couplings and large CP phases, like those in Ref. [@ellis-n]. However, as we now turn to discuss, the FV contribution, underestimated by previous analyses, could drastically enhance the $d_i$ upper estimate.
Flavour violating contributions
-------------------------------
In the following we isolate the potentially most important products of $\d$’s and give an estimate of their relative magnitude with respect to the FC amplitude. The relevant approximations for the flavour violating elements, $i \neq j$, of the $\d$’s are: m\_L m\_R \^[LR]{}\_[ij]{} & = & a\_0 m\_[i]{} \[ - 2 t\_3 [N]{}\_[ij]{} + \_[a]{} F\_A(a , a )\_[ij]{} + \_[a>b]{} t\_a t\_b F\_A(a,b)\_[ij]{} \] \[eq\_Aess\]\
m\^2\_R \^[RR]{}\_[ij]{} & = & \_[a]{} F\_R( a , a )\_[ij]{} + \_[a>b]{} t\_a t\_b F\_R(a,b)\_[ij]{} \[eq\_Rss\]\
m\^2\_L \^[LL]{}\_[ij]{} &=& - (6 02 +2 02 ) t\_3 [N]{}\_[ij]{} + \_[a]{} F\_L( a , a )\_[ij]{} + \_[a>b]{} t\_a t\_b F\_L(a,b)\_[ij]{} \[eq\_Lss\] where $a,b=1,2,3$ , the matrix $\cal N$ is defined as $ t_3 {\cal N} \equiv t_3 \3 + t_2 \2 +t_1 \1 $ and F\_A(a,b) &=& 15 { E , } - 5 { E, } + 12 { , } + 4 \[, u\_E\^[(b)]{} \]\
&+& 2 ( ( \_[(d)]{} + D\_e ) +( \_[(d)]{} + D\_e ) ) +4(D\_\^[(a)]{}/a\_0 +D\_\^[(a)]{})\
&+& \[,E\] + 4 \[,\] + 7 \[E,\] ,\
F\_R(a,b) & = & 8 \[ (6 02 + 4 02) y\_e y\_e\^- (6 02 +2 02 ) y\_e y\_e\^\] \[FR\_ss\]\
F\_L(a,b) & =& 2 (6 02 +2 02 ) ( { 3 + E, } + 2 D\_\^[(a)]{} + \[,u\_E\^[(b)]{}\] ) +2 [m\^2]{}’\_[H\_[u]{}]{}\
& +& 4 02 ( { 3 +E , }+{ E,}) + 2 (G\_L+4 a\_0 D\_\^[(a)]{}) \[FL\_ss\] with the Yukawas in the r.h.s. evaluated at the scale $\Lambda$. The subscript $(d)$ indicates to take only the diagonal elements of the matrix, $u_E^{(a)}$ is defined through $[u_E^{(a)},E]=\{E, 3 E+ \a + D_e \}$ and the definition of all the other quantities can be found in Appendix A.
By means of the above expressions, the predictions for the imaginary part of the various product of $\d$’s can be studied. An imaginary part in the products $(\d^{LL} \d^{LR})_{ii}$, $(\d^{LR} \d^{RR})_{ii}$, ..., only arises at third order in at least two different $t$’s.
### The LL RR contribution
Let us firstly discuss the LL RR double insertion, ${\cal I}m(\d^{LL} \mu \eta_\ell m_\ell \d^{RR})_{ii}$, which turns out to be the quantitatively most interesting one. Since in general $\eta_\ell \approx \i3$ and the phase of $\mu$ - if any - is extremely small so that it can be safely neglected in the discussion, ${\cal I}m(\d^{LL} \mu \eta_\ell m_\ell \d^{RR})_{ii} \approx \mu {\cal I}m(\d^{LL}m_\ell \d^{RR})_{ii}$. From eqs. (\[eq\_Rss\], \[eq\_Lss\]) it appears that $\d^{RR}_{ij}$, $\d^{LL}_{ij}$ are respectively of second and first order in the $t$’s. Thus, the lowest order is the cubic and the LL RR contribution reads: m(\^[LL]{} m\_\^[RR]{})\_[ii]{} = 8 m\_[i]{} \_[a>b]{} t\_a t\_b (t\_a+t\_b) [I]{}m(N\_a E N\_b)\_[ii]{} . \[ss\_LLRR\] Notice that ${\cal I}m(N_a E N_b)_{ii} = \tan^2 \beta {\cal I}m(N_a m_\ell^2 N_b)_{ii} /m_t^2$, where $m_t$ is the top mass. Due to the hierarchy in $m_\ell$, a potentially important effect could only come from ${\cal I}m({N_a}_{i3} {N_b}_{3i})$. Notice also that, on the contrary of the FC contribution, (\[ss\_LLRR\]) doesn’t strongly depend on $a_0 $.
The relative importance between this amplitude and the FC one can be easily appreciated by neglecting the terms proportional to $t_1$, which, as already mentioned, is justified when the lightest right-handed neutrino has the smallest Yukawa couplings: = (t\_2 +t\_3) . For realistic values, the ratio of the two loop functions is slightly smaller than one. To obtain a rule of thumb, let us take $m_0 \sim a_0 \sim m_{L,R}$ and the reference ratio $\Lambda/M_2 = 2~10^{6}$. Then, unless ad hoc fine-tunings in the structure of $N$, ${\cal I}m(\3 m_\ell^2 \2)_{ii} \sim m_\tau^2 {\cal I}m(\3 \2)_{ii}$, and one finds \~0.5 [(rule of thumb),]{} \[rot\] where $|\mu_{ew}|^2 \approx 0.5 m_R^2 + 20 \ti M_1^2$ is the value of $\mu$ accounting for radiative electroweak breaking in mSUGRA. Despite being of third order in the $t$’s, the FV amplitude can take over the FC one thanks to its $\tan^3 \beta$ dependence. The precise value of this ratio is displayed in figs. \[Fss\_R\] in the plane $(\ti M_1, m_R)$. Cases a) and b) are separately displayed so that the behavior for any situation in between can be easily extrapolated. The relevant generalizations are also reminded.
(-300,235)[ Theory: $\frac{ d_i^{FV_{LLRR}} } { d_i^{FC} }$ for Seesaw]{} (-335,200)[ a) $a_0^2=2 m_0^2$]{} (-120,200)[ b) $a_0^2= \ti M_{1/2}^2$]{} (-320,62)[ $ \times \frac{\tan^3 \beta}{10^3} \frac{\mu}{\mu_{ew}} \frac{t_2+t_3}{0.1}$]{} (-100,62)[ $\times \frac{\tan^3 \beta}{10^3} \frac{\mu}{\mu_{ew}} \frac{t_2+t_3}{0.1}$ ]{}
1 cm
The plots show that the rule of thumb is quite reliable and allow to extract, for each point of the plane, the value of $\tgb$ for which the FV amplitude takes over the FC one. In both cases a) and b), this happens for $\tan \beta \gtrsim 10$ - the only exception being the region with $r \approx 1$ for case a), where $\tgb \gtrsim 20$ is required.
### The other contributions
The amplitudes with the other products of $\d$’s are less important than the LL RR one. Consider for instance ${\cal I}m(\d^{LR} \d^{RR})_{ii}$. The flavour violating elements in $\d^{RR}$ and $\d^{LR}$ are respectively of second and first order, so that the product appears at third order: m ( \^[LR]{} \^[RR]{} )\_[ii]{} = 8 \_[a>b]{} t\_a t\_b (t\_a+t\_b) [I]{}m(N\_a E N\_b)\_[ii]{} . It vanishes in the limit $a_0 \rightarrow 0$, as the FC contribution. Neglecting the terms proportional to $t_1$, the ratio of the LR RR amplitude and the FC one reads = \^2 (t\_2 +t\_3) . \[LRRR\] It is easy to check that this ratio is smaller than one (unless ad hoc fine-tunings in the structure of $N$): the ratio of the two loop functions is slightly smaller than one for realistic values of the supersymmetric parameters and, taking for instance $m_0=a_0$ and $\Lambda/M_2 = 2 ~10^{6}$, (\[LRRR\]) should not exceed $\sim~10^{-4} \tan^2 \beta$.
For ${\cal I}m(\d^{LL} \d^{LR})_{ii}$, no imaginary part can arise at second order in $t$’s because both $ \d^{LL}$ and $\d^{LR}$ are proportional to $t_3 {\cal N}$. At third order there are many contributions and it is lengthy but straightforward to check that they are proportional to at least two different $t$’s. This contribution is also proportional to $a_0$ and could be comparable to the FC one but is in any case smaller than the LL RR one. The expression for the double LR insertion is also quite involved. It is proportional to $a_0^2$ and, being also suppressed by a factor $m_\ell^2 /m_L^2$ with respect to the LL RR one, it can be safely neglected.
Predicted range and constrains on Yukawas
-----------------------------------------
For values of $\tan \beta \gtrsim 10$, for which the EDMs are enhanced and thus most likely to be observed, the FV amplitude with the LL RR double insertion is generically dominant with respect to all other amplitudes. Then, when $\tgb \gtrsim 10$, all the considerations made previously for $d_i^{FC}$ actually apply to $d_i$ when strengthened by the rule of thumb factor (\[rot\]). To give an example, if $t_2+t_3 \sim 0.1$ and $\mu \sim \mu_{ew}$, $d_e$ cannot exceed $\sim 0.5~10^{-28(-27)} ~\tan^3 \beta/10^3 $ e cm and $d_\mu$ $\sim 10^{-26(-25)}~\tan^3 \beta/10^3$ e cm when $m_R \sim 500 (200)$ GeV. Planned experimental sensitivities to $d_\mu$ could then test the seesaw radiative contribution for models with small $m_R$ and/or large $\tan \beta$, in which case the factor (\[rot\]) could be up to $\sim 50 - 60$ so that $d_\mu$ should not exceed $O(10^{-23})$ e cm. On the contrary, the range of the seesaw induced $d_e$ already overlaps with the present experimental limit for values of $m_R$ up to $1/2$ TeV when $\tgb \sim 30$.
Barring unnatural cancellations, planned searches for $d_e$ could thus test each term of the sum in (\[ss\_LLRR\]), namely each [^10] t\_a t\_b (t\_a+t\_b) [I]{}m(N\_a N\_b)\_[11]{} (a>b) . The effect of an eventual two orders of magnitude improvement for $d_e$ on the upper limit on ${\cal I}m(N_a m^2_\ell N_b)_{11}/m^2_\tau$ is displayed in fig. \[Fss\_INN\] by taking as reference values $\tgb =30$ and $t_a t_b (t_a+t_b) = 2~10^{-4}$. It turns out that planned limits could be severe enough to test models with hierarchical neutrino Yukawa couplings. This cannot be done by present limits. For any given seesaw model, it is straightforward to extrapolate from the plot the level of experimental sensitivity required to test it. Notice also that for $d_\mu$, limits on ${\cal I}m(N_a m^2_\ell N_b)_{22}/m^2_\tau$ as strong as the present ones on ${\cal I}m(N_a m^2_\ell N_b)_{11}/m^2_\tau$ would require a sensitivity to $d_\mu$ at the level of $2~10^{-25}$ e cm.
(-200,245)[ [Upper bound on]{} ${\cal I}m({N_a} \frac{m_\ell^2}{m_\tau^2} {N_b})_{11}$ ]{} (-160,53)[ $\times \frac{d_e[ {\rm e ~cm} ]}{10^{-29}} \frac{30^3}{\tan^3 \beta}
\frac{\mu_{ew}}{\mu} \frac{2 ~10^{-4}}{t_a t_b (t_a + t_b)} $ ]{}
Lepton EDMs, Seesaw and $SU(5)$ Triplets
=========================================
We now add to the supersymmetric seesaw model a stage of a minimal $SU(5)$ GUT above the gauge couplings unification scale, $M_{GUT} \sim 2~ 10^{16}$ GeV. Namely, we include the contribution of the higgs triplets Yukawa interactions to the RGE evolution of slepton masses from $M_{Pl}$ down to their threshold decoupling scale $M_T$, which is very likely to be bigger than $M_3$. Notations are defined in Appendix B.
It is not restrictive to work (at any scale) in the basis where $(y_d^T=)y_e$ and $M_R$ are real and diagonal and $y_u = V^T d_u \phi_u V$, where $d_u$ are the (real and positive) eigenvalues of $y_u$, $V$ is the CKM matrix in the standard parameterization (more on this later) and $\phi_u$ is a diagonal $SU(3)$ matrix. The RGE for the radiatively induced misalignments are written in eqs. (\[rsu5\_y\]) to (\[rsu5\_A\]). At first order in $t_{1,2,3}$ defined in eq. (\[thresholds\]) and t\_T , their solutions at the scale $M_1$ reads: m\^2\_R \^[RR]{} &=& - (6 02 +2 02 ) (2 t\_[1]{} y\_e\^2 + 3 t\_T U\^\*)\
m\^2\_L \^[LL]{} &=& - (6 02 + 2 02 ) ( (3 t\_T + t\_[1]{} ) y\_e\^2 + t\_3 [N]{} ) \[eq\_dsu5\]\
m\_L m\_R \^[RL]{} &=& - a\_0 ( 6 t\_T U\^\* m\_+ 2 m\_t\_3 [N]{} ) where the matrix $\cal N$ is defined as $t_3 {\cal N} \equiv t_3 \3 + t_2 \2 +t_1 \1 $. It is understood that all the quantities in the r.h.s. of (\[eq\_dsu5\]) are evaluated at $\Lambda$. The small effects of the subsequent evolution from $M_1$ down to $m_{susy}$ can be neglected in the following discussion. We explicitly write only the first order terms [^11] in the $t$’s because, contrarily to the seesaw case, they already produce a potential imaginary part for the FV contribution, eq. (\[FV\]). Instead, for the diagonal part of $A_e$ this is not the case, as we now turn to discuss.
Flavour conserving contribution
-------------------------------
As anticipated in the simplified discussion of Section \[sec2\], a potential candidate for ${\cal I}m(A_{ii})$, proportional to ${\cal I}m(U^* y_e N)_{ii}$, could show up at $O(t_T^2)$. It is lengthy but straightforward to see that such term is exactly canceled by the effect of rotating the basis. For the same reason, in the general case with different thresholds $M_{1,2,3}$, $M_T$, the overall coefficient of ${\cal I}m(U^* y_e {\cal N})_{ii}$ is zero. Therefore, the second order contribution to ${\cal I}m(A_{ii})$ is just m(A\_[ii]{}) = 8 a\_0 y\_[e\_i]{} \[ ( t\_T + t\_3 ) t\_2 [I]{}m( )\_[ii]{} + ( t\_T + t\_3 ) t\_1 [I]{}m( )\_[ii]{}\
+ t\_2 t\_1 [I]{}m( )\_[ii]{} \] , \[ae\_ss+su5\] namely the pure seesaw one discussed in the previous section, eq. (\[ae\_ss\]), with the substitution $t_3 \rightarrow (3/2~ t_T + ~t_3 )$, due to the fact that above $M_T$ also the triplets circulate in the loop renormalizing the wave functions. $t_T$ is naturally expected to be small (triplets will not decouple much below $10^{16}$ GeV) so that eventual higher order contributions involving $t_T$ are expected to be negligible with respect to (\[ae\_ss+su5\]). As a result, a stage of $SU(5)$-like grand unification, cannot enhance by much the FC contribution with respect to the pure seesaw case.
Flavour violating contribution
------------------------------
On the contrary, products of two $\delta$’s have an imaginary part proportional to $t_3 t_T $ ${\cal I}m(U^* y_e {\cal N})_{ii}$ and the FV contribution to $d_i$ is potentially bigger than the FC one. Most interestingly, the predicted range for the radiatively induced $d_e$ turns out to have been already sizeably excluded by the present experimental bounds. Planned searches for $d_\mu$ would also get close to test the range corresponding to radiatively induced misalignments.
Let us consider in turn the predictions for the imaginary part of the products of $\d$’s, eq. (\[FV\]), from their expressions given in eq. (\[eq\_dsu5\]). As before, the most important contribution comes out from the LL RR double insertion. Since $\eta_{\ell} \approx \i3$ and the phase of $\mu$ - if any - is experimentally small enough to be safely neglected in the present discussion, one has at the lowest relevant order in the $t$’s m(\^[LL]{} m\_\^[RR]{})\_[ii]{} = 3 t\_T t\_3 [I]{}m( [N]{} m\_U\^\* )\_[ii]{} . \[su5\_LLRR\] This FV contribution is potentially much bigger than the FC one because, apart from Yukawas and numerical coefficients, it is enhanced by a factor $(m_\tau \mu \tgb) /(m_i a_0)$.
The other FV combinations in (\[FV\]) are: m(\^[LR]{} m\_\^[LR]{})\_[ii]{} &=& 12 t\_T t\_3 \[su5\_LRLR\]\
[I]{}m(\^[LR]{} \^[RR]{})\_[ii]{}& = & 6 t\_T t\_3 \[su5\_LRRR\]\
[I]{}m(\^[LL]{} \^[LR]{})\_[ii]{}& = & m(\^[LR]{} \^[RR]{})\_[ii]{} . \[su5\_LLLR\] The only contribution which does not vanish in the limit $a_0 \rightarrow 0$ is the LL RR one. To compare the amplitudes, (\[su5\_LLRR\]), (\[su5\_LRLR\]) and (\[su5\_LRRR\]), (\[su5\_LLLR\]) have to be multiplied respectively by $\mu \tgb$ and $m_L m_R$ and also by the appropriate loop functions, which have the same sign and in general are of the same order of magnitude (for more details see [@ms1]). Then, if $\mu > 0$ the four FV amplitudes have the same sign. However, the double LR insertion is always negligible with respect to the LL RR one because of the suppression factor $m_\ell^2 / m_L^2$. For the other amplitudes (\[su5\_LRRR\]),(\[su5\_LLLR\]) the suppression factor with respect to (\[su5\_LLRR\]) is $m_{R,L}/(\mu \tgb)$ (actually smaller due to the numerical coefficients). Then, even in the case of $\mu < 0$, a reduction of the LL RR amplitude due to accidental cancellations seems unrealistic. Of course, also the contributions due to different thresholds in the right-handed neutrino spectrum are present, in exact analogy to what has been discussed in the previous section.
It is instructive to focus on the magnitude and dependencies of the combination ${\cal I}m({\cal N} m_\ell U^* )_{ii}$. For $d_e$ and $d_\mu$, neglecting subleading terms proportional to $y_c^2$, $y_u^2$ and defining $V_{td} \equiv |V_{td}| e^{i \phi_{td}}$: m([N]{} m\_U\^\*)\_[22]{} & & m\_ y\_t\^2 V\_[ts]{} ( [I]{}m([N]{}\_[23]{}) - |V\_[td]{}| [I]{}m( e\^[i \_[td]{}]{} [N]{}\_[12]{}))\
[I]{}m([N]{} m\_U\^\*)\_[11]{} & & m\_ y\_t\^2 |V\_[td]{}| ( [I]{}m(e\^[-i \_[td]{}]{} [N]{}\_[13]{}) + V\_[ts]{} [I]{}m( e\^[-i \_[td]{}]{} [N]{}\_[12]{}) ) where we exploited the fact that $V_{ts}$ is real in the standard parameterization. The latter is convenient to stress that the CP phases involved in the above combinations could be naturally large - as is indeed the case for $\phi_{td}$ - but, of course, any other choice must give equivalent results [^12]. The contribution proportional to ${\cal I}m({\cal N}_{12})$ has important suppression factors. Moreover $|{\cal N}_{12}|$ is independently constrained to be quite small from the present limits on $\meg$. A plot of the present upper limit on $|C_{12}|$, with $C \equiv (4 \pi)^2 t_3 {\cal N}$, in the plane $(\ti M_1, m_R)$ can be found in Ref. [@imsusy02]. The limit were derived for the seesaw but also applies without significant modifications to the case of $SU(5)$ plus seesaw. As a result, once fixed $M_T$, experimental searches for $d_e$ and $d_\mu$ represent a test for ${\cal I}m(e^{-i \phi_{td}} {\cal N}_{13})$ and ${\cal I}m({\cal N}_{23})$ respectively. Although present searches for $\tmg$ ($\tau \rightarrow e \gamma$) are not able by now to interestingly constrain $|C_{23}|$ ($|C_{13}|$), eventual experimental improvements would have an impact on $d_\mu$ ($d_e$) [@inprog].
Notice also that the naive scaling relation is violated according to = \[scsu5\] and that the combinations of Yukawas relevant for $d_i$ are independent on the phases of the diagonal $SU(3)$ matrix, $\phi_u$. On the contrary, the latter affects (see for instance Ref. [@gotonihei]) the proton decay lifetime due to d=5 operators, whose most important decay mode in the case of minimal $SU(5)$ is $p \rightarrow K^+ \bar \nu$.
Predicted range and constraints on Yukawas
------------------------------------------
To understand how close to the experimental sensitivity is the radiatively induced EDM range, we plot in figs. \[Fsu5\_de\] and \[Fsu5\_dmu\] the upper estimate for the most important products of $\d$’s. We consider a degenerate spectrum of right-handed neutrinos to pick out just the effect of the triplets, substitute the $M_{Pl}$-values for $y_t$ ($\approx .7$) and the relevant CKM elements and choose as reference values $\Lambda / M_T =10^2$ and $\Lambda / M_3 = 2~ 10^3$. Then, the upper estimate follows by requiring perturbativity, ${\cal I}m(N_{23}) \le 1$, ${\cal I}m(e^{-i \phi_{td}} N_{13}) \le 1$. Solid and dashed lines refers to cases a) and b) respectively. We don’t show case b) for ${\cal I}m(\d^{LL} m_\ell \d^{RR})_{ii}/m_\tau$, because the predicted value is essentially flat. The upper estimate for ${\cal I}m(\d^{LL} \d^{LR})_{ii}$ is not shown, being closely related to that on ${\cal I}m(\d^{LR} \d^{RR})_{ii}$ (see eq. (\[su5\_LLLR\])). The bounds on ${\cal I}m(\d^{LR} m_\ell \d^{LR})_{ii}/m_\tau$ are not displayed because they are too small to be of potential interest.
For an easy comparison, we have taken from the sleptonarium [@ms1] the experimental limits on the same quantities, figs. \[Fedm\_de\] and \[Fedm\_dmu\] . The experimental limits on ${\cal I}m(\d^{LR} \d^{RR})_{ii}$ are close to those on ${\cal I}m(\d^{LL} \d^{LR})_{ii}$ and mildly depend on $m_L$, which has been fixed as in mSUGRA in the plots. On the contrary, the limits on ${\cal I}m(\d^{LL} m_\ell \d^{RR})_{ii}/m_\tau$ are proportional to $1/ (\mu \tan \beta)$. For definiteness, $\tan \beta =10$ and $\mu=\mu_{ew}$ have been assumed. The experimental bounds shown in figs. \[Fedm\_de\] and \[Fedm\_dmu\] correspond to the present bound $d_e < 10^{-27}$ e cm and to the planned sensitivity $d_\mu < 10^{-24}$ e cm. Since the experimental bounds are proportional to the bound on $d_i$, it is straightforward to extrapolate the sensitivity to $d_\mu$ and $d_e$ required to test the radiatively induced $d_e$ and $d_\mu$.
.3 cm
Experiment
.2 cm
(-320,60)[$\times \frac{d_e [{\rm e~cm}]}{10^{-27}} $]{} (-115,60)[$\times \frac{d_e [{\rm e~cm}]}{10^{-27}} \frac{\mu_{ew}}{\mu} \frac{10}{\tgb} $]{}
.5 cm
(-320,230)[ Theory: $SU(5)$ with Seesaw ]{} (-350,200)[ ${\cal I}m(\d^{LR} \d^{RR})_{11}$]{} (-140,200)[ ${\cal I}m(\d^{LL} \frac{m_\ell}{m_\tau} \d^{RR})_{11}$]{} (-300,60)[$\times \frac{t_T t_3}{1.4~ 10^{-3}} $ ]{} (-130,40)[ b) $ (1 \pm 0.2)~10^{-3} $ ]{} (-80,60)[$\times \frac{t_T t_3}{1.4~ 10^{-3}} $ ]{}
(-155,270)[Upper limit on]{} (-150,240)[${\cal I}m(e^{-i \phi_{td}} {\cal N}_{13})$]{} (-145,60)[$\times \frac{1.4~ 10^{-3}}{t_T t_3} \frac{10}{\tgb} \frac{\mu_{ew}}{\mu}
\frac{d_e [{\rm e~ cm}]}{10^{-27}}$]{}
Let us firstly discuss the $d_e$ range. For our reference values, the upper estimate for ${\cal I}m(\d^{LR} \d^{RR})_{11}$ is $O(10^{-6})$ and the present bound on $d_e$ already constrains it to be smaller in a large region of the plane. Although we already know that they are not dominant, it is worth to discuss the LR RR amplitude and the similar LL LR one because, as already mentioned, they are quite model independent. Allowing for higher triplet masses, however, these amplitudes shift below the present experimental sensitivity. This is not the case for the LL RR amplitude. The maximum value allowed for ${\cal I}m(\d^{LL} m_\ell \d^{RR})_{11}/m_\tau$ is displayed in the right panel of fig. \[Fsu5\_de\] for case a). In case b) it is $\approx 10^{-3}$ everywhere. Then, the present experimental bound on $d_e$ has already explored the radiative range for roughly $2-3$ orders of magnitude.
This can be translated into an upper limit on ${\cal I}m(e^{-i \phi_{td}} {\cal N}_{13})$, as in fig. \[Fsu5\_N13\]. Notice that this limit is indeed very strong and can be hardly evaded even looking for less favorable parameters than those taken as reference. To check this, keep e.g. $t_3$ fixed and try to worsen the limit: a reduction of $\mu$ with respect to $\mu_{ew}$ is unlikely to reduce it more than one order of magnitude; a suppression by another factor $10$ would require a $\Lambda / M_T \approx 1.6$ ; on the other hand, for values of $\tgb$ larger than 10 the limit linearly improves. As a result, in the framework of the seesaw accompanied by $SU(5)$, the limit on ${\cal I}m(e^{-i \phi_{td}} {\cal N}_{13})$ can be considered robust.
.3 cm
Experiment
.2 cm
(-320,60)[$\times \frac{d_\mu [{\rm e~cm}]}{10^{-24}} $]{} (-115,60)[$\times \frac{d_\mu [{\rm e~cm}]}{10^{-24}} \frac{\mu_{ew}}{\mu} \frac{10}{\tgb}$]{}
1.3 cm
(-300,230)[Theory: $SU(5)$ with Seesaw ]{} (-350,200)[ ${\cal I}m(\d^{LR} \d^{RR})_{22}$]{} (-150,200)[ ${\cal I}m(\d^{LL} \frac{m_\ell}{m_\tau} \d^{RR})_{22}$]{} (-300,60)[$\times \frac{t_T t_3}{1.4~ 10^{-3}} $ ]{} (-130,40)[ b) $ \approx (4 \pm 0.5)~10^{-3} $ ]{} (-80,60)[$\times \frac{t_T t_3}{1.4~ 10^{-3}} $ ]{}
(-155,270)[Upper limit on]{} (-140,240)[${\cal I}m({\cal N}_{23})$]{} (-150,60)[$\times \frac{1.4~ 10^{-3}}{t_T t_3} \frac{10}{\tgb} \frac{\mu_{ew}}{\mu}
\frac{d_\mu [{\rm e~ cm}]}{5~10^{-26}}$]{}
1 cm
Let us now turn to discuss $d_\mu$. By comparing figs. \[Fedm\_dmu\] and \[Fsu5\_dmu\], it turns out that the upper bound on the LR RR insertion would require a sensitivity to $d_\mu$ at $O(10^{-26})$ e cm, at the very limit of planned experimental improvements. Quantitatively, the upper estimate for ${\cal I}m(\d^{LL} m_\ell \d^{RR})_{22}/m_\tau$ which, for case b) is $\approx 4 ~10^{-3}$, is more promising. The major part of the plane in fig. \[Fsu5\_dmu\] could be tested with $d_\mu$ at the level of $10^{-24}$–$10^{-25}$ e cm and in general $d_\mu$ should not exceed $O(10^{-23})$ e cm. As a result, the eventual presence of triplets doesn’t enhance by much the range for $d_\mu$ with respect to the pure seesaw case. Nevertheless, here too the possibility of constraining ${\cal I}m({\cal N}_{23})$ can be envisaged, as shown in fig. \[Fsu5\_N23\]. Notice that, due to (\[scsu5\]), a limit on ${\cal I}m({\cal N}_{23})$ comparable to the present one on ${\cal I}m(e^{-i \phi_{td}} {\cal N}_{13})$ would require to improve the $d_\mu$ sensitivity down to $5~ 10^{-27}$ e cm.
1 cm
Conclusions
===========
Planned experiments might significantly strengthen the limit on $d_e$ [@deexpf; @lam] and $d_\mu$ [@dmuexpf; @dmuexpff]. Their eventual discovery could be interpreted as an indirect manifestation of supersymmetry but could not reveal [*which source*]{} of CP (and possibly flavour) violation is actually responsible for the measured effect. Clearly, all sources in principle able to give the lepton EDM even at an higher level would be automatically constrained while those which fail in giving the lepton EDM at the desired level would be automatically excluded from the list of possible candidates.
In this work, we have estimated the ranges for the lepton EDMs induced by the Yukawa interactions of the heavy neutrinos, both alone and with the simultaneous presence of the heavy $SU(5)$ triplets. It turns out that the FV LLRR amplitude is in general larger or comparable to the FC one. So, EDMs are enhanced for large values of $\tan \beta$ and do not strongly depend on $a_0$.
The pure seesaw, even with large $\tgb$ and very hierarchical right-handed neutrinos, cannot account for $d_\mu$ above $10^{-23}$ e cm. Its eventual discovery above this level would then signal the presence of some source of CP and LF violation other than the neutrino Yukawa couplings. The heavy triplets Yukawa couplings would be excluded from the list of possible sources because their additional presence do not significantly enhance the predicted range for $d_\mu$. Notice however that in the latter case a hierarchical right-handed neutrino spectrum is no more essential to end up with $d_\mu$ at an interesting level for planned searches. From the theoretical point of view, finding $d_\mu$ above $10^{-23}$ e cm would indeed have a remarkable impact.
Interestingly enough, the present experimental sensitivity to $d_e$ is already testing the simultaneous presence of triplets and right-handed neutrinos. Correspondingly, constrains on ${\cal I}m(e^{- i \phi_{td}} {\cal N}_{13})$ have been derived which are significant even for quite large values of the triplet mass and unfavorable supersymmetric masses. Without the triplets, the radiatively-induced $d_e$ is close to the present experimental sensitivity only in models with large $\tgb$ and small slepton masses. Therefore, an experimental improvement would eventually provide interesting limits on the imaginary part of the relevant combination of neutrino Yukawa couplings and right-handed neutrino masses.
In the present discussion we have been looking for results as general as possible. Indeed, the specification of any particular seesaw model has been avoided and the attention has rather focused on the dependencies on the supersymmetric masses and heavy thresholds. Although some relevant seesaw models deserve a dedicated analysis [@inprog], figs. \[Fss\_INN\], \[Fsu5\_N13\] and \[Fsu5\_N23\] are suitable for a quick check of the status of any given seesaw model with respect to the present and planned experimental limits on lepton EDMs.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank C.A. Savoy for useful discussions and collaboration in the early stage of this work. I.M. acknowledge the CNRS and the “A. Della Riccia” Foundation for support and the SPhT, CEA-Saclay, for kind hospitality. We also acknowledge M. Peskin and Y. Farzan for pointing out a wrong numerical coefficient in the previous version of eq. (\[FR\_ss\]); the impact of this correction is quantitatively negligible and the subsequent analysis and results stay unaffected.
2 cm
Appendix: Seesaw
================
In the basis where charged fermion and right handed Majorana neutrino masses are diagonal u\^[cT]{} y\_u Q H\_u + d\^[cT]{} y\_d Q H\_d + e\^[cT]{} y\_e L H\_d + \^[cT]{} y\_L H\_u + \^[cT]{} M\_R \^c \[W\] where $Q=(u ~d)^T$, $L=(\nu ~e)^T$ and $\langle H_{d(u)}^0 \rangle = v_{d(u)}$. Soft scalar masses are defined as \_[soft]{} & &u\_R\^m\^2\_u u\_R + d\_R\^m\^2\_d d\_R +Q\^m\^2\_Q Q + e\_R\^m\^2\_e e\_R +L\^m\^2\_L L +\_R\^m\^2\_\_R\
& +& (u\_R\^A\_u u\_L v\_u + d\_R\^A\_d d\_L v\_d + e\_R\^A\_e e\_L v\_d + \_R\^A\_\_L v\_u +h.c.) Let us also introduce the following notations: y\_x\^y\_x X y\_x y\_x\^X (x=e,u,d)\
P\_a y\_y\_\^[(a)]{} y\_\^P\_a y\_ P\_a y\_y\_\^P\_a (a=1,2,3) where $P_2$, $P_1$ project out $M_3$ and $M_{3,2}$ respectively, $P_2={\rm diag}(1,1,0) $, $ P_1={\rm diag}(1,0,0)$, and $P_3 = \i3$.
Running
-------
Defining $t \equiv \frac{1}{(4 \pi)^2} \ln Q$, the running of the Yukawa coupling constants is governed by: &=& y\_\^[(a)]{} \[3 + E + D\_\^[(a)]{}\]\
&=& y\_e \[3 E + + D\_e\]\
&=& y\_u \[3 U + D + D\_u\^[(a)]{}\]\
&=& y\_d \[3 D + U + D\_d\]\
&=& 2 \[ P\_a M\_R \^T + M\_R P\_a\] where D\_\^[(a)]{}& =& \[Tr(3 U + ) - (3 g\_2\^2 + g\_1\^2)\] i3\
D\_e &=& \[Tr(3 D + E ) - (3 g\_2\^2 + g\_1\^2)\] i3\
D\_u\^[(a)]{}& =& \[Tr(3 U + ) - ( g\_3\^2 +3 g\_2\^2 + g\_1\^2 )\] i3\
D\_d &=& \[Tr(3 D + E ) - ( g\_3\^2 +3 g\_2\^2 + g\_1\^2)\] i3 .For the trilinear couplings, defining $P_a A_\nu \equiv A_\nu^{(a)}$ &=& 4 A\_\^[(a)]{} + 5 A\_\^[(a)]{} +2 y\_\^[(a)]{} y\_e\^A\_e + A\_\^[(a)]{} E + D\_\^[(a)]{} A\_\^[(a)]{} + 2 D\_\^[(a)]{} y\_\^[(a)]{}\
&=& 4 E A\_e + 5 A\_e E +2 y\_e [y\_\^[(a)]{} ]{}\^A\_\^[(a)]{} +A\_e + D\_e A\_e + 2 D\_e y\_e \[Ae\_ss\]\
&=& 4 U A\_u + 5 A\_u U +2 y\_u y\_d\^A\_d +A\_u D + D\_u\^[(a)]{} A\_u + 2 D\_u\^[(a)]{} y\_u\
&=& 4 D A\_d + 5 A\_d D +2 y\_d y\_u\^A\_u +A\_d U + D\_d A\_d + 2 D\_d y\_d where D\_\^[(a)]{} & =& \[Tr(3 y\_u\^A\_u+ [y\_\^[(a)]{}]{}\^A\_\^[(a)]{} ) - (3 g\_2\^2 M\_2 + g\_1\^2 M\_1)\] i3\
D\_e &=& \[Tr(3 y\_d\^A\_d+ y\_e\^A\_e ) - (3 g\_2\^2 M\_2 + g\_1\^2 M\_1)\] i3\
D\_u\^[(a)]{} &=& \[Tr(3 y\_u\^A\_u+ [y\_\^[(a)]{}]{}\^A\_\^[(a)]{} ) - (g\_3\^2 M\_3 + 3 g\_2\^2 M\_2 + g\_1\^2 M\_1)\] i3\
D\_d & =& \[Tr(3 y\_d\^A\_d+ y\_e\^A\_e ) - (g\_3\^2 M\_3+3 g\_2\^2 M\_2 + g\_1\^2 M\_1)\] i3 .For soft scalars, defining $P_a m^2_\nu P_a \equiv {m^2_\nu}^{(a)}$ &=& { m\^2\_L , E + } +2 ( y\_e\^m\^2\_e y\_e + m\^2\_[H\_d]{} E + A\_e\^A\_e) + 2 ( [y\_\^[(a)]{}]{}\^\^[(a)]{} y\_\^[(a)]{} + m\^2\_[H\_u]{} + [A\_\^[(a)]{}]{}\^A\_\^[(a)]{}) + G\_L\
&=& 2 { m\^2\_e , E } +4 ( y\_e m\^2\_L y\_e\^+ m\^2\_[H\_d]{} E + A\_e A\_e\^) + G\_e\
&=& 2 { [m\^2\_]{}\^[(a)]{} , } +4 ( y\_\^[(a)]{} m\^2\_L [y\_\^[(a)]{}]{}\^+ m\^2\_[H\_u]{} + A\_\^[(a)]{} [A\_\^[(a)]{}]{}\^)\
&=& { m\^2\_Q , U + D } +2 ( y\_u\^m\^2\_u y\_u + m\^2\_[H\_u]{} U + A\_u\^A\_u) + 2 ( y\_d\^m\^2\_d y\_d + m\^2\_[H\_d]{} D + A\_d\^A\_d) + G\_Q\
&=& 2 { m\^2\_u , U } +4 ( y\_u m\^2\_Q y\_u\^+ m\^2\_[H\_u]{} U + A\_u A\_u\^) + G\_u\
&=& 2 { m\^2\_d , D } +4 ( y\_d m\^2\_Q y\_d\^+ m\^2\_[H\_d]{} D + A\_d A\_d\^) + G\_d where G\_L = -( g\_1\^2 M\_1\^2 +6 g\_2\^2 M\_2\^2) i3 G\_e = -( g\_1\^2 M\_1\^2) i3\
G\_Q = -( g\_1\^2 M\_1\^2 +6 g\_2\^2 M\_2\^2 + g\_3\^2 M\_3\^2) i3 \
G\_u = -( g\_1\^2 M\_1\^2 + g\_3\^2 M\_3\^2) i3 G\_d = -( g\_1\^2 M\_1\^2 + g\_3\^2 M\_3\^2) i3 .Finally, &=& 6 Tr(y\_d m\^2\_Q y\_d\^+y\_d\^m\^2\_d y\_d + m\^2\_[H\_d]{} D + A\_d\^A\_d) + 2 Tr(y\_e m\^2\_L y\_e\^+y\_e\^m\^2\_e y\_e + m\^2\_[H\_d]{} E + A\_e\^A\_e) + G\_L\
&=& 6 Tr(y\_u m\^2\_Q y\_u\^+y\_u\^m\^2\_u y\_u + m\^2\_[H\_u]{} U + A\_u\^A\_u)\
&+& 2 Tr(y\_\^[(a)]{} m\^2\_L [y\_\^[(a)]{}]{}\^+[y\_\^[(a)]{}]{}\^\^[(a)]{} y\_\^[(a)]{} + m\^2\_[H\_u]{} + [A\_\^[(a)]{}]{}\^A\_\^[(a)]{}) + G\_L and we define $ {m^2}'_{H_{d(u)}} \equiv d m^2_{H_{d(u)}} / dt $.
For energy scales between $\Lambda$ and $M_3$, one has to take $a=3$; below $M_3$ and above $M_2$, $a=2$; while below $M_2$ and above $M_1$, $a=1$.
Appendix: $SU(5)$ + See-saw
===========================
We adopt the following notation to write matter and Higgs superfields: \_[10]{}= ( ) \_[|5]{} = ( ) \_1 = \^c |H = ( ) H = ( ) where $< H_{2d(2u)}^0 > \equiv v_{d(u)}$. The superpotential W [\^[AB]{}]{} y\_u \^[CD]{} H\^E \_[ABCDE]{} + [\^[AB]{}]{} y\_e \_A |H\_B + \_1 y\_ \_A H\^A + \_1 M\_R \_1 where $A,B, ... =1,...,5$ and flavour indices are understood, gives rise to (\[W\]) with $y_d=y_e^T$ and $y_u=y_u^T$. For the soft breaking part of the Lagrangian \_[soft]{}& =& \^m\^2\_ + \^m\^2\_+ \^m\^2\_+ m\^2\_h h\^h + m\^2\_[|h]{} [|h]{}\^ + ( M\_5 \_5 \_5 + [h.c.]{} )\
&+& ( u\^[cT]{} A\_u u v\_u + d\^[cT]{} A\_e\^T d v\_d + e\^[cT]{} A\_e e v\_d + \^[cT]{} A\_ v\_u ) + [h.c.]{} where the scalar fields are $\ti \psi=(\ti u^c, \ti u, \ti d, \ti e^c)$, $\ti \phi=(\ti d^c, \ti e, \ti \nu)$ and $\ti \eta = \ti \nu^c$ and gauginos are denoted with $\lambda_5$. In this way, in the scalar lepton mass matrix, $m^2_{LL}=m^2_\phi$, $m^2_{RR}=m^{2*}_\psi$.
Running
-------
Setting $t \equiv \frac{1}{(4 \pi)^2} \ln Q$, $d g_5/dt= -3 g_5^3$, $ d M_5/dt= -6 g_5^2 M_5$ and &=& 6 y\_e E + 3 U y\_e + y\_e N +G\_e y\_e\
&=& 6 y\_u U + 2 E y\_u + 2 y\_u D +G\_u y\_u\
&=& 6 y\_N + 4 y\_E + G\_y\_\[rsu5\_y\] where $G_e= -84/5 g_5^2 + 4 Tr(E)$, $G_u= -96/5 g_5^2 + Tr(3 U+ N)$, $G_\nu= -48/5 g_5^2 + Tr(3 U+N)$. For scalar masses &=& { m\^[2\*]{}\_, 2 E+ 3 U } + 4 (y\_e m\^2\_y\_e\^ + m\^2\_[|h]{} E + A\_e A\_e\^) + 6 (y\_u m\^2\_y\_u\^ + m\^2\_[ h]{} U + A\_u A\_u\^) + G\_\
&=& { m\^[2]{}\_, 4 E+ N } + 8 (y\_e\^m\^[2\*]{}\_y\_e + m\^2\_[|h]{} D + A\_e\^A\_e ) + 2 (y\_\^m\^[2\*]{}\_y\_ + m\^2\_[ h]{} N + A\_\^A\_) + G\_\
&=& 5 { m\^[2\*]{}\_, N } + 10 (y\_m\^[2]{}\_y\_\^+ m\^2\_h N + A\_A\_\^) \[rsu5\_m\] where $G_\psi = -144/5 g_5^2 M_5^2 \i3$, $G_\phi = -96/5 g_5^2 M_5^2 \i3$. For trilinear couplings &=& 10 E A\_e + 3 U A\_e + 8 A\_e E + 6 A\_u y\_u\^y\_e + A\_e N + 2 y\_e y\_\^A\_+ G\_e A\_e +2 G\_e y\_e \[Ae\_su5\]\
&=& 9 U A\_u + 2 E A\_u + 9 A\_u U + 4 A\_e y\_e\^y\_u + 2 A\_u D + 4 y\_u y\_d\^A\_d + G\_u A\_u +2 G\_u y\_u\
&=& 7 N A\_+ 11 A\_N + 4 A\_E + 8 y\_y\_e\^A\_e + G\_A\_+2 G\_y\_ \[rsu5\_A\] where $\ti G_e= -84/5 g_5^2 M_5 + 4 Tr(y_e^\dagger A_e)$, $\ti G_u= -96/5 g_5^2 M_5 + Tr(3 y_u^\dagger A_u+ y_\nu^\dagger A_\nu)$, $\ti G_\nu= -48/5 g_5^2 M_5 + Tr(3 y_u^\dagger A_u + y_\nu^\dagger A_\nu)$. Finally, for scalar higgses &=&Tr(6 U +2 N ) m\^2\_h + 6 Tr(y\_u m\^2\_y\_u\^ + y\_u\^m\^[2\*]{}\_y\_u + A\_u A\_u\^)\
&+& 2 Tr(y\_m\^2\_y\_\^+ y\_\^T m\^2\_y\_\^\* + A\_A\_\^) - g\_5\^2 M\_5\^2\
&=& 8 Tr( E ) m\^2\_[|h]{} + 8 Tr(y\_e m\^2\_y\_e\^ + y\_e\^m\^[2\*]{}\_y\_e + A\_e A\_e\^) - g\_5\^2 M\_5\^2 .
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work in progress.
[^1]: Laboratoire de la Direction des Sciences de la Matière du Commissariat à l’Énergie Atomique et Unité de Recherche Associée au CNRS (URA 2306).
[^2]: Of course, several effects of different origin could be simultaneously present, but there would be no reason for a destructive interference among them to occur.
[^3]: In Ref. [@ellis-2] a dependence on $\tgb$ arises due to the particular class of seesaw textures studied.
[^4]: E.g. a contribution to $d_i^{FC}$ of the form of (\[FV\]), with $\d^{LR} \rightarrow $ $(A_{\ell \ell}^* v_d - \mu \tan\beta m_{\ell})/(m_L m_R)$, is in principle present, but it is negligible with respect to (\[FC\]).
[^5]: For dedicated studies on cancellations between amplitudes, also in more general frameworks, see e.g. [@prs; @cancell].
[^6]: Here and in the following Dirac mass terms are always written as $\bar f_R m_f f_L$.
[^7]: Actually, (\[ss\_ae\]) is an approximation by excess and improves as the involved Yukawa couplings become small. However, for our estimates it is reliable up to $O(1)$ Yukawa couplings.
[^8]: As far as the comparison with the expressions in Refs. [@ellis-1; @ellis-2; @ellis-n] is possible, we find agreement with the last papers up to the coefficient of ${\cal I}m( \1 \2 )_{ii}$.
[^9]: The function $I_B$ in eq. (\[FC\]) contains a factor $|\mu|^2$ that cancels the one in the overall coefficient. The exact expressions for $I_B$ can be found in [@ms1], as well as approximate ones suitable for various pattern of supersymmetric masses.
[^10]: Needless to say, this remains true even if the FC were the dominant amplitude, in which case the bound would be stronger.
[^11]: Of course eqs. (\[eq\_dsu5\]) overestimate the misalignment. However, for our estimates here they are reliable up to $y_t \sim y_{\nu_3} \sim 1$.
[^12]: Had we exploited the freedom of parameterizing $V$ in such a way that $V_{ti}$ are real numbers, then the CKM phase of $V$ would have been hidden in the redefinition of ${\cal N}$.
|
---
abstract: 'Optically detected magnetic resonance (ODMR) is a way to characterize the NV$^-$ centers. Recently, a remarkably sharp dip was observed in the ODMR with a high-density ensemble of NV centers, and this was reproduced by a theoretical model in \[Zhu [*[et al]{}*]{}., Nature Communications [**[5]{}**]{}, 3424 (2014)\], showing that the dip is a consequence of the spin-1 properties of the NV$^-$ centers. Here, we present much more details of analysis to show how this model can be applied to investigate the properties of the NV$^-$ centers.'
author:
- Yuichiro Matsuzaki
- Hiroki Morishita
- Takaaki Shimooka
- Toshiyuki Tashima
- Kosuke Kakuyanagi
- Kouichi Semba
- 'W. J. Munro'
- Hiroshi Yamaguchi
- Norikazu Mizuochi
- Shiro Saito
title: 'Optically detected magnetic resonance of high-density ensemble of NV$^-$ centers in diamond.'
---
Introduction
============
Recently, a remarkably sharp dip has been observed around 2870 MHz in the ODMR with zero applied magnetic fields [@kubo2010strongetal; @simanovskaia2013sidebands; @zhudark2014]. the ODMR results observed in [@kubo2010strongetal; @simanovskaia2013sidebands; @zhudark2014] cannot be well reproduced by such a fitting [@kubo2010strongetal], and no theoretical model can explain the dip until a new approach is suggested in [@zhudark2014]. The model described in [@zhudark2014] contains spin-1 properties of the NV$^-$ centers while most of the previous models assume the NV$^-$ center to be a spin-half system . By including the strain distributions, randomized magnetic fields, and homogeneous width of the NV$^-$ centers, the sharp dip in the ODMR has been reproduced in [@zhudark2014]. Moreover, this dip is shown to be the cause of a long-lived collective dark state observed in a spectroscopy of superconductor diamond hybrid system, and so this dip could be useful if we will use the collective dark state for a long-lived quantum memory of a superconducting qubit [@zhudark2014].
In this paper, we present the details about how the model suggested in [@zhudark2014] can be applied to investigate the properties of an ensemble of NV$^-$ centers. The rest of this paper is organized as follows. In section 2, we explain the experimental setup. In section 3, we introduce the theoretical model introduced in [@zhudark2014]. In section 4, we show the ODMR results and explain how these experimental results can be reproduced by our theoretical model. Finally, section 5 contains a summary of our results.
Experimental setup
==================
We begin by describing how we generate the NV$^-$ centers in diamond. To create the NV$^-$ center ensemble, we performed ion implantation of $^{12}$C$^{2+}$ and we annealed the sample in high vacuum [@zhu2011coherent]. The density of the NV$^-$ centers is approximately $5\times 10^{17}$ cm$^{-3}$, and we have the NV$^-$ centers over the depth of 1$\mu$m from the surface of the diamond.
![ NV$^-$ center consists of a nitrogen atom (N) and a vacancy (V) in the adjacent site. Since NV$^-$ center is a spin-1 system, we have three states of $|0\rangle $, $|1\rangle $, and $|-1\rangle$. We can characterize the NV$^-$ center by an optically detected magnetic resonance (ODMR) spectrum, and we perform the ODMR with an applied magnetic field of $B=0,1,2$ mT, along the \[111\] direction. []{data-label="config"}](2nvconfig.eps)
The ODMR was performed on the diamond sample by a confocal microscope with a magnetic resonance system at room temperature [@mizuochi2009coherence]. The magnetic field of $0$, $1$, or $2$ mT was applied along the \[111\] axis. With zero or weak applied magnetic field, a quantization axis of the NV$^-$ center is determined by the direction from the vacancy to the nitrogen, which we call an NV$^-$ axis. This axis is along one of four possible crystallographic axes. The NV$^-$ centers usually occupy these four directions equally. The applied magnetic field along \[111\] is aligned with one of these four axes as shown in Fig. \[config\]. In this case, the Zeeman splitting of the NV$^-$ centers having the NV$^-$ axis of \[111\] is larger than that of the NV$^-$ centers having the other three NV$^-$ axes.
Model
=====
We describe the model to simulate the ODMR of the NV$^-$ center ensemble, which was introduced in [@zhudark2014]. The Hamiltonian of the NV$^-$ centers is as follows.
In a rotating frame defined by $U=e^{-i\omega
\hat{S}_z^2t/\hbar}$, we can perform the rotating wave approximation, and we obtain the simplified Hamiltonian. $$\begin{aligned}
H\simeq \hbar
\sum_{k=1}^{N}\Big{\{}(D_k-\omega )\hat{S}^2_{z,k}+E_{1}^{(k)}(\hat{S}_{x,k}^2-\hat{S}_{y,k}^2)
\ \ \ \ \ \ \ \ \ \ \
\nonumber \\
+E_{2}^{(k)}(\hat{S}_{x,k}\hat{S}_{y,k}+\hat{S}_{y,k}\hat{S}_{x,k})
+g_e\mu _BB^{(k)}_z\hat{S}_z+\frac{\lambda}{2}\hat{S}^{(k)}_{x}
\Big{\}}\nonumber\end{aligned}$$ If the number of excitations in the spin ensemble is much smaller than the number of spins, we can consider the spin ensemble as a number of harmonic oscillators. In this case, we can replace the spin ladder operators as creational operators of the harmonic oscillators such as $\hat{b}^{\dagger }_k\simeq |B\rangle _k\langle 0|$, $\hat{d}^{\dagger
}_k\simeq |D\rangle _k\langle 0|$ where $|B\rangle _k=\frac{1}{\sqrt{2}}(|1\rangle_k
+|-1\rangle _k)$, $|D\rangle _k=\frac{1}{\sqrt{2}}(|1\rangle_k
-|-1\rangle _k)$. By using this approximation, we can simplify the Hamiltonian as follows [@zhudark2014]. $$\begin{aligned}
H&\simeq &\hbar
\sum_{k=1}^{N}
\Big{\{}(\omega ^{(k)}_{b}-\omega )\hat{b}^{\dagger }_k\hat{b}_k
+(\omega ^{(k)}_{d}-\omega )\hat{d}^{\dagger }_k\hat{d}_k\nonumber \\
&+&J_k(\hat{b}^{\dagger
}_k\hat{d}_k+\hat{b}_k\hat{d}^{\dagger }_k
)
+iJ'_k(\hat{b}^{\dagger
}_k\hat{d}_k-\hat{b}_k\hat{d}^{\dagger }_k
)+\frac{\lambda }{2}(\hat{b}_k+\hat{b}^{\dagger }_k)
\Big{\}}\nonumber\end{aligned}$$ where $\omega ^{(k)}_b=D_k -E^{(k)}_{1}$, $\omega ^{(k)}_d=D_k +E^{(k)}_{1}$, $J_k=g\mu _BB^{(k)}_z$, and $J'_k=E^{(k)}_{2}$.
The inhomogeneous broadening can be included in this model as following. We use Lorentzian distributions to include an inhomogeneous effect of $(k=1,2,\cdots ,N)$. It is worth mentioning that the Lorentzian distributions have been typically used to describe the inhomogeneous broadening of the NV$^-$ centers [@zhu2011coherent; @zhudark2014; @kubo2011hybridetal; @kubo2012electronetal]. For an inhomogeneous magnetic field $B^{(k)}_z$, we need to consider the following two effect. First, since there is an electron spin-half bath in the environment due to the centers, NV$^-$ centers are affected by randomized magnetic fields. Second, a hyperfine coupling of the nitrogen nuclear spin splits the energy of the NV$^-$ center into three levels. So we use a random distribution of the magnetic fields with the form of the mixture of three Lorentzian functions. Here, each peak of the Lorentzian is separated with $2\pi \times
2.3 $ MHz that corresponds to the hyperfine interaction with nuclear spin [@kubo2011hybridetal; @saito2013towards]. It is worth mentioning that, since the frequency shift of $D_k$ is almost two-orders of magnitude smaller than that of $E^{(1)}_k$ and $E^{(2)}_k$ [@dolde2011electric], we consider the effect of inhomogeneity of $D_k$ as this order in this paper.
We can describe the dynamics of the NV$^-$ centers by using the Heisenberg equation as follows. $$\begin{aligned}
\frac{d\hat{b}_k}{dt}&=&-i(\omega ^{(k)}_b-i\Gamma _b)\hat{b}_k-iJ_k
\hat{d}_k+J'_k\hat{d}_k-i\lambda \nonumber \\
\frac{d\hat{d}_k}{dt}&=&-i(\omega ^{(k)}_d-i\Gamma _d)\hat{d}_k-iJ_k
\hat{b}_k-J'_k\hat{b}_k \end{aligned}$$ where $\Gamma _b(=\Gamma _d)$ denotes the homogeneous width of the NV center. We assume that the initial state is a vacuum state. Since we consider a steady state after a long time, we can set the time derivative as zero. In this condition, we obtain $$\begin{aligned}
\langle \hat{b}^{\dagger }_{k,t=\infty } \hat{b}_{k,t=\infty
}\rangle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\nonumber \\
= |\frac{\lambda(\omega -\omega ^{(k)}_d+i\Gamma _d)}{(\omega -\omega ^{(k)}_b+i\Gamma
_b)(\omega -\omega ^{(k)}_d+i\Gamma _d) -(|J_k|^2
+|J'_k|^2)}|^2\label{brightex} \\
\langle \hat{d}^{\dagger }_{k,t=\infty }\hat{d}_{k,t=\infty }\rangle
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\nonumber
\\
=| \frac{\lambda(J_k-iJ'_k)}{(\omega -\omega ^{(k)}_b+i\Gamma
_b)(\omega -\omega ^{(k)}_d+i\Gamma _d) -(|J_k|^2 +|J'_k|^2)}|^2 \label{darkex}\end{aligned}$$ $$\begin{aligned}
P_e= \frac{1}{N}(\sum_{k=1}^{N} \langle \hat{b}^{\dagger }_{k,t=\infty
}\hat{b}_{k,t=\infty }\rangle
+\langle \hat{d}^{\dagger }_{k,t=\infty }\hat{d}_{k,t=\infty }\rangle
) \label{probabilitye}.\end{aligned}$$
In the actual experiment, if we excite the NV$^-$ centers by the microwave pulses, the intensity of the photons emitted from the NV$^-$ centers will be changed from
Main results
============
Reproducing the experimental results
------------------------------------
![ ODMR with zero applied magnetic fields. $\delta (g\mu _BB _z)/2\pi =1.96$ MHz (HWHM), $\delta E_1/2\pi =\delta E_2/2\pi=0.73$ MHz (HWHM), $\delta D_k=0.01$ MHz (HWHM), $\lambda /2\pi=2$ MHz, $\Gamma _b/2\pi=\Gamma _d/2\pi= 0.3$ MHz. Also, we assume a Nitrogen hyperfine coupling of $2\pi \times 2.3$ MHz. The red line denotes a numerical simulation and blue dots denote the experimental results. []{data-label="zero"}](2odmrzerob.eps)
![ ODMR with an applied magnetic field of 1 mT. We use the same parameters as the Fig. \[zero\]. The red line denotes a numerical simulation and blue dots denote the experimental results. []{data-label="onemt"}](odmroneb.eps)
![ ODMR with an applied magnetic field of 2 mT. We use the same parameters as the Fig. \[zero\]. The red line denotes a numerical simulation and blue dots denote the experimental results. []{data-label="twomt"}](2odmrtwob.eps)
By using the model described above, we have reproduced the ODMR signals when we apply $B=0,1,2$ mT, as shown in Figs \[zero\], \[onemt\], and \[twomt\]. A sharp dip is observed around $2870$ MHz for the case of $B=0$ mT, and our simulation can reproduce this. Two peaks are observed in the ODMR with zero applied magnetic field as shown in Fig \[zero\], which corresponds to the transition between the state . If we consider a single NV$^-$ center, the frequency difference between the two exited states is $\delta \omega =2\sqrt{(g\mu _BB_z)^2 +(E_1)^2+(E_2)^2}$. Since we consider an ensemble of the NV$^-$ center, this frequency difference variates depending on the position of the NV$^-$ center. For simplicity, we use a dimensionless variable for $B_z$, $E_1$, and $E_2$, defined as $\tilde{B}_z=g\mu _BB_z/\gamma $, $\tilde{E}_1=E_1/\gamma $, and $\tilde{E}_2=E_2/\gamma $ where $\gamma $ denotes a damping rate with an unit of the frequency. To calculate a probability that the are degenerate ($\delta \omega =0$), we define probability density functions of $\tilde{B}_z$, $\tilde{E}_1$, and $\tilde{E}_2$ as $P_a(\tilde{B}_z)$, $P_b(\tilde{E}_1)$, and $P_c(\tilde{E}_2)$, respectively. The joint probability is calculated as $$\begin{aligned}
&& P(\tilde{B}_z,\tilde{E_1},\tilde{E_2})\Delta \tilde{B}_z\Delta
\tilde{E}_1\Delta \tilde{E}_2\nonumber \\
&=&P_a(\tilde{B}_z=0)\Delta \tilde{B}_z\cdot P_b(\tilde{E}_1=0)\Delta
\tilde{E}_1\cdot
P_c(\tilde{E}_2=0)\Delta \tilde{E}_2.\nonumber\end{aligned}$$ where $\Delta \tilde{B}_z$, $\Delta E_1$, and $\Delta E_2$ denote a finite range of each variable. We assume $P(\tilde{B}_z,\tilde{E_1},\tilde{E_2})=P_a(\tilde{B}_z)P_b(\tilde{E}_1)P_c(\tilde{E}_2)$ because these are independent. By using spherical coordinates where $\tilde{B}_z=r \sin \theta \cos
\phi $, $\tilde{E}_1=r \sin \theta \sin \phi$, $\tilde{E_2}=r \cos
\theta $ with $r=\sqrt{|\tilde{B}_z|^2+|\tilde{E}_1|^2+|\tilde{E}_2|^2}$, we rewrite this as $$\begin{aligned}
&& P(\tilde{B}_z,\tilde{E_1},\tilde{E_2})\Delta \tilde{B}_Z\Delta
\tilde{E}_1\Delta \tilde{E}_2\nonumber \\
&=&P_a(\tilde{B}_z=0) P_b(\tilde{E}_1=0) P_c(\tilde{E}_2=0)r^2 \sin
\theta \Delta r \Delta \theta \Delta \phi. \nonumber\end{aligned}$$ This shows that, even if we consider a finite range $\Delta \tilde{B}_z$, $\Delta E_1$, and $\Delta E_2$, the probability for the two energy to be exactly degenerate ($r=0$) is zero. This means that, if homogeneous broadening is negligible, the two peaks to denote the two energy of each NV$^-$ center should be always separated in the ODMR so that the ODMR signal at the frequency of $D/2\pi =2870$ MHz should be the same as the base line. However, due to the effect of the homogeneous broadening, small signals deviated from the base line can be observed at the frequency of $D/2\pi =2870$ MHz. This is the cause of the sharp dip observed around the frequency of $2\pi \times 2870$ MHz in the ODMR.
With an applied magnetic field, four peaks are observed in the ODMR where two of them are larger than the other two, as shown in Figs \[onemt\] and \[twomt\]. The two smaller peaks correspond to the energy of the NV$^-$ centers with an NV$^-$ axe along \[111\], which is aligned with the applied magnetic field. A quarter of the NV$^-$ centers in the ensemble have such an NV$^-$ axis. The other larger peaks come from the other NV$^-$ centers where the applied magnetic field is not aligned with the NV$^-$ axis. Three-quarters NV$^-$ center have such axes. In this case, the Zeeman splitting of these is smaller than that of the NV$^-$ centers with the \[111\] axis. It is worth mentioning that a small dip is observed in the 1mT ODMR around $2\pi \times 2870$ MHz due to the mechanism explained above. On the other hand, such a dip is not clearly observed in the 2mT ODMR, because the NV$^-$ centers are considered to be as approximate two-level systems in this regime.
The behavior of the ODMR against the change in the parameters
-------------------------------------------------------------
We perform a numerical simulation with several parameters to understand the behavior of the sharp dip. In the Fig \[zeroebg\] a, we change the parameter $\Gamma _{b}$ while we fix the other parameters. Similarly, in the Fig \[zeroebg\] b (c), we change the parameter $\delta B_k$ ($\delta E_k$) while we fix the other parameters. We have found that the sharp dip is very sensitive against the change in $\Gamma _{b}$, while the dip is relatively insensitive against the change in $\delta E_k$ and $\delta
B_k$.
![ Numerical simulation of ODMR with zero applied magnetic fields. Here, $x$ axis denotes the microwave frequency, $y$ axis denotes $\gamma /2\pi $ (for the figure a) or $\delta (g\mu
_BB_z)/2\pi $ (for the figure b) or $\delta E/2\pi $ (for the figure c), and $z$ axis denotes the ODMR signal intensity. Other than the inhomogeneous width of the inhomogeneous width, we use the same parameters as the Fig. \[zero\]. []{data-label="zeroebg"}](2threefigures.eps)
Also, we perform a numerical simulation with several parameters for the ODMR with an applied magnetic field. In the Figs \[bb\], we have plotted one of the peaks of the ODMR with an applied magnetic field of 2mT. This peak corresponds to a transition between $|0\rangle $ and $|-1\rangle $ of the NV$^-$ center with an axis of \[111\].
![ Numerical simulation of ODMR with an applied magnetic field of 2 mT. Here, $x$ axis denotes the microwave frequency, $y$ axis denotes $\delta E/2\pi $ for the left figure while $y$ axis denotes $\delta (g\mu _BB)/2\pi $ for the right figure, and $z$ axis denotes the ODMR signal intensity. Other than $\delta E$ or $\delta (g\mu
_BB_z)/2\pi $, we use the same parameters as the Fig. \[twomt\]. []{data-label="bb"}](2be-bchange.eps)
From the numerical simulations, we have found that this ODMR signals with applied magnetic field is robust against the strain variations $\delta E_k$, while the peak will be broadened due to the effect of the randomized magnetic field $\delta B_k$. The frequency difference between the ground state and can be calculated as $\delta \omega '=D_k-
\sqrt{E^{(k)}_{1}+E^{(k)}_{2}+(g_e\mu _BB_z^{(k)})^2}$. If the applied magnetic field is large, we obtain $\delta \omega '\simeq g_e \mu _B B^{(k)}_z\pm \frac{|E^{(k)}_{1}|^2+|E^{(k)}_{2}|^2}{2g_e \mu
_B B^{(k)}_z}$. This means that the effect of the strain is insignificant in this regime while the inhomogeneous magnetic field from the environment can easily change this frequency. These can explain the simulation results shown in Figs \[bb\] Such an effect to suppress the strain distributions by an applied magnetic field was mentioned in [@acosta2013optical], and was recently demonstrated in a vacuum Rabi oscillation between a superconducting flux qubit and NV$^-$ centers in [@matsuzaki2015improvingpra]. Our results here are consistent with these previous results.
Parameter estimation
--------------------
An ensemble of NV$^-$ centers is affected by inhomogeneous magnetic fields, inhomogeneous strain distributions, and homogeneous broadening. In the ODMR, the observed peaks contain the information of the total width that is a composite effect of three noise mentioned above, and so it was not straightforward to separate these three effects for the estimation about how individual noise contributes to the width.
Summary
=======
In conclusion, we have studied an ODMR with a high-density ensemble of NV$^-$ centers. Our model succeeds to reproduce the ODMR with and without applied magnetic field. Also, we have shown that our model is useful to determine the typical parameters of the ensemble NV$^-$ centers such as strain distributions, inhomogeneous magnetic fields, and homogeneous broadening width. Such a parameter estimation is essential for the use of NV$^-$ centers to realize diamond-based quantum information processing.
Y.M thanks K. Nemoto and H. Nakano for discussion. This work was supported by JSPS KAKENHI No. 15K17732, JSPS KAKENHI Grant No. 25220601, and the Commissioned Research of NICT.
Here, we consider the effect of inhomogeneous microwave amplitude. If we have such an inhomogeneity, by solving the Heisenberg equation, we obtain $$\begin{aligned}
&&\langle \hat{b}^{\dagger }_{k,t=\infty } \hat{b}_{k,t=\infty
}\rangle\nonumber \\
&=& |\frac{\lambda_k(\omega -\omega ^{(k)}_d+i\Gamma _d)}{(\omega -\omega ^{(k)}_b+i\Gamma
_b)(\omega -\omega ^{(k)}_d+i\Gamma _d) -(|J_k|^2
+|J'_k|^2)}|^2\nonumber \\
&&\langle \hat{d}^{\dagger }_{k,t=\infty }\hat{d}_{k,t=\infty }\rangle
\nonumber \\
&=&| \frac{\lambda_k(J_k-iJ'_k)}{(\omega -\omega ^{(k)}_b+i\Gamma
_b)(\omega -\omega ^{(k)}_d+i\Gamma _d) -(|J_k|^2 +|J'_k|^2)}|^2\nonumber \end{aligned}$$ where the value of $\lambda $ differs depending on the position of the NV$^-$ centers. If we define an average probability of the NV$^-$ center in the bright (dark) state as $P_b$ ($P_d$), we obtain $$\begin{aligned}
P_b&=&\sum_{k=1}^{N}\frac{|\frac{\lambda_k(\omega -\omega ^{(k)}_d+i\Gamma _d)}{(\omega -\omega ^{(k)}_b+i\Gamma
_b)(\omega -\omega ^{(k)}_d+i\Gamma _d) -(|J_k|^2
+|J'_k|^2)}|^2}{N}\nonumber \\
P_d&=&\frac{\sum_{k=1}^{N}| \frac{\lambda_k(J_k-iJ'_k)}{(\omega -\omega ^{(k)}_b+i\Gamma
_b)(\omega -\omega ^{(k)}_d+i\Gamma _d) -(|J_k|^2
+|J'_k|^2)}|^2}{N}\nonumber \\\end{aligned}$$ Since inhomogeneity of $\lambda $ is independent from the inhomogeneity of $\omega _b$, $\omega _d$, $J$, and $J'$, we can rewrite these probabilities for a large number of NV$^-$ centers as follows $$\begin{aligned}
P_b&\simeq& \frac{1}{N}\sum_{j=1}^{m}|\lambda
_j|^2\sum_{k=1}^{\text{floor}(\frac{N}{m})}
p^{(b)}_{k}\nonumber \\
&=&(\frac{1}{m}\sum_{j=1}^{m}|\lambda
_j|^2)(\frac{1}{(\frac{N}{m})}\sum_{k=1}^{\text{floor}(\frac{N}{m})}
p^{(b)}_{k})\end{aligned}$$ $$\begin{aligned}
P_d&\simeq &\frac{1}{N}\sum_{j=1}^{m}|\lambda
_j|^2\sum_{k=1}^{\text{floor}(\frac{N}{m})}
p^{(d)}_{k}\nonumber \\
&=&(\frac{1}{m}\sum_{j=1}^{m}|\lambda
_j|^2)(\frac{1}{(\frac{N}{m})} \sum_{k=1}^{\text{floor}(\frac{N}{m})}
p^{(d)}_{k})\end{aligned}$$ where $$\begin{aligned}
p^{(b)}_{k}=|\frac{(\omega -\omega ^{(k)}_d+i\Gamma _d)}{(\omega -\omega ^{(k)}_b+i\Gamma
_b)(\omega -\omega ^{(k)}_d+i\Gamma _d) -(|J_k|^2
+|J'_k|^2)}|^2\ \ \ \ \ \ \nonumber \\
p^{(d)}_{k}=| \frac{(J_k-iJ'_k)}{(\omega -\omega ^{(k)}_b+i\Gamma
_b)(\omega -\omega ^{(k)}_d+i\Gamma _d) -(|J_k|^2
+|J'_k|^2)}|^2\nonumber
\end{aligned}$$ Therefore, we obtain $$\begin{aligned}
P_b&\simeq& |\lambda |^2_{\text{av}}(\frac{1}{N'}\sum_{k=1}^{N'}
p^{(b)}_{k})
\\
P_d&\simeq &
|\lambda |^2_{\text{av}}(\frac{1}{N'}\sum_{k=1}^{N'}
p^{(d)}_{k})
\end{aligned}$$ where $|\lambda _{\text{av}}|^2=(\frac{1}{m}\sum_{j=1}^{m}|\lambda
_j|^2)$ and $N'=\text{floor}(\frac{N}{m})$. The probability of the NV$^-$ center in the ground states can be calculated as $$\begin{aligned}
P_e\simeq P_b+P_d
\end{aligned}$$ and this is the same form as the probability of the homogeneous microwave amplitude case described in the Eq. (\[probabilitye\]) where $N$ ($\lambda ^2$) is replaced by $N'$ ($|\lambda |^2_{\text{av}}$). So the inhomogeneous microwave amplitude does not affect the theoretical prediction of ODMR signals.
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---
author:
- Jakub Kopřiva
title: 'On the homotopy transfer of $A_\infty$ structures'
---
*[Dedicated to the memory of Martin Doubek]{}*
The present article is devoted to the study of transfers for $A_\infty$ structures, their maps and homotopies, as developed in [@Markl06]. In particular, we supply the proofs of claims formulated therein and provide their extension by comparing them with the former approach based on the homological perturbation lemma.
[**Key words:**]{} $A_\infty$ structures, transfer, homological perturbation lemma.
[**MSC classification:**]{} 18D10, 55S99 .
Introduction
============
The notion of strongly homotopy associative or $A_{\infty }$ algebras is a generalization of the concept of differential graded algebras. These algebras were introduced by J. Stasheff with the aim of a characterization of (de)looping and bar construction in the category of topological spaces. Since then they found many applications ranging from algebraic topology and operads to quantum theories in theoretical physics.
We consider the following situation: let $(V, \partial_V)$ and $(W, \partial_W)$ be two chain complexes of modules, and $f: (V, \partial_V) \to (W, \partial_W)$ and $g: (W, \partial_W) \to (V, \partial_V)$ two mappings of chain complexes such that $gf$ is homotopic to the identity map on $V$ and $(V, \partial_V)$ is equipped with $A_\infty$ algebra structure. Then a natural question arises $-$ can $A_\infty$ structure be transferred to $(W, \partial_W)$ and secondly, what is its explicit form in terms of $A_\infty$ algebra structure on $(V, \partial_V)$ and and in which sense it is unique?
While the existence of a transfer follows from general model structure considerations, an unconditional and elaborate answer producing explicit formulas for the transferred objects was formulated in [@Markl06]. The present article contributes to the problem of transfer of $A_\infty$ structures. Its modest aim is to supply detailed proofs of many claims omitted in the original article [@Markl06], thereby facilitating complete subtle proofs to a reader interested in this topic. This exposition also extends the results of the aforementioned article in several ways, and sheds a light on its relationship with the homological perturbation lemma.
The content of our article goes as follows. In the Section \[sec:2\] we recall a well-known correspondence between $A_\infty$ algebras and codifferentials on reduced tensor coalgebras. This allows us to simplify the proofs in Section \[sec:3\] considerably. The Section \[sec:3\] is devoted to the problem of homotopy transfer of $A_\infty$ algebras. We first derive the formulas introduced in [@Markl06], and then give their self-contained proofs. Here we achieve a substantial simplification of all proofs due to the reduction of sign factors. We also comment on another remark in [@Markl06], namely, the relationship between the homological perturbation lemma and homotopy transfer of $A_\infty$ algebras. We prove that on certain assumptions the explicit formulas in [@Markl06] do coincide with those coming from the homological perturbation lemma.
We shall work in the category of $\mathbb{Z}$-graded modules over an arbitrary commutative unital ring $R$, and their graded $R$-homomorphisms.
We first briefly recall the concepts of $A_\infty$ algebra, $A_\infty$ morphism of $A_\infty$ algebras and $A_\infty$ homotopy of $A_\infty$ morphisms, cf. [@Markl06], [@Keller01].
Let $(V, \partial_V)$ be a chain complex of modules indexed by $\mathbb{Z}$, i.e. $(V, \partial_V)$ is a $\mathbb{Z}$-graded modules $V = \bigoplus_{i = -\infty}^{\infty}V_i$ with $\partial_V(V_i) \subset V_{i-1}$ and $\partial_V \circ \partial_V = 0$. Let $\mu_n: V^{\otimes n} \to V$ be a collection of linear mappings of degree $n-2$ ($n \geq 2$), satisfying $$\begin{aligned}
\label{ainftyrelations}
\partial_V & \mu_n - \sum_{i = 1}^{n} (-1)^{n} \mu_n\left(\mathbb{1}_V^{\otimes i-1}
\otimes \partial_V \otimes \mathbb{1}_V^{\otimes n-i}\right) \nonumber \\
& = \sum_{A(n)} (-1)^{i(\ell+1) + n} \mu_k \left(\mathbb{1}_V^{\otimes i-1} \otimes
\mu_\ell \otimes \mathbb{1}_V^{\otimes k - i}\right)\end{aligned}$$ for all $n \geq 2$ and $A(n) = \{k, \ell\in{\mathbb N}|\, k+\ell = n+1, k,\ell \geq 2, 1 \leq i \leq k\}$. The structure $(V, \partial_V, \mu_2, \mu_3, \dots)$ is called $A_\infty$ algebra.
Throughout the article, we use the Koszul sign convention. This means that for $U$, $V$ a $W$ graded modules and $f: U \to V$, $g: U \to V$, $h: V \to W$ and $i: V \to W$ linear maps of degrees $|f|$, $|g|$, $|h|$ and $|i|$, respectively, holds $$(h \otimes i)(f \otimes g) = (-1)^{|f||i|} hf \otimes ig.$$ Similarly for $u_1, u_2 \in U$ of degree $|u_1|$ and $|u_2|$, respectively, holds $$(f \otimes g)(u_1 \otimes u_2) = (-1)^{|u_1||g|}f(u_1) \otimes g(u_2).$$
Let $(V, \partial_V, \mu_2, \dots)$ and $(W, \partial_W, \nu_2, \dots)$ be $A_\infty$ algebras. Then the set $\{f_n: V^{\otimes n} \to W, |f_n| = n-1\}_{n \geq 1}$ is called $A_\infty$ morphism if $$\begin{aligned}
& \partial_Wf_n + \sum_{B(n)}(-1)^{\vartheta(r_1, \dots, r_k)} \nu_k(f_{r_1} \otimes \dots \otimes f_{r_k})
\nonumber \\
& = f_1\mu_n - \sum_{i = 1}^{n} (-1)^{n} f_n\left(\mathbb{1}_V^{\otimes i-1} \otimes \partial_V \otimes \mathbb{1}_V^{\otimes n-i}\right)
\nonumber \\
& -\sum_{A(n)} (-1)^{i(\ell+1) + n} f_k \left(\mathbb{1}_V^{\otimes i-1} \otimes \mu_\ell \otimes \mathbb{1}_V^{\otimes k - i}\right)\end{aligned}$$ holds for all $n \geq 1$ with $B(n) = \{k, r_1, \mbox{\dots}, r_k\in{\mathbb N}|\, k \geq 2, r_1, \mbox{\dots}, r_k \geq 1,
r_1 + \mbox{\dots} + r_k = n\}$ and $\vartheta(r_1, \dots, r_k) = \sum_{1 \leq i < j \leq k} r_i(r_j + 1)$.
Morphisms of $A_\infty$ algebras can be composed: for $(U, \partial_U, \varrho_2, \dots)$, $(V, \partial_V, \mu_2, \dots)$ and $(W, \partial_W, \nu_2, \dots)$ $A_\infty$ algebras, $\{f_n: U^{\otimes n} \to V\}_{n \geq 1}$ and $\{g_n: V^{\otimes n} \to W\}_{n \geq 1}$ $A_\infty$ morphisms, their composition $\{(gf)_n: U^{\otimes n} \to W\}_{n \geq 1}$ is defined as $$\begin{aligned}
\label{composition}
(gf)_n = g_1f_n + \sum_{B(n)}(-1)^{\vartheta(r_1, \dots, r_k)} g_k(f_{r_1} \otimes \dots \otimes f_{r_k}).\end{aligned}$$
\[ainftyhomotopy\] Let $\{f_n: V^{\otimes n} \to W\}_{n \geq 1}$ and $\{g_n: V^{\otimes n} \to W\}_{n \geq 1}$ be morphisms between $A_\infty$ algebras $(V, \partial_V, \mu_2, \dots)$ and $(W, \partial_W, \nu_2, \dots)$. The set of linear mappings $\{h_n: V^{\otimes n} \to W, |h_n| = n\}_{n \geq 1}$ is an $A_\infty$ homotopy between $A_\infty$ morphisms $\{f_n: V^{\otimes n} \to W\}_{n \geq 1}$ and $\{g_n: V^{\otimes n} \to W\}_{n \geq 1}$ provided $$\begin{aligned}
& f_n - g_n = h_1\mu_n - \sum_{i = 1}^{n} (-1)^{n} h_n\left(\mathbb{1}_V^{\otimes i-1} \otimes \partial_V \otimes \mathbb{1}_V^{\otimes n-i}\right)
\nonumber \\
& -\sum_{A(n)} (-1)^{i(\ell+1) + n} h_k \left(\mathbb{1}_V^{\otimes i-1} \otimes \mu_\ell \otimes \mathbb{1}_V^{\otimes k - i}\right) + \delta_W h_n
\nonumber \\
& + \sum_{B(n)}\sum_{1 \leq i \leq k}(-1)^{\vartheta(r_1, \dots, r_k)} \nu_k(f_{r_1} \otimes \dots \otimes f_{r_{i-1}} \otimes h_{r_i} \otimes g_{r_{i+1}} \otimes \dots \otimes g_{r_k}),\end{aligned}$$ is true for all $n \geq 1$ with $B(n) = \{k, r_1, \mbox{\dots}, r_k\in{\mathbb N}|\, k \geq 2, r_1, \mbox{\dots}, r_k \geq 1, r_1 + \mbox{\dots} + r_k = n\}$.
Reduced tensor coalgebras {#sec:2}
=========================
In the present section we introduce a bijective correspondence between $A_\infty$ algebras and codifferentials on reduced tensor coalgebras, cf. [@Keller01]. We retain the notation $V = \bigoplus_{i = -\infty}^{\infty} V_i$ for $\mathbb{Z}-$ graded modules as well as $$\tag{A}
\label{A}
A(n) = \{k, \ell\in{\mathbb N}|\, k+\ell = n+1, k,\ell \geq 2, 1 \leq i \leq k\},$$ $$\tag{B}
\label{B}
B(n) = \{k, r_1, \mbox{\dots}, r_k\in{\mathbb N}|\, k \geq 2, r_1, \mbox{\dots}, r_k \geq 1, r_1 + \mbox{\dots} + r_k = n\}$$ for $n \in \mathbb{N}$, and $A(1) = A(2) = B(1) = \emptyset.$ We use a few natural variations on this notation, e.g. $A'(n) = \{k', \ell'\in{\mathbb N}|\, k'+\ell' = n+1, k',\ell' \geq 2, 1 \leq i' \leq k'\}$.
Codiferentials on tensor coalgebras
-----------------------------------
\[k1d1\] Let $\overline{T}V = \bigoplus_{n=1}^\infty V^{\otimes n}$, where the elements in $V^{\otimes i}$ have degree (or homogeneity) $i$, and let the mapping $C: \overline{T}V \to \overline{T}V \otimes \overline{T}V$ be defined in such a way that $C: v \mapsto 0$ for $v \in V^{\otimes 1} = V$ and $$\begin{aligned}
C: v_1 \otimes \mbox{\dots} \otimes v_n \mapsto \sum_{i=1}^{n-1} (v_1 \otimes \mbox{\dots} \otimes v_i) \otimes (v_{i+1} \otimes \mbox{\dots} \otimes v_n),\end{aligned}$$ for $n \geq 2$ and $v_1, \mbox{\dots}, v_n \in V$. The pair $(\overline{T}V, C)$ is called the reduced tensor coalgebra.
\[k1d2\] A linear mapping $\delta: \overline{T}V \to \overline{T}V$ of degree $-1$ is called coderivation if $C \circ \delta = (\delta \otimes \mathbb{1} + \mathbb{1} \otimes \delta) \circ C$. Moreover, if $\delta$ satisfies $\delta \circ \delta = 0$, it is called codifferential.
We notice that $C$ is coassociative, $(\mathbb{1} \otimes C) \circ C = (C \otimes \mathbb{1}) \circ C$. For all $v \in \overline{T}V$ holds $C(v) = 0$ if and only if $v$ is of homogeneity $1$. For all maps $\varphi: V^{\otimes n} \to \overline{T}W$, $n \geq 1$, holds $C_{\overline{T}W} \circ \varphi = 0$ if and only if $\varphi\left(V^{\otimes n}\right) \subseteq W$. For all $v = v_1 \otimes \mbox{\dots} \otimes v_n \in \overline{T}V$ and $w = w_1 \otimes \mbox{\dots} \otimes w_m \in \overline{T}V$, we have $$C(v \otimes w) = \sum_{i = 1}^{n-1} \left(v_{1,i}\right) \otimes \left(v_{i+1, n} \otimes w\right) + \left(v\right)\otimes \left(w\right) + \sum_{i = 1}^{m-1} \left(v \otimes w_{1,i}\right) \otimes \left(w_{i+1,m}\right),$$ with $v_{i,j} = v_i \otimes \mbox{\dots} \otimes v_j$, $i \leq j$, $i,j \in \{1, \dots, n\}$, and analogously for $w_{i,j}$. This little calculation expresses a fact that $\overline{T}V$ is a bialgebra which is, as a conilpotent coalgebra, cogenerated by $V$.
\[k1l3\] Let $E: \overline{T}V \to \overline{T}W$ be a linear mapping for which there exist $\{e_n: V^{\otimes n} \to W\}_{n \geq 1}$ with $E|_{V^{\otimes n}} = e_n + \sum_{B(n)} e_{r_1} \otimes \mbox{\dots} \otimes e_{r_{k}}$, and $B(n)$ given in . Then $$\begin{aligned}
C_{\overline{T}W} \circ E|_{V^{\otimes n}} = \sum_{i=1}^{n-1} \left(E|_{V^{\otimes i}}\right)
\otimes \left(E|_{V^{\otimes n-i}}\right).\end{aligned}$$
Obviously, we can write $E|_{V^{\otimes n}} = e_n + \sum_{i=1}^{n-1} e_{i} \otimes E|_{V^{\otimes n-i}}$. The proof is by induction on $n$: the claim holds for $n=1$ and we assume it is true foll all natural numbers less than $n$. Then $$C_{\overline{T}W} \circ E|_{V^{\otimes n}} = C_{\overline{T}W} \circ \left( e_n + \sum_{i=1}^{n-1} e_{i} \otimes E|_{V^{\otimes n-i}}\right)$$ $$= C_{\overline{T}W} \circ \left(\sum_{i=1}^{n-1} e_{i} \otimes E|_{V^{\otimes n-i}}\right)$$ $$= \sum_{i=1}^{n-1} \left(e_i \right) \otimes \left(E|_{V^{\otimes n-i}}\right) + \sum_{i=1}^{n-1}\sum_{j=1}^{n-1-i} \left(e_i \otimes E|_{V^{\otimes j}}\right) \otimes \left(E|_{V^{\otimes n-i-j}}\right)$$ $$=\sum_{i=1}^{n-1} \left(e_i \right) \otimes \left(E|_{V^{\otimes n-i}}\right) + \sum_{\ell=2}^{n-1}\sum_{j=1}^{\ell-1} \left(e_j \otimes E|_{V^{\otimes \ell-j}}\right) \otimes \left(E|_{V^{\otimes n-\ell}}\right)$$ $$= \left(e_1 \right) \otimes \left(E|_{V^{\otimes n-1}}\right) + \sum_{\ell=2}^{n-1}\left(e_\ell + \sum_{j=1}^{\ell-1} e_j \otimes E|_{V^{\otimes \ell-j}}\right) \otimes \left(E|_{V^{\otimes n-\ell}}\right),$$ and the proof follows by induction hypothesis from $E|_{V^{\otimes \ell}} = e_\ell + \sum_{i=1}^{\ell-1} e_{i} \otimes E|_{V^{\otimes \ell-i}}.$
\[k1v4\] Let $E: \overline{T}V \to \overline{T}W$ a $G: \overline{T}V \to \overline{T}W$ be linear mappings for which there exist linear mappings $\{e_n: V^{\otimes n} \to W\}_{n \geq 1}$, $\{g_n: V^{\otimes n} \to W\}_{n \geq 1}$ such that $E|_{V^{\otimes n}} = e_n + \sum_{B(n)} e_{r_1} \otimes \mbox{\dots} \otimes e_{r_{k}}$ and $G|_{V^{\otimes n}} = g_n + \sum_{B(n)} g_{r_1} \otimes \mbox{\dots} \otimes g_{r_{k}}$ with $B(n)$ given in . Given a linear mapping $F: \overline{T}V \to \overline{T}W$, the following conditions are equivalent:
1. $C_{\overline{T}W} \circ F = \left(E \otimes F + F \otimes G\right) \circ C_{\overline{T}V}$,
2. there exist linear mappings $\{f_n: V^{\otimes n} \to W\}_{n \geq 1}$ such that
$$F|_{V^{\otimes n}} = f_n + \sum_{B(n)}\sum_{1 \leq i \leq k} e_{r_1} \otimes \mbox{\dots} \otimes e_{r_{i-1}} \otimes f_{r_i} \otimes g_{r_{i+1}} \otimes \mbox{\dots} \otimes g_{r_k}.$$
$(2) \Rightarrow (1)$: We have $F|_{V^{\otimes n}} = f_n + \sum_{i=1}^{n-1} E|_{V^{\otimes i}} \otimes f_{n-i} + \sum_{i=1}^{n-1} f_i \otimes G|_{V^{\otimes n-i}} + \sum_{i=1}^{n-1}\sum_{j = 1}^{n-i-1} E|_{V^{\otimes j}} \otimes f_{i} \otimes G|_{V^{\otimes n-i-j}}$ for all $n \geq 1$. We now verify $1.$ by expanding both sides: $$\left(E \otimes F + F \otimes G\right) \circ C_{\overline{T}V}|_{V^{\otimes n}} = \left(E \otimes F + F \otimes G\right) \circ \sum_{i=1}^{n-1} \left(\mathbb{1}_V^{\otimes n-i}\right) \otimes \left(\mathbb{1}_V^{\otimes i}\right)$$ $$= \sum_{i=1}^{n-1} \left[\left(E|_{V^{\otimes n-i}}\right) \otimes \left(F|_{V^{\otimes i}}\right) + \left(F|_{V^{\otimes n-i}}\right) \otimes \left(G|_{V^{\otimes i}}\right)\right],$$ and by Lemma \[k1l3\] we get $$C_{\overline{T}W} \circ \left(\sum_{i=1}^{n-1} E|_{V^{\otimes i}} \otimes f_{n-i}\right)$$ $$= \sum_{i=1}^{n-1}\left(E|_{V^{\otimes n-i}}\right) \otimes \left(f_{i}\right) + \sum_{i=1}^{n-1} \sum_{j=1}^{n-1-i}\left(E|_{V^{\otimes n-i-j}} \right) \otimes \left(E|_{V^{\otimes j}} \otimes f_{i}\right),$$ $$C_{\overline{T}W} \circ \left(\sum_{i=1}^{n-1} f_i \otimes G|_{V^{\otimes n-i}}\right)$$ $$=\sum_{i=1}^{n-1} \left(f_i\right) \otimes \left(G|_{V^{\otimes n-i}}\right) + \sum_{i=1}^{n-1} \sum_{j=1}^{n-1-i}\left(f_i \otimes G|_{V^{\otimes j}}\right) \otimes \left(G|_{V^{\otimes n-i-j}}\right),$$ $$C_{\overline{T}W} \circ \left(\sum_{i=1}^{n-1}\sum_{j = 1}^{n-i-1} E|_{V^{\otimes j}} \otimes f_{i} \otimes G|_{V^{\otimes n-i-j}}\right)$$ $$=\sum_{i=1}^{n-1}\sum_{j = 1}^{n-i-1} \left(E|_{V^{\otimes n-i-j}}\right) \otimes \left(f_{i} \otimes G|_{V^{\otimes j}}\right) + \sum_{i=1}^{n-1}\sum_{j = 1}^{n-i-1} \left(E|_{V^{\otimes j}} \otimes f_{i}\right) \otimes \left(G|_{V^{\otimes n-i-j}}\right)$$ $$+ \sum_{i=1}^{n-1}\sum_{j = 1}^{n-i-1}\sum_{k=1}^{j-1} \left(E|_{V^{\otimes n-i-j-k}}\right) \otimes \left(E|_{V^{\otimes j}} \otimes f_{i} \otimes G|_{V^{\otimes k}}\right)$$ $$+\sum_{i=1}^{n-1}\sum_{j = 1}^{n-i-1} \sum_{k=1}^{j-1} \left(E|_{V^{\otimes j}} \otimes f_{i} \otimes G|_{V^{\otimes k}}\right) \otimes \left(G|_{V^{\otimes n-i-j-k}}\right).$$ The summation in the variables $i+j$ and $i+j+k$, respectively, yields $$C_{\overline{T}W} \circ \left(\sum_{i=1}^{n-1} E|_{V^{\otimes i}} \otimes f_{n-i}\right)$$ $$= \sum_{i=1}^{n-1}\left(E|_{V^{\otimes n-i}}\right) \otimes \left(f_{i}\right) + \sum_{\ell=2}^{n-1} \sum_{j=1}^{\ell-1}\left(E|_{V^{\otimes n-\ell}} \right) \otimes \left(E|_{V^{\otimes \ell-j}} \otimes f_{j}\right),$$ $$C_{\overline{T}W} \circ \left(\sum_{i=1}^{n-1} f_i \otimes G|_{V^{\otimes n-i}}\right)$$ $$=\sum_{i=1}^{n-1} \left(f_i\right) \otimes \left(G|_{V^{\otimes n-i}}\right) + \sum_{\ell=2}^{n-1} \sum_{j=1}^{\ell-1}\left(f_j \otimes G|_{V^{\otimes \ell - j}}\right) \otimes \left(G|_{V^{\otimes n-\ell}}\right),$$ $$C_{\overline{T}W} \circ \left(\sum_{i=1}^{n-1}\sum_{j = 1}^{n-i-1} E|_{V^{\otimes j}} \otimes f_{i} \otimes G|_{V^{\otimes n-i-j}}\right)$$ $$=\sum_{\ell=2}^{n-1} \sum_{j=1}^{\ell-1} \left(E|_{V^{\otimes n-\ell}}\right) \otimes \left(f_{j} \otimes G|_{V^{\otimes \ell-j}}\right) + \sum_{\ell=2}^{n-1} \sum_{j=1}^{\ell-1} \left(E|_{V^{\otimes \ell-j}} \otimes f_{j}\right) \otimes \left(G|_{V^{\otimes n-\ell}}\right)$$ $$+ \sum_{\ell=3}^{n-1}\sum_{j = 1}^{m-1} \sum_{i=1}^{j-1} \left(E|_{V^{\otimes n-\ell}}\right) \otimes \left(E|_{V^{\otimes \ell-j}} \otimes f_{i} \otimes G|_{V^{\otimes j-i}}\right)$$ $$+\sum_{\ell=3}^{n-1}\sum_{j = 1}^{m-1} \sum_{i=1}^{j-1} \left(E|_{V^{\otimes j-i}} \otimes f_{i} \otimes G|_{V^{\otimes \ell-j}}\right) \otimes \left(G|_{V^{\otimes n-\ell}}\right).$$ Taking all terms of the form $\left(E|_{V^{\otimes n-i}}\right) \otimes \star\,$ and $\,\star \otimes \left(G|_{V^{\otimes n-i}}\right)$ results in $$C_{\overline{T}W} \circ F|_{V^{\otimes n}} = \sum_{i=1}^{n-1} \left[\left(E|_{V^{\otimes n-i}}\right) \otimes \left(F|_{V^{\otimes i}}\right) + \left(F|_{V^{\otimes n-i}}\right) \otimes \left(G|_{V^{\otimes i}}\right)\right]$$ and the implication is proved. Notice that we also proved, on the assumption $F|_{V^{\otimes m}} = f_n + \sum_{B(m)}\sum_{1 \leq i \leq k} e_{r_1} \otimes \mbox{\dots} \otimes e_{r_{i-1}} \otimes f_{r_i} \otimes g_{r_{i+1}} \otimes \mbox{\dots} \otimes g_{r_k}$ for $n > m \geq 1$, that $$C_{\overline{T}W} \circ \Bigg(\sum_{i=1}^{n-1} E|_{V^{\otimes i}} \otimes f_{n-i} + \sum_{i=1}^{n-1} f_i \otimes G|_{V^{\otimes n-i}} + \sum_{i=1}^{n-1}\sum_{j = 1}^{n-i-1}\!E|_{V^{\otimes j}} \otimes f_{i} \otimes G|_{V^{\otimes n-i-j}}\Bigg)$$ $$= \sum_{i=1}^{n-1} \left[\left(E|_{V^{\otimes n-i}}\right) \otimes \left(F|_{V^{\otimes i}}\right) + \left(F|_{V^{\otimes n-i}}\right) \otimes \left(G|_{V^{\otimes i}}\right)\right].$$
$(1) \Rightarrow (2)$: The proof is again by induction. For all $v \in V$ holds $C_{\overline{T}W} \circ F(v) = 0$, which gives $F(V) \subset W$ and so there exists a linear mapping $f_1: V \to W$ such that $F|_V = f_1$. Assume now the claim of the implication is true for all natural numbers less than $n$, i.e. $F|_{V^{\otimes m}} = f_m + \sum_{B(m)}\sum_{1 \leq i \leq k} e_{r_1} \otimes \mbox{\dots} \otimes e_{r_{i-1}} \otimes f_{r_i} \otimes g_{r_{i+1}} \otimes \mbox{\dots} \otimes g_{r_k},$ for $n > m \geq 1$. The proof of the previous implication claims for $F|_{V^{\otimes m}} = f_m + \sum_{B(m), r_i > 0} e_{r_1} \otimes \mbox{\dots} \otimes e_{r_{i-1}} \otimes f_{r_i} \otimes g_{r_{i+1}} \otimes \mbox{\dots} \otimes g_{r_k}$ with $n > m \geq 1$, that $$C_{\overline{T}W} \circ F|_{V^{\otimes n}}=\sum_{i=1}^{n-1} \left[\left(E|_{V^{\otimes n-i}}\right) \otimes \left(F|_{V^{\otimes i}}\right) + \left(F|_{V^{\otimes n-i}}\right) \otimes \left(G|_{V^{\otimes i}}\right)\right]$$ $$=C_{\overline{T}W} \circ \Bigg(\sum_{i=1}^{n-1} E|_{V^{\otimes i}} \otimes f_{n-i} + \sum_{i=1}^{n-1} f_i \otimes G|_{V^{\otimes n-i}} + \sum_{i=1}^{n-1}\sum_{j = 1}^{n-i-1}\!E|_{V^{\otimes j}} \otimes f_{i} \otimes G|_{V^{\otimes n-i-j}}\Bigg).$$ Because $C_{\overline{T}W}$ is linear, $F|_{V^{\otimes n}}$ differs from $\sum_{i=1}^{n-1} E|_{V^{\otimes i}} \otimes f_{n-i} + \sum_{i=1}^{n-1} f_i \otimes G|_{V^{\otimes n-i}} + \sum_{i=1}^{n-1}\sum_{j = 1}^{n-i-1}\!E|_{V^{\otimes j}} \otimes f_{i} \otimes G|_{V^{\otimes n-i-j}}$ by a linear map $f_n: V^{\otimes n} \to W$. This means $F|_{V^{\otimes n}}$ is of the required form and the proof is complete.
\[k1v5\] A linear mapping $\delta: \overline{T}V \to \overline{T}V$ of degree $-1$ fulfills $C \circ \delta = (\delta \otimes {\mathbb 1}_V + \mathbb{1}_V \otimes \delta) \circ C$ if and only if there exist a set of maps $\{\delta_n: V^{\otimes n} \to V\}_{n \geq 1}$ of degree $-1$ such that $\delta|_V = \delta_1$ and for $n\geq 2$ holds $\delta|_{V^{\otimes n}} = \delta_n + \sum_{i = 1}^{n} \mathbb{1}_V^{\otimes i-1} \otimes \delta_1 \otimes \mathbb{1}_V^{\otimes n-i} + \sum_{A(n)} \mathbb{1}_V^{\otimes i-1} \otimes \delta_\ell \otimes \mathbb{1}_V^{\otimes k - i},$ where $A(n)$ is given by .
In Theorem \[k1v4\] we take $E=G=\mathbb{1}_V$, where $e_1 = g_1 = \mathbb{1}_{V}$ and $e_n = g_n = 0$ for $n \geq 2$.
\[k1t6\] Let $\delta: \overline{T}V \to \overline{T}V$ be a linear map of degree $-1$ such that $\delta|_V = \delta_1$ and for $n\geq 2$ holds $\delta|_{V^{\otimes n}} = \delta_n + \sum_{i = 1}^{n} \mathbb{1}_V^{\otimes i-1} \otimes \delta_1 \otimes \mathbb{1}_V^{\otimes n-i} + \sum_{A(n)} \mathbb{1}_V^{\otimes i-1} \otimes \delta_\ell \otimes \mathbb{1}_V^{\otimes k - i}$. Then the following conditions are equivalent:
1. $\delta \circ \delta = 0$,
2. $\delta_1 \circ \delta_1 = 0$ and for all $n \geq 2$ we have $$\label{T6}
\begin{split}
\delta_1(\delta_n) + \sum_{i = 1}^{n} \delta_n \left(\mathbb{1}_V^{\otimes i-1} \otimes \delta_1 \otimes \mathbb{1}_V^{\otimes n-i}\right) \\
+ \sum_{A(n)} \delta_k\left(\mathbb{1}_V^{\otimes i-1} \otimes \delta_\ell \otimes \mathbb{1}_V^{\otimes k - i}\right) = 0,
\end{split}$$
where $A(n)$ is given by .
$(1) \Rightarrow (2)$: The proof goes by induction. By assumption we have for $v \in V$ $\delta(\delta_1(v)) = 0,$ so $\delta_1: V \to V$ implies $\delta_1(\delta_1(v)) = 0.$ Now assume is true for all natural numbers less than $n$. Then $$\delta^2|_{V^{\otimes n}} = \delta_1(\delta_n) + \sum_{i = 1}^{n} \delta|_{V^{\otimes n}} \left(\mathbb{1}_V^{\otimes i-1} \otimes \delta_1 \otimes \mathbb{1}_V^{\otimes n-i}\right)$$$$+ \sum_{A(n)} \delta|_{V^{\otimes k}}\left(\mathbb{1}_V^{\otimes i-1} \otimes \delta_\ell \otimes \mathbb{1}_V^{\otimes k - i}\right).$$ Schematically, this means $$\delta^2|_{V^{\otimes n}} = \delta_1(\delta_n) + \sum_{i = 1}^{n} \delta_n \left(\mathbb{1}_V^{\otimes i-1} \otimes \delta_1 \otimes \mathbb{1}_V^{\otimes n-i}\right)$$$$+ \sum_{A(n)} \delta_k\left(\mathbb{1}_V^{\otimes i-1} \otimes \delta_\ell \otimes \mathbb{1}_V^{\otimes k - i}\right) + \sum \mathbb{1}_V^{\otimes a} \otimes \delta_{b+d+1}\left(\mathbb{1}_V^{\otimes b} \otimes \delta_c \otimes \mathbb{1}_V^{\otimes d}\right) \otimes \mathbb{1}_V^{\otimes e}$$ $$+ \sum \mathbb{1}_V^{\otimes a} \otimes \delta_b \otimes \mathbb{1}_V^{\otimes c} \otimes \delta_d \otimes \mathbb{1}_V^{\otimes e} - \sum \mathbb{1}_V^{\otimes a} \otimes \delta_b \otimes \mathbb{1}_V^{\otimes c} \otimes \delta_d \otimes \mathbb{1}_V^{\otimes e},$$ where the last row is a consequence of the Koszul sign convention: $$\left(\mathbb{1}_V^{\otimes a} \otimes \delta_b \otimes \mathbb{1}_V^{\otimes c + 1 + e}\right)\left(\mathbb{1}_V^{\otimes a + b + c} \otimes \delta_{d} \otimes \mathbb{1}_V^{\otimes e}\right) = \mathbb{1}_V^{\otimes a} \otimes \delta_b \otimes \mathbb{1}_V^{\otimes c} \otimes \delta_d \otimes \mathbb{1}_V^{\otimes e},$$ $$\left(\mathbb{1}_V^{\otimes a + 1 + c} \otimes \delta_{d} \otimes \mathbb{1}_V^{\otimes e}\right)\left(\mathbb{1}_V^{\otimes a} \otimes \delta_{b} \otimes \mathbb{1}_V^{\otimes c+ d + e}\right) = (-1)^{\left|\delta_b \right|\left|\delta_d\right|}\mathbb{1}_V^{\otimes a} \otimes \delta_b \otimes \mathbb{1}_V^{\otimes c} \otimes \delta_d \otimes \mathbb{1}_V^{\otimes e}$$ with $\left|\delta_n\right| = -1$ for all $n \in \mathbb{N}$. The term $\sum \mathbb{1}_V^{\otimes a} \otimes \delta_{b+d+1}\left(\mathbb{1}_V^{\otimes b} \otimes \delta_c \otimes \mathbb{1}_V^{\otimes d}\right) \otimes \mathbb{1}_V^{\otimes e}$ can be written as $$\mathbb{1}_V^{\otimes a} \otimes \delta_{b+d+1}\left(\mathbb{1}_V^{\otimes b} \otimes \delta_c \otimes \mathbb{1}_V^{\otimes d}\right) \otimes \mathbb{1}_V^{\otimes e} = \left(\mathbb{1}_V^{\otimes a} \otimes \delta_{b+d+1} \otimes \mathbb{1}_V^{\otimes e} \right)\left(\mathbb{1}_V^{\otimes a+b} \otimes \delta_c \otimes \mathbb{1}_V^{\otimes d+e}\right).$$ We have $a+b+c+d+e = n$, choose arbitrary $a,e \geq 0$, $1 \leq a + e < n$ and sum over all $b,c,d$ such that $0 \leq b,d \leq n-a-e$ and $1 \leq c \leq n-a-e$ such that $b+c+d = n -e - a$: $$\sum_{b,c,d} \delta_{b+d+1}\left(\mathbb{1}_V^{\otimes b} \otimes \delta_c \otimes \mathbb{1}_V^{\otimes d}\right)
=\delta_1(\delta_{n'}) + \sum_{i = 1}^{n'} \delta_{n'} \left(\mathbb{1}_V^{\otimes i-1} \otimes \delta_1 \otimes \mathbb{1}_V^{\otimes n'-i}\right)$$$$+ \sum_{A(n')} \delta_k\left(\mathbb{1}_V^{\otimes i-1} \otimes \delta_\ell \otimes \mathbb{1}_V^{\otimes k - i}\right),$$ where $n'= n - a - e$. By induction hypothesis, the last display is equal to $0$, and we have $$\sum \mathbb{1}_V^{\otimes a} \otimes \delta_{b+d+1}\left(\mathbb{1}_V^{\otimes b} \otimes \delta_c \otimes \mathbb{1}_V^{\otimes d}\right) \otimes \mathbb{1}_V^{\otimes e}$$ $$= \sum_{a,e} \mathbb{1}_V^{\otimes a} \otimes\left(\sum_{b,c,d} \delta_{b+d+1}\left(\mathbb{1}_V^{\otimes b} \otimes \delta_c \otimes \mathbb{1}_V^{\otimes d}\right)\right) \otimes \mathbb{1}_V^{\otimes e} = \sum_{a,e} \mathbb{1}_V^{\otimes a} \otimes\, 0\, \otimes \mathbb{1}_V^{\otimes e} = 0.$$ Consequently, is true for $n$ and $$\delta_1(\delta_n) + \sum_{i = 1}^{n} \delta_n \left(\mathbb{1}_V^{\otimes i-1} \otimes \delta_1 \otimes \mathbb{1}_V^{\otimes n-i}\right)$$$$+ \sum_{A(n)} \delta_k\left(\mathbb{1}_V^{\otimes i-1} \otimes \delta_\ell \otimes \mathbb{1}_V^{\otimes k - i}\right) = \delta^2|_{V^{\otimes n}} = 0.$$
$(2) \Rightarrow (1)$: The second implication can be easily deduced from the first one.
Morphisms and homotopies
------------------------
\[k1d7\] Let $\delta^V$ be a codifferential on $(\overline{T}V, C)$ and $\delta^W$ be a codifferential on $(\overline{T}W, C)$. A linear mapping $F: \left(\overline{T}V, C, \delta^V\right) \to \left(\overline{T}W, C, \delta^W\right)$ of degree $0$ is called morphism provided $C_{\overline{T}W} \circ F = \left(F \otimes F\right) \circ C_{\overline{T}V}$ and $\delta^W \circ F = F \circ \delta^V.$
\[k1t8\] Let $F: \left(\overline{T}V, \delta^V\right) \to \left(\overline{T}W, \delta^W\right)$ be a linear map of degree $0$. Then the following claims are equivalent:
1. $C_{\overline{T}W} \circ F = \left(F \otimes F\right) \circ C_{\overline{T}V}$,
2. there is a set of linear mappings $\{f_n: V^{\otimes n} \to W\}_{n \geq 1}$ of degree $0$ such that $F|_{V^{\otimes n}} = f_n + \sum_{B(n)} f_{r_1} \otimes \mbox{\dots} \otimes f_{r_k}$, with $B(n)$ given in .
$(2) \Rightarrow (1)$: A consequence of Lemma \[k1l3\].\
$(1) \Rightarrow (2)$ The proof goes by induction. For $v \in V$ we have $C(v) = 0,$ which implies $0 = (F \otimes F) \circ C_{\overline{T}V} = C_{\overline{T}W} \circ F$ and so $F(v) \in W.$\
Assuming the claim is true for all natural numbers less than $n$, $$(F \otimes F) \circ C|_{V^{\otimes n}} = (F \otimes F) \circ \sum_{i = 1}^{n-1} \left(\mathbb{1}^{\otimes i}\right) \otimes \left(\mathbb{1}^{\otimes n-i}\right)$$$$= \sum_{i = 1}^{n-1} \left(F|_{V^{\otimes i}}\right) \otimes \left(F|_{V^{\otimes n-i}}\right)$$ and by induction hypothesis $F|_{V^{\otimes m}} = f_m + \sum_{B(m)} f_{r_1} \otimes \mbox{\dots} \otimes f_{r_k}$ for all $n > m \geq 1$. Lemma \[k1l3\] gives $$\sum_{i = 1}^{n-1} \left(F|_{V^{\otimes i}}\right) \otimes \left(F|_{V^{\otimes n-i}}\right) = C_{\overline{T}W} \circ \left(\sum_{i = 1}^{n-1} f_i \otimes F|_{V^{\otimes n-i}}\right)$$ and because $C_{\overline{T}W}$ is linear, $F|_{V^{\otimes n}}$ differs from $\sum_{i = 1}^{n-1} f_i \otimes F|_{V^{\otimes n-i}}$ by a linear map $f_n: V^{\otimes n} \to W$. Then $F|_{V^{\otimes n}}$ is of the required form and the proof is complete.
\[k1t9\] Let $F: \left(\overline{T}V, \delta^V\right) \to \left(\overline{T}W, \delta^W\right)$ be a linear map of degree $0$ such that $F|_{V^{\otimes n}} = f_n + \sum_{B(n)} f_{r_1} \otimes \mbox{\dots} \otimes f_{r_k},$ with all $\{f_n: V^{\otimes n} \to W\}_{n \geq 1}$ linear of degree $0$. Then the following are equivalent:
1. $\delta^W \circ F = F \circ \delta^V$,
2. for all $n \geq 1$ holds $$\label{T9}
\begin{split}
& \delta^W_1(f_n) + \sum_{B(n)} \delta_{k}^W\left(f_{r_{1}} \otimes \mbox{\dots} \otimes f_{r_{k}} \right)= f_1\left(\delta^V_{n}\right) \\
& +\sum_{i = 1}^{n} f_{n}\left(\mathbb{1}_V^{\otimes i-1} \otimes \delta^V_1 \otimes \mathbb{1}_V^{\otimes n-i}\right) + \sum_{A(n)} f_k\left(\mathbb{1}_V^{\otimes i-1} \otimes \delta^V_\ell \otimes \mathbb{1}_V^{\otimes k - i}\right).
\end{split}$$
$(1) \Rightarrow (2)$: The proof goes by induction. The restriction to $V$, $\delta^W \circ F|_{V} = F \circ \delta^V|_{V}$, corresponds to $\delta_1^W \circ f_1 = f_1 \circ \delta_1^V.$ We now assume applies to all natural numbers less than $n$. We expand both sides of , $$\delta^W \circ F|_{V^{\otimes n}} =$$ $$\delta_1^W\left(f_n\right) + \sum_{B(n)} \sum_{a,b} f_{r_1} \otimes \mbox{\dots} \otimes f_{r_a} \otimes \delta_{b}^W\left(f_{r_{a+1}} \otimes \mbox{\dots} \otimes f_{r_{a+b}}\right) \otimes f_{r_{a+b+1}} \otimes \mbox{\dots} \otimes f_{r_k},$$ $$F \circ \delta^V|_{V^{\otimes n}} =$$ $$f_1\left(\delta^V_1\right) + \sum_{B(n)} \sum_{j, \ell} f_{r_1} \otimes \mbox{\dots} \otimes f_{r_{i-1}} \otimes f_{r_i}\left(\mathbb{1}_V^{\otimes j} \otimes \delta^V_\ell \otimes \mathbb{1}_V^{\otimes r_i - j - 1}\right) \otimes f_{r_{i+1}} \otimes \mbox{\dots} \otimes f_{r_k}$$ and compare the terms of same homogeneities. We fix $j \geq 1$ and $r_1, \mbox{\dots}, r_j \geq 1, r_1+ \mbox{\dots} + r_j < n$ and $0 \leq m \leq j$, and focus on terms of the form $f_{r_1} \star \mbox{\dots} \otimes f_{r_{i-1}} \otimes \star \otimes f_{r_{i}} \otimes \mbox{\dots} \otimes f_{r_j}$, where $\star$ is an expression of the form $\delta_{\star}^W\left(f_{{\star}} \otimes \mbox{\dots} \otimes f_{\star}\right)$ or $f_{\star}\left(\mathbb{1}_V^{\otimes \star} \otimes \delta^V_\star \otimes \mathbb{1}_V^{\otimes \star}\right)$.\
\
Terms on the right hand side of the form $f_{r_1} \otimes \mbox{\dots} \otimes f_{r_{i-1}} \otimes \delta_{\star}^W\left(f_{{\star}} \otimes \mbox{\dots} \otimes f_{\star}\right) \otimes f_{r_{i}} \otimes \mbox{\dots} \otimes f_{r_j}$ correspond to $$f_{r_1} \otimes \mbox{\dots} \otimes f_{r_{m}} \otimes\left(\delta^W_1(f_n') + \sum_{B(n')} \delta_{k}^W\left(f_{r'_{1}} \otimes \mbox{\dots} \otimes f_{r'_{k}}\right) \right)\otimes f_{r_{m+1}} \otimes \mbox{\dots} \otimes f_{r_j},$$ while the terms of the form $f_{r_1} \otimes \mbox{\dots} \otimes f_{r_{i-1}} \otimes f_{\star}\left(\mathbb{1}_V^{\otimes \star} \otimes \delta^V_\star \otimes \mathbb{1}_V^{\otimes \star}\right) \otimes f_{r_{i}} \otimes \mbox{\dots} \otimes f_{r_j}$ correspond to $$f_{r_1} \otimes \mbox{\dots} \otimes f_{r_{m}} \otimes$$ $$\otimes\Bigg(f_1\left(\delta^V_{n'}\right) + \sum_{i = 1}^{n'} f_{n'}\left(\mathbb{1}_V^{\otimes i-1} \otimes \delta^V_1 \otimes \mathbb{1}_V^{\otimes n'-i}\right)$$$$+ \sum_{A(n')} f_k\left(\mathbb{1}_V^{\otimes i-1} \otimes \delta^V_\ell \otimes \mathbb{1}_V^{\otimes k - i}\right)\Bigg) \otimes \otimes f_{r_{m+1}} \otimes \mbox{\dots} \otimes f_{r_j}$$ with $n' = n - r_1+ \mbox{\dots} + r_j.$ Because $n' < n,$ they fulfill the equality and hence are equal. Subtracting from both sides all elements of homogeneity greater than $1$, we arrive at $$\delta^W_1(f_n) + \sum_{B(n)} \delta_{k}^W\left(f_{r_{1}} \otimes \mbox{\dots} \otimes f_{r_{k}} \right)$$ $$= f_1\left(\delta^V_{n}\right) + \sum_{i = 1}^{n} f_{n}\left(\mathbb{1}_V^{\otimes i-1} \otimes \delta^V_1 \otimes \mathbb{1}_V^{\otimes n-i}\right)$$$$+ \sum_{A(n)} f_k\left(\mathbb{1}_V^{\otimes i-1} \otimes \delta^V_\ell \otimes \mathbb{1}_V^{\otimes k - i}\right).$$ However, this equality is true by for $n$.\
$(2) \Rightarrow (1)$: This implication can be again reduced to the previous one.
\[k1d10\] Let $\delta^V$ be a codifferential on $(\overline{T}V, C)$ and $\delta^W$ be a codifferential on $(\overline{T}W, C)$. Let $F: \left(\overline{T}V, C, \delta^V\right) \to \left(\overline{T}W, C, \delta^W\right)$ and $G: \left(\overline{T}V, C, \delta^V\right) \to \left(\overline{T}W, C, \delta^W\right)$ be morphisms. $F$ and $G$ are homotopy equivalent provided there exist linear maps $H: \overline{T}V \to \overline{T}W$ of degree $1$ such that $C_{\overline{T}W} \circ H = \left(F \otimes H + H \otimes G\right) \circ C_{\overline{T}V}$ and $F - G = H\delta^V + \delta^W H.$ The map $H$ is a homotopy between $F$ a $G$.
Theorem \[k1v4\] implies that $H: \overline{T}V \to \overline{T}W$ of degree $1$ fulfills $C_{\overline{T}W} \circ H = \left(F \otimes H + H \otimes G\right) \circ C_{\overline{T}V}$ if and only if there is a set of maps $\{h_n: V^{\otimes n} \to W\}_{n \geq 1}$ of degree $1$ such that $H|_{V^{\otimes n}} = h_n + \sum_{B(n), r_i > 0} f_{r_1} \otimes \mbox{\dots} \otimes f_{r_{i-1}} \otimes h_{r_i} \otimes g_{r_{i+1}} \otimes \mbox{\dots} \otimes g_{r_k}.$
\[k1v11\] We retain the assumptions of Definition \[k1d10\], and in addition assume the existence of the set of linear maps $\{e_n: V^{\otimes n} \to W\}_{n \geq 1}$, $\{g_n: V^{\otimes n} \to W\}_{n \geq 1}$ of even degree $d$ such that $E|_{V^{\otimes n}} = e_n + \sum_{B(n)} e_{r_1} \otimes \mbox{\dots} \otimes e_{r_{k}}$ and $G|_{V^{\otimes n}} = g_n + \sum_{B(n)} g_{r_1} \otimes \mbox{\dots} \otimes g_{r_{k}}$. Let $F: \overline{T}V \to \overline{T}W$ be a linear mapping for which there exists a set of linear maps $\{f_n: V^{\otimes n} \to W\}_{n \geq 1}$ of odd degree $d+1$ fulfilling $$F|_{V^{\otimes n}} = f_n + \sum_{B(n), r_i > 0} e_{r_1} \otimes \mbox{\dots} \otimes e_{r_{i-1}} \otimes f_{r_i} \otimes g_{r_{i+1}} \otimes \mbox{\dots} \otimes g_{r_k}.$$ Then the following assertions are equivalent:
1. $E - G = F\delta^V + \delta^W F$,
2. $e_n - g_n = f_1(\delta^V_n) + \sum_{i=1}^{n}f_n(\mathbb{1}_V^{\otimes i-1} \otimes \delta^V_1 \otimes \mathbb{1}_V^{\otimes n-i}) + \sum_{A(n)} f_k(\mathbb{1}_V^{\otimes i-1} \otimes \delta^V_\ell \otimes \mathbb{1}_V^{\otimes k-i}) + \delta_1^W(f_n) + \sum_{B(n), r_i > 0} \delta^W_k(e_{r_1} \otimes \mbox{\dots} \otimes e_{r_{i-1}} \otimes f_{r_i} \otimes g_{r_{i+1}} \otimes \mbox{\dots} \otimes g_{r_k})$ for all ${n \geq 1}$.
The proof can be done along the same lines as the proofs of Lemma \[k1t6\] and Lemma \[k1t9\].
Codifferentials and $A_\infty$ algebras
---------------------------------------
\[k1d11\] For $V$ graded we define $sV$ in such a way that $\left(sV\right)_i = V_{i-1}$. The graded modules $V$ and $sV$ are canonically isomorphic: $s: V \to sV$ is a linear map of degree $1$ called suspension, $\omega : sV \to V$ is a linear map of degree $-1$ called desuspension.
We have $s^{\otimes n} \otimes \omega^{\otimes n} = (-1)^{\frac{n(n-1)}{2}}$ by the Koszul sign convention.
\[k1v13\] The following claims are equivalent:
1. $\{\mu_n: V^{\otimes n} \to V; |\mu_n| = n-2\}_{n \geq 1}$ is $A_\infty$ structure on $V$,
2. The linear maps $\delta_n = s \circ \mu_n \circ \omega^{\otimes n}$ are of degree $-1$, and are the components of a codifferential on $\overline{T}sV$ in the sense of Theorem \[k1v5\].
$(2) \Rightarrow (1)$: $\delta_n = s \circ \mu_n \circ \omega^{\otimes n}$ are the components of a codifferential, and so we have for all $n \geq 1$ $$\delta_1(\delta_n) + \sum_{i = 1}^{n} \delta_n \left(\mathbb{1}_V^{\otimes i-1} \otimes \delta_1 \otimes \mathbb{1}_V^{\otimes n-i}\right)$$$$+ \sum_{A(n)} \delta_k\left(\mathbb{1}_V^{\otimes i-1} \otimes \delta_\ell \otimes \mathbb{1}_V^{\otimes k - i}\right) = 0.$$ This can be rewritten, by Koszul sign convention, as $$\delta_1(\delta_n) = s \circ \mu_1 \circ \omega \circ s \circ \mu_n \circ \omega^{\otimes n} = s \circ \mu_1(\mu_n) \circ \omega^{\otimes n},$$ $$\sum_{i = 1}^{n} \delta_n \left(\mathbb{1}_V^{\otimes i-1} \otimes \delta_1 \otimes \mathbb{1}_V^{\otimes n-i}\right)
= \sum_{i = 1}^{n} s \circ \mu_n \circ \omega^{\otimes n} \left(\mathbb{1}_V^{\otimes i-1} \otimes s \circ \mu_1 \circ \omega \otimes \mathbb{1}_V^{\otimes n-i}\right)$$ $$=\sum_{i = 1}^{n} (-1)^{n-i} s \circ \mu_n\left(\omega^{\otimes i-1} \otimes \mu_1 \circ \omega \otimes \omega^{\otimes n-i}\right)$$ $$= \sum_{i = 1}^{n} (-1)^{n-i}(-1)^{i-1} s \circ \mu_n\left(\mathbb{1}_V^{\otimes i-1} \otimes \mu_1 \otimes \mathbb{1}_V^{\otimes n-i}\right) \circ \omega^{\otimes n},$$ $$\sum_{A(n)} \delta_k\left(\mathbb{1}_V^{\otimes i-1} \otimes \delta_\ell \otimes \mathbb{1}_V^{\otimes k - i}\right)$$$$= \sum_{A(n)} s \circ \mu_k \circ \omega^{\otimes k}\left(\mathbb{1}_V^{\otimes i-1} \otimes s \circ \mu_\ell \circ \omega^{\otimes \ell} \otimes \mathbb{1}_V^{\otimes k - i}\right)$$ $$= \sum_{A(n)} (-1)^{k-i}s \circ \mu_k \left(\omega^{\otimes i-1} \otimes \mu_\ell \circ \omega^{\otimes \ell} \otimes \omega^{\otimes k - i}\right)$$ $$= \sum_{A(n)} (-1)^{k-i} (-1)^{\ell(i-1)}s \circ \mu_k \left(\mathbb{1}_V^{\otimes i-1} \otimes \mu_\ell \otimes \mathbb{1}_V^{\otimes k - i}\right)\circ \omega^{\otimes n}.$$ The mappings $s$ and $\omega$ are linear, hence $$s \circ \left(\mu_1(\mu_n) + \sum_{i = 1}^{n} (-1)^{n-1} \mu_n\left(\mathbb{1}_V^{\otimes i-1} \otimes \mu_1 \otimes \mathbb{1}_V^{\otimes n-i}\right) \right.$$ $$+\left. \sum_{A(n)} (-1)^{i(\ell+1) + n-1} \mu_k \left(\mathbb{1}_V^{\otimes i-1} \otimes \mu_\ell \otimes \mathbb{1}_V^{\otimes k - i}\right)\right) \circ \omega^{\otimes n} = 0.$$ $(1) \Rightarrow (2)$: This can be easily reduced to the proof of the previous implication.
\[k1v14\] The following claims are equivalent:
1. $\{\varphi_n: V^{\otimes n} \to W; |\varphi_n| = n-1\}_{n \geq 1}$ is $A_\infty$ morphism from $(V, \pmb{\mu})$ to $(W, \pmb{\nu})$,
2. the mappings $$f_n = s_W \circ \varphi_n \circ \omega_V^{\otimes n}$$ are of degree $0$, and are the components of $A_\infty$ morphism from $(\overline{T}sV, \delta^V)$ to $(\overline{T}sW, \delta^W)$ in the sense of Lemma \[k1t8\]. The codifferentials are given by $A_\infty$ structures on $V$ and $W$, respectively, via Theorem \[k1v13\].
The following claims are equivalent:
1. $\{h_n: V^{\otimes n} \to W; |h_n| = n\}_{n \geq 1}$ is $A_\infty$ homotopy between $A_\infty$ morphisms $\pmb{\varphi}$ with components $\{\varphi_n: V^{\otimes n} \to W; |\varphi_n| = n-1\}_{n \geq 1}$ and $\pmb{\psi}$ with components $\{\psi_n: V^{\otimes n} \to W; |\psi_n| = n-1\}_{n \geq 1}$, respectively, from $(V, \pmb{\mu})$ to $(W, \pmb{\nu})$,
2. $$h_n = s_W \circ h_n \circ \omega_V^{\otimes n}$$ are of degree $1$, and are the components of $A_\infty$ homotopy between morphisms $\pmb{F}$ and $\pmb{G}$ from $(\overline{T}sV, \delta^V)$ to $(\overline{T}sW, \delta^W)$, where $\pmb{F}$ corresponds to $\pmb{\varphi}$ and $\pmb{G}$ corresponds to $\pmb{\psi}$ in the sense of the first equivalence in the theorem. The codifferentials are given by $A_\infty$ structures on $V$ and $W$, respectively, as in Theorem \[k1v13\].
The proof goes along the same lines as in Theorem \[k1v13\].
Homotopy transfer of $A_\infty$ algebras {#sect:hotr}
========================================
\[sec:3\]
The starting point for the present section are the chain complexes $(V, \partial_V)$ and $(W, \partial_W)$, $f: V \to W$, $g: W \to V$ their morphisms such that $gf$ is homotopy equivalent to $\mathbb{1}_V$ by a homotopy $h$. Let $(V, \partial_V)$ be equipped with $A_\infty$ algebra structure, which means that there is a set of multilinear maps $\pmb{\mu} = (\mu_2, \mu_3, \mbox{\dots})$ satisfying the relations . We would like to induce $A_\infty$ structure $(W, \partial_W, \nu_2, \nu_3, \mbox{\dots})$ on $(W, \partial_W)$ by transferring $(V, \partial_V, \mu_2, \mu_3, \mbox{\dots})$, as well as the morphisms of $A_\infty$ algebras $\pmb{\psi} = (g, \psi_2, \psi_3, \mbox{\dots})$ from $(W, \partial_W, \pmb{\nu})$ to $(V, \partial_V, \pmb{\mu})$ and $\pmb{\varphi} = (f, \varphi_2, \varphi_3, \mbox{\dots})$ acting in the opposite direction such that their composition $\pmb{\psi \varphi}$ is $A_\infty$ homotopy equivalent with the identity map via $\pmb{H}= (h, H_2, H_3, \mbox{\dots})$.
The strategy to solve this problem, cf. [@Markl06], suggests to construct the set of maps $\{\pmb{p}_n: V^{\otimes n} \to V\}_{n \geq 2}$ of degree $n-2$ called $\pmb{p}-$kernels, and the set of maps $\{\pmb{q}_n: V^{\otimes n} \to V\}_{n \geq 1}$ of degree $n-1$ called $\pmb{q}-$kernels in such a way that $\nu_n, \varphi_n, \psi_n$ and $H_n$ defined by $$\label{Antanz}
\nu_n := f \circ \pmb{p}_n \circ g^{\otimes n}, \;\;\; \varphi_n := f \circ \pmb{q}_n, \;\;\; \psi_n := h \circ \pmb{p}_n \circ g^{\otimes n}, \;\;\; H_n = h \circ \pmb{q}_n,$$ fulfill the transfer problem of $A_\infty$ algebra as discussed in the previous paragraph.
We shall first introduce the $\pmb{p}-$kernels and based on them we introduce the $\pmb{q}-$kernels later on. Apart from a , we shall rely on the notation (cf., [@Markl06]) $$\tag{C}
\label{C}
\begin{split}
C(n) = \{k,i, r_1, \mbox{\dots}, r_i\in{\mathbb N}| \, 2 \leq k \leq n, 1 \leq i \leq k,\\ r_1, \mbox{\dots}, r_i \geq 1, r_1 + \mbox{\dots} + r_i + k - i = n\},
\end{split}$$ for $n \in \mathbb{N}$, and $$\tag{$\vartheta$}
\label{theta}
\vartheta(u_1, \mbox{\dots}, u_k) = \sum_{1 \leq i < j \leq k} u_i(u_j + 1),$$ for arbitrary $u_1, \mbox{\dots}, u_k, k \in \mathbb{N}$.
p-kernels
---------
\[k2t15\] The $\pmb{p}-$kernels together with $\partial_W$ constitute an $A_\infty$ structure on $(W, \partial_W)$ via if and only if for all $n \geq 2$ holds $$f \circ \left(\partial_V\pmb{p}_n - \sum_{u=1}^{n}(-1)^{n} \pmb{p}_n(\mathbb{1}_V^{\otimes u-1} \otimes \partial_V \otimes \mathbb{1}_V^{\otimes n-u}) \right.$$ $$\left. - \sum_{A(n)}(-1)^{i(\ell+1) + n}\pmb{p}_k(\mathbb{1}_V^{\otimes i-1} \otimes gf \circ \pmb{p}_\ell \otimes \mathbb{1}_V^{\otimes k - i})\right) \circ g^{\otimes n} = 0$$
$(W, \partial_W, \nu_2 \mbox{\dots})$ is an $A_\infty$ algebra if we have for all $n \geq 1$ $$\partial_W \nu_n - \sum_{u=1}^{n}(-1)^{n} \nu_n(\mathbb{1}_W^{\otimes u-1} \otimes \partial_W \otimes \mathbb{1}_W^{\otimes n-u})$$ $$- \sum_{A(n)}(-1)^{i(\ell+1) + n}\nu_k(\mathbb{1}_W^{\otimes i-1} \otimes \nu_\ell \otimes \mathbb{1}_W^{\otimes k - i})= 0.$$ This is true for $n = 1$, because $(W, \partial_W)$ is the chain complex ($f \circ \partial_V = \partial_W \circ f$ and analogously for $g$.) Now expand $\nu_n$ following : $$\partial_W \nu_n - \sum_{u=1}^{n}(-1)^{n} \nu_n(\mathbb{1}_W^{\otimes u-1} \otimes \partial_W \otimes \mathbb{1}_W^{\otimes n-u})$$$$- \sum_{A(n)}(-1)^{i(\ell+1) + n}\nu_k(\mathbb{1}_W^{\otimes i-1} \otimes \nu_\ell \otimes \mathbb{1}_W^{\otimes k - i})$$ $$= \partial_W \left(f \circ \pmb{p}_n \circ g^{\otimes n} \right) - \sum_{u=1}^{n}(-1)^{n} \left(f \circ \pmb{p}_n \circ g^{\otimes n} \right)(\mathbb{1}_W^{\otimes u-1} \otimes \partial_W \otimes \mathbb{1}_W^{\otimes n-u})$$ $$- \sum_{A(n)}(-1)^{i(\ell+1) + n}\left(f \circ \pmb{p}_k \circ g^{\otimes k} \right)(\mathbb{1}_W^{\otimes i-1} \otimes \left(f \circ \pmb{p}_\ell \circ g^{\otimes \ell} \right) \otimes \mathbb{1}_W^{\otimes k - i}).$$ Because both $f$ and $g$ are linear maps of degree $0$, this equals to $$f\circ \left(\partial_V \circ \pmb{p}_n \right) \circ g^{\otimes n} - f \circ \left(\sum_{u=1}^{n}(-1)^{n} \pmb{p}_n(g^{\otimes u-1} \otimes g \circ \partial_W \otimes g^{\otimes n-u})\right)$$ $$- f \circ\left(\sum_{A(n)}(-1)^{i(\ell+1) + n}\pmb{p}_k(g^{\otimes i-1} \otimes gf \circ \pmb{p}_\ell \circ g^{\otimes \ell} \otimes g^{\otimes k - i})\right),$$ which is $$f\circ \left(\partial_V \circ \pmb{p}_n \right) \circ g^{\otimes n} - f \circ \left(\sum_{u=1}^{n}(-1)^{n} \pmb{p}_n(\mathbb{1}_V^{\otimes u-1} \otimes \partial_V \otimes \mathbb{1}_V^{\otimes n-u})\right) \circ g^{\otimes n}$$ $$- f \circ\left(\sum_{A(n)}(-1)^{i(\ell+1) + n}\pmb{p}_k(\mathbb{1}_V^{\otimes i-1} \otimes gf \circ \pmb{p}_\ell \otimes \mathbb{1}_V^{\otimes k - i})\right)\circ g^{\otimes n} = 0.$$
\[k2t16\] Let us assume that $\pmb{p}-$kernels induce the transfer of $A_\infty$ algebra as formulated above, and they fulfill ($n \geq 2$) $$\label{T16}
\begin{split}
\partial_V\pmb{p}_n - \sum_{u=1}^{n}(-1)^{n} \pmb{p}_n(\mathbb{1}_V^{\otimes u-1} \otimes \partial_V \otimes \mathbb{1}_V^{\otimes n-u})\\
-\sum_{A(n)}(-1)^{i(\ell+1) + n}\pmb{p}_k(\mathbb{1}_V^{\otimes i-1} \otimes gf \circ \pmb{p}_\ell \otimes \mathbb{1}_V^{\otimes k - i})= 0.
\end{split}$$ Then $$\pmb{p}_n \circ g^{\otimes n}= \left(\sum_{B(n)} (-1)^{\vartheta(r_1, \mbox{\dots}, r_k)}\mu_k(h \circ \pmb{p}_{r_1} \otimes \mbox{\dots} \otimes h \circ \pmb{p}_{r_k})\right) \circ g^{\otimes n},$$ where we define $h \circ \pmb{p}_1 = \mathbb{1}_V$.
According to these $\pmb{p}-$kernels induce $A_\infty$ structure on $(W, \partial_W)$ by Lemma \[k2t15\]. It remains to verify that they give $A_\infty$ morphism from $(V, \partial_V, \pmb{\mu})$ to $(W, \partial_W, \pmb{\nu})$, i.e. $$\partial_V \psi_n + \sum_{B(n)} (-1)^{\vartheta(r_1, \mbox{\dots}, r_k)}\mu_k(\psi_{r_1} \otimes \mbox{\dots} \otimes \psi_{r_k})$$ $$= \psi_1 \nu_n -\sum_{u=1}^{n}(-1)^{n} \psi_n(\mathbb{1}_W^{\otimes u-1} \otimes \partial_W \otimes \mathbb{1}_W^{\otimes n-u})$$$$- \sum_{A(n)}(-1)^{i(\ell+1) + n}\psi_k(\mathbb{1}_W^{\otimes i-1} \otimes \nu_\ell \otimes \mathbb{1}_W^{\otimes k - i}),$$ which by can be formulated as $$\partial_V h \circ \pmb{p}_n \circ g^{\otimes n} + \left(\sum_{B(n)} (-1)^{\vartheta(r_1, \mbox{\dots}, r_k)}\mu_k(h \circ \pmb{p}_{r_1} \otimes \mbox{\dots} \otimes h \circ \pmb{p}_{r_k})\right) \circ g^{\otimes n}$$ $$= gf \circ \pmb{p}_n \circ g^{\otimes n} -h \circ \left(\sum_{u=1}^{n}(-1)^{n} \pmb{p}_n(\mathbb{1}_V^{\otimes u-1} \otimes \partial_V \otimes \mathbb{1}_V^{\otimes n-u})\right) \circ g^{\otimes n}$$ $$- h \circ \left(\sum_{A(n)}(-1)^{i(\ell+1) + n}\pmb{p}_k(\mathbb{1}_V^{\otimes i-1} \otimes gf \circ \pmb{p}_\ell \otimes \mathbb{1}_V^{\otimes k - i})\right)\circ g^{\otimes n}.$$ Due to $gf - \mathbb{1}_V = \partial_V h + h \partial_V,$ we have $$\left(\sum_{B(n)} (-1)^{\vartheta(r_1, \mbox{\dots}, r_k)}\mu_k(h \circ \pmb{p}_{r_1} \otimes \mbox{\dots} \otimes h \circ \pmb{p}_{r_k})\right) \circ g^{\otimes n}$$ $$= \pmb{p}_n \circ g^{\otimes n} -h \circ \left(-\partial_V \pmb{p}_n + \sum_{u=1}^{n}(-1)^{n} \pmb{p}_n(\mathbb{1}_V^{\otimes u-1} \otimes \partial_V \otimes \mathbb{1}_V^{\otimes n-u})\right) \circ g^{\otimes n}$$ $$- h \circ \left(\sum_{A(n)}(-1)^{i(\ell+1) + n}\pmb{p}_k(\mathbb{1}_V^{\otimes i-1} \otimes gf \circ \pmb{p}_\ell \otimes \mathbb{1}_V^{\otimes k - i})\right)\circ g^{\otimes n}.$$ By assumption , we obtain $$h \circ \left(-\partial_V \pmb{p}_n + \sum_{u=1}^{n}(-1)^{n} \pmb{p}_n(\mathbb{1}_V^{\otimes u-1} \otimes \partial_V \otimes \mathbb{1}_V^{\otimes n-u})\right) \circ g^{\otimes n}$$ $$+ h \circ \left(\sum_{A(n)}(-1)^{i(\ell+1) + n}\pmb{p}_k(\mathbb{1}_V^{\otimes i-1} \otimes gf \circ \pmb{p}_\ell \otimes \mathbb{1}_V^{\otimes k - i})\right) \circ g^{\otimes n} = 0,$$ which reduces to $$\left(\pmb{p}_n-\sum_{B(n)} (-1)^{\vartheta(r_1, \mbox{\dots}, r_k)}\mu_k(h \circ \pmb{p}_{r_1} \otimes \mbox{\dots} \otimes h \circ \pmb{p}_{r_k})\right) \circ g^{\otimes n}= 0.$$
The assumption of Lemma \[k2t16\] can be weaken to $$\label{T16g}
\begin{split}
\partial_V\pmb{p}_n\circ g^{\otimes n} - \sum_{u=1}^{n}(-1)^{n} \pmb{p}_n(\mathbb{1}_V^{\otimes u-1} \otimes \partial_V \otimes \mathbb{1}_V^{\otimes n-u})\circ g^{\otimes n} \\
-\sum_{A(n)}(-1)^{i(\ell+1) + n}\pmb{p}_k(\mathbb{1}_V^{\otimes i-1} \otimes gf \circ \pmb{p}_\ell \otimes \mathbb{1}_V^{\otimes k - i}) \circ g^{\otimes n} = 0,
\end{split}$$ where is fulfilled if the $\pmb{p}-$kernels define $A_\infty$ structure on $(W, \partial_W)$, and $f$ is a monomorphism. In the situation of interest is $f$, however, assumed to be an epimorphism.\
\[k2d17\] We define for each $n \geq 2$: $$\begin{aligned}
\label{pkernels}
\pmb{p}_n = \sum_{B(n)} (-1)^{\vartheta(r_1, \mbox{\dots}, r_k)}\mu_k(h \circ \pmb{p}_{r_1} \otimes \mbox{\dots} \otimes h \circ \pmb{p}_{r_k}),\end{aligned}$$ where $h \circ \pmb{p}_1 = \mathbb{1}_V$, with $B(n)$ given in and $\vartheta(r_1, \mbox{\dots}, r_k)$ given in .
For $\pmb{p}-$kernels there exists a non-inductive explicit expression. Each term in the $\pmb{p}-$kernel can be represented by a rooted plane tree, and there is a function which associates to a rooted plane tree a sign corresponding to our inductive definition.
\[k2v18\] The $\pmb{p}-$kernels introduced in [@Markl06] satisfy $$\tag{\ref{T16}}\label{pkernelform}
\begin{split}
\partial_V\pmb{p}_n - \sum_{u=1}^{n}(-1)^{n} \pmb{p}_n(\mathbb{1}_V^{\otimes u-1} \otimes \partial_V \otimes \mathbb{1}_V^{\otimes n-u})\\
-\sum_{A(n)}(-1)^{i(\ell+1) + n}\pmb{p}_k(\mathbb{1}_V^{\otimes i-1} \otimes gf \circ \pmb{p}_\ell \otimes \mathbb{1}_V^{\otimes k - i})= 0,
\end{split}$$ for all $n \geq 2$.
Let us first simplify our situation by passing to the suspension $\overline{T}sV$ with the induced codifferential $\delta$. Because $s$ and $\omega$ are by Definition \[k1d11\] izomorphisms, is true if and only if $$s\circ \left(\partial_V\pmb{p}_n - \sum_{u=1}^{n}(-1)^{n} \pmb{p}_n(\mathbb{1}_V^{\otimes u-1} \otimes \partial_V \otimes \mathbb{1}_V^{\otimes n-u})\right)\circ \omega^{\otimes n}$$ $$= s \circ \left(\sum_{A(n)}(-1)^{i(\ell+1) + n}\pmb{p}_k(\mathbb{1}_V^{\otimes i-1} \otimes gf \circ \pmb{p}_\ell \otimes \mathbb{1}_V^{\otimes k - i})\right) \circ \omega^{\otimes n}.$$ Introducing $\hat{\pmb{p}}_m = s \circ \pmb{p}_m \circ \omega^{\otimes m},$ $\hat{g} = s \circ g \circ \omega$ and $\hat{f} = s \circ f \circ \omega$ ($|\hat{\pmb{p}}_m| = -1$, $|\hat{g}| = |\hat{f}| = 0$), we have $$\delta_1\hat{\pmb{p}}_n + \sum_{u=1}^{n}\hat{\pmb{p}}_n(\mathbb{1}_V^{\otimes u-1} \otimes \delta_1 \otimes \mathbb{1}_V^{\otimes n-u}) + \sum_{A(n)}\hat{\pmb{p}}_k(\mathbb{1}_V^{\otimes i-1} \otimes \hat{g}\hat{f} \circ \hat{\pmb{p}}_\ell \otimes \mathbb{1}_V^{\otimes k - i}) = 0.$$ The proof of the last claim goes by induction. The case $n = 2$ corresponds to $$\delta_1 \delta_2 + \delta_2(\mathbb{1}_V \otimes \delta_1) + \delta_2(\delta_1 \otimes \mathbb{1}_V) = 0,$$ which is certainly true because $\{\delta_n: V^{\otimes n} \to V\}_{n \geq 1}$ are the components of the codifferential on $\overline{T}sV$ (cf., for $n = 2$ in Lemma \[k1t6\].)\
By induction hypothesis, we assume the claim is true for all natural numbers less than $n$. The proof is naturally divided into three steps:\
**I.** We shall first expand the term $\delta_1 \hat{\pmb{p}}_n$: we have $\hat{\pmb{p}}_n = s \circ \pmb{p}_n \circ \omega^{\otimes n},$ so by Definition \[k2d17\] $$\hat{\pmb{p}}_n = s \circ \left(\sum_{B(n)} (-1)^{\vartheta(r_1, \mbox{\dots}, r_k)}\mu_k(h \circ \pmb{p}_{r_1} \otimes \mbox{\dots} \otimes h \circ \pmb{p}_{r_k})\right) \circ \omega^{\otimes n}$$$$=\sum_{B(n)} (-1)^{\vartheta(r_1, \mbox{\dots}, r_k)}(-1)^{\sigma}s \circ \mu_k(\omega \circ s \circ h \circ \pmb{p}_{r_1} \circ \omega^{\otimes r_1} \otimes \mbox{\dots}$$$$\mbox{\dots} \otimes \omega \circ s \circ h \circ \pmb{p}_{r_k} \circ \omega^{\otimes r_k})$$ with $\sigma = \sum_{1 \leq i < j \leq k} r_i(r_j + 1)$. However, $|s \circ h \circ \pmb{p}_{r_i} \circ \omega^{\otimes r_i}| = 1 + 1 + (r_i - 2) - r_i = 0$, so the last display equals to $$\begin{aligned}
\nonumber
\sum_{B(n)} s \circ \mu_k \circ \omega^{\otimes k}(s \circ h \circ \omega \circ s \circ \pmb{p}_{r_1} \circ \omega^{\otimes r_1} \otimes \mbox{\dots} \otimes s \circ h \circ \omega \circ s \circ \pmb{p}_{r_k} \circ \omega^{\otimes r_k}).\end{aligned}$$ Consequently, $$\label{p-kernel}
\hat{\pmb{p}}_n = \sum_{B(n)} \delta_k(\hat{h} \circ \hat{\pmb{p}}_{r_1} \otimes \mbox{\dots} \otimes \hat{h} \circ \hat{\pmb{p}}_{r_k}),\quad \hat{h} = s \circ h \circ \omega\quad (|\hat{h}| =1),$$ and so $$\begin{aligned}
\delta_1 \hat{\pmb{p}}_n =& \sum_{B(n)} \delta_1\delta_{k}(\hat{h} \circ \hat{\pmb{p}}_{r_1} \otimes \mbox{\dots} \otimes \hat{h} \circ \hat{\pmb{p}}_{r_{k}})
\nonumber \\
=& -\sum_{B(n)} \left(\sum_{i = 1}^{k} \delta_k \left(\mathbb{1}_V^{\otimes i-1} \otimes \delta_1 \otimes \mathbb{1}_V^{\otimes k-i}\right)\right)(\hat{h} \circ \hat{\pmb{p}}_{r_1} \otimes \mbox{\dots} \otimes \hat{h} \circ \hat{\pmb{p}}_{r_k})
\nonumber \\ \nonumber
&- \sum_{B(n)}\left(\sum_{A(k)} \delta_{k'}\left(\mathbb{1}_V^{\otimes i-1} \otimes \delta_\ell \otimes \mathbb{1}_V^{\otimes k' - i}\right) \right)(\hat{h} \circ \hat{\pmb{p}}_{r_1} \otimes \mbox{\dots} \otimes \hat{h} \circ \hat{\pmb{p}}_{r_k}).\end{aligned}$$ The last summation can be rewritten as $$\sum_{B(n)}\left(\sum_{A(k)} \delta_{k'}\left(\mathbb{1}_V^{\otimes i-1} \otimes \delta_\ell \otimes \mathbb{1}_V^{\otimes k' - i}\right) \right)(\hat{h} \circ \hat{\pmb{p}}_{r_1} \otimes \mbox{\dots} \otimes \hat{h} \circ \hat{\pmb{p}}_{r_k})$$ $$= \sum_{B(n)}\sum_{A(k)} \delta_{k'}\left(\hat{h} \circ \hat{\pmb{p}}_{r_1} \otimes \mbox{\dots} \otimes \delta_\ell(\hat{h} \circ \hat{\pmb{p}}_{r_i} \otimes \mbox{\dots} \otimes \hat{h} \circ \hat{\pmb{p}}_{r_{i+\ell}}) \otimes \mbox{\dots} \otimes \hat{h} \circ \hat{\pmb{p}}_{r_k}\right)$$ $$= \sum_{B(n),\, r_i > 1} \delta_{k}(\hat{h} \circ \hat{\pmb{p}}_{r_1} \otimes \mbox{\dots} \otimes \hat{\pmb{p}}_{r_i} \otimes \mbox{\dots} \otimes \hat{h} \circ \hat{\pmb{p}}_{r_{k}}),$$ where the last equality comes from the summation over all $r_1,\ldots,r_{i+l}$ with $r_i + \mbox{\dots} + r_{i+\ell}$ fixed. We conclude $$\begin{aligned}
\delta_1 \hat{\pmb{p}}_n = & -\sum_{B(n), r_i > 1} \delta_{k}(\hat{h} \circ \hat{\pmb{p}}_{r_1} \otimes \mbox{\dots} \otimes (\delta_1\hat{h} + \mathbb{1}_V)\hat{\pmb{p}}_{r_i} \otimes \mbox{\dots} \otimes \hat{h} \circ \hat{\pmb{p}}_{r_{k}})
\nonumber \\ \nonumber
& -\sum_{B(n), r_i = 1} \delta_{k}(\hat{h} \circ \hat{\pmb{p}}_{r_1} \otimes \mbox{\dots} \otimes \delta_1\hat{h}\circ \hat{\pmb{p}}_{r_i} \otimes \mbox{\dots} \otimes \hat{h} \circ \hat{\pmb{p}}_{r_{k}}).\end{aligned}$$
**II.** We shall apply the induction hypothesis to $\delta_1 \hat{\pmb{p}}_n$. We remind the formal equality $\hat{h} \circ \hat{\pmb{p}}_{1} = \mathbb{1}_V$ and also $gf - \mathbb{1}_V = \partial_Vh + h\partial_V$ equivalent to $\delta_1 \hat{h} + \mathbb{1}_V = \hat{g}\hat{f} - \hat{h} \delta_1$. Then $$\begin{aligned}
\label{V24_1}
\delta_1 \hat{\pmb{p}}_n &= \sum_{B(n), r_i > 1} \delta_{k}(\hat{h} \circ \hat{\pmb{p}}_{r_1} \otimes
\mbox{\dots} \otimes (\hat{h}\delta_1 - \hat{g}\hat{f})\hat{\pmb{p}}_{r_i} \otimes \mbox{\dots} \otimes \hat{h} \circ \hat{\pmb{p}}_{r_{k}}) \nonumber \\
&-\sum_{B(n), r_i = 1} \delta_{k}(\hat{h} \circ \hat{\pmb{p}}_{r_1} \otimes \mbox{\dots} \otimes \delta_1\hat{h}\circ \hat{\pmb{p}}_{r_i} \otimes \mbox{\dots} \otimes \hat{h} \circ \hat{\pmb{p}}_{r_{k}}).\end{aligned}$$ The second part of the first term on the right hand side equals $$-\sum_{B(n)} \delta_{k}(\hat{h} \circ \hat{\pmb{p}}_{r_1} \otimes \mbox{\dots} \otimes \hat{g}\hat{f} \circ \hat{\pmb{p}}_{r_i} \otimes \mbox{\dots} \otimes \hat{h} \circ \hat{\pmb{p}}_{r_{k}})$$ $$= \sum_{B(n)} \delta_{k}(\hat{h} \circ \hat{\pmb{p}}_{r_1} \otimes \mbox{\dots} \otimes \hat{h} \circ \hat{\pmb{p}}_{1} \otimes \mbox{\dots} \otimes \hat{h} \circ \hat{\pmb{p}}_{r_{k}})(\mathbb{1}_V^{\otimes s} \otimes \hat{g}\hat{f} \circ\hat{\pmb{p}}_{r_i} \otimes \mathbb{1}_V^{\otimes n - s - r_i}),$$ where $s = \sum_{j < i} r_j.$ The second term in equals $$-\sum_{B(n), r_i = 1} \delta_{k}(\hat{h} \circ \hat{\pmb{p}}_{r_1} \otimes \mbox{\dots} \otimes \delta_1\hat{h}\circ \hat{\pmb{p}}_{r_i} \otimes \mbox{\dots} \otimes \hat{h} \circ \hat{\pmb{p}}_{r_{k}})$$ $$= -\sum_{B(n), r_i = 1} \delta_{k}(\hat{h} \circ \hat{\pmb{p}}_{r_1} \otimes \mbox{\dots} \otimes \hat{h}\circ \hat{\pmb{p}}_{r_i} \otimes \mbox{\dots} \otimes \hat{h} \circ \hat{\pmb{p}}_{r_{k}})(\mathbb{1}_V^{\otimes s} \otimes \delta_1 \otimes \mathbb{1}_V^{\otimes n - s - r_i}).$$ By induction hypothesis, we have for all $m < n$ $$\delta_1\hat{\pmb{p}}_m = -\sum_{u=1}^{m}\hat{\pmb{p}}_m(\mathbb{1}_V^{\otimes u-1} \otimes \delta_1 \otimes \mathbb{1}_V^{\otimes m-u}) - \sum_{A(m)}\hat{\pmb{p}}_k(\mathbb{1}_V^{\otimes i-1} \otimes \hat{g}\hat{f} \circ \hat{\pmb{p}}_\ell \otimes \mathbb{1}_V^{\otimes k - i}).$$ Finally, the first part of the first term equals $$\sum_{B(n), r_i > 1} \delta_{k}(\hat{h} \circ \hat{\pmb{p}}_{r_1} \otimes \mbox{\dots} \otimes \hat{h} \circ\delta_1 \hat{\pmb{p}}_{r_i} \otimes \mbox{\dots} \otimes \hat{h} \circ \hat{\pmb{p}}_{r_{k}})$$ $$=-\sum_{B(n), r_i > 1} \delta_{k}\left(\hat{h} \circ \hat{\pmb{p}}_{r_1} \otimes \mbox{\dots} \otimes \sum_{u=1}^{r_i}\hat{h} \circ \hat{\pmb{p}}_{r_i}(\mathbb{1}_V^{\otimes u-1} \otimes \delta_1 \otimes \mathbb{1}_V^{\otimes r_i-u}) \otimes \mbox{\dots} \otimes \hat{h} \circ \hat{\pmb{p}}_{r_{k}}\right)$$ $$- \sum_{B(n), r_i > 1} \delta_{k}\!\!\left(\hat{h} \circ \hat{\pmb{p}}_{r_1} \otimes \!\mbox{\dots}\! \otimes \! \sum_{A(r_i)}\hat{h} \circ \hat{\pmb{p}}_k(\mathbb{1}_V^{\otimes i-1} \otimes \hat{g}\hat{f} \circ \hat{\pmb{p}}_\ell \otimes \mathbb{1}_V^{\otimes k - i}) \otimes \!\mbox{\dots}\! \otimes \hat{h} \circ \hat{\pmb{p}}_{r_{k}}\right)\!\!.$$ **III.** Now we pair up the contributions appearing in the previous step: the right hand side of can be rewritten as $$\begin{aligned}
\tag{P1}
\label{P1}
&-\sum_{B(n), r_i > 1} \delta_{k}\!\!\left(\hat{h} \circ \hat{\pmb{p}}_{r_1} \otimes \mbox{\dots} \otimes \sum_{u=1}^{r_i}\hat{h} \circ \hat{\pmb{p}}_{r_i}(\mathbb{1}_V^{\otimes u-1} \otimes \delta_1 \otimes \mathbb{1}_V^{\otimes r_i-u}) \otimes \mbox{\dots} \otimes \hat{h} \circ \hat{\pmb{p}}_{r_{k}}\right)\\
\tag{P2}
\label{P2}
&-\sum_{B(n), r_i > 1} \delta_{k}\!\!\left(\hat{h} \circ \hat{\pmb{p}}_{r_1}\! \otimes \!\mbox{\dots} \! \otimes \! \sum_{A(r_i)}\hat{h} \circ \hat{\pmb{p}}_k(\mathbb{1}_V^{\otimes i-1} \otimes \hat{g}\hat{f} \circ \hat{\pmb{p}}_\ell \! \otimes \! \mathbb{1}_V^{\otimes k - i}) \otimes \! \mbox{\dots} \! \otimes \hat{h} \circ \hat{\pmb{p}}_{r_{k}}\right)\\
\tag{P3}
\label{P3}
&-\sum_{B(n)} \delta_{k}(\hat{h} \circ \hat{\pmb{p}}_{r_1} \otimes \mbox{\dots} \otimes \hat{h} \circ \hat{\pmb{p}}_{1} \otimes \mbox{\dots} \otimes \hat{h} \circ \hat{\pmb{p}}_{r_{k}})(\mathbb{1}_V^{\otimes s} \otimes \hat{g}\hat{f} \circ\hat{\pmb{p}}_{r_i} \otimes \mathbb{1}_V^{\otimes n - s - r_i}) \\
\tag{P4}
\label{P4}
&-\sum_{B(n), r_i = 1} \delta_{k}(\hat{h} \circ \hat{\pmb{p}}_{r_1} \otimes \mbox{\dots} \otimes \hat{h}\circ \hat{\pmb{p}}_{r_i} \otimes \mbox{\dots} \otimes \hat{h} \circ \hat{\pmb{p}}_{r_{k}})(\mathbb{1}_V^{\otimes s} \otimes \delta_1 \otimes \mathbb{1}_V^{\otimes n - s - r_i}),\end{aligned}$$ with $s = \sum_{j < i} r_j$, and we get $$\begin{aligned}
\eqref{P1} + \eqref{P4} &= -\sum_{u=1}^{n}\hat{\pmb{p}}_n(\mathbb{1}_V^{\otimes u-1} \otimes \delta_1 \otimes \mathbb{1}_V^{\otimes n-u}),\\
\eqref{P2} + \eqref{P3} &=- \sum_{A(n)}\hat{\pmb{p}}_k(\mathbb{1}_V^{\otimes i-1} \otimes \hat{g}\hat{f} \circ \hat{\pmb{p}}_\ell \otimes \mathbb{1}_V^{\otimes k - i}).\end{aligned}$$
Theorem \[k2v18\] implies that the $\pmb{p}-$kernels in [@Markl06] fulfill .
q-kernels
---------
\[k2t19\] The $\pmb{q}-$kernels constitute $A_\infty$ morphism $\pmb{\varphi} = (f, \varphi_2, \varphi_3, \mbox{\dots})$, $\varphi_n = f \circ \pmb{q}_n$ and $\nu_n = f \circ \pmb{p}_n \circ g^{\otimes n}$, from $(V, \partial_V, \mu_2, \mu_3, \mbox{\dots})$ to $(W, \partial_W, \nu_2,$$\nu_3, \mbox{\dots})$ if and only if for all $n \geq 2$: $$f \circ \left(\partial_V\pmb{q}_n + \sum_{u=1}^{n}(-1)^{n} \pmb{q}_n(\mathbb{1}_V^{\otimes u-1} \otimes \partial_V \otimes \mathbb{1}_V^{\otimes n-u}) \right.$$ $$+ \sum_{B(n)}(-1)^{\vartheta(r_1, \mbox{\dots}, r_k)}\pmb{p}_k(gf \circ \pmb{q}_{r_1}\! \otimes \mbox{\dots} \otimes gf \circ \pmb{q}_{r_k})$$ $$\left. +\sum_{A(n)}(-1)^{i(\ell + 1) + n}\pmb{q}_{k}(\mathbb{1}_V^{\otimes i-1} \otimes \mu_{\ell} \otimes \mathbb{1}_V^{\otimes n-k}) - \pmb{q}_1\mu_n\!\right)\! = 0.$$
The proof easily follows from the explicit expansion of $A_\infty$ morphism $\pmb{\varphi} = (f, \varphi_2, \varphi_3, \mbox{\dots})$, which maps $(V, \partial_V, \mu_2, \mu_3, \mbox{\dots})$ to $(W, \partial_W,$$ \nu_2, \nu_3, \mbox{\dots})$ for $\varphi_n = f \circ \pmb{q}_n$ and $\nu_n = f \circ \pmb{p}_n \circ g^{\otimes n}$ (cf. ).
\[k2t20\] Let the $\pmb{q}-$kernels fulfill $$\begin{aligned}
\label{T20}
& \partial_V\pmb{q}_n +\sum_{u=1}^{n}(-1)^{n} \pmb{q}_n(\mathbb{1}_V^{\otimes u-1} \otimes \partial_V \otimes \mathbb{1}_V^{\otimes n-u}) \nonumber \\
& + \!\sum_{B(n)}(-1)^{\vartheta(r_1, \mbox{\dots}, r_k)}\pmb{p}_k(gf \circ \pmb{q}_{r_1}\! \otimes \mbox{\dots} \otimes gf \circ \pmb{q}_{r_k}) \\ \nonumber
& + \!\sum_{A(n)}(-1)^{i(\ell + 1) + n}\pmb{q}_{k}(\mathbb{1}_V^{\otimes i-1} \otimes \mu_{\ell} \otimes \mathbb{1}_V^{\otimes n-k}) - \pmb{q}_1\mu_n= 0.\end{aligned}$$ for all $n \geq 2$. Then we have $$\pmb{q}_n = \sum_{C(n)} (-1)^{n+r_i + \vartheta(r_1, \mbox{\dots}, r_i)}\mu_k((\pmb{\psi}\pmb{\varphi})_{r_1} \otimes \mbox{\dots} \otimes (\pmb{\psi}\pmb{\varphi})_{r_{i-1}} \otimes h \circ \pmb{q}_{r_i} \otimes \mathbb{1}_V^{\otimes k-i}),$$ for all $n \geq 2$, where $A_\infty$ morphisms $\pmb{\varphi}$ and $\pmb{\psi}$ are given by $\pmb{p}-$kernels and $\pmb{q}-$kernels, . We also used the notation $C(n)$ as in and $\vartheta(r_1, \mbox{\dots}, r_k)$ as in .
Assuming , the set of $\pmb{q}-$kernels constitutes by Lemma \[k2t19\] $A_\infty$ morphism $\pmb{\varphi} = (f, \varphi_2, \varphi_3, \mbox{\dots})$ from $(V, \partial_V, \mu_2, \mu_3, \mbox{\dots})$ to $(W, \partial_W,$$ \nu_2, \nu_3, \mbox{\dots})$. We also demand the set of maps $H_n = h \circ \pmb{q}_n$ gives $A_\infty$ homotopy $\pmb{H}= (h, H_2, H_3, \mbox{\dots})$ between $\pmb{\psi \varphi}$ and $\mathbb{1}$. This is equivalent by Definition \[ainftyhomotopy\] to $$\partial_VH_n - \sum_{u=1}^{n}(-1)^{n} H_n(\mathbb{1}_V^{\otimes u-1} \otimes \partial_V \otimes \mathbb{1}_V^{\otimes n-u})$$ $$+\sum_{C(n)} (-1)^{n+r_i + \vartheta(r_1, \mbox{\dots}, r_i)}\mu_k((\pmb{\psi}\pmb{\varphi})_{r_1} \otimes \mbox{\dots} \otimes (\pmb{\psi}\pmb{\varphi})_{r_{i-1}} \otimes H_{r_i} \otimes \mathbb{1}_V^{\otimes k-i}) + H_1\mu_n$$ $$= \sum_{A(n)}(-1)^{i(\ell + 1) + n} H_k(\mathbb{1}_V^{\otimes i-1} \otimes \mu_\ell \otimes \mathbb{1}_V^{\otimes n-k}) + (\pmb{\psi}\pmb{\varphi})_{n} - (\mathbb{1})_{n}$$ for all $n \geq 2$. According to , we have $$(\pmb{\psi}\pmb{\varphi})_m = \psi_1\varphi_m + \sum_{B(m)} (-1)^{\vartheta(r_1, \mbox{\dots}, r_k)}\psi_k(\varphi_{r_1} \otimes \mbox{\dots} \otimes \varphi_{r_k}),$$ and so we can write the composition of $A_\infty$ morphisms in terms of $\pmb{p}-$kernels and $\pmb{q}-$kernels: $$\label{p,q-slozeni}
(\pmb{\psi}\pmb{\varphi})_m = gf\circ \pmb{q}_m + \sum_{B(m)} (-1)^{\vartheta(r_1, \mbox{\dots}, r_k)}h \circ \pmb{p}_k(gf \circ \pmb{q}_{r_1} \otimes \mbox{\dots} \otimes gf \circ \pmb{q}_{r_k}).$$ By Definition \[ainftyhomotopy\], the $A_\infty$ homotopy $\pmb{H}= (h, H_2, H_3, \mbox{\dots})$ can be rewritten in terms of $\pmb{p}-$kernels and $\pmb{q}-$kernels (we use again $\partial_Vh = gf - \mathbb{1}_V - h\partial_V$ and $(\mathbb{1})_n=0$): $$gf\circ\pmb{q}_n - \pmb{q}_n - h\partial_V\pmb{q}_n - \sum_{u=1}^{n}(-1)^{n} h \circ \pmb{q}_n(\mathbb{1}_V^{\otimes u-1} \otimes \partial_V \otimes \mathbb{1}_V^{\otimes n-u})$$ $$+\sum_{C(n)} (-1)^{n+r_i + \vartheta(r_1, \mbox{\dots}, r_i)}\mu_k((\pmb{\psi}\pmb{\varphi})_{r_1} \otimes \mbox{\dots} \otimes (\pmb{\psi}\pmb{\varphi})_{r_{i-1}} \otimes h \circ \pmb{q}_{r_i} \otimes \mathbb{1}_V^{\otimes k-i}) + h \circ \pmb{q}_1 \mu_n$$ $$= \sum_{A(n)}(-1)^{i(\ell + 1) + n} h\circ\pmb{q}_k(\mathbb{1}_V^{\otimes i-1} \otimes \mu_\ell \otimes \mathbb{1}_V^{\otimes n-k}) + gf\circ \pmb{q}_n$$ $$+ \sum_{B(n)} (-1)^{\vartheta(r_1, \mbox{\dots}, r_k)}h \circ \pmb{p}_k(gf \circ \pmb{q}_{r_1} \otimes \mbox{\dots} \otimes gf \circ \pmb{q}_{r_k}).$$ We subtract from both sides of the last display $gf\circ \pmb{q}_n$, and by conclude $$- h\partial_V\pmb{q}_n - \sum_{u=1}^{n}(-1)^{n} h \circ \pmb{q}_n(\mathbb{1}_V^{\otimes u-1} \otimes \partial_V \otimes \mathbb{1}_V^{\otimes n-u}) + h \circ \pmb{q}_1 \mu_n=$$ $$= \sum_{A(n)}(-1)^{i(\ell + 1) + n} h\circ\pmb{q}_k(\mathbb{1}_V^{\otimes i-1} \otimes \mu_\ell \otimes \mathbb{1}_V^{\otimes n-k}) +$$ $$+ \sum_{B(n)} (-1)^{\vartheta(r_1, \mbox{\dots}, r_k)}h \circ \pmb{p}_k(gf \circ \pmb{q}_{r_1} \otimes \mbox{\dots} \otimes gf \circ \pmb{q}_{r_k}),$$ which finally results in $$\pmb{q}_n = \sum_{C(n)} (-1)^{n+r_i + \vartheta(r_1, \mbox{\dots}, r_i)}\mu_k((\pmb{\psi}\pmb{\varphi})_{r_1} \otimes \mbox{\dots} \otimes (\pmb{\psi}\pmb{\varphi})_{r_{i-1}} \otimes h \circ \pmb{q}_{r_i} \otimes \mathbb{1}_V^{\otimes k-i}).$$
The assumption is fulfilled as soon as the $\pmb{q}-$kernels give a $A_\infty$ morphism $\pmb{\varphi} = (f, \varphi_2, \varphi_3, \mbox{\dots})$ from $(V, \partial_V, \mu_2, \mu_3, \mbox{\dots})$ to $(W, \partial_W, \nu_2, \nu_3, \mbox{\dots})$ and $f$ is a monomorphism.
\[k2d21\] Let $n \geq 2$ and define $\pmb{q}_1 := \mathbb{1}_V$. We define $\pmb{q}-$kernels inductively by $$\pmb{q}_n = \sum_{C(n)} (-1)^{n+r_i + \vartheta(r_1, \mbox{\dots}, r_i)}\mu_k((\pmb{\psi}\pmb{\varphi})_{r_1} \otimes \mbox{\dots} \otimes (\pmb{\psi}\pmb{\varphi})_{r_{i-1}} \otimes h \circ \pmb{q}_{r_i} \otimes \mathbb{1}_V^{\otimes k-i}),$$ where $(\pmb{\psi}\pmb{\varphi})_m = gf\circ \pmb{q}_m + \sum_{B(m)} (-1)^{\vartheta(r_1, \mbox{\dots}, r_k)}h \circ \pmb{p}_k(gf \circ \pmb{q}_{r_1} \otimes \mbox{\dots} \otimes gf \circ \pmb{q}_{r_k})$ (cf., ), $\pmb{p}-$kernels were introduced in \[k2d17\], with $C(n)$ given in and $\vartheta(u_1, \mbox{\dots}, u_k)$ in .
There is an explicit description of the $\pmb{q}-$kernels in terms of rooted plane trees, but it is much more complicated when compared to the analogous description for the $\pmb{p}-$kernels.
We shall now prove that the $\pmb{q}-$kernels introduced in Definition \[k2d21\] satisfy . Let us consider again the suspension $\overline{T}sV$ with the induced codifferential $\delta$ such that $\delta_1 = s \circ \partial_V \circ \omega$ and $\delta_n = s \circ \mu_n \circ \omega^{\otimes n}$, $n \geq 2$. Then $\hat{\pmb{q}}_m = s \circ \pmb{q}_m \circ \omega^{\otimes m},$ $\hat{\pmb{\psi}}_m = s \circ \psi_m \circ \omega^{\otimes m}$ and $\hat{\pmb{\varphi}}_m = s \circ \varphi_m \circ \omega^{\otimes m}$ for $m \geq 2$ ($|\hat{\pmb{q}}_m| = |\hat{\pmb{\varphi}}_m| = |\hat{\pmb{\psi}}_m| = 0$), and is equivalent to $$\begin{split}
\notag
\delta_1\hat{\pmb{q}}_n + \sum_{B(n)}\hat{\pmb{p}}_k(\hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_1} \otimes \mbox{\dots} \otimes \hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_k}) & \\ =\sum_{u=1}^{n}\hat{\pmb{q}}_n(\mathbb{1}_V^{\otimes u-1} \otimes \delta_1 \otimes \mathbb{1}_V^{\otimes n-u}) + \sum_{A(n)}\hat{\pmb{q}}_{k}(\mathbb{1}_V^{\otimes i-1} \otimes &\delta_{\ell} \otimes \mathbb{1}_V^{\otimes n-k}) + \hat{\pmb{q}}_1\delta_n.
\end{split}$$ In the following two lemmas we prove that $\hat{\pmb{\psi}}\hat{\pmb{\varphi}}$ is an $A_\infty$ morphism.
\[k2l22\] Let us assume is true for all $n \leq m$. Then the $\pmb{p}-$kernels in Definition \[k2d17\] and the $\pmb{q}-$kernels in Definition \[k2d21\] fulfill $$\delta_1(\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_m = \sum_{u=1}^{m}(\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_m(\mathbb{1}_V^{\otimes u-1} \otimes \delta_1 \otimes \mathbb{1}_V^{\otimes m-u})$$$$+ \sum_{A(m)}(\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_k(\mathbb{1}_V^{\otimes i-1} \otimes \delta_{\ell} \otimes \mathbb{1}_V^{\otimes n-k}) + (\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_1\delta_m$$$$-\sum_{B(m)}\hat{\pmb{p}}_k(\hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_1} \otimes \mbox{\dots} \otimes \hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_k})$$ for all $m \geq 2$.
We shall first expand the composition of morphisms in the suspended form as in , and also use the homotopy $\hat{h}$ between $\hat{g}\hat{f}$ and $\mathbb{1}_V$: $$\delta_1(\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_m = \hat{g}\hat{f}\circ \delta_1 \pmb{q}_m + \sum_{B(m)}\delta_1 \hat{h} \circ \hat{\pmb{p}}_k(\hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_1} \otimes \mbox{\dots} \otimes \hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_k})$$ $$\label{L22_1}
= \hat{g}\hat{f}\circ \delta_1 \pmb{q}_m + \sum_{B(m)}(\hat{g}\hat{f} - \mathbb{1}_V - \hat{h}\delta_1) \circ \hat{\pmb{p}}_k(\hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_1} \otimes \mbox{\dots} \otimes \hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_k}).$$ By Theorem \[k2v18\] $$\sum_{B(m)}\hat{h} \circ\delta_1 \hat{\pmb{p}}_k(\hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_1} \otimes \mbox{\dots} \otimes \hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_k})$$ $$\label{L22_2}
= -\sum_{B(m)}\sum_{u=1}^{k}\hat{h} \circ\hat{\pmb{p}}_k(\mathbb{1}_V^{\otimes u-1} \otimes \delta_1 \otimes \mathbb{1}_V^{\otimes k-u})(\hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_1} \otimes \mbox{\dots} \otimes \hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_k})$$ $$\label{L22_3}
-\sum_{B(m)}\sum_{A'(k)}\hat{h} \circ\hat{\pmb{p}}_{k'}(\mathbb{1}_V^{\otimes i'-1} \otimes \hat{g}\hat{f} \circ \hat{\pmb{p}}_{\ell'} \otimes \mathbb{1}_V^{\otimes k-k'})(\hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_1} \otimes \mbox{\dots} \otimes \hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_k})$$ and as $\hat{g}, \hat{f}$ and $\hat{\pmb{q}}_m$ are of degree $0$, we have $$\eqref{L22_2} = -\sum_{B(m)}\sum_{u=1}^{k}\hat{h} \circ\hat{\pmb{p}}_k(\hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_1} \otimes \mbox{\dots} \otimes \hat{g}\hat{f} \circ \delta_1\hat{\pmb{q}}_{r_u} \otimes \mbox{\dots} \otimes \hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_k}).$$ By for $n \leq m$, we expand the terms of the form $\delta_1 \hat{\pmb{q}}_\bullet$ as $$-\sum_{B(m)}\sum_{u=1}^{k}\hat{\pmb{p}}_k(\hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_1} \otimes \mbox{\dots} \otimes \sum_{v=1}^{r_u} \hat{g}\hat{f} \circ\hat{\pmb{q}}_{r_u}(\mathbb{1}_V^{\otimes v-1} \otimes \delta_1 \otimes \mathbb{1}_V^{\otimes r_u -v}) \otimes \mbox{\dots} \otimes \hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_k})$$ $$-\sum_{B(m)}\sum_{u=1}^{k}\hat{\pmb{p}}_k(\hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_1} \otimes \mbox{\dots} \otimes \sum_{A'(r_u)} \hat{g}\hat{f} \circ\hat{\pmb{q}}_{k'}(\mathbb{1}_V^{\otimes i'-1} \otimes \delta_{\ell'} \otimes \mathbb{1}_V^{\otimes r_u-k'}) \otimes \mbox{\dots} \otimes \hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_k})$$ $$-\sum_{B(m)}\sum_{u=1}^{k}\hat{\pmb{p}}_k(\hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_1} \otimes \mbox{\dots} \otimes \hat{g}\hat{f} \circ\hat{\pmb{q}}_{1}\delta_{r_u} \otimes \mbox{\dots} \otimes \hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_k})$$ $$+\sum_{B(m)}\sum_{u=1}^{k}\hat{\pmb{p}}_k(\hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_1} \otimes \mbox{\dots} \otimes \sum_{B'(r_u)} \hat{g}\hat{f} \circ\hat{\pmb{p}}_{k'}(\hat{g}\hat{f} \circ \hat{\pmb{q}}_{r'_1} \otimes \mbox{\dots} \otimes \hat{g}\hat{f} \circ \hat{\pmb{q}}_{r'_{k'}}) \otimes \mbox{\dots} \otimes \hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_k}).$$ In the second contribution , which equals to $$-\sum_{B(m)}\sum_{A'(k)}\hat{\pmb{p}}_{k'}(\hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_1} \otimes \mbox{\dots} \otimes$$ $$\otimes \hat{g}\hat{f} \circ \hat{\pmb{p}}_{\ell'}(\hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_{i}} \otimes \mbox{\dots} \otimes \hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_{i + \ell'-1}}) \otimes \mbox{\dots} \otimes \hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_{k'}}),$$ we sum over all inner positions of $\hat{\pmb{p}}_{k'}(\hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_1} \otimes \mbox{\dots} \otimes \bullet \otimes \mbox{\dots} \otimes \hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_{k'}})$ and get $$\eqref{L22_3} = -\sum_{B(m)}\sum_{u=1}^{k}\hat{\pmb{p}}_k(\hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_1} \otimes \mbox{\dots} \otimes$$ $$\otimes \sum_{B'(r_u)} \hat{g}\hat{f} \circ\hat{\pmb{p}}_{k'}(\hat{g}\hat{f} \circ \hat{\pmb{q}}_{r'_1} \otimes \mbox{\dots} \otimes \hat{g}\hat{f} \circ \hat{\pmb{q}}_{r'_{k'}}) \otimes \mbox{\dots} \otimes \hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_k}).$$ Up to a sign, this is the same expression as the expression on the fourth line of the expansion . We substitute into for $\sum_{B(m)}\hat{h} \circ\delta_1 \hat{\pmb{p}}_k(\hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_1} \otimes \mbox{\dots} \otimes \hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_k})$ the combination $\eqref{L22_2} + \eqref{L22_3}$ and also substitute for $\delta_1\hat{\pmb{q}}_m$ according to : $$\delta_1(\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_m = -\sum_{B(m)}\hat{g}\hat{f} \circ \hat{\pmb{p}}_k(\hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_1} \otimes \mbox{\dots} \otimes \hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_k})$$$$+ \sum_{u=1}^{m}\hat{g}\hat{f} \circ\hat{\pmb{q}}_m(\mathbb{1}_V^{\otimes u-1} \otimes \delta_1 \otimes \mathbb{1}_V^{\otimes m-u})$$ $$+ \sum_{A(m)}\hat{g}\hat{f} \circ\hat{\pmb{q}}_{k}(\mathbb{1}_V^{\otimes i-1} \otimes \delta_{\ell} \otimes \mathbb{1}_V^{\otimes n-k}) + \hat{g}\hat{f} \circ \hat{\pmb{q}}_1\delta_m$$ $$+ \sum_{B(m)}(\hat{g}\hat{f} - \mathbb{1}_V)\circ \hat{\pmb{p}}_k(\hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_1} \otimes \mbox{\dots} \otimes \hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_k})$$ $$+ \sum_{B(m)}\sum_{u=1}^{k}\hat{\pmb{p}}_k(\hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_1} \otimes \mbox{\dots} \otimes \sum_{v=1}^{r_u} \hat{g}\hat{f} \circ\hat{\pmb{q}}_{r_u}(\mathbb{1}_V^{\otimes v-1} \otimes \delta_1 \otimes \mathbb{1}_V^{\otimes r_u -v}) \otimes \mbox{\dots} \otimes \hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_k})$$ $$+ \sum_{B(m)}\sum_{u=1}^{k}\hat{\pmb{p}}_k(\hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_1} \otimes \mbox{\dots} \otimes \sum_{A'(r_u)} \hat{g}\hat{f} \circ\hat{\pmb{q}}_{k'}(\mathbb{1}_V^{\otimes i'-1} \otimes \delta_{\ell'} \otimes \mathbb{1}_V^{\otimes r_u-k'}) \otimes \mbox{\dots} \otimes \hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_k})$$ $$+ \sum_{B(m)}\sum_{u=1}^{k}\hat{\pmb{p}}_k(\hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_1} \otimes \mbox{\dots} \otimes \hat{g}\hat{f} \circ\hat{\pmb{q}}_{1}\delta_{r_u} \otimes \mbox{\dots} \otimes \hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_k}).$$ This completes the proof.
\[k2l23\] The $\pmb{p}-$kernels in Definition \[k2d17\] and the $\pmb{q}-$kernels in Definition \[k2d21\] fulfill $$\sum_{B(m)}\delta_k((\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_{r_1} \otimes \mbox{\dots} \otimes (\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_{r_k}) = \sum_{B(m)}\hat{\pmb{p}}_k(\hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_1} \otimes \mbox{\dots} \otimes \hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_k})$$ for all $m \geq 2$.
By , we have $$\sum_{B(m)}\hat{\pmb{p}}_k(\hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_1} \otimes \mbox{\dots} \otimes \hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_k})$$ $$= \sum_{B(m)} \sum_{B'(k)} \delta_{k'}(h \circ \hat{\pmb{p}}_{r'_1} \otimes \mbox{\dots} \otimes h \circ \hat{\pmb{p}}_{r'_{k'}})(\hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_1} \otimes \mbox{\dots} \otimes \hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_k}).$$ Taking into account that $\hat{g}, \hat{f}$ and $\hat{\pmb{q}}_m$ are of degree $0$, the last display equals to $$\sum_{B(m)} \sum_{B'(k)} \delta_{k'}(h \circ \hat{\pmb{p}}_{r'_1}(\hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_1} \otimes \mbox{\dots} \otimes \hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_{r'_1}}) \otimes \mbox{\dots}$$ $$\mbox{\dots} \otimes h \circ \hat{\pmb{p}}_{r'_{k'}}(\hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_{{r_{k'-1}} + 1}} \otimes \mbox{\dots} \otimes \hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_{k'}}))$$ and the summation over the terms $\delta_k(\star_{r_1} \otimes \mbox{\dots} \otimes \star_{r_k})$ in all possible indices ($\star_j$ denoting a map $V^{\otimes j} \to V$) gives $$\sum_{B(m)}\delta_k\left[\left(\hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_1} + \sum_{B'(r_1)}\hat{\pmb{p}}_{k'}(\hat{g}\hat{f} \circ \hat{\pmb{q}}_{r'_1} \otimes \mbox{\dots} \otimes \hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_{k'}})\right) \otimes \mbox{\dots}\right.$$ $$\left. \mbox{\dots} \otimes \left(\hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_{k}} +\sum_{B'(r_k)}\hat{\pmb{p}}_{k'}(\hat{g}\hat{f} \circ \hat{\pmb{q}}_{r'_1} \otimes \mbox{\dots}
\otimes \hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_{k'}})\right)\right].$$ However this is already composed with the suspension, and the proof is complete.
Because the formula for the $\pmb{q}-$kernels in Lemma \[k2t20\] was based on the assumption , we have to prove that it is fulfilled by the $\pmb{q}-$kernels in Definition \[k2d21\].
\[k2v24\] The $\pmb{p}-$kernels in Definition \[k2d17\] and the $\pmb{q}-$kernels in Definition \[k2d21\] fulfill , i.e. $$\notag
\begin{split}
\partial_V\pmb{q}_n + \sum_{u=1}^{n}(-1)^{n} \pmb{q}_n(\mathbb{1}_V^{\otimes u-1} \otimes \partial_V \otimes \mathbb{1}_V^{\otimes n-u}) &\\ +\!\sum_{B(n)}(-1)^{\vartheta(r_1, \mbox{\dots}, r_k)}\pmb{p}_k(gf \circ \pmb{q}_{r_1}\! \otimes \mbox{\dots} \otimes gf \circ \pmb{q}_{r_k}) &\\ +\!\sum_{A(n)}(-1)^{i(\ell + 1) + n}\pmb{q}_{k}(\mathbb{1}_V^{\otimes i-1} \otimes \mu_{\ell} \otimes \mathbb{1}_V^{\otimes n-k}) - \pmb{q}_1\mu_n&= 0.
\end{split}$$ This means that the objects introduced in solve the problem of the transfer of $A_\infty$ structure.
We shall prove an equivalent assertion: $$\begin{split}
\notag
\delta_1\hat{\pmb{q}}_n + \sum_{B(n)}\hat{\pmb{p}}_k(\hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_1} \otimes \mbox{\dots} \otimes \hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_k}) &=\\
=\sum_{u=1}^{n}\hat{\pmb{q}}_n(\mathbb{1}_V^{\otimes u-1} \otimes \delta_1 \otimes \mathbb{1}_V^{\otimes n-u}) + \sum_{A(n)}\hat{\pmb{q}}_{k}(\mathbb{1}_V^{\otimes i-1} \otimes &\delta_{\ell} \otimes \mathbb{1}_V^{\otimes n-k}) + \hat{\pmb{q}}_1\delta_n,
\end{split}$$ with suspended $\pmb{q}-$kernels given by Definition \[k2d21\]: $$\label{q-kernel}
\hat{\pmb{q}}_n = \sum_{C(n)}\delta_k((\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_{r_1} \otimes \mbox{\dots} \otimes (\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_{r_{i-1}} \otimes \hat{h} \circ \hat{\pmb{q}}_{r_i} \otimes \mathbb{1}_V^{\otimes k-i}).$$ The proof goes by induction on $n$: for $n = 2$, we have by (for $n=2$) and : $$\delta_1\hat{\pmb{q}}_2 = \delta_1(\delta_2(\hat{g}\hat{f} \otimes \hat{h}) + \delta_2(\hat{h} \otimes \mathbb{1}_V))$$ $$= -\delta_2(\delta_1\otimes \mathbb{1}_V)(\hat{g}\hat{f} \otimes \hat{h}) -\delta_2(\mathbb{1}_V \otimes \delta_1)(\hat{g}\hat{f} \otimes \hat{h})$$ $$-\delta_2(\delta_1\otimes \mathbb{1}_V)(\hat{h} \otimes \mathbb{1}_V) -\delta_2(\mathbb{1}_V \otimes \delta_1)(\hat{h} \otimes \mathbb{1}_V).~$$ By the Koszul sign convention $$\begin{aligned}
\delta_2(\delta_1\otimes \mathbb{1}_V)(\hat{g}\hat{f} \otimes \hat{h}) &= (-1)^{|\delta_1||\hat{h}|}\delta_2(\hat{g}\hat{f} \otimes \hat{h})(\delta_1\otimes \mathbb{1}_V),\\
\delta_2(\mathbb{1}_V \otimes \delta_1)(\hat{g}\hat{f} \otimes \hat{h}) &= \delta_2(\hat{g}\hat{f} \otimes \hat{g}\hat{f} - \mathbb{1}_V -\hat{h}\delta_1) =\\
&= \delta_2(\hat{g}\hat{f} \otimes \hat{g}\hat{f}) - \delta_2(\hat{g}\hat{f} \otimes \mathbb{1}_V) - \delta_2(\hat{g}\hat{f} \otimes \hat{h})(\mathbb{1}_V \otimes \delta_1),\\
\delta_2(\delta_1\otimes \mathbb{1}_V)(\hat{h} \otimes \mathbb{1}_V) &= \delta_2(\hat{g}\hat{f} - \mathbb{1}_V -\hat{h}\delta_1 \otimes \mathbb{1}_V) =\\
&= \delta_2(\hat{g}\hat{f} \otimes \mathbb{1}_V) - \delta_2(\mathbb{1}_V \otimes \mathbb{1}_V) - \delta_2(\hat{h} \otimes \mathbb{1}_V)(\delta_1 \otimes \mathbb{1}_V),\\
\delta_2(\mathbb{1}_V \otimes \delta_1)(\hat{h} \otimes \mathbb{1}_V) &= (-1)^{|\delta_1||\hat{h}|}\delta_2(\hat{h} \otimes \mathbb{1}_V)(\mathbb{1}_V \otimes \delta_1),\end{aligned}$$ where $(-1)^{|\delta_1||\hat{h}|} = -1$ is a consequence of $|\hat{h}| = |\delta_1| = 1$, and so $$\delta_1\hat{\pmb{q}}_2 = \delta_2(\hat{g}\hat{f} \otimes \hat{h})(\delta_1\otimes \mathbb{1}_V) + \delta_2(\hat{h} \otimes \mathbb{1}_V)(\delta_1 \otimes \mathbb{1}_V) + \delta_2(\hat{g}\hat{f} \otimes \hat{h})(\mathbb{1}_V \otimes \delta_1)$$ $$+ \delta_2(\hat{h} \otimes \mathbb{1}_V)(\mathbb{1}_V \otimes \delta_1) + \delta_2(\mathbb{1}_V \otimes \mathbb{1}_V) - \delta_2(\hat{g}\hat{f} \otimes \hat{g}\hat{f})$$ $$=\hat{\pmb{q}}_2(\delta_1 \otimes \mathbb{1}_V) + \hat{\pmb{q}}_2(\mathbb{1}_V \otimes \delta_1) + \hat{\pmb{q}}_1\delta_2 - \hat{\pmb{p}}_2(\hat{g}\hat{f} \otimes \hat{g}\hat{f}).$$ The induction step is divided into three steps:\
**I.** We first expand the term $\delta_1 \hat{\pmb{q}}_n$: by $$\delta_1\hat{\pmb{q}}_n = \sum_{C(n)}\delta_1\delta_k((\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_{r_1} \otimes \mbox{\dots} \otimes (\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_{r_{i-1}} \otimes \hat{h} \circ \hat{\pmb{q}}_{r_i} \otimes \mathbb{1}_V^{\otimes k-i})$$ $$= -\sum_{C(n)}\left(\sum_{u = 1}^{k} \delta_k \left(\mathbb{1}_V^{\otimes u-1} \otimes \delta_1 \otimes \mathbb{1}_V^{\otimes k-u}\right)\right)((\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_{r_1} \otimes \mbox{\dots} \otimes (\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_{r_{i-1}} \otimes \hat{h} \circ \hat{\pmb{q}}_{r_i} \otimes \mathbb{1}_V^{\otimes k-i})$$ $$- \sum_{C(n)}\left(\sum_{A'(k)} \delta_{k'}\left(\mathbb{1}_V^{\otimes i'-1} \otimes \delta_{\ell'} \otimes \mathbb{1}_V^{\otimes k' - i'}\right) \right)((\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_{r_1} \otimes \mbox{\dots} \otimes (\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_{r_{i-1}} \otimes \hat{h} \circ \hat{\pmb{q}}_{r_i} \otimes \mathbb{1}_V^{\otimes k-i}).$$ The first summation can be rewritten as $$-\sum_{C(n)}\left(\sum_{u = 1}^{k} \delta_k \left(\mathbb{1}_V^{\otimes u-1} \otimes \delta_1 \otimes \mathbb{1}_V^{\otimes k-u}\right)\right)((\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_{r_1} \otimes \mbox{\dots} \otimes (\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_{r_{i-1}} \otimes \hat{h} \circ \hat{\pmb{q}}_{r_i} \otimes \mathbb{1}_V^{\otimes k-i})$$ $$\begin{aligned}
\tag{Q1.1}
\label{Q1.1}
&=-\sum_{C(n)}\sum_{u = 1}^{i-1} \delta_k((\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_{r_1} \otimes \mbox{\dots} \otimes \delta_1(\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_{r_{u}} \otimes \mbox{\dots} \otimes (\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_{r_{i-1}} \otimes \hat{h} \circ \hat{\pmb{q}}_{r_i} \otimes \mathbb{1}_V^{\otimes k-i}) \\
\tag{Q1.2}
\label{Q1.2}
&-\sum_{C(n)}\delta_k((\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_{r_1} \otimes \mbox{\dots} \otimes (\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_{r_{i-1}} \otimes \delta_1\hat{h} \circ \hat{\pmb{q}}_{r_i} \otimes \mathbb{1}_V^{\otimes k-i}) \\
\tag{Q1.3}
\label{Q1.3}
&-(-1)^{|\delta_1||\hat{h}|}\! \sum_{C(n)}\sum_{u=i+1}^{k}\!\!\delta_k((\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_{r_1}\!\! \otimes \! \mbox{\dots} \! \otimes \! (\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_{r_{i-1}}\!\! \otimes \hat{h} \circ \hat{\pmb{q}}_{r_i}\! \otimes \mathbb{1}_V^{\otimes u-1}\! \otimes \delta_1 \! \otimes \mathbb{1}_V^{\otimes k-u-i}),\end{aligned}$$\
while the second as $$-\sum_{C(n)}\left(\sum_{A'(k)} \delta_{k'}\left(\mathbb{1}_V^{\otimes i'-1} \otimes \delta_{\ell'} \otimes \mathbb{1}_V^{\otimes k' - i'}\right) \right) \circ$$ $$\circ ((\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_{r_1} \otimes \mbox{\dots} \otimes (\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_{r_{i-1}} \otimes \hat{h} \circ \hat{\pmb{q}}_{r_i} \otimes \mathbb{1}_V^{\otimes k-i})$$ $$\begin{aligned}
\tag{Q2.1}
\label{Q2.1}
&=- \sum_{C(n)}\delta_k((\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_{r_1} \otimes \mbox{\dots} \otimes \delta_{\star}((\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_\star \otimes \mbox{\dots} \otimes (\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_\star ) \otimes \mbox{\dots}\\
\notag
&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\mbox{\dots} \otimes (\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_{r_{i-1}} \otimes \hat{h} \circ \hat{\pmb{q}}_{r_i} \otimes \mathbb{1}_V^{\otimes k-i})\\
\tag{Q2.2}
\label{Q2.2}
&- \sum_{C(n)}\delta_k((\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_{r_1} \otimes \mbox{\dots} \otimes \delta_{\star}((\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_\star \otimes \mbox{\dots} \otimes (\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_{r_{i-1}} \otimes \hat{h} \circ \hat{\pmb{q}}_{r_i} \otimes \mathbb{1}_V^{\otimes \star}) \otimes \mathbb{1}_V^{\otimes k-\star}\\
\tag{Q2.3}
\label{Q2.3}
&-(-1)^{|\delta_{\ell'}||\hat{h}|}\!
\sum_{C(n)}\sum_{A'(k-i)}\!\!\delta_{k'+ i}((\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_{r_1}\!\! \otimes\! \mbox{\dots}\! \otimes\! (\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_{r_{i-1}}\!\! \otimes \! \hat{h} \circ \hat{\pmb{q}}_{r_i} \!\! \otimes \mathbb{1}_V^{\otimes i'-1} \! \otimes \delta_{\ell'} \! \otimes \mathbb{1}_V^{\otimes k' - i'}).\end{aligned}$$ The summation over all indices in terms of the form $\delta_k((\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_{r_1} \otimes \mbox{\dots} \otimes \star \otimes \mbox{\dots} \otimes (\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_{r_{i-1}} \otimes \hat{h} \circ \hat{\pmb{q}}_{r_i} \otimes \mathbb{1}_V^{\otimes k-i})$ leads to $$\eqref{Q2.1} = -\sum_{C(n)} \sum_{u = 1}^{i-1} \delta_k((\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_{r_1} \otimes \mbox{\dots} \otimes \sum_{B'(r_u)}\delta_{k'}((\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_{r'_1} \otimes \mbox{\dots} \otimes (\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_{r_{k'}}) \otimes \mbox{\dots}$$ $$\mbox{\dots} \otimes (\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_{r_{i-1}} \otimes \hat{h} \circ \hat{\pmb{q}}_{r_i} \otimes \mathbb{1}_V^{\otimes k-i}).$$ Analogously, the summation over all indices in terms of the form $\delta_k((\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_{r_1} \otimes \mbox{\dots} \otimes \star \otimes \mathbb{1}_V^{\otimes k-j})$ gives $$\eqref{Q2.2} = -\sum_{C(n)}\sum_{r_i>1} \delta_k((\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_{r_1} \otimes \mbox{\dots} \otimes (\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_{r_{i-1}} \otimes \hat{\pmb{q}}_{r_i} \otimes \mathbb{1}_V^{\otimes k-i}).$$
**II.** By Lemma \[k2l23\]: $$\eqref{Q2.1} = -\sum_{C(n)} \sum_{u = 1}^{i-1} \delta_k((\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_{r_1} \otimes \mbox{\dots} \otimes \sum_{B'(r_u)}\hat{\pmb{p}}_{k'}(\hat{g}\hat{f} \circ \hat{\pmb{q}}_{r'_1} \otimes \mbox{\dots} \otimes \hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_{k'}}) \otimes \mbox{\dots}$$ $$\mbox{\dots} \otimes (\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_{r_{i-1}} \otimes \hat{h} \circ \hat{\pmb{q}}_{r_i} \otimes \mathbb{1}_V^{\otimes k-i}),$$ and Lemma \[k2l22\] for $(\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_m$ (Definition \[k2d21\] and definition of $C(n)$ in imply that $m$ is strictly less than $n$, so that assumptions of Lemma \[k2l22\] are fulfilled by our induction hypothesis) gives $$\eqref{Q1.1} + \eqref{Q2.1} =$$ $$\tag{Q1.1 + 2.1a}
\label{Q1.1 + 2.1a}
\sum_{C(n)}\sum_{u = 1}^{i-1}\sum_{r_u = 1} \delta_k((\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_{r_1} \otimes \mbox{\dots} \otimes (-1)^{|\hat{h}||\delta_1|}(\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_{r_{u}}\delta_1 \otimes \mbox{\dots} \otimes (\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_{r_{i-1}} \otimes \hat{h} \circ \hat{\pmb{q}}_{r_i} \otimes \mathbb{1}_V^{\otimes k-i})$$ $$\notag
+ \sum_{C(n)}\sum_{u = 1}^{i-1}\sum_{r_u > 1} \delta_k((\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_{r_1} \otimes \mbox{\dots} \otimes \sum_{u=1}^{r_i}(-1)^{|\hat{h}||\delta_1|}(\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_{r_i}(\mathbb{1}_V^{\otimes u-1} \otimes \delta_1 \otimes \mathbb{1}_V^{\otimes r_i-u}) \otimes \mbox{\dots}$$ $$\tag{Q1.1 + 2.1b}\label{Q1.1 + 2.1b}\mbox{\dots} \otimes (\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_{r_{i-1}} \otimes \hat{h} \circ \hat{\pmb{q}}_{r_i} \otimes \mathbb{1}_V^{\otimes k-i})$$ $$\notag+ \sum_{C(n)}\sum_{u = 1}^{i-1}\sum_{r_u > 1} \delta_k((\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_{r_1} \otimes \mbox{\dots} \otimes \sum_{A'(r_u)}(-1)^{|\hat{h}||\delta_{\ell'}|}(\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_{k'}(\mathbb{1}_V^{\otimes i'-1} \otimes \delta_{\ell'} \otimes \mathbb{1}_V^{\otimes r_u-k'}) \otimes \mbox{\dots}$$ $$\tag{Q1.1 + 2.1c}\label{Q1.1 + 2.1c}\mbox{\dots} \otimes (\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_{r_{i-1}} \otimes \hat{h} \circ \hat{\pmb{q}}_{r_i} \otimes \mathbb{1}_V^{\otimes k-i})$$ $$\tag{Q1.1 + 2.1d}\label{Q1.1 + 2.1d}+ \sum_{C(n)}\sum_{u = 1}^{i-1}\sum_{r_u > 1} \delta_k((\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_{r_1} \otimes \mbox{\dots} \otimes (-1)^{|\hat{h}||\delta_{r_u}|}(\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_{1}\delta_{r_u} \otimes \mbox{\dots} \otimes (\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_{r_{i-1}} \otimes \hat{h} \circ \hat{\pmb{q}}_{r_i} \otimes \mathbb{1}_V^{\otimes k-i})$$ $$\notag+\sum_{C(n)}\sum_{u = 1}^{i-1}\sum_{r_u > 1} \delta_k((\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_{r_1} \otimes \mbox{\dots} \otimes (-1)^{|\hat{h}||\delta_{r_u}|}(\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_{1}\delta_{r_u} \otimes \mbox{\dots} \otimes (\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_{r_{i-1}} \otimes \hat{h} \circ \hat{\pmb{q}}_{r_i} \otimes \mathbb{1}_V^{\otimes k-i})$$ $$\tag{\ref{Q2.1}}-\sum_{C(n)}\sum_{u = 1}^{i-1}\sum_{r_u > 1} \delta_k((\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_{r_1} \otimes \mbox{\dots} \otimes (-1)^{|\hat{h}||\delta_{r_u}|}(\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_{1}\delta_{r_u} \otimes \mbox{\dots} \otimes (\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_{r_{i-1}} \otimes \hat{h} \circ \hat{\pmb{q}}_{r_i} \otimes \mathbb{1}_V^{\otimes k-i}),$$ where the first five terms come from by application of Lemma \[k2l22\], and the fifth one cancels out when combined with . Recall that we have $|\delta_\ell| = -1$ for all $\ell$, and so $(-1)^{|\hat{h}||\delta_\ell|} = -1$ as well as $$\eqref{Q1.2} + \eqref{Q2.2} = \sum_{C(n), r_i > 1} \delta_k((\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_{r_1} \otimes \mbox{\dots} \otimes (\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_{r_{i-1}} \otimes (-\delta_1\hat{h} - \mathbb{1}_V)\hat{\pmb{q}}_{r_i} \otimes \mathbb{1}_V^{\otimes k-i})$$ $$+\sum_{C(n), r_i = 1} \delta_k((\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_{r_1} \otimes \mbox{\dots} \otimes (\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_{r_{i-1}} \otimes -\delta_1\hat{h}\circ \hat{\pmb{q}}_{r_i} \otimes \mathbb{1}_V^{\otimes k-i})$$ $$= \sum_{C(n), r_i > 1} \delta_k((\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_{r_1} \otimes \mbox{\dots} \otimes (\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_{r_{i-1}} \otimes (\hat{h}\delta_1 - \hat{g}\hat{f})\hat{\pmb{q}}_{r_i} \otimes \mathbb{1}_V^{\otimes k-i})$$ $$+ \sum_{C(n), r_i = 1} \delta_k((\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_{r_1} \otimes \mbox{\dots} \otimes (\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_{r_{i-1}} \otimes (\hat{h}\delta_1 - \hat{g}\hat{f} + \mathbb{1}_V)\hat{\pmb{q}}_{r_i} \otimes \mathbb{1}_V^{\otimes k-i}).$$ Thanks to the induction hypothesis we substitute for $\delta_1\hat{\pmb{q}}_\star$ and the last display turns into $$-\sum_{C(n), r_i > 1} \delta_k((\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_{r_1} \otimes \mbox{\dots} \otimes (\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_{r_{i-1}} \otimes$$ $$\otimes \sum_{B'(r_i)}\hat{h} \circ \hat{\pmb{p}}_{k'}(\hat{g}\hat{f} \circ \hat{\pmb{q}}_{r'_1} \otimes \mbox{\dots} \otimes \hat{g}\hat{f} \circ \hat{\pmb{q}}_{r'_{k'}})\otimes \mathbb{1}_V^{\otimes k-i})$$ $$-\sum_{C(n), r_i > 1} \delta_k((\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_{r_1} \otimes \mbox{\dots} \otimes (\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_{r_{i-1}} \otimes \hat{g}\hat{f}\circ \hat{\pmb{q}}_{r_i}\otimes \mathbb{1}_V^{\otimes k-i})$$ $$\tag{Q1.2 + 2.2a}
\label{Q1.2 + 2.2a}
+ \sum_{C(n), r_i > 1} \delta_k((\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_{r_1} \otimes \mbox{\dots} \otimes (\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_{r_{i-1}} \otimes \sum_{u=1}^{r_i}\hat{\pmb{q}}_{r_i}(\mathbb{1}_V^{\otimes u-1} \otimes \delta_1 \otimes \mathbb{1}_V^{\otimes r_i-u})\otimes \mathbb{1}_V^{\otimes k-i})$$ $$\tag{Q1.2 + 2.2b}
\label{Q1.2 + 2.2b}
+\sum_{C(n), r_i > 1} \delta_k((\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_{r_1} \otimes \mbox{\dots} \otimes (\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_{r_{i-1}} \otimes \sum_{A'(r_i)}\hat{\pmb{q}}_{k'}(\mathbb{1}_V^{\otimes i'-1} \otimes \delta_{\ell'} \otimes \mathbb{1}_V^{\otimes r_i-k'}) + \hat{\pmb{q}}_1\delta_{r_i}\otimes \mathbb{1}_V^{\otimes k-i})$$ $$\tag{Q1.2 + 2.2c}
\label{Q1.2 + 2.2c}
+\sum_{C(n), r_i = 1} \delta_k((\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_{r_1} \otimes \mbox{\dots} \otimes (\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_{r_{i-1}} \otimes \hat{h}\circ\hat{\pmb{q}}_{r_i}\delta_1 \otimes \mathbb{1}_V^{\otimes k-i})$$ $$+ \sum_{C(n), r_i = 1} \delta_k((\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_{r_1} \otimes \mbox{\dots} \otimes (\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_{r_{i-1}} \otimes (-\hat{g}\hat{f} + \mathbb{1}_V)\hat{\pmb{q}}_{r_i} \otimes \mathbb{1}_V^{\otimes k-i}).$$ The non-numbered terms (first, second and sixth) can be further simplified. We notice $$-\sum_{C(n), r_i > 1} \delta_k((\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_{r_1} \otimes \mbox{\dots} \otimes (\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_{r_{i-1}} \otimes \sum_{B'(r_i)}\hat{h} \circ \hat{\pmb{p}}_{k'}(\hat{g}\hat{f} \circ \hat{\pmb{q}}_{r'_1} \otimes \mbox{\dots} \otimes \hat{g}\hat{f} \circ \hat{\pmb{q}}_{r'_{k'}})\otimes \mathbb{1}_V^{\otimes k-i})$$ $$-\sum_{C(n), r_i > 1} \delta_k((\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_{r_1} \otimes \mbox{\dots} \otimes (\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_{r_{i-1}} \otimes \hat{g}\hat{f}\circ \hat{\pmb{q}}_{r_i}\otimes \mathbb{1}_V^{\otimes k-i})$$ $$+ \sum_{C(n), r_i = 1} \delta_k((\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_{r_1} \otimes \mbox{\dots} \otimes (\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_{r_{i-1}} \otimes (-\hat{g}\hat{f} + \mathbb{1}_V)\hat{\pmb{q}}_{r_i} \otimes \mathbb{1}_V^{\otimes k-i})$$ $$= -\sum_{C(n)} \delta_k((\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_{r_1} \otimes \mbox{\dots} \otimes (\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_{r_{i-1}} \otimes (\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_{r_i} \otimes \mathbb{1}_V^{\otimes k-i})$$ $$+ \sum_{C(n)} \delta_k((\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_{r_1} \otimes \mbox{\dots} \otimes (\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_{r_{i-1}} \otimes \mathbb{1}_V^{\otimes k-i+1})$$ $$= -\sum_{B(n)} \delta_k((\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_{r_1} \otimes \mbox{\dots} \otimes (\hat{\pmb{\psi}}\hat{\pmb{\varphi}})_{r_{k}}) + \delta_n(\mathbb{1}_V^{\otimes n}).$$ By Lemma \[k2l23\], this expression equals to $$\tag{Q1.2 + 2.2d}
\label{Q1.2 + 2.2d}
-\sum_{B(n)}\hat{\pmb{p}}_k(\hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_1} \otimes \mbox{\dots} \otimes \hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_k}) + \hat{\pmb{q}}_1\delta_n.$$
**III.** In the last step we pair various contributions together: the first step can be written as $$\delta_1\hat{q}_n = \eqref{Q1.1} + \eqref{Q1.2} + \eqref{Q1.3} + \eqref{Q2.1} + \eqref{Q2.2} + \eqref{Q2.3},$$ while the second step as $$\eqref{Q1.1} + \eqref{Q2.1} = \eqref{Q1.1 + 2.1a} + \eqref{Q1.1 + 2.1b} + \eqref{Q1.1 + 2.1c} + \eqref{Q1.1 + 2.1d}$$ and $$\eqref{Q1.2} + \eqref{Q2.2} = \eqref{Q1.2 + 2.2a} + \eqref{Q1.2 + 2.2b} + \eqref{Q1.2 + 2.2c} + \eqref{Q1.2 + 2.2d}.$$ Taken altogether, $$\begin{aligned}
\eqref{Q1.3} + \eqref{Q1.1 + 2.1a} + &\eqref{Q1.1 + 2.1b} + \eqref{Q1.2 + 2.2a} + \eqref{Q1.2 + 2.2c} =\\
&=\sum_{u=1}^{n}\hat{\pmb{q}}_n(\mathbb{1}_V^{\otimes u-1} \otimes \delta_1 \otimes \mathbb{1}_V^{\otimes n-u}),\\
\eqref{Q2.3} + \eqref{Q1.1 + 2.1c} + &\eqref{Q1.1 + 2.1d} + \eqref{Q1.2 + 2.2b} =\\
&=\sum_{A(n)}\hat{\pmb{q}}_{k}(\mathbb{1}_V^{\otimes i-1} \otimes \delta_{\ell} \otimes \mathbb{1}_V^{\otimes n-k}),\\
\eqref{Q1.2 + 2.2d} &= - \sum_{B(n)}\hat{\pmb{p}}_k(\hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_1} \otimes \mbox{\dots} \otimes \hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_k}) + \hat{\pmb{q}}_1\delta_n.\end{aligned}$$ The proof is complete.
Homotopy transfer and the homological perturbation lemma {#sec:4}
========================================================
In the present section we discuss a motivation to find explicit formulas for the transfer of $A_\infty$ algebra structure presented in an apparently arbitrary form in . In the following, we recall the homological perturbation lemma and show that it gives a recipe to search for the transfer problem exactly in the form . This is the approach with which we develop and formalize [@Markl06 Remark $4$].
\[k2l25\] Let $(V, \partial_V)$ and $(W, \partial_W)$ be chain complexes together with quasi-isomorphisms $f: V \to W$ and $g: W \to V$ such that $gf - \mathbb{1}_V = \partial_Vh + h\partial_V$ for a linear map $h: V \to V$. Let $\pmb{\mu}: V \to V$ be a linear map of the same degree as $\partial_V$ such that $(\partial_V + \pmb{\mu})^2 = 0$ and the linear map $\mathbb{1}_V - \pmb{\mu} h$ is invertible ($\pmb{\mu}$ is called in this context perturbation.) We define $$\begin{aligned}
\pmb{\nu} = \partial_W + fAg, \;\;\;\; \pmb{\psi} = g + hAg, \;\;\;\; \pmb{\varphi} = f + fAh,
\;\;\;\; \pmb{H} = h + hAh,\end{aligned}$$ where $A = (\mathbb{1}_V - \pmb{\mu}h)^{-1}\pmb{\mu}$. Then $(V, \partial_V + \pmb{\mu})$ and $(W, \pmb{\nu})$ are chain complexes and $\pmb{\varphi}: V \to V$, $\pmb{\psi}: W \to W$ their quasi-isomorphisms with $\pmb{\psi\varphi} - \mathbb{1}_V = (\partial_V + \pmb{\mu})\pmb{H} + \pmb{H}(\partial_V + \pmb{\mu}).$
In our case, on $(V, \partial_V)$ we have an additional $A_\infty$ structure given by a collection of multilinear maps $\pmb{\mu} = (\mu_2, \mu_3, \mbox{\dots})$ fulfilling certain axioms. In order to regard $\pmb{\mu}$ as a perturbation, we have to pass to the (suspended) tensor algebra generated by $V$. Let us consider $\overline{T}sV$ with a coderivation $\delta_V$ and $\overline{T}sW$ with a coderivation $\delta_W$, $\hat{F}$ and $\hat{G}$ morphisms and $\hat{H}$ a homotopy between $\hat{G}\hat{F}$ and the identity on $\overline{T}sV$. Here $\delta_V$ is given by components $\{s \circ \partial_V \circ \omega: sV \to sV\} \cup
\{0: sV^{\otimes n} \to sV\}_{n \geq 2}$ in the sense of Theorem \[k1v5\], and it is codifferential by Lemma \[k1t6\] because $\partial_V$ is a differential on $V$. Analogous conclusions do apply to $\delta_W$. The map $\hat{F}: (\overline{T}sV, \delta_V) \to (\overline{T}sW, \delta_W)$ is given by components $\{s \circ f \circ \omega: sV \to sW\} \cup \{0: (sV)^{\otimes n} \to sW\}_{n \geq 2}$ (Lemma \[k1t8\]). By Lemma \[k1t9\], $\hat{F}$ is a morphism ($f$ is a map of chain complexes), i.e. $\hat{F}|_{(sV)^{\otimes n}} = \hat{f}^{\otimes n}$ for $\hat{f} = s \circ f \circ \omega$. Analogous conclusions apply to $\hat{G}$ as well. Homotopy $\hat{H}: \overline{T}sV \to \overline{T}sV$ is a map given by $\{\hat{g}\hat{f}: sV \to sV\} \cup \{0: (sV)^{\otimes n} \to sV\}_{n \geq 2}$ on the left, $\{s \circ h \circ \omega: sV \to sV\} \cup \{0: (sV)^{\otimes n} \to sV\}_{n \geq 2}$ in the middle and $\{\mathbb{1}_V: sV \to sV\} \cup \{0: (sV)^{\otimes n} \to sV\}_{n \geq 2}$ on the right in the sense of Theorem \[k1v4\]. Because $h$ is a homotopy between $gf$ and $\mathbb{1}_V$, $\hat{H}$ is a homotopy between $\hat{G}\hat{F}$ and the identity on $\overline{T}sV$ according to Theorem \[k1v11\]; the notation is $\hat{h} = s \circ h \circ \omega.$
Let $\delta_{\pmb{\mu}}$ be a coderivation on $\overline{T}sV$ corresponding to $\pmb{\mu}$, whose components are given by $\{0: sV \to sV\} \cup \{s \circ \mu_n \circ \omega^{\otimes n}: (sV)^{\otimes n} \to sV\}_{n \geq 2}$ in the sense of Theorem \[k1v13\]. Because $\partial_V$ and $\pmb{\mu}$ form an $A_\infty$ structure on $V$, $(\delta_V + \delta_{\pmb{\mu}})^2 = 0$ by Theorem \[k1v13\]; we use the notation $\delta_n = s \circ \mu_n \circ \omega^{\otimes n}.$
The remaining assumption in Lemma \[k2l25\] is the invertibility of the map $\mathbb{1}-\delta_{\pmb{\mu}}\hat{H}$. We know $$\hat{H}|_{(sV)^{\otimes n}} = \sum_{\substack{i+j = n-1,\\ i,j \geq 0}} (\hat{g}\hat{f})^{\otimes i} \otimes \hat{h} \otimes \mathbb{1}_V^{\otimes j}$$ so that $\hat{H}((sV)^{\otimes n}) \subseteq (sV)^{\otimes n}$ for all $n \geq 1$, and also $\delta_{\pmb{\mu}}|_{sV} = 0$ implies $$\delta_{\pmb{\mu}}|_{(sV)^{\otimes n}} = \delta_n + \sum_{A(n)} \mathbb{1}_V^{\otimes i-1} \otimes \delta_\ell \otimes \mathbb{1}_V^{\otimes n-k}$$ for all $n \geq 2$ with $A(n)$ as in . Consequently, for all $n \geq 2$ [holds]{} $\delta_{\pmb{\mu}}((sV)^{\otimes n}) \subseteq sV \oplus \mbox{\dots} \oplus (sV)^{\otimes n-1},$ and its iteration results in $(\delta_{\pmb{\mu}} \hat{H})^{n-1}((sV)^{\otimes n}) \subseteq sV$, $(\delta_{\pmb{\mu}} \hat{H})^{n}((sV)^{\otimes n}) = 0$.
By previous discussion and in accordance with Remark $2.3$, [@Crainic04], $$(\mathbb{1}-\delta_{\pmb{\mu}}\hat{H})^{-1}|_{sV \oplus \mbox{\dots} \oplus (sV)^{\otimes n}} = \mathbb{1} + \sum_{i = 1}^{n-1}(\delta_{\pmb{\mu}} \hat{H})^{n},$$ which means that $\mathbb{1}-\delta_{\pmb{\mu}}\hat{H}$ is invertible. Now all assumptions of Lemma \[k2l25\] are fulfilled and we can write $$\begin{aligned}
\notag
& \delta_W + \delta_{\pmb{\nu}} = \delta_W + \hat{F}\left(\delta_{\pmb{\mu}}\sum_{n \geq 0}(\hat{H}\delta_{\pmb{\mu}})^{n}\right)\hat{G},\quad \hat{\pmb{\psi}} = \hat{G} + \hat{H}\left(\delta_{\pmb{\mu}}\sum_{n \geq 0}(\hat{H}\delta_{\pmb{\mu}})^{n}\right)\hat{G},\\
\notag
& \hat{\pmb{\varphi}} = \hat{F} + \hat{F}\left(\sum_{n \geq 1}(\delta_{\pmb{\mu}}\hat{H})^{n}\right),\quad
\hat{\pmb{H}} = \hat{H} + \hat{H}\left(\sum_{n \geq 1}(\delta_{\pmb{\mu}}\hat{H})^{n}\right).\end{aligned}$$ Here we see immediately the motivation for : $\delta_{\pmb{\mu}}\sum_{n \geq 0}(\hat{H}\delta_{\pmb{\mu}})^{n}$ corresponds to the $\hat{\pmb{p}}-$kernels and $\sum_{n \geq 1}(\delta_{\pmb{\mu}}\hat{H})^{n}$ corresponds to the $\hat{\pmb{q}}-$kernels. For our purposes it is more convenient to write $$\label{PLrozpis}
\begin{split}
&\delta_W + \delta_{\pmb{\nu}} = \delta_W + \hat{F}\delta_{\pmb{\mu}}\hat{G} +\hat{F}\left(\sum_{n \geq 1}(\delta_{\pmb{\mu}}\hat{H})^{n}\right)\delta_{\pmb{\mu}}\hat{G},\\
&\hat{\pmb{\psi}} = \hat{G} + \hat{H}\delta_{\pmb{\mu}}\hat{G} +\hat{H}\left(\sum_{n \geq 1}(\delta_{\pmb{\mu}}\hat{H})^{n}\right)\delta_{\pmb{\mu}}\hat{G},\\
&\hat{\pmb{\varphi}} = \hat{F} + \hat{F}\delta_{\pmb{\mu}}\hat{H} + \hat{F}\left(\sum_{n \geq 1}(\delta_{\pmb{\mu}}\hat{H})^{n}\right)\delta_{\pmb{\mu}}\hat{H},\\
&\hat{\pmb{H}} = \hat{H} + \hat{H}\delta_{\pmb{\mu}}\hat{H} + \hat{H}\left(\sum_{n \geq 1}(\delta_{\pmb{\mu}}\hat{H})^{n}\right)\delta_{\pmb{\mu}}\hat{H}.
\end{split}$$ There is a drawback related to these formulas, however: by a direct inspection we see that $\delta_W + \delta_{\pmb{\nu}}$ is not a coderivation in the sense of Theorem \[k1v5\], $\hat{\pmb{\varphi}}$ and $\hat{\pmb{\psi}}$ do not define a morphism in the sense of Lemma \[k1t8\], and $\hat{\pmb{H}}$ does not fulfill the first part of morphism definition in the sense of Theorem \[k1v4\].
In what follows we prove that on the additional assumptions (see [@Markl06 Remark $4$]): $$\label{Okraj}
\hat{f}\hat{g} = \mathbb{1}, \;\;\;\; \hat{f}\hat{h} = 0, \;\;\;\; \hat{h}\hat{g} = 0, \;\;\;\; \hat{h}\hat{h} = 0,$$ the homological perturbation lemma gives the results compatible with Section \[sect:hotr\].
\[k2l26\] Let us assume the formulas in are satisfied. Then
1. $\hat{\pmb{q}}_n \circ \hat{g}^{\otimes n} = 0$ for $n \geq 2$,
2. $\hat{\pmb{q}}_{i+1+j} \circ ((\hat{g}\hat{f})^{\otimes i} \otimes \hat{h} \otimes \mathbb{1}_V^{\otimes j}) = 0$ for all $i,j \geq 0$, $i+j \geq 1$.
$(1)$: The proof goes by induction. By definition $\hat{\pmb{q}}_2 = \delta_2(\hat{g}\hat{f} \otimes \hat{h}) + \delta_{2}(\hat{h} \otimes \mathbb{1}_V)$ for $n = 2$, so that $\hat{\pmb{q}}_2 \otimes \hat{g}^{\otimes 2} = \delta_2(\hat{g}\hat{f}\hat{g} \otimes \hat{h}\hat{g})
+ \delta_{2}(\hat{h}\hat{g} \otimes \hat{g})$ and the claim follows from ($\hat{h}\hat{g} = 0.$)
We assume the assertion is true for all natural numbers less than $n\in{\mathbb N}$ ($n\geq 2$.) By definition $$\hat{\pmb{q}}_n\circ \hat{g}^{\otimes n} = \sum_{C(n)}\delta_k([\![\hat{\pmb{\psi}}\hat{\pmb{\varphi}}]\!]_{r_1} \circ \hat{g}^{\otimes r_1} \otimes \mbox{\dots} \otimes [\![\hat{\pmb{\psi}}\hat{\pmb{\varphi}}]\!]_{r_{i-1}} \circ \hat{g}^{\otimes r_{i-1}}\otimes \hat{h} \circ \hat{\pmb{q}}_{r_i} \circ \hat{g}^{\otimes r_i} \otimes \hat{g}^{\otimes k-i}),$$ where $$\label{p,q-symbol}
[\![\hat{\pmb{\psi}}\hat{\pmb{\varphi}}]\!]_{m} = \hat{g}\hat{f}\circ \hat{\pmb{q}}_m + \sum_{B(m)}\hat{h}\circ\hat{\pmb{p}}_k(\hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_1} \otimes \mbox{\dots} \otimes \hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_k})$$ with $[\![\hat{\pmb{\psi}}\hat{\pmb{\varphi}}]\!]_1 = \hat{g}\hat{f}.$ In the case $r_i > 1$, the composition $\hat{h} \circ \hat{\pmb{q}}_{r_i} \circ \hat{g}^{\otimes r_i}$ is trivial by the induction hypothesis. If $r_i = 1$, $\hat{h} \circ \hat{\pmb{q}}_1 \circ \hat{g} = \hat{h}\hat{g}$ is trivial by .
$(2)$: The proof is by induction on $n = i+1+j$. For $n = 2$ we prove $$\hat{\pmb{q}}_2(\hat{h} \otimes \mathbb{1}_V) = 0, \;\;\;\;\hat{\pmb{q}}_2(\hat{g}\hat{f} \otimes \hat{h}) = 0.$$ As we know $\hat{\pmb{q}}_2(\hat{h} \otimes \mathbb{1}_V) = (-1)^{|\hat{h}||\hat{h}|}\delta_2(\hat{g}\hat{f}\hat{h} \otimes \hat{h}) + \delta_2(\hat{h}\hat{h} \otimes \mathbb{1}_V)$ and $\hat{\pmb{q}}_2(\hat{g}\hat{f} \otimes \hat{h}) = \delta_2(\hat{g}\hat{f}\hat{g}\hat{f} \otimes \hat{h}\hat{h}) + \delta_2(\hat{h}\hat{g}\hat{f} \otimes \hat{h})$, the claim follows thanks to .
Let the claim hold for $m \geq 2$ and all natural numbers less than $n$, we prove it is true for $n$. First of all, for $n > i'+j'+1 \geq 2$ we have $$[\![\hat{\pmb{\psi}}\hat{\pmb{\varphi}}]\!]_{i'+1+j'} \circ ((\hat{g}\hat{f})^{\otimes i'} \otimes \hat{h} \otimes \mathbb{1}_V^{\otimes j'}) = 0$$ and also $\hat{g}\hat{f} \circ \hat{\pmb{q}}_1 \circ \hat{h} = \hat{g}\hat{f} \circ \hat{h} = 0.$ By definition $$[\![\hat{\pmb{\psi}}\hat{\pmb{\varphi}}]\!]_{i'+1+j'} \circ ((\hat{g}\hat{f})^{\otimes i'} \otimes \hat{h} \otimes \mathbb{1}_V^{\otimes j'}) = \hat{g}\hat{f} \circ \hat{\pmb{q}}_{i' + 1 + j'}\circ ((\hat{g}\hat{f})^{\otimes i'} \otimes \hat{h} \otimes \mathbb{1}_V^{\otimes j'})$$ $$+ \sum_{B(i' + 1 + j')}\hat{h} \circ \hat{\pmb{p}}_k(\hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_1} \otimes \mbox{\dots} \otimes \hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_k})\circ ((\hat{g}\hat{f})^{\otimes i'} \otimes \hat{h} \otimes \mathbb{1}_V^{\otimes j'}),$$ and by induction hypothesis $\hat{\pmb{q}}_{i' + 1 + j'}\circ ((\hat{g}\hat{f})^{\otimes i'} \otimes \hat{h} \otimes \mathbb{1}_V^{\otimes j'}) = 0.$ The last summation can be conveniently rewritten as $$\sum_{B(i' + 1 + j')}\hat{h} \circ\hat{\pmb{p}}_k(\hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_1} \otimes \mbox{\dots} \otimes \hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_k})\circ ((\hat{g}\hat{f})^{\otimes i'} \otimes \hat{h} \otimes \mathbb{1}_V^{\otimes j'})$$ $$= \sum_{B(i' + 1 + j')}\hat{h} \circ \hat{\pmb{p}}_k(\hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_1} \circ (\hat{g}\hat{f})^{\otimes r_1} \otimes \mbox{\dots}$$ $$\mbox{\dots} \otimes \hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_u} \circ ((\hat{g}\hat{f})^{\otimes \star} \otimes \hat{h} \otimes \mathbb{1}_V^{\otimes \star})\otimes \mbox{\dots} \otimes \hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_k}),$$ and the induction implies $\hat{\pmb{q}}_{r_u} \circ ((\hat{g}\hat{f})^{\otimes \star} \otimes \hat{h} \otimes \mathbb{1}_V^{\otimes \star}) = 0$ for $r_u > 1$. We already showed $\hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_u} \circ ((\hat{g}\hat{f})^{\otimes \star} \otimes \hat{h} \otimes \mathbb{1}_V^{\otimes \star}) = \hat{g}\hat{f} \circ \hat{\pmb{q}}_1 \circ \hat{h} = 0$ for $r_u = 1$.
We now return back to the main thread of the proof and show $\hat{\pmb{q}}_{n} \circ ((\hat{g}\hat{f})^{\otimes i} \otimes \hat{h} \otimes \mathbb{1}_V^{\otimes j}) = 0$. We consider $k,i',r_1, \mbox{\dots}, r_{i'-1}, r_{i'}$ in $C(n)$ given by , and compute $$\delta_k([\![\hat{\pmb{\psi}}\hat{\pmb{\varphi}}]\!]_{r_1} \otimes \mbox{\dots} \otimes [\![\hat{\pmb{\psi}}\hat{\pmb{\varphi}}]\!]_{r_{i'-1}} \otimes \hat{h} \circ \hat{\pmb{q}}_{r_{i'}} \otimes \mathbb{1}_V^{\otimes k-i'})\circ ((\hat{g}\hat{f})^{\otimes i} \otimes \hat{h} \otimes \mathbb{1}_V^{\otimes j}).$$ After substitution for $[\![\hat{\pmb{\psi}}\hat{\pmb{\varphi}}]\!]$, there are the following three possibilities for indices $i$ a $i'$:
: Then there exist $1 \leq u < i$ such that $[\![\hat{\pmb{\psi}}\hat{\pmb{\varphi}}]\!]_{r_u}\circ ((\hat{g}\hat{f})^{\otimes \star} \otimes \hat{h} \otimes \mathbb{1}_V^{\otimes \star})$. For $r_u \geq 2$ we already proved $[\![\hat{\pmb{\psi}}\hat{\pmb{\varphi}}]\!]_{r_u}\circ ((\hat{g}\hat{f})^{\otimes \star} \otimes \hat{h} \otimes \mathbb{1}_V^{\otimes \star}) = 0$, for $r_u = 1$ we have $[\![\hat{\pmb{\psi}}\hat{\pmb{\varphi}}]\!]_{r_u}\circ ((\hat{g}\hat{f})^{\otimes \star} \otimes \hat{h} \otimes \mathbb{1}_V^{\otimes \star}) = \hat{g}\hat{f} \circ \hat{h} = 0$.
: In the tensor product there is a term of the form $\hat{h} \circ \hat{\pmb{q}}_{r_{i'}} \circ ((\hat{g}\hat{f})^{\otimes \star} \otimes \hat{h} \otimes \mathbb{1}_V^{\otimes \star}),$ which is by the induction hypothesis $0$ for $r_{i'} > 1$. If $r_{i'} = 1$, then $\hat{h} \circ \hat{\pmb{q}}_{r_{i'}} \circ ((\hat{g}\hat{f})^{\otimes \star} \otimes \hat{h} \otimes \mathbb{1}_V^{\otimes \star}) = \hat{h}\hat{h}$ equals to $0$ by .
: In this case we get in the tensor product the term of the form $\hat{h} \circ \hat{\pmb{q}}_{r_{i'}} \circ (\hat{g}\hat{f})^{\otimes r_{i'}} = \hat{h} \circ \hat{\pmb{q}}_{r_{i'}} \circ \hat{g}^{\otimes r_{i'}} \circ \hat{f}^{\otimes r_{i'}}$, which is trivial for $r_{i'} \geq 2$ by $(1)$ of the lemma. If $r_{i'} = 1$, then $\hat{h} \circ \hat{\pmb{q}}_{r_{i'}} \circ (\hat{g}\hat{f})^{\otimes r_{i'}} = \hat{h} \circ \hat{g}\hat{f}$ equals to zero again by .
Because $k,i',r_1, \mbox{\dots}, r_{i'-1}, r_{i'}$ in $C(n)$ was chosen arbitrarily, we get $$\sum_{C(n)} \delta_k([\![\hat{\pmb{\psi}}\hat{\pmb{\varphi}}]\!]_{r_1} \otimes \mbox{\dots} \otimes [\![\hat{\pmb{\psi}}\hat{\pmb{\varphi}}]\!]_{r_{i'-1}} \otimes \hat{h} \circ \hat{\pmb{q}}_{r_{i'}} \otimes \mathbb{1}_V^{\otimes k-i'})\circ ((\hat{g}\hat{f})^{\otimes i} \otimes \hat{h} \otimes \mathbb{1}_V^{\otimes j}) = 0,$$ and so finally $\hat{\pmb{q}}_{n} \circ ((\hat{g}\hat{f})^{\otimes i} \otimes \hat{h} \otimes \mathbb{1}_V^{\otimes j}) = 0.$
\[remarkpsifi\] We easily observe:
1. For all $n \geq 2$ and for linear mappings $\{\pmb{a}_n: (sV)^{\otimes n} \to sV\}_{n \geq 1}$, $$\sum_{B(n)} \pmb{a}_{r_1} \otimes \mbox{\dots} \otimes \pmb{a}_{r_k} = \sum_{\substack{B(n),\\ r_k > 1}} \pmb{a}_{r_1} \otimes \mbox{\dots} \otimes \pmb{a}_{r_k}$$$$+ \sum_{u = 1}^{n-3}\sum_{\substack{B(n-u),\\ r_k > 1}}\pmb{a}_{r_1} \otimes \mbox{\dots} \otimes \pmb{a}_{r_k} \otimes \pmb{a}_1^{\otimes u} + \sum_{u = 2}^{n-1}\pmb{a}_u \otimes \pmb{a}_1^{\otimes n-u} + \pmb{a}_1^{\otimes n},$$ where $B(n)$ given as in ,
2. For all $n \geq 2$, we have $$[\![\hat{\pmb{\psi}}\hat{\pmb{\varphi}}]\!]_{n} \circ \hat{g}^{\otimes n}= \hat{h} \circ \hat{\pmb{p}}_n \circ \hat{g}^{\otimes n},$$ and if $\hat{h} \circ \hat{\pmb{p}}_1 = \mathbb{1}_V$ (Definition \[k2d17\]) the formula is true for $n = 1$ as well.
3. For all $n \geq 1$ and $0 \leq u \leq n-1$: $$[\![\hat{\pmb{\psi}}\hat{\pmb{\varphi}}]\!]_{n} \circ ((\hat{f}\hat{g})^{\otimes u} \otimes \hat{h} \otimes \mathbb{1}_V^{\otimes n-1-u}) = 0.$$
\[k2t27\] Let us assume is true for $n \geq 2$. Then $$\label{T27}
\hat{\pmb{p}}_n \circ \hat{g}^{\otimes n} = \delta_n \circ \hat{g}^{\otimes n} + \sum_{i = 2}^{n-1} \hat{\pmb{q}}_{n-i+1}\circ\left(\sum_{u = 0}^{n-i}\mathbb{1}_V^{\otimes u} \otimes \delta_{i} \otimes \mathbb{1}_V^{\otimes n-i-u}\right)\circ \hat{g}^{\otimes n}.$$
The proof goes by induction on $n$. As for $n = 2$ we have $\hat{\pmb{p}}_2 = \delta_2$, hence the claim follows.
We now assume the assertion holds for all natural numbers greater than $1$ and less than $n$. Let us consider $2 \leq m < n$ and $k,i,r_1, \mbox{\dots}, r_{i-1}, r_i$ as given in $C(m)$, so that $$\delta_k([\![\hat{\pmb{\psi}}\hat{\pmb{\varphi}}]\!]_{r_1} \otimes \mbox{\dots} \otimes [\![\hat{\pmb{\psi}}\hat{\pmb{\varphi}}]\!]_{r_{i-1}} \otimes \hat{h} \circ \hat{\pmb{q}}_{r_{i}} \otimes \mathbb{1}_V^{\otimes k-i})\circ (\hat{g}^{\otimes u} \otimes \delta_\ell \otimes \hat{g}^{\otimes m-1-u})= 0$$ whenever $u < r_1 + \mbox{\dots} + r_{i - 1}$ or $r_1 + \mbox{\dots} + r_{i} \leq u$ because $\hat{h} \circ \hat{\pmb{q}}_{r_i} \circ \hat{g}^{\otimes r_i} = 0$ for all $r_i \geq 1$ by Lemma \[k2l26\].
We fix $n-1 \geq k \geq 2, k \geq i \geq 1$ and $r_1, \mbox{\dots}, r_{i-1} \geq 1$ as in . As follows from the previous observation, all terms in $$\sum_{i = 2}^{n-1} \hat{\pmb{q}}_{n-i+1}\circ\left(\sum_{u = 0}^{n-i}\mathbb{1}_V^{\otimes u} \otimes \delta_{i} \otimes \mathbb{1}_V^{\otimes n-i-u}\right)\circ \hat{g}^{\otimes n}$$ are of the form $\delta_k([\![\hat{\pmb{\psi}}\hat{\pmb{\varphi}}]\!]_{r_1}\circ \hat{g}^{\otimes r_1} \otimes \mbox{\dots} \otimes [\![\hat{\pmb{\psi}}\hat{\pmb{\varphi}}]\!]_{r_{i-1}}\circ \hat{g}^{\otimes r_{i-1}} \otimes \star \otimes \hat{g}^{\otimes k-i})$ with $\star$ representing a mapping $(sV)^{\otimes \star} \to sV$ (the $\hat{\pmb{q}}-$kernels are given by . We can rewrite them in the form $$\delta_k([\![\hat{\pmb{\psi}}\hat{\pmb{\varphi}}]\!]_{r_1}\circ \hat{g}^{\otimes r_1} \otimes \mbox{\dots} \otimes [\![\hat{\pmb{\psi}}\hat{\pmb{\varphi}}]\!]_{r_{i-1}} \circ \hat{g}^{\otimes r_{i-1}} \otimes$$ $$\otimes \left[\delta_{n'} \circ \hat{g}^{\otimes r_1} + \sum_{i = 2}^{n'-1} \hat{\pmb{q}}_{n'-i+1}\circ\left(\sum_{u = 0}^{n'-i}\mathbb{1}_V^{\otimes u} \otimes \delta_{i} \otimes \mathbb{1}_V^{\otimes n'-i-u}\right)\circ \hat{g}^{\otimes n'}\right] \otimes \hat{g}^{\otimes k-i}),$$ where $n' = n + i - k - (r_1 + \mbox{\dots} + r_{i-1})$, $n' > 1$. Applying the second point of Remark \[remarkpsifi\] to $[\![\hat{\pmb{\psi}}\hat{\pmb{\varphi}}]\!]_{\star}\circ \hat{g}^{\otimes \star}$, the inducing hypothesis reduces the last display to $$\label{T27_1}
\delta_k(\hat{h} \circ \hat{\pmb{p}}_{r_1} \circ \hat{g}^{\otimes r_1} \otimes \mbox{\dots} \otimes \hat{h} \circ \hat{\pmb{p}}_{r_{i-1}} \circ \hat{g}^{\otimes r_{i-1}} \otimes \hat{h} \circ \hat{\pmb{p}}_{n'} \circ \hat{g}^{\otimes n'} \otimes \hat{g}^{\otimes k-i})$$ (we write $\hat{g} = \hat{h}\circ\hat{\pmb{p}}_1 \circ \hat{g}.$) By the first point of Remark \[remarkpsifi\], $$\sum_{i = 2}^{n-1} \hat{\pmb{q}}_{n-i+1}\circ\left(\sum_{u = 0}^{n-i}\mathbb{1}_V^{\otimes u} \otimes \delta_{i} \otimes \mathbb{1}_V^{\otimes n-i-u}\right)\circ \hat{g}^{\otimes n}$$ $$= \sum_{C(n), r_i > 1}\delta_k(\hat{h} \circ \hat{\pmb{p}}_{r_1} \circ \hat{g}^{\otimes r_1} \otimes \mbox{\dots} \otimes \hat{h} \circ \hat{\pmb{p}}_{r_{i-1}} \circ \hat{g}^{\otimes r_{i-1}} \otimes \hat{h} \circ \hat{\pmb{p}}_{r_i} \circ \hat{g}^{\otimes r_i} \otimes \hat{g}^{\otimes k-i})$$ $$= \sum_{B(n), k \neq n}\delta_k(\hat{h} \circ \hat{\pmb{p}}_{r_1} \circ \hat{g}^{\otimes r_1} \otimes \mbox{\dots} \otimes \hat{h} \circ \hat{\pmb{p}}_{r_k} \circ \hat{g}^{\otimes r_k}),$$ so that for each term in the sum there exists $u$, $r_u > 1$ (they are of the form of terms in with $n' > 1$.) Adding the remaining term $\delta_n \circ \hat{g}^{\otimes n}$ and using the formula for the $\hat{\pmb{p}}-$kernels, the proof concludes.
\[k2l28\] Let us assume , and also $$\label{L28}
\hat{\pmb{q}}_n = \delta_n \circ \hat{H}|_{(sV)^{\otimes n}} + \sum_{i = 2}^{n-1} \hat{\pmb{q}}_{n-i+1}\circ\left(\sum_{u = 0}^{n-i}\mathbb{1}_V^{\otimes u} \otimes \delta_{i} \otimes \mathbb{1}_V^{\otimes n-i-u}\right)\circ \hat{H}|_{(sV)^{\otimes n}}$$ to be true for all $2 \leq m \leq n$. Then $$[\![\hat{\pmb{\psi}}\hat{\pmb{\varphi}}]\!]_{n} - \hat{h} \circ \hat{\pmb{p}}_n \circ (\hat{g}\hat{f})^{\otimes n} =$$ $$= [\![\hat{\pmb{\psi}}\hat{\pmb{\varphi}}]\!]_{1} \circ \delta_n \circ \hat{H}|_{(sV)^{\otimes n}}$$$$+ \sum_{i = 2}^{n-1} [\![\hat{\pmb{\psi}}\hat{\pmb{\varphi}}]\!]_{n-i+1}\circ\left(\sum_{u = 0}^{n-i}\mathbb{1}_V^{\otimes u} \otimes \delta_{i} \otimes \mathbb{1}_V^{\otimes n-i-u}\right)\circ \hat{H}|_{(sV)^{\otimes n}}.$$
By , we have for all $m \geq 2$ $$[\![\hat{\pmb{\psi}}\hat{\pmb{\varphi}}]\!]_{m} = \hat{g}\hat{f}\circ \hat{\pmb{q}}_m + \sum_{B(m)}\hat{h}\circ\hat{\pmb{p}}_k(\hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_1} \otimes \mbox{\dots} \otimes \hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_k}),$$ (and $[\![\hat{\pmb{\psi}}\hat{\pmb{\varphi}}]\!]_{1} = \hat{g}\hat{f}.$) We can split $$[\![\hat{\pmb{\psi}}\hat{\pmb{\varphi}}]\!]_{1} \circ \delta_n \circ \hat{H}|_{(sV)^{\otimes n}}$$$$+ \sum_{i = 2}^{n-1} [\![\hat{\pmb{\psi}}\hat{\pmb{\varphi}}]\!]_{n-i+1}\circ\left(\sum_{u = 0}^{n-i}\mathbb{1}_V^{\otimes u} \otimes \delta_{i} \otimes \mathbb{1}_V^{\otimes n-i-u}\right)\circ \hat{H}|_{(sV)^{\otimes n}}$$ in two components and write $$\begin{aligned}
\label{L28_1}
\hat{g}\hat{f} &\circ \hat{\pmb{q}}_1 \circ \delta_n \circ \hat{H}|_{(sV)^{\otimes n}}\\ \notag+ \sum_{i = 2}^{n-1} \hat{g}\hat{f} \circ \hat{\pmb{q}}_{n-i+1}\circ&\left(\sum_{u = 0}^{n-i}\mathbb{1}_V^{\otimes u} \otimes \delta_{i} \otimes \mathbb{1}_V^{\otimes n-i-u}\right)\circ \hat{H}|_{(sV)^{\otimes n}}\end{aligned}$$ $$\label{L28_2}
+ \sum_{i = 2}^{n-1}\sum_{B(n-i+1)}\hat{h}\circ\hat{\pmb{p}}_k(\hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_1} \otimes \mbox{\dots} \otimes \hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_k}) \circ \left(\sum_{u = 0}^{n-i}\mathbb{1}_V^{\otimes u} \otimes \delta_{i} \otimes \mathbb{1}_V^{\otimes n-i-u}\right)\circ \hat{H}|_{(sV)^{\otimes n}}.$$ Because $\hat{\pmb{q}}_n = \mathbb{1}_{V}$, we have $\eqref{L28_1} = \hat{g}\hat{f} \circ \hat{\pmb{q}}_n$ thanks to . As for the second component , consider $k, r_1, \mbox{\dots}, r_k \in B(n-i+1)$ for some $i \geq 2$ with $B(n-i+1)$ as in . Then $$\hat{h}\circ\hat{\pmb{p}}_k(\hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_1} \otimes \mbox{\dots} \otimes \hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_k}) \circ \left(\sum_{u = 0}^{n-i}\mathbb{1}_V^{\otimes u} \otimes \delta_{i} \otimes \mathbb{1}_V^{\otimes n-i-u}\right)\circ \hat{H}|_{(sV)^{\otimes n}}$$ $$= \sum_{u=1}^{k} \hat{h}\circ\hat{\pmb{p}}_k(\hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_1} \circ (\hat{g}\hat{f})^{\otimes r_1} \otimes \mbox{\dots} \otimes \hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_{u-1}} \circ (\hat{g}\hat{f})^{\otimes r_{u-1}} \otimes$$ $$\otimes \left[\hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_u} \circ \left(\sum_{v = 0}^{r_u-1}\mathbb{1}_V^{\otimes v} \otimes \delta_{i} \otimes \mathbb{1}_V^{\otimes r_u-1-v}\right)\circ \hat{H}|_{(sV)^{\otimes r_u-1+i}}\right] \otimes \mbox{\dots} \otimes \hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_k}).$$ The reason for the appearance of such terms is that when $r_\star \geq 2$ and $\hat{h}$ were in any other $\hat{\pmb{q}}-$kernel than $\delta_i$, we would get $\hat{\pmb{q}}_{r_\star}\circ ((gf)^{\otimes \star} \otimes h \otimes \mathbb{1}_V^{\otimes \star})$ which is trivial by Lemma \[k2l26\]. If $r_\star = 1$, we get $\hat{g}\hat{f} \circ \hat{\pmb{q}}_1 \circ \hat{h} = 0$ because $\hat{\pmb{q}}_1 = \mathbb{1}_V$ and $\hat{f}\hat{h} = 0$ by . Thus we have for $i \geq 2$: $$\sum_{B(n-i+1)}\hat{h}\circ\hat{\pmb{p}}_k(\hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_1} \otimes \mbox{\dots} \otimes \hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_k})\circ\left(\sum_{u = 0}^{n-i}\mathbb{1}_V^{\otimes u} \otimes \delta_{i} \otimes \mathbb{1}_V^{\otimes n-i-u}\right)\circ \hat{H}|_{(sV)^{\otimes n}}$$ $$= \sum_{B(n-i+1)}\hat{h}\circ\hat{\pmb{p}}_k(\underline{\hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_1}} \otimes \mbox{\dots} \otimes \hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_k})$$ $$+\sum_{u = 1}^{n-i-1}\sum_{B(n-i+1-u)}\hat{h}\circ\hat{\pmb{p}}_{u + k}((\hat{g}\hat{f} \circ \hat{\pmb{q}}_1)^{\otimes u} \otimes \underline{\hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_1}} \otimes \mbox{\dots} \otimes \hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_k})$$ $$+ \sum_{r_1 = 1}^{n-1} \hat{h} \circ \hat{\pmb{p}}_{n-r_1 + 1}((\hat{g}\hat{f} \circ \hat{\pmb{q}}_1)^{\otimes n- r_1} \otimes \underline{\hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_1}}),$$ where $\underline{\hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_1}} = \hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_1} \circ \left(\sum_{v = 0}^{r_1-1}\mathbb{1}_V^{\otimes v} \otimes \delta_{i} \otimes \mathbb{1}_V^{\otimes r_1-1-v}\right)\circ \hat{H}|_{(sV)^{\otimes r_1 - 1 + i}}$. We notice $\hat{\pmb{q}}_m \circ (\hat{g}\hat{f})^{\otimes m} \neq 0$ if and only if $m = 1$ (cf., Lemma \[k2l26\].) Therefore, we expand the second contribution into $$\eqref{L28_2} = \sum_{i = 2}^{n-1}\sum_{B(n-i+1)}\hat{h}\circ\hat{\pmb{p}}_k(\underline{\hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_1}} \otimes \mbox{\dots} \otimes \hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_k})$$ $$+\sum_{i = 2}^{n-1}\sum_{u = 1}^{n-i-1}\sum_{B(n-i+1-u)}\hat{h}\circ\hat{\pmb{p}}_{u + k}((\hat{g}\hat{f} \circ \hat{\pmb{q}}_1)^{\otimes u} \otimes \underline{\hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_1}} \otimes \mbox{\dots} \otimes \hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_k})$$ $$+ \sum_{i = 2}^{n-1}\sum_{r_1 = 1}^{n-1} \hat{h} \circ \hat{\pmb{p}}_{n-r_1 + 1}((\hat{g}\hat{f} \circ \hat{\pmb{q}}_1)^{\otimes n- r_1} \otimes \underline{\hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_1}})$$ and for fixed $k, i, r_2, \mbox{\dots}, r_i$ sum up all terms of the form $\hat{h} \circ \hat{\pmb{p}}_k((\hat{g}\hat{f} \circ \hat{\pmb{q}}_1)^{\otimes k-i} \otimes \underline{\hat{g}\hat{f} \circ \hat{\pmb{q}}_{\star}} \otimes \hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_2} \otimes \mbox{\dots} \otimes \hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_i})$ in : $$\hat{h} \circ \hat{\pmb{p}}_k((\hat{g}\hat{f} \circ \hat{\pmb{q}}_1)^{\otimes k-i} \otimes \sum_{r_1 = 1}^{r'}\underline{\hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_1}} \otimes \hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_2} \otimes \mbox{\dots} \otimes \hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_i}),$$ where $r' = n - k + i - (r_2 + \mbox{\dots} + r_i).$ We recall $\underline{\hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_1}} = \hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_1}\circ\left(\sum_{u = 0}^{r_1 - 1}\mathbb{1}_V^{\otimes u} \otimes \delta_{r' - r_1 + 1} \otimes \mathbb{1}_V^{\otimes r_1-1-u}\right)\circ \hat{H}|_{(sV)^{\otimes r_1 - 1 + i}}$ and use to get $$\label{L28_3}
\hat{h} \circ \hat{\pmb{p}}_k((\hat{g}\hat{f} \circ \hat{\pmb{q}}_1)^{\otimes k-i} \otimes \hat{g}\hat{f} \circ \hat{\pmb{q}}_{r'} \otimes \hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_2} \otimes \mbox{\dots} \otimes \hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_i}).$$ Clearly $r' > 1$, and the summation over all terms in leads to $$\eqref{L28_2} = \sum_{B(n), r_1 > 1}\hat{h} \circ \hat{\pmb{p}}_k(\hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_1} \otimes \mbox{\dots} \otimes \hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_k})$$ $$+ \sum_{u = 1}^{n-3} \sum_{B(n-u), r_1 > 1} \hat{h} \circ \hat{\pmb{p}}_k((\hat{g}\hat{f} \circ \hat{\pmb{q}}_1)^{\otimes u} \otimes \hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_2} \otimes \mbox{\dots} \otimes \hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_k})$$ $$+ \sum_{r = 2}^{n-1}\hat{h} \circ \hat{\pmb{p}}_{n-r+1}((\hat{g}\hat{f} \circ \hat{\pmb{q}}_1)^{\otimes n-r} \otimes \hat{g}\hat{f} \circ \hat{\pmb{q}}_{r}).$$ We conclude $$\eqref{L28_2} = \sum_{B(n), k \neq n}\hat{h} \circ \hat{\pmb{p}}_k(\hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_1} \otimes \mbox{\dots} \otimes \hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_k}),$$ because all terms are as those in and there is always at least one $u$ such that $r_u > 1$ (this is equivalent to $r' > 1$ in .) Recall we started with $$[\![\hat{\pmb{\psi}}\hat{\pmb{\varphi}}]\!]_{1} \circ \delta_n \circ \hat{H}|_{(sV)^{\otimes n}} + \sum_{i = 2}^{n-1} [\![\hat{\pmb{\psi}}\hat{\pmb{\varphi}}]\!]_{n-i+1}\circ\left(\sum_{u = 0}^{n-i}\mathbb{1}_V^{\otimes u} \otimes \delta_{i} \otimes \mathbb{1}_V^{\otimes n-i-u}\right)\circ \hat{H}|_{(sV)^{\otimes n}}$$ $$= \eqref{L28_1} + \eqref{L28_2}$$ and showed $$\eqref{L28_1} + \eqref{L28_2} = \hat{g}\hat{f} \circ \hat{\pmb{q}}_n + \sum_{B(n), k \neq n}\hat{h} \circ \hat{\pmb{p}}_k(\hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_1} \otimes \mbox{\dots} \otimes \hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_k}).$$ Taking into account the definition of $[\![\hat{\pmb{\psi}}\hat{\pmb{\varphi}}]\!]_{n}$ in , the desired conclusion follows immediately.
\[k2t29\] Let us assume to be true. Then for all $n \geq 2$ $$\label{T29}
\hat{\pmb{q}}_n = \delta_n \circ \hat{H}|_{(sV)^{\otimes n}} + \sum_{i = 2}^{n-1} \hat{\pmb{q}}_{n-i+1}\circ\left(\sum_{u = 0}^{n-i}\mathbb{1}_V^{\otimes u} \otimes \delta_{i} \otimes \mathbb{1}_V^{\otimes n-i-u}\right)\circ \hat{H}|_{(sV)^{\otimes n}}.$$
The proof is by the induction hypothesis on $n$. For $n = 2$, by we have $\delta_2(\hat{g}\hat{f} \otimes \hat{h}) + \delta_2(\hat{h} \otimes \mathbb{1}_V)
= \delta_2 \circ (\hat{g}\hat{f} \otimes \hat{h} + \hat{h} \otimes \mathbb{1}_V)$ which is certainly true.
We assume the claim is true for all natural numbers greater than $1$ and strictly less than $n$. Let us consider $2 \leq j < n$ and $k,i,r_1, \mbox{\dots}, r_{i-1}, r_i$ as given in $C(n-j+1)$. The same reasoning as in Lemma \[k2l28\] leads to $$\delta_k([\![\hat{\pmb{\psi}}\hat{\pmb{\varphi}}]\!]_{r_1} \otimes \mbox{\dots} \otimes [\![\hat{\pmb{\psi}}\hat{\pmb{\varphi}}]\!]_{r_{i-1}} \otimes \hat{h} \circ \hat{\pmb{q}}_{r_{i}} \otimes \mathbb{1}_V^{\otimes k-i})\circ$$ $$\circ \left(\sum_{u = 0}^{n-j}\mathbb{1}_V^{\otimes u} \otimes \delta_{j} \otimes \mathbb{1}_V^{\otimes n-j-u}\right)\circ \hat{H}|_{(sV)^{\otimes n}}$$ $$\label{T29_1}
\begin{split}
&= \delta_k(\underline{[\![\hat{\pmb{\psi}}\hat{\pmb{\varphi}}]\!]_{r_1}} \otimes \mbox{\dots} \otimes [\![\hat{\pmb{\psi}}\hat{\pmb{\varphi}}]\!]_{r_{i-1}} \otimes \hat{h} \circ \hat{\pmb{q}}_{r_{i}} \otimes \mathbb{1}_V^{\otimes k-i}) \\
&+ \delta_k(\hat{h} \circ \hat{\pmb{p}}_{r_1} \circ (\hat{g}\hat{f})^{\otimes r_1} \otimes \underline{[\![\hat{\pmb{\psi}}\hat{\pmb{\varphi}}]\!]_{r_2}} \otimes \mbox{\dots} \otimes [\![\hat{\pmb{\psi}}\hat{\pmb{\varphi}}]\!]_{r_{i-1}} \otimes \hat{h} \circ \hat{\pmb{q}}_{r_{i}} \otimes \mathbb{1}_V^{\otimes k-i}) \\
& + \delta_k(\hat{h} \circ \hat{\pmb{p}}_{r_1} \circ (\hat{g}\hat{f})^{\otimes r_1} \otimes \mbox{\dots} \otimes \hat{h} \circ \hat{\pmb{p}}_{r_{i-2}} \circ (\hat{g}\hat{f})^{\otimes r_{i-2}} \otimes \underline{[\![\hat{\pmb{\psi}}\hat{\pmb{\varphi}}]\!]_{r_{i-1}}} \otimes \hat{h} \circ \hat{\pmb{q}}_{r_{i}} \otimes \mathbb{1}_V^{\otimes k-i}) \\
&+ \delta_k(\hat{h} \circ \hat{\pmb{p}}_{r_1} \circ (\hat{g}\hat{f})^{\otimes r_1} \otimes \mbox{\dots} \otimes \hat{h} \circ \hat{\pmb{p}}_{r_{i-1}} \circ (\hat{g}\hat{f})^{\otimes r_{i-1}} \otimes \underline{\hat{h} \circ \hat{\pmb{q}}_{r_{i}}} \otimes \mathbb{1}_V^{\otimes k-i}) \\
&+ \delta_k(\hat{h} \circ \hat{\pmb{p}}_{r_1} \circ (\hat{g}\hat{f})^{\otimes r_1} \otimes \mbox{\dots} \otimes \hat{h} \circ \hat{\pmb{p}}_{r_{i-1}} \circ (\hat{g}\hat{f})^{\otimes r_{i-1}} \otimes \overline{\hat{h} \circ \hat{\pmb{q}}_{r_{i}}} \otimes \hat{H}|_{(sV)^{\otimes k -i}}).
\end{split}$$ Hereby we expanded a general summand in the definition of $\hat{\pmb{q}}_{n-j+1}$ as in , where $$\underline{[\![\hat{\pmb{\psi}}\hat{\pmb{\varphi}}]\!]_{r_\ell}} = [\![\hat{\pmb{\psi}}\hat{\pmb{\varphi}}]\!]_{r_1}\left(\sum_{u = 0}^{r_\ell - 1}\mathbb{1}_V^{\otimes u} \otimes \delta_{j} \otimes \mathbb{1}_V^{\otimes r_\ell - 1 -u}\right)\circ \hat{H}|_{(sV)^{\otimes r_i - 1 + j}},$$ $$\underline{\hat{h} \circ \hat{\pmb{q}}_{r_{i}}} = \hat{h} \circ \hat{\pmb{q}}_{r_{i}}\left(\sum_{u = 0}^{r_i - 1}\mathbb{1}_V^{\otimes u} \otimes \delta_{j} \otimes \mathbb{1}_V^{\otimes r_i - 1 - u}\right)\circ \hat{H}|_{(sV)^{\otimes r_i - 1 + j}},$$ $$\overline{\hat{h} \circ \hat{\pmb{q}}_{r_{i}}} = \hat{h} \circ \hat{\pmb{q}}_{r_{i}}\left(\sum_{u = 0}^{r_i - 1}\mathbb{1}_V^{\otimes u} \otimes \delta_{j} \otimes \mathbb{1}_V^{\otimes r_i - 1 - u}\right)\circ (\hat{g}\hat{f})^{\otimes r_i - 1 + j}.$$ In the previous formulas there are no signs whatsoever, because $\hat{h} \circ \hat{\pmb{q}}_{r_i}$ pass through the terms of degree $0$, and $\hat{h}$ in $\hat{H}$ and $\delta_j$ are of degree $1$ and $-1$, respectively, so that their sign contributions cancel out.
In the next few steps we show how the terms are organized:
**I.** Let us choose $k, i, r_1, \mbox{\dots}, r_i$ given in $C(n)$ such that $r_i > 1$, and sum up all terms of the form $\delta_k(\hat{h} \circ \hat{\pmb{p}}_{r_1} \circ (\hat{g}\hat{f})^{\otimes r_1} \otimes \mbox{\dots} \otimes \hat{h} \circ \hat{\pmb{p}}_{r_{i-1}} \circ (\hat{g}\hat{f})^{\otimes r_{i-1}} \otimes \underline{\hat{h} \circ \hat{\pmb{q}}_r} \otimes \mathbb{1}_V^{\otimes k-i})$ out of the summation $$\delta_n \circ \hat{H}|_{(sV)^{\otimes n}} + \sum_{i = 2}^{n-1} \hat{\pmb{q}}_{n-i+1}\circ\left(\sum_{u = 0}^{n-i}\mathbb{1}_V^{\otimes u} \otimes \delta_{i} \otimes \mathbb{1}_V^{\otimes n-i-u}\right)\circ \hat{H}|_{(sV)^{\otimes n}}$$ for all allowable $r$. We get $$\delta_k(\hat{h} \circ \hat{\pmb{p}}_{r_1} \circ (\hat{g}\hat{f})^{\otimes r_1} \otimes \mbox{\dots} \otimes \hat{h} \circ \hat{\pmb{p}}_{r_{i-1}} \circ (\hat{g}\hat{f})^{\otimes r_{i-1}} \otimes \sum_{r = 1}^{r_i}\underline{\hat{h} \circ \hat{\pmb{q}}_{r}} \otimes \mathbb{1}_V^{\otimes k-i}),$$ where $$\underline{\hat{h}\circ \hat{\pmb{q}}_{r}} = \hat{h}\circ \hat{\pmb{q}}_{r} \circ\left(\sum_{u = 0}^{r - 1}\mathbb{1}_V^{\otimes u} \otimes \delta_{r_i - r + 1} \otimes \mathbb{1}_V^{\otimes r-1-u}\right)\circ \hat{H}|_{(sV)^{\otimes r_i}}.$$ Because $r_i < n$, we get by the induction hypothesis $$\label{terma}
\delta_k(\hat{h} \circ \hat{\pmb{p}}_{r_1} \circ (\hat{g}\hat{f})^{\otimes r_1} \otimes \mbox{\dots} \otimes \hat{h} \circ \hat{\pmb{p}}_{r_{i-1}} \circ (\hat{g}\hat{f})^{\otimes r_{i-1}} \otimes \hat{h} \circ \hat{\pmb{q}}_{r_i} \otimes \mathbb{1}_V^{\otimes k-i}).$$ If $r_{i - 1} > 1$, we sum up all terms of the form $\delta_k(\hat{h} \circ \hat{\pmb{p}}_{r_1} \circ (\hat{g}\hat{f})^{\otimes r_1} \otimes \mbox{\dots} \otimes\underline{[\![\hat{\pmb{\psi}}\hat{\pmb{\varphi}}]\!]_{\star}} \otimes \hat{h} \circ \hat{\pmb{q}}_{r_i} \otimes \mathbb{1}_V^{\otimes k-i})$: $$\delta_k(\hat{h} \circ \hat{\pmb{p}}_{r_1} \circ (\hat{g}\hat{f})^{\otimes r_1} \otimes \mbox{\dots} \otimes \sum_{r = 1}^{r_{i-1}}\underline{[\![\hat{\pmb{\psi}}\hat{\pmb{\varphi}}]\!]_{r}} \otimes \hat{h} \circ \hat{\pmb{q}}_{r_i} \otimes \mathbb{1}_V^{\otimes k-i}),$$ with $$\underline{[\![\hat{\pmb{\psi}}\hat{\pmb{\varphi}}]\!]_{r}} = [\![\hat{\pmb{\psi}}\hat{\pmb{\varphi}}]\!]_{r} \circ\left(\sum_{u = 0}^{r - 1}\mathbb{1}_V^{\otimes u} \otimes \delta_{r_{i-1} - r + 1} \otimes \mathbb{1}_V^{\otimes r-1-u}\right)\circ \hat{H}|_{(sV)^{\otimes r_{i-1}}}.$$ Because $r_i < n$, by the induction hypothesis is Lemma \[k2l28\] fulfilled and the last display reduces to $$\delta_k(\hat{h} \circ \hat{\pmb{p}}_{r_1} \circ (\hat{g}\hat{f})^{\otimes r_1} \otimes \mbox{\dots} \otimes \hat{h} \circ \hat{\pmb{p}}_{r_{i-2}} \circ (\hat{g}\hat{f})^{\otimes r_{i-2}} \otimes$$ $$\otimes \left[ [\![\hat{\pmb{\psi}}\hat{\pmb{\varphi}}]\!]_{r_{i-1}} - \hat{h} \circ \hat{\pmb{p}}_{r_{i-1}} \circ (\hat{g}\hat{f})^{r_{i-1}} \right]\otimes \hat{h} \circ \hat{\pmb{q}}_{r_i} \otimes \mathbb{1}_V^{\otimes k-i}).$$ The sum of the last display and results in $$\delta_k(\hat{h} \circ \hat{\pmb{p}}_{r_1} \circ (\hat{g}\hat{f})^{\otimes r_1} \otimes \mbox{\dots} \otimes \hat{h} \circ \hat{\pmb{p}}_{r_{i-2}} \circ (\hat{g}\hat{f})^{\otimes r_{i-2}} \otimes [\![\hat{\pmb{\psi}}\hat{\pmb{\varphi}}]\!]_{r_{i-1}}\otimes \hat{h} \circ \hat{\pmb{q}}_{r_i} \otimes \mathbb{1}_V^{\otimes k-i}),$$ which is the same expression as for $r_{i-1} = 1$ because $[\![\hat{\pmb{\psi}}\hat{\pmb{\varphi}}]\!]_{r_{i-1}} = \hat{h} \circ \hat{\pmb{p}}_{r_{i-1}} \circ (\hat{g}\hat{f})^{r_{i-1}}$ in this case. Repeating this procedure, we arrive at $\delta_k([\![\hat{\pmb{\psi}}\hat{\pmb{\varphi}}]\!]_{r_{1}} \otimes \mbox{\dots} \otimes [\![\hat{\pmb{\psi}}\hat{\pmb{\varphi}}]\!]_{r_{i-1}}\otimes \hat{h} \circ \hat{\pmb{q}}_{r_i} \otimes \mathbb{1}_V^{\otimes k-i}).$
We summarize the previous considerations: for $k, i, r_1, \mbox{\dots}, r_i$ as in $C(n)$ such that $r_i > 1$, we have $$\label{T29_2}
\begin{split}
&\delta_k([\![\hat{\pmb{\psi}}\hat{\pmb{\varphi}}]\!]_{r_1} \otimes \mbox{\dots} \otimes [\![\hat{\pmb{\psi}}\hat{\pmb{\varphi}}]\!]_{r_{i-1}} \otimes \hat{h} \circ \hat{\pmb{q}}_{r_{i}} \otimes \mathbb{1}_V^{\otimes k-i})\\
&= \delta_k(\hat{h} \circ \hat{\pmb{p}}_{r_1} \circ (\hat{g}\hat{f})^{\otimes r_1} \otimes \mbox{\dots} \otimes \hat{h} \circ \hat{\pmb{p}}_{r_{i-1}} \circ (\hat{g}\hat{f})^{\otimes r_{i-1}} \otimes \sum_{r = 1}^{r_i}\underline{\hat{h} \circ \hat{\pmb{q}}_{r}} \otimes \mathbb{1}_V^{\otimes k-i}) \\
&+ \delta_k(\hat{h} \circ \hat{\pmb{p}}_{r_1} \circ (\hat{g}\hat{f})^{\otimes r_1} \otimes \mbox{\dots} \otimes \sum_{r = 1}^{r_{i-1}}\underline{[\![\hat{\pmb{\psi}}\hat{\pmb{\varphi}}]\!]_{r_{i-i}}} \otimes \hat{h} \circ \hat{\pmb{q}}_{r_i} \otimes \mathbb{1}_V^{\otimes k-i}) + \mbox{\dots}\\
& + \delta_k(\hat{h} \circ \hat{\pmb{p}}_{r_1} \circ (\hat{g}\hat{f})^{\otimes r_1} \otimes \sum_{r = 1}^{r_2}\underline{[\![\hat{\pmb{\psi}}\hat{\pmb{\varphi}}]\!]_{r}}
\otimes \mbox{\dots} \otimes [\![\hat{\pmb{\psi}}\hat{\pmb{\varphi}}]\!]_{r_{i-i}} \otimes \hat{h} \circ \hat{\pmb{q}}_{r_i} \otimes \mathbb{1}_V^{\otimes k-i}) \\
&+ \delta_k(\sum_{r = 1}^{r_1}\underline{[\![\hat{\pmb{\psi}}\hat{\pmb{\varphi}}]\!]_{r}} \otimes [\![\hat{\pmb{\psi}}\hat{\pmb{\varphi}}]\!]_{r_2}
\otimes \mbox{\dots} \otimes [\![\hat{\pmb{\psi}}\hat{\pmb{\varphi}}]\!]_{r_{i-i}} \otimes \hat{h} \circ \hat{\pmb{q}}_{r_i} \otimes \mathbb{1}_V^{\otimes k-i}).
\end{split}$$ **II.** Let us choose $k, i, r_1, \mbox{\dots}, r_i$ as in $C(n)$ such that $i > 1$, $r_i = 1$, and there exists $1 \leq u \leq i - 1$ such that $r_u > 1$ and $r_{u + 1} = \mbox{\dots} = r_{i-1} = 1$. Then $$\label{T29_3}
\begin{split}
& \delta_k([\![\hat{\pmb{\psi}}\hat{\pmb{\varphi}}]\!]_{r_1}
\otimes \mbox{\dots} \otimes [\![\hat{\pmb{\psi}}\hat{\pmb{\varphi}}]\!]_{r_{u}} \otimes(\hat{g}\hat{f})^{\otimes i - u + 1}\otimes \hat{h} \otimes \mathbb{1}_V^{\otimes k-i}) \\
& = \delta_k(\sum_{r = 1}^{r_1}\underline{[\![\hat{\pmb{\psi}}\hat{\pmb{\varphi}}]\!]_{r}} \otimes [\![\hat{\pmb{\psi}}\hat{\pmb{\varphi}}]\!]_{r_2}
\otimes \mbox{\dots} \otimes [\![\hat{\pmb{\psi}}\hat{\pmb{\varphi}}]\!]_{r_{u}} \otimes(\hat{g}\hat{f})^{\otimes i - u + 1}\otimes \hat{h} \otimes \mathbb{1}_V^{\otimes k-i}) \\
& + \delta_k(\hat{h} \circ \hat{\pmb{p}}_{r_1} \circ (\hat{g}\hat{f})^{\otimes r_1} \otimes \sum_{r = 1}^{r_2}\underline{[\![\hat{\pmb{\psi}}\hat{\pmb{\varphi}}]\!]_{r}}
\otimes \mbox{\dots} \otimes [\![\hat{\pmb{\psi}}\hat{\pmb{\varphi}}]\!]_{r_{u}} \otimes\\
&\quad \otimes(\hat{g}\hat{f})^{\otimes i - u + 1}\otimes \hat{h} \otimes \mathbb{1}_V^{\otimes k-i}) + \mbox{\dots}\\
& + \delta_k(\hat{h} \circ \hat{\pmb{p}}_{r_1} \circ (\hat{g}\hat{f})^{\otimes r_1} \otimes \mbox{\dots} \otimes \hat{h} \circ \hat{\pmb{p}}_{r_{u-1}} \circ (\hat{g}\hat{f})^{\otimes r_{u-1}}
\otimes \sum_{r = 1}^{r_u}\underline{[\![\hat{\pmb{\psi}}\hat{\pmb{\varphi}}]\!]_{r}} \otimes\\
& \quad\otimes (\hat{g}\hat{f})^{\otimes i - u + 1}\otimes \hat{h} \otimes \mathbb{1}_V^{\otimes k-i})\\
& + \delta_k(\hat{h} \circ \hat{\pmb{p}}_{r_1} \circ (\hat{g}\hat{f})^{\otimes r_1} \otimes \mbox{\dots} \otimes \hat{h} \circ \hat{\pmb{p}}_{r_{u-1}} \circ (\hat{g}\hat{f})^{\otimes r_{u-1}} \otimes \sum_{r = 1}^{r_u}\overline{\hat{h} \circ \hat{\pmb{q}}_{r}} \otimes\\
& \quad\otimes (\hat{g}\hat{f})^{\otimes i - u + 1}\otimes \hat{h} \otimes \mathbb{1}_V^{\otimes k-i})
\end{split}$$ with $$\overline{\hat{h} \circ \hat{\pmb{q}}_{r}} = \hat{h} \circ \hat{\pmb{q}}_{r} \circ \left(\sum_{v = 0}^{r - 1}\mathbb{1}_V^{\otimes v} \otimes \delta_{r_{u} - r + 1} \otimes \mathbb{1}_V^{\otimes r-1-v}\right)\circ (\hat{g}\hat{f})^{\otimes r_u}.$$ By Lemma \[k2t27\] $$\sum_{r = 1}^{r_u}\overline{\hat{h} \circ \hat{\pmb{q}}_{r}} = \hat{h} \circ \hat{\pmb{p}}_{r_u} \circ (\hat{g}\hat{f})^{\otimes r_u},$$ which can be justified in the same way as in the first step **I.**
We expand all terms in the summation (denoted ) $$\sum_{i = 2}^{n-1} \hat{\pmb{q}}_{n-i+1}\circ\left(\sum_{u = 0}^{n-i}\mathbb{1}_V^{\otimes u} \otimes \delta_{i} \otimes \mathbb{1}_V^{\otimes n-i-u}\right)\circ \hat{H}|_{(sV)^{\otimes n}}$$ and use a to rewrite terms in the definition of $\hat{\pmb{q}}_n$: $$\sum_{i = 2}^{n-1} \hat{\pmb{q}}_{n-i+1}\circ\left(\sum_{u = 0}^{n-i}\mathbb{1}_V^{\otimes u} \otimes \delta_{i} \otimes \mathbb{1}_V^{\otimes n-i-u}\right)\circ \hat{H}_{(sV)^{\otimes n}}$$ $$=\sum_{C(n), k \neq n}\delta_k([\![\hat{\pmb{\psi}}\hat{\pmb{\varphi}}]\!]_{r_1} \otimes \mbox{\dots} \otimes [\![\hat{\pmb{\psi}}\hat{\pmb{\varphi}}]\!]_{r_{i-1}} \otimes \hat{h} \circ \hat{\pmb{q}}_{r_i} \otimes \mathbb{1}_V^{\otimes k-i}).$$ Certainly, $$\delta_n \circ \hat{H}_{(sV)^{\otimes n}} = \sum_{C(n), k = n}\delta_k([\![\hat{\pmb{\psi}}\hat{\pmb{\varphi}}]\!]_{r_1} \otimes \mbox{\dots} \otimes [\![\hat{\pmb{\psi}}\hat{\pmb{\varphi}}]\!]_{r_{i-1}} \otimes \hat{h} \circ \hat{\pmb{q}}_{r_i} \otimes \mathbb{1}_V^{\otimes k-i}),$$ which together with completes the proof.
Adopting slight changes in the proofs, our claims can be reformulated as follows:
[Lemma \[k2t27\]]{}:
: [*On the assumption holds for all $n \geq 2$*]{} $$\label{T27*}
\begin{split}
\left(\sum_{B(n)} \hat{h} \circ \hat{\pmb{p}}_{r_1} \circ \hat{g}^{\otimes r_1} \otimes \mbox{\dots} \otimes \hat{h} \circ \hat{\pmb{p}}_{r_k}\right) \circ \hat{g}^{\otimes r_k} = \hat{g}^{\otimes n} \\
+ \sum_{i = 2}^{n-1} \sum_{C(n-i+1)}[\![\hat{\pmb{\psi}}\hat{\pmb{\varphi}}]\!]_{r_1} \otimes \mbox{\dots} \otimes [\![\hat{\pmb{\psi}}\hat{\pmb{\varphi}}]\!]_{r_{i-1}} \otimes \hat{h} \circ \hat{\pmb{q}}_{r_i} \otimes \mathbb{1}_V^{\otimes k-i}\circ\\
\circ\left(\sum_{u = 0}^{n-i}\mathbb{1}_V^{\otimes u} \otimes \delta_{i} \otimes \mathbb{1}_V^{\otimes n-i-u}\right)\circ \hat{g}^{\otimes n},
\end{split}$$ where we write $\hat{h} \circ \hat{\pmb{p}}_{r_1} \circ \hat{g}^{\otimes r_1} \otimes \mbox{\dots} \otimes \hat{h} \circ \hat{\pmb{p}}_{r_k}$ instead of $\delta_k(\hat{h} \circ \hat{\pmb{p}}_{r_1} \circ \hat{g}^{\otimes r_1} \otimes \mbox{\dots} \otimes \hat{h} \circ \hat{\pmb{p}}_{r_k})$ and $[\![\hat{\pmb{\psi}}\hat{\pmb{\varphi}}]\!]_{r_1} \otimes \mbox{\dots} \otimes [\![\hat{\pmb{\psi}}\hat{\pmb{\varphi}}]\!]_{r_{i-1}} \otimes \hat{h} \circ \hat{\pmb{q}}_{r_i} \otimes \mathbb{1}_V^{\otimes k-i}$ instead of $\delta_k([\![\hat{\pmb{\psi}}\hat{\pmb{\varphi}}]\!]_{r_1} \otimes \mbox{\dots} \otimes [\![\hat{\pmb{\psi}}\hat{\pmb{\varphi}}]\!]_{r_{i-1}} \otimes \hat{h} \circ \hat{\pmb{q}}_{r_i} \otimes \mathbb{1}_V^{\otimes k-i})$;
[Lemma \[k2l28\]]{}:
: [*On the assumption holds for all $n \geq 2$*]{} $$\label{L28*}
\begin{split}
\hat{f} \circ \hat{\pmb{q}}_n + \sum_{B(n)}\hat{f} \circ \hat{\pmb{q}}_{r_1} \otimes \mbox{\dots} \otimes\hat{f} \circ \hat{\pmb{q}}_{r_k} - \hat{f}^{\otimes n} = \hat{f} \circ \delta_n \circ \hat{H}|_{(sV)^{\otimes n}} \\
+ \sum_{i = 2}^{n-1} \left(\hat{f} \circ \hat{\pmb{q}}_{n-i+1} + \sum_{B(n-i+1)}\hat{f} \circ \hat{\pmb{q}}_{r_1} \otimes \mbox{\dots} \otimes\hat{f} \circ \hat{\pmb{q}}_{r_k}\right)\circ\\
\circ \left(\sum_{u = 0}^{n-i}\mathbb{1}_V^{\otimes u} \otimes \delta_{i} \otimes \mathbb{1}_V^{\otimes n-i-u}\right)\circ \hat{H}|_{(sV)^{\otimes n}},
\end{split}$$ where we write $\hat{f} \circ \hat{\pmb{q}}_{r_1} \otimes \mbox{\dots} \otimes\hat{f} \circ \hat{\pmb{q}}_{r_k}$ instead of $\hat{h} \circ \hat{\pmb{p}}_k(\hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_1} \otimes \mbox{\dots} \otimes \hat{g}\hat{f} \circ \hat{\pmb{q}}_{r_k})$;
[Lemma \[k2t29\]]{}:
: [*On the assumption , we have for all $n \geq 2$*]{} $$\label{T29*}
\begin{split}
\sum_{C(n)}[\![\hat{\pmb{\psi}}\hat{\pmb{\varphi}}]\!]_{r_1} \otimes \mbox{\dots} \otimes [\![\hat{\pmb{\psi}}\hat{\pmb{\varphi}}]\!]_{r_{i-1}} \otimes \hat{h} \circ \hat{\pmb{q}}_{r_i} \otimes \mathbb{1}_V^{\otimes k-i} = \hat{H}|_{(sV)^{\otimes n}}\\
+ \sum_{i = 2}^{n-1} \sum_{C(n-i+1)}[\![\hat{\pmb{\psi}}\hat{\pmb{\varphi}}]\!]_{r_1} \otimes \mbox{\dots} \otimes [\![\hat{\pmb{\psi}}\hat{\pmb{\varphi}}]\!]_{r_{i-1}} \otimes \hat{h} \circ \hat{\pmb{q}}_{r_i} \otimes \mathbb{1}_V^{\otimes k-i}\circ
\\
\circ\left(\sum_{u = 0}^{n-i}\mathbb{1}_V^{\otimes u} \otimes \delta_{i} \otimes \mathbb{1}_V^{\otimes n-i-u}\right)\circ \hat{H}|_{(sV)^{\otimes n}},
\end{split}$$ where we write $[\![\hat{\pmb{\psi}}\hat{\pmb{\varphi}}]\!]_{r_1} \otimes \mbox{\dots} \otimes [\![\hat{\pmb{\psi}}\hat{\pmb{\varphi}}]\!]_{r_{i-1}} \otimes \hat{h} \circ \hat{\pmb{q}}_{r_i} \otimes \mathbb{1}_V^{\otimes k-i}$ instead of $\delta_k([\![\hat{\pmb{\psi}}\hat{\pmb{\varphi}}]\!]_{r_1} \otimes \mbox{\dots} \otimes [\![\hat{\pmb{\psi}}\hat{\pmb{\varphi}}]\!]_{r_{i-1}} \otimes \hat{h} \circ \hat{\pmb{q}}_{r_i} \otimes \mathbb{1}_V^{\otimes k-i})$.
On the assumption , the formulas produced by the homological perturbation lemma fulfill $$\tag{\textit{{1}}}
\label{V30a}
\delta_{\pmb{\nu}}|_{(sV)^{\otimes n}} = \hat{f} \circ \hat{\pmb{p}}_n \circ \hat{g}^{\otimes n} + \sum_{A(n)} \mathbb{1}_W^{\otimes i - 1} \otimes \hat{f} \circ \hat{\pmb{p}}_\ell \circ \hat{g}^{\otimes \ell} \otimes \mathbb{1}_W^{\otimes n-k},$$ $$\tag{\textit{{2}}}
\label{V30b}
\hat{\pmb{\psi}}|_{(sV)^{\otimes n}} = \hat{h} \circ \hat{\pmb{p}}_n \circ \hat{g}^{\otimes n} + \sum_{B(n)} \hat{h} \circ \hat{\pmb{p}}_{r_1} \circ \hat{g}^{\otimes r_1} \otimes \mbox{\dots} \otimes \hat{h} \circ \hat{\pmb{p}}_{r_k} \circ \hat{g}^{\otimes r_k},$$ $$\tag{\textit{{3}}}
\label{V30c}
\hat{\pmb{\varphi}}|_{(sV)^{\otimes n}} = \hat{f} \circ \hat{\pmb{q}}_n + \sum_{B(n)} \hat{f} \circ \hat{\pmb{q}}_{r_1} \otimes \mbox{\dots} \otimes \hat{f} \circ \hat{\pmb{q}}_{r_k},$$ $$\tag{\textit{{4}}}
\label{V30d}
\hat{\pmb{H}}|_{(sV)^{\otimes n}} = \hat{h} \circ \hat{\pmb{q}}_n + \sum_{C(n)}[\![\hat{\pmb{\psi}}\hat{\pmb{\varphi}}]\!]_{r_1} \otimes \mbox{\dots} \otimes [\![\hat{\pmb{\psi}}\hat{\pmb{\varphi}}]\!]_{r_{i-1}} \otimes \hat{h} \circ \hat{\pmb{q}}_{r_i} \otimes \mathbb{1}_V^{\otimes k-i}$$ for all $n \geq 2$. In particular, $\delta_W + \delta_{\pmb{\nu}}$ is a codifferential, $\hat{\pmb{\psi}}$, $\hat{\pmb{\varphi}}$ are morphisms and $\hat{\pmb{H}}$ is a homotopy between $\hat{\pmb{\psi}}\hat{\pmb{\varphi}}$ and $\mathbb{1}$. When expressed in terms of $A_\infty$ algebras, the relevant objects fulfill .
We already noticed $$(\delta_{\pmb{\mu}}\hat{H})((sV)^{\otimes n}) \subseteq sV \oplus \mbox{\dots} \oplus (sV)^{\otimes n-1},$$ $$(\delta_{\pmb{\mu}} \hat{H})^{n-1}((sV)^{\otimes n}) \subseteq sV, \;\;\; (\delta_{\pmb{\mu}} \hat{H})^{n}((sV)^{\otimes n}) = 0.$$ & : We prove by the induction hypothesis . For $n = 2$, we get by $$\hat{\pmb{\varphi}}|_{(sV)^{\otimes 2}} = \hat{F}|_{(sV)^{\otimes 2}} + \hat{F}\delta_{\pmb{\mu}} \hat{H}|_{(sV)^{\otimes 2}} = \hat{f} \otimes \hat{f} + \hat{f}\circ (\delta_2(\hat{g}\hat{f} \otimes \hat{h}) + \delta_2(\hat{h} \otimes \mathbb{1}_V))$$ $$= \hat{f} \circ \hat{\pmb{q}}_1 \otimes \hat{f} \circ \hat{\pmb{q}}_1 + \hat{f} \circ \hat{\pmb{q}}_2.$$ Let us assume holds for all natural number greater than $1$ and less than $n$. Because $\delta_{\pmb{\mu}} \hat{H}$ decreases the homogeneity, $$\hat{\pmb{\varphi}}|_{(sV)^{\otimes n}} = \hat{F}|_{(sV)^{\otimes n}} + \hat{F}\delta_{\pmb{\mu}} \hat{H}|_{(sV)^{\otimes n}} + \hat{F}\left(\sum_{m = 1}^{n-2}(\delta_{\pmb{\mu}} \hat{H})^{m}\right)\delta_{\pmb{\mu}} \hat{H}|_{(sV)^{\otimes n}}$$ $$= \hat{F}|_{(sV)^{\otimes n}} + \hat{F} \circ\left(\sum_{u = 0}^{n-i}\mathbb{1}_V^{\otimes u} \otimes \delta_{i} \otimes \mathbb{1}_V^{\otimes n-i-u}\right)\circ \hat{H}|_{(sV)^{\otimes n}}$$ $$+ \hat{F}\left(\sum_{m = 1}^{n-2}(\delta_{\pmb{\mu}} \hat{H})^{m}\right)\circ\left(\sum_{u = 0}^{n-i}\mathbb{1}_V^{\otimes u} \otimes \delta_{i} \otimes \mathbb{1}_V^{\otimes n-i-u}\right)\circ \hat{H}|_{(sV)^{\otimes n}}.$$ The mapping $\left(\sum_{u = 0}^{n-i}\mathbb{1}_V^{\otimes u} \otimes \delta_{i} \otimes \mathbb{1}_V^{\otimes n-i-u}\right)\circ \hat{H}|_{(sV)^{\otimes n}}$ is of homogeneity $n-i+1$, so allows us to rewrite the last result as $$\hat{F}|_{(sV)^{\otimes n}} + \hat{f} \circ \delta_n \circ \hat{H}|_{(sV)^{\otimes n}}$$ $$+ \sum_{i = 2}^{n-1} \hat{\pmb{\varphi}}|_{(sV)^{\otimes n-i+1}}\circ\left(\sum_{u = 0}^{n-i}\mathbb{1}_V^{\otimes u} \otimes \delta_{i} \otimes \mathbb{1}_V^{\otimes n-i-u}\right)\circ \hat{H}|_{(sV)^{\otimes n}}$$ and the combination of induction hypothesis $\hat{\pmb{\varphi}}|_{(sV)^{\otimes n-i+1}}$ and Lemma \[k2t29\], , gives the required form $$\hat{F}|_{(sV)^{\otimes n}} + \hat{f} \circ \hat{\pmb{q}}_n + \sum_{B(n)}\hat{f} \circ \hat{\pmb{q}}_{r_1} \otimes \mbox{\dots} \otimes\hat{f} \circ \hat{\pmb{q}}_{r_k} - \hat{f}^{\otimes n}.$$ Let us remark that gives for all $n \geq 2$ $$\delta_{\pmb{\nu}}|_{(sV)^{\otimes n}} = \hat{f} \circ \delta_n \circ \hat{g}^{\otimes n}$$$$+ \sum_{i = 2}^{n-1} \hat{\pmb{\varphi}}|_{(sV)^{\otimes n-i+1}}\circ\left(\sum_{u = 0}^{n-i}\mathbb{1}_V^{\otimes u} \otimes \delta_{i} \otimes \mathbb{1}_V^{\otimes n-i-u}\right)\circ \hat{g}^{\otimes n}.$$ Choosing $2 \leq j \leq n-1$, Lemma \[k2l26\] implies $$\hat{\pmb{\varphi}}|_{(sV)^{\otimes n-i+1}}\circ\left(\sum_{u = 0}^{n-i}\mathbb{1}_V^{\otimes u} \otimes \delta_{i} \otimes \mathbb{1}_V^{\otimes n-i-u}\right)\circ \hat{g}^{\otimes n}$$ $$= \hat{f} \circ \hat{\pmb{q}}_{n-i+1} \circ\left(\sum_{u = 0}^{n-i}\mathbb{1}_V^{\otimes u} \otimes \delta_{i} \otimes \mathbb{1}_V^{\otimes n-i-u}\right)\circ \hat{g}^{\otimes n}$$ $$+\sum_{u = 1}^{n-i+1} (\hat{f}\hat{g})^{\otimes u - 1} \otimes \hat{f} \circ \delta_i \circ \hat{g}^{\otimes i} \otimes (\hat{f}\hat{g})^{\otimes n-i+1-u}$$ $$+ \sum_{A(n-i+1)} (\hat{f}\hat{g})^{\otimes i - 1} \otimes \hat{f} \circ \hat{\pmb{q}}_{\ell} \circ\left(\sum_{u = 0}^{\ell-1}\mathbb{1}_V^{\otimes u} \otimes \delta_{i} \otimes \mathbb{1}_V^{\otimes \ell-1 - u}\right) \circ \hat{g}^{\otimes \ell - 1 + i} \otimes (\hat{f}\hat{g})^{\otimes n-i+1-k}.$$ We take into account , $\hat{f}\hat{g} = \mathbb{1}_W$, and sum up over all terms of the form $\mathbb{1}_W^{\otimes \star} \otimes \star \otimes \mathbb{1}_W^{\otimes \star}$: $$\delta_{\pmb{\nu}}|_{(sV)^{\otimes n}} = \hat{f} \circ \delta_n \circ \hat{g}^{\otimes n} + \sum_{i = 2}^{n-1} \hat{f} \circ \hat{\pmb{q}}_{n-i+1} \circ\left(\sum_{u = 0}^{n-i}\mathbb{1}_V^{\otimes u} \otimes \delta_{j} \otimes \mathbb{1}_V^{\otimes n-i-u}\right)\circ \hat{g}^{\otimes n}$$ $$+ \sum_{A(n)} (\hat{f}\hat{g})^{\otimes i - 1} \otimes \Bigg[ \hat{f} \circ \delta_{\ell} \circ \hat{g}^{\otimes \ell}$$ $$+ \sum_{i = 2}^{\ell - 1} \hat{f} \circ \hat{\pmb{q}}_{\ell - i + 1} \circ\left(\sum_{u = 0}^{\ell-i}\mathbb{1}_V^{\otimes u} \otimes \delta_{i} \otimes \mathbb{1}_V^{\otimes \ell-i-u}\right) \circ \hat{g}^{\otimes \ell}\Bigg] \otimes (\hat{f}\hat{g})^{\otimes n-k}.$$ The application of Lemma \[k2t27\] concludes the proof.
& : Similarly to the previous part of the proof, we first concentrate on and then derive . For $n = 2$, it follows from $$\hat{\pmb{H}}|_{(sV)^{\otimes 2}} = \hat{H}|_{(sV)^{\otimes 2}} + \hat{H}\delta_{\pmb{\mu}} \hat{H}|_{(sV)^{\otimes 2}} = \hat{g}\hat{f} \otimes \hat{h} + \hat{h} \otimes \mathbb{1}_V + \hat{h}\circ (\delta_2(\hat{g}\hat{f} \otimes \hat{h}) + \delta_2(\hat{h} \otimes \mathbb{1}_V))$$ $$= [\![\hat{\pmb{\psi}}\hat{\pmb{\varphi}}]\!] \otimes \hat{h} \circ \hat{\pmb{q}}_1 + \hat{h} \circ \hat{\pmb{q}}_1 \otimes \mathbb{1}_V + \hat{h} \circ \hat{\pmb{q}}_2.$$ By the induction hypothesis, we assume holds for all natural numbers greater than $1$ and less than $n$. We can write $$\hat{\pmb{H}}|_{(sV)^{\otimes n}} = \hat{H}|_{(sV)^{\otimes n}} + \hat{h} \circ \delta_n \circ \hat{H}|_{(sV)^{\otimes n}}$$ $$+ \sum_{i = 2}^{n-1} \hat{\pmb{H}}|_{(sV)^{\otimes n-i+1}}\circ\left(\sum_{u = 0}^{n-i}\mathbb{1}_V^{\otimes u} \otimes \delta_{i} \otimes \mathbb{1}_V^{\otimes n-i-u}\right)\circ \hat{H}|_{(sV)^{\otimes n}}$$ Thanks to the induction hypothesis we can expand $\hat{\pmb{H}}|_{(sV)^{\otimes n-i+1}}$, and apply Lemma \[k2t29\] together with : $$\hat{h} \circ \hat{\pmb{q}}_n + \sum_{C(n)}[\![\hat{\pmb{\psi}}\hat{\pmb{\varphi}}]\!]_{r_1} \otimes \mbox{\dots} \otimes [\![\hat{\pmb{\psi}}\hat{\pmb{\varphi}}]\!]_{r_{i-1}} \otimes \hat{h} \circ \hat{\pmb{q}}_{r_i} \otimes \mathbb{1}_V^{\otimes k-i},$$ which completes the proof of the first assertion. Now we use again for $n \geq 2$: $$\hat{\pmb{\psi}}|_{(sV)^{\otimes n}} = \hat{G}|_{(sV)^{\otimes n}} + \hat{h} \circ \delta_n \circ \hat{G}_{(sV)^{\otimes n}}$$ $$+ \sum_{i = 2}^{n-1} \hat{\pmb{H}}|_{(sV)^{\otimes n-i+1}}\circ\left(\sum_{u = 0}^{n-i}\mathbb{1}_V^{\otimes u} \otimes \delta_{i} \otimes \mathbb{1}_V^{\otimes n-i-u}\right)\circ \hat{G}|_{(sV)^{\otimes n}}.$$ A consequence of Lemma \[k2l26\], $2 \leq j \leq n-1$ arbitrary, is $$\hat{\pmb{H}}|_{(sV)^{\otimes n-j+1}}\circ\left(\sum_{u = 0}^{n-j}\mathbb{1}_V^{\otimes u} \otimes \delta_{j} \otimes \mathbb{1}_V^{\otimes n-j-u}\right)\circ \hat{G}|_{(sV)^{\otimes n}}$$ $$= \hat{h} \circ \hat{\pmb{q}}_{n-j+1} \circ\left(\sum_{u = 0}^{n-j}\mathbb{1}_V^{\otimes u} \otimes \delta_{j} \otimes \mathbb{1}_V^{\otimes n-j-u}\right)\circ \hat{G}|_{(sV)^{\otimes n}}$$ $$+ \sum_{C(n-j+1)}\hat{h} \circ \hat{\pmb{p}}_{r_1} \circ \hat{g}^{\otimes r_1} \otimes \mbox{\dots} \otimes \hat{h} \circ \hat{\pmb{p}}_{r_{i-1}} \circ \hat{g}^{\otimes r_{i-1}} \otimes$$ $$\otimes \hat{h} \circ \hat{\pmb{q}}_{r_i} \circ\left(\sum_{u = 0}^{r_i - 1}\mathbb{1}_V^{\otimes u} \otimes \delta_{j} \otimes \mathbb{1}_V^{\otimes r_i - 1-u}\right)\circ \hat{G} \otimes (\hat{h} \circ \hat{\pmb{p}}_1 \circ \hat{g})^{\otimes k-i},$$ because $\hat{h} \circ \hat{\pmb{q}}_m \circ g^{\otimes m} = 0$ for all $m \geq 1$. In other words, if $\delta_\star$ in the last summation would not fit into $\hat{h} \circ \hat{\pmb{q}}_\star$ the corresponding term will be trivial. The summation then leads to $$\hat{\pmb{\psi}}|_{(sV)^{\otimes n}} = \hat{G}|_{(sV)^{\otimes n}} + \hat{h} \circ \delta_n \circ \hat{G}_{(sV)^{\otimes n}}$$ $$+ \sum_{i = 2}^{n-1} \hat{h} \circ \hat{\pmb{q}}|_{(sV)^{\otimes n-i+1}}\circ\left(\sum_{u = 0}^{n-i}\mathbb{1}_V^{\otimes u} \otimes \delta_{i} \otimes \mathbb{1}_V^{\otimes n-i-u}\right)\circ \hat{G}|_{(sV)^{\otimes n}}$$ $$+ \sum_{C(n)}\hat{h} \circ \hat{\pmb{p}}_{r_1} \circ \hat{g}^{\otimes r_1} \otimes \mbox{\dots} \otimes \hat{h} \circ \hat{\pmb{p}}_{r_{i-1}} \circ \hat{g}^{\otimes r_{i-1}} \otimes$$ $$\otimes \left[ \hat{h} \circ \delta_{\ell} \circ \hat{G} + \sum_{i = 2}^{\ell - 1} \hat{h} \circ \hat{\pmb{q}}_{\ell - i + 1} \circ\left(\sum_{u = 0}^{i-1}\mathbb{1}_V^{\otimes u} \otimes \delta_{i} \otimes \mathbb{1}_V^{\otimes i-u}\right) \circ \hat{G}\right] \otimes (\hat{h} \circ \hat{\pmb{p}}_1 \circ \hat{g})^{\otimes k-i}.$$ In order to finish the proof, we remind the equality $\hat{h} \circ \hat{\pmb{p}}_1 = \mathbb{1}_V$ and use Lemma \[k2t27\].
Acknowledgments {#acknowledgments .unnumbered}
---------------
The present article is based on the thesis by the author. The author wants to thank Petr Somberg very much for his great help in preparing this text for publication $-$ if it hadn’t been for it, this article would have never come to being.
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|
---
abstract: |
Suppose a finite group acts on a scheme $X$ and a finite-dimensional Lie algebra ${\ensuremath{\mathfrak{g}}}$. The corresponding *equivariant map algebra* is the Lie algebra ${\mathfrak{M}}$ of equivariant regular maps from $X$ to ${\ensuremath{\mathfrak{g}}}$. We classify the irreducible finite-dimensional representations of these algebras. In particular, we show that all such representations are tensor products of evaluation representations and one-dimensional representations, and we establish conditions ensuring that they are all evaluation representations. For example, this is always the case if ${\mathfrak{M}}$ is perfect.
Our results can be applied to multiloop algebras, current algebras, the Onsager algebra, and the tetrahedron algebra. Doing so, we easily recover the known classifications of irreducible finite-dimensional representations of these algebras. Moreover, we obtain previously unknown classifications of irreducible finite-dimensional representations of other types of equivariant map algebras, such as the generalized Onsager algebra.
address:
- 'E. Neher: University of Ottawa, Ottawa, Ontario, Canada'
- 'E. Savage: University of Ottawa, Ottawa, Ontario, Canada'
- 'P. Senesi: The Catholic University of America, Washington, D.C.'
author:
- Erhard Neher
- Alistair Savage
- Prasad Senesi
bibliography:
- 'neher-savage-senesi-biblist.bib'
date: 'July 11, 2010'
title: 'Irreducible finite-dimensional representations of equivariant map algebras'
---
Introduction {#introduction .unnumbered}
============
When studying the category of representations of a (possibly infinite-dimensional) Lie algebra, the irreducible finite-dimensional representations often play an important role. Let $X$ be a scheme and let ${\ensuremath{\mathfrak{g}}}$ be a finite-dimensional Lie algebra, both defined over an algebraically closed field $k$ of characteristic zero and both equipped with the action of a finite group $\Gamma$ by automorphisms. The *equivariant map algebra* ${\mathfrak{M}}=M(X,{\ensuremath{\mathfrak{g}}})^\Gamma$ is the Lie algebra of regular maps $X\to {\ensuremath{\mathfrak{g}}}$ which are equivariant with respect to the action of $\Gamma$. Denoting by $A$ the coordinate ring of $X$, an equivariant map algebra can also be realized as the fixed point Lie algebra ${\mathfrak{M}}=({\ensuremath{\mathfrak{g}}}{\otimes}A)^\Gamma$ with respect to the diagonal action of $\Gamma$ on ${\ensuremath{\mathfrak{g}}}\otimes A$. The purpose of the current paper is to classify the irreducible finite-dimensional representations of such algebras.
One important class of examples of equivariant map algebras are the (twisted) loop algebras which play a crucial role in the theory of affine Lie algebras. The description of the irreducible finite-dimensional representations of loop algebras goes back to the work of Chari and Pressley [@C], [@CPunitary], [@CPtw]. Their work has had a long-lasting impact. Generalizations and more precise descriptions of their work have appeared in many papers, for example in Batra [@Bat], Chari-Fourier-Khandai [@CFK], Chari-Fourier-Senesi [@CFS], Chari-Moura [@CM], Feigin-Loktev [@FL], Lau [@Lau], Li [@Li], and Rao [@R; @R2]. Other examples of equivariant map algebras whose irreducible finite-dimensional representations have been classified are the Onsager algebra [@DR] and the tetrahedron algebra (or three-point $\mathfrak{sl}_2$ loop algebra) [@Ha]. In all these papers it was proven, sometimes using complicated combinatorial or algebraic arguments and sometimes without explicitly stating so, that all irreducible finite-dimensional representations are evaluation representations.
In the current paper, we provide a complete classification of the irreducible finite-dimensional representations of an arbitrary equivariant map algebra. This class of Lie algebras includes all the aforementioned examples and we obtain classification results in these cases with greatly simplified proofs. However, the class of Lie algebras covered by the results of this paper is infinitely larger than this set of examples. To demonstrate this, we work out some previously unknown classifications of the irreducible finite-dimensional representations of other Lie algebras such as the generalized Onsager algebra.
If ${\mathfrak{M}}=M(X,{\ensuremath{\mathfrak{g}}})^\Gamma$ is an equivariant map algebra, ${\alpha}\in
{\mathfrak{M}}$, and $x$ is a $k$-rational point of $X$, the image ${\alpha}(x)$ is not an arbitrary element of ${\ensuremath{\mathfrak{g}}}$, but rather an element of ${\ensuremath{\mathfrak{g}}}^x =
\{ u \in {\ensuremath{\mathfrak{g}}}: g\cdot u = u \mbox{ for all }g\in \Gamma_x\}$ where $\Gamma_x= \{g \in \Gamma : g \cdot x = x\}$. For a finite subset ${\mathbf{x}}$ of $k$-rational points of $X$ and finite-dimensional representations $\rho_x : {\ensuremath{\mathfrak{g}}}^x \to \operatorname{End}_k V_x$, $x \in {\mathbf{x}}$, we define the associated *evaluation representation* as the composition $$\textstyle
{\mathfrak{M}}\stackrel{\operatorname{ev}_{\mathbf{x}}}{\longrightarrow} \bigoplus_{x \in {\mathbf{x}}}
{\ensuremath{\mathfrak{g}}}^x \xrightarrow{\otimes_{x \in {\mathbf{x}}} \rho_x}
\operatorname{End}_k (\bigotimes_{x \in {\mathbf{x}}} V_x),$$ where $\operatorname{ev}_{\mathbf{x}}: \alpha \mapsto (\alpha(x))_{x \in {\mathbf{x}}}$ is evaluation at ${\mathbf{x}}$. Our definition is slightly more general than the classical definition of evaluation representations, which require the $V_x$ to be representations of ${\ensuremath{\mathfrak{g}}}$ instead of ${\ensuremath{\mathfrak{g}}}^x$. We believe our definition to be more natural and it leads to a simplification of the classification of irreducible finite-dimensional representations in certain cases. For instance, the Onsager algebra is an equivariant map algebra where $X=\operatorname{Spec}k[t^{\pm 1}]$, ${\ensuremath{\mathfrak{g}}}={\ensuremath{\mathfrak{sl}}}_2(k)$, and $\Gamma=\{1, {\sigma}\}$ with ${\sigma}$ acting on $X$ by ${\sigma}\cdot x = x^{-1}$ and on ${\ensuremath{\mathfrak{g}}}$ by the Chevalley involution. For the fixed points $x={\pm 1}\in X$, the subalgebras ${\ensuremath{\mathfrak{g}}}^x$ are one-dimensional, in fact Cartan subalgebras, while ${\ensuremath{\mathfrak{g}}}^x={\ensuremath{\mathfrak{g}}}$ for $x\ne \pm 1$. In the classification of the irreducible finite-dimensional representations of the Onsager algebra given in [@DR], not all irreducible finite-dimensional representations are evaluation representations since the more restrictive definition is used. Instead, a discussion of *type* is needed to reduce all irreducible finite-dimensional representations (via an automorphism of the enveloping algebra) to evaluation representations. However, under the more general definition of evaluation representation given in the current paper, all irreducible finite-dimensional representations of the Onsager algebra are evaluation representations and no discussion of type is needed. Additionally, contrary to what has been imposed before (for example in the multiloop case), we allow the representations $\rho_x$ to be non-faithful. This will provide us with greater flexibility. Finally, we note that we use the term *evaluation representation* even when we evaluate at more than one point (i.e. when $|{\mathbf{x}}| > 1$). Such representations are often called tensor products of evaluation representations in the literature, where evaluation representations are at a single point.
By the above, the main question becomes: When does an irreducible finite-dimensional representation of ${\mathfrak{M}}$ factor through an evaluation map $\operatorname{ev}_{\mathbf{x}}$? Surprisingly, the answer is not always. We see that for $\Gamma \ne \{1\}$, the Lie algebra may have one-dimensional representations that are not evaluation representations. Any one-dimensional representation corresponds to a linear form ${\lambda}\in {\mathfrak{M}}^*$ vanishing on $[{\mathfrak{M}}, {\mathfrak{M}}]$. In some cases, all such linear forms are evaluation representations but in other cases this is not true. Our main result (Theorem \[thm:fd-reps=eval-reps+linear-form\]) is that *any irreducible finite-dimensional representation of ${\mathfrak{M}}$ is a tensor product of an evaluation representation and a one-dimensional representation.*
Obviously, if ${\mathfrak{M}}=[{\mathfrak{M}}, {\mathfrak{M}}]$ is perfect then our theorem implies that every irreducible finite-dimensional representation of ${\mathfrak{M}}$ is an evaluation representation. For example, this is so in the case of a multiloop algebra which is an equivariant map algebra ${\mathfrak{M}}=M(X,{\ensuremath{\mathfrak{g}}})^\Gamma$ for $X= \operatorname{Spec}k[t^{\pm 1}_1, \ldots, t_n^{\pm
1}]$, ${\ensuremath{\mathfrak{g}}}$ simple, and $\Gamma={\Bbb Z}/(m_1{\Bbb Z}) \times \cdots \times
{\Bbb Z}/(m_n{\Bbb Z})$ with the $i$-th factor of $\Gamma$ acting on the $i$-th coordinate of $X$ by a primitive $m_i$-th root of unity. Note that in this case $\Gamma$ acts freely on $X$ and therefore all ${\ensuremath{\mathfrak{g}}}^x={\ensuremath{\mathfrak{g}}}$. But perfectness of ${\mathfrak{M}}$ is by no means necessary for all irreducible finite-dimensional representations to be evaluation representations. In fact, in Section \[sec:classification\] we provide an easy criterion for this to be the case, which in particular can be applied to the Onsager algebra to see that all irreducible finite-dimensional representations are evaluation representations in the more general sense. On the other hand, we also provide conditions under which not all irreducible finite-dimensional representations of an equivariant map algebra are evaluation representations (see Proposition \[prop:infinite-tildeX\]).
An important feature of our classification is the fact that isomorphism classes of evaluation representations are parameterized in a natural and uniform fashion. Specifically, for a rational point $x \in X_{\mathrm{rat}}$, let $\mathcal{R}_x$ be the set of isomorphism classes of irreducible finite-dimensional representations of ${\ensuremath{\mathfrak{g}}}^x$ and set $\mathcal{R}_X = \bigsqcup_{x \in X_{\mathrm{rat}}} \mathcal{R}_x$. Then there is a canonical $\Gamma$-action on $\mathcal{R}_X$ and isomorphism classes of evaluation representations are naturally enumerated by finitely-supported $\Gamma$-equivariant functions $\Psi : X_{\mathrm{rat}}\to \mathcal{R}_X$ such that $\Psi(x) \in
\mathcal{R}_x$. Thus we see that evaluation representations “live on orbits”. This point of view gives a natural geometric explanation for the somewhat technical algebraic conditions that appear in previous classifications (such as for the multiloop algebras).
We shortly describe the contents of the paper. After a review of some results in the representation theory of Lie algebras in Section \[sec:review\], we introduce map algebras ($\Gamma=\{1\}$) and equivariant map algebras (arbitrary $\Gamma$) in Sections \[sec:mapal\] and \[sec:equiva\] and discuss old and new examples. We introduce the formalism of evaluation representations in Section \[sec:eval-reps\] and classify the irreducible finite-dimensional representations of equivariant map algebras in Section \[sec:classification\]. We show that in general not all irreducible finite-dimensional representations are evaluation representations and we derive a sufficient criterion for this to nevertheless be the case, as well as a necessary condition. Finally, in Section \[sec:applications\] we apply our general theorem to some specific cases of equivariant map algebras, recovering previous results as well as obtaining new ones.
Notation {#notation .unnumbered}
--------
Throughout this paper, $k$ is an algebraically closed field of characteristic zero. For schemes, we use the terminology of [@EH]. In particular, an affine scheme $X$ is the (prime) spectrum of a commutative associative $k$-algebra $A$. Note that we do *not* assume that $A$ is finitely generated in general. We say that $X$ is an affine variety if $A$ is finitely generated and reduced, in which case we identify $X$ with the maximal spectrum of $A$. For an arbitrary scheme $X$, we set $A =
\mathcal{O}_X(X)$, except when the possibility of confusion exists (for instance, when more than one scheme is being considered), in which case we use the notation $A_X$. We let $X_{\mathrm{rat}}$ denote the set of $k$-rational points of $X$. Recall ([@EH p. 45]) that $x
\in X$ is a *$k$-rational point of $X$* if its residue field is $k$. The symbol ${\ensuremath{\mathfrak{g}}}$ will always denote a finite-dimensional Lie algebra (but see Remark \[rem:generalization\]). All tensor products will be over $k$, unless indicated otherwise. If a group $\Gamma$ acts on a vector space $V$ we denote by $V^\Gamma=\{v\in V
: g\cdot v = v \mbox{ for all }g\in \Gamma\}$ the subspace of fixed points.
Acknowledgements {#acknowledgements .unnumbered}
----------------
The authors thank Michael Lau for having explained the results of his preprint [@Lau] to them before it was posted, and Oliver Schiffmann and Mark Haiman for helpful discussions concerning the geometry of algebraic varieties. They would also like to thank Geordie Williamson, Ben Webster and Daniel Juteau for useful conversations. The first two authors gratefully acknowledge support from NSERC through their respective Discovery grants. The third author was partially supported by the Discovery grants of the first two authors.
Review of some results on representations of Lie algebras {#sec:review}
=========================================================
For easier reference we review some mostly known results about representations of Lie algebras. Let $L$ be a Lie algebra, not necessarily of finite dimension. The first items concern representations of the Lie algebra $$L= L_1 \oplus \cdots \oplus L_n$$ where $L_1, \ldots , L_n$ are ideals of $L$. Recall that if $V_i$, $i=1, \ldots, n$, are $L_i$-modules, then $V_1 \otimes \dots \otimes
V_n$ is an $L$-module with action $$(u_1, \dots , u_n) \cdot (v_1 \otimes \dots \otimes v_n)
= \sum_{i=1}^n (v_1 \otimes
\dots \otimes v_{i-1} \otimes u_i \cdot v_i \otimes v_{i+1}
\otimes \dots \otimes v_n).$$ We will always use this $L$-module structure on $V=V_1 \otimes
\cdots \otimes V_n$. It will sometimes be useful to denote this representation by $\rho_1 \otimes \cdots \otimes \rho_n$ where $\rho_i : L_i \to \operatorname{End}_k(V_i)$ are the individual representations. We call an irreducible $L$-module $V$ *absolutely irreducible* if $\operatorname{End}_L (V) = k \operatorname{Id}$. By Schur’s Lemma, an irreducible $L$-module of countable dimension is absolutely irreducible.
We collect in the following proposition various well-known facts that we will need in the current paper.
\[prop:basic-facts\]
1. \[item:tensor-prods\] Let $V_i$, $1\le i \le n$, be non-zero $L_i$-modules and let $V= V_1
\otimes \cdots \otimes V_n$. Then $V$ is completely reducible (respectively absolutely irreducible) if and only if all $V_i$ are completely reducible (respectively absolutely irreducible). [For a proof, see for example [@Bo:A8 §7.4, Theorem 2] and use the fact that the universal enveloping algebra of $L$ is $\operatorname{U}(L) =
\operatorname{U}(L_1) \otimes \cdots \otimes \operatorname{U}(L_n)$, or see [@Li Lemma 2.7].]{}
2. \[item:n=2-case\] Let $n=2$, thus $L = L_1 \oplus L_2$, and let $V$ be an irreducible $L$-module. Suppose that $V$, considered as $L_1$-module, contains an absolutely irreducible $L_1$-submodule $V_1$. Then there exists an irreducible $L_2$-module $V_2$ such that $V\cong V_1 \otimes V_2$. The isomorphism classes of $V_1$ and $V_2$ are uniquely determined by $V$. [For a proof in case $\dim_k V <
\infty$ see, for example, [@Bo:A8 §7.7, Proposition 8] and observe that irreducibility of $V_2$ follows from . For a proof in general see [@Li Lemma 2.7].]{}
3. Suppose $L_2, \ldots, L_n$ are finite-dimensional semisimple Lie algebras (note that there are no assumptions on $L_1$) and let $V$ be an irreducible finite-dimensional $L$-module. Then there exist irreducible finite-dimensional $L_i$-modules $V_i$, $1\le i \le n$, such that $V\cong V_1 \otimes \cdots \otimes V_n$. The isomorphism classes of the $L_i$-modules $V_i$ are uniquely determined by $V$. [This follows from by induction.]{}
4. \[item:faithful-reps\] Let $L$ be a finite-dimensional Lie algebra over $k$.
1. $L$ has a finite-dimensional faithful completely reducible representation if and only if $L$ is reductive [([@Bou:Lie1 §6.4, Proposition 5])]{}.
2. $L$ has a finite-dimensional faithful irreducible representation if and only if $L$ is reductive and its center is at most one-dimensional [([@Bou:Lie1 §6, Exercise 20])]{}.
5. \[item:one-dim-reps\] We denote by $L^* = \operatorname{Hom}_k(L,k)$ the dual space of $L$ and identify $(L/[L,L])^*= \{ {\lambda}\in L^*:
{\lambda}([L,L])=0\}$. Any ${\lambda}\in (L/[L, L])^*$ is a one-dimensional, hence irreducible, representation of the Lie algebra $L$, and every one-dimensional representation of $L$ is of this form. Two linear forms ${\lambda}, \mu \in (L/[L,L])^*$ are isomorphic as representations if and only if ${\lambda}=\mu$. [The proof of these facts is immediate.]{}
We combine some of these facts in the following lemma.
\[lem:deco\] Let $L$ be a (possibly infinite-dimensional) Lie algebra and let $\rho : L \to \operatorname{End}_k(V)$ be an irreducible finite-dimensional representation. Then there exist unique irreducible finite-dimensional representations $\rho_i : L \to
\operatorname{End}_k(V_i)$, $i=1,2$ such that $\rho = \rho_1 \otimes \rho_2$ and
1. $\rho_1 = {\lambda}$ for a unique ${\lambda}\in (L/[L,L])^*$ as in Proposition \[prop:basic-facts\] (hence $V=V_2$ as $k$-vector spaces), and
2. $L/\operatorname{Ker}\rho_2$ is finite-dimensional semisimple.
The representation $\rho$ factors as $$\xymatrix@!C{
L \ar@{->>}[dr]_\pi \ar[rr]^{\rho} & & \operatorname{End}_k(V)\\
& \bar L \ar@{^(->}[ur]_{\overline{\rho}}
}$$ where $\bar L = L /\operatorname{Ker}\rho$, $\pi$ is the canonical epimorphism and $\bar \rho$ is a faithful representation of $\bar
L$, whence $\bar L$ is finite-dimensional. By Proposition \[prop:basic-facts\], $\bar
L$ is reductive. If $\bar L$ is semisimple, let ${\lambda}= 0$ and $\rho=\rho_2$. Otherwise, by Proposition \[prop:basic-facts\] again, $\bar L$ has a one-dimensional center $\bar L_{\mathfrak{z}}$. Let $\bar
L_{\mathfrak{s}}= [\bar L, \, \bar L]$ be the semisimple part of $\bar L$. Since $[\bar \rho(\bar L_{\mathfrak{z}}), \, \bar \rho (\bar L_{\mathfrak{s}})] = 0$, it follows that ${\bar \rho}|_{{\bar L}_{\mathfrak{s}}}$ is irreducible. By Burnside’s Theorem, see for example [@BAII Chapter 4.3], the associative subalgebra of $\operatorname{End}_k (V)$ generated by $\bar \rho (\bar
L_{\mathfrak{s}})$ is therefore equal to $\operatorname{End}_k(V)$. Consequently, $\bar \rho
(\bar L_{\mathfrak{z}}) \subseteq k \operatorname{Id}_V$. Because $\bar \rho$ is faithful, we actually have $\bar \rho(\bar L_{\mathfrak{z}}) = k \operatorname{Id}_V$. We identify $k\operatorname{Id}_V \equiv k$ and then define $\rho_1 (x) = \bar \rho(\bar
x_{\mathfrak{z}})$ where $\bar x_{\mathfrak{z}}$ is the $\bar L_{\mathfrak{z}}$-component of $\bar
x = \pi(x) \in \bar L$. By Proposition \[prop:basic-facts\], $\rho_1
= {\lambda}$ for a unique $0 \ne {\lambda}\in (L/[L,L])^*$. Finally, we define $\rho_2 : L \to \operatorname{End}_k(V)$ by $\rho_2(x) = \bar \rho(\bar x_{\mathfrak{s}})$ where $\bar x_{\mathfrak{s}}$ is the $\bar L_{\mathfrak{s}}$-component of $\bar x$. Since $\bar \rho |_{{\bar L}_{\mathfrak{s}}}$ is faithful, it follows that $\operatorname{Ker}\rho_2 = \pi^{-1}(\bar L_{\mathfrak{z}})$, whence $L/\operatorname{Ker}\rho_2 \cong
\bar L_{\mathfrak{s}}$ is semisimple. This proves existence of the decomposition. Uniqueness follows from the construction above.
Map algebras {#sec:mapal}
============
In this and the next section we define our main object of study – the Lie algebra of (equivariant) maps from a scheme to another Lie algebra – and discuss several examples. We remind the reader that $X$ is a scheme defined over $k$ and ${\ensuremath{\mathfrak{g}}}$ is a finite-dimensional Lie algebra over $k$. Then ${\ensuremath{\mathfrak{g}}}$ is naturally equipped with the structure of an affine algebraic scheme, namely the affine $n$-space, where $n=\dim {\ensuremath{\mathfrak{g}}}$. Addition and multiplication on ${\ensuremath{\mathfrak{g}}}$ give rise to morphisms of schemes ${\ensuremath{\mathfrak{g}}}\times_k {\ensuremath{\mathfrak{g}}}\to {\ensuremath{\mathfrak{g}}}$ and multiplication by a fixed scalar yields a morphism of schemes ${\ensuremath{\mathfrak{g}}}\to {\ensuremath{\mathfrak{g}}}$.
\[Defmap\] We denote by $M(X,{\ensuremath{\mathfrak{g}}})$ the Lie algebra of regular functions on $X$ with values in ${\ensuremath{\mathfrak{g}}}$ (equivalently, the set of morphisms of schemes $X \to {\ensuremath{\mathfrak{g}}}$), called the *untwisted map algebra* or the *Lie algebra of currents* ([@FL]). The multiplication in $M(X,{\ensuremath{\mathfrak{g}}})$ is defined pointwise. That is, for $\alpha, \beta \in
M(X,{\ensuremath{\mathfrak{g}}})$, we define $[\alpha, \beta] \in M(X,{\ensuremath{\mathfrak{g}}})$ to be the composition $$X \xrightarrow{(\alpha,\beta)} {\ensuremath{\mathfrak{g}}}\times_k {\ensuremath{\mathfrak{g}}}\xrightarrow{[ \cdot,
\cdot ]} {\ensuremath{\mathfrak{g}}}.$$ The addition and scalar multiplication are defined similarly.
\[lem:M-isom-O\] There is an isomorphism $$M(X,{\ensuremath{\mathfrak{g}}}) \cong {\ensuremath{\mathfrak{g}}}\otimes A$$ of Lie algebras over $A$ and hence also over $k$. The product on ${\ensuremath{\mathfrak{g}}}\otimes A$ is given by $\left[ u \otimes f, v \otimes g \right] = \left[ u,v
\right] \otimes fg$ for $x,y \in {\ensuremath{\mathfrak{g}}}$ and $f,g \in A$.
Because of Lemma \[lem:M-isom-O\], whose proof is routine, we will sometimes identify $M(X,{\ensuremath{\mathfrak{g}}})$ and ${\ensuremath{\mathfrak{g}}}\otimes A$ in what follows.
\[categ\] If $\phi : X \to Y$ is a morphism of schemes then $\phi^* : M(Y,{\ensuremath{\mathfrak{g}}}) \to M(X,{\ensuremath{\mathfrak{g}}})$ given by $\phi^*(\alpha)
= \alpha \circ \phi$ is a Lie algebra homomorphism. For example, if $\iota: X \hookrightarrow Y$ is an inclusion of schemes, then $\iota^*: M(Y, {\ensuremath{\mathfrak{g}}}) \rightarrow M(X, {\ensuremath{\mathfrak{g}}})$ is the restriction $\phi^*:\alpha \mapsto \alpha|_{X}$. The assignments $X \mapsto
M(X,{\ensuremath{\mathfrak{g}}})$, $\phi \mapsto \phi^*$ are easily seen to define a contravariant functor from the category of schemes to the category of Lie algebras. Analogously, for fixed $X$, the assignment ${\ensuremath{\mathfrak{g}}}\mapsto M(X,{\ensuremath{\mathfrak{g}}})$ is a covariant functor from the category of finite-dimensional Lie algebras to the category of Lie algebras.
\[eg:current\] Let $X$ be the $n$-dimensional affine space. Then $A
\cong k[t_1,\dots,t_n]$ is a polynomial algebra in $n$ variables and $M(X,{\ensuremath{\mathfrak{g}}}) \cong {\ensuremath{\mathfrak{g}}}\otimes k[t_1,\dots,t_n]$ is the so-called *current algebra*.
\[eg:multiloop\] Let $X= \operatorname{Spec}A$, where $A=k[t_1^{\pm 1},\dots, t_n^{\pm 1}]$ is the $k$-algebra of Laurent polynomials in $n$ variables. Then $M(X,{\ensuremath{\mathfrak{g}}})
\cong {\ensuremath{\mathfrak{g}}}\otimes k[t_1^{\pm 1}, \dots, t_n^{\pm n}]$ is an *untwisted multiloop algebra*. In the case $n=1$, it is usually called the *untwisted loop algebra of ${\ensuremath{\mathfrak{g}}}$*.
\[eg:threepoint\] If $X$ is the variety $k \setminus \{0,1\}$, then $A \cong k[t, t^{-1}, (t-1)^{-1}]$ and $M(X,{\ensuremath{\mathfrak{sl}}}_2) \cong
{\ensuremath{\mathfrak{sl}}}_2 \otimes k[t,t^{-1},(t-1)^{-1}]$ is the *three point $\mathfrak{sl}_2$ loop algebra*. Removing any two distinct points of $k$ results in an algebra isomorphic to $M(X,{\ensuremath{\mathfrak{sl}}}_2)$, and so there is no loss in generality in assuming the points are 0 and 1. It was shown in [@HT07] that $M(X,{\ensuremath{\mathfrak{sl}}}_2)$ is isomorphic to the *tetrahedron Lie algebra* and to a direct sum of three copies of the *Onsager algebra* (see Example \[eg:onsager\]). We refer the reader to [@HT07] and the references cited therein for further details.
Equivariant map algebras {#sec:equiva}
========================
Recall that we assume ${\ensuremath{\mathfrak{g}}}$ is a finite-dimensional Lie algebra. We denote the group of Lie algebra automorphisms of ${\ensuremath{\mathfrak{g}}}$ by $\operatorname{Aut}_k
{\ensuremath{\mathfrak{g}}}$. Any Lie algebra automorphism of ${\ensuremath{\mathfrak{g}}}$, being a linear map, can also be viewed as an automorphism of ${\ensuremath{\mathfrak{g}}}$ considered as a scheme. An action of a group $\Gamma$ on ${\ensuremath{\mathfrak{g}}}$ and on a scheme $X$ will always be assumed to be by Lie algebra automorphisms of ${\ensuremath{\mathfrak{g}}}$ and scheme automorphisms of $X$. Recall that there is an induced $\Gamma$-action on $A$ given by $$g \cdot f = f g^{-1},\quad f \in A,
\quad g \in \Gamma,$$ where on the right hand side we view $g^{-1}$ as the corresponding automorphism of $X$.
\[def:EMA-geometric\] Let $\Gamma$ be a group acting on a scheme $X$ and a Lie algebra ${\ensuremath{\mathfrak{g}}}$ by automorphisms. Then $\Gamma$ acts on $M(X, {\ensuremath{\mathfrak{g}}})$ by automorphisms: For $g\in \Gamma$ and $\alpha \in M(X,{\ensuremath{\mathfrak{g}}})$ the map $g\cdot \alpha$ is defined by $$\begin{aligned}
g \cdot \alpha = g \alpha g^{-1},
\quad \alpha \in M(X,{\ensuremath{\mathfrak{g}}}), \quad g \in \Gamma, \label{eq:acmgdef}\end{aligned}$$ where on the right hand side $g$ and $g^{-1}$ are viewed as automorphisms of ${\ensuremath{\mathfrak{g}}}$ and $X$ respectively. We define $M(X,{\ensuremath{\mathfrak{g}}})^\Gamma$ to be the set of fixed points under this action. That is, $$M(X,{\ensuremath{\mathfrak{g}}})^\Gamma = \{\alpha \in M(X,{\ensuremath{\mathfrak{g}}}) : \alpha g = g \alpha \
\forall\ g \in \Gamma\}$$ is the subalgebra of $M(X,{\ensuremath{\mathfrak{g}}})$ consisting of $\Gamma$-equivariant maps from $X$ to ${\ensuremath{\mathfrak{g}}}$. We call $M(X,{\ensuremath{\mathfrak{g}}})^\Gamma$ an *equivariant map algebra*.
Suppose $X$ is a discrete (hence finite) variety. Let $X'$ be a subset of $X$ obtained by choosing one element from each $\Gamma$-orbit of $X$. Then $$M(X,{\ensuremath{\mathfrak{g}}})^\Gamma \cong \prod_{x \in X'} {\ensuremath{\mathfrak{g}}}^{\Gamma_x},\quad \alpha \mapsto (\alpha(x))_{x
\in X'},\quad \alpha \in M(X,{\ensuremath{\mathfrak{g}}})^\Gamma,$$ is an isomorphism of Lie algebras, where $\Gamma_x = \{g \in \Gamma
: g \cdot x = x\}$ is the stabilizer subgroup of $x$.
\[lem:conjugate-actions\] Let $\Gamma$ be a group acting on a scheme $X$ and a Lie algebra ${\ensuremath{\mathfrak{g}}}$. Suppose $\tau_1 \in \operatorname{Aut}_k {\ensuremath{\mathfrak{g}}}$ and $\tau_2 \in \operatorname{Aut}X$. Then we can define a second action of $\Gamma$ on ${\ensuremath{\mathfrak{g}}}$ and $X$ by declaring $g \in \Gamma$ to act by $\tau_1 g \tau_1^{-1}$ on ${\ensuremath{\mathfrak{g}}}$ by by $\tau_2 g \tau_2^{-1}$ on $X$. Let ${\mathfrak{M}}$ and ${\mathfrak{M}}'$ be the equivariant map algebras with respect to these two actions. That is, $$\begin{aligned}
{\mathfrak{M}}&= \{\alpha \in M(X,{\ensuremath{\mathfrak{g}}}) : \alpha g = g \alpha\ \forall\ g \in \Gamma \}, \\
{\mathfrak{M}}' &= \{\beta \in M(X,{\ensuremath{\mathfrak{g}}}) : \beta \tau_2 g \tau_2^{-1} = \tau_1 g
\tau_1^{-1} \beta \ \forall\ g \in \Gamma\}.\end{aligned}$$ Then ${\mathfrak{M}}\cong {\mathfrak{M}}'$ as Lie algebras.
One easily checks that $\alpha \mapsto \tau_1 \circ \alpha \circ
\tau_2^{-1}$ intertwines the two $\Gamma$-actions and thus yields the desired automorphism.
The group $\Gamma$ acts naturally on ${\ensuremath{\mathfrak{g}}}\otimes A$ by extending the map $g \cdot (u \otimes f) = (g \cdot u) \otimes (g \cdot f)$ by linearity. Define $$({\ensuremath{\mathfrak{g}}}\otimes A)^\Gamma := \{\alpha \in {\ensuremath{\mathfrak{g}}}\otimes A : g \cdot \alpha = \alpha \ \forall\ g \in \Gamma\}$$ to be the subalgebra of ${\ensuremath{\mathfrak{g}}}\otimes A$ consisting of elements fixed by $\Gamma$. The proof of the following lemma is immediate.
\[lem:invM-iso-O\] Let $\Gamma$ be a group acting on a scheme $X$ and a Lie algebra ${\ensuremath{\mathfrak{g}}}$. Then the isomorphism $M(X,{\ensuremath{\mathfrak{g}}}) \cong {\ensuremath{\mathfrak{g}}}\otimes A$ of Lemma \[lem:M-isom-O\] is $\Gamma$-equivariant. In particular, under this isomorphism $({\ensuremath{\mathfrak{g}}}\otimes A)^\Gamma$ corresponds to $M(X,{\ensuremath{\mathfrak{g}}})^\Gamma$.
\[remark on generalizations\] Let $V = \operatorname{Spec}A$, an affine scheme but not necessarily an affine variety. By assumption, $\Gamma$ acts on $X$, hence on $A$. Thus $\Gamma$ acts on $V$ by [@EH I-40]. Since $A_V = A_X = A$, we have $M(X,{\ensuremath{\mathfrak{g}}})^\Gamma \cong ({\ensuremath{\mathfrak{g}}}\otimes A)^\Gamma \cong
M(V,{\ensuremath{\mathfrak{g}}})^\Gamma$. Therefore, we lose no generality in assuming that $X$ is an affine scheme and we will often do so in the sequel.
If $\Gamma$ acts trivially on ${\ensuremath{\mathfrak{g}}}$, then $M(X,{\ensuremath{\mathfrak{g}}})^\Gamma \cong
M(\operatorname{Spec}(A^\Gamma),{\ensuremath{\mathfrak{g}}})$ and hence $M(X,{\ensuremath{\mathfrak{g}}})^\Gamma$ is isomorphic to an (untwisted) map algebra.
The proof is straightforward.
Let $\Gamma$ be a finite group. Recall that any $\Gamma$-module $B$ decomposes uniquely as a direct sum $B= B^\Gamma \oplus B_\Gamma$ of the two $\Gamma$-submodules $B^\Gamma$ and $B_\Gamma = \operatorname{Span}\{
g\cdot m - m : g\in \Gamma, m\in B\}$. We let $\sharp : B \to
B^\Gamma$ be the canonical $\Gamma$-module epimorphism, given by $m\mapsto m^\sharp = \frac{1}{|\Gamma|} \sum_{g\in \Gamma}\, g\cdot
m$. If $B$ is a (possibly nonassociative) algebra and $\Gamma$ acts by automorphisms, then $$\label{eg:perfrm1}
[B^\Gamma,\, B^\Gamma] \subseteq B^\Gamma, \quad [B^\Gamma, B_\Gamma] \subseteq B_\Gamma,
\quad [m_0, m^\sharp] = [m_0, m]^\sharp,$$ for $m_0 \in B^\Gamma$ and $m\in B$. Since ${\ensuremath{\mathfrak{g}}}{\otimes}A = ({\ensuremath{\mathfrak{g}}}^\Gamma
{\otimes}A^\Gamma) \oplus ({\ensuremath{\mathfrak{g}}}^\Gamma {\otimes}A_\Gamma) \oplus ({\ensuremath{\mathfrak{g}}}_\Gamma {\otimes}A^\Gamma) \oplus ({\ensuremath{\mathfrak{g}}}_\Gamma {\otimes}A_\Gamma)$, we get $${\mathfrak{M}}= ({\ensuremath{\mathfrak{g}}}^\Gamma {\otimes}A^\Gamma) \oplus ({\ensuremath{\mathfrak{g}}}_\Gamma {\otimes}A_\Gamma)^\Gamma,$$ and $$[{\ensuremath{\mathfrak{g}}}^\Gamma \otimes A^\Gamma, {\ensuremath{\mathfrak{g}}}^\Gamma \otimes A^\Gamma] \subseteq
{\ensuremath{\mathfrak{g}}}^\Gamma \otimes A^\Gamma,\quad [{\ensuremath{\mathfrak{g}}}^\Gamma \otimes A^\Gamma,
({\ensuremath{\mathfrak{g}}}_\Gamma \otimes A_\Gamma)^\Gamma] \subseteq ({\ensuremath{\mathfrak{g}}}_\Gamma \otimes
A_\Gamma)^\Gamma.$$
\[lem:perfrm\] We suppose that $\Gamma$ is a finite group and use the notation above.
1. \[item:perfect-cond1\] If $[{\ensuremath{\mathfrak{g}}}^\Gamma, {\ensuremath{\mathfrak{g}}}_\Gamma] = {\ensuremath{\mathfrak{g}}}_\Gamma$, then $$[{\ensuremath{\mathfrak{g}}}^\Gamma, {\ensuremath{\mathfrak{g}}}^\Gamma] {\otimes}A^\Gamma \subseteq [{\mathfrak{M}}, {\mathfrak{M}}]
= \big( [{\mathfrak{M}}, {\mathfrak{M}}] \cap ({\ensuremath{\mathfrak{g}}}^\Gamma {\otimes}A^\Gamma)\big) \oplus ({\ensuremath{\mathfrak{g}}}_\Gamma {\otimes}A_\Gamma)^\Gamma$$ In particular, if $[{\ensuremath{\mathfrak{g}}}^\Gamma, {\ensuremath{\mathfrak{g}}}]={\ensuremath{\mathfrak{g}}}$ then ${\mathfrak{M}}$ is perfect.
2. If ${\ensuremath{\mathfrak{g}}}$ is simple, ${\ensuremath{\mathfrak{g}}}^\Gamma\ne \{0\}$ is perfect and the representation of ${\ensuremath{\mathfrak{g}}}^\Gamma$ on ${\ensuremath{\mathfrak{g}}}_\Gamma$ does not have a trivial non-zero subrepresentation, then ${\ensuremath{\mathfrak{g}}}=[{\ensuremath{\mathfrak{g}}}^\Gamma,{\ensuremath{\mathfrak{g}}}]$ and hence ${\mathfrak{M}}$ is perfect.
<!-- -->
1. The inclusion $ [{\ensuremath{\mathfrak{g}}}^\Gamma, {\ensuremath{\mathfrak{g}}}^\Gamma] {\otimes}A^\Gamma \subseteq [{\mathfrak{M}}, {\mathfrak{M}}]$ is obvious since ${\ensuremath{\mathfrak{g}}}^\Gamma \otimes 1 \subseteq {\mathfrak{M}}$. For $u_0\in {\ensuremath{\mathfrak{g}}}^\Gamma$ and $v\in {\ensuremath{\mathfrak{g}}}_\Gamma$ we have, using (\[eg:perfrm1\]), $([u_0, v] {\otimes}f)^\sharp = [u_0 {\otimes}1, (v{\otimes}f)^\sharp] \in [{\mathfrak{M}}, {\mathfrak{M}}]$, whence $({\ensuremath{\mathfrak{g}}}_\Gamma {\otimes}A_\Gamma)^\Gamma = ({\ensuremath{\mathfrak{g}}}_\Gamma {\otimes}A_\Gamma)^\sharp \subseteq [{\mathfrak{M}}, {\mathfrak{M}}]$ by linearity of $\sharp$. The same argument also shows that $[{\ensuremath{\mathfrak{g}}}^\Gamma, {\ensuremath{\mathfrak{g}}}]={\ensuremath{\mathfrak{g}}}$ implies that ${\mathfrak{M}}$ is perfect.
2. Since ${\ensuremath{\mathfrak{g}}}^\Gamma$ is reductive by [@Bou:Lie78 VII, §5.1, Proposition 14] and perfect by assumption, ${\ensuremath{\mathfrak{g}}}^\Gamma$ is semisimple. Thus the representation of ${\ensuremath{\mathfrak{g}}}^\Gamma$ on ${\ensuremath{\mathfrak{g}}}_\Gamma$ is a direct sum of irreducible representations. If $U \subseteq
{\ensuremath{\mathfrak{g}}}_\Gamma$ is an irreducible ${\ensuremath{\mathfrak{g}}}^\Gamma$-submodule then $[{\ensuremath{\mathfrak{g}}}^\Gamma,
U]\subseteq U$ is a submodule, which is non-zero by assumption on $(\operatorname{ad}{\ensuremath{\mathfrak{g}}}^\Gamma)|_{{\ensuremath{\mathfrak{g}}}_\Gamma}$. Hence $[{\ensuremath{\mathfrak{g}}}^\Gamma, U]=U$ and thus $[{\ensuremath{\mathfrak{g}}}^\Gamma, {\ensuremath{\mathfrak{g}}}_\Gamma]= {\ensuremath{\mathfrak{g}}}_\Gamma$. But then ${\ensuremath{\mathfrak{g}}}= [{\ensuremath{\mathfrak{g}}}^\Gamma,
{\ensuremath{\mathfrak{g}}}^\Gamma] \oplus [{\ensuremath{\mathfrak{g}}}^\Gamma, {\ensuremath{\mathfrak{g}}}_\Gamma]= [{\ensuremath{\mathfrak{g}}}^\Gamma, {\ensuremath{\mathfrak{g}}}]$ follows, so that we can apply .
\[eq:abeli\] Since the groups in several of the examples discussed later will be abelian, it is convenient to discuss the case of an arbitrary abelian group. We assume that the action of $\Gamma$ on ${\ensuremath{\mathfrak{g}}}$ and on $A$ is diagonalizable – a condition which is always fulfilled if $\Gamma$ is finite. Hence, denoting by $\Xi=\Xi(\Gamma)$ the character group of $\Gamma$, the action of $\Gamma$ on ${\ensuremath{\mathfrak{g}}}$ induces a $\Xi$-grading of ${\ensuremath{\mathfrak{g}}}$, i.e., $$\textstyle {\ensuremath{\mathfrak{g}}}= \bigoplus_{\xi \in \Xi} {\ensuremath{\mathfrak{g}}}_\xi, \quad
[{\ensuremath{\mathfrak{g}}}_\xi, {\ensuremath{\mathfrak{g}}}_\zeta] \subseteq {\ensuremath{\mathfrak{g}}}_{\xi + \zeta},$$ where ${\ensuremath{\mathfrak{g}}}_\xi = \{ u \in {\ensuremath{\mathfrak{g}}}:g\cdot u = \xi(g) u \mbox{ for all }
g\in \Gamma\}$ and $\xi,\zeta \in \Xi$. Thus ${\ensuremath{\mathfrak{g}}}_0 = {\ensuremath{\mathfrak{g}}}^\Gamma$ and $\bigoplus_{0\ne \xi} {\ensuremath{\mathfrak{g}}}_\xi = {\ensuremath{\mathfrak{g}}}_\Gamma$ in the notation of above. We have a similar decomposition for $A$. The fixed point subalgebra of the diagonal action of $\Gamma$ on ${\ensuremath{\mathfrak{g}}}{\otimes}A$ is therefore $$({\ensuremath{\mathfrak{g}}}{\otimes}A)^\Gamma = \textstyle \bigoplus_{\xi \in \Xi}
{\ensuremath{\mathfrak{g}}}_\xi {\otimes}A_{-\xi},$$ where $-\xi$ corresponds to the representation dual to $\xi$. The assumption $[{\ensuremath{\mathfrak{g}}}^\Gamma, {\ensuremath{\mathfrak{g}}}_\Gamma]={\ensuremath{\mathfrak{g}}}^\Gamma$ means $[{\ensuremath{\mathfrak{g}}}_0,
{\ensuremath{\mathfrak{g}}}_\xi]={\ensuremath{\mathfrak{g}}}_\xi$ for all $0\ne \xi$, and if this is fulfilled we have $$[{\mathfrak{M}}, {\mathfrak{M}}] = \left(\sum_\xi ([{\ensuremath{\mathfrak{g}}}_\xi, {\ensuremath{\mathfrak{g}}}_{-\xi}] {\otimes}A_\xi
A_{-\xi}\right) \oplus \bigoplus_{0\ne \xi} {\ensuremath{\mathfrak{g}}}_\xi \otimes
A_{-\xi}$$ As a special case, let $\Gamma=\{1,{\sigma}\}$ be a group of order $2$. Then ${\ensuremath{\mathfrak{g}}}= {\ensuremath{\mathfrak{g}}}_0 \oplus {\ensuremath{\mathfrak{g}}}_1$ is a ${\Bbb Z}_2$-grading where ${\ensuremath{\mathfrak{g}}}_s = \{ u
\in {\ensuremath{\mathfrak{g}}}: {\sigma}\cdot u= (-1)^s u \}$ for $s= 0,1\in {\Bbb Z}_2$. Thus ${\mathfrak{M}}= ({\ensuremath{\mathfrak{g}}}{\otimes}A)^\Gamma = ({\ensuremath{\mathfrak{g}}}_0 {\otimes}A_0) \, \oplus \, ({\ensuremath{\mathfrak{g}}}_1 {\otimes}A_1)$ where $A = A_0 \oplus A_1$ is the ${\Bbb Z}_2$-grading of $A$ induced by ${\sigma}$. Hence $$[{\mathfrak{M}}, \, {\mathfrak{M}}]= ([{\ensuremath{\mathfrak{g}}}_0,{\ensuremath{\mathfrak{g}}}_0] {\otimes}A_0 + [{\ensuremath{\mathfrak{g}}}_1, {\ensuremath{\mathfrak{g}}}_1] {\otimes}A_1^2)
\, \oplus \,([{\ensuremath{\mathfrak{g}}}_0, {\ensuremath{\mathfrak{g}}}_1]{\otimes}A_1).$$ In particular, if ${\ensuremath{\mathfrak{g}}}$ is simple and ${\sigma}\ne \operatorname{Id}$, then $${\ensuremath{\mathfrak{g}}}= [{\ensuremath{\mathfrak{g}}},{\ensuremath{\mathfrak{g}}}] = ([{\ensuremath{\mathfrak{g}}}_0,{\ensuremath{\mathfrak{g}}}_0] +[{\ensuremath{\mathfrak{g}}}_1,{\ensuremath{\mathfrak{g}}}_1])\oplus([{\ensuremath{\mathfrak{g}}}_0,{\ensuremath{\mathfrak{g}}}_1]).$$ This implies $[{\ensuremath{\mathfrak{g}}}_0,{\ensuremath{\mathfrak{g}}}_1]={\ensuremath{\mathfrak{g}}}_1$. Also, since ${\ensuremath{\mathfrak{g}}}_1 + [{\ensuremath{\mathfrak{g}}}_1,{\ensuremath{\mathfrak{g}}}_1]$ is the ideal generated by ${\ensuremath{\mathfrak{g}}}_1$, which must be all of ${\ensuremath{\mathfrak{g}}}$, we have $[{\ensuremath{\mathfrak{g}}}_1,{\ensuremath{\mathfrak{g}}}_1]={\ensuremath{\mathfrak{g}}}_0$. Thus $$\label{eq:derivedM}
[{\mathfrak{M}}, \, {\mathfrak{M}}]= ([{\ensuremath{\mathfrak{g}}}_0,{\ensuremath{\mathfrak{g}}}_0] {\otimes}A_0 + {\ensuremath{\mathfrak{g}}}_0 {\otimes}A_1^2)
\, \oplus \,({\ensuremath{\mathfrak{g}}}_1{\otimes}A_1) \quad\mbox{($|\Gamma|=2$, ${\ensuremath{\mathfrak{g}}}$ simple).}$$ Therefore, in this case ${\mathfrak{M}}$ is perfect as soon as ${\ensuremath{\mathfrak{g}}}_0$ is perfect, i.e., semisimple, or $A_0 = A_1^2$, i.e., $A=A_0 \oplus
A_1$ is a strong ${\Bbb Z}_2$-grading.
\[eg:onsager\] Let $X=k^\times = \operatorname{Spec}k[t,t^{-1}]$, ${\ensuremath{\mathfrak{g}}}$ be a simple Lie algebra, and $\Gamma=\{1,\sigma\}$ be a group of order $2$. We choose a set of Chevalley generators $\{e_i, f_i, h_i\}$ for ${\ensuremath{\mathfrak{g}}}$ and let $\Gamma$ act on ${\ensuremath{\mathfrak{g}}}$ by the standard Chevalley involution, i.e., $$\sigma(e_i) = -f_i,\ \sigma(f_i) = -e_i,\ \sigma(h_i) = -h_i.$$ Let $\Gamma$ act on $ k[t,t^{-1}]$ by ${\sigma}\cdot t = t^{-1}$, inducing an action of $\Gamma$ on $X$. We define the *generalized Onsager algebra ${\EuScript{O}}({\ensuremath{\mathfrak{g}}})$* to be the equivariant map algebra associated to these data: $${\EuScript{O}}({\ensuremath{\mathfrak{g}}}) := M(k^\times,{\ensuremath{\mathfrak{g}}})^\Gamma \cong ({\ensuremath{\mathfrak{g}}}\otimes k[t,t^{-1}])^\Gamma$$ These algebras have been considered by G. Benkart and M. Lau. The action of $\sigma$ on ${\ensuremath{\mathfrak{g}}}$ interchanges the positive and negative root spaces and thus the dimension of the fixed point subalgebra ${\ensuremath{\mathfrak{g}}}^\Gamma$ is equal to the number of positive roots. This fact, together with the classification of automorphisms of order two (see, for example, [@Helga01 Chapter X, §5, Tables II and III]) determines ${\ensuremath{\mathfrak{g}}}_0={\ensuremath{\mathfrak{g}}}^\Gamma$ as follows.
Type of ${\ensuremath{\mathfrak{g}}}$ (Type of) ${\ensuremath{\mathfrak{g}}}_0$
--------------------------------------- -------------------------------------------------------------
$A_n$ $\mathfrak{so}_{n+1}$
$B_n$ ($n \ge 2$) $\mathfrak{so}_n \oplus \mathfrak{so}_{n+1}$
$C_n$ ($n \ge 2$) $\mathfrak{gl}_n = k \oplus {\ensuremath{\mathfrak{sl}}}_n$
$D_n$ ($n \ge 4$) $\mathfrak{so}_n \oplus \mathfrak{so}_n$
$E_6$ $C_4$
$E_7$ $A_7$
$E_8$ $D_8$
$F_4$ $C_3 \oplus A_1$
$G_2$ $A_1 \oplus A_1$
Since $\mathfrak{so}_2$ is one-dimensional, we see that ${\ensuremath{\mathfrak{g}}}^\Gamma$ is semisimple in all cases, except ${\ensuremath{\mathfrak{g}}}= {\ensuremath{\mathfrak{sl}}}_2$ (type $A_1$), ${\ensuremath{\mathfrak{g}}}=\mathfrak{so}_5$ (type $B_2$) and ${\ensuremath{\mathfrak{g}}}=\mathfrak{sp}_n$ (type $C_n$). Hence, using (\[eq:derivedM\]), we have $$\label{eq:derived-onsager}
[{\EuScript{O}}({\ensuremath{\mathfrak{g}}}),\, {\EuScript{O}}({\ensuremath{\mathfrak{g}}})] =
\begin{cases}
({\ensuremath{\mathfrak{g}}}_0 {\otimes}A_1^2) \oplus ({\ensuremath{\mathfrak{g}}}_1 {\otimes}A_1) & \mbox{if ${\ensuremath{\mathfrak{g}}}={\ensuremath{\mathfrak{sl}}}_2$,} \\
(\mathfrak{so}_3 \otimes A_0 + {\ensuremath{\mathfrak{g}}}_0 \otimes A_1^2) \oplus
({\ensuremath{\mathfrak{g}}}_1 \otimes A_1) & \mbox{if ${\ensuremath{\mathfrak{g}}}=\mathfrak{so}_5$,} \\
({\ensuremath{\mathfrak{sl}}}_n \otimes A_0 + {\ensuremath{\mathfrak{g}}}_0 \otimes A_1^2) \oplus ({\ensuremath{\mathfrak{g}}}_1
\otimes A_1) & \mbox{if ${\ensuremath{\mathfrak{g}}}=\mathfrak{sp}_n$, and} \\
{\EuScript{O}}({\ensuremath{\mathfrak{g}}}) & \mbox{otherwise.}
\end{cases}$$ Therefore $$\label{eq:quotient-onsager}
{\EuScript{O}}({\ensuremath{\mathfrak{g}}})/[{\EuScript{O}}({\ensuremath{\mathfrak{g}}}),\, {\EuScript{O}}({\ensuremath{\mathfrak{g}}})] \cong
\begin{cases}
A_0/A_1^2 & \mbox{if ${\ensuremath{\mathfrak{g}}}={\ensuremath{\mathfrak{sl}}}_2$,
$\mathfrak{so}_5$, or $\mathfrak{sp}_n$, and} \\
0 & \mbox{otherwise.}
\end{cases}$$
\[eg:general-involution\] One can consider the following situation, even more general than Example \[eg:onsager\]. Namely, we consider the same setup except that we allow $\sigma$ to act by an *arbitrary* involution of ${\ensuremath{\mathfrak{g}}}$. Again, by the classification of automorphisms of order two, it is known that ${\ensuremath{\mathfrak{g}}}_0$ is either semisimple or has a one-dimensional center. Thus we have $$\label{eq:quotient-general-involution}
{\mathfrak{M}}/[{\mathfrak{M}},\, {\mathfrak{M}}] \cong
\begin{cases}
0 & \mbox{if ${\ensuremath{\mathfrak{g}}}_0$ is semisimple, and} \\
A_0/A_1^2 & \mbox{otherwise.}
\end{cases}$$
For $k={\mathbb{C}}$ it was shown in [@Roan91] that ${\EuScript{O}}({\ensuremath{\mathfrak{sl}}}_2)$ is isomorphic to the usual *Onsager algebra*. This algebra was a key ingredient in Onsager’s original solution of the 2D Ising model. The algebra ${\EuScript{O}}({\ensuremath{\mathfrak{sl}}}_n)$, $k={\mathbb{C}}$, was introduced in [@UI96], although the definition given there differs slightly from the one given in Example \[eg:onsager\]. For ${\ensuremath{\mathfrak{g}}}={\ensuremath{\mathfrak{sl}}}_n$, the Chevalley involution of Example \[eg:onsager\] is given by $\sigma \cdot u =
-u^t$, while the involution used in [@UI96] is given by $E_{ij}
\mapsto (-1)^{i+j+1}E_{ji}$, where $E_{ij}$ is the standard elementary matrix with $(i,j)$ entry equal to one and all other entries equal to zero. This involution is equal to $\tau_1 \sigma
\tau_1^{-1}$, where $\tau_1 (u) = DuD^{-1}$ for $D= \mathrm{diag}
(\sqrt{-1},+1,\sqrt{-1},\dots)$. Furthermore, the involution of $X$ considered in [@UI96] is $x \mapsto (-1)^n x^{-1}$, which is equal to $\sigma$ if $n$ is even and to $\tau_2 \sigma \tau_2^{-1}$, where $\tau_2 \cdot x = \sqrt{-1}x$, if $n$ is odd. Therefore it follows from Lemma \[lem:conjugate-actions\] that the two versions are isomorphic.
\[eg:twistedmultiloop\] \[eg:twisted-multiloop\] Fix positive integers $n, m_1, \dots,
m_n$. Let $$\Gamma = \langle g_1,\dots, g_n : g_i^{m_i}=1,\
g_i g_j = g_j g_i,\ \forall\ 1 \le i,j \le n
\rangle.$$ Then $\Xi = \Xi(\Gamma) \cong {\mathbb{Z}}/m_1{\mathbb{Z}}\times \cdots \times {\mathbb{Z}}/m_n{\mathbb{Z}}\cong \Gamma$. Suppose $\Gamma$ acts on ${\ensuremath{\mathfrak{g}}}$. Note that this is equivalent to specifying commuting automorphisms $\sigma_i$, $i=1,\dots,n$, of ${\ensuremath{\mathfrak{g}}}$ such that $\sigma_i^{m_i}={\mathrm{id}}$. For $i =
1,\dots, n$, let $\xi_i$ be a primitive $m_i$-th root of unity. As in Example \[eq:abeli\] we then see that ${\ensuremath{\mathfrak{g}}}$ has a $\Xi$-grading for which the homogenous subspace of degree $\bar{\mathbf{k}}$, $\mathbf{k}=(k_1, \ldots, k_n)\in {\Bbb Z}^n$, is given by $${\ensuremath{\mathfrak{g}}}_{\bar{\mathbf{k}}} = \{u \in {\ensuremath{\mathfrak{g}}}: \sigma_i(u) = \xi_i^{k_i}u\ \forall\
i=1,\dots,n\}.$$ Let $X=(k^\times)^n$ and define an action of $\Gamma$ on $X$ by $$g_i \cdot (z_1, \dots, z_n) = (z_1, \dots, z_{i-1},
\xi_i z_i, z_{i+1}, \dots, z_n).$$ Then $$\label{eq:twisted-multiloop-def}
M({\ensuremath{\mathfrak{g}}},\sigma_1,\dots,\sigma_n,m_1,\dots,m_n) := M(X,{\ensuremath{\mathfrak{g}}})^\Gamma$$ is the *multiloop algebra* of ${\ensuremath{\mathfrak{g}}}$ relative to $(\sigma_1,
\dots, \sigma_n)$ and $(m_1, \ldots, m_n)$. If all ${\sigma}_i=\operatorname{Id}$ we recover the untwisted multiloop algebra of Example \[eg:multiloop\]. If not all ${\sigma}_i=\operatorname{Id}$, the algebra $M$ is therefore sometimes called the *twisted loop algebra*.
With $\Gamma$ and $X$ as above, the induced action of $\Gamma$ on $A
= k[t_1^{\pm 1}, \dots, t_n^{\pm 1}]$ is given by $$\sigma_i \cdot f(t_1, \dots, t_n) = f(t_1, \dots, t_{i-1},
\xi_i^{-1} t_i, t_{i+1}, \dots, t_n),\quad f \in
k[t_1^{\pm 1},\dots,t_n^{\pm 1}].$$ Hence $$\label{eq:twisted-multiloop-diagonal}
M({\ensuremath{\mathfrak{g}}},\sigma_1,\dots,\sigma_n,m_1,\dots,m_n)
\cong ({\ensuremath{\mathfrak{g}}}\otimes k[t_1^{\pm 1}, \dots, t_n^{\pm 1}])^\Gamma
\cong
\bigoplus_{\mathbf{k} \in {\mathbb{Z}}^n} {\ensuremath{\mathfrak{g}}}_{\bar {\mathbf{k}}}
\otimes k t^{\mathbf{k}},$$ where $t^{\mathbf{k}} = t_1^{k_1} \cdots t_n^{k_n}$.
Loop and multiloop algebras play an important role in the theory of affine Kac-Moody algebras [@Kac90], extended affine Lie algebras, and Lie tori. The connection between the last two classes of Lie algebras is the following: The core and the centerless core of an extended affine Lie algebra is a Lie torus or centerless Lie torus respectively, every Lie torus arises in this way, and there is a precise construction of extended affine Lie algebras in terms of centerless Lie tori [@Neh]. Any centerless Lie torus whose grading root system is not of type A can be realized as a multiloop algebra [@abfp2].
While most classification results involving (special cases of) equivariant map algebras in the literature use abelian groups, we will see that the general classification developed in the current paper only assumes that the group $\Gamma$ is finite. Therefore, for illustrative purposes, we include here an example of an equivariant map algebra where the group $\Gamma$ is not abelian. We will see in Section \[subsec:applications-nonabelian\] that the representation theory of this algebra is quite interesting.
\[eg:nonabelian\] Let $\Gamma = S_3$, the symmetric group on 3 objects, $X=
\mathbb{P}^1 \setminus \{0,1,\infty\}$ and ${\ensuremath{\mathfrak{g}}}=\mathfrak{so}_8$, the simple Lie algebra of type $D_4$. The symmetry group of the Dynkin diagram of type $D_4$ is isomorphic to $S_3$, and so $\Gamma$ acts naturally on ${\ensuremath{\mathfrak{g}}}$ by diagram automorphisms (see Remark \[rem:diagram-autom-case\]). Now, given points $x_i, y_i$, $i=1,2,3$, of $\mathbb{P}^1$, there is a unique Möbius transformatoin of $\mathbb{P}^1$ mapping $x_i$ to $y_i$, $i =
1,2,3$. Thus, for any permutation $\sigma$ of the points $\{0,1,\infty\}$, there is a unique Möbius transformation of $\mathbb{P}^1$ which induces $\sigma$ on the set $\{0,1,\infty\}$. Hence each permutation $\sigma$ naturally corresponds to an automorphism of $X$, which we also denote by $\sigma$. Therefore we can form the equivariant map algebra ${\mathfrak{M}}=M(X,{\ensuremath{\mathfrak{g}}})^\Gamma$. Now, the subalgebra of ${\ensuremath{\mathfrak{g}}}$ fixed by the unique order three subgroup of $\Gamma$ is the simple Lie algebra of type $G_2$ (see [@Kac90 Proposition 8.3]). One easily checks that this subalgebra is fixed by all of $\Gamma$. Therefore ${\ensuremath{\mathfrak{g}}}^\Gamma$ is perfect. By [@Kac90 Proposition 8.3], the representation of ${\ensuremath{\mathfrak{g}}}^\Gamma$ on ${\ensuremath{\mathfrak{g}}}_\Gamma$ is a direct sum of two $7$-dimensional irreducible representations. Thus, by Lemma \[lem:perfrm\], ${\mathfrak{M}}$ is perfect.
Evaluation representations {#sec:eval-reps}
==========================
From now on we assume that $\Gamma$ is a *finite* group, acting on an affine scheme $X$ and a (finite-dimensional) Lie algebra ${\ensuremath{\mathfrak{g}}}$ by automorphisms. We abbreviate ${\mathfrak{M}}=M(X,{\ensuremath{\mathfrak{g}}})^\Gamma$.
Let $Y$ be subscheme of $X$. Then, as in Remark \[categ\], we have the *restriction* Lie algebra homomorphism $$\operatorname{Res}^X_Y : M(X,{\ensuremath{\mathfrak{g}}})^\Gamma \to M(Y,{\ensuremath{\mathfrak{g}}}),\quad \operatorname{Res}^X_Y(\alpha) =
\alpha|_Y,\quad \alpha \in M(X,{\ensuremath{\mathfrak{g}}})^\Gamma.$$ If $Y$ is a $\Gamma$-invariant subscheme, the image of $\operatorname{Res}^X_Y$ is contained in $M(Y,{\ensuremath{\mathfrak{g}}})^\Gamma$.
Given a finite subset ${\mathbf{x}}\subseteq X_{\mathrm{rat}}$, we define the corresponding *evaluation map* $$\textstyle
\operatorname{ev}_{\mathbf{x}}: M(X,{\ensuremath{\mathfrak{g}}})^\Gamma \to \bigoplus_{x \in {\mathbf{x}}} {\ensuremath{\mathfrak{g}}},\quad
\alpha \mapsto (\alpha(x))_{x \in {\mathbf{x}}},\quad \alpha \in
M(X,{\ensuremath{\mathfrak{g}}})^\Gamma.$$
For a subset $Z$ of $X$ we define $$\Gamma_Z = \{g \in \Gamma : g \cdot z = z \ \forall\ z \in Z\} \quad
\text{and} \quad {\ensuremath{\mathfrak{g}}}^Z = {\ensuremath{\mathfrak{g}}}^{\Gamma_Z}.$$ Obviously, $\Gamma_Z$ is a subgroup of $\Gamma$ and ${\ensuremath{\mathfrak{g}}}^Z$ is a subalgebra of ${\ensuremath{\mathfrak{g}}}$. In particular, for any $x \in X$, we put $\Gamma_x = \Gamma_{\{x\}}$ and ${\ensuremath{\mathfrak{g}}}^x = {\ensuremath{\mathfrak{g}}}^{\{x\}}$.
\[lem:restriction-image\]
1. \[lem-item1:restriction-image\] Let $Z$ be a subset of $X_{\mathrm{rat}}$. Then ${\alpha}(Z) \subseteq {\ensuremath{\mathfrak{g}}}^Z$ for all ${\alpha}\in {\mathfrak{M}}(X,{\ensuremath{\mathfrak{g}}})^\Gamma$. In particular, ${\alpha}(x) \in {\ensuremath{\mathfrak{g}}}^x$ for all $x\in X_{\mathrm{rat}}$.
2. \[lem-item2:restriction-image\] Let $Z$ be a $\Gamma$-invariant subscheme of $X$ for which the restriction $A_X \to A_Z$, $f \mapsto f|_Z$, is surjective. Then the restriction map $\operatorname{Res}^X_Z : M(X,{\ensuremath{\mathfrak{g}}})^\Gamma \to M(Z,{\ensuremath{\mathfrak{g}}})^\Gamma$ is also surjective.
Part is immediate from the definitions. In , finite-dimensionality of ${\ensuremath{\mathfrak{g}}}$ implies that the restriction $M(X,{\ensuremath{\mathfrak{g}}})
\to M(Z,{\ensuremath{\mathfrak{g}}})$ is surjective. Since $\Gamma$ acts completely reducibly on $M(X,{\ensuremath{\mathfrak{g}}})$ and $M(Z,{\ensuremath{\mathfrak{g}}})$, the restriction is then also surjective on the subalgebras of $\Gamma$-invariants.
We denote by $X_n$ the set of $n$-element subsets ${\mathbf{x}}\subseteq
X_{\mathrm{rat}}$ consisting of $k$-rational points and having the property that $y \not \in \Gamma \cdot x$ for distinct $x,y \in {\mathbf{x}}$.
\[cor:twisted-surjectivity\] For ${\mathbf{x}}\in X_n$ the image of $\operatorname{ev}_{\mathbf{x}}$ is $\bigoplus_{x \in {\mathbf{x}}}
{\ensuremath{\mathfrak{g}}}^x$.
Let $Z= \bigcup_{x \in {\mathbf{x}}} \Gamma \cdot x$. Then $Z$ is a $\Gamma$-invariant closed subvariety. Hence $A_X \to A_Z$ is surjective, and therefore $\operatorname{Res}^X_Z : M(X,{\ensuremath{\mathfrak{g}}})^\Gamma \to
M(Z,{\ensuremath{\mathfrak{g}}})^\Gamma$ is also surjective by Lemma \[lem:restriction-image\]. That $\operatorname{ev}_{\mathbf{x}}: M(Z,{\ensuremath{\mathfrak{g}}})^\Gamma
\to \bigoplus_{x \in {\mathbf{x}}} {\ensuremath{\mathfrak{g}}}^x$ is surjective, is immediate.
\[def:evaluation-rep\] Fix a finite subset ${\mathbf{x}}\subseteq X_{\mathrm{rat}}$ and let $\rho_x : {\ensuremath{\mathfrak{g}}}^x
\to \operatorname{End}_k V_x$, $x \in {\mathbf{x}}$, be representations of ${\ensuremath{\mathfrak{g}}}^x$ on the vector spaces $V_x$. Then define $\operatorname{ev}_{\mathbf{x}}(\rho_x)_{x \in {\mathbf{x}}} $ to be the composition $$\textstyle
M(X,{\ensuremath{\mathfrak{g}}})^\Gamma \xrightarrow{\operatorname{ev}_{\mathbf{x}}} \bigoplus_{x \in {\mathbf{x}}}
{\ensuremath{\mathfrak{g}}}^x \xrightarrow{\otimes_{x \in {\mathbf{x}}} \rho_x}
\operatorname{End}_k (\bigotimes_{x \in {\mathbf{x}}} V_x).$$ This defines a representation of $M(X,{\ensuremath{\mathfrak{g}}})^\Gamma$ on $\bigotimes_{x
\in {\mathbf{x}}} V_x$ called a *(twisted) evaluation representation*.
\[rem:new-eval-rep\] We note some important distinctions between Definition \[def:evaluation-rep\] and other uses of the term *evaluation representation* in the literature. First of all, some authors reserve the term *evaluation representation* for the case $n=1$ and would refer to the more general case as a tensor product of evaluation representations. Furthermore, traditionally the $\rho_x$ are representations of ${\ensuremath{\mathfrak{g}}}$ instead of ${\ensuremath{\mathfrak{g}}}^x$ and are required to be faithful. In the case that ${\ensuremath{\mathfrak{g}}}^x = {\ensuremath{\mathfrak{g}}}$ for all $x \in
X$ (for instance, if $\Gamma$ acts freely on $X$), this of course makes no difference. However, we will see in the sequel that the more general definition of evaluation representation given above allows for a more uniform classification of irreducible finite-dimensional representations.
\[prop:eval-irred\] Let ${\mathbf{x}}\in X_n$ and for $x \in {\mathbf{x}}$ let $\rho_x : {\ensuremath{\mathfrak{g}}}^x \to \operatorname{End}_k
V_x$ be an irreducible finite-dimensional representation of ${\ensuremath{\mathfrak{g}}}^x$. Then the evaluation representation $\operatorname{ev}_{\mathbf{x}}(\rho_x)_{x \in {\mathbf{x}}}$ is an irreducible finite-dimensional representation of $M(X,{\ensuremath{\mathfrak{g}}})^\Gamma$.
Since $\operatorname{ev}_{\mathbf{x}}$ is surjective, this follows from Proposition \[prop:basic-facts\].
If ${\mathbf{x}}$ is in $X^n$ (but not necessarily in $X_n$), ${\ensuremath{\mathfrak{g}}}^x$ is semisimple for all $x \in {\mathbf{x}}$, and $\rho_x$ is an arbitrary finite-dimensional representation of ${\ensuremath{\mathfrak{g}}}^x$ for each $x \in {\mathbf{x}}$, then the evaluation representation $\operatorname{ev}_{\mathbf{x}}(\rho_x)_{x \in {\mathbf{x}}}$ is completely reducible.
This follows from Proposition \[prop:eval-irred\] and complete reducibility of finite-dimensional representations of each ${\ensuremath{\mathfrak{g}}}^x$.
By abuse of notation, we will sometimes denote a representation of ${\ensuremath{\mathfrak{g}}}$ by the underlying vector space $V$. Then $\operatorname{ev}_x V$, $x \in X$, will denote the corresponding evaluation representation of $M(X,{\ensuremath{\mathfrak{g}}})^\Gamma$. Note that, with the notation of Definition \[def:evaluation-rep\], we have $$\label{eq:tensor-prod-evals}
\operatorname{ev}_{\mathbf{x}}(\rho_x)_{x \in {\mathbf{x}}} \cong \bigotimes_{x \in {\mathbf{x}}} \operatorname{ev}_x V_x.$$
Note that $X_{\mathrm{rat}}$ is $\Gamma$-invariant, and $X_{\mathrm{rat}}=X$ if $X$ is an affine variety. Let $x \in X_{\mathrm{rat}}$ and $g\in \Gamma$. Since $\Gamma_{g \cdot x} = g \Gamma_x g^{-1}$ we see that ${\ensuremath{\mathfrak{g}}}^{g\cdot x}
= g \cdot {\ensuremath{\mathfrak{g}}}^x$. Hence if $\rho$ is a representation of ${\ensuremath{\mathfrak{g}}}^x$, then $\rho \circ g^{-1}$ is a representation of ${\ensuremath{\mathfrak{g}}}^{g\cdot x}$. Let $\mathcal{R}_x$ denote the set of isomorphism classes of irreducible finite-dimensional representations of ${\ensuremath{\mathfrak{g}}}^x$, and put $\mathcal{R}_X=\bigsqcup_{x \in X_{\mathrm{rat}}} \mathcal{R}_x$. Then $\Gamma$ acts on $\mathcal{R}_X$ by $$\Gamma \times \mathcal{R}_X \to \mathcal{R}_X,\quad (g,[\rho]) \mapsto g
\cdot [\rho] := [\rho \circ g^{-1}] \in \mathcal{R}_{g \cdot x},$$ where $[\rho] \in \mathcal{R}_x$ denotes the isomorphism class of a representation $\rho$ of ${\ensuremath{\mathfrak{g}}}^x$.
\[def:E\] Let $\mathcal{E}$ denote the set of finitely supported $\Gamma$-equivariant functions $\Psi : X_{\mathrm{rat}}\to \mathcal{R}_X$ such that $\Psi(x) \in \mathcal{R}_x$. Here the support $\operatorname{supp}\Psi$ of $\Psi \in \mathcal{E}$ is the set of all $x \in X_{\mathrm{rat}}$ for which $\Psi(x) \ne 0$, where $0$ denotes the isomorphism class of the trivial representation.
For isomorphic representations $\rho$ and $\rho'$ of ${\ensuremath{\mathfrak{g}}}^x$, the evaluation representations $\operatorname{ev}_x \rho$ and $\operatorname{ev}_x \rho'$ are isomorphic. Therefore, for $[\rho] \in \mathcal{R}_x$, we can define $\operatorname{ev}_x [\rho]$ to be the isomorphism class of $\operatorname{ev}_x \rho$ and this is independent of the representative $\rho$. Similarly, for a finite subset ${\mathbf{x}}\subseteq X_{\mathrm{rat}}$ and representations $\rho_x$ of ${\ensuremath{\mathfrak{g}}}^x$ for $x \in {\mathbf{x}}$, we define $\operatorname{ev}_{\mathbf{x}}([\rho_x])_{x \in {\mathbf{x}}}$ to be the isomorphism class of $\operatorname{ev}_{\mathbf{x}}(\rho_x)_{x \in {\mathbf{x}}}$.
\[rem:diagram-autom-case\] Suppose ${\ensuremath{\mathfrak{g}}}$ is a semisimple Lie algebra. In the case that $\Gamma$ is cyclic and acts on ${\ensuremath{\mathfrak{g}}}$ by admissible diagram automorphisms (no edge joins two vertices in the same orbit), there exists a simple description of $\mathcal{E}$ which can be seen as follows. Let $I$ be the set of vertices of the Dynkin diagram of ${\ensuremath{\mathfrak{g}}}$. An action of $\Gamma$ on this Dynkin diagram gives rise to an action of $\Gamma$ on ${\ensuremath{\mathfrak{g}}}$ via $$g \cdot h_i = h_{g \cdot i},\quad g \cdot e_i = e_{g \cdot
i},\quad g \cdot f_i = f_{g \cdot i},\quad g \in \Gamma,$$ where $\{h_i, e_i, f_i\}_{i \in I}$ is a set of Chevalley generators of ${\ensuremath{\mathfrak{g}}}$. We then have a natural action of $\Gamma$ on the weight lattice $P$ of ${\ensuremath{\mathfrak{g}}}$ given by $$g \cdot \omega_i = \omega_{g \cdot i},\quad g \in \Gamma,$$ where $\{\omega_i\}_{i \in I}$ is the set of fundamental weights of ${\ensuremath{\mathfrak{g}}}$. Now, isomorphism classes of irreducible finite-dimensional representations of ${\ensuremath{\mathfrak{g}}}$ are naturally enumerated by the set of dominant weights $P^+$ by associating to $\lambda \in P^+$ the isomorphism class of the irreducible highest weight representation of highest weight $\lambda$. Let $\tilde{\mathcal{E}}$ denote the set of $\Gamma$-equivariant functions $X_{\mathrm{rat}}\to P^+$ with finite support. It follows that for $\Psi \in \tilde{\mathcal{E}}$ and $x
\in X_{\mathrm{rat}}$, we have $\Psi(x) \in (P^+)^{\Gamma_x}$, where $(P^+)^{\Gamma_x}$ denotes the set of $\Gamma_x$-invariant elements of $P^+$. There is a canonical bijection between $(P^+)^{\Gamma_x}$ and the positive weight lattice of ${\ensuremath{\mathfrak{g}}}^x$ (see [@Lus93 Proposition 14.1.2]) and so we can associate to $\Psi(x)$ the isomorphism class of the corresponding representation of ${\ensuremath{\mathfrak{g}}}^x$. Thus, we have a natural bijection between $\tilde{\mathcal{E}}$ and $\mathcal{E}$. Therefore, in the case that $\Gamma$ acts on ${\ensuremath{\mathfrak{g}}}$ by admissible diagram automorphisms, the evaluation representations are naturally enumerated by $\Gamma$-equivariant maps from $X_{\mathrm{rat}}$ to the positive weight lattice $P^+$ of ${\ensuremath{\mathfrak{g}}}$. In the case that $\Gamma$ acts freely on $X$, we can drop the assumption that the diagram automorphisms be admissible.
\[lem:twisted-eval-invariance\] Suppose $\Psi \in \mathcal{E}$ and $x \in \operatorname{supp}\Psi$. Then for all $g \in \Gamma$, $$\operatorname{ev}_x \Psi (x) = \operatorname{ev}_{g \cdot x} \left( g \cdot \Psi(x) \right) =
\operatorname{ev}_{g \cdot x} \Psi(g \cdot x).$$
For any $g \in \Gamma$ and representation $\rho$ of ${\ensuremath{\mathfrak{g}}}^x$, the following diagram commutes: $$\xymatrix{ & {\ensuremath{\mathfrak{g}}}^x \ar[dd]^g \ar[dr]^\rho & \\
M(X,{\ensuremath{\mathfrak{g}}})^\Gamma \ar[ur]^{\operatorname{ev}_x} \ar[dr]_{\operatorname{ev}_{g \cdot x}} & & \operatorname{End}_k V \\
& {\ensuremath{\mathfrak{g}}}^{g \cdot x} \ar[ur]_{\rho \circ g^{-1}} & }$$ Thus, $\operatorname{ev}_x \rho = \operatorname{ev}_{g \cdot x} (\rho \circ g^{-1})$ and the result follows.
\[def:eval-rep\] For $\Psi \in \mathcal{E}$, we define $\operatorname{ev}_\Psi = \operatorname{ev}_{\mathbf{x}}(\Psi(x))_{x \in {\mathbf{x}}}$ where ${\mathbf{x}}\in X_n$ contains one element of each $\Gamma$-orbit in $\operatorname{supp}\Psi$. By Lemma \[lem:twisted-eval-invariance\], $\operatorname{ev}_\Psi$ is independent of the choice of ${\mathbf{x}}$. If $\Psi$ is the map that is identically 0 on $X$, we define $\operatorname{ev}_\Psi$ to be the isomorphism class of the trivial representation of ${\mathfrak{M}}$. Thus $\Psi \mapsto \operatorname{ev}_\Psi$ defines a map $\mathcal{E} \to {\mathcal{S}}$, where ${\mathcal{S}}$ denotes the set of isomorphism classes of irreducible finite-dimensional representations of ${\mathfrak{M}}$.
\[prop:inj\] The map $\mathcal{E} \to {\mathcal{S}}$, $\Psi \mapsto \operatorname{ev}_\Psi$, is injective.
Suppose $\Psi \ne \Psi' \in \mathcal{E}$. Then there exists $x \in
X_{\mathrm{rat}}$ such that $\Psi(x) \ne \Psi'(x)$. Without loss of generality, we may assume $\Psi(x) \ne 0$. Let $$m = (\dim \operatorname{ev}_\Psi)/(\dim \Psi(x)),\qquad m' = (\dim
\operatorname{ev}_{\Psi'})/(\dim \Psi'(x))$$ where the dimension of an isomorphism class of representations is simply the dimension of any representative of that class and $m'=\dim \operatorname{ev}_{\Psi'}$ if $\Psi'(x) = 0$. By , $m$ and $m'$ are positive integers. By Corollary \[cor:twisted-surjectivity\], there exists a subalgebra $\mathfrak{a}$ of ${\mathfrak{M}}$ such that $\operatorname{ev}_x (\mathfrak{a}) = {\ensuremath{\mathfrak{g}}}^x$ and $\operatorname{ev}_{x'} (\mathfrak{a})=0$ for all $x' \in \left( \operatorname{supp}\Psi \cup
\operatorname{supp}\Psi' \right) \setminus \{\Gamma \cdot x\}$. Then $$\operatorname{ev}_\Psi|_{\mathfrak{a}} = \Psi(x)^{\oplus m},\quad
\operatorname{ev}_{\Psi'}|_{\mathfrak{a}}= \Psi'(x)^{\oplus m'}.$$ Since $\Psi(x) \ne \Psi'(x)$, we have $\operatorname{ev}_\Psi \ne \operatorname{ev}_{\Psi'}$. In the above, we have used the convention that the restriction of an isomorphism class is the isomorphism class of the restriction of any representative and a direct sum of isomorphism classes is the isomorphism class of the corresponding direct sum of representatives.
\[lem:AGammamod\] Let ${\mathfrak{K}}$ be an ideal of ${\mathfrak{M}}$. If ${\mathfrak{M}}/{\mathfrak{K}}$ does not contain non-zero solvable ideals, e.g. if ${\mathfrak{M}}/{\mathfrak{K}}$ is finite-dimensional semisimple, then ${\mathfrak{K}}$ is an $A^\Gamma$-submodule of ${\mathfrak{M}}$.
Let $f\in A^\Gamma$. Since $A^\Gamma$ is contained in the centroid of ${\mathfrak{M}}$, the $k$-subspace $f{\mathfrak{K}}$ is an ideal of ${\mathfrak{M}}$: $[{\mathfrak{M}},
f{\mathfrak{K}}] = f[{\mathfrak{M}}, {\mathfrak{K}}]\subseteq f{\mathfrak{K}}$. Let $\pi : {\mathfrak{M}}\to
{\mathfrak{M}}/{\mathfrak{K}}$ be the canonical epimorphism. Since $[f{\mathfrak{K}}+ {\mathfrak{K}},
f{\mathfrak{K}}+ {\mathfrak{K}}] \subseteq [{\mathfrak{K}}+ f {\mathfrak{K}}+ f^2{\mathfrak{K}}, {\mathfrak{K}}] \subseteq
{\mathfrak{K}}$, the ideal $\pi(f{\mathfrak{K}})$ of ${\mathfrak{M}}/{\mathfrak{K}}$ is abelian, whence $f{\mathfrak{K}}\subseteq {\mathfrak{K}}$.
\[prop:fixed-point-alg-eval\] Suppose $I$ is an ideal of $\mathfrak{M}$ such that $\mathfrak{M}/I = N_1
\oplus \dots \oplus N_s$ with $N_i$ a finite-dimensional simple Lie algebra for $i=1,\dots,s$. Let $\pi : \mathfrak{M} \to
\mathfrak{M}/I$ denote the canonical projection and for $i=1,\dots,s$, let $\pi_i : \mathfrak{M} \to N_i$ denote the map $\pi$ followed by the projection from $\mathfrak{M}/I$ to $N_i$. Then there exist $x_1, \dots, x_s \in X_{\mathrm{rat}}$ such that $$\pi(f \alpha) = (f(x_1) \pi_1(\alpha), \dots, f(x_s)
\pi_s(\alpha)) \ \forall\ f \in A^\Gamma,\ \alpha \in \mathfrak{M}.$$
It suffices to show that for $i=1,\dots,s$, $f \in A^\Gamma$ and $\alpha \in \mathfrak{M}$ we have $\pi_i(f \alpha) = f(x_i)
\pi_i(\alpha)$ for some $x_i \in X$. Since $N_i$ is simple, the action of $\mathfrak{M}$ on $N_i$ induced by the adjoint action is irreducible. By Lemma \[lem:AGammamod\], $N_i$ is a $A^\Gamma$-module and $\pi_i$ is a $A^\Gamma$-module homomorphism. Since the action of $A^\Gamma$ commutes with the action of $\mathfrak{M}$, we have that $A^\Gamma$ must act by scalars and thus as a character $\chi : A^\Gamma \to k$. This character corresponds to evaluation at a point $\tilde x_i \in X{{/\!/}}\Gamma := \operatorname{Spec}A^\Gamma$. Choosing any $x_i$ in the preimage of $\tilde x_i$ under the canonical projection $X \to X{{/\!/}}\Gamma$, the result follows.
If $s=1$, then in the graded setting these types of maps have been studied extensively in [@abfp]. There $\rho$ is a character of the full centroid of ${\mathfrak{M}}$, while in the above $A^\Gamma$ is a priori only a subalgebra of the centroid of ${\mathfrak{M}}$.
\[prop:lifting\] If $\operatorname{supp}\Psi \subseteq \{x \in X : {\ensuremath{\mathfrak{g}}}^x = {\ensuremath{\mathfrak{g}}}\}$, then any evaluation in the isomorphism class $\operatorname{ev}_\Psi$ of $M(X,{\ensuremath{\mathfrak{g}}})^\Gamma$ is obtained by restriction from an evaluation representation of the untwisted map algebra $M(X,{\ensuremath{\mathfrak{g}}})$. In particular, the restriction map from the set of isomorphism classes of evaluation representations of $M(X,{\ensuremath{\mathfrak{g}}})$ to the set of isomorphism classes of evaluation representations of $M(X,{\ensuremath{\mathfrak{g}}})^\Gamma$ is surjective if ${\ensuremath{\mathfrak{g}}}^x = {\ensuremath{\mathfrak{g}}}$ for all $x \in X$.
The proof is immediate.
Classification of irreducible finite-dimensional representations {#sec:classification}
================================================================
We consider a finite group $\Gamma$, acting on a finite-dimensional Lie algebra ${\ensuremath{\mathfrak{g}}}$ and an affine scheme $X$. We put ${\mathfrak{M}}=
M(X,{\ensuremath{\mathfrak{g}}})^\Gamma$.
\[lem:ideal-of-subring\] [([@bou:A II, Corollaire 2 de la Proposition 6, §3.6])]{} Let $S$ be a subring of a ring $R$ and $I$ an ideal of $S$. Then $R \otimes_S (S/I) \cong R/(RI)$.
Note that $A^\Gamma$ is the coordinate ring of the quotient $X{{/\!/}}\Gamma$. For a point $\left[ x\right] \in X{{/\!/}}\Gamma$, let $\mathfrak{m}_{\left[ x\right]}$ denote the corresponding maximal ideal of $A^\Gamma$ and define $A_{\left[ x\right]} = A
\otimes_{A^\Gamma} (A^\Gamma/\mathfrak{m}_{\left[ x\right]})$. We will sometimes view $\left[ x\right]$ as an orbit in $X$.
\[prop:evaluation-factoring\] Suppose ${\mathfrak{K}}$ is an ideal of ${\mathfrak{M}}$ such that the quotient algebra ${\mathfrak{M}}/{\mathfrak{K}}$ is finite-dimensional and simple. Then there exists a point $x \in X_{\mathrm{rat}}$ such that the canonical epimorphism $\pi : {\mathfrak{M}}\to {\mathfrak{M}}/{\mathfrak{K}}$ factors through the evaluation map $\operatorname{ev}_x : {\mathfrak{M}}\to {\ensuremath{\mathfrak{g}}}^x$.
We have shown in Proposition \[prop:fixed-point-alg-eval\] that there exists a character $\chi : A^\Gamma \to k$ such that $\pi({\alpha}f) = \pi({\alpha}) \chi(f)$ holds for all ${\alpha}\in {\mathfrak{M}}$ and $f\in
A^\Gamma$. Let ${\mathfrak{n}}= \operatorname{Ker}\chi \in \operatorname{Spec}(A^\Gamma)$ be the corresponding $k$-rational point. Temporarily viewing ${\mathfrak{M}}$ as a $A^\Gamma$-module, it follows that ${\mathfrak{n}}$ annihilates ${\mathfrak{M}}/{\mathfrak{K}}$, whence ${\mathfrak{n}}{\mathfrak{M}}\subseteq {\mathfrak{K}}$ and we can factor $\pi$ through obvious maps: $$\xymatrix
{ {\mathfrak{M}}\ar@{->>}[r]^(0.4)\pi \ar@{->>}[d] & {\mathfrak{M}}/{\mathfrak{K}}\\
{\mathfrak{M}}/{\mathfrak{n}}{\mathfrak{M}}\ar@{->>}[ru] }$$ Observe $$\label{thm:fac1}
{\mathfrak{M}}/{\mathfrak{n}}{\mathfrak{M}}\cong {\mathfrak{M}}{\otimes}_{A^\Gamma} (A^\Gamma/{\mathfrak{n}})
= ({\ensuremath{\mathfrak{g}}}{\otimes}_k A)^\Gamma {\otimes}_{A^\Gamma} (A^\Gamma/{\mathfrak{n}})
= \big({\ensuremath{\mathfrak{g}}}{\otimes}_k A {\otimes}_{A^\Gamma} (A^\Gamma/{\mathfrak{n}}) \big)^\Gamma$$ where in the last equality we used that $\Gamma$ acts trivially on $A^\Gamma/{\mathfrak{n}}$. Also note, by Lemma \[lem:ideal-of-subring\], $$\label{thm:fac2}
A {\otimes}_{A^\Gamma} (A^\Gamma/{\mathfrak{n}}) \cong A / I \quad \hbox{for } I=A
{\mathfrak{n}}.$$ Hence, putting these canonical isomorphisms together, we get a new factorization: $$\xymatrix
{ {\mathfrak{M}}\ar@{->>}[r]^\pi \ar@{->>}[d] & {\mathfrak{M}}/{\mathfrak{K}}\\
{\mathfrak{M}}/{\mathfrak{n}}{\mathfrak{M}}\ar[r]^<<<<\cong &({\ensuremath{\mathfrak{g}}}{\otimes}_k \big(A/I)\big)^\Gamma
\ar@{->>}[u]_\psi }$$ It now remains to show that $\psi$ factors through $\operatorname{ev}_x$ for an appropriate $x\in X$. To this end, let $[{\mathfrak{n}}] = \{ {\mathfrak{m}}\in \operatorname{Spec}A
: {\mathfrak{m}}\cap A^\Gamma = {\mathfrak{n}}\}$. One knows ([@Bou:ACb V, §2]) that $[{\mathfrak{n}}]$ is a nonempty set of $k$-rational points on which $\Gamma$ acts transitively. Let $[{\mathfrak{n}}] = \{ {\mathfrak{m}}_1, \ldots, {\mathfrak{m}}_s\}$. We claim $$\label{thm:fac3}
\sqrt{I} = {\mathfrak{m}}_1 \cap \dots \cap {\mathfrak{m}}_s.$$ Indeed, any ${\mathfrak{p}}\in V(I) \subseteq \operatorname{Spec}A$ satisfies ${\mathfrak{p}}\cap
A^\Gamma = {\mathfrak{n}}$, so that $V(I) = [{\mathfrak{n}}]$. Now (\[thm:fac3\]) follows from [@Bou:ACa II, §2.6, Corollaire de la Proposition 13].
We have an exact sequence of algebras and $\Gamma$-modules $$\label{eq:pro1}
\textstyle
0 \to {\ensuremath{\mathfrak{g}}}{\otimes}_k \big((\bigcap_i {\mathfrak{m}}_i)/I \big) \to {\ensuremath{\mathfrak{g}}}{\otimes}_k (A/I)
\to {\ensuremath{\mathfrak{g}}}{\otimes}_k ( A /\bigcap_i {\mathfrak{m}}_i) \to 0$$ where $$\textstyle \label{eq:pro2}
{\ensuremath{\mathfrak{g}}}{\otimes}_k(A / \bigcap_i {\mathfrak{m}}_i)
\cong \bigoplus_i \big({\ensuremath{\mathfrak{g}}}{\otimes}_k (A/{\mathfrak{m}}_i)) $$ since $A/\bigcap_i {\mathfrak{m}}_i \cong \bigoplus_i A/{\mathfrak{m}}_i$. Now observe that the summands on the right hand side of (\[eq:pro2\]) are permuted by the action of $\Gamma$. Thus, if we fix $x={\mathfrak{m}}\in [
{\mathfrak{n}}]$ we have $$\textstyle
\big( {\ensuremath{\mathfrak{g}}}{\otimes}_k (A / \bigcap_i {\mathfrak{m}}_i ) \big)^\Gamma
\cong ( {\ensuremath{\mathfrak{g}}}{\otimes}_k (A/{\mathfrak{m}})^{\Gamma_x} = {\ensuremath{\mathfrak{g}}}^x.$$ Therefore, taking $\Gamma$-invariants in (\[eq:pro1\]) we get an epimorphism $$\zeta : \big( {\ensuremath{\mathfrak{g}}}{\otimes}_k (A/I)\big)^\Gamma
\twoheadrightarrow {\ensuremath{\mathfrak{g}}}^x$$ with kernel $\big({\ensuremath{\mathfrak{g}}}{\otimes}_k (\bigcap_i {\mathfrak{m}}_i)/I \big)^\Gamma$.
Any ${\alpha}\in \operatorname{Ker}\zeta$ is a finite sum ${\alpha}= \sum_j u_j {\otimes}\bar
f_j$ where every $\bar f_j\in (\bigcap_i {\mathfrak{m}}_i)/I$ is nilpotent by (\[thm:fac3\]). The ideal $J$ of $\big({{\ensuremath{\mathfrak{g}}}}{\otimes}_k
(A/I)\big)^\Gamma$ generated by ${\alpha}$ is therefore nilpotent. Since ${\mathfrak{M}}/{\mathfrak{K}}$ is simple, it does not contain a nonzero nilpotent ideal. Thus $\psi(J) = 0$. Therefore $\psi$ factors through $\zeta$ and we get the following commutative diagram: $$\xymatrix{
\mathfrak{M} \ar[rr]^\pi \ar@{->>}[d]_p && {\mathfrak{M}}/{\mathfrak{K}}\\
\big( {\ensuremath{\mathfrak{g}}}{\otimes}_k (A/I)\big)^\Gamma \ar@{->>}[rru]^\psi \ar@{->>}[rr]_\zeta && {\ensuremath{\mathfrak{g}}}^x \ar[u]
}$$ Since $\zeta \circ p = \operatorname{ev}_x$, the result follows.
\[rem:generalization\] Proposition \[prop:evaluation-factoring\] remains true for $k$ an arbitrary algebraically closed field whose characteristic does not divide the order of $\Gamma$ and for ${\ensuremath{\mathfrak{g}}}$ an arbitrary finite-dimensional (not necessarily Lie or associative) algebra.
\[cor:small-reps\]
1. \[cor-item1:small-reps\] Suppose $\varphi : \mathfrak{M} \to \operatorname{End}_k V$ is an irreducible finite-dimensional representation such that $\mathfrak{M}/\ker
\varphi$ is a semisimple Lie algebra. Then $\varphi$ is an irreducible finite-dimensional evaluation representation.
2. \[cor-item2:small-reps\] The irreducible finite-dimensional representations of ${\mathfrak{M}}$ are precisely the representations of the form ${\lambda}{\otimes}\varphi$ where ${\lambda}\in
({\mathfrak{M}}/[{\mathfrak{M}},{\mathfrak{M}}])^*$ and $\varphi$ is an irreducible finite-dimensional evaluation representation with ${\mathfrak{M}}/\operatorname{Ker}\varphi$ semisimple. The factors ${\lambda}$ and $\varphi$ are uniquely determined.
Part follows from Proposition \[prop:evaluation-factoring\] and the results of Section \[sec:review\]. Part is then a simple application of Lemma \[lem:deco\].
We now state our main theorem which gives a classification of the irreducible finite-dimensional representations of an arbitrary equivariant map algebra. Recall the definitions of $\mathcal{E}$ (Definition \[def:E\]) and ${\mathcal{S}}$ (Definition \[def:eval-rep\]).
\[thm:fd-reps=eval-reps+linear-form\] Suppose $\Gamma$ is a finite group acting on an affine scheme $X$ and a finite-dimensional Lie algebra ${\ensuremath{\mathfrak{g}}}$. Then the map $$({\mathfrak{M}}/[{\mathfrak{M}},{\mathfrak{M}}])^* \times \mathcal{E} \to {\mathcal{S}},\quad
(\lambda, \Psi) \mapsto \lambda \otimes \operatorname{ev}_\Psi,\qquad
\lambda \in ({\mathfrak{M}}/[{\mathfrak{M}},{\mathfrak{M}}])^*,\quad \Psi \in \mathcal{E},$$ is surjective. In particular, all irreducible finite-dimensional representations of ${\mathfrak{M}}$ are tensor products of an evaluation representation and a one-dimensional representation.
Furthermore, we have that $\lambda \otimes \operatorname{ev}_\Psi = \lambda'
\otimes \operatorname{ev}_{\Psi'}$ if and only if there exists $\Phi \in
\mathcal{E}$ such that $\dim \operatorname{ev}_\Phi=1$, $\lambda' = \lambda -
\operatorname{ev}_\Phi$ and $\operatorname{ev}_{\Psi'} = \operatorname{ev}_{\Psi \otimes \Phi}$. Here $\Psi
\otimes \Phi \in \mathcal{E}$ is given by $(\Psi \otimes \Phi)(x) =
\Psi(x) \otimes \Phi(x)$, where $\Psi(x)$ (respectively $\Phi(x)$) is the one-dimensional trivial representation if $x \not \in \operatorname{supp}\Psi$ (respectively $x \not \in \operatorname{supp}\Phi$). In particular, the restriction of the map $(\lambda, \Psi) \mapsto \lambda \otimes
\operatorname{ev}_{\Psi}$ to either factor (times the zero element of the other) is injective.
This follows from Corollary \[cor:small-reps\] and Lemma \[lem:deco\].
Since $\lambda, \mu \in ({\mathfrak{M}}/[{\mathfrak{M}},{\mathfrak{M}}])^*$ are isomorphic as representations if and only if they are equal as linear functions, in Theorem \[thm:fd-reps=eval-reps+linear-form\] we have identified elements of $({\mathfrak{M}}/[{\mathfrak{M}},{\mathfrak{M}}])^*$ with isomorphism classes of representations.
Note that the evaluation representations of $M(\operatorname{Spec}A,{\ensuremath{\mathfrak{g}}})^\Gamma$ are the same as the evaluation representations of $M(\operatorname{Spec}(A/\operatorname{rad}A),{\ensuremath{\mathfrak{g}}})^\Gamma$ since one evaluates at rational points. However, the one-dimensional representations of these two Lie algebras can be different in general. Thus, we do not assume that the scheme $X$ is reduced.
\[cor:perfectreps\] If ${\mathfrak{M}}$ is perfect, then the map $\Psi \mapsto \operatorname{ev}_\Psi$ is a bijection between $\mathcal{E}$ and ${\mathcal{S}}$. In particular, this is true if any one of the following conditions holds:
1. $[{\ensuremath{\mathfrak{g}}}^\Gamma,{\ensuremath{\mathfrak{g}}}] = {\ensuremath{\mathfrak{g}}}$, \[perfect-cond1\]
2. ${\ensuremath{\mathfrak{g}}}$ is simple, ${\ensuremath{\mathfrak{g}}}^\Gamma \ne \{0\}$ is perfect and acts on ${\ensuremath{\mathfrak{g}}}_\Gamma$ without a trivial non-zero submodule, or \[perfect-cond2\]
3. $\Gamma$ acts on ${\ensuremath{\mathfrak{g}}}$ by diagram automorphisms. \[perfect-cond3\]
If ${\mathfrak{M}}$ is perfect, then $[{\mathfrak{M}},{\mathfrak{M}}]={\mathfrak{M}}$ and the first statement follows immediately from Theorem \[thm:fd-reps=eval-reps+linear-form\]. Conditions or imply that ${\mathfrak{M}}$ is perfect by Lemma \[lem:perfrm\]. It remains to show that Condition implies that ${\ensuremath{\mathfrak{g}}}^\Gamma$ is perfect. It suffices to consider the case where ${\ensuremath{\mathfrak{g}}}$ is simple. If $\Gamma$ acts on ${\ensuremath{\mathfrak{g}}}$ by diagram automorphisms then there are two possibilities: either $\Gamma$ is a cyclic group generated by a single diagram automorphism or ${\ensuremath{\mathfrak{g}}}$ is of type $D_4$ and $\Gamma
\cong S_3$. If $\Gamma$ is generated by a single diagram automorphism, it is well known that ${\ensuremath{\mathfrak{g}}}^\Gamma$ is a simple Lie algebra and hence perfect (see [@Kac90 §8.2]). The case $\Gamma \cong S_3$ was described in Example \[eg:nonabelian\], where it was shown that ${\ensuremath{\mathfrak{g}}}^\Gamma$ is simple as well.
Note that the three conditions in Corollary \[cor:perfectreps\] depend only on the action of $\Gamma$ on ${\ensuremath{\mathfrak{g}}}$ and not on the scheme $X$ or its $\Gamma$-action.
\[rem:untwisted\] If $\Gamma$ is trivial (or, more generally, acts trivially on ${\ensuremath{\mathfrak{g}}}$), we have ${\ensuremath{\mathfrak{g}}}^\Gamma = {\ensuremath{\mathfrak{g}}}$. Thus the map $\mathcal{E} \to {\mathcal{S}}$, $\Psi \mapsto \operatorname{ev}_\Psi$, is a bijection if and only if ${\ensuremath{\mathfrak{g}}}$ is perfect. In the case when $\Gamma$ is trivial, ${\ensuremath{\mathfrak{g}}}$ is a finite-dimensional simple Lie algebra, and $A$ is finitely generated, a similar statement has recently been made in [@CFK].
\[comp-red\] Suppose that all irreducible finite-dimensional representations of ${\mathfrak{M}}$ are evaluation representations (e.g. ${\mathfrak{M}}$ is perfect), and that all ${\ensuremath{\mathfrak{g}}}^x$, $x\in X_{\mathrm{rat}}$, are semisimple. Then a finite-dimensional ${\mathfrak{M}}$-module $V$ is completely reducible if and only if there exists ${\mathbf{x}}\in X_n$ for some $n\in {\Bbb N}$, $n>0$, such that $\operatorname{Ker}\operatorname{ev}_{\mathbf{x}}\subseteq \operatorname{Ann}_{\mathfrak{M}}V := \{\alpha \in {\mathfrak{M}}:
\alpha \cdot v =0 \ \forall\ v \in V\}$.
For the case of the current algebra ${\mathfrak{M}}={\ensuremath{\mathfrak{g}}}{\otimes}{\mathbb{C}}[t]$, this corollary is proven in [@cha-gre Prop. 3.9(iii)].
Let $V$ be a completely reducible ${\mathfrak{M}}$-module, hence a finite direct sum of irreducible finite-dimensional representations $V^{(i)}$. By assumption, every $V^{(i)}$ is an evaluation representation, given by some ${\mathbf{x}}^{(i)} \in X_{n_i}$. Then $\operatorname{Ann}_{\mathfrak{M}}V = \bigcap_i
\operatorname{Ann}_{\mathfrak{M}}V^{(i)} \supseteq \bigcap_i \operatorname{Ker}\operatorname{ev}_{{\mathbf{x}}^{(i)}}= \operatorname{Ker}\operatorname{ev}_{\mathbf{y}}$ for ${\mathbf{y}} = \bigcup_i {\mathbf{x}}^{(i)}$. Since $\operatorname{Ker}\operatorname{ev}_x=\operatorname{Ker}\operatorname{ev}_{g\cdot x}$, we can replace ${\mathbf{y}}$ by some ${\mathbf{x}}\in X_n$ satisfying $\operatorname{Ker}\operatorname{ev}_{\mathbf{y}} = \operatorname{Ker}\operatorname{ev}_{\mathbf{x}}$.
Conversely, if $\operatorname{Ker}\operatorname{ev}_{\mathbf{x}}\subseteq \operatorname{Ann}_{\mathfrak{M}}(V)$, then the representation of ${\mathfrak{M}}$ on $V$ factors through the semisimple Lie algebra $\bigoplus_{x\in {\mathbf{x}}} {\ensuremath{\mathfrak{g}}}^x$ and is therefore completely reducible.
While Theorem \[thm:fd-reps=eval-reps+linear-form\] classifies all the irreducible finite-dimensional representations of an arbitrary equivariant map algebra, in case ${\mathfrak{M}}$ is not perfect it leaves open the possibility that not all irreducible finite-dimensional representations are evaluation representations. We see that ${\mathfrak{M}}$ has irreducible finite-dimensional representations that are not evaluation representations precisely when it has one-dimensional representations that are not evaluation representations. We therefore turn our attention to one-dimensional evaluation representations. Let $$\tilde X = \{x \in X_{\mathrm{rat}}: [{\ensuremath{\mathfrak{g}}}^x, {\ensuremath{\mathfrak{g}}}^x] \ne {\ensuremath{\mathfrak{g}}}^x\}.$$ Note that $\tilde X$ is a $\Gamma$-invariant subset of $X$ (i.e. $\tilde X$ is a union of $\Gamma$-orbits).
\[lem:one-dim-evals\] If $\operatorname{ev}_\Psi$ is (the isomorphism class of) a one-dimensional representation, then $\operatorname{supp}\Psi \subseteq \tilde X$.
This follows easily from the fact that for $x \in X \setminus \tilde
X$, we have that ${\ensuremath{\mathfrak{g}}}^x$ is perfect and thus the one-dimensional representations of ${\ensuremath{\mathfrak{g}}}^x$ are trivial.
Let $${\mathfrak{M}}^d = \{\alpha \in {\mathfrak{M}}: \alpha(x) \in [{\ensuremath{\mathfrak{g}}}^x,{\ensuremath{\mathfrak{g}}}^x]\ \forall\ x
\in X_{\mathrm{rat}}\} = \{\alpha \in {\mathfrak{M}}: \alpha(x) \in [{\ensuremath{\mathfrak{g}}}^x,{\ensuremath{\mathfrak{g}}}^x]\ \forall\
x \in \tilde X\}.$$ Then it is easy to see that $[{\mathfrak{M}},{\mathfrak{M}}] \subseteq {\mathfrak{M}}^d$. The proof of the following lemma is straightforward.
The Lie algebra ${\mathfrak{M}}$ is perfect if and only if ${\mathfrak{M}}^d =
[{\mathfrak{M}},{\mathfrak{M}}]$ and $\tilde X = \emptyset$.
Now assume that $|\tilde X| < \infty$. Let $\mathbf{x}$ be a set of representatives of the $\Gamma$-orbits comprising $\tilde X$ and consider the composition $$\label{eq:tildeX-composition}
\xymatrix{
{\mathfrak{M}}\ar@{->>}[r]^(.35){\operatorname{ev}_{\mathbf{x}}} & \bigoplus_{x \in \mathbf{x}}
{\ensuremath{\mathfrak{g}}}^x \ar@{->>}[r]^(.3){\pi} & \bigoplus_{x \in \mathbf{x}}
{\mathfrak{z}}^x,\quad {\mathfrak{z}}^x := {\ensuremath{\mathfrak{g}}}^x/[{\ensuremath{\mathfrak{g}}}^x,{\ensuremath{\mathfrak{g}}}^x],
}$$ where the $x$-component of $\pi$ is the canonical projection ${\ensuremath{\mathfrak{g}}}^x
\to {\ensuremath{\mathfrak{g}}}^x/[{\ensuremath{\mathfrak{g}}}^x,{\ensuremath{\mathfrak{g}}}^x]$. If ${\ensuremath{\mathfrak{g}}}$ is reductive, then so is every ${\ensuremath{\mathfrak{g}}}^x$ and we can identify ${\mathfrak{z}}^x$ with the center $Z({\ensuremath{\mathfrak{g}}}^x)$ of ${\ensuremath{\mathfrak{g}}}^x$. However, we will not assume that ${\ensuremath{\mathfrak{g}}}$ is reductive. The kernel of is precisely ${\mathfrak{M}}^d$ and thus the composition factors through ${\mathfrak{M}}/[{\mathfrak{M}}, {\mathfrak{M}}]$, yielding the following commutative diagram: $$\label{eq:center-surjection}
\xymatrix{
{\mathfrak{M}}\ar@{->>}[r]^(.35){\operatorname{ev}_{\mathbf{x}}} \ar@{->>}[dr] & \bigoplus_{x \in \mathbf{x}}
{\ensuremath{\mathfrak{g}}}^x \ar@{->>}[r]^{\pi} & \bigoplus_{x \in \mathbf{x}}
{\mathfrak{z}}^x \\
& {\mathfrak{M}}/[{\mathfrak{M}},{\mathfrak{M}}] \ar@{->>}[ur]_{\gamma} &
}$$ We then have an isomorphism of vector spaces $$\label{eq:linear-form-decomp} \textstyle
({\mathfrak{M}}/[{\mathfrak{M}},{\mathfrak{M}}])^* \cong (\ker \gamma)^* \oplus \left(\bigoplus_{x \in \mathbf{x}}
{\mathfrak{z}}^x \right)^*.$$
\[prop:one-dim-eval-reps\] If $|\tilde X| < \infty$ and $\mathbf{x}$ is a set of representatives of the $\Gamma$-orbits comprising $\tilde X$, then there is a natural identification $$\textstyle
\left(\bigoplus_{x \in \mathbf{x}} {\mathfrak{z}}^x \right)^* \cong \{\operatorname{ev}_\Psi
: \Psi \in \mathcal{E},\ \dim \operatorname{ev}_\Psi = 1 \}.$$
Choose $\lambda \in \left(\bigoplus_{x \in \mathbf{x}} {\mathfrak{z}}^x
\right)^*$. To $\lambda$ we associate the evaluation representation $$\xymatrix{
{\mathfrak{M}}\ar@{->>}[r]^(.35){\operatorname{ev}_{\mathbf{x}}} & \bigoplus_{x \in \mathbf{x}}
{\ensuremath{\mathfrak{g}}}^x \ar@{->>}[r]^(.47){\pi} & \bigoplus_{x \in \mathbf{x}}
{\mathfrak{z}}^x \ar[r]^(.6){\lambda} & k.
}$$ By Lemma \[lem:one-dim-evals\], this gives the desired bijective correspondence.
We can now refine Theorem \[thm:fd-reps=eval-reps+linear-form\] as follows.
\[thm:finite-tilde-X\] Suppose $\Gamma$ is a finite group acting on an affine scheme $X$ and a finite-dimensional Lie algebra ${\ensuremath{\mathfrak{g}}}$ and assume that $|\tilde
X| < \infty$. If $\gamma$ is defined as in , then the map $$(\lambda, \Psi) \mapsto \lambda \otimes \operatorname{ev}_\Psi,\quad \lambda \in
(\ker \gamma)^*,\quad \Psi \in \mathcal{E}$$ is a bijection between $(\ker \gamma)^* \times \mathcal{E}$ and ${\mathcal{S}}$.
This follows from Theorem \[thm:fd-reps=eval-reps+linear-form\], and Proposition \[prop:one-dim-eval-reps\].
\[cor:non-ss-reps\] Assume $|\tilde X| < \infty$. Then $[{\mathfrak{M}},{\mathfrak{M}}]={\mathfrak{M}}^d$ if and only if all irreducible finite-dimensional representations are evaluation representations.
By Theorem \[thm:finite-tilde-X\], all irreducible finite-dimensional representations are evaluation representations if and only if $\gamma$ is injective (and hence an isomorphism, since it is surjective). Then the result follows from the commutative diagram since the kernel of is ${\mathfrak{M}}^d$.
Note that if ${\ensuremath{\mathfrak{g}}}$ is perfect and $\Gamma$ acts on $X$ in such a way that there are only a finite number of points of $X$ that have a non-trivial stabilizer, then $|\tilde X| < \infty$ and so the hypotheses of Theorem \[thm:finite-tilde-X\] are satisfied.
In Section \[subsec:onsager\] we will see that the Onsager algebra is an equivariant map algebra which is not perfect but for which $\gamma$ is injective and thus all irreducible finite-dimensional representations are nonetheless evaluation representations.
Having considered the case when all irreducible finite-dimensional representations are evaluation representations, we now examine the opposite situation: equivariant map algebras for which there exist irreducible finite-dimensional representations that are not evaluation representations.
\[prop:infinite-tildeX\] Suppose $X$ is a Noetherian affine scheme and $\tilde X$ is infinite. Then ${\mathfrak{M}}$ has a one-dimensional representation that is not an evaluation representation.
We first set up some notation for one-dimensional evaluation representations. Let ${\mathbf{x}}\in X_n$ and let $\rho_x : {{\ensuremath{\mathfrak{g}}}}^x \to
\operatorname{End}_k(V_x)$, $x \in {\mathbf{x}}$, be representations such that $\operatorname{ev}_{\mathbf{x}}(\rho_x)_{x \in {\mathbf{x}}}$ is a one-dimensional representation. Necessarily $\dim V_x = 1$, say $V_x = k v_x$, so $V= \bigotimes_{x
\in {\mathbf{x}}} V_x = k {\mathbf{v}}$ for ${\mathbf{v}}= \bigotimes_{x \in {\mathbf{x}}} v_x$. We can assume that all $\rho_x \ne 0$, whence ${\mathbf{x}}\subseteq \tilde X$. Let $\tilde \rho_x \in ({\ensuremath{\mathfrak{g}}}^x)^*$ be defined by $\rho_x(u) (v_x) = \tilde
\rho_x(u) v_x$ for $u\in {\ensuremath{\mathfrak{g}}}^x$. For ${\alpha}= \sum_i u_i {\otimes}f_i\in
{\mathfrak{M}}$ we then have $$\big( (\operatorname{ev}_{\mathbf{x}}(\rho_x)_{x \in {\mathbf{x}}})({\alpha})\big) ({\mathbf{v}})
= \left(\sum_{x \in {\mathbf{x}},\, i} \tilde \rho_x(u_i) f_i(x)\right) {\mathbf{v}}.$$ Suppose that there exist ${\alpha}\in {\mathfrak{M}}$ and $f \in A^\Gamma$ such that $$\{ {\alpha}f^m : m\in {\Bbb N}\} \mbox{ is linearly independent and }
\operatorname{Span}\{ {\alpha}f^m : m\in {\Bbb N}\} \cap [{\mathfrak{M}},{\mathfrak{M}}] =
0. \eqno{(*)}$$ Then there exists ${\lambda}\in {\mathfrak{M}}^*$ such that ${\lambda}([{\mathfrak{M}}, {\mathfrak{M}}])=0$ and ${\lambda}({\alpha}f^m)\in k$, $m \ge 1$, is a root of an irreducible rational polynomial $p_m$ of degree $m$, for example $p_m(z) = z^m
-2$. Now suppose that the corresponding one-dimensional representation is an evaluation representation. Writing ${\alpha}=\sum_i
u_i {\otimes}f_i$ with $u_i \in {\ensuremath{\mathfrak{g}}}$ and $f_i \in A$, we get from the equation above $${\lambda}({\alpha}f^m) = \sum_{x \in {\mathbf{x}},\, i} \tilde \rho_x(u_i) (f_i f^m)(x)
= \sum_{x \in {\mathbf{x}},\, i} \tilde \rho_x(u_i) f_i(x) f(x)^m.$$ But this is a contradiction since the elements $\tilde \rho_x(u_i)
f_i(x) f(x)^m\in k$ all lie in the $\mathbb{Q}$-subalgebra of $k$ generated by the finitely many elements $\tilde \rho_x(u_i), f_i(x),
f(x)$ of $k$, while the elements ${\lambda}(\alpha f^m)$, $m\in {\Bbb N}$, do not lie in such a subalgebra.
We will now construct ${\alpha}\in {\mathfrak{M}}$ and $f\in A^\Gamma$ satisfying (\*). For a subgroup $H$ of $\Gamma$, let $$X_H = \{x \in X_{\mathrm{rat}}: \Gamma_x = H\}.$$ Since $$\tilde X = \bigcup_{H\, :\, {\ensuremath{\mathfrak{g}}}^H \ne [{\ensuremath{\mathfrak{g}}}^H,{\ensuremath{\mathfrak{g}}}^H]} X_H,$$ there exists a subgroup $H$ of $\Gamma$, with ${\ensuremath{\mathfrak{g}}}^H \ne
[{\ensuremath{\mathfrak{g}}}^H,{\ensuremath{\mathfrak{g}}}^H]$, such that $X_H$ is infinite. Observe that $X_H = X^H
\setminus \bigcup_{K\supsetneq H} X^K$ is open in the closed subset $X^H$ and that $X^H$ is a Noetherian affine scheme since $X$ is so. Because $X^H$ has only finitely many irreducible components, there exists an irreducible component $Y$ of $X^H$ such that $\tilde Y = Y
\cap X_H$ is infinite.
Let $R : A_X^\Gamma \to A_Y$ be the restriction map. Choose an infinite set $\{y_1, y_2, \dots\}$ of points of $Y$, no two of which are in the same $\Gamma$-orbit. It follows as in the proof of Corollary \[cor:twisted-surjectivity\] that for all $j \in
\mathbb{N}$ there exists $f_j \in A_X^\Gamma$ such that $f_j(y_j)\ne
0$ and $f_j(y_i) = 0$ for $i<j$. Since the set $\{R(f_j) : j\in
{\Bbb N}\}$ is linearly independent, the image of $R$ is infinite-dimensional. Because $A_X^\Gamma$ is finitely generated, so is the image of $R$. Therefore, by the Noether normalization lemma, this image contains an element $f_Y$ such that the set $\{1,f_Y,f_Y^2,\dots,\}$ is linearly independent. Choose $f \in
R^{-1}(f_Y)$. It follows that $\{1,f,f^2,\dots,\}$ is also linearly independent. Since $\tilde Y$ is open in the irreducible $Y$ we can choose $\tilde y \in \tilde Y$ such that $f(\tilde y) \ne 0$.
Let $\{u_i\}_{i=1}^l$ be a basis of $[{\ensuremath{\mathfrak{g}}}^H, {\ensuremath{\mathfrak{g}}}^H]$ and complete it to a basis $\{u_i\}_{i=1}^m$ of ${\ensuremath{\mathfrak{g}}}^H$. Then complete this to a basis $\{u_i\}_{i=1}^n$ of ${\ensuremath{\mathfrak{g}}}$. Again by Corollary \[cor:twisted-surjectivity\] there exists ${\alpha}\in {\mathfrak{M}}$ such that ${\alpha}(\tilde y) = u_m$. Observe that ${\alpha}_Y = {\alpha}|_Y$ can be written in the form ${\alpha}_Y = \sum_{i=1}^m u_i \otimes f_i$ for some $f_i \in A_Y$. Since ${\alpha}(\tilde y) = u_m$, we have $f_m(\tilde y) = 1$. Therefore $f_m(y) \ne 0$ for all $y$ in some dense subset of $Y$. Now suppose $$\sum_{j=0}^\infty d_j \alpha f^j \in [{\mathfrak{M}},{\mathfrak{M}}]$$ for some $d_j \in k$ with $d_j=0$ for all but finitely many $j$. We then have $$\left. \left( \sum_{j=0}^\infty d_j \alpha f^j \right) \right|_Y
= \alpha_Y \sum_{j=0}^\infty d_j f_Y^j = \left( \sum_{i=1}^m u_i
\otimes f_i \right) \sum_{j=0}^\infty d_j f_Y^j = \sum_{i=1}^m u_i
\otimes \left( f_i \sum_{j=0}^\infty d_j f_Y^j \right).$$ Since $f_m(y)\ne 0$ for $y$ in a dense subset of $Y$, we must have $\sum_{j=0}^\infty d_j f_Y^j=0$. Because $\{1, f_Y, f_Y^2,\dots\}$ is linearly independent, we must have $d_j = 0$ for all $j$. Therefore $$\operatorname{Span}\{ {\alpha}f^m : m\in {\Bbb N}\} \cap [{\mathfrak{M}},{\mathfrak{M}}] =
0.$$ Now observe that the preceding argument also shows that $\{\alpha,
\alpha f, \alpha f^2, \dots\}$ is linearly independent: Suppose $\sum_{j=1}^\infty c_j \alpha f^j=0$ for some $c_j \in k$ with $c_j
= 0$ for all but finitely many $j$. Then, since $0\in [{\mathfrak{M}}, {\mathfrak{M}}]$, all $c_j = 0$ by what we have just shown. Thus (\*) holds, finishing the proof of the proposition.
\[ex:infinite-tildeX\] We give an example in which the assumptions of Proposition \[prop:infinite-tildeX\] are fulfilled. Let $\Gamma=\{1, {\sigma}\}$ be the group of order two acting on ${\ensuremath{\mathfrak{g}}}={\ensuremath{\mathfrak{sl}}}_2(k)$ by the Chevalley involution with respect to some ${\ensuremath{\mathfrak{sl}}}_2$-triple and let $X$ be the affine space $k^2$ with ${\sigma}$ acting on $X$ by fixing the first coordinate of points in $X$ while multiplying the second by $-1$. For points $(x_1,x_2) \in X$ with $x_2 \ne 0$, the isotropy subalgebra ${\ensuremath{\mathfrak{g}}}^x = {\ensuremath{\mathfrak{g}}}$, while for $(x_1,0)
\in X$, the subalgebra ${\ensuremath{\mathfrak{g}}}^x$ is the fixed point subalgebra of $\sigma$, which is one-dimensional. Therefore $\tilde X = \{(x_1,0)
\in X\}$ is infinite.
Proposition \[prop:infinite-tildeX\] says that when $X$ is an affine variety, a necessary condition for all irreducible finite-dimensional representations to be evaluation representations is that $\tilde X$ be finite. We now show that this condition is not sufficient.
Let ${\ensuremath{\mathfrak{g}}}= \mathfrak{sl}_2(k)$ and $$X = Z(y^2-x^3) =\{(y,x) : y^2=x^3\} \subseteq k^2,$$ an affine variety. Then $A = k[y,x]/(y^2-x^3)$. Let $\Gamma =
\langle \sigma \rangle = {\mathbb{Z}}_2$ act on $k^2$ by $\sigma \cdot y = -y$ and $\sigma \cdot x = x$. Since this action fixes $y^2-x^3$, we have an induced action of $\Gamma$ on $X$ and the only fixed point is the origin. In particular, $\tilde X$ only contains the origin and thus is finite. We let $\sigma$ act on ${\ensuremath{\mathfrak{g}}}$ by a Chevalley involution. We have $$A_1 = y k[y^2,x]/(y^2-x^3),$$ and so $$A_1^2 = y^2 k[y^2,x]/(y^2-x^3) \cong x^3 k[x],$$ while $$A_0 = A^\Gamma = k[y^2,x]/(y^2-x^3) \cong k[x].$$ Recall that ${\ensuremath{\mathfrak{g}}}_0$ is one-dimensional. Then $$[{\mathfrak{M}},{\mathfrak{M}}] = ([{\ensuremath{\mathfrak{g}}}_0,{\ensuremath{\mathfrak{g}}}_0]\otimes A_0 + {\ensuremath{\mathfrak{g}}}_0 \otimes
A_1^2) \oplus ({\ensuremath{\mathfrak{g}}}_1 \otimes A_1) = \left( {\ensuremath{\mathfrak{g}}}_0 \otimes x^3
k[x] \right) \oplus \left( {\ensuremath{\mathfrak{g}}}_1 \otimes A_1 \right),$$ while $${\mathfrak{M}}^d = \left( {\ensuremath{\mathfrak{g}}}_0 \otimes x k[x] \right) \oplus \left(
{\ensuremath{\mathfrak{g}}}_1 \otimes A_1 \right).$$ Therefore $${\mathfrak{M}}^d/[{\mathfrak{M}},{\mathfrak{M}}] \cong x k[x]/(x^3)$$ and by Theorem \[thm:finite-tilde-X\], ${\mathfrak{M}}$ has irreducible finite-dimensional representations that are not evaluation representations.
Applications {#sec:applications}
============
In this section we use our classification to describe the irreducible finite-dimensional representations of certain equivariant map algebras. The classification of these representations for the multiloop, tetrahedron, or Onsager algebra ${\EuScript{O}}(\mathfrak{sl}_2)$ by the results obtained in Section \[sec:classification\] provide a simplified and unified interpretation of results previously obtained. For example, the classification of the irreducible finite-dimensional representations of $L^\sigma({\mathfrak{g}})$ (as found in [@CPweyl] and [@CFS]) via Drinfeld polynomials requires two distinct treatments for the untwisted ($\sigma = \operatorname{Id}$) and twisted ($\sigma \neq \operatorname{Id}$) cases – and for these twisted cases, the twisted loops $L^\sigma({\mathfrak{g}})$ with ${\mathfrak{g}}$ of type $A_{2n}$ require special attention. The classification resulting from our approach, however, is uniform. As we will see, the identification of isomorphism classes of representations with equivariant maps $X_{\mathrm{rat}}\to \mathcal{R}_X$ also provides a simple explanation for many of the technical conditions appearing in previous classifications.
We first note that from Remark \[rem:untwisted\] we immediately obtain the classification of irreducible finite-dimensional representations of the current algebras (Example \[eg:current\]), of the untwisted loop and multiloop algebras (Example \[eg:multiloop\]), and of the $n$-point algebras $M(X,{\ensuremath{\mathfrak{g}}})$, where $X = \mathbb{P}^1 \backslash \left\{ c_1, \ldots,
c_n \right\}$. This includes the tetrahedron algebra, which is isomorphic to the three-point $\mathfrak{sl}_2$ loop algebra $\mathfrak{sl}_2 \otimes \mathbb{C}\left[ t, t^{-1},
(t-1)^{-1}\right]$ (Example \[eg:threepoint\]), and we recover the classification found in [@Ha]. In particular, we easily recover in all of these cases the fact that for ${\mathbf{x}}= \{x_1, \ldots, x_l\}
\subseteq X$ and irreducible representations $V_1, \ldots, V_l$ of ${\ensuremath{\mathfrak{g}}}$, the evaluation representation $\operatorname{ev}_{{\mathbf{x}}}(\otimes V_i)$ is irreducible if and only if $x_i \neq x_j$ for $i \neq j$.
Multiloop algebras
------------------
If ${\mathfrak{M}}= M({\ensuremath{\mathfrak{g}}},\sigma_1,\dots,\sigma_n,m_1,\dots,m_n)$ is a multiloop algebra (Example \[eg:twistedmultiloop\]), then ${\mathfrak{M}}$ is perfect (${\mathfrak{M}}$ is an iterated loop algebra; see [[@abp Lemma 4.9]]{}). Therefore, by Corollary \[cor:perfectreps\] we have the following classification:
\[cor:multiloop-reps\] The map $\mathcal{E} \to {\mathcal{S}}$, $\Psi \mapsto \operatorname{ev}_\Psi$, is a bijection. In particular, all irreducible finite-dimensional representations are evaluation representations.
The irreducible finite-dimensional representations of an arbitrary multiloop algebra have been discussed in [@Bat] and [@Lau]. With Corollary \[cor:multiloop-reps\] we recover the recent results in [@Lau], which subsume all previous classifications of irreducible finite-dimensional representations of loop algebras. We note that these previous classifications involved some rather complicated algebraic conditions on points of evaluation (see, for example, [@Lau Theorem 5.7]). However, in the approach of Theorem \[thm:fd-reps=eval-reps+linear-form\], such conditions are not necessary. In fact, we see that the presence of these algebraic conditions arises from the description of evaluation representations in terms of individual points rather than as equivariant maps (i.e. elements of $\mathcal{E}$). For instance, if ${\mathfrak{M}}=
M({\ensuremath{\mathfrak{g}}},\sigma_1,\dots,\sigma_n,m_1,\dots,m_n)$ is an arbitrary multiloop algebra and $\Psi \in \mathcal{E}$, then $\operatorname{ev}_{\Psi}$ is the isomorphism class $\operatorname{ev}_{{\mathbf{x}}}(\Psi(x_i))_{i=1}^l$ where ${\mathbf{x}}= \{x_1,\dots,x_l\} \in
X_l$ contains one element from each $\Gamma$-orbit in $\operatorname{supp}\Psi$ and this class is independent of the choice of ${\mathbf{x}}$ (Definition \[def:evaluation-rep\]). It is immediate that such an ${\mathbf{x}}$ must satisfy the condition that $x_1^m, \ldots, x_l^m$ (where $x_i^m =
(x_{i1}^{m_1}, \dots, x_{in}^{m_n})$ for $x_i=(x_{i1}, \dots,
x_{in}$)) are pairwise distinct in $(k^\times)^n$. We therefore recover the conditions on the points $x_i$ found in [@Lau Theorem 5.7] which are necessary and sufficient for $\operatorname{ev}_{x_1, \ldots, x_l}(\otimes V_i)$ to be irreducible. The other conditions found there are similarly explained.
Connections to Drinfeld polynomials
-----------------------------------
In [@CPweyl], [@CM], [@CFS] and [@S], the isomorphism classes of irreducible finite-dimensional representations of loop algebras $L({\ensuremath{\mathfrak{g}}})$, $L^\sigma({\ensuremath{\mathfrak{g}}})$ are parameterized by certain collections of polynomials, sometimes referred to as *Drinfeld polynomials*. We explain here the relationship between this parametrization and ours. For better comparison with the existing literature, we assume in this subsection that $k=\mathbb{C}$.
Denote by ${\mathcal{P}}$ the set of of $n$-tuples of polynomials with constant term 1: $${\mathcal{P}}= \left\{ {{\mbox {\boldmath $\pi$}}}= \left( \pi_1(u), \ldots, \pi_n(u) \right) :
\pi_i \in \mathbb{C}\left[ u \right], \; \; \pi_i(0) = 1 \right\}.$$ Then the set ${\mathcal{P}}$ is in bijective correspondence with the isomorphism classes of irreducible finite-dimensional representations of $L({\ensuremath{\mathfrak{g}}})$ ([@CPweyl Proposition 2.1]); to the element ${{\mbox {\boldmath $\pi$}}}\in {\mathcal{P}}$ we associate an irreducible representation $V({{\mbox {\boldmath $\pi$}}})$ (the construction of $V({{\mbox {\boldmath $\pi$}}})$ is given in [@CPweyl]). We describe this correspondence.
Fix a simple finite-dimensional Lie algebra ${\ensuremath{\mathfrak{g}}}$, denote by $n$ its rank, and fix a Cartan decomposition ${\ensuremath{\mathfrak{g}}}= {\mathfrak{n}}^- \oplus {\mathfrak{h}}
\oplus {\mathfrak{n}}^+$ with Cartan subalgebra ${\mathfrak{h}} = \oplus_{i=1}^n
\mathbb{C} h_i \subseteq {\ensuremath{\mathfrak{g}}}$, and weight lattice $P = \oplus_{i=1}^n
\mathbb{Z} \omega_i$, with fundamental weights $\left\{ \omega_i
\right\}_{i=1}^n$, $\omega_i(h_j) = \delta_{ij}$.
Let ${{\mbox {\boldmath $\pi$}}}= (\pi_1, \ldots, \pi_n) \in {\mathcal{P}}$, and $\left\{ x_i
\right\}_{i=1}^l = \bigcup_{j=1}^n \left\{ z \in \mathbb{C}^\times :
\pi_j(z^{-1}) = 0 \right\}$. Then each $\pi_j$ can be written uniquely in the form $$\pi_j(u) = \prod_{i=1}^l(1 - x_i u)^{N_{ij}}, \; \; \; N_{ij} \in \mathbb{N}.$$ Let $\mathbf{x} = \{x_1, \ldots, x_l\}$. For $i = 1, \ldots, l$, define $\lambda_i \in P^+$ by $\lambda_i(h_j) = N_{ij}$, and let $\rho_i : {\ensuremath{\mathfrak{g}}}\rightarrow \operatorname{End}_k(V(\lambda_i))$ be the corresponding irreducible finite-dimensional representation of ${\ensuremath{\mathfrak{g}}}$. Then $V({{\mbox {\boldmath $\pi$}}})$ is isomorphic as an $L({\ensuremath{\mathfrak{g}}})$-module to the evaluation representation $$\textstyle
\operatorname{ev}_{\mathbf{x}}(\rho_i)_{i=1}^l : L({\ensuremath{\mathfrak{g}}}) \stackrel{\operatorname{ev}_{\mathbf{x}}}{\longrightarrow}
{\ensuremath{\mathfrak{g}}}^{\oplus l} \xrightarrow{\bigotimes_{i=1}^l \rho_i}
\operatorname{End}_k \left( \bigotimes_{i=1}^l V_i \right).$$ To produce an element ${{\mbox {\boldmath $\pi$}}}\in {\mathcal{P}}$ from an irreducible representation $V$ of $L({\ensuremath{\mathfrak{g}}})$, we first find an evaluation representation $\operatorname{ev}_{\mathbf{x}}(\rho_i)_{i=1}^l : L({\ensuremath{\mathfrak{g}}})
\longrightarrow \operatorname{End}_k( \otimes_{i=1}^l V(\lambda_i))$ isomorphic to $V$ ([@R Theorem 2.14] or Corollary \[cor:multiloop-reps\]). Next, for $i = 1, \ldots, l$, we define elements $ {{\mbox {\boldmath $\pi$}}}_{\lambda_i,
x_i} \in {\mathcal{P}}$ by $${{\mbox {\boldmath $\pi$}}}_{\lambda_i, x_i} = \left( (1 - x_i u)^{\lambda_i(h_1)}, \ldots,
(1 - x_i u)^{\lambda_i(h_n)} \right),$$ and define ${{\mbox {\boldmath $\pi$}}}=
\prod_{i=1}^l {{\mbox {\boldmath $\pi$}}}_{\lambda_i, x_i}$, where multiplication of $n$-tuples of polynomials occurs componentwise.
Given an element ${{\mbox {\boldmath $\pi$}}}\in \mathcal{P}$, we can uniquely decompose ${{\mbox {\boldmath $\pi$}}}= \prod_{i=1}^l {{\mbox {\boldmath $\pi$}}}_{\lambda_i, x_i}$, $x_i \neq x_j$, and we define $$\Psi_{{{\mbox {\boldmath $\pi$}}}}
:= \left\{ x_i \mapsto \left[ V(\lambda_i) \right] \right\} \in
\mathcal{E}.$$ Then $V({{\mbox {\boldmath $\pi$}}})$ is a representative of $\operatorname{ev}_{\Psi_{{{\mbox {\boldmath $\pi$}}}}}$.
In [@CFS], there is a similar parametrization of the irreducible finite-dimensional representations of $L^\sigma({\ensuremath{\mathfrak{g}}})$, where $\sigma$ is a nontrivial diagram automorphism of ${\ensuremath{\mathfrak{g}}}$, but in this case the bijective correspondence is between isomorphism classes of irreducible finite-dimensional $L^\sigma({\ensuremath{\mathfrak{g}}})$-modules and the set $\mathcal{P}^\sigma$ of $m$-tuples of polynomials ${\mbox {\boldmath
$\pi$}^\sigma} = (\pi_1, \ldots, \pi_m)$, $\pi_i(0)=1$, where $m$ is the rank of the fixed-point subalgebra ${\ensuremath{\mathfrak{g}}}_0 \subseteq {\ensuremath{\mathfrak{g}}}$. One feature of this classification is the fact that every irreducible finite-dimensional $L^\sigma({\ensuremath{\mathfrak{g}}})$-module is the restriction of an irreducible finite-dimensional $L({\ensuremath{\mathfrak{g}}})$-module (see [@CFS Theorem 2]). This fact follows immediately from Proposition \[prop:lifting\] once we note that in the setup of multiloop algebras, the action of $\Gamma$ on $X={\mathbb{C}}^\times$ is via multiplication by roots of unity and hence is free. Thus ${\ensuremath{\mathfrak{g}}}^{x} =
{\ensuremath{\mathfrak{g}}}^{\Gamma_x} = {\ensuremath{\mathfrak{g}}}^{\left\{ \operatorname{Id}\right\}} = {\ensuremath{\mathfrak{g}}}$ for all $x \in X$. Of course, the approach of the current paper yields an enumeration by elements of $\mathcal{E}$. The induced identification of $\mathcal{E}$ with $\mathcal{P}^\sigma$ is somewhat technical and will not be described here, but can found in [@Ssurvey].
The generalized Onsager algebra {#subsec:onsager}
-------------------------------
Our results also provide a classification of the irreducible finite-dimensional representations of the generalized Onsager algebra ${\EuScript{O}}({\ensuremath{\mathfrak{g}}})$ introduced in Example \[eg:onsager\] (in fact, for the more general equivariant map algebra of Example \[eg:general-involution\]). For ${\ensuremath{\mathfrak{g}}}\ne {\ensuremath{\mathfrak{sl}}}_2$ this classification was previously unknown.
\[prop:onsager-reps\] Let ${\ensuremath{\mathfrak{g}}}$ be a simple Lie algebra, $X=\operatorname{Spec}k[t^{\pm 1}]$, and $\Gamma=\{1, \sigma\}$ be a group of order two. Suppose $\sigma$ acts on $X$ by $\sigma \cdot x = x^{-1}$, $x \in X$, and on ${\ensuremath{\mathfrak{g}}}$ by an automorphism of order two. Then the map $\mathcal{E} \to {\mathcal{S}}$, $\Psi \mapsto \operatorname{ev}_\Psi$, is a bijection. In particular, all irreducible finite-dimensional representations are evaluation representations. In particular, this is true for the generalized Onsager algebra ${\EuScript{O}}({\ensuremath{\mathfrak{g}}})$.
Recall that ${\ensuremath{\mathfrak{g}}}_0 = {\ensuremath{\mathfrak{g}}}^\Gamma$ is either semisimple or has a one-dimensional center. In the case when ${\ensuremath{\mathfrak{g}}}_0$ is semisimple, the result follows from Corollary \[cor:perfectreps\] and . We thus assume that $Z({\ensuremath{\mathfrak{g}}}_0) \cong {\ensuremath{\mathfrak{g}}}_0/[{\ensuremath{\mathfrak{g}}}_0,{\ensuremath{\mathfrak{g}}}_0]$ is one-dimensional. By , we have ${\mathfrak{M}}/[{\mathfrak{M}},{\mathfrak{M}}]
\cong A_0/A_1^2$, a Lie algebra with trivial Lie bracket. Now, $A_0
= k[t+t^{-1}]$ and $A_1=(t-t^{-1})A_0$. Thus, setting $z=t+t^{-1}$, we have $$A_0/A_1^2 \cong k[z]/\langle z^2-4 \rangle,$$ which is a two-dimensional vector space. The points 1 and $-1$ are each $\Gamma$-fixed points and so we must take $\mathbf{x} = \{\pm
1\}$ in . Therefore $\bigoplus_{x \in
\mathbf{x}} Z({\ensuremath{\mathfrak{g}}}^x)$ is also a two-dimensional vector space and so the map $\gamma$ in is injective (since it is surjective). The result then follows from Theorem \[thm:finite-tilde-X\].
In the special case ${\ensuremath{\mathfrak{g}}}={\ensuremath{\mathfrak{sl}}}_2$, $k = {\mathbb{C}}$, the irreducible finite-dimensional representations of ${\EuScript{O}}({\ensuremath{\mathfrak{sl}}}_2)$ were described in [@DR] as follows. Let $\{e,h,f\}$ be an ${\ensuremath{\mathfrak{sl}}}_2$-triple and define $X = (e+f)\otimes 1$, and $Y = e\otimes t + f \otimes t^{-1}
$. Then ${\EuScript{O}}({\ensuremath{\mathfrak{sl}}}_2)$ is generated by $X,Y$. Furthermore, if $V$ is an irreducible finite-dimensional representation of ${\EuScript{O}}({\ensuremath{\mathfrak{sl}}}_2)$, then $X$ and $Y$ are diagonalizable on $V$, and there exists an integer $d \geq 0$ and scalars $\gamma, \gamma^* \in k$ such that the set of distinct eigenvalues of X (resp. Y ) on $V$ is $\left\{d
- 2i + \gamma : 0 \leq i \leq d\right\}$ (resp. $\left\{d - 2i +
\gamma^* : 0 \leq i \leq d\right\}$) [[@Ha Corollary 2.7]]{}. The ordered pair $(\gamma, \gamma^*)$ is called the *type* of $V$. Replacing $X, Y$ by $X - \gamma I, Y - \gamma^*I$ (in the universal enveloping algebra $U({\EuScript{O}}({\ensuremath{\mathfrak{sl}}}_2))$ of ${\EuScript{O}}({\ensuremath{\mathfrak{sl}}}_2)$) the type becomes $(0, 0)$.
Let $\operatorname{ev}_{x_1}V_1, \ldots , \operatorname{ev}_{x_n}V_n$ denote a finite sequence of evaluation modules for ${\EuScript{O}}({\ensuremath{\mathfrak{sl}}}_2)$, and $V$ the evaluation module $\operatorname{ev}_{x_1} V_1 \otimes \cdots \otimes \operatorname{ev}_{x_n} V_n$. Any module that can be obtained from $V$ by permuting the order of the factors and replacing any number of the $x_i$’s with their multiplicative inverses will be called *equivalent* to $V$. The classification of irreducible finite-dimensional ${\EuScript{O}}({\ensuremath{\mathfrak{sl}}}_2)$-modules of type $(0, 0)$ is described in [@DR] as follows.
1. [[@DR Theorem 6]]{} Every nontrivial irreducible finite-dimensional ${\EuScript{O}}({\ensuremath{\mathfrak{sl}}}_2)$-module of type $(0, 0)$ is isomorphic to a tensor product of evaluation modules.
2. [[@DR Proposition 5]]{} Let $\operatorname{ev}_{x_1} V_1, \ldots,
\operatorname{ev}_{x_n} V_n$ denote a finite sequence of evaluation modules for ${\EuScript{O}}({\ensuremath{\mathfrak{sl}}}_2)$, and consider the ${\EuScript{O}}({\ensuremath{\mathfrak{sl}}}_2)$-module $\operatorname{ev}_{x_1} V_1
\otimes \cdots \otimes \operatorname{ev}_{x_n} V_n$. This module is irreducible if and only if $x_1, x_1{^{-1}},
\ldots, x_n, x_n{^{-1}}$ are pairwise distinct.
3. [[@DR Proposition 5]]{} Let $U$ and $V$ denote tensor products of finitely many evaluation modules for ${\EuScript{O}}({\ensuremath{\mathfrak{sl}}}_2)$. Assume each of $U$, $V$ is irreducible as an ${\EuScript{O}}({\ensuremath{\mathfrak{sl}}}_2)$-module. Then the ${\EuScript{O}}({\ensuremath{\mathfrak{sl}}}_2)$-modules $U$ and $V$ are isomorphic if and only if they are equivalent.
These results are immediate consequences of Proposition \[prop:onsager-reps\]. The type $(0,0)$ representations are precisely the evaluation representations $\operatorname{ev}_\Psi$ such that $\left\{ \pm 1 \right\} \cap \operatorname{supp}\Psi =
\emptyset$. We note that under the previous definition of evaluation representations appearing in the literature (see Remark \[rem:new-eval-rep\]), *only* type $(0,0)$ representations are evaluation representations. However, using Definition \[def:evaluation-rep\], *all* irreducible finite-dimensional representations (of arbitrary type) are evaluation representations. The key is that we allow one-dimensional representations of ${\ensuremath{\mathfrak{g}}}^x$ for $x \in \{\pm 1\}$, where ${\ensuremath{\mathfrak{g}}}^x$ is one-dimensional. Thus our definition allows for a more uniform description of the representations. The condition that the points $x_1, x_1^{-1}, \dots, x_n, x_n^{-1}$ be pairwise distinct also follows automatically as in the case of multiloop algebras.
A nonabelian example {#subsec:applications-nonabelian}
--------------------
Let ${\ensuremath{\mathfrak{g}}}= \mathfrak{so}_8$, $X=\mathbb{P}^1\setminus \{0,1,\infty\}$ and $\Gamma=S_3$ as in Example \[eg:nonabelian\], and let ${\mathfrak{M}}=
M(X,{\ensuremath{\mathfrak{g}}})^\Gamma$. Since ${\mathfrak{M}}$ is perfect, by Corollary \[cor:perfectreps\] all irreducible finite-dimensional representations of ${\mathfrak{M}}$ are evaluation representations and these are naturally enumerated by $\mathcal{E}$. We identify $\Gamma=S_3$ with the permutations of the set $\{0,1,\infty\}$ and use the usual cycle notation for permutations. For instance, $(0\, \infty)$ denotes the permutation given by $0 \mapsto \infty$, $\infty \mapsto
0$, $1 \mapsto 1$. A straightforward computation shows that the points with nontrivial stabilizer are listed in the table below (note that $\{0,1,\infty\} \not \in X$). Each ${\ensuremath{\mathfrak{g}}}^x$ is the fixed point algebra of a diagram automorphism of ${\ensuremath{\mathfrak{g}}}$ and thus a simple Lie algebra of type $B_3$ or $G_2$.
--------------------------------------------------------------------------------------------------------------------------
$x$ $\Gamma_x$ Type of ${\ensuremath{\mathfrak{g}}}^x$
------------------- ------------------------------------------------------------ -----------------------------------------
$-1$ $\{\operatorname{Id}, (0\, \infty)\} \cong {\mathbb{Z}}_2$ $B_3$
2 $\{\operatorname{Id}, (1\, \infty)\} \cong {\mathbb{Z}}_2$ $B_3$
$\frac{1}{2}$ $\{\operatorname{Id}, (0\, 1)\} \cong {\mathbb{Z}}_2$ $B_3$
$e^{\pm \pi i/3}$ $\{\operatorname{Id}, (0\, 1\, \infty), (0\, $G_2$
\infty\, 1)\} \cong {\mathbb{Z}}_3$
--------------------------------------------------------------------------------------------------------------------------
Furthermore, the sets $\{-1,2,\frac{1}{2}\}$ and $\{e^{\pi i/3},
e^{-\pi i/3}\}$ are $\Gamma$-orbits. We thus see that the representation theory of ${\mathfrak{M}}$ is quite rich. Elements of $\mathcal{E}$ can assign to the three-element orbit (the isomorphism class of) any irreducible representation of the simple Lie algebra of type $B_3$, to the two-element orbit any irreducible representation of the simple Lie algebra of type $G_2$ and to any of the other (six-element) orbits, any irreducible representation of the simple Lie algebra ${\ensuremath{\mathfrak{g}}}$ of type $D_4$.
|
---
abstract: 'This article studies the degenerate warped products of singular semi-Riemannian manifolds. One main result is that a degenerate warped product of semi-regular semi-Riemannian manifolds with the warping function satisfying a certain condition is a semi-regular semi-Riemannian manifold. The main invariants of the warped product are expressed in terms of those of the factor manifolds. Examples of singular semi-Riemannian manifolds which are semi-regular are constructed as warped products. Degenerate warped products are used to define spherical warped products. As applications, cosmological models and black holes solutions with semi-regular singularities are constructed. Such singularities are compatible with the densitized version of Einstein’s equation, and don’t block the time evolution. In following papers we will apply the technique developed here to resolve the singularities of the Friedmann-Lemaître-Robertson-Walker, Schwarzschild, Reissner-Nordström and Kerr-Newman spacetimes.'
author:
- Cristi Stoica
date: 'June 2, 2010'
title: Warped Products of Singular SemiRiemannian Manifolds
---
[^1]
Introduction {#introduction .unnumbered}
============
The warped product provides a way to construct new semi-Riemannian manifolds from known ones [@BON69; @BEP82; @ONe83]. This construction has useful applications in General Relativity, in the study cosmological models and black holes. In such models, singularities are usually present, and at such points the warping function becomes $0$. The metric of the product manifold in this case becomes degenerate, and we need to apply the tools of singular semi-Riemannian geometry.
This article continues the study of singular semi-Riemannian manifolds [@Kup87b; @Kup96], [@Sto11a], extending it to warped products. After a brief recall of notions related to product manifolds in [§\[ss\_prod\_man\]]{}, basic notions of singular semi-Riemannian geometry and the main ideas from [@Sto11a], which will be applied here, are remembered in [§\[ss\_singular\_semi\_riem\]]{}. Then, in [§\[s\_deg\_wp\_ssr\]]{} we define the degenerate warped products of singular [[semiRiemannian]{}]{} maniolds, and study the Koszul form of the warped product in terms of the Koszul form of the factors. The main results known from the literature about the non-degenerate warped products of [[semiRiemannian]{}]{} maniolds are recalled in [§\[s\_non\_deg\]]{}. Then, in [§\[s\_semi\_reg\_semi\_riem\_man\_warped\]]{} we show that the warped products of [[radicalstationary]{}]{} manifolds are also [[radicalstationary]{}]{}, if the warping function satisfies a certain condition. Then we prove a similar result for [[semiregular]{}]{} manifolds, which ensures the smoothness of the Riemann curvature tensor. In [§\[s\_riemann\_wp\_deg\]]{} we express the Riemann curvature of [[semiregular]{}]{} warped products in terms of the factor manifolds. Then, in [§\[s\_riemann\_wp\_spherical\]]{}, we introduce the polar and spherical warped products, which allows us to construct singular [[semiRiemannian]{}]{} manifolds with radial or spherical symmetry. We conclude in [§\[s\_riemann\_wp\_deg\_applications\]]{} by giving some examples of [[semiregular]{}]{} warped products, and some applications to General Relativity. Cosmological models having the Big Bang singularity [[semiregular]{}]{}, are proposed. Spherical solutions with [[semiregular]{}]{} singularities are constructed in a general way. [[Semiregular]{}]{} singularities are compatible with a densitized version of Einstein’s equation, and they don’t block the time evolution.
This article is part of a series developing the Singular [[SemiRiemannian]{}]{} Geometry and its applications to the study of the singularities occurring in General Relativity.
Preliminaries {#s_prelim}
=============
Product manifolds {#ss_prod_man}
-----------------
We first recall some elementary notions about the *product manifold* $B\times F$ of two differentiable manifolds $B$ and $F$ [([*cf.* ]{}[*e.g.* ]{}[[@ONe83], p. 24–25]{})]{}.
At each point $p=(p_1,p_2)$ of the manifold $M_1\times M_2$ the tangent space decomposes as $$T_{(p_1,p_2)}(M_1\times M_2)\cong T_{(p_1,p_2)}(M_1)\oplus T_{(p_1,p_2)}(M_2),$$ where $T_{(p_1,p_2)}(M_1):=T_{(p_1,p_2)}(M_1\times p_2)$ and $T_{(p_1,p_2)}(M_2):=T_{(p_1,p_2)}(p_1\times M_2)$.
Let $\pi_i:M_1\times M_2\to M_i$, for $i\in\{1,2\}$, be the canonical projections. The *lift of the scalar field* $f_i\in{{\mathscr{F}}(M_i)}$ is the scalar field $\tilde f_i:=f_i\circ\pi_i\in{{{\mathfrak{X}}}(M_1\times M_2)}$. The *lift of the vector field* $X_i\in{{{\mathfrak{X}}}(M_i)}$ is the unique vector field $\tilde X_i$ on $M_1\times M_2$ satisfying ${\textnormal{d}}\pi_i(\tilde X_i)=X_i$. We denote the set of all vector fields $X\in {{{\mathfrak{X}}}(M_1\times M_2)}$ which are lifts of vector fields $X_i\in {{{\mathfrak{X}}}(M_i)}$ by ${{\mathfrak{L}}(M,M_i)}$. The *lift of a covariant tensor* $T\in{{\mathcal{T}}{}^{0}_{s}M_i}$ is given by $\tilde T\in {{\mathcal{T}}{}^{0}_{s}(M_1 \times M_2)}$, $\tilde T:=\pi^*_i(T)$. The *lift of a tensor* $T\in{{\mathcal{T}}{}^{1}_{s}M_i}$ is given, for any $X_1,\ldots,X_s\in {{{\mathfrak{X}}}(M_1 \times M_2)}$, by $\tilde T\in {{\mathcal{T}}{}^{1}_{s}(M_1 \times M_2)}$, $\tilde T(X_1,\ldots,X_s) =\tilde X$, where $\tilde X \in {{{\mathfrak{X}}}(M_1 \times M_2)}$ is the lifting of the vector field $X\in{{{\mathfrak{X}}}(M_i)}$, $X=T(\pi_i(X_1),\ldots,\pi_i(X_s))$.
Singular [[semiRiemannian]{}]{} manifolds {#ss_singular_semi_riem}
-----------------------------------------
We recall here some notions about singular semi-Riemannian manifolds, and some of the main results from [@Sto11a], which will be used in the rest of the article.
(also see [@Kup87b]) \[def\_sing\_semiRiemm\_man\] A *singular [[semiRiemannian]{}]{} manifold* $(M,g)$ is a differentiable manifold $M$ endowed with a symmetric bilinear form $g\in \Gamma(T^*M \odot_M T^*M)$ named *metric*. The manifold $(M,g)$ is said to be with *constant signature* if the signature of $g$ is fixed, otherwise, $(M,g)$ is said to be with *variable signature*. Particular cases are the *[[semiRiemannian]{}]{} manifolds*, when the metric is [[nondegenerate]{}]{}, and *Riemannian manifolds*, when $g$ is positive definite.
(also see [@Bej95][1]{}, [@Kup96][3]{} and [@ONe83][53]{}) If $(V,g)$ is a finite dimensional inner product space with an inner product $g$ which may be degenerate, then we call the totally degenerate space ${{V{}_{\circ}{}}}:=V^\perp$ the *radical* of $V$. The inner product $g$ on $V$ is [[nondegenerate]{}]{} if and only if ${{V{}_{\circ}{}}}=\{0\}$.
(see [@Kup87b][261]{}, [@Pam03][263]{}) The *radical of $TM$*, denoted by ${{T{}_{\circ}{}}}M$, is defined by ${{T{}_{\circ}{}}}M=\cup_{p\in M}{{(T_pM){}_{\circ}{}}}$. We denote by ${{{\mathfrak{X}}}_\circ(M)}$ the set of vector fields on $M$ for which $W_p\in{{(T_pM){}_{\circ}{}}}$. They form a vector space over ${\mathbb{R}}$ and a module over ${{\mathscr{F}}(M)}$.
The remaining of this section recalls very briefly the main notions and results on singular semi-Riemannian manifolds, as presented in [@Sto11a].
We define $${{T{}^{\bullet}{}}}M=\bigcup_{p\in M}{{(T_pM){}^{\bullet}{}}}$$ where ${{(T_pM){}^{\bullet}{}}} \subseteq T^*_pM$ is the space of covectors at $p$ of the form $\omega_p(X_p)={\langleY_p,X_p\rangle}$ for some vectors $Y_p\in T_p M$ and any $X_p\in T_p M$. We define sections of ${{T{}^{\bullet}{}}}M$ by $${{{{\mathcal{A}}{}^{\bullet}{}}}(M)}:=\{\omega\in{{\mathcal{A}}^{1}(M)}|\omega_p\in{{(T_pM){}^{\bullet}{}}}{\textnormal}{ for any }p\in M\}.$$
\[def\_co\_inner\_product\] On ${{T{}^{\bullet}{}}}M$ there is a a unique [[nondegenerate]{}]{} inner product ${{{g{}_{\bullet}{}}}}$, defined by ${{{\langle\!\langle\omega,\tau\rangle\!\rangle{}_{\bullet}{}}}}:={{{g{}_{\bullet}{}}}}(\omega,\tau):={\langleX,Y\rangle}$, where ${{X{}^{\bullet}{}}}=\omega$, ${{Y{}^{\bullet}{}}}=\tau$, $X,Y\in{{{\mathfrak{X}}}(M)}$.
\[def\_radix\_annih\_tensor\_field\] A tensor $T$ of type $(r,s)$ is named *[[radicalannihilator]{}]{}* in the $l$-th covariant slot if $T\in {{\mathcal{T}}{}^{r}_{l-1}M}\otimes_M{{T{}^{\bullet}{}}}M\otimes_M {{\mathcal{T}}{}^{0}_{s-l}M}$.
\[def\_contraction\_covariant\] We now show how to define uniquely the *covariant contraction* or *covariant trace*. We define it first on tensors $T\in{{T{}^{\bullet}{}}}M\otimes_M{{T{}^{\bullet}{}}}M$, by $C_{12}T={{{g{}_{\bullet}{}}}}^{ab}T_{ab}$. This definition does not depend on the basis, because ${{{g{}_{\bullet}{}}}}\in{{T{}^{\bullet}{}}}^*M\otimes_M{{T{}^{\bullet}{}}}^*M$. This operation can be extended by linearity to any tensors which are radical in two covariant indices. For a tensor field $T$ we define the contraction $C_{kl} T$ by $$T(\omega_1,\ldots,\omega_r,v_1,\ldots,{{{}_\bullet}},\ldots,{{{}_\bullet}},\ldots,v_s).$$
If the metric is non-degenerate, we can define the covariant derivative of a vector field $Y$ in the direction of a vector field $X$, where $X,Y\in{{{\mathfrak{X}}}(M)}$, by the *Koszul formula* (see [*e.g.* ]{}[@ONe83][61]{}). If the metric is degenerate, we cannot extract the covariant derivative from the Koszul formula. We define the *Koszul form* as a shorthand for the long right part of the Koszul formula and were emphasized some of its properties.
Let’s recall the definition of the Koszul form and its properties, without proof, from [@Sto11a].
\[def\_Koszul\_form\] *The Koszul form* is defined as $${{\mathcal{K}}}:{{{\mathfrak{X}}}(M)}^3\to{\mathbb{R}},$$ $$\label{eq_Koszul_form}
\begin{array}{llll}
{{\mathcal{K}}}(X,Y,Z) &:=&{\displaystyle}{\frac 1 2} \{ X {\langleY,Z\rangle} + Y {\langleZ,X\rangle} - Z {\langleX,Y\rangle} \\
&&\ - {\langleX,[Y,Z]\rangle} + {\langleY, [Z,X]\rangle} + {\langleZ, [X,Y]\rangle}\}.
\end{array}$$
\[thm\_Koszul\_form\_props\] Properites of the Koszul form of a singular [[semiRiemannian]{}]{} manifold $(M,g)$:
1. \[thm\_Koszul\_form\_props\_linear\] Additivity and ${\mathbb{R}}$-linearity in each of its arguments.
2. \[thm\_Koszul\_form\_props\_flinearX\] ${{\mathscr{F}}(M)}$-linearity in the first argument:
${{\mathcal{K}}}(fX,Y,Z) = f{{\mathcal{K}}}(X,Y,Z).$
3. \[thm\_Koszul\_form\_props\_flinearY\] The *Leibniz rule*:
${{\mathcal{K}}}(X,fY,Z) = f{{\mathcal{K}}}(X,Y,Z) + X(f) {\langleY,Z\rangle}.$
4. \[thm\_Koszul\_form\_props\_flinearZ\] ${{\mathscr{F}}(M)}$-linearity in the third argument:
${{\mathcal{K}}}(X,Y,fZ) = f{{\mathcal{K}}}(X,Y,Z).$
5. \[thm\_Koszul\_form\_props\_commutYZ\] It is *metric*:
${{\mathcal{K}}}(X,Y,Z) + {{\mathcal{K}}}(X,Z,Y) = X {\langleY,Z\rangle}$.
6. \[thm\_Koszul\_form\_props\_commutXY\] It is *symmetric*:
${{\mathcal{K}}}(X,Y,Z) - {{\mathcal{K}}}(Y,X,Z) = {\langle[X,Y],Z\rangle}$.
7. \[thm\_Koszul\_form\_props\_commutZX\] Relation with the Lie derivative of $g$:
${{\mathcal{K}}}(X,Y,Z) + {{\mathcal{K}}}(Z,Y,X) = ({{\mathcal{L}}}_Y g)(Z,X)$.
8. \[thm\_Koszul\_form\_props\_commutX2Y\]
${{\mathcal{K}}}(X,Y,Z) + {{\mathcal{K}}}(Y,Z,X) = Y{\langleZ,X\rangle} + {\langle[X,Y],Z\rangle}$.
for any $X,Y,Z\in{{{\mathfrak{X}}}(M)}$ and $f\in{{\mathscr{F}}(M)}$.
\[def\_l\_cov\_der\] Let $X,Y\in\in{{{\mathfrak{X}}}(M)}$. The *lower covariant derivative* of $Y$ in the direction of $X$ is defined as the differential $1$-form ${{{{\nabla}^{\flat}}_{X}}{Y}} \in {{\mathcal{A}}^{1}(M)}$ $$\label{eq_l_cov_der_vect}
{({{{{\nabla}^{\flat}}_{X}}{Y}})(Z)} := {{\mathcal{K}}}(X,Y,Z)$$ for any $Z\in{{{\mathfrak{X}}}(M)}$. We also define the *lower covariant derivative operator* $${{\nabla}^{\flat}}:{{{\mathfrak{X}}}(M)} \times {{{\mathfrak{X}}}(M)} \to {{\mathcal{A}}^{1}(M)}$$ which associates to each $X,Y\in{{{\mathfrak{X}}}(M)}$ the differential $1$-form ${{{\nabla}^{\flat}}_{X}}Y$.
\[def\_radical\_stationary\_manifold\] A singular [[semiRiemannian]{}]{} manifold $(M,g)$ is *[[radicalstationary]{}]{}* if it satisfies the condition $$\label{eq_radical_stationary_manifold}
{{\mathcal{K}}}(X,Y,\_)\in{{{{\mathcal{A}}{}^{\bullet}{}}}(M)},$$ for any $X,Y\in{{{\mathfrak{X}}}(M)}$.
\[def\_cov\_der\_covect\] Let $X\in{{{\mathfrak{X}}}(M)}$, $\omega\in{{{{\mathcal{A}}{}^{\bullet}{}}}(M)}$, where $(M,g)$ is [[radicalstationary]{}]{}. The covariant derivative of $\omega$ in the direction of $X$ is defined as $${\nabla}:{{{\mathfrak{X}}}(M)} \times {{{{\mathcal{A}}{}^{\bullet}{}}}(M)} \to {A_d{}^{1}(M)}$$ $$\left({\nabla}_X\omega\right)(Y) := X\left(\omega(Y)\right) - {{{\langle\!\langle{{{{\nabla}^{\flat}}_{X}}{Y}},\omega\rangle\!\rangle{}_{\bullet}{}}}},$$ where ${A_d{}^{1}(M)}$ denotes the set of $1$-forms which are smooth on the regions of constant signature.
\[def\_cov\_der\_smooth\] If the [[semiRiemannian]{}]{} manifold $(M,g)$ is [[radicalstationary]{}]{}, we define: $${{{{\mathscr{A}}{}^{\bullet}{}}}{}^{1}(M)} = \{\omega\in{{{{\mathcal{A}}{}^{\bullet}{}}}(M)}|(\forall X\in{{{\mathfrak{X}}}(M)})\ {\nabla}_X\omega\in{{{{\mathcal{A}}{}^{\bullet}{}}}(M)}\},$$ $${{{{\mathscr{A}}{}^{\bullet}{}}}{}^{k}(M)} := \bigwedge^k_M{{{{\mathscr{A}}{}^{\bullet}{}}}{}^{1}(M)}.$$
\[def\_riemann\_curvature\] The *Riemann curvature tensor* is defined as $$R: {{{\mathfrak{X}}}(M)}\times {{{\mathfrak{X}}}(M)}\times {{{\mathfrak{X}}}(M)}\times {{{\mathfrak{X}}}(M)} \to {\mathbb{R}},$$ $$\label{eq_riemann_curvature}
R(X,Y,Z,T) := ({{\nabla}_{X}} {{{{\nabla}^{\flat}}_{Y}}}Z - {{\nabla}_{Y}} {{{{\nabla}^{\flat}}_{X}}}Z - {{{\nabla}^{\flat}}_{[X,Y]}}Z)(T)$$ for any vector fields $X,Y,Z,T\in{{{\mathfrak{X}}}(M)}$.
\[def\_semi\_regular\_semi\_riemannian\] A singular [[semiRiemannian]{}]{} manifold $(M,g)$ satisfying $${{{\nabla}^{\flat}}_{X}} Y \in{{{{\mathscr{A}}{}^{\bullet}{}}}{}^{1}(M)}$$ for any vector fields $X,Y\in{{{\mathfrak{X}}}(M)}$ is called *[[semiregular]{}]{} [[semiRiemannian]{}]{} manifold*.
\[thm\_sr\_cocontr\_kosz\] A [[radicalstationary]{}]{} [[semiRiemannian]{}]{} manifold $(M,g)$ is [[semiregular]{}]{} if and only if for any $X,Y,Z,T\in{{{\mathfrak{X}}}(M)}$ $${{\mathcal{K}}}(X,Y,{{{}_\bullet}}){{\mathcal{K}}}(Z,T,{{{}_\bullet}}) \in {{\mathscr{F}}(M)}.$$
\[s\_semi\_reg\_semi\_riem\_man\_example\_diagonal\] We will construct a useful example of [[semiregular]{}]{} metric [@Sto11a]. Let’s consider that there is a coordinate chart in which the metric is diagonal. The components of the Koszul form are in this case the Christoffel’s symbols of the first kind, which are of the form $\pm\frac 1 2\partial_a g_{bb}$, because the metric is diagonal. Assume that $g=\sum_a\varepsilon_a\alpha_a^2{\textnormal{d}}x^a\otimes {\textnormal{d}}x^a$, $\varepsilon_a\in\{-1,1\}$. Then the metric is [[semiregular]{}]{} if there is a smooth function $f_{abc}\in{{\mathscr{F}}(M)}$ with ${{\textnormal}{supp}(f_{abc})}\subseteq{{\textnormal}{supp}(\alpha_c)}$ for any $a,b\in\{1,\ldots,n\}$ and $c\in\{a,b\}$, and $$\label{eq_diagonal_metric:semireg}
\partial_a\alpha_b^2=f_{abc}\alpha_c.$$ If $c=b$, $\partial_a\alpha_b^2=2\alpha_b\partial_a\alpha_b$ implies that the function is $f_{abb}=2\partial_a\alpha_b$. In addition, this has to satisfy the condition $\partial_a\alpha_b=0$ whenever $\alpha_b=0$. We require the condition ${{\textnormal}{supp}(f_{abc})}\subseteq{{\textnormal}{supp}(\alpha_c)}$ because for being [[semiregular]{}]{}, a manifold has to be [[radicalstationary]{}]{}.
\[thm\_riemann\_curvature\_semi\_regular\] The Riemann curvature of a [[semiregular]{}]{} [[semiRiemannian]{}]{} manifold $(M,g)$ is a smooth tensor field $R\in{{\mathcal{T}}{}^{0}_{4}M}$.
\[thm\_riemann\_curvature\_tensor\_koszul\_formula\] The Riemann curvature of a [[semiregular]{}]{} [[semiRiemannian]{}]{} manifold $(M,g)$ satisfies $$\begin{array}{lll}
R(X,Y,Z,T) &=& X\left({({{{{\nabla}^{\flat}}_{Y}}{Z}})(T)}\right) - Y\left({({{{{\nabla}^{\flat}}_{X}}{Z}})(T)}\right) - {({{{{\nabla}^{\flat}}_{[X,Y]}}{Z}})(T)} \\
&& + {{{\langle\!\langle{{{{\nabla}^{\flat}}_{X}}{Z}},{{{{\nabla}^{\flat}}_{Y}}{T}}\rangle\!\rangle{}_{\bullet}{}}}} - {{{\langle\!\langle{{{{\nabla}^{\flat}}_{Y}}{Z}},{{{{\nabla}^{\flat}}_{X}}{T}}\rangle\!\rangle{}_{\bullet}{}}}} \\
\end{array}$$ and $$\label{eq_riemann_curvature_tensor_koszul_formula}
\begin{array}{lll}
R(X,Y,Z,T)&=& X {{\mathcal{K}}}(Y,Z,T) - Y {{\mathcal{K}}}(X,Z,T) - {{\mathcal{K}}}([X,Y],Z,T)\\
&& + {{\mathcal{K}}}(X,Z,{{{}_\bullet}}){{\mathcal{K}}}(Y,T,{{{}_\bullet}}) - {{\mathcal{K}}}(Y,Z,{{{}_\bullet}}){{\mathcal{K}}}(X,T,{{{}_\bullet}})
\end{array}$$ for any vector fields $X,Y,Z,T\in{{{\mathfrak{X}}}(M)}$.
Degenerate warped products of singular [[semiRiemannian]{}]{} manifolds {#s_deg_wp_ssr}
=======================================================================
The warped product is defined in general between two ([[nondegenerate]{}]{}) [[semiRiemannian]{}]{} manifolds, ([*cf.* ]{}[@BON69], [@BEP82], [@ONe83][204–211]{}. It is straightforward to extend the definition to singular [[semiRiemannian]{}]{} manifolds, as it is done in this section.
\[def\_wp\] Let $(B,g_B)$ and $(F,g_F)$ be two singular [[semiRiemannian]{}]{} manifolds, and $f\in{{\mathscr{F}}(B)}$ a smooth function. The *warped product* of $B$ and $F$ with *warping function* $f$ is the [[semiRiemannian]{}]{} manifold $$B\times_f F:=\big(B\times F, \pi^*_B(g_B) + (f\circ \pi_B)\pi^*_F(g_F)\big),$$ where $\pi_B:B\times F \to B$ and $\pi_F: B \times F \to F$ are the canonical projections. It is customary to call $B$ the *base* and $F$ the *fiber* of the warped product $B\times_f F$.
We will use for all vector fields $X_B,Y_B\in{{{\mathfrak{X}}}(B)}$ and $X_F,Y_F\in{{{\mathfrak{X}}}(F)}$ the notation ${\langleX_B,Y_B\rangle}_B := g_B(X_B,Y_B)$ and ${\langleX_F,Y_F\rangle}_F := g_F(X_F,Y_F)$. The inner product on $B\times_f F$ takes, for any point $p\in B\times F$ and for any pair of tangent vectors $x,y\in T_p(B\times F)$, the explicit form $$\label{eq_wp_metric}
{\langlex,y\rangle}={\langle{\textnormal{d}}\pi_B(x),{\textnormal{d}}\pi_B(y)\rangle}_B + f^2(p){\langle{\textnormal{d}}\pi_F(x),{\textnormal{d}}\pi_F(y)\rangle}_F.$$
Definition \[def\_wp\] is a generalization of the warped product definition, which is usually given for the case when both $g_B$ and $g_F$ are [[nondegenerate]{}]{} and $f>0$ (see [@BON69], [@BEP82] and [@ONe83]). In our definition these restrictions are dropped.
For any $p_B\in B$, $\pi_B^{-1}(p_B)=p_B\times F$ is named the *fiber* through $p_B$ and it is a [[semiRiemannian]{}]{} manifold. $\pi_F|_{p_B\times F}$ is a (possibly degenerate) homothety onto $F$. For each $p_F\in F$, $\pi_F^{-1}(p_F)=B\times p_F$ is a [[semiRiemannian]{}]{} manifold named the *leave* through $p_F$. $\pi_B|_{B\times p_F}$ is an isometry onto $B$. For each $(p_B,p_F)\in B\times F$, $B \times p_F$ and $p_B \times F$ are orthogonal at $(p_B,p_F)$. For simplicity, if a vector field is a lift, we will use sometimes the same notation if they can be distinguished from the context. For example, we will be using ${\langleV,W\rangle}_F:={\langle\pi_F(V),\pi_F(W)\rangle}_F$ for $V,W\in{{\mathfrak{L}}(B \times F,F)}$.
The following proposition recalls some evident facts used repeatedly in the proofs of the properties of warped products in [@ONe83][24–25, 206]{}.
\[thm\_wp\_deg\_fundam\] Let $B \times_f F$ be a warped product, and let be the vector fields $X,Y,Z\in{{\mathfrak{L}}(B \times F,B)}$ and $U,V,W\in{{\mathfrak{L}}(B \times F,F)}$. Then
1. \[thm\_wp\_deg\_fundam:mixed\_metric\] ${\langleX,V\rangle}=0$.
2. \[thm\_wp\_deg\_fundam:mixed\_lie\_bracket\] $[X,V]=0$.
3. \[thm\_wp\_deg\_fundam:metric\_B\_constant\_F\] $V{\langleX,Y\rangle}=0$.
4. \[thm\_wp\_deg\_fundam:metric\_F\_var\_B\] $X{\langleV,W\rangle}=2f{\langleV,W\rangle}_F X(f)$.
and are evident because the manifold is $B\times F$.
${\langleX,Y\rangle}={\langleX,Y\rangle}_B$ is constant on fibers, and $V{\langleX,Y\rangle}=0$ because $V$ is vertical.
$X{\langleV,W\rangle}=X(f^2{\langleV,W\rangle}_F)=2f{\langleV,W\rangle}_F X(f)$.
The following proposition generalizes the properties of the Levi-Civita connection for the warped product of ([[nondegenerate]{}]{}) [[semiRiemannian]{}]{} manifolds [([*cf.* ]{}[*e.g.* ]{}[[@ONe83], p. 206]{})]{}, to the degenerate case. We preferred to express them in terms of the Koszul form, and to give the proof explicitly, because for degenerate metric the Levi-Civita connection is not defined, and we need to avoid the index raising.
\[thm\_wp\_deg\_koszul\] Let $B \times_f F$ be a warped product, and let be the vector fields $X,Y,Z\in{{\mathfrak{L}}(B \times F,B)}$ and $U,V,W\in{{\mathfrak{L}}(B \times F,F)}$. Let ${{\mathcal{K}}}$ be the Koszul form on $B \times_f F$, and ${{\mathcal{K}}}_B, {{\mathcal{K}}}_F$ the lifts of the Koszul forms on $B$, respectively $F$. Then
1. \[thm\_wp\_deg\_koszul:BBB\] ${{\mathcal{K}}}(X,Y,Z)={{\mathcal{K}}}_B(X,Y,Z)$.
2. \[thm\_wp\_deg\_koszul:BBF\] ${{\mathcal{K}}}(X,Y,W) = {{\mathcal{K}}}(X,W,Y) = {{\mathcal{K}}}(W,X,Y) = 0$.
3. \[thm\_wp\_deg\_koszul:BFF\] ${{\mathcal{K}}}(X,V,W) = {{\mathcal{K}}}(V,X,W) = -{{\mathcal{K}}}(V,W,X) = f {\langleV,W\rangle}_F X(f)$.
4. \[thm\_wp\_deg\_koszul:FFF\] ${{\mathcal{K}}}(U,V,W)=f^2{{\mathcal{K}}}_F(U,V,W)$.
and follow from properties of the lifts of vector fields, the Definition \[def\_Koszul\_form\] of the Koszul form, and the equation .
By Definition \[def\_Koszul\_form\], $$\begin{array}{llll}
{{\mathcal{K}}}(X,Y,W) &=&{{\displaystyle}{\frac{1}{2}}} \{ X {\langleY,W\rangle} + Y {\langleW,X\rangle} - W {\langleX,Y\rangle} \\
&&\ - {\langleX,[Y,W]\rangle} + {\langleY, [W,X]\rangle} + {\langleW, [X,Y]\rangle}\}
\end{array}$$ We apply the Proposition \[thm\_wp\_deg\_fundam\]. From the relation , ${\langleY,W\rangle}={\langleW,X\rangle}={\langleW,[X,Y]\rangle}=0$, from the relation $[Y,W]=[W,X]=0$, from the relation $W {\langleX,Y\rangle}=0$. Therefore ${{\mathcal{K}}}(X,Y,W)=0$.
From of the Theorem \[thm\_Koszul\_form\_props\] we obtain that $${{\mathcal{K}}}(X,W,Y) = X {\langleW,Y\rangle} - {{\mathcal{K}}}(X,Y,W) = 0.$$
From of the Theorem \[thm\_Koszul\_form\_props\] and from Proposition \[thm\_wp\_deg\_fundam\] we obtain that $${{\mathcal{K}}}(W,X,Y) = {{\mathcal{K}}}(X,W,Y) - {\langle[X,W],Y\rangle}= 0.$$
$$\begin{array}{lrl}
\tag{\ref{thm_wp_deg_koszul:BFF}} {{\mathcal{K}}}(X,V,W) &:=&{{\displaystyle}{\frac{1}{2}}} \{ X {\langleV,W\rangle} + V {\langleW,X\rangle} - W {\langleX,V\rangle} \\
&&\ - {\langleX,[V,W]\rangle} + {\langleV, [W,X]\rangle} + {\langleW, [X,V]\rangle}\}\\
&=& {{\displaystyle}{\frac{1}{2}}} X {\langleV,W\rangle}
\end{array}$$
from Proposition \[thm\_wp\_deg\_fundam\], using it as in the property of the present Proposition. By applying the property we have ${{\mathcal{K}}}(X,V,W) = f {\langleV,W\rangle}_F X(f)$. From Theorem \[thm\_Koszul\_form\_props\] property , $${{\mathcal{K}}}(V,X,W)={{\mathcal{K}}}(X,V,W)-{\langle[X,V],W\rangle},$$ but since $[X,V]=0$, ${{\mathcal{K}}}(V,X,W) = f {\langleV,W\rangle}_F X(f)$ as well.
From Theorem \[thm\_Koszul\_form\_props\] property , $${{\mathcal{K}}}(V,W,X) = V{\langleW,X\rangle} - {{\mathcal{K}}}(V,X,W),$$ but since ${\langleW,X\rangle}=0$, the property of the present Proposition shows that $${{\mathcal{K}}}(V,W,X) = -f{\langleV,W\rangle}_F X(f).$$
Further, we will study some properties of the warped products, in situations when the warping function $f$ is allowed to cancel or to become negative, and when $(B,g_B)$ and $(F,g_F)$ are allowed to be singular and with variable signature. But first, we need to recall what we know about [[nondegenerate]{}]{} warped products of non-singular [[semiRiemannian]{}]{} manifolds.
Non-degenerate warped products {#s_non_deg}
==============================
Here we recall for comparison and without proofs some fundamental properties of non-degenerate warped products between non-singular [[semiRiemannian]{}]{} manifolds. The main reference is [@ONe83][204–211]{}. Here, $(B,g_B)$ and $(F,g_F)$ are [[semiRiemannian]{}]{} manifolds, $f\in{{\mathscr{F}}(B)}$ a smooth function so that $f>0$, and $B\times_f F$ the warped product of $B$ and $F$.
For the proofs of the next propositions, see for example [@ONe83].
\[thm\_wp\_nondeg\_conn\] Let $B \times_f F$ be a warped product, and let be the vector fields $X,Y\in{{\mathfrak{L}}(B \times F,B)}$ and $V,W\in{{\mathfrak{L}}(B \times F,F)}$. Let $\nabla, \nabla^B, \nabla^F$ be the Levi-Civita connections on $B \times_f F$, $B$, respectively $F$. Then
1. $\nabla_X Y$ is the lift of $\nabla^B_X Y$.
2. $\nabla_X V = \nabla_V X = {{\displaystyle}{\frac{Xf}{f}}} V$.
3. $\nabla_V W = -{{\displaystyle}{\frac{{\langleV,W\rangle}}{f}}} {\textnormal{grad }}f + \widetilde{\nabla^F_V W}$, where $\widetilde{\nabla^F_V W}$ is the lift of $\nabla^F_V W$.
\[thm\_wp\_nondeg\_riemm\] Let $B \times_f F$ be a warped product, and $R_B, R_F$ the lifts of the Riemann curvature tensors on $B$ and $F$. Let be the vector fields $X,Y,Z\in{{\mathfrak{L}}(B \times F,B)}$ and $U,V,W\in{{\mathfrak{L}}(B \times F,F)}$, and let $H^f$ be the *Hessian* of $f$, $H^f(X,Y)={\langle\nabla_X({\textnormal{grad }}f),Y\rangle}_B$. Then
1. $R(X,Y)Z\in{{\mathfrak{L}}(B \times F,B)}$ is the lift of $R_B(X,Y)Z$.
2. $R(V,X)Y = -{{\displaystyle}{\frac{H^f(X,Y)}{f}}}V$.
3. $R(X,Y)V = R(V,W)X = 0$.
4. $R(X,V)W = -{{\displaystyle}{\frac{{\langleV,W\rangle}}{f}}}\nabla_X({\textnormal{grad }}f)$.
5. $\begin{aligned}[t]
R(V,W)U =& \,R_F(V,W)U \\
&+ {{\displaystyle}{\frac{{\langle{\textnormal{grad }}f,{\textnormal{grad }}f\rangle}}{f^2}}}\big({\langleV,U\rangle}W - {\langleW,U\rangle}V\big).
\end{aligned}$
\[thm\_wp\_nondeg\_ric\] Let $B \times_f F$ be a warped product, with $\dim F>1$, and let be the vector fields $X,Y\in{{\mathfrak{L}}(B \times F,B)}$ and $V,W\in{{\mathfrak{L}}(B \times F,F)}$. Then
1. ${{\textnormal}{Ric}}(X,Y) = {{\textnormal}{Ric}}_B(X,Y) + {{\displaystyle}{\frac{\dim F}{f}}}H^f(X,Y)$.
2. ${{\textnormal}{Ric}}(X,V) = 0$.
3. $\begin{aligned}[t]
{{\textnormal}{Ric}}(V,W) =&\, {{\textnormal}{Ric}}_F(V,W)\\
&+ \({{\displaystyle}{\frac{\Delta f}{f}}} + (\dim F-1){{\displaystyle}{\frac{{\langle{\textnormal{grad }}f,{\textnormal{grad }}f\rangle}}{f^2}}}\){\langleV,W\rangle}.
\end{aligned}$
\[thm\_wp\_nondeg\_scal\] Let $B \times_f F$ be a warped product, with $\dim F>1$. Then, the scalar curvature $s$ of $B \times_f F$ is related to the scalar curvatures $s_B$ and $s_F$ of $B$ and $F$ by $$s = s_B + \frac {s_F}{f^2} + 2\dim F{{\displaystyle}{\frac{\Delta f}{f}}} + \dim F(\dim F - 1){{\displaystyle}{\frac{{\langle{\textnormal{grad }}f, {\textnormal{grad }}f\rangle}_B}{f^2}}}.$$
Warped products of [[semiregular]{}]{} [[semiRiemannian]{}]{} manifolds {#s_semi_reg_semi_riem_man_warped}
=======================================================================
In the following we will provide the condition for a degenerate warped product of [[semiregular]{}]{} [[semiRiemannian]{}]{} manifolds to be a [[semiregular]{}]{} [[semiRiemannian]{}]{} manifold.
\[thm\_rad\_stat\_semi\_riem\_man\_warped\] Let $(B,g_B)$ and $(F,g_F)$ be two [[radicalstationary]{}]{} [[semiRiemannian]{}]{} manifolds, and $f\in{{\mathscr{F}}(B)}$ a smooth function so that ${\textnormal{d}}f\in{{{{\mathcal{A}}{}^{\bullet}{}}}(B)}$. Then, the warped product manifold $B \times_f F$ is a [[radicalstationary]{}]{} [[semiRiemannian]{}]{} manifold.
We have to show that ${{\mathcal{K}}}(X,Y,W)=0$ for any $X,Y\in{{{\mathfrak{X}}}(B \times_f F)}$ and $W\in{{{\mathfrak{X}}}_\circ(B \times_f F)}$. It is enough to check this for vector fields which are lifts of vector fields $X_B,Y_B,W_B\in {{\mathfrak{L}}(B \times F,B)}$, $X_F,Y_F,W_F\in {{\mathfrak{L}}(B \times F,F)}$, where $W_B,W_F\in{{{\mathfrak{X}}}_\circ(B \times_f F)}$. Then, from the Proposition \[thm\_wp\_deg\_koszul\]:
1. ${{\mathcal{K}}}(X_B,Y_B,W_B)={{\mathcal{K}}}_B(X_B,Y_B,W_B)=0$,
2. ${{\mathcal{K}}}(X_B,Y_B,W_F) = {{\mathcal{K}}}(X_B,Y_F,W_B) = {{\mathcal{K}}}(X_F,Y_B,W_B) = 0$,
3. ${{\mathcal{K}}}(X_B,Y_F,W_F) = {{\mathcal{K}}}(Y_F,X_B,W_F) = f {\langleY_F,W_F\rangle}_F X_B(f) = 0$, because ${\langleY_F,W_F\rangle}_F=0$, and\
${{\mathcal{K}}}(X_F,Y_F,W_B) = -f {\langleX_F,Y_F\rangle}_F W_B(f) = 0$, from $W_B(f)=0$,
4. ${{\mathcal{K}}}(X_F,Y_F,W_F)=f^2{{\mathcal{K}}}_F(X_F,Y_F,W_F) = 0$.
\[thm\_semi\_reg\_semi\_riem\_man\_warped\] Let $(B,g_B)$ and $(F,g_F)$ be two [[semiregular]{}]{} [[semiRiemannian]{}]{} manifolds, and $f\in{{\mathscr{F}}(B)}$ a smooth function so that ${\textnormal{d}}f\in{{{{\mathscr{A}}{}^{\bullet}{}}}{}^{1}(B)}$. Then, the warped product manifold $B \times_f F$ is a [[semiregular]{}]{} [[semiRiemannian]{}]{} manifold.
All contractions of the form ${{\mathcal{K}}}(X,Y,{{{}_\bullet}}){{\mathcal{K}}}(Z,T,{{{}_\bullet}})$ are well defined, according to Theorem \[thm\_rad\_stat\_semi\_riem\_man\_warped\]. From Proposition \[thm\_sr\_cocontr\_kosz\], it is enough to show that they are smooth. It is enough to check this for vector fields which are lifts of vector fields $X_B,Y_B,Z_B,T_B\in {{\mathfrak{L}}(B \times F,B)}$, $X_F,Y_F,Z_F,T_F\in {{\mathfrak{L}}(B \times F,F)}$. Let’s denote by ${{{}_\bullet}}_B$ and ${{{}_\bullet}}_F$ the symbol for the covariant contraction on $B$, respectively $F$. Then, from the Proposition \[thm\_wp\_deg\_koszul\]: $$\begin{array}{lll}
{{\mathcal{K}}}(X_B,Y_B,{{{}_\bullet}}){{\mathcal{K}}}(Z_B,T_B,{{{}_\bullet}})&=&{{\mathcal{K}}}(X_B,Y_B,{{{}_\bullet}}_B){{\mathcal{K}}}(Z_B,T_B,{{{}_\bullet}}_B) \\
&&+{{\mathcal{K}}}(X_B,Y_B,{{{}_\bullet}}_F){{\mathcal{K}}}(Z_B,T_B,{{{}_\bullet}}_F) \\
&=&{{\mathcal{K}}}_B(X_B,Y_B,{{{}_\bullet}}_B){{\mathcal{K}}}_B(Z_B,T_B,{{{}_\bullet}}_B) \\
&\in& {{\mathscr{F}}(B \times_f F)}. \\
{{\mathcal{K}}}(X_B,Y_B,{{{}_\bullet}}){{\mathcal{K}}}(Z_F,T_B,{{{}_\bullet}})&=&{{\mathcal{K}}}(X_B,Y_B,{{{}_\bullet}}){{\mathcal{K}}}(Z_B,T_F,{{{}_\bullet}})\\
&=&{{\mathcal{K}}}(X_B,Y_B,{{{}_\bullet}}_B){{\mathcal{K}}}(Z_B,T_F,{{{}_\bullet}}_B) \\
&&+{{\mathcal{K}}}(X_B,Y_B,{{{}_\bullet}}_F){{\mathcal{K}}}(Z_B,T_F,{{{}_\bullet}}_F) = 0. \\
{{\mathcal{K}}}(X_B,Y_B,{{{}_\bullet}}){{\mathcal{K}}}(Z_F,T_F,{{{}_\bullet}})&=&{{\mathcal{K}}}(X_B,Y_B,{{{}_\bullet}}_B){{\mathcal{K}}}(Z_F,T_F,{{{}_\bullet}}_B) \\
&&+{{\mathcal{K}}}(X_B,Y_B,{{{}_\bullet}}_F){{\mathcal{K}}}(Z_F,T_F,{{{}_\bullet}}_F) \\
&=&-{{\mathcal{K}}}_B(X_B,Y_B,{{{}_\bullet}}_B)f {\langleZ_F,T_F\rangle}_F {\textnormal{d}}f({{{}_\bullet}}_B) \\
&=&-f{\langleZ_F,T_F\rangle}_F (\nabla^B_{X_B}{Y_B})({\textnormal{d}}f) \\
&\in& {{\mathscr{F}}(B \times_f F)}. \\
{{\mathcal{K}}}(X_B,Y_F,{{{}_\bullet}}){{\mathcal{K}}}(T_F,Z_B,{{{}_\bullet}})&=&{{\mathcal{K}}}(X_B,Y_F,{{{}_\bullet}}){{\mathcal{K}}}(Z_B,T_F,{{{}_\bullet}})\\
&=&{{\mathcal{K}}}(X_B,Y_F,{{{}_\bullet}}_B){{\mathcal{K}}}(Z_B,T_F,{{{}_\bullet}}_B) \\
&&+{{\mathcal{K}}}(X_B,Y_F,{{{}_\bullet}}_F){{\mathcal{K}}}(Z_B,T_F,{{{}_\bullet}}_F) \\
&=&f {\langleY_F,{{{}_\bullet}}_F\rangle}_F X_B(f){{\mathcal{K}}}(Z_B,T_F,{{{}_\bullet}}_F) \\
&=& f^3 X_B(f){{\mathcal{K}}}_F(Z_B,T_F,Y_F) \\
&\in& {{\mathscr{F}}(B \times_f F)}. \\
{{\mathcal{K}}}(X_B,Y_F,{{{}_\bullet}}){{\mathcal{K}}}(Z_F,T_F,{{{}_\bullet}})&=&{{\mathcal{K}}}(X_B,Y_F,{{{}_\bullet}}_B){{\mathcal{K}}}(Z_F,T_F,{{{}_\bullet}}_B) \\
&&+{{\mathcal{K}}}(X_B,Y_F,{{{}_\bullet}}_F){{\mathcal{K}}}(Z_F,T_F,{{{}_\bullet}}_F) \\
&=&f^3 X_B(f) {\langleY_F,{{{}_\bullet}}_F\rangle}_F{{\mathcal{K}}}_F(Z_F,T_F,{{{}_\bullet}}_F) \\
&=&f^3 X_B(f) {{\mathcal{K}}}_F(Z_F,T_F,Y_F) \\
&\in& {{\mathscr{F}}(B \times_f F)}. \\
\end{array}$$
Even though $(B,g_B)$ and $(F,g_F)$ are [[nondegenerate]{}]{} [[semiRiemannian]{}]{} manifolds, if the function $f$ becomes $0$, the warped product manifold $B\times_f F$ is a singular [[semiRiemannian]{}]{} manifold.
Let’s consider that $(B,g_B)$ is a [[nondegenerate]{}]{} [[semiRiemannian]{}]{} manifold, and let $f\in{{\mathscr{F}}(B)}$. If $(F,g_F)$ is [[radicalstationary]{}]{}, then the warped product $B\times_f F$ also is [[radicalstationary]{}]{}. If $(F,g_F)$ is [[semiregular]{}]{}, then the warped product $B\times_f F$ also is [[semiregular]{}]{}. In particular, if both manifolds $(B,g_B)$ and $(F,g_F)$ are [[nondegenerate]{}]{}, and the warping function $f\in{{\mathscr{F}}(B)}$, then $B\times_f F$ is [[semiregular]{}]{}.
If the manifold $(B,g_B)$ is [[nondegenerate]{}]{}, then any function $f\in{{\mathscr{F}}(B)}$ also satisfies ${\textnormal{d}}f\in{{{{\mathcal{A}}{}^{\bullet}{}}}(B)}$ and ${\textnormal{d}}f\in{{{{\mathscr{A}}{}^{\bullet}{}}}{}^{1}(B)}$. Then the corollary follows from Theorems \[thm\_rad\_stat\_semi\_riem\_man\_warped\] and \[thm\_semi\_reg\_semi\_riem\_man\_warped\].
$B\times_0 F$ is a singular [[semiRiemannian]{}]{} manifold with degenerate metric of constant ${\textnormal{rank }}g=\dim B$.
The proof can be found in [@Kup87b][287]{}. In fact, Kupeli does even more in [@Kup87b], by showing that any [[radicalstationary]{}]{} [[semiRiemannian]{}]{} manifold is locally a warped product of the form $B\times_0 F$.
The warped product of [[nondegenerate]{}]{} [[semiRiemannian]{}]{} manifolds stays [[nondegenerate]{}]{} for $f>0$. If $f\to 0$, we can see for example from [@ONe83] that the connection $\nabla$ ([@ONe83][206–207]{}), the Riemann curvature $R_\nabla$ ([@ONe83][209–210]{}), the Ricci tensor ${{\textnormal}{Ric}}$ and the scalar curvature $s$ ([@ONe83][211]{}) diverge in general.
Riemann curvature of [[semiregular]{}]{} warped products {#s_riemann_wp_deg}
========================================================
In this section we will assume $(B,g_B)$ and $(F,g_F)$ to be [[semiregular]{}]{} [[semiRiemannian]{}]{} manifolds, $f\in{{\mathscr{F}}(B)}$ a smooth function so that ${\textnormal{d}}f\in{{{{\mathscr{A}}{}^{\bullet}{}}}{}^{1}(B)}$, and $B\times_f F$ the warped product of $B$ and $F$. The central point is to find the relation between the Riemann curvature $R$ of $B\times_f F$ and those on $(B,g_B)$ and $(F,g_F)$. The relations are similar to those for the [[nondegenerate]{}]{} case [([*cf.* ]{}[[@ONe83], p. 210–211]{})]{} for the Riemann curvature operator $R(\_,\_)$, but since this operator is not well defined and is divergent for degenerate metric, we need to use the Riemann curvature tensor $R(\_,\_,\_,\_)$. The proofs given here are based only on formulae which work for the degenerate case as well.
\[def\_hessian\] Let $(M,g)$ be a [[semiregular]{}]{} [[semiRiemannian]{}]{} manifold. The *Hessian* of a scalar field $f$ satisfying ${\textnormal{d}}f\in{{{{\mathscr{A}}{}^{\bullet}{}}}{}^{1}(M)}$ is the smooth tensor field $H^f\in{{\mathcal{T}}{}^{0}_{2}M}$ defined by $$H^f(X,Y) := \left({\nabla}_X{\textnormal{d}}f\right)(Y)$$ for any $X,Y\in{{{\mathfrak{X}}}(M)}$.
\[thm\_wp\_nondeg\_riemm\_tens\] Let $B \times_f F$ be a degenerate warped product of [[semiregular]{}]{} [[semiRiemannian]{}]{} manifolds with $f\in{{\mathscr{F}}(B)}$ a smooth function so that ${\textnormal{d}}f\in{{{{\mathscr{A}}{}^{\bullet}{}}}{}^{1}(B)}$, and $R_B, R_F$ the lifts of the Riemann curvature tensors on $B$ and $F$. Let $X,Y,Z,T\in{{\mathfrak{L}}(B \times F,B)}$, $U,V,W,Q\in{{\mathfrak{L}}(B \times F,F)}$, and let $H^f$ be the *Hessian* of $f$ (which exists because ${\textnormal{d}}f\in{{{{\mathscr{A}}{}^{\bullet}{}}}{}^{1}(B)}$, see Definition \[def\_hessian\]. Then:
1. \[thm\_wp\_nondeg\_riemm\_tens:BBBB\] $R(X,Y,Z,T) = R_B(X,Y,Z,T)$
2. \[thm\_wp\_nondeg\_riemm\_tens:BBBF\] $R(X,Y,Z,Q) = 0$
3. \[thm\_wp\_nondeg\_riemm\_tens:BBFF\] $R(X,Y,W,Q) = 0$
4. \[thm\_wp\_nondeg\_riemm\_tens:FFBF\] $R(U,V,Z,Q) = 0$
5. \[thm\_wp\_nondeg\_riemm\_tens:BFFB\] $R(X,V,W,T) = -fH^f(X,T){\langleV,W\rangle}_F$
6. \[thm\_wp\_nondeg\_riemm\_tens:FFFF\] $\begin{aligned}[t]
R(U,V,W,Q)=&R_F(U,V,W,Q) \\
&+ f^2 {{{\langle\!\langle{\textnormal{d}}f,{\textnormal{d}}f\rangle\!\rangle{}_{\bullet}{}}}}_B\big({\langleU,W\rangle}_F{\langleV,Q\rangle}_F \\
&- {\langleV,W\rangle}_F{\langleU,Q\rangle}_F\big)
\end{aligned}$
the other cases being obtained by the symmetries of the Riemann curvature tensor.
In order to prove these identities, we will use the Koszul formula for the Riemann curvature from equation . We will denote the covariant contraction with ${{{}_\bullet}}$ on $B \times_f F$, and with $\stackrel B{{{{}_\bullet}}}$ and $\stackrel F{{{{}_\bullet}}}$ on $B$, respectively $F$.
$$\begin{array}{llll}
\eqref{thm_wp_nondeg_riemm_tens:BBBB}&
R(X,Y,Z,T)&=& X {{\mathcal{K}}}(Y,Z,T) - Y {{\mathcal{K}}}(X,Z,T) - {{\mathcal{K}}}([X,Y],Z,T)\\
&&& + {{\mathcal{K}}}(X,Z,{{{}_\bullet}}){{\mathcal{K}}}(Y,T,{{{}_\bullet}}) - {{\mathcal{K}}}(Y,Z,{{{}_\bullet}}){{\mathcal{K}}}(X,T,{{{}_\bullet}}) \\
&&=& X {{\mathcal{K}}}(Y,Z,T) - Y {{\mathcal{K}}}(X,Z,T) - {{\mathcal{K}}}([X,Y],Z,T)\\
&&& + {{\mathcal{K}}}(X,Z,\stackrel B{{{{}_\bullet}}}){{\mathcal{K}}}(Y,T,\stackrel B{{{{}_\bullet}}}) - {{\mathcal{K}}}(Y,Z,\stackrel B{{{{}_\bullet}}}){{\mathcal{K}}}(X,T,\stackrel B{{{{}_\bullet}}}) \\
&&=& R_B(X,Y,Z,T)
\end{array}$$
where we applied from the Proposition \[thm\_wp\_deg\_koszul\]. $$\begin{array}{llll}
\eqref{thm_wp_nondeg_riemm_tens:BBBF}&
R(X,Y,Z,Q)&=& X {{\mathcal{K}}}(Y,Z,Q) - Y {{\mathcal{K}}}(X,Z,Q) - {{\mathcal{K}}}([X,Y],Z,Q)\\
&&& + {{\mathcal{K}}}(X,Z,{{{}_\bullet}}){{\mathcal{K}}}(Y,Q,{{{}_\bullet}}) - {{\mathcal{K}}}(Y,Z,{{{}_\bullet}}){{\mathcal{K}}}(X,Q,{{{}_\bullet}}) \\
&&=& {{\mathcal{K}}}(X,Z,{{{}_\bullet}}){{\mathcal{K}}}(Y,Q,{{{}_\bullet}}) - {{\mathcal{K}}}(Y,Z,{{{}_\bullet}}){{\mathcal{K}}}(X,Q,{{{}_\bullet}}) \\
&&=& {{\mathcal{K}}}(X,Z,\stackrel B{{{{}_\bullet}}}){{\mathcal{K}}}(Y,Q,\stackrel B{{{{}_\bullet}}}) - {{\mathcal{K}}}(Y,Z,\stackrel B{{{{}_\bullet}}}){{\mathcal{K}}}(X,Q,\stackrel B{{{{}_\bullet}}}) \\
&&=& 0
\end{array}$$ by the same property, which also leads to $$\begin{array}{llll}
\eqref{thm_wp_nondeg_riemm_tens:BBFF}&
R(X,Y,W,Q)&=& X {{\mathcal{K}}}(Y,W,Q) - Y {{\mathcal{K}}}(X,W,Q) - {{\mathcal{K}}}([X,Y],W,Q)\\
&&& + {{\mathcal{K}}}(X,W,{{{}_\bullet}}){{\mathcal{K}}}(Y,Q,{{{}_\bullet}}) - {{\mathcal{K}}}(Y,W,{{{}_\bullet}}){{\mathcal{K}}}(X,Q,{{{}_\bullet}}) \\
&&=& {{\mathcal{K}}}(X,W,{{{}_\bullet}}){{\mathcal{K}}}(Y,Q,{{{}_\bullet}}) - {{\mathcal{K}}}(Y,W,{{{}_\bullet}}){{\mathcal{K}}}(X,Q,{{{}_\bullet}}) \\
&&=& 0.
\end{array}$$ $$\begin{array}{llll}
\eqref{thm_wp_nondeg_riemm_tens:FFBF}&
R(U,V,Z,Q)&=& U {{\mathcal{K}}}(V,Z,Q) - V {{\mathcal{K}}}(U,Z,Q) - {{\mathcal{K}}}([U,V],Z,Q)\\
&&& + {{\mathcal{K}}}(U,Z,{{{}_\bullet}}){{\mathcal{K}}}(V,Q,{{{}_\bullet}}) - {{\mathcal{K}}}(V,Z,{{{}_\bullet}}){{\mathcal{K}}}(U,Q,{{{}_\bullet}}) \\
&&=& U \left(f{\langleV,Q\rangle}_FZ(f)\right) - V \left(f{\langleU,Q\rangle}_FZ(f)\right) \\
&&&- f{\langle[U,V],Q\rangle}_FZ(f) \\
&&& + {{\mathcal{K}}}(U,Z,\stackrel B{{{{}_\bullet}}}){{\mathcal{K}}}(V,Q,\stackrel B{{{{}_\bullet}}}) - {{\mathcal{K}}}(V,Z,\stackrel B{{{{}_\bullet}}}){{\mathcal{K}}}(U,Q,\stackrel B{{{{}_\bullet}}}) \\
&&& + {{\mathcal{K}}}(U,Z,\stackrel F{{{{}_\bullet}}}){{\mathcal{K}}}(V,Q,\stackrel F{{{{}_\bullet}}}) - {{\mathcal{K}}}(V,Z,\stackrel F{{{{}_\bullet}}}){{\mathcal{K}}}(U,Q,\stackrel F{{{{}_\bullet}}}) \\
&&=& f Z(f) \left(U {\langleV,Q\rangle}_F - V {\langleU,Q\rangle}_F - {\langle[U,V],Q\rangle}_F\right) \\
&&& + {{\mathcal{K}}}(U,Z,\stackrel F{{{{}_\bullet}}}){{\mathcal{K}}}(V,Q,\stackrel F{{{{}_\bullet}}})_F - {{\mathcal{K}}}(V,Z,\stackrel F{{{{}_\bullet}}}){{\mathcal{K}}}(U,Q,\stackrel F{{{{}_\bullet}}})_F \\
&&=& f Z(f) \left(U {\langleV,Q\rangle}_F - V {\langleU,Q\rangle}_F - {\langle[U,V],Q\rangle}_F\right) \\
&&& + f{\langleU,\stackrel F{{{{}_\bullet}}}\rangle}_FZ(f) {{\mathcal{K}}}(V,Q,\stackrel F{{{{}_\bullet}}})_F \\
&&& -f{\langleV,\stackrel F{{{{}_\bullet}}}\rangle}_FZ(f){{\mathcal{K}}}(U,Q,\stackrel F{{{{}_\bullet}}})_F \\
&&=& f Z(f) (U {\langleV,Q\rangle}_F - V {\langleU,Q\rangle}_F - {\langle[U,V],Q\rangle}_F ) \\
&&& + {{\mathcal{K}}}(V,Q,U)_F - {{\mathcal{K}}}(U,Q,V))_F \\
&&=& 0 \\
\end{array}$$ where we used and from the Proposition \[thm\_wp\_deg\_koszul\], together with the Definition \[def\_Koszul\_form\]. We also used the property that the covariant contraction on $F$ cancels the coefficient $f^2$ of ${{\mathcal{K}}}(U,V,W)_F$.
$$\begin{array}{llll}
\eqref{thm_wp_nondeg_riemm_tens:BFFB}&
R(X,V,W,T)&=& X {{\mathcal{K}}}(V,W,T) - V {{\mathcal{K}}}(X,W,T) - {{\mathcal{K}}}([X,V],W,T)\\
&&& + {{\mathcal{K}}}(X,W,{{{}_\bullet}}){{\mathcal{K}}}(V,T,{{{}_\bullet}}) - {{\mathcal{K}}}(V,W,{{{}_\bullet}}){{\mathcal{K}}}(X,T,{{{}_\bullet}}) \\
&&=& - X \left(f T(f) {\langleV,W\rangle}_F\right) \\
&&& - {{\mathcal{K}}}(V,W,\stackrel B{{{{}_\bullet}}}){{\mathcal{K}}}(X,T,\stackrel B{{{{}_\bullet}}}) \\
&&& + {{\mathcal{K}}}(X,W,\stackrel F{{{{}_\bullet}}}){{\mathcal{K}}}(V,T,\stackrel F{{{{}_\bullet}}})_F \\
&&=& - X \left(f T(f) {\langleV,W\rangle}_F\right) \\
&&& + f{\langleV,W\rangle}_F{\textnormal{d}}f({{{}_\bullet}}) {{\mathcal{K}}}(X,T,\stackrel B{{{{}_\bullet}}})_B \\
&&& + X(f) {\langleW,\stackrel F{{{{}_\bullet}}}\rangle}_F T(f) {\langleV,\stackrel F{{{{}_\bullet}}}\rangle}_F \\
&&=& - X(f) T(f) {\langleV,W\rangle}_F - fX(T(f)) {\langleV,W\rangle}_F \\
&&& + f{\langleV,W\rangle}_F {{\mathcal{K}}}(X,T,\stackrel B{{{{}_\bullet}}})_B{\textnormal{d}}f(\stackrel B{{{{}_\bullet}}}) \\
&&& + X(f) T(f) {\langleW,V\rangle}_F \\
&&=& f{\langleV,W\rangle}_F \left[{{\mathcal{K}}}(X,T,\stackrel B{{{{}_\bullet}}})_B{\textnormal{d}}f(\stackrel B{{{{}_\bullet}}}) - X(T(f))\right] \\
&&=& f{\langleV,W\rangle}_F \left[{{\mathcal{K}}}(X,T,\stackrel B{{{{}_\bullet}}})_B{\textnormal{d}}f(\stackrel B{{{{}_\bullet}}}) - X{\langleT,{\textnormal{grad }}f\rangle}_B\right] \\
&&=& -f H^f(X,T){\langleV,W\rangle}_F \\
\end{array}$$
where we applied the definition of the Hessian for [[semiregular]{}]{} [[semiRiemannian]{}]{} manifolds, for $f$ so that ${\textnormal{d}}f\in{{{{\mathscr{A}}{}^{\bullet}{}}}{}^{1}(B)}$, and the properties of the Koszul derivative of warped products, as in the Proposition \[thm\_wp\_deg\_koszul\].
$$\begin{array}{llll}
\eqref{thm_wp_nondeg_riemm_tens:FFFF}&
R(U,V,W,Q)&=& U {{\mathcal{K}}}(V,W,Q) - V {{\mathcal{K}}}(U,W,Q) - {{\mathcal{K}}}([U,V],W,Q)\\
&&& + {{\mathcal{K}}}(U,W,{{{}_\bullet}}){{\mathcal{K}}}(V,Q,{{{}_\bullet}}) - {{\mathcal{K}}}(V,W,{{{}_\bullet}}){{\mathcal{K}}}(U,Q,{{{}_\bullet}}) \\
&&=& R_F(U,V,W,Q)\\
&&& + {{\mathcal{K}}}(U,W,\stackrel B{{{{}_\bullet}}}){{\mathcal{K}}}(V,Q,\stackrel B{{{{}_\bullet}}}) - {{\mathcal{K}}}(V,W,\stackrel B{{{{}_\bullet}}}){{\mathcal{K}}}(U,Q,\stackrel B{{{{}_\bullet}}}) \\
&&=& R_F(U,V,W,Q)\\
&&& + f^2 {\langleU,W\rangle}_F{\textnormal{d}}f(\stackrel B{{{{}_\bullet}}}){\langleV,Q\rangle}_F{\textnormal{d}}f(\stackrel B{{{{}_\bullet}}}) \\
&&& - f^2 {\langleV,W\rangle}_F{\textnormal{d}}f(\stackrel B{{{{}_\bullet}}}){\langleU,Q\rangle}_F{\textnormal{d}}f(\stackrel B{{{{}_\bullet}}}) \\
&&=& R_F(U,V,W,Q)\\
&&& + f^2 {{{\langle\!\langle{\textnormal{d}}f,{\textnormal{d}}f\rangle\!\rangle{}_{\bullet}{}}}}_B\big({\langleU,W\rangle}_F{\langleV,Q\rangle}_F \\
&&&- {\langleV,W\rangle}_F{\langleU,Q\rangle}_F\big) \\
\end{array}$$
Despite the fact that the Riemann tensor $R(\_,\_)$ is divergent when the warping function converges to $0$ even for warped products of [[nondegenerate]{}]{} metrics ([@ONe83][209–210]{}), Theorem \[thm\_wp\_nondeg\_riemm\_tens\] shows again that the Riemann curvature tensor $R(\_,\_,\_,\_)$ is smooth.
Polar and spherical warped products {#s_riemann_wp_spherical}
===================================
In the following, we use the degenerate inner product of [[semiregular]{}]{} manifolds to construct other manifolds. We start by providing a recipe to obtain from warped products spherical solutions of various dimension.
Polar warped products {#s_riemann_wp_spherical:polar:general}
---------------------
Let $\mu,\rho\in{{\mathscr{F}}({\mathbb{R}})}$ be smooth real functions so that $\mu^2(-r)=\mu^2(r)$ and $\rho^2(-r)=\rho^2(r)$ for any $r\in{\mathbb{R}}$, $i\in\{1,2\}$. We can construct the following warped products between the spaces $({\mathbb{R}},\pm\mu^2{\textnormal{d}}r\otimes{\textnormal{d}}r)$ and $S^1$: $$({\mathbb{R}}\times_rS^1, \pm\mu^2 {\textnormal{d}}r\otimes{\textnormal{d}}r + \rho^2{\textnormal{d}}\vartheta\otimes{\textnormal{d}}\vartheta).$$
We define on ${\mathbb{R}}\times_rS^1$ the equivalence relation $(r_1,\vartheta_1)\sim(r_2,\vartheta_2)$ if and only if either $r_1=r_2$ and $\vartheta_1=\vartheta_2$, or $r_1=-r_2$ and $\vartheta_1\equiv(\vartheta_2 + \pi) \mod 2\pi$.
The manifold $(M,g):=({\mathbb{R}},\pm\mu^2{\textnormal{d}}r\otimes{\textnormal{d}}r)\times_\rho S^1/\sim$ is named the *polar warped product* between $({\mathbb{R}},\pm\mu^2{\textnormal{d}}r\otimes{\textnormal{d}}r)$ and $S^1$.
The manifold $M$ is diffeomorphic to ${\mathbb{R}}^2$.
We are looking for conditions which ensure the smoothness of the metric $g$ on $M$.
The metric $g$ on $M$ is smooth if and only if the following limit exists and is smooth: $$\label{eq_polar_wp_condition}
{\displaystyle}{\lim_{r\to 0} \frac{\pm\mu^2r^2 - \rho^2}{r^4}}.$$
The metric on ${\mathbb{R}}^2-\{(0,0)\}$ is, in Cartesian coordinates: $$\begin{array}{lll}
g&=&
{{\displaystyle}{\frac{1}{r^2}}}
\left(
\begin{array}{ll}
r\cos\vartheta & -\sin\vartheta \\
r\sin\vartheta & \cos\vartheta
\end{array}
\right)
\left(
\begin{array}{ll}
\pm\mu^2 & 0 \\
0 & \rho^2
\end{array}
\right)
\left(
\begin{array}{ll}
r\cos\vartheta & r\sin\vartheta \\
-\sin\vartheta & \cos\vartheta
\end{array}
\right) \\
&=&
{{\displaystyle}{\frac{1}{r^2}}}\left(
\begin{array}{ll}
\pm\mu^2r^2\cos^2\vartheta + \rho^2\sin^2\vartheta & (\pm\mu^2r^2-\rho^2)\sin\vartheta\cos\vartheta \\
(\pm\mu^2r^2-\rho^2)\sin\vartheta\cos\vartheta & \pm\mu^2r^2\sin^2\vartheta + \rho^2\cos^2\vartheta
\end{array}
\right) \\
&=&
{{\displaystyle}{\frac{1}{r^2}}}\left(
\begin{array}{ll}
\pm\mu^2r^2 - (\pm\mu^2r^2 - \rho^2)\sin^2\vartheta & (\pm\mu^2r^2-\rho^2)\sin\vartheta\cos\vartheta \\
(\pm\mu^2r^2-\rho^2)\sin\vartheta\cos\vartheta & \pm\mu^2r^2 - (\pm\mu^2r^2 - \rho^2)\cos^2\vartheta
\end{array}
\right) \\
&=&
{{\displaystyle}{\frac{1}{r^2}}}\left(
\begin{array}{ll}
\pm\mu^2r^2 - {{\displaystyle}{\frac{\pm\mu^2r^2 - \rho^2}{r^2}}} y^2 & {{\displaystyle}{\frac{\pm\mu^2r^2 - \rho^2}{r^2}}} x y \\
{{\displaystyle}{\frac{\pm\mu^2r^2 - \rho^2}{r^2}}} x y & \pm\mu^2r^2 - {{\displaystyle}{\frac{\pm\mu^2r^2 - \rho^2}{r^2}}} x^2
\end{array}
\right)\\
\end{array}$$ Hence, $g$ is smooth if and only if the limit exists and is smooth.
The smoothness of $g$ on $M$ is ensured by the condition that $\rho^2(r)=\pm\mu^2r^2 + u(r) r^4$ for some smooth function $u:{\mathbb{R}}\to{\mathbb{R}}$.
The metric becomes, in Cartesian coordinates, $$\label{eq_polar_metric_u}
\begin{array}{lll}
g&=&
\left(
\begin{array}{ll}
\pm\mu^2 + u y^2 & -u x y \\
-u x y & \pm\mu^2 + u x^2
\end{array}
\right)\\
\end{array}$$ The determinant of the metric is $$\det g = \mu^4 \pm u \mu^2 r^2,$$ and it follows that the metric becomes degenerate if $\mu=0$ or $\mu^2 = \pm u r^2$.
If we want the metric to be [[semiregular]{}]{}, we need to make sure that the equation is respected. Since the coefficients $\mu$ and $\rho$ depend only on $r$, it suffices that ${{\textnormal}{supp}(\partial_r\mu)}\subseteq{{\textnormal}{supp}(\mu)}$ and that there exists a smooth function $f\in{{\mathscr{F}}({\mathbb{R}})}$ so that ${{\textnormal}{supp}(f)}\subseteq{{\textnormal}{supp}(\mu)}$ and $$\label{eq_diagonal_metric:semireg:polar}
{{\displaystyle}{\frac{\partial\rho^2(r)}{\partial r}}}=f(r)\mu(r).$$
The next example shows how we can obtain the Euclidean plane ${\mathbb{R}}^2$ from a degenerate warped product.
\[s\_riemann\_wp\_spherical:polar\] The flat metric on ${\mathbb{R}}^2-\{(0, 0)\}$ can be expressed in polar coordinates $(r,\vartheta)$ as $$\label{eq_riemann_wp_deg_applications:polar}
g={\textnormal{d}}r\otimes{\textnormal{d}}r + r^2{\textnormal{d}}\vartheta\otimes{\textnormal{d}}\vartheta.$$ The manifold ${\mathbb{R}}^2-\{(0, 0)\}$ can be obtained as the [[nondegenerate]{}]{} warped product ${\mathbb{R}}^+\times_rS^1$, where ${\mathbb{R}}^+=(0,\infty)$, with the natural metric ${\textnormal{d}}r^2$, and $S^1$ is the unit circle parameterized by $\vartheta$, with the metric ${\textnormal{d}}\vartheta^2$. The metric of ${\mathbb{R}}^+\times_rS^1$ becomes degenerate at the point $r=0$. We can use the degenerate warped product ${\mathbb{R}}\times_rS^1$, where the metric has the same form as in equation , and obtain a cylinder whose metric becomes degenerate at the points $r=0$. The coordinate $r$ is allowed here to become $0$ or negative. The polar warped product $M= {\mathbb{R}}\times_rS^1/\sim$ is isometric to the Euclidean space ${\mathbb{R}}^2$.
The following example shows how we can obtain the sphere $S^2$ from a degenerate warped product.
\[s\_riemann\_wp\_spherical:sphere\] Let’s rename the coordinate $r$ to $\varphi$, let’s take instead of $\rho(r)$ the function $\sin\varphi$, and let’s make the metric on ${\mathbb{R}}$ to be ${\textnormal{d}}\varphi\otimes{\textnormal{d}}\varphi$ (hence $\mu^2(\varphi)=1$). Since $\sin\varphi = \varphi - {{\displaystyle}{\frac{\varphi^3}{3!}}} + {{\displaystyle}{\frac{\varphi^5}{5!}}} - \ldots$, it follows that $\sin\varphi = \varphi - \varphi^3 h(\varphi)$, where $h$ is a smooth function. Hence, $\sin^2\varphi=\varphi^2 + u(\varphi)\varphi^4$, where $u(\varphi) = -2 h(\varphi) + \varphi^2 h^2(\varphi)$ is a smooth function, and the smoothness of the metric $g$ at $(0,0)$ is ensured. Let us now use instead the equivalence relation from [§\[s\_riemann\_wp\_spherical:polar:general\]]{}, the relation defined by $(\varphi_1,\vartheta_1)\sim(\varphi_2,\vartheta_2)$ if and only if either $\varphi_1\equiv \varphi_2 \mod 2\pi$ and $\vartheta_1=\vartheta_2$, or $\varphi_1\equiv -\varphi_2 \mod 2\pi$ and $\vartheta_1\equiv(\vartheta_2 + \pi) \mod 2\pi$. We obtain the sphere $S^2\cong{\mathbb{R}}\times_{\sin\varphi} S^1/\sim$, having the metric $$\label{eq_metric_sphere}
g_{S^2} = {\textnormal{d}}\varphi\otimes{\textnormal{d}}\varphi + \sin^2\varphi{\textnormal{d}}\vartheta\otimes{\textnormal{d}}\vartheta.$$ The usual *spherical coordinates* can be obtained by restraining the coordinates $(\vartheta,\varphi)$ to the domain $[0,2\pi)\times[0,\pi]$.
Spherical warped products {#s_riemann_wp_spherical:spherical:general}
-------------------------
In a similar manner as in [§\[s\_riemann\_wp\_spherical:polar:general\]]{}, we can define *spherical warped products*. We will work on ${\mathbb{R}}\times_{\rho}S^2$, where the sphere $S^2$ has the metric and parameterization as in the Example \[s\_riemann\_wp\_spherical:sphere\]. The equivalence relation is defined as $(r_1,\vartheta_1,\varphi_1)\sim(r_2,\vartheta_2,\varphi_2)$ if and only if either $r_1=r_2$ and $\vartheta_1=\vartheta_2$ and $\varphi_1=\varphi_2$, or $r_1=-r_2$ and $\vartheta_1\equiv(\vartheta_2 + \pi) \mod 2\pi$ and $\varphi_1=\varphi_2$. We start with real smooth functions $\mu,\rho\in{{\mathscr{F}}({\mathbb{R}})}$ so that $\mu^2(-r)=\mu^2(r)$ and $\rho^2(-r)=\rho^2(r)$ for any $r\in{\mathbb{R}}$, $i\in\{1,2\}$, exactly as in the polar case. We can construct the following warped products, between the spaces $({\mathbb{R}},\pm\mu^2{\textnormal{d}}r\otimes{\textnormal{d}}r)$ and $S^2$: $$\left({\mathbb{R}}\times{\rho}S^2, \pm\mu^2 {\textnormal{d}}r\otimes{\textnormal{d}}r + \rho^2({\textnormal{d}}\varphi\otimes{\textnormal{d}}\varphi + \sin^2\varphi{\textnormal{d}}\vartheta\otimes{\textnormal{d}}\vartheta)\right).$$
Let $(M,g)=({\mathbb{R}},\pm\mu^2{\textnormal{d}}r\otimes{\textnormal{d}}r)\times_\rho S^2/\sim$. The manifold is $M={\mathbb{R}}^3$. From [§\[s\_riemann\_wp\_spherical:polar:general\]]{} it follows that for any plane of $M$ containing the axis ${\mathbb{R}}\times(0,0)$ the smoothness results from the condition $\rho^2(r)=\pm\mu^2r^2 + u(r) r^4$ for some function $u:{\mathbb{R}}\to{\mathbb{R}}$. The smoothness of $g$ in these planes ensures its smoothness on the entire $M$. Moreover, by similar considerations it follows that $M$ is [[semiregular]{}]{} from the same condition given by the equation .
The same method can be used to obtain *$n$-spherical warped products*, by factoring the warped product ${\mathbb{R}}\times_{\rho}S^n$.
\[s\_riemann\_wp\_spherical:spherical\] As a direct application we can obtain the Euclidean space ${\mathbb{R}}^3$ in spherical coordinates from the degenerate warped product ${\mathbb{R}}\times_r S^2$.
\[s\_riemann\_wp\_spherical:hypersphere\] Similar to the Example \[s\_riemann\_wp\_spherical:sphere\], we can define an equivalence $\sim$ so that the $3$-sphere $S^3$ can be obtained as the spherical warped product $S^3\cong{\mathbb{R}}\times_{\sin\gamma} S^2/\sim$, having the metric $$\label{eq_metric_hypersphere}
g_{S^3} = {\textnormal{d}}\gamma\otimes{\textnormal{d}}\gamma + \sin^2\gamma({\textnormal{d}}\varphi\otimes{\textnormal{d}}\varphi + \sin^2\varphi{\textnormal{d}}\vartheta\otimes{\textnormal{d}}\vartheta).$$
\[s\_riemann\_wp\_spherical:pseudohypersphere\] If in equation we replace $\sin^2\gamma$ with $\sinh^2\gamma$, and $\sim$ with the equivalence relation defined at the beginning of this section, we obtain the hyperbolic $3$-space $H^3$ of constant sectional curvature $-1$.
Applications of [[semiregular]{}]{} warped products {#s_riemann_wp_deg_applications}
===================================================
In this section we apply the spherical warped product to construct cosmological models and to model black holes having [[semiregular]{}]{} singularities.
The degenerate warped product allows the warping function to become $0$ at some points. Under the hypothesis of the Theorem \[thm\_riemann\_curvature\_semi\_regular\] the Riemann curvature still remains well-defined and smooth. As we shown in [@Sto11a], for a smooth Riemann curvature tensor of a four-dimensional [[semiregular]{}]{} manifold we can write a *densitized version of Einstein’s equation* which remains smooth, and which reduces to the standard version if the metric is non-degenerate: $$\label{eq_einstein_idx:densitized}
G\det g + \Lambda g\det g = \kappa T\det g,$$ where $G={{\textnormal}{Ric}}- \frac 1 2 sg$, $T$ is the stress-energy tensor, $\kappa:={{\displaystyle}{\frac{8\pi {\mathcal{G}}}{c^4}}}$, ${\mathcal{G}}$ is Newton’s constant and $c$ the speed of light.
The generalization of the warped product we propose here provides a powerful method to resolve singularities in cosmology. If we show that a singularity can be obtained as a degenerate warped product of [[semiregular]{}]{} (in particular non-degenerate) manifolds, it follows that the densitized version of the Einstein equation is smooth at that singularity.
Cosmological models {#s_riemann_wp_deg_applications:FLRW}
-------------------
In the following we propose a generalization of the *Friedmann-Lemaître-Robertson-Walker* spacetime, allowing singularities.
If $(\Sigma,g_{\Sigma})$ is a connected three-dimensional Riemannian manifold of constant sectional curvature $k\in\{-1,0,1\}$ ([*i.e.* ]{}$H^3$, ${\mathbb{R}}^3$ or $S^3$) and $a\in (t_1,t_2)$, $-\infty\leq t_1 < t_2\leq \infty$, $a\geq 0$, then the warped product $(t_1,t_2) \times_a \Sigma$ is called a *Friedmann-Lemaître-Robertson-Walker* spacetime: $$g = -{\textnormal{d}}t\otimes{\textnormal{d}}t + a^2(t) g_{\Sigma}$$
For more generality, we use instead of the constant metric on ${\mathbb{R}}$, a metric $-\mu^2{\textnormal{d}}t\otimes{\textnormal{d}}t$, where $\mu\in{{\mathscr{F}}({\mathbb{R}})}$, and we allow $a$ to become $0$ or negative. The warped product becomes then $$g = -\mu^2{\textnormal{d}}t\otimes{\textnormal{d}}t + a^2(t) g_{\Sigma}$$ The singularities are [[semiregular]{}]{} if there exists a smooth function $f\in{{\mathscr{F}}({\mathbb{R}})}$ so that ${{\textnormal}{supp}(f)}\subseteq{{\textnormal}{supp}(\mu)}$ and $$\label{eq_diagonal_metric:semireg:flrw}
{{\displaystyle}{\frac{\partial a^2(t)}{\partial t}}}=f(t)\mu(t).$$
\[s\_riemann\_wp\_deg\_applications:cyclic\_cosmology\] By allowing $a$ to become $0$, we can study cosmological models in which the evolution equation can pass through the singularities [^2].
\[s\_riemann\_wp\_deg\_applications:semireg\_big\_bang\] Another possible application of the [[semiregular]{}]{} warped products to the Friedmann-Lemaître-Robertson-Walker spacetime is to make the Big Bang singularity to be [[semiregular]{}]{}. The model is obtained as an $n$-spherical warped product, as in the section [§\[s\_riemann\_wp\_spherical:spherical:general\]]{}. The initial singularity is smooth if $a$ is smooth on ${\mathbb{R}}$, and $a^2(t)=-\mu^2t^2 + u(t) t^4$ for some function $u$. It is [[semiregular]{}]{} if the condition from equation is satisfied at $t=0$. The condition $a^2(t)=-\mu^2t^2 + u(t) t^4$ shows that near $t=0$, $a^2(t)< 0$, and the metric becomes negative definite. This model is resemblant to the Hartle-Hawking one [@HH83], except that in the former the spacelike directions change the signature, while in the latter, the time direction changes it. In order to keep the metric Lorentzian, we have to make sure that $a^2\geq 0$. For this, the function $\mu$ has to be of the form $\mu=\widetilde\mu t$ for some smooth function $\widetilde\mu$, and since $$\label{eq_cond_a}
a^2=(u-\widetilde\mu^2)t^4$$ there has to exist a smooth function $\widetilde a\geq 0$ satisfying $$\label{eq_cond_smooth_round_lorentz_2}
\widetilde a^2=u-\widetilde\mu^2.$$ and $$\label{eq_cond_smooth_round_lorentz}
a = \widetilde a t^2.$$
Spherical black holes {#s_riemann_wp_deg_applications:black_holes}
---------------------
The spherically symmetric solutions $(M,g)$ in General Relativity are usually of the form $M={\mathbb{R}}\times{\mathbb{R}}^+\times S^2$, and the metric $g$ has the form $$g = -\alpha_t^2(t,r){\textnormal{d}}t\otimes{\textnormal{d}}t + \alpha_r^2(t,r){\textnormal{d}}r\otimes{\textnormal{d}}r + \rho^2(t,r)g_{S^2},$$ where $\alpha_t, \alpha_r, \rho \in {{\mathscr{F}}({\mathbb{R}}\times{\mathbb{R}}^+)}$ and $g_{S^2}$ is given by the equation . This spacetime is in fact the warped product $$({\mathbb{R}}\times{\mathbb{R}}^+)\times_{\rho(t,r)} S^2$$ between the [[semiRiemannian]{}]{} manifold $$\({\mathbb{R}}\times{\mathbb{R}}^+,-\alpha_t^2(t,r){\textnormal{d}}t\otimes{\textnormal{d}}t + \alpha_r^2(t,r){\textnormal{d}}r\otimes{\textnormal{d}}r\)$$ and the Riemannian manifold $S^2$, with warping function $\rho(t,r)$. This example is known for [[nondegenerate]{}]{} metric [([*cf.* ]{}[*e.g.* ]{}[[@Rin06], p. 228]{})]{}, and it usually has singularities at $r=0$.
Let $\widetilde M$ be a manifold ${\mathbb{R}}^2\times_\rho S^2$, where the functions $\alpha_t,\alpha_r,\rho\in{{\mathscr{F}}({\mathbb{R}}^2)}$ are taken so that $\alpha_t^2,\alpha_r^2$ and $\rho$ are symmetric in $r$. We can choose $\alpha_t,\alpha_r$ and $\rho$ so that, by identifying each point $(t, r, \vartheta, \varphi)$ with $(t, -r, (\vartheta+\pi)\mod 2\pi, \varphi)$, we obtain a smooth singular [[semiRiemannian]{}]{} manifold $M$. By appropriately choosing them, we can make the singularities at $r=0$ of $\widetilde M$ to be [[semiregular]{}]{}, or we can even remove them, similar to the examples from [§\[s\_riemann\_wp\_spherical\]]{}.
If at some points $\alpha_t$ or $\alpha_r$ becomes $0$, the metric on ${\mathbb{R}}^2$ is degenerate. From the condition it follows that if for any $a,b\in\{t,r\}$, $c\in\{a,b\}$, there are some functions $f_{abc}\in{{\mathscr{F}}({\mathbb{R}}^2)}$ satisfying ${{\textnormal}{supp}(f_{abc})}\subseteq{{\textnormal}{supp}(\alpha_c)}$, so that $$\label{eq_s_riemann_wp_deg_applications:spherically_symmetric:semireg}
\partial_a \alpha_b^2=f_{abc}\alpha_c,$$ then the manifold $\({\mathbb{R}}^2, -\alpha_t^2(t,r){\textnormal{d}}t\otimes{\textnormal{d}}t + \alpha_r^2(t,r){\textnormal{d}}r\otimes{\textnormal{d}}r\)$ is [[semiregular]{}]{}. Let us now find a condition ensuring that ${\textnormal{d}}\rho\in{{{{\mathscr{A}}{}^{\bullet}{}}}{}^{1}({\mathbb{R}}^2)}$ (and, by the Theorem \[thm\_semi\_reg\_semi\_riem\_man\_warped\], that the manifold $M$ is [[semiregular]{}]{}). From Definition \[def\_cov\_der\_covect\], ${\textnormal{d}}\rho\in{{{{\mathscr{A}}{}^{\bullet}{}}}{}^{1}({\mathbb{R}}^2)}$ is equivalent to ${\textnormal{d}}\rho\in{{{{\mathcal{A}}{}^{\bullet}{}}}{}^{1}({\mathbb{R}}^2)}$ and the condition $$\label{eq_s_riemann_wp_deg_applications:spherically_symmetric:cocontr}
{{{\langle\!\langle{{{{\nabla}^{\flat}}_{\partial_a}}{\partial_b}},{\textnormal{d}}\rho\rangle\!\rangle{}_{\bullet}{}}}} \in {{\mathscr{F}}({\mathbb{R}}^2)}$$ for any $a,b\in\{t,r\}$. From $({\textnormal{d}}\rho)(X)=X(\rho)$ it follows that the equation is equivalent to the existence of some functions $h_{abc}\in{{\mathscr{F}}({\mathbb{R}}^2)}$ satisfying ${{\textnormal}{supp}(h_{abc})}\subseteq{{\textnormal}{supp}(\alpha_c)}$, so that $$\label{eq_s_riemann_wp_deg_applications:spherically_symmetric:cocontr:coord}
{{\mathcal{K}}}_{abc}\partial_c\rho = h_{abc}{\alpha_c^2}.$$ Remember that the components of the Koszul form in a coordinate chart reduces to Christoffel’s symbols of the first kind, which, when the metric is diagonal, are of the form $\pm\frac 1 2\partial_a\alpha_b^2$. The condition becomes equivalent to a condition of the form $$\partial_a \alpha_b^2\partial_c\rho=h_{abc}\alpha_c^2$$ for all $a,b\in\{t,r\}$, $c\in\{a,b\}$. If there is a function $F_{c}\in{{\mathscr{F}}({\mathbb{R}}^2)}$ so that $$\label{eq_s_riemann_wp_deg_applications:spherically_symmetric:F}
\partial_c\rho=F_{c}\alpha_c,$$ from , we can take $h_{abc}=f_{abc}F_{c}$. This ensures that ${\textnormal{d}}\rho\in{{{{\mathscr{A}}{}^{\bullet}{}}}{}^{1}({\mathbb{R}}^2)}$, and it follows that $M$ is [[semiregular]{}]{}.
This discussion suggests a possibility to construct black holes having [[semiregular]{}]{} singularities, and being therefore compatible with the densitized version of Einstein’s equation. Such singularities don’t block the time evolution. They are compatible with a unitary evolution, and Hawking’s information loss paradox [@Haw73; @Haw76] may be avoided. We will develop this viewpoint in the following articles [^3].
[10]{}
J. K. Beem, P. E. Ehrlich, and Th. G. Powell, *[Warped Product Manifolds in Relativity]{}*, [Th. M. Rassias, G. M. Rassias (Eds.), Selected Studies, North-Holland, Amsterdam]{}, 1982, pp. 41–56.
A. Bejancu and K.L. Duggal, *[Lightlike Submanifolds of Semi-Riemannian Manifolds]{}*, Acta Appl. Math. **38** (1995), no. 2, 197–215.
R. L. Bishop and B. O’Neil, *[Manifolds of Negative Curvature]{}*, Trans. Amer. Math. Soc. (1969), no. 145, 1–49.
J.B. Hartle and S.W. Hawking, *[Wave Function of the Universe]{}*, Physical Review D **28** (1983), no. 12, 2960.
S. Hawking, *[Particle Creation by Black Holes]{}*, Comm. Math. Phys. (1973), no. 33, 323.
S. Hawking, *[Breakdown of Predictability in Gravitational Collapse]{}*, Phys. Rev. D (1976), no. 14, 2460.
D. Kupeli, *[Degenerate Manifolds]{}*, Geom. Dedicata **23** (1987), no. 3, 259–290.
D. Kupeli, *[Singular Semi-Riemannian Geometry]{}*, Kluwer Academic Publishers Group, 1996.
B. O’Neill, *[[Semi-Riemannian]{} Geometry with Applications to Relativity]{}*, Pure Appl. Math. (1983), no. 103, 468.
A. Pambira, *[Harmonic Morphisms Between Segenerate Semi-Riemannian Manifolds]{}*, Contributions to Algebra and Geometry **46** (2005), no. 1, 261–281.
Wolfgang Rindler, *[Relativity: Special, General, and Cosmological]{}*, [Oxford University Press, USA]{}, May 2006.
C. [Stoica]{}, *[[On Singular Semi-Riemannian Manifolds]{}]{}*, arXiv:math.DG /1105.0201 (2011), 36.
C. [Stoica]{}, *[[Schwarzschild Singularity is Semi-Regularizable]{}]{}*, arXiv:gr-qc /1111.4837 (2011), 8.
C. [Stoica]{}, *[[Analytic Reissner-Nordstrom Singularity]{}]{}*, arXiv:gr-qc /1111.4332 (2011), 11.
C. [Stoica]{}, *[[Kerr-Newman Solutions with Analytic Singularity and no Closed Timelike Curves]{}]{}*, arXiv:gr-qc /1111.7082 (2011), 9.
C. [Stoica]{}, *[[Big Bang singularity in the Friedmann-Lemaitre-Robertson-Walker spacetime]{}]{}*, arXiv:gr-qc /1112.4508 (2011), 11.
[^1]: Partially supported by Romanian Government grant PN II Idei 1187.
[^2]: In the meantime, in [@Sto11h], we applied the technique presented here in detail and we proved that the Friedmann-Lemaître-Robertson-Walker spacetime can be extended before the Big Bang singularity, and the densitized version of the Einstein equation remains smooth.
[^3]: In the meantime, we applied the ideas introduced in this article to resolve the singularities of the Schwarzschild [@Sto11e], Reissner-Nordström [@Sto11f] and Kerr-Newman [@Sto11g] black holes.
|
---
abstract: |
It is a classical important problem of differential topology by Thom; for a homology class of a compact manifold, can we realize this by a closed submanifold with no boundary? This is true if the degree of the class is smaller or equal to the half of the dimension of the outer manifold under the condition that the coefficient ring is $\mathbb{Z}/2\mathbb{Z}$. If the degree of the class is smaller or equal to $6$ or equal to $k-2$ or $k-1$ under the condition that the coefficient ring is $\mathbb{Z}$ where $k$ is the dimension of the manifold, then this is also true. As a specific study, for $4$-dimensional closed manifolds, the topologies (genera) of closed and connected surfaces realizing given 2nd homology classes have been actively studied, for example.
In the present paper, we consider the following similar problem; can we realize a homology class of a compact manifold by a homology class of an explicit closed manifold embedded in the (interior of the) given compact manifold? This problem is considered as a variant of previous problems. We present an affirmative answer via important theory in the singularity theory of differentiable maps: lifting a given smooth map to an embedding or obtaining an embedding such that the composition of this with the canonical projection is the given map. Presenting this application of lifting smooth maps and related fundamental propositions is also a main purpose of the present paper.
address: |
Institute of Mathematics for Industry, Kyushu University, 744 Motooka, Nishi-ku Fukuoka 819-0395, Japan\
TEL (Office): +81-92-802-4402\
FAX (Office): +81-92-802-4405
author:
- Naoki Kitazawa
title: 'Realizing a homology class of a compact manifold by a homology class of an explicit closed submanifold–a new approach to Thom’s works on homology classes of submanifolds-'
---
Introduction and fundamental notation and terminologies. {#sec:1}
========================================================
The following problem, essentially launched by Thom, is a classical important problem in differential topology ([@thom]). We discuss the problem and related problems here in the smooth category. Let $A$ be a module.
For a homology class $c \in H_j(X;A)$ of a closed manifold $X$, can we realize this by a closed submanifold $Y$ with no boundary or can we represent $c$ as $i_{\ast}({\nu}_Y)=c$ for a generator ${\nu}_{Y} \in H_{\dim Y}(Y;A)$ of the module $H_{\dim Y}(Y;A)$ and the inclusion $i:Y \rightarrow X$?
This is true if the degree of the class is smaller or equal to the half of the dimension of the outer manifold under the condition that the coefficient $A$ is the ring $\mathbb{Z}/2\mathbb{Z}$. If the degree of the class is smaller or equal to 6 or equal to $\dim X-2$ or $\dim X-1$ under the conditions that the outer manifold $X$ is orientable and that the coefficient is the ring $\mathbb{Z}$, then this is also true. [@bohrhankekotschick] is on classes we cannot represent in this way.
For a 2nd homology class of a $4$-dimensional closed manifold, how about the orientability and the genus of a closed and connected surface realizing this?
This gives various, explicit, important and interesting problems. See [@kronheimer] for example. In low dimensional differential topology, these kinds of problems are actively studied via technique on low dimensional topology, gauge theory and so on.
Problems studied in this paper related to these problems by Thom
----------------------------------------------------------------
In this paper, related to the problems before, we consider the following problem.
\[prob:3\] For a homology class $c \in H_j(X;A)$ of a compact manifold $X$, can we realize the homology class by a homology class of a closed manifold $Y$ satisfying $\partial Y=\emptyset$ embedded in the (interior of the) manifold or can we represent $c$ as $i_{\ast}(c^{\prime})=c$ for a class $c^{\prime} \in H_{j}(Y;A)$ of the module $H_{j}(Y;A)$ and the inclusion $i:Y \rightarrow X$? Moreover, can we obtain $Y$ in a constructive way?
Note that we consider manifolds of arbitrary dimensions in the present paper.
Fold maps and Reeb spaces
-------------------------
### Fold maps
We introduce terminologies on differentiable maps. A [*singular*]{} point of a differentiable map $c:X \rightarrow Y$ is a point at which the rank of the differential of the map drops or the image of the differential is smaller than the dimension of $Y$. A [*singular value*]{} of the map is a point realized as a value at a singular point of the map. The set $S(c)$ of all singular points is the [*singular set*]{} of the map. The [*singular value set*]{} is the image $c(S(c))$ of the singular set. The [*regular value set*]{} of the map is the complementary set $Y-c(S(c))$ of the singular value set and a [*regular value*]{} is a point in the regular value set.
Manifolds are assumed to be smooth or differentiable and of $C^{\infty}$ and so are maps between manifolds. A diffeomorphism on a manifold is always smooth and the [*diffeomorphism group*]{} of it is defined as the group of all diffeomorphisms on it.
\[def:1\] Let $m \geq n \geq 1$ be integers. A smooth map from an $m$-dimensional smooth manifold with no boundary into an $n$-dimensional smooth manifold ｗｉｔｈ no boundary is said to be a [*fold*]{} map if at each singular point $p$, the form is $$(x_1, \cdots, x_m) \mapsto (x_1,\cdots,x_{n-1},\sum_{k=n}^{m-i}{x_k}^2-\sum_{k=m-i+1}^{m}{x_k}^2)$$ for some coordinates and an integer $0 \leq i(p) \leq \frac{m-n+1}{2}$.
For a fold map in Definition \[def:1\], we can obtain the following two properties.
1. For any singular point $p$, the $i(p)$ is unique [(]{}$i(p)$ is called the [*index*]{} of $p$[)]{}.
2. The set consisting of all singular points of a fixed index of the map is a closed submanifold of dimension $n-1$ with no boundary of the domain and the restriction to the singular set is an immersion of codimension $1$.
A fold map is a [*special generic*]{} map if $i(p)=0$ for all singular points. A Morse function with exactly two singular points on a homotopy sphere and canonical projections of unit spheres are simplest examples of special generic maps. If $m=n$, then a fold map is always special generic.
### Reeb spaces
The [*Reeb space*]{} of a continuous map is defined as the space of all connected components of preimages of the map.
\[def:2\] Let $X$ and $Y$ be topological spaces. For a continuous map $c:X \rightarrow Y$, we define a relation on $X$ as $p_1 {\sim}_c p_2$ if and only if $p_1$ and $p_2$ are in a same connected component of $c^{-1}(p)$ for some $p \in Y$. Thus ${\sim}_{c}$ is an equivalence relation on $X$ and we denote the quotient space $X/{\sim}_c$ by $W_c$ and call it the [*Reeb space*]{} of $c$.
We denote the induced quotient map from $X$ into $W_c$ by $q_c$ and we can define $\bar{c}: W_c \rightarrow Y$ uniquely by $c=\bar{c} \circ q_c$ See also [@reeb] for example.
\[prop:1\] For a fold map, the Reeb space is a polyhedron whose dimension is equal to that of the target manifold.
For suitable classes of these maps, Reeb spaces inherit topological properties of the manifolds admitting the maps. See [@kitazawa], [@kitazawa2], [@kitazawa3] and [@saekisuzuoka] for example.
Organization of the present paper
---------------------------------
This paper concerns Problem \[prob:3\] before and also presents a new answer to important problems on differentiable maps between manifolds, lifting of maps, which is a key ingredient in an explicit problem related to Problem \[prob:3\]. The organization of the paper is as the following. In the next section, we introduce a class of [*standard-spherical*]{} fold maps. It is defined as a fold map such that indices of all singular points are $0$ or $1$, that preimages of regular values are points or disjoint unions of standard spheres: if the codimension of the map $f:M \rightarrow N$ is $n-m=-1$, then assume also that the domain is an orientable manifold. A [*simple*]{} fold map $f$ is a fold map such that ${q_f}{\mid}_{S(f)}$ is injective. Special generic maps form a proper subclass of these classes.
In the third section, we present examples of simple standard-spherical fold maps. After that, we consider lifting these maps to embeddings or obtaining embeddings such that the compositions of the embeddings with the canonical projections are the given maps. This is a fundamental, important and interesting studies in the theory of singularity of differentiable maps and this gives strong tools in obtaining main results of the present paper. We also present a new answer for a kind of these problems as Theorem \[thm:1\]: this is also an important ingredient in main results. The last section is devoted to main results or explicit answers to Problem \[prob:3\] via tools and theory in the third section.
[^1]
Special generic maps, standard-spherical fold maps and simple fold maps {#sec:2}
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Throughout this paper, let $m \geq n \geq 1$ be integers, $M$ be a closed and connected manifold of dimension $m$, $N$ be a manifold of dimension $n$ without boundary and $f:M \rightarrow N$ be a smooth map unless otherwise stated. In addition, the structure groups of bundles such that the fibers are manifolds are assumed to be (subgroups of) diffeomorphism groups unless otherwise stated or these bundles are assumed to be so-called [*smooth*]{} bundles: [*PL*]{} bundles are discussed as exceptional cases. A [*linear*]{} bundle is a smooth bundle whose fiber is a ($k+1$)-dimensional (closed) unit disc or the $k$-dimensional unit sphere in ${\mathbb{R}}^{k+1}$ and whose structure group is a subgroup of the ($k+1$)-dimensional one $O(k+1)$ acting linearly in a canonical way. A [*PL*]{} bundle is a bundle whose fiber is a polyhedron and whose structure group is a group consisting of PL homeomorphisms on the fiber.
A fold map $f:M \rightarrow N$ is said to be [*standard-spherical*]{} if the following properties hold (we can easily know the definitions of a [*crossing*]{} and a [*normal*]{} crossing of a smooth immersion and we omit the definition).
1. The restriction map $f {\mid}_{S(f)}$ is an immersion whose crossings are normal.
2. Indices of singular points are $0$ or $1$.
3. If $m-n=1$ and there exists a singular point of index $1$ of $f$, then $M$ is orientable.
4. Preimages of regular values are disjoint unions of points or standard spheres.
A special generic map $f$ is a map of this class. Under the condition $m=n$ its Reeb space $W_f$ is regarded as a manifold diffeomorphic to $M$. Under the condition $m>n$ its Reeb space $W_f$ is an $n$-dimensional compact manifold we can smoothly immerse into $N$: $f$ is represented as the composition of $q_f$ with the immersion and furthermore, $q_f(S(f))=\partial W_f$ and ${q_f} {\mid}_{S(f)}$ is injective.
We say a fold map $f$ is [*simple*]{} if the restriction of $q_f$ to the singular set is injective.
Special generic maps are simple. Let $m>n$. For a simple standard-spherical map $f$, for the complementary set of the union of the regular neighborhoods, the restriction of $q_f$ to the preimage gives a smooth bundle whose fiber is $S^{m-n}$ and the fiber of the bundle $N(C)$ is Y-shaped and the bundle ${q_f}^{-1}(N(C))$ is a smooth bundle whose fiber is diffeomorphic to a manifold obtained by removing the interior of a disjoint union of three standard closed discs of dimension $m-n$ smoothly and disjointly embedded into $S^{m-n}$ for each connected component $C$ consisting of singular points of index $1$. For this, see also [@saeki] for example. For each connected component $C$ consisting of singular points of index $0$, the fiber of the PL bundle is a closed interval and the smooth bundle is a linear bundle and this holds for general fold maps by [@saeki2].
Under the condition $m>n$, generally, for a simple fold map $f$, for each connected component $C$ of $q_f(S(f))$, its small regular neighborhood $N(C)$ is represented as a PL bundle over $C$ whose fiber is a closed interval or a $Y$-shaped $1$-dimensional polyhedron and the composition of the restriction of the map $q_f$ to the preimage ${q_f}^{-1}(N(C))$ of the total space $N(C)$ of the bundle with the canonical projection to $C$ is a smooth bundle.
Under the condition $m=n$, for each connected component $C$ of the singular set, we can take a closed tubular neighborhood $N(C) \subset M$ and the canonical projection to $C$ gives a linear bundle whose fiber is a closed interval.
In the case of a simple standard-spherical fold map $f$, we do not assume that the restriction map $f {\mid}_{S(f)}$ is an immersion whose crossings are normal: we may assume this but the assumption is not essential.
S-trivial standard-spherical fold maps and lifting these maps to embeddings. {#sec:3}
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In section \[sec:2\], if in the case $m>n$ the PL bundle $N(C)$ over $C$ and the smooth bundle ${q_f}^{-1}(N(C))$ over $C$ are trivial for each $C$, then the simple fold map $f$ is said to be [*S-trivial*]{}. If in the case $m=n$, the linear bundle $N(C)$ over $C$ is trivial for each $C$, then the fold map $f$ is said to be [*S-trivial*]{}.
We present several examples of S-trivial standard-spherical fold maps.
\[ex:2\]
1. A special generic map $f:M \rightarrow N$ between equidimensional manifolds such that the following properties hold.
1. $M$ is orientable.
2. $S(f)$ is orientable.
For example, let $M$ be a homotopy sphere for example. [@eliashberg] implies that for any homotopy class of continuous maps between $S^m$ and $S^n$ where $m=n$, we can find a fold map satisfying the properties before.
2. \[ex:2.2\] Projections of bundles whose fibers are standard spheres: they are also special generic maps having no singular point.
3. \[ex:2.3\] [@costantinothurston], [@ishikawakoda] and [@saeki3] present S-trivial standard-spherical fold maps. Through them, we can know that 3-dimensional closed and orientable manifolds of a class (the class of [*graph manifolds*]{}) admit such maps into surfaces. For such a map, we can regard the Reeb space $W_f$ as the [*shadow*]{} of the manifold. Roughly speaking, a [*shadow*]{} is a polyhedron at each point of which there exists a small regular neighborhood PL homeomorphic to a regular neighborhood of a point of a Reeb space of a simple fold map into the plane as before. Moreover, the space of all points the regular neighborhoods of which are $2$-dimensional discs containing the points in the interiors is a $2$-dimensional manifold and an integer called a [*gleam*]{} is assigned to each connected component of this. For the definition, see [@ishikawakoda] and see also [@turaev] and [@turaev2]. See also Remark \[rem:1\].
4. \[ex:2.4\] [@kitazawa5] and [@kitazawa7] present construction of infinite families of such maps and infinite families of closed and connected manifolds admitting them starting from fundamental examples including special generic maps and examples in [@kitazawa], [@kitazawa2] and [@kitazawa4] for example via surgery operations to the maps and the manifolds.
\[rem:1\] In fact, in Example \[ex:2\] (\[ex:2.3\]) a [*shadow*]{} is a $2$-dimensional polyhedron of a wider class in fact and an integer or a rational number of the form $\frac{k}{2}$ where $k$ is an integer is assigned to each connected component of the $2$-dimensional manifold as before: the number is said to be a [*gleam*]{}. Similarly, from a fold map $f$ such that the restriction map $f {\mid}_{S(f)}$ is an immersion whose crossings are normal on a $3$-dimensional closed and orientable manifold into a surface, which always exists, and the Reeb space $W_f$, we can obtain a shadow.
The following proposition is a fundamental and key proposition in the present paper. ${\rm Emd}(X,Y)$ denotes the space of all the smooth embeddings of a manifold $X$ into a manifold $Y$ endowed with the so-called [*$C^{\infty}$ Whitney topology*]{} (see [@golubitskyguillemin]).
\[fact:1\] ${\rm Emd}(X,Y)$ is $\max\{0,\min \{2\dim
Y-3\dim X-4,\dim Y-dim X-2\}\}$-connected if $\dim Y \geq \dim X+2 \geq 3$ is assumed.
See also [@nishioka] for Fact \[fact:1\]. The following is for example shown in Lemma 1 of [@kitazawa6].
\[lem:1\] For a Morse function $\bar{f}$ on a $k$-dimensional compact manifold satisfying $k \geq 1$ and either of the following properties, we can represent this as a composition of an embedding into ${\mathbb{R}}^{k+1}$ with a canonical projection.
1. A Morse function is a function on a closed unit disc of dimension $k$ with exactly one singular point in the interior such that the preimage of one of the two extrema and the boundary coincide,
2. $k>1$ and a Morse function is a function on a compact manifold obtained by removing the interior of the union of three disjointly smoothly embedded $k$-dimensional standard closed discs in $S^k$ with exactly one singular point in the interior such that the preimages of the two extrema are a connected component of the boundary and the disjoint union of the remaining two connected components of the boundary, respectively.
The former case follows by the definition of a singular point of index $0$ of a fold map. The latter case follows by the argument as the following.
1. Decompose the $k$-dimensional manifold into two $k$-dimensional compact submanifolds with corners along ($k-1$)-dimensional submanifolds: one is a disjoint union of two copies of the total space of a product bundle over a closed interval whose fiber is a standard closed disc of dimension $k-1$ and the other is diffeomorphic to the product of a closed interval and a standard closed disc of dimension $k-1$, which is so-called a $1$-handle of the Morse function.
2. On each connected component of the disjoin union of the total spaces of the product bundles, the original function is regarded as a projection of the bundle by virtue of a kind of Ehresmann’s fibration theorem ([@ehresmann]).
\[prop:2\] For an S-trivial standard-spherical fold map $f:M \rightarrow N$, there exists an embedding $F$ such that $f={\pi}_{N,k} \circ F$ where ${\pi}_{N,k}:N \times {\mathbb{R}}^k \rightarrow N$ is the canonical projection onto $N$ if $n+k \geq \max\{\frac{3m+3}{2},m+n+1\}$.
For each connected component $C$ of $q_f(S(f))$, consider a small regular neighborhood $N(C)$ and the local smooth map represented as the composition of ${q_f} {\mid}_{{q_f}^{-1}(N(C))} $ with ${\bar{f}} {\mid}_{N(C)}$, we can consider an embedding so that the composition with the composition of the canonical projection $N(C) \times {\mathbb{R}}^k$ onto $N(C)$ with ${\bar{f}} {\mid}_{N(C)}$ is the original map: the assumptions that the map is S-trivial together with arguments related to so-called Thom’s isotopy theorem and Lemma \[lem:1\] and that $n+k \geq m+n+1$ enable us to do this. We extend this over the complementary set of the union of $N(C)$ in $W_f$. The essential assumption is that the space ${\rm Emb}(S^{m-n},{\mathbb{R}}^k)$ is ($n-1$)-connected together with the relation $n+k \geq m+n+1$. We consider suitable cell decompositions of $W_f$ and the complementary set. On each $1$-cell in the complemantary set, the original map is regarded as the projection of a trivial smooth bundle and we can construct an embedding as before. We can extend this as projections over $2$-cells. We can do this inductively over $k$-cells where $2 \leq k\leq n$. This completes the proof.
We explain about the fact that ${\rm Emb}(S^{m-n},{\mathbb{R}}^k)$ is ($n-1$)-connected. In Fact \[fact:1\], the inequality $2k-3(m-n)-4 \geq 2(\frac{3m+3}{2}-n)-3(m-n)-4=n-1$ completes the proof.
Nishioka studied only cases of special generic maps on closed, connected and orientable manifolds into Euclidean spaces satisfying the relation $m>n$. For the methods here, see also [@saekitakase] for example. They are on a hot topic of the singularity theory of differentiable maps: lifting a smooth map to a smooth map of a suitable class into a higher dimensional space or finding a representation of the original map by a composition of a map of the suitable class with a canonical projection. Note also that in these problems, as the author know, the target manifolds of smooth maps are Euclidean spaces.
The following theorem presents another new explicit answer for a kind of these problems and important in a main result or Theorem \[thm:3\] later.
\[thm:1\] For an S-trivial standard-spherical fold map $f:M \rightarrow N$ from a $3$-dimensional closed, connected and orientable manifold into a $2$-dimensional manifold $N$ with no boundary such that for the shadow defined from the map and the Reeb space in Example \[ex:2\] [(]{}\[ex:2.3\][)]{} and Remark \[rem:1\], all gleams are even, there exists an embedding $F:M \rightarrow N \times {\mathbb{R}}^k$ such that $f={\pi}_{N,k} \circ F$ where ${\pi}_{N,k}:N \times {\mathbb{R}}^k \rightarrow N$ is the canonical projection onto $N$ for $k \geq 3$.
Around each connected component $C$ of $q_f(S(f))$ and each connected component of $W_f-{\bigcup}_C N(C)$ where we abuse notation in the proof of Proposition \[prop:2\], we can construct smooth map into ${\mathbb{R}}^{2+k}$ similarly. By gluing them together, we obtain a desired embedding. The assumption that the gleams of the shadow are always even guarantees this. We explain this shortly to complete the proof: to know more precisely, we need to know about terminologies and notions on shadows more and see the cited articles.
First we investigate the case where each connected component $O$ of $W_f-{\bigcup}_C N(C)$ is not a closed surface. The restriction of $f$ to the preimage ${q_f}^{-1}(N(C))$ is, for suitable coordinates, represented as the composition of a product map of either of the following Morse functions and the identity map ${\rm id}_{C}$ with a suitable immersion into $N$.
1. A Morse function on a closed unit disc of dimension $2$ with exactly one singular point in the interior such that the preimage of one of the two extrema and the boundary coincide.
2. A Morse function on a compact, connected and orientable surface whose genus is $0$ and whose boundary consists of exactly three connected components with exactly one singular point in the interior such that the preimage of the two extrema are a connected component of the boundary and the disjoint union of the remaining two connected components of the boundary.
In the situation of the proof of Proposition \[prop:2\], they are represented as the compositions of embeddings into $I \times {\mathbb{R}}^3$ with the canonical projection to the first component where $I$ is a closed interval, regarded as a fiber of a suitable trivial bundle over $C$. We can consider functions obtained by considering the 1-dimensional higher versions of these two functions as the second Morse function in Lemma \[lem:1\]. They are also represented as the compositions of embeddings into $I \times {\mathbb{R}}^3$ with the canonical projection to the first component where $I$ is a closed interval, regarded as a fiber of the trivial bundle over $C$, in the same situation. Furthermore, the original functions are obtained as restrictions of the corresponding functions on the $3$-dimensional closed unit disc or the $3$-dimensional compact manifold obtained by removing the interior of the union of three disjointly smoothly embedded $3$-dimensional standard closed discs in $S^3$. Moreover, the boundary of the domain of each function on the surface is obtained as a closed submanifold with no boundary of the boundary of the domain of the Morse function on the $3$-dimensional manifold so that each connected component, diffeomorphic to a circle, is an equator of each of the three $2$-dimensional standard spheres in the boundary of the $3$-dimensional manifold. Distinct circles are embedded as equators in distinct $2$-dimensional spheres in the boundary of the $3$-dimensional manifold.
On each connected component $O$ of $W_f-{\bigcup}_C N(C)$, the map $f$ is regarded as the composition of the projection of a linear bundle whose fiber is a circle with a suitable immersion into $N$. We can represent the projection as the composition of an embedding into $O \times {\mathbb{R}}^3$ with the canonical projection to $O$. We can also construct a map regarded as the composition of the projection of a linear bundle whose fiber is a $2$-dimensional standard sphere with a suitable immersion into $N$ and we can represent this as the composition of the composition of an embedding into $O \times {\mathbb{R}}^3$ with the canonical projection to $O$ with the last immersion. Furthermore, the original projection is obtained as the restriction of the corresponding projection of the total space of the linear bundle whose fiber is the $2$-dimensional sphere by restricting each fiber to the equator of the sphere. In other words, the original bundle is regarded as a subbundle.
We can glue the local maps on $3$-dimensional and $4$-dimensional manifolds together to obtain global embeddings into $N \times {\mathbb{R}}^{2+k}$ and a smooth map into $N$. Note for example that we obtain the map on the $3$-dimensional closed manifold $M$ into $N$ as the given map $f$. Connected components of the boundaries where we glue the maps together are, regarded as total spaces of trivial linear bundles over circles whose fibers are $S^1$ or $S^2$ and the restrictions of the map $q_f$ to the spaces give the projections. Note that these fibers are in a preimage of the projection ${\pi}_{N,k}:N \times {\mathbb{R}}^k \rightarrow N$ and $S^2$ is embedded as a so-called [*unknot*]{} in ${\mathbb{R}}^k$ in the smooth category. $S^1$ is, as explained, embedded in $S^2$ as an equator. The assumption on the gleams imply that the way we glue the maps yields global maps. Regard each copy of $S^1 \times S^2$ as a trivial linear bundle over $S^1$ equipped with a projection regarded as a canonical projection to the first component. The bundle isomorphisms between two of these bundles is regarded as a product map of diffeomorphisms between the base spaces, regarded as $S^1 \times \{{\ast}_1\}$ and fibers, regarded as $\{{\ast}_2\} \times S^2$.
Last we investigate the case where there exists no singular point of the map $f$ and this completes the proof. We can show the statement similarly by the assumption on the gleam or the discussion on the attachments of the local maps and spaces just before (there exists exactly one connected component a gleam is assigned to). Note also that this case accounts for the case which we cannot regard as the first case.
This completes the proof.
Exmaple \[ex:3\] presents infinitely many examples of maps and $3$-dimensional closed, connected and orientable manifolds to which we can apply Theorem \[thm:1\]. We omit fundamental notions on general $k$-dimensional linear bundles such that [*oriented*]{} linear bundles and the Euler classes of them, defined as $k$-th integral cohomology classes.
\[ex:3\] Projections of bundles whose fibers are circles over closed surfaces, which are also regarded as $2$-dimensional linear bundles, such that the Euler classes are divisible by $2$ (if it is oriented), are simplest examples satisfying the assumption of Theorem \[thm:1\].
Main theorems
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\[def:6\] For a homology class of a compact manifold, if for the homology group of this, the coefficient is a commutative ring $R$ having the unique identity element $1 \neq 0 \in R$ and a homology class $c \neq 0$ satisfies the following properties, then it is said to be a [*UFG*]{}.
1. If $rc=0$ for $r \in R$, then $r=0$.
2. For any element $r \in R$ which is not a unit and any homology class $c^{\prime}$, $c$ is never represented as $rc^{\prime}$.
\[def:7\] Let $R$ be a commutative ring having the unique identity element $1 \neq 0 \in R$. For a UFG homology class $c \in H_j(X;R)$ of a compact manifold $X$, the class $c^{\ast} \in H^j(X;R)$ satisfying the following properties is said to be the [*dual*]{} $c$.
1. $c^{\ast}(c)=1$
2. For any submodule $B$ of $H_j(X;R)$ such that the internal direct sum of the submodule generated by $c$ and $B$ is $H_j(X;R)$, $c^{\ast}(B)=0$.
We denote the dual of a UFG homology class $c$ as in Definition \[def:7\]. Hereafter, we assume that $R:=\mathbb{Z}, \mathbb{Q}, \mathbb{Z}/k\mathbb{Z}$ where $k>1$ is an integer. We denote the Poincaré dual to a homology class or cohomology class $c$ by ${\rm PD}_R(c)$ where $X$ is a closed, connected and orientable manifold. If $R:=\mathbb{Z}/2\mathbb{Z}$, then we do not assume the orientability of $X$ or $X$ may not be orientable.
For two closed, connected, oriented and equidimensional manifolds $X$ and $Y$, we denote by ${\nu}_X \in H_{\dim X}(X;\mathbb{Z})$ and ${\nu}_Y \in H_{\dim Y}(Y;\mathbb{Z})$ the [*fundamental classes*]{} of the manifolds or the generators compatible with the orientations: the groups are isomorphic to $\mathbb{Z}$. The [*mapping degree*]{} of a continuous map $c:X \rightarrow Y$ is the integer $d(c)$ satisfying ${\nu}_Y=d(c) c_{\ast}({\nu}_X)$. For two closed, connected and equidimensional manifolds $X$ and $Y$, we denote by ${\nu}_X \in H_{\dim X}(X;\mathbb{Z}/2\mathbb{Z})$ and ${\nu}_Y \in H_{\dim Y}(Y;\mathbb{Z}/2\mathbb{Z})$ the [*$\mathbb{Z}/2\mathbb{Z}$ fundamental classes*]{} of the manifolds or the generators of the groups, isomorphic to $\mathbb{Z}/2\mathbb{Z}$. The [*$\mathbb{Z}/2\mathbb{Z}$ mapping degree*]{} of a continuous map $c:X \rightarrow Y$ is the integer $d(c)=0,1$ satisfying ${\nu}_Y=d(c) c_{\ast}({\nu}_X)$.
Let $X$ be a compact manifold. Let $Y$ be a closed and connected manifold satisfying $\dim Y<\dim X$. A class $c_X \in H_j(X;R)$ is [*realized*]{} by a class $c_Y \in H_j(Y;R)$ if for an embedding $i_{Y,X}:Y \rightarrow X$ whose image is in ${\rm Int} X$ $c_X=i_{Y,X} {\ast}(c_Y)$ holds.
\[thm:2\] Let $R:=\mathbb{Z}, \mathbb{Q}, \mathbb{Z}/k\mathbb{Z}$. Let $X$ be a compact manifold and let $c \in H_{n}(X;R)$ be realized by a closed and connected manifold $N$ of dimension $n>0$. Let a closed and connected manifold $M$ of dimension $m \geq n$ admit a smooth map $f:M \rightarrow N$ satisfying the following properties.
1. There exists a preimage $F$ of a regular value representing a class $c_F \in H_{m-n}(M;R)$.
2. There exists an integer $k$ and a UFG $c_{F,0} \in H_{m-n}(M;R)$ satisfying $c_F=kc_{F_0}$ and $H_{m-n}(M;R)$ is the internal direct sum of the submodule generated by the one element set $\{c_{F_0}\}$ and a suitable submodule.
3. $f$ is an S-trivial standard-spherical fold map.
We also assume the following conditions.
1. If $R$ is not isomorphic to $\mathbb{Z}/2\mathbb{Z}$, then $M$ and $N$ are orientable and $N$ is oriented so that the fundamental class is ${\nu}_N \in H_n(N;R)$. We also use ${\nu}_N \in H_n(N;R)$ for the $\mathbb{Z}/2\mathbb{Z}$ fundamental class where $R$ is isomorphic to $\mathbb{Z}/2\mathbb{Z}$.
2. We can take an embedding $i_{N,X}:N \rightarrow X$ satisfying $i_{N,X}(N) \subset {\rm Int} X
$ and $c={i_{N,X}}_{\ast}({\nu}_N)$ so that the normal bundle of the image is trivial.
3. $\dim X \geq \max\{\frac{3m+3}{2},m+n+1\}$.
In this situation, $kc$ is realized by the class $k{\rm PD}_{R}({c_{F,0}}^{\ast})$.
In the situation of Proposition \[prop:2\], we consider a map $i_{N,X} \circ f:M \rightarrow i_{N,X}(N)$ and take the total space of the normal bundle of the image, which is a trivial linear bundle over the image, instead of “$N \times {\mathbb{R}}^k$” in the situation of Proposition \[prop:2\]. By the definitions of $c_F$ and the embedding $i_{N,X}$, we have the result.
\[thm:3\] Let $R:=\mathbb{Z}, \mathbb{Q}, \mathbb{Z}/k\mathbb{Z}$. Let $X$ be a compact and spin manifold and let $c \in H_{2}(X;R)$ be realized by a closed, connected and orientable manifold $N_0$ of dimension $n=2$. Let $M$ be a closed and connected manifold of dimension $m \geq n=2$. Let $N$ be a closed, connected and orientable manifold $N$ of dimension $n=2$. If $R$ is not isomorphic to $\mathbb{Z}/2\mathbb{Z}$, then $M$ is assumed to be orientable and $N$ and $N_0$ are oriented so that the fundamental classes are ${\nu}_N \in H_2(N;R)$ and ${\nu}_{N_0} \in H_2(N_0;R)$, respectively. In this case, we also assume that there exists a smooth map $c_N:N \rightarrow N_0$ of mapping degree $k_2$. We also use ${\nu}_N \in H_2(N;R)$ and ${\nu}_{N_0} \in H_2(N_0;R)$ for $\mathbb{Z}/2\mathbb{Z}$ fundamental classes where $R$ is isomorphic to $\mathbb{Z}/2\mathbb{Z}$ and in this case we also assume that there exists a smooth map $c_N:N \rightarrow N_0$ of $\mathbb{Z}/2\mathbb{Z}$ mapping degree $k_2$. Suppose that a smooth map $f:M \rightarrow N$ satisfying the following properties exists.
1. There exists a preimage $F$ of a regular value representing a class $c_F \in H_{m-2}(M;R)$.
2. There exists an integer $k$ and a UFG $c_{F,0} \in H_{m-2}(M;R)$ satisfying $c_F=kc_{F_0}$ and $H_{m-2}(M;R)$ is the internal direct sum of the submodule generated by the one element set $\{c_{F_0}\}$ and a suitable submodule.
3. $f$ is an S-trivial standard-spherical fold map.
Last we also assume the following conditions.
1. We can take an embedding $i_{N_0,X}:N_0 \rightarrow X$ satisfying $i_{N_0,X}(N_0) \subset {\rm Int} X
$ and $c={i_{N_0,X}}_{\ast}({\nu}_{N_0})$, which is assumed in the beginning.
2. $\dim X \geq \max\{\frac{3m+3}{2},m+3\}$.
3. If $f$ is an S-trivial map as in Theorem \[thm:1\] with $(m,n)=(3,2)$, then $\dim X \geq 5$.
In this situation, $k k_2 c$ is realized by the class $k k_2 {\rm PD}_{R}({c_{F,0}}^{\ast})$.
First by the assumption, $\dim X \geq 2n+1$ holds. The existence of $c_N:N \rightarrow N_0$ yields the existence of an embedding $i_{N,X}:N \rightarrow X$ satisfying $i_{N,X}(N) \subset {\rm Int} X
$ and $k_2 c=k_2 {i_{N,X}}_{\ast}({\nu}_N)$. The assumptions that $X$ is spin and that $N$ and $N_0$ are orientable guarantee that the normal bundle of the image is trivial together with fundamental arguments on characteristic classes of linear bundles over manifolds related to smooth embeddings of smooth manifolds. This follows by Theorem \[thm:2\] (and the proof) together with Theorem \[thm:1\] (and the proof).
\[thm:4\] Let $R:=\mathbb{Z}, \mathbb{Q}, \mathbb{Z}/k\mathbb{Z}$. Let $X$ be a compact and spin manifold and let $c \in H_{3}(X;R)$ be realized by a closed, connected and orientable manifold $N_0$ of dimension $n=3$. Let $M$ be a closed and connected manifold of dimension $m \geq n=3$. Let $N$ be a closed, connected and orientable manifold $N$ of dimension $n=3$. If $R$ is not isomorphic to $\mathbb{Z}/2\mathbb{Z}$, then $M$, $N$ and $N_0$ are orientable and $N$ and $N_0$ are oriented so that the fundamental classes are ${\nu}_N \in H_3(N;R)$ and ${\nu}_{N_0} \in H_3(N_0;R)$, respectively. In this case, we also assume that there exists a smooth map $c_N:N \rightarrow N_0$ of mapping degree $k_3$. We also use ${\nu}_N \in H_3(N;R)$ and ${\nu}_{N_0} \in H_3(N_0;R)$ for $\mathbb{Z}/2\mathbb{Z}$ fundamental classes where $R$ is isomorphic to $\mathbb{Z}/2\mathbb{Z}$ and in this case we also assume that there exists a smooth map $c_N:N \rightarrow N_0$ of $\mathbb{Z}/2\mathbb{Z}$ mapping degree $k_3$. Suppose that a smooth map $f:M \rightarrow N$ satisfying the following properties exists.
1. There exists a preimage $F$ of a regular value representing a class $c_F \in H_{m-3}(M;R)$.
2. There exists an integer $k$ and a UFG $c_{F,0} \in H_{m-3}(M;R)$ satisfying $c_F=kc_{F_0}$ and $H_{m-3}(M;R)$ is the internal direct sum of the submodule generated by the one element set $\{c_{F_0}\}$ and a suitable submodule.
3. $f$ is an S-trivial standard-spherical fold map.
We also assume the following conditions.
1. We can take an embedding $i_{N_0,X}:N_0 \rightarrow X$ satisfying $i_{N_0,X}(N_0) \subset {\rm Int} X
$ and $c={i_{N_0,X}}_{\ast}({\nu}_{N_0})$, which is assumed in the beginning.
2. $\dim X \geq \max\{\frac{3m+3}{2},m+4\}$.
In this situation, $k k_3 c$ is realized by the class $k k_3 {\rm PD}_{R}({c_{F,0}}^{\ast})$.
First by the assumption, $\dim X \geq 2n+1$ holds. The existence of $c_N:N \rightarrow N_0$ yields the existence of an embedding $i_{N,X}:N \rightarrow X$ satisfying $i_{N,X}(N) \subset {\rm Int} X
$ and $k_3 c=k_3 {i_{N,X}}_{\ast}({\nu}_N)$. The assumptions that $X$ is spin and that $N$ and $N_0$ are orientable guarantee that the normal bundle of the image is trivial together with fundamental arguments on characteristic classes of linear bundles over manifolds related to smooth embeddings of smooth manifolds. Theorem \[thm:2\] (and the proof) completes the proof.
Last we present important facts in constructing explicit cases explaining these theorems well.
1. For two closed, connected, oriented and equidimensional manifolds $X$ and $Y$ where $Y$ is a standard sphere of dimension larger than $0$ and an arbitrary integer $k$, there exists a smooth map $c:X \rightarrow Y$ whose mapping degree is $k$. This is important as an example for $c_N$ in Theorems \[thm:3\] and \[thm:4\].
2. For two closed, connected and equidimensional manifolds $X$ and $Y$ where $Y$ is a standard sphere of dimension larger than $0$ and an arbitrary element $k \in \mathbb{Z}/2\mathbb{Z}$, there exists a smooth map $c:X \rightarrow Y$ whose $\mathbb{Z}/2\mathbb{Z}$ mapping degree is $k$. This is important as an example for $c_N$ in Theorems \[thm:3\] and \[thm:4\].
3. The projections of product bundles over closed and connected manifolds whose fibers are closed and connected manifolds present infinitely many examples for maps $f$ in Theorems \[thm:2\], \[thm:3\] and \[thm:4\].
4. The projections of linear bundles over standard spheres whose fibers are unit spheres present infinitely many examples for maps $f$ in Theorems \[thm:2\], \[thm:3\] and \[thm:4\] especially for the case $R=\mathbb{Z}/k\mathbb{Z}$.
In addition, Example \[ex:2\] is also important.
[30]{} C. Bohr, B. Hanke and D. Kotschick, *Cycles, submanifolds, and structures on normal bundles*, manuscripta mathematica 108 (2002), 483–494, arXiv:math/0011178. R. Budney, *A family of embeddings spaces*, Geometry and Topology Monographs 13 (2008), 41–83. F. Costantino and D. Thurston, *$3$-manifolds efficiently bound $4$-manifolds*, J. Topol. 1 (2008), 703–745. C. Ehresmann, *Les connexions infinitesimales dans un espace fibre differentiable*, Colloque de Topologie, Bruxelles (1950), 29–55.
Y. Eliashberg, *On singularities of folding type*, Math. USSR Izv. 4 (1970). 1119–1134. M. Golubitsky and V. Guillemin, *Stable mappings and their singularities*, Graduate Texts in Mathematics (14), Springer-Verlag (1974). M. Ishikawa and Y. Koda, *Stable maps and branched shadows of $3$-manifolds*, Mathematische Annalen 367 (2017), no. 3, 1819–1863, arXiv:1403.0596. N. Kitazawa, *On round fold maps* (in Japanese), RIMS Kôkyûroku Bessatsu B38 (2013), 45–59. N. Kitazawa, *On manifolds admitting fold maps with singular value sets of concentric spheres*, Doctoral Dissertation, Tokyo Institute of Technology (2014). N. Kitazawa, *Fold maps with singular value sets of concentric spheres*, Hokkaido Mathematical Journal Vol.43, No.3 (2014), 327–359. N. Kitazawa, *Round fold maps and the topologies and the differentiable structures of manifolds admitting explicit ones*, submitted to a refereed journal, arXiv:1304.0618 (the title has changed). N. Kitazawa, *Constructing fold maps by surgery operations and homological information of their Reeb spaces*, submitted to a refereed journal, arxiv:1508.05630 (the title has been changed). N. Kitazawa, *Lifts of spherical Morse functions*, submitted to a refereed journal, arxiv:1805.05852. N. Kitazawa, *Notes on fold maps obtained by surgery operations and algebraic information of their Reeb spaces*, arxiv:1811.04080. P. B. Kronheimer, *Embedded surfaces and gauge theory in three and four dimensions* (www.math.harvard.edu/\~kronheim/jdg96.pdf), Harvard University, Cambridge MA 02138. M. Nishioka, *Desingularizing special generic maps into $3$-dimensional space*, PhD Thesis (Kyushu Univ.), arxiv:1603.04520. G. Reeb, *Sur les points singuliers dùne forme de Pfaff completement integrable ou d’une fonction numerique*, -C. R. A. S. Paris 222 (1946), 847–849. O. Saeki, *Notes on the topology of folds*, J. Math. Soc. Japan Volume 44, Number 3 (1992), 551–566. O. Saeki, *Topology of special generic maps of manifolds into Euclidean spaces*, Topology Appl. 49 (1993), 265–293. O. Saeki, *Simple stable maps of 3-manifolds into surfaces*, Topology 35 (1996), 671–698. O. Saeki and K. Suzuoka, *Generic smooth maps with sphere fibers* J. Math. Soc. Japan Volume 57, Number 3 (2005), 881–902. O. Saeki and M. Takase, *Desingularizing special generic maps*, Journal of Gokova Geometry Topology 7 (2013), 1–24. M. Shiota, *Thom’s conjecture on triangulations of maps*, Topology 39 (2000), 383–399. R. Thom, *Quelques propriétés globales des variétés différentiables*, Commentarii mathematici Helvetici (1954), Volume 28, 17–86. Vladimir G. Turaev, *Topology of shadows*, Preprint, 1991. Vladimir G. Turaev, *Shadow links and face models of statistical mechanics*, J. Differential Geom. 36 (1992), 35–74.
[^1]: The author is a member of and supported by the project Grant-in-Aid for Scientific Research (S) (17H06128 Principal Investigator: Osamu Saeki) “Innovative research of geometric topology and singularities of differentiable mappings” ( https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-17H06128/ ).
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abstract: 'We calculate the one-photon loop radiative corrections to the charged pion-pair production process $\pi^-\gamma\to\pi^+\pi^-\pi^-$. In the low-energy region this reaction is governed by the chiral pion-pion interaction. The pertinent set of 42 irreducible photon-loop diagrams is calculated by using the package FeynCalc. Electromagnetic counterterms with two independent low-energy constants $\widehat k_1$ and $\widehat k_2$ are included in order to remove the ultraviolet divergences generated by the photon-loops. Infrared finiteness of the virtual radiative corrections is achieved by including soft photon radiation below an energy cut-off $\lambda$. The purely electromagnetic interaction of the charged pions mediated by one-photon exchange is also taken into account. The radiative corrections to the total cross section (in the isospin limit) vary between $+10\%$ close to threshold and about $-1\%$ at a center-of-mass energy of $7m_\pi$. The largest contribution comes from the simple one-photon exchange. Radiative corrections to the $\pi^+\pi^-$ and $\pi^-\pi^-$ mass spectra are studied as well. The Coulomb singularity of the final-state interaction produces a kink in the dipion mass spectra. The virtual radiative corrections to elastic $\pi^-\pi^-$ scattering are derived additionally.'
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[[^1] ]{}
N. Kaiser and S. Petschauer\
Introduction and summary
========================
The pions ($\pi^+,\pi^0,\pi^-$) are the Goldstone bosons of spontaneous chiral symmetry breaking in QCD. Their strong interaction dynamics at low energies can therefore be calculated systematically (and accurately) with chiral perturbation theory in the form of a loop expansion based on an effective chiral Lagrangian. The very accurate two-loop predictions [@cola] for the S-wave $\pi\pi$-scattering lengths, $a_0=(0.220\pm 0.005)m_\pi^{-1}$ and $a_2=(-0.044
\pm 0.001)m_\pi^{-1}$, have been confirmed experimentally by analyzing the $\pi\pi$ final-state interaction effects occurring in various (rare) charged kaon decay modes [@bnl; @batley; @cusp; @batgamow]. Electromagnetic processes with pions offer further possibilities to test chiral perturbation theory. For example, pion Compton scattering $\pi^- \gamma \to\pi^- \gamma$ allows one to extract the electric and magnetic polarizabilities ($\alpha_\pi$ and $\beta_\pi$) of the charged pion. Chiral perturbation theory at two-loop order gives for the dominant pion polarizability difference the firm prediction $\alpha_\pi-\beta_\pi
=(5.7\pm1.0)\cdot 10^{-4}\,$fm$^3$ [@gasser]. It is however in conflict with the existing experimental results from Serpukhov $\alpha_\pi-\beta_\pi=(15.6\pm
7.8)\cdot 10^{-4}\,$fm$^3$ [@serpukov1; @serpukov2] and MAMI $\alpha_\pi-
\beta_\pi=(11.6\pm 3.4)\cdot 10^{-4}\,$fm$^3$ [@mainz], which amount to values more than twice as large. In that contradictory situation it is promising that the COMPASS experiment [@private] at CERN aims at measuring the pion polarizabilities, $\alpha_\pi$ and $\beta_\pi$, with high statistics using the Primakoff effect. The scattering of high-energy negative pions in the Coulomb-field of a heavy nucleus (of charge $Z$) gives access to cross sections for $\pi^-\gamma$ reactions through the equivalent photon method [@pomer]. The theoretical framework to extract the pion polarizabilities from the measured cross sections for low-energy pion Compton scattering $\pi^- \gamma \to\pi^- \gamma$ or the primary pion-nucleus bremsstrahlung process $\pi^- Z \to\pi^- Z \gamma$ has been described (in the one-loop approximation) in refs.[@picross; @comptcor]. In addition to the strong interaction effects, the QED radiative corrections to real and virtual pion Compton scattering $\pi^-\gamma^{(*)} \to \pi^- \gamma$ have been calculated in refs.[@comptcor; @bremscor]. The relative smallness of the pion-structure effects in low-energy pion Compton scattering [@picross] makes it necessary to include such higher order electromagnetic corrections.
The COMPASS experiment is set up to detect simultaneously various (multi-particle) hadronic final-states which are produced in the Primakoff scattering of high-energy pions. The cross sections of the $\pi^-\gamma\to 3\pi$ reactions in the low-energy region offer new possibilities to test the strong interaction dynamics of the pions as predicted by chiral perturbation theory. In a recent analysis by the COMPASS collaboration [@private; @compass] the total cross section for the process $\pi^-\gamma\to \pi^+\pi^- \pi^-$ has been extracted in the region from threshold up to a $3\pi$-invariant mass of about $5m_\pi$, and good agreement with the prediction of chiral perturbation theory has been found. The analysis of the $\pi^-\pi^0 \pi^0$-channel is ongoing [@private] and the corresponding experimental results are eagerly awaited. On the theoretical side the production amplitudes for $\pi^- \gamma \to \pi^-\pi^0 \pi^0$ and $\pi^- \gamma \to \pi^+\pi^-\pi^-$ have been calculated at one-loop order in chiral perturbation theory [@3pion]. It has been found that the next-to-leading order chiral corrections enhance sizeably (by a factor $1.5 -1.8$) the total cross section for neutral pion-pair production $\pi^-\gamma \to\pi^-\pi^0\pi^0$, but leave the one for charged pion-pair production $\pi^-\gamma \to\pi^+\pi^-\pi^-$ almost unchanged in comparison to the tree approximation. Let us note that these calculations would be implicitly contained in the work by Ecker and Unterdorfer [@ecker], where the processes $\gamma^*
\to 4\pi$ have been studied in chiral effective field theory. In particular, the chiral resonance theory used in that work offers the possibility to extend the description of the $\pi^-\gamma \to 3\pi$ reactions to higher energies.
The QED radiative corrections to neutral pion-pair production $\pi^-\gamma \to
\pi^- \pi^0\pi^0$ have been computed recently in ref.[@neutral]. This calculation was simplified by the fact that the virtual photon-loops could be represented by a multiplicative correction factor $R\sim \alpha/2\pi$ to the tree-amplitude and the number of contributing diagrams was limited to one dozen. The purpose of the present work is to extend the calculation of QED radiative corrections to the (more complex) charged pion-pair production process $\pi^-\gamma
\to \pi^+ \pi^-\pi^-$. We use the package FeynCalc [@feyncalc] to evaluate the difficult set of 42 irreducible photon-loop diagrams. Other contributions of the same order in $\alpha$, such as the one-photon exchange and the soft-photon bremsstrahlung, can still be given in concise analytical formulas. As a result we find that the radiative corrections to the total cross section and the dipion mass spectra in the isospin limit are of the magnitude of a few percent. The largest contribution is provided by the simple one-photon exchange, which reaches up to $8\%$ close to threshold. It is however partly compensated by the leading isospin-breaking correction arising from the charged and neutral pion mass difference. The Coulomb singularity of the $\pi^\pm \pi^-$ final-state interaction causes a kink in the invariant mass spectra.
Let us clarify that the present analysis is not a complete calculation of all effects of order ${\cal O}(e^2p^2)$ in chiral perturbation theory, since only photon-loops are considered but not the electromagnetic effects induced in pion-loops via the charged and neutral pion mass difference. The latter (subleading) isospin-breaking corrections are expected to be of similar size as the “genuine” radiative corrections studied in this work.
Charged pion-pair production: T-matrix and cross section
========================================================
We start out with recalling the kinematical and dynamical description [@3pion] of the charged pion-pair production process: $\pi^-(p_1)+
\gamma(k,\epsilon\,) \to\pi^+(p_2) +\pi^-(q_1)+\pi^-(q_2)$. It is advantageous to choose for the transversal real photon the Coulomb-gauge in the center-of-mass frame, which entails the conditions $\epsilon \cdot p_1
=\epsilon \cdot k= 0$. These subsidiary conditions imply that all diagrams for which the photon $\gamma(k,\epsilon\,)$ couples solely to the incoming negative pion $\pi^-(p_1)$ vanish identically. Under this valid specification the T-matrix has the following general form: $$T_{\rm cm} = {2e \over f_\pi^2} \Big[ \vec \epsilon \cdot \vec q_1
\, A_1 + \vec \epsilon \cdot \vec q_2 \, A_2 \Big] \,,$$ where $f_\pi= 92.4$MeV denotes the pion decay constant and $e$ is the elementary charge. In the above decomposition $A_1$ and $A_2$ are two (dimensionless) production amplitudes, which depend on five independent (dimensionless) Mandelstam variables $(s,s_1,s_2,t_1,t_2)$, defined as: $$\begin{aligned}
s\,m_\pi^2 = (p_1+k)^2 \,, &&s_1m_\pi^2=(p_2+q_1)^2\,,
\quad t_1m_\pi^2=(q_1-k)^2\,, \nonumber \\ && s_2m_\pi^2=(p_2+q_2)^2
\,, \quad t_2m_\pi^2=(q_2-k)^2\,. \end{aligned}$$ In this adapted notation $\sqrt{s}\,m_\pi$ is the total center-of-mass energy of the process, with $m_\pi=139.57\,$MeV the charged pion mass. The introduced set of variables is particularly suitable for describing the permutation of the two identical negative pions in the final state, via the interchanges $(s_1\leftrightarrow s_2,\, t_1\leftrightarrow t_2)$. The second amplitude $A_2$ introduced in eq.(1) is determined by the crossing relation: $$A_2(s,s_1,s_2,t_1,t_2)= A_1(s,s_2,s_1,t_2,t_1)\,,$$ and therefore it is sufficient to specify only the first amplitude $A_1(s,s_1,s_2,t_1,t_2)$.
At low energies the reaction $\pi^- \gamma \to \pi^+ \pi^-\pi^-$ is governed by the chiral pion-pion interaction at leading order [@3pion]. It is advantageous to parameterize the special-unitary matrix-field $U$ in the chiral Lagrangian ${\cal L}_{\pi\pi}$ through an interpolating pion-field $\vec \pi$ in the form $U =\sqrt{1-\vec \pi^{\,2} /f_\pi^2} + i \vec \tau \cdot \vec \pi/f_\pi$. This has the consequence that no $\gamma 4\pi$ and $\gamma\gamma 4\pi$ contact-vertices exist at leading order. The tree amplitude of chiral perturbation theory reads [@3pion]: $$A_1^{(\rm tree)} = {2s-2-s_1-s_2+t_1+t_2\over 3-s-t_1-t_2} +{s-s_1
-s_2+t_2 \over t_1-1} \,,$$ where in each term the numerator stems from the (off-shell) $\pi\pi$-interaction in the isospin limit (proportional to $f_\pi^{-2}$) and the denominator from a pion-propagator. The respective tree diagrams are shown in Fig.1 of ref.[@3pion] and these coincide with the diagrams in Fig.2 of the present paper when deleting the external self-energy corrections. Note that the prefactor $2e/f_\pi^2$ in eq.(1) collects all occurring coupling constants.
Applying the flux factor $[2m_\pi^2(s-1)]^{-1}$ and a symmetry factor $1/2$, the total cross section for the reaction $\pi^- \gamma \to \pi^+ \pi^-\pi^-$ is obtained by integrating the (polarization-averaged) squared transversal T-matrix over the three-pion phase space: $$\sigma_{\rm tot}(s)={\alpha\, m_\pi^2\over 32\pi^3 f_\pi^4 (s-1)}
\int\limits_{z^2<1}\!\!\!\!\!\int\!d\omega_1 d\omega_2 \int_{-1}^1\!dx\int_0^\pi \!
d\phi \, \big|\hat k \times (\vec q_1 A_1+\vec q_2 A_2)\big|^2 \,.$$ Here, $\omega_1$ and $\omega_2$ are the center-of-mass energies of the outgoing negative pions divided by $m_\pi$. In terms of the directional cosines $x=\hat k \cdot \hat q_1,\,y= \hat k \cdot \hat q_2,\, z= \hat q_1 \cdot \hat q_2$ the squared cross products in eq.(5) take the form: $$(\hat k \times \vec q_1)^2 = q_1^2(1-x^2)\,, \quad (\hat k
\times \vec q_2)^2 = q_2^2(1-y^2)\,, \quad (\hat k \times \vec q_1)\cdot
(\hat k \times \vec q_2)= q_1 q_2(z-x y)\,,$$ with $q_{1,2} = \sqrt{\omega_{1,2}^2 -1}$ the momenta of the outgoing negative pions divided by $m_\pi$, and the subsidiary relations: $$q_1q_2 \,z =\omega_1 \omega_2-\sqrt{s}(\omega_1 +\omega_2)
+{s+1 \over 2} \,, \quad\quad y = xz +\sqrt{(1-x^2)(1-z^2)} \cos\phi \,.$$ The Mandelstam variables $s_1,\,s_2,\,t_1,\,t_2$ follow in the center-of-mass frame as: $$\begin{aligned}
&& s_1 = s+1-2 \sqrt{s}\, \omega_2\,, \quad s_2 =
s+1-2 \sqrt{s} \,\omega_1\,,\nonumber \\ && t_1 = 1+{1-s\over
\sqrt{s}} (\omega_1-q_1 x) \,,\quad t_2 = 1+{1-s\over \sqrt{s}}
(\omega_2-q_2 y)\,. \end{aligned}$$ The (bounded) integration region in the $\omega_1\omega_2$ plane is determined by the inequality $z^2<1$. It is straightforward to solve at fixed $\omega_1$ the quadratic equation for the upper and lower limit $\omega_2^\pm$ of the $\omega_2$-integration:
$$\omega_2^\pm = {1\over 2} \Bigg(\sqrt{s}-\omega_1\pm q_1 \,
\sqrt{s-2\sqrt{s} \omega_1-3 \over s-2\sqrt{s} \omega_1+1} \,\Bigg)\,, \qquad
{\rm for} \quad 1< \omega_1 < {s-3 \over 2\sqrt{s}}\,.$$
The right hand side of eq.(5) allows to calculate also the dipion mass spectra of the process $\pi^- \gamma \to \pi^+ \pi^-\pi^-$ by omitting one integration over an energy variable.
Radiative corrections
=====================
In this section we present the radiative corrections of relative order $\alpha
=e^2/4\pi=1/137.036$ to the charged pion-pair production process $\pi^- \gamma
\to \pi^+ \pi^-\pi^-$. These include the purely electromagnetic interaction of the charged pions mediated by one-photon exchange and the one-photon loop corrections to the tree level diagrams of chiral perturbation theory. Electromagnetic counterterms [@knecht] will be needed in order to eliminate the ultraviolet divergences generated by the virtual photon-loops. Infrared finiteness of the radiative corrections is achieved in the standard way by including soft-photon bremsstrahlung below an energy cutoff.
One-photon exchange
-------------------
\
The simplest electromagnetic correction to the process $\pi^- \gamma
\to \pi^+ \pi^-\pi^-$ at low energies is given by describing the $\pi^-\pi^-$ interaction in terms of the one-photon exchange. A representative subset of the 10 corresponding tree diagrams is shown in Fig.1. The diagrams (in the upper row) involving the $\pi\pi\gamma\gamma$ contact-vertex of scalar quantum electrodynamics lead to the following production amplitude: $$A_1^{(\gamma\gamma)} = 4\pi \alpha\, {f_\pi^2 \over m_\pi^2} \bigg\{
{2\over s_1}+ {1\over s_2}+ {1\over 2-s+s_2-t_1}\bigg\}\,,$$ while the additional diagrams with three ordinary $\pi\pi\gamma$ vertices (in the lower row) give rise to the production amplitude: $$\begin{aligned}
A_1^{(\gamma)} &=& 4\pi \alpha\, {f_\pi^2 \over m_\pi^2} \Bigg\{
{1\over 3-s-t_1-t_2}\bigg[{2s+1-s_1-2s_2+t_1 \over s-2-s_1+t_2}+{2s+1-2s_1-s_2
+t_2 \over s-2-s_2+t_1} \bigg] \nonumber \\ &&+{1\over t_1-1}
\bigg[ {1-2s+2s_1+s_2-t_1-2t_2 \over s_2}+ {s+1-s_1-2s_2+t_1+t_2 \over
s-2-s_1+t_2}\bigg] \Bigg\}\,. \end{aligned}$$ The prefactor $f_\pi^2/m_\pi^2$ is a consequence of the normalization introduced in eq.(1).
Photon-loop diagrams and electromagnetic counterterms
-----------------------------------------------------
The virtual radiative corrections to $\pi^-\gamma \to \pi^+ \pi^-\pi^-$ are obtained by dressing the (three) tree diagrams with a photon-loop in all possible ways. It is helpful to divide these loop diagrams into three classes: self-energy corrections on external pion-lines (I), vertex corrections to the pion-photon coupling (II), and “irreducible” photon-loop diagrams (III). We use dimensional regularization to treat both ultraviolet and infrared divergences (where the latter are caused by the masslessness of the photon). Divergent pieces of one-loop integrals show up in the form of the composite constant: $$\xi = {1\over d-4} + {1\over 2}(\gamma_E-\ln 4\pi) +
\ln{m_\pi \over \mu_r} \,,$$ containing a simple pole at $d=4$. The arbitrary mass scale $\mu_r$ is introduced in order to keep (via a prefactor $\mu_r^{4-d}$) the mass dimension of one-loop integrals independent of $d$. Ultraviolet (UV) and infrared (IR) divergences are distinguished by the feature of whether $d<4$ or $d>4$ is the condition for convergence of the $d$-dimensional integral. We discriminate them in the notation by putting appropriate subscripts, i.e. $\xi_{UV}$ and $\xi_{IR}$. In order to simplify all calculations we employ the Feynman gauge, where the photon propagator is directly proportional to the Minkowski metric tensor $g_{\mu\nu}$.
Fig.2 shows a representative subset of the 12 photon-loop diagrams with a self-energy correction on an external pion-line. The corresponding production amplitude $A_1^{(\rm I)}$ is the tree amplitude in eq.(4) multiplied with twice the (electromagnetic) wave function renormalization factor $Z_2^{(\gamma)}-1$ of the charged pion: $$A_1^{(\rm I)} = A_1^{(\rm tree)} \, {2 \alpha \over \pi }
(\xi_{IR} -\xi_{UV})\,.$$
\
$\vcenter{\hbox{\includegraphics[scale=0.8,clip]{blob1.ps}}}\ \Large{=}\
\vcenter{\hbox{\includegraphics[scale=0.8,clip]{blob2.ps}}}\ \Large{\bf +}\
\vcenter{\hbox{\includegraphics[scale=0.8,clip]{blob3.ps}}} \Large{\bf +}\
\vcenter{\hbox{\includegraphics[scale=0.8,clip]{blob4.ps}}} \ \Large{ +} \
\vcenter{\hbox{\includegraphics[scale=0.8,clip]{blob5.ps}}}$
Fig.3 shows (reducible) photon-loop diagrams including vertex corrections to the pion-photon coupling. The blob introduced in the upper row stands for the sum of the four diagrams depicted in the lower row. This whole class of diagrams leads to a vanishing production amplitude: $$A_1^{(\rm II)} =0\,,$$ where the zero results from the sum: $$\begin{aligned}
0 &=& {\alpha \over \pi} \bigg[-\xi_{UV}+1 -{u+1 \over 2u}
\ln(1-u)\bigg] + {\alpha \over 8\pi}(6\xi_{UV}-7) \nonumber \\ && + {\alpha
\over 8\pi} \bigg[6\xi_{UV}-6-{1\over u} +{u-1 \over u^2}(3u+1) \ln(1-u)\bigg]
\nonumber \\ &&+ {\alpha \over 8\pi} \bigg[-4\xi_{UV}+5+{1\over u} +{u^2+6u+1
\over u^2} \ln(1-u)\bigg] \,. \end{aligned}$$ Here, each of the four summands corresponds to a diagram in the lower row of Fig.3 (in the order shown) and $u$ abbreviates the variable $(p_2-k)^2/m_\pi^2$, $(q_1-k)^2/m_\pi^2$, or $(q_2-k)^2/m_\pi^2$ depending on which outgoing pion-line the vertex corrections take place. Note that the first term in eq.(15) is the once-subtracted (off-shell) self-energy of the pion. Nevertheless, it is most advantageous to combine it with the (proper) vertex corrections in Fig.3 in order to obtain the zero-sum. The same pattern of cancellation has been observed in section2 of ref.[@neutral].
\
In the remaining loop diagrams the virtual photon connects two charged pions with each other. Fig.4 shows for a selected pion-pair the 7 diagrams which result from all possible couplings of the incoming photon $\gamma(k,\epsilon\,)$. Given the 6 possible pion-pairs, there are in total $6 \cdot 7 =42$ such “irreducible” photon-loop diagrams for the process $\pi^-\gamma \to \pi^+
\pi^-\pi^-$. We have evaluated these diagrams by using the Mathematica package FeynCalc [@feyncalc]. After reduction to basic scalar loop-functions the corresponding production amplitude $A_1$ consists of several hundreds of terms, which prohibits a reproduction of the explicit analytical expression in this paper. A good check of the completeness of diagrams in the automatized calculation is provided by the crossing relation between $A_2$ and $A_1$ in eq.(3). The irreducible photon-loop diagrams generate also an ultraviolet divergent contribution. We have extracted this piece and after combining it with the $\xi_{UV}$-term from classI written in eq.(13) one gets in total the following ultraviolet divergent contribution from the virtual photon-loops: $$A_1^{(\rm UV-div)} = {3 \alpha \over 2\pi}\, \xi_{UV}\bigg[
{2s-s_1-s_2+t_1+t_2 \over 3-s-t_1-t_2} +{s+2-s_1-s_2+t_2 \over t_1-1} \bigg]
\,.$$ Note that $A_1^{(\rm UV-div)}$ is not proportional to the tree amplitude $A_1^{(\rm tree)}$ written in eq.(4). Since chiral perturbation theory with inclusion of virtual photons is a non-renormalizable effective field theory, the cancellation of ultraviolet divergences in radiative corrections requires the consideration of additional electromagnetic counterterms. The complete Lagrangian ${\cal L}_{e^2p^2}$ of order ${\cal O}(e^2p^2)$ consists of 11 different terms and has been given in eq.(3.6) of ref.[@knecht]. We have extracted from ${\cal L}_{e^2p^2}$ all those vertices which are relevant for the process $\pi^-\gamma \to \pi^+ \pi^-\pi^-$ considered here. After evaluation of the tree diagrams shown in Fig.5 one obtains the following contribution to the production amplitude $A_1$ from the electromagnetic counterterms: $$\begin{aligned}
A_1^{(\rm ct)}&=& {\alpha \over 2\pi}\bigg\{ (3 \xi_{UV}+
\widehat k_1 ) \bigg[{s_1+s_2-2s-t_1-t_2 \over 3-s-t_1-t_2}+{s_1+s_2-s-t_2-2
\over t_1-1} \bigg] \nonumber \\ && \qquad - \,\widehat k_2 \,\bigg(
{1\over 3-s-t_1-t_2}+{1\over t_1-1} \bigg)\bigg\} \,, \end{aligned}$$ with the two linear combinations of low-energy constants: $$\widehat k_1= {16 \pi^2 \over 9}(10 k_1^r -26k_2^r-54k_3^r
-27 k_4^r)-{3\over 2}-3\ln {m_\pi \over \mu_r}\,,$$ $$\widehat k_2= {64 \pi^2 \over 9}(18 k_3^r +9k_4^r-5k_5^r
+31 k_6^r -k_7^r+36k_8^r)\,.$$ The last term $-3/2 -3 \ln(m_\pi/\mu_r)$ in eq.(18) comes from matching our convention for the ultraviolet divergence $\xi_{UV}$ to that of ref.[@knecht]. One sees that in the sum $A_1^{(\rm UV-div)}+ A_1^{(\rm ct)}$ the ultraviolet divergence $\xi_{UV}$ drops out. This exact cancellation serves as a further important check of our calculation. It should be noted that the coefficients $\sigma_i$ written in eq.(3.11) of ref.[@knecht] (which determine the divergent part of an individual counterterm) are taken here consistently for $Z=0$, since we do not consider the additional electromagnetic effects induced in pion-loops via the charged and neutral pion mass difference. Actually, in the complete ChPT calculation to order ${\cal O}(e^2p^2)$ the low-energy constant $\widehat k_2$ in eq.(19) would be split into two pieces where one part is used to express the numerator of the chiral tree-amplitude in terms of the physical neutral pion mass square $m^2_{\pi^0}$ (see eq.(3.13) in ref.[@knecht]).
\
Soft photon bremsstrahlung
--------------------------
In the next step we have to consider the infrared divergent terms proportional to $\xi_{IR}$ present in eq.(13) and in the production amplitude $A_1$ from the irreducible photon-loops. At the level of measurable cross sections these get eliminated by contributions from (undetected) soft-photon bremsstrahlung. In its final effect, soft-photon radiation off the in- or outgoing charged pions multiplies the tree-level differential cross section for $\pi^-\gamma \to \pi^+
\pi^-\pi^-$ by the correction factor: $$\begin{aligned}
\delta_{\rm soft}&=& \alpha\, \mu^{4-d}\!\!\!\int\limits_{|\vec
l\,|<\lambda}\!\!\!{d^{d-1}l \over (2\pi)^{d-2}\, l_0} \bigg\{ {2p_1\cdot q_1
\over p_1 \cdot l \, q_1 \cdot l} +{2p_2\cdot q_1 \over p_2\cdot l\,q_1 \cdot l}
+{2p_1\cdot q_2 \over p_1 \cdot l \,q_2 \cdot l} +{2p_2\cdot q_2 \over p_2
\cdot l \, q_2 \cdot l}\nonumber \\ && - {2p_1\cdot p_2 \over p_1 \cdot l \,
p_2 \cdot l} - {2q_1\cdot q_2 \over q_1 \cdot l \, q_2 \cdot l} - {m_\pi^2 \over
(p_1 \cdot l)^2}- {m_\pi^2 \over (p_2 \cdot l)^2} - {m_\pi^2 \over (q_1 \cdot l)^2}
- {m_\pi^2 \over (q_2 \cdot l)^2}\bigg\} \,, \end{aligned}$$ which depends on a small energy cut-off $\lambda$. Working out this momentum space integral by the method of dimensional regularization (with $d>4$) one finds two contributions. The first (universal) contribution includes the infrared divergence $\xi_{IR}$ and it has a logarithmic dependence on the cut-off $\lambda$. The detailed expression for $\delta_{\rm soft}^{(\rm uni)}$ reads: $$\begin{aligned}
\delta_{\rm soft}^{(\rm uni)}&=& {4\alpha\over \pi} \bigg(
\ln{m_\pi \over 2\lambda} -\xi_{IR}\bigg)\bigg[(s_1+s_2+1-s-t_1-t_2)\,{\bf L}(
s_1+s_2-1-s-t_1-t_2) \nonumber \\ && +(2-s_1)\,{\bf L}(s_1-4) +(2-s_2)\,
{\bf L}(s_2-4)+ (s_2-s-t_1)\,{\bf L}(s+t_1-s_2-2) \nonumber \\ &&
+(s_1-s-t_2)\,{\bf L}(s+t_2-s_1-2)+(s+1-s_1-s_2)\,{\bf L}(s-1-s_1-s_2) +1\bigg]
\,,\nonumber \\ \end{aligned}$$ with the logarithmic function: $${\bf L}(z)={1\over \sqrt{z(4+z)}} \,\ln {\sqrt{z}+\sqrt{4+z}
\over 2}\,.$$ Here, each of the six terms of the form $(\dots)\,{\bf L}(\dots)$ comes from an interference term in eq.(20) and the $1$ at the end from the sum of last four squares. The second contribution is specific for evaluating the soft-photon correction factor $\delta_{\rm soft}$ in the center-of-mass system and imposing an infrared cut-off, $|\vec l\,|<\lambda$, in this reference frame. Its explicit expression reads: $$\begin{aligned}
\delta_{\rm soft}^{(\rm cm)}&=&{\alpha\over 2\pi} \Bigg\{
{s+1 \over s-1}
\ln s +\sum_{j=1}^3{2\omega_j \over \sqrt{\omega_j^2-1}}\,\ln\Big(\omega_j+\sqrt{
\omega_j^2-1}\,\Big) \nonumber \\ && +\sum_{j=1}^6 \int_0^1 dx \,{V_j C_j
\over D_j \sqrt{C_j^2 -4s D_j}}\, \ln { C_j+ \sqrt{C_j^2 -4s D_j} \over C_j-
\sqrt{C_j^2 -4s D_j} }\Bigg\}\,, \end{aligned}$$ with $$\omega_1 = {s+1-s_2 \over 2\sqrt{s}}\,,\qquad \omega_2 =
{s+1-s_1 \over 2\sqrt{s}} \,,\qquad \omega_3= \sqrt{s}- \omega_1-\omega_2\,,$$ the center-of-mass energies of the outgoing pions divided by $m_\pi$, and the abbreviations: $$V_1 = s+1-s_1-s_2\,, \qquad C_1 = s+1-s_1+(s_1-s_2)x \,,$$$$D_1= 1+x(1-x)(s-1-s_1-s_2)\,,$$ $$V_2 = s_1+s_2+1-s-t_1-t_2\,, \qquad C_2 = s+1+(s_1+s_2-s-3)x \,,$$ $$D_2= 1+x(1-x)(s_1+s_2-1-s-t_1-t_2)\,,$$ $$V_3 = s_2-s-t_1\,, \qquad C_3 = s+1 - s_2\, x \,,
\qquad D_3= 1+x(1-x)(s+t_1-s_2-2)\,,$$ $$V_4 = 2-s_1\,, \qquad C_4 = s+1-s_2+(s_1+2s_2-s-3)x \,,
\qquad D_4= 1+x(1-x)(s_1-4)\,,$$ $$V_5 = s_1-s-t_2\,, \qquad C_5 = s+1-s_1\, x \,,
\qquad D_5= 1+x(1-x)(s+t_2-s_1-2)\,,$$ $$V_6 = 2-s_2\,, \qquad C_6 = s+1-s_1+(2s_1+s_2-s-3)x \,,
\qquad D_6= 1+x(1-x)(s_2-4)\,,$$ for linear polynomials in the dimensionless Mandelstam variables $(s,s_1,s_2,t_1,t_2)$.
Results and discussion
======================
In this section we present and discuss the numerical results for the radiative corrections to the charged pion-pair production process $\pi^-\gamma \to \pi^+
\pi^-\pi^-$. We study in detail the radiative corrections to the total cross section $\sigma_{\rm tot}(s)$ and to the $\pi^+\pi^-$ and $\pi^-\pi^-$ invariant mass spectra. At the level of the production amplitudes the radiative corrections are given in eq.(5) by the real parts of interference terms between $A_{1,2}^{\rm (tree)}$ and $A_{1,2}^{\rm (rad-cor)} \sim \alpha$. The lengthy expressions generated by FeynCalc have been evaluated numerically with the help of the routine LoopTools [@looptools]. By assigning different values to the parameter $\xi_{IR}$ in the code the exact cancellation of infrared divergences from virtual photon-loops and soft-photon radiation (see eq.(21)) has been verified.
Radiative corrections to total cross section
--------------------------------------------
For orientation, we reproduce in Fig.6 the total cross section $\sigma_{\rm tot}(s)$ of the reaction $\pi^-\gamma \to \pi^+ \pi^-\pi^-$ calculated at tree-level in chiral perturbation theory [@picross; @3pion]. As demonstrated in ref.[@3pion] $\sigma_{\rm tot}(s)$ remains almost unchanged in the region $3<\sqrt{s}<6$ after inclusion of the next-to-leading order chiral corrections (from pion-loops and chiral counterterms in the isospin limit). The dashed line in Fig.6 is obtained by taking into account the leading isospin-breaking correction due to the difference of the charged and neutral pion mass (see eq.(2.6) in ref.[@knecht]). Isospin-breaking modifies the tree amplitude $A_1^{(\rm tree)}$ in such a way that the constant $\delta_{\rm ib} = 2(m_{\pi^0}/m_\pi)^2 -2 = -0.1295$ is added to both numerators in eq.(4). The relative correction to the total cross section $\sigma_{\rm tot}(s)$ arising from this effect amounts to $-7.5\%,\, -5.0\%,\, -2.6\%,\, -1.6\%,\, -1.1\%$ at $\sqrt{s} =3.5,\,
4,\,5,\,6,\,7$, respectively (see Fig.8).
Fig.7 shows in percent the radiative corrections to the total cross section $\sigma_{\rm tot}(s)$ in the region $3<\sqrt{s}<7$. The (lower) dashed-dotted and dashed curves display the effects of the soft-photon bremsstrahlung, separated into the universal contribution proportional to $\ln(m_\pi/2\lambda)$ and the contribution specific for imposing the infrared cut-off in the center-of-mass frame. As in refs.[@comptcor; @neutral] the value $\lambda =5\,$MeV has been chosen, which is in the order of magnitude as it appears in the kinematics measured at the COMPASS experiment [@janhabil]. The (upper) full line in Fig.7 shows the effect of the virtual photon-loops. Surprisingly, this radiative correction which requires a huge effort for its computation is almost constant in the region $3.5<\sqrt{s}<7$. With an average value of $1.6\%$ it is also rather close to the analogous result (about $2.2\%$) for neutral pion-pair production $\pi^-\gamma \to \pi^- \pi^0\pi^0$ shown in Fig.5 of ref.[@neutral]. The behavior close to threshold $3<\sqrt{s}<3.5$ is governed by the Coulomb singularity in the $\pi^\pm\pi^-$ final-state interaction, i.e. the Gamow factor $\pm \pi \alpha/\beta$ with $\beta$ the relative pion velocity [@batgamow]. Since the $\pi^+ \pi^-\pi^-$ final-state gives rise to two pion-pairs with opposite charges and only one with equal charges the attractive effect (i.e. an enhancement of the cross section) prevails. The effects of the Coulomb singularity will be analyzed in more detail in subsection4.2 and in the appendix.
Fig.8 shows by the (upper) dashed line the electromagnetic corrections arising from the one-photon exchange and the sum of the previous contributions (in Fig.7) is reproduced by the (lower) full curve. With values ranging between $8\pi \alpha f_\pi^2/m_\pi^2 = 8.04\%$ at threshold and $2.1\%$ at $\sqrt{s}=7$ the simple one-photon exchange constitutes the largest correction of relative order $\alpha$. The horizontal grey band shows the additional effects of the electromagnetic counterterms. We have varied the low-energy constants $ \widehat k_1$ and $ \widehat k_2$ in the range $ -1\leq \widehat k_{1,2} \leq 1$, which is expected to cover electromagnetic counterterms of natural size [@knecht]. Evidently, if one allows for a larger range of the low-energy constants $\widehat k_{1,2}$ the band will broaden accordingly. The total sum of the radiative corrections is shown by the dropping band in Fig.8. The dashed-dotted line is finally obtained by complementing this sum (for $\widehat k_{1,2}=0$) by the leading isospin-breaking correction. Putting aside the one-photon exchange contribution which is special for the process $\pi^-\gamma \to \pi^+ \pi^-\pi^-$, one can conclude that the radiative corrections calculated here are of similar size as in the case of the neutral pion-pair production process $\pi^-\gamma \to \pi^- \pi^0\pi^0$ studied in ref.[@neutral]. Moreover, one finds that the one-photon exchange and the leading isospin-breaking correction given by $\delta_{\rm ib} = 2(m_{\pi^0}/m_\pi)^2
-2$ compensate each other partly.
Radiative corrections to dipion mass spectra
--------------------------------------------
In this subsection we study the radiative corrections to more exclusive observables of the process $\pi^-\gamma \to \pi^+ \pi^-\pi^-$, namely the dipion mass spectra. We start with the $\pi^+\pi^-$ mass spectrum. In terms of the Mandelstam variables the $\pi^+\pi^-$ invariant mass is $\sqrt{s_2}\,m_\pi $. Exploiting the relation $s_2= s+1-2 \sqrt{s}\omega_1$, one sees that the differential cross section $d\sigma/d\sqrt{s_2}$ is obtained from the right hand side of eq.(5) by leaving out the integration over $\omega_1$ and multiplying with a factor $\sqrt{s_2/s}$. Again for orientation, we reproduce in Fig.9 the $\pi^+\pi^-$ mass spectrum calculated at tree-level in the isospin limit [@3pion]. The numbers next to the curves are the values of $\sqrt{s}$. The radiative corrections arising from soft-photon bremsstrahlung and virtual photon-loops are shown by the dashed and solid curves in Fig.10. Note that these corrections are given here in absolute units of microbarn, without dividing by the respective mass spectra at tree-level. The pattern of radiative corrections is completed in Fig.11, where the effect the one-photon exchange and the sum of all contributions are shown. Throughout, one observes positive corrections to $d\sigma/d\sqrt{s_2}$ from the virtual photon-loops and the one-photon exchange and negative corrections from the soft-photon radiation. In the total sum of all contributions oscillations and sign-changes occur, which get more pronounced with increasing $\sqrt{s}$. The overlying bands produced by the variation of the low-energy constants $\widehat k_1$ and $\widehat k_2$ have not been included for reasons of a cleaner presentation.
A striking feature visible in Figs.10,11 is that the full curves are not smooth, but have kinks at intermediate values of $\sqrt{s_2}$. These kinks are by no means numerical artifacts, but they have a clear physical origin in the Coulomb singularity of the electromagnetic $\pi^\pm\pi^-$ final-state interaction. Let us elaborate on this close relationship in more detail. At fixed $s_2$ the range of the (integration) variable $s_1$ is $s_1^- < s_1<s_1^+$ with: $$s_1^\pm = {1\over 2} \Bigg( 3+s-s_2 \pm \sqrt{s_2-4\over s_2}
\,\sqrt{(s+1-s_2)^2-4s}\,\Bigg)\,,$$ where $4<s_2<(\sqrt{s}-1)^2$. The isolated Coulomb singularity $1/\sqrt{s_1-4}$ leads to a dipion mass spectrum of the form: $$F(\sqrt{s_2}\,)= \int_{s_1^-}^{s_1^+}ds_1\, {1\over
\sqrt{s_1-4}} = 2\sqrt{s_1^+ -4}-2\sqrt{s_1^- -4}\,.$$ The just constructed function $F(\sqrt{s_2}\,)$ is shown by the full lines in Fig.12 for $\sqrt{s}=4,5,6,7$. One observes a sharp kink at the position $s_2=(s-1)/2$, which is determined by the solution of the equation $s_1^- =4$. At this point the integral in eq.(32) extends fully into the inverse square-root singularity. One finds by a simple calculation that the left and right derivative of $F(\sqrt{s_2}\,)$ with respect to the variable $\sqrt{s_2}$ at this point are different with values: $\sqrt{2/(s-9)}\big[\pm 8-\sqrt{2(s-1)}\,\big]$. Therefore, this example demonstrates that the Coulomb singularity causes inevitably a kink in the two-pion mass spectrum. Actually, it is evidence for the accuracy of the employed numerical methods when this kink is visible in the $\pi^+\pi^-$ mass spectra calculated from a large number of terms. It is also interesting to consider the effects of the resummed Coulomb singularity by evaluating eq.(32) with the integrand $\mp(1-G)/\pi\alpha$, where $G=\eta/(e^\eta-1)$ is the Gamow function [@batgamow] with $\eta=\mp 2\pi \alpha/\sqrt{s_1-4}$. In this treatment the dashed and dashed-dotted curves in Fig.12 result from the (singular part of the) Coulomb interaction between two pions with opposite and equal charges, respectively. In order to make the higher order electromagnetic effects better visible the fine-structure constant $\alpha $ has been increased by a factor 10 to $\alpha =1/13.7$. One observes that in the attractive case $(-)$ the kink in the dipion mass spectrum becomes more pronounced whereas in the repulsive case $(+)$ it gets apparently smoothened out.
We continue with the discussion of the $\pi^-\pi^-$ mass spectrum of the process $\pi^- \gamma \to \pi^+\pi^-\pi^-$. The $\pi^-\pi^-$ invariant mass is denoted by $\mu\,m_\pi $, where the (dimensionless) variable $\mu$ fulfills the relation $\mu^2 = s+3-s_1-s_2= 1-s+2
\sqrt{s}(\omega_1+\omega_2)$. We introduce the sum $ \omega_+=\omega_1+\omega_2$ and half-difference $ \omega_-=(\omega_1-\omega_2)/2$. After this change of variables, $\omega_{1,2}=\omega_+/2\pm \omega_-$, the differential cross section $d\sigma/d\mu$ is obtained from the right hand side of eq.(5) by leaving out the integration over $\omega_+$ and multiplying with a factor $\mu /\sqrt{s}$. The limits for the remaining integration over $\omega_-$ are $\pm \sqrt{\mu^2-4} \sqrt{[s-(\mu+1)^2][s-(\mu-1)^2]}
/(4\mu \sqrt{s})$. For the purpose of comparison the $\pi^-\pi^-$ mass spectra calculated in tree approximation are reproduced in Fig.13. Note that these have a completely different shape than the $\pi^+\pi^-$ mass spectra displayed in Fig.9. The radiative corrections to the differential cross section $d\sigma/d\mu$ as they arise from soft-photon bremsstrahlung, virtual photon-loop and the one-photon exchange are shown in Figs.14,15, together with the total sum of these three contributions. The kinks at $\mu =
\sqrt{(s-1)/2}$ caused by the Coulomb singularity in the $\pi^+\pi^-$ final-state interaction appear very prominently in Fig.15. The overlying bands produced by the variation of the low-energy constants $\widehat k_1$ and $\widehat k_2$ have been omitted for reasons of a cleaner presentation.
Altogether, the radiative corrections to the total cross section and dipion mass spectra of the reaction $\pi^-\gamma \to \pi^+ \pi^- \pi^-$ are of the order of a few percent, with the exception of the region $3<\sqrt{s}<4$ close to threshold. The electromagnetic corrections are indeed below the $5\%$ level as assumed in the analysis of the COMPASS data in ref.[@compass]. However, with the results of the present work the radiative corrections (as well as the leading isospin-breaking effect) can be consistently taken into account in the analysis of a future high statistics experiment.
Appendix: Virtual radiative corrections to elastic [$\pi^- \pi^-$]{} scattering {#appendix-virtual-radiative-corrections-to-elastic-pi--pi--scattering .unnumbered}
===============================================================================
In this appendix we study the virtual radiative corrections to elastic $\pi^- \pi^-$ scattering. Since $\pi^- \pi^- \to \pi^- \pi^-$ is the strong interaction process underlying the charged pion-pair production $\pi^- \gamma
\to \pi^+\pi^-\pi^-$, it is most instructive to investigate separately the radiative corrections for this simpler subprocess. An essential advantage is that these can be given in closed analytical form. Based on the chiral $\pi\pi$ contact-vertex at leading order, there are 10 associated photon-loop diagrams. For the four diagrams with external self-energy corrections the chiral tree amplitude $T_-^{(\rm tree)} =m_\pi^2(2-s)/f_\pi^2$ in the isospin limit gets multiplied with twice the (electromagnetic) wave function renormalization factor of the charged pion: $$T_-^{(\rm I)} = {2 \alpha \,m_\pi^2 \over \pi f_\pi^2} (2-s)
(\xi_{IR} -\xi_{UV})\,.$$ The two (equal) diagrams with a photon-loop in the $s$-channel (either in the initial or the final state) give together rise to a contribution to the $\pi^-\pi^-$ scattering amplitude, whose real part reads: $$\begin{aligned}
{\rm Re}\,T_-^{(\rm II)}&=&{\alpha\, m_\pi^2 \over \pi f_\pi^2}
(2-s)\Bigg\{4\xi_{UV}-{9 \over 2} +{4-s+4 (s-2)\xi_{IR} \over \sqrt{s^2-4s}}
\nonumber \\ &&\times \ln {\sqrt{s-4}+\sqrt{s} \over 2} +(2-s)
-\hspace{-0.47cm}\int_4^\infty\! {dx \over x-s}\, { \ln(x-4) \over \sqrt{x^2-4x}}
\Bigg\}\,. \end{aligned}$$ Here, we have used the (concise) spectral function representation of the scalar loop integral involving one photon and two pion propagators. In the same way one finds from the two photon-loop diagrams in the $t$-channel: $$\begin{aligned}
T_-^{(\rm III)}&=&{\alpha\, m_\pi^2 \over \pi f_\pi^2}\Bigg\{
\xi_{UV}\bigg( s+{3t \over 2}-5 \bigg) +{13-3s \over 2} -2t +\bigg[ 4s t +{3t^2
\over 2} -8s\nonumber \\ && -10 t +16 +4(s-2)(2-t)\xi_{IR}\bigg] {1 \over
\sqrt{t^2-4t}} \ln {\sqrt{4-t}+\sqrt{-t} \over 2} \nonumber \\ && +(s-2)(2-t)
\int_4^\infty\! {dx \over x-t}\, { \ln(x-4) \over \sqrt{x^2-4x}} \Bigg\}\,, \end{aligned}$$ and the contribution from the photon-loops in the $u$-channel is immediately obtained via the substitution $t\to u$: $$\begin{aligned}
T_-^{(\rm IV)}&=&{\alpha\, m_\pi^2 \over \pi f_\pi^2}\Bigg\{
\xi_{UV}\bigg( s+{3u \over 2}-5 \bigg) +{13-3s \over 2} -2u +\bigg[ 4s u +{3u^2
\over 2} -8s\nonumber \\ && -10 u +16 +4(s-2)(2-u)\xi_{IR}\bigg] {1 \over
\sqrt{u^2-4u}} \ln {\sqrt{4-u}+\sqrt{-u} \over 2} \nonumber \\ && +(s-2)(2-u)
\int_4^\infty\! {dx \over x-u}\, { \ln(x-4) \over \sqrt{x^2-4x}} \Bigg\}\,. \end{aligned}$$
Note that we have expressed here the elastic $\pi^-\pi^-$ scattering amplitude $T_-$ in terms of dimensionless Mandelstam variables $(s,t,u)$, which satisfy the constraint $s+t+u= 4$ and the inequalities $s>4$ and $t,u<0$ hold in the physical region [@cola]. The total ultraviolet divergence resulting from the sum of all 10 photon-loop diagrams is: $T_-^{(\rm UV-div)}=-(\alpha\, m_\pi^2 /2\pi f_\pi^2)\,
3s\, \xi_{UV}$. This piece gets eliminated by the electromagnetic counterterms of ref.[@knecht]. The corresponding contribution to the $\pi^-\pi^-$ scattering amplitude reads: $$T_-^{(\rm ct)} = {\alpha \,m_\pi^2 \over 2\pi f_\pi^2} \Big[
( 3\xi_{UV}+\widehat k_1 )s + \widehat k_2 \Big]\,,$$ with $\widehat k_1$ and $\widehat k_2$ the same linear combinations of low-energy constants as written in eqs.(18,19). Let us note that the virtual photon-loops in charged pion-pion scattering have been calculated earlier by Knecht and Nehme [@nehme]. We find perfect agreement with the pertinent terms proportional to $e^2$ written in eqs.(12,13) of ref.[@nehme]. The infrared divergence is identified as $\xi_{IR} = \ln(m_\pi/m_\gamma)$ with $m_\gamma$ a regulator photon mass. The detailed comparison shows also that the term proportional to the low-energy constant $\widehat k_2$ in eq.(37) needs to be split up into two pieces and one part has been used in ref.[@nehme] to express the chiral tree amplitude $T_-^{(\rm tree)}
=[m_\pi^2(4-s)-2m^2_{\pi^0}]/f_\pi^2$ in terms of the physical neutral pion mass.
Of particular interest is the threshold behavior of the last loop-function appearing in eq.(34). By expanding this principal-value integral around $s=4$ for $s>4$ one finds: $$\begin{aligned}
&& -\hspace{-0.46cm}\int_4^\infty\! {dx \over x-s}\, { \ln(x-4)
\over \sqrt{x^2-4x}} = {\pi^2 \over 2 \sqrt{s-4}} -1 -{\pi^2 \over 16}
\sqrt{s-4}+{\cal O}(s-4) \nonumber \\ && ={\pi^2 \over 4 v}(1-v^2) -1
+{\cal O}(v^2)= {\pi^2 \over 4 \beta}(2-\beta^2) -1
+{\cal O}(\beta^2) \,, \end{aligned}$$ where the first term corresponds to the well-known Coulomb singularity proportional to the inverse pion velocity $v = \sqrt{1-4/s}$ in the center-of-mass frame or the inverse relative velocity $\beta = 2v/(1+v^2)= \sqrt{s(s-4)}/ (s-2)$. In non-relativistic quantum mechanics the same electromagnetic initial- or final-state interaction effect is described by the Gamow factor $\pm \pi \alpha/\beta $.
The full line in Fig.16 shows the radiative corrections to the total cross section $\sigma_{\rm tot}(s)=(64\pi s\,m_\pi^2)^{-1} \int_{-1}^1\!dz\,|T_-|^2$ for elastic $\pi^-\pi^-$ scattering as they arise from the virtual photon-loops calculated in eqs.(34-36). We have discarded the $\xi_{UV}$ and $\xi_{IR}$ terms which get eliminated by the electromagnetic counterterm and the soft-photon bremsstrahlung. One observes negative corrections of the order of a several percent in the region $2<\sqrt{s}<6$. The behavior close to threshold is governed by the Coulomb singularity $-2\pi \alpha(s-4)^{-1/2}$, which is shown separately by the dashed line in Fig.16. The dashed-dotted line gives for comparison the (repulsive) Gamow factor $-2\pi \alpha/\beta$ with $\beta =
\sqrt{s(s-4)}/ (s-2)$ the relative velocity of both pions. One is instructed that the result of the complete calculation in relativistic quantum field theory lies in between these two approximations. Interestingly, the pure Coulomb singularity $-2\pi \alpha(s-4)^{-1/2}$ provides the better approximation. Although the sign and $\sqrt{s}$-dependence are different, the magnitude of these virtual radiative corrections is comparable to the ones shown by the full line in Fig.7. Note that the one-photon exchange amplitude $T_-^{(1\gamma)}= 4\pi \alpha[(s-u)t^{-1}+(s-t)u^{-1}]$ for $\pi^-\pi^-\to\pi^-\pi^-$ has not been considered here since it would spoil the discussion in terms of the total cross section.
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[^1]: This work has been supported in part by DFG and NSFC (CRC110).
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abstract: 'We present electric noise measurements of nanoscale biepitaxial YBa$_2$Cu$_3$O$_{7-\delta}$ (YBCO) Josephson junctions fabricated by two different lithographic methods. The first (conventional) technique defines the junctions directly by ion milling etching through an amorphous carbon mask. The second (soft patterning) method makes use of the phase competition between the superconducting YBCO (Y123) and the insulating Y$_2$BaCuO$_5$ (Y211) phase at the grain boundary interface on MgO (110) substrates. The voltage noise properties of the two methods are compared in this study. For all junctions (having a thickness of 100 nm and widths of 250-500 nm) we see a significant amount of individual charge traps. We have extracted an approximate value for the effective area of the charge traps from the noise data. From the noise measurements we infer that the soft patterned junctions with a grain boundary (GB) interface manifesting a large c-axis tunneling component have a uniform barrier and a SIS like behavior. The noise properties of soft patterned junctions having a GB interface dominated by transport parallel to the ab-planes are in accordance with a resonant tunneling barrier model. The conventionally patterned junctions, instead, have suppressed superconducting transport channels with an area much less than the nominal junction area. These findings are important for the implementation of nanosized Josephson junctions in quantum circuits.'
author:
- 'D. Gustafsson'
- 'F. Lombardi'
- 'T. Bauch'
title: Noise properties of nanoscale YBCO Josephson junctions
---
*This document is the Accepted Manuscript version of a Published Work that appeared in final form in Physical Review B, copyright ©American Physical Society. To access the final edited and published work see http://prb.aps.org/abstract/PRB/v84/i18/e184526*
\[sec:level1\]Introduction
==========================
Micrometer-sized grain boundary (GB) Josephson junctions (JJs) made of High Critical Temperature Superconductors (HTS) are commonly used for the realization of superconducting devices operated in a wide temperature range up to the boiling temperature of liquid nitrogen. A prominent example is the Superconducting QUantum Interference Device (SQUID) for sensitive magnetic flux detection [@Koelle1999]. Nevertheless, GB JJs are still a fundamental tool for the exploration of the complex physics inherent to HTS materials [@Alff1998; @Tsuei2000; @Jesper; @Bauch2005; @Bauch2006]. Their implementation in superconducting circuits operated in the quantum limit, such as quantum bits or single electron transistors, are expected to give further useful hints on the unresolved nature of superconductivity in HTS materials [@Bauch2009]. Recent advances in the thin film technology and nano-fabrication of HTS made it possible to observe macroscopic quantum phenomena in YBa$_2$Cu$_3$O$_{7-\delta}$ (YBCO) biepitaxial grain boundary Josephson junctions [@Bauch2005; @Bauch2006] opening the way for the realization of HTS quantum circuits. For typical JJ-based devices, which operate in the quantum limit (typically at temperatures below 100 mK), the requirements on junction critical currents and capacitances are met for lateral dimensions on a length scale of 100 nm [@Bauch2009; @Gustafsson2010].
The realization of reproducible HTS JJs at the nanoscale can also be instrumental to fabricate sensors with a quantum limited sensitivity like nano-SQUIDs, which can allow the detection of magnetic nano-particles in a much wider temperature and magnetic field range compared to its low critical temperature superconductor (LTS) counterpart.
In this respect it is of particular importance to understand the microscopic properties and dynamics of charge transport across nano-sized GB JJs. Here, the investigation of low frequency electric noise is a very useful tool to study the dynamics of both cooper pair and quasiparticle charge transport, revealing among other things information about the nature of the GB interface and its homogeneity. Still after numerous experimental studies on HTS GB junctions during the last decades the underlying physical transport mechanisms across the GB interface are subject of recurring discussion.
A large number of noise studies have been performed on wide bicrystal and biepitaxial GB JJs with junction widths ranging from one to several tens of micrometers [@Kawasaki1992; @Miklich; @Marx1995a; @Marx1995b; @Marx1998; @Liatti2006]. Only a few electric transport studies have been performed on sub-micrometer bicrystal GBs, where single charge trapping states, responsible for the low frequency fluctuations of the transparency of the GB barrier, could be resolved [@Herbstritt].
In this article we compare two methods to fabricate YBCO Josephson Junctions at the nano scale and their respective noise properties. These methods are based on biepitaxial grain boundaries created in single layer YBCO films. Both a conventional technique, where the nanosized junctions are patterned by electron beam lithography and ion milling, and a new technique, where the junctions are formed as a result of phase competition between superconducting and insulating phases at the grain boundary interface, will be described. We have previously shown that the two methods give Josephson junctions with fundamentally different critical current density $j_C$ and resistivity $\rho_N$ values[@Davidnanoletter]. In this paper we compare the noise data of soft nanopatterned GB junctions to various electrical transport models, which allows us to determine the nature of the biepitaxial GB barriers. Moreover, from the analysis of single charge trap states in the GB barriers we are able to qualitatively and quantitatively assess the detrimental effect of ion milling on GBs during the conventional fabrication.
This paper is organized as follows. In section \[sec:Sample fabrication\] we describe both the conventional and soft nano-structuring of biepitaxial YBCO JJs. Section \[sec:Comparison\] is dedicated to the comparison of the dc transport properties between junctions fabricated with the two nano-patterning methods. The noise models applicable to GB JJs are introduced in section \[sec:NoiseTheory\]. In section \[sec:Results\] we present the results and discussion of noise measurements on JJs fabricated with the two nano-lithographic methods.
Sample fabrication {#sec:Sample fabrication}
==================
Conventional nanostructuring
----------------------------
The conventional way to fabricate deep submicron biepitaxial Josephson junctions is to use electron beam lithography in combination with a hard mask and ion beam milling. This procedure, with amorphous carbon as hard mask, is well established and has been proven to work well for the realization of various kinds of submicron HTS Josephson junctions, for example ramp type[@Komissinski], bicrystal[@tobias] and biepitaxial[@DaniellaJAP]. In this work we have fabricatated deep submicron Josephson junctions by the biepitaxial technique. Details on the fabrication procedure can be found elsewhere[@DaniellaJAP; @DaniellaIEEE]. Here we only summarize the main steps. First a 30 nm thick SrTiO$_3$ (STO) layer is deposited on a MgO (110) substrate using Pulsed Laser Deposition (PLD). Next an amorphous carbon mask is deposited and then patterned using e-beam lithography and oxygen plasma. Part of the seed layer is then removed using Ion milling. Then a 100-120 nm thick YBCO film is grown by PLD at a temperature of 790$^\circ$C . The film will grow (001) oriented on the MgO substrate and (103) on the STO seed layer. The YBCO film is then patterned using ion milling through an amorphous carbon mask defined by e-beam lithography. Even though junctions with widths smaller than 100 nm can, in principle, be fabricated with this procedure, the damage caused by the ion milling process will effectively limit the smallest possible width. The damaged grain boundary region on both sides of the junction constitutes a significant part of the total junction width, which strongly affects the superconducting properties. We have therefore engineered an alternative way to nanostructure HTS Josephson junctions, which is described in the following section.
Soft nanostructuring
--------------------
We have developed a new soft patterning method that allows fabricating biepitaxial grain boundary junctions at the nanoscale without significant lateral damaging effects due to the ion milling. The procedure is based on the competition between the nucleation of the superconducting and insulating phases at the grain boundary. To fabricate the junctions we use the fact that for certain deposition conditions secondary insulating phases like Y$_2$BaCuO$_5$ (Y211, also called greenphase) can nucleate on MgO(110) in addition to the superconducting YBa$_2$Cu$_3$O$_{7-\delta}$ (Y123). The amount of greenphase increases for non optimal deposition conditions and in the presence of grain boundaries[@Green; @Gustafsson2010; @DiChiara]. Nano sized superconducting Y123 connections embedded in a greenphase matrix are expected to be formed at the grain boundary, see Figure \[Fig1\] (a). These connections can be isolated using a Focused Ion Beam (FIB), see Figure \[Fig1\] (b) and (c).
We first fabricate 10 $\mu$m wide grain boundary junctions using the conventional method. A deposition temperature of 740$^\circ$C is used for the YBCO film. The grain boundary is then examined using atomic force microscopy (AFM) and scanning electron microscopy (SEM). A suitable superconductive nanoconnection is selected and then isolated by using FIB. By leaving greenphase regions of at least 300 nm on each side of the Y123 connection, nanosized Josephson junction with no lateral damage are created since the Ga ions will only get implanted into the greenphase layer. Figure \[Fig1\] (c) shows a final device, where we have isolated an approximately 200 nm wide junction protected on both sides by greenphase.
![(a) SEM image of an interface between a (001) and (103) YBCO film. A significant amount of greenphase is present near the grain boundary. In a different study[@Davidnanoletter] transmission electron microscopy and energy dispersive x-ray analysis was used to confirm that the precipitate at the grain boundary is greenphase. (b) AFM scan of a 10 $\mu$m wide grain boundary interface before the FIB procedure. (c) SEM image of the same interface after the unwanted YBCO have been removed by the FIB leaving only one or two connections. []{data-label="Fig1"}](SEM_AFM4.eps)
Comparison between the transport properties of conventional and soft nanopatterned junctions {#sec:Comparison}
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Electrical properties such as critical current density ($j_C$), specific resistance ($\rho_N$) and critical current ($I_C$) vs magnetic field ($B$) have been extensively examined[@Davidnanoletter] and have shown significant differences for the two fabrication methods. Figure \[Fig2\] shows the current voltage characteristics (IVC) for (a) a soft nanopatterned junction (200 nm wide), (b) a 300 nm wide conventionally patterned junction and (c) a 200 nm wide conventionally patterned junction. A recurring pattern is seen here: the soft nanostructured junctions have an order of magnitude or more higher $j_C$ and one or several orders of magnitude lower $\rho_N$ when compared to conventionally fabricated samples. Conventionally fabricated junctions with a width of 200 nm or less have high resistive nonlinear IVCs with a suppressed Josephson current. Only junctions with widths 300 nm or more showed a Josephson current.
![Current voltage characteristics for (a) A nanojunction fabricated using the soft nanopatterning technique. The width of the junction extracted from AFM is 200 nm. (b) a sample fabricated with the conventional nanopatterning technique. The second switch is because this specific sample was designed to have 2 Josephson junctions in series to allow study of charging effects. Here the nominal junction width is 300 nm. (c) A sample fabricated using the conventional method, 200 nm wide, with a coulomb blockade like behavior and no critical current. The measurements were done at 271, 16 and 22 mK, respectively.[]{data-label="Fig2"}](IV5.eps)
$I_C$ vs $B$ measurements revealed significant differences in the modulation period for the two fabrication methods. The period of the magnetic field modulation ($\Delta B$) of the Josephson current can be used to approximate the width ($w$) of the region exhibiting Josephson coupling in the junctions. Figure \[Fig3\] (a) shows the $I_C$ vs. $B$ for a 10 $\mu$m wide grain boundary junction before the FIB cut to isolate the nanojunction; The behavior of the magnetic pattern is that of several parallel junctions[@Tinkham; @Likharev]. Figure \[Fig3\] (b) shows the magnetic pattern of a softpatterned junction after the FIB cut, leaving only one or two connections.
Depending on the electrode geometry we use the two expressions for $\Delta$B as a function of the junction width $w_j$ from Rosenthal and coworkers[^1][@Rosenthal]: For the soft patterned junctions having wide electrodes $w_e\simeq 10\mu$m we use the thick electrode limit expression $$\label{Rosenthal1}
\Delta B=\frac{\Phi_0t}{1.2w_j^2(\lambda_{103}+\lambda_{001} +d)}$$ valid for $\lambda_{001,103}^2/t < w_e$. Here $\lambda_{001}$ and $\lambda_{103}$ are the London penetration depth in the (001) and (103) electrode, respectively. $\Phi_0$ is the magnetic flux quantum, $t$ is the thickness of the film and $d$ is the thickness of the junction barrier. For the conventionally nanopatterned junctions the width of the electrodes is equal to the width of the junctions $w_e \simeq 300-500$ nm. Here the thin electrode limit ($\lambda_{001,103}^2/t \geq w_e$) applies $$\label{Rosenthal2}
\Delta B=\frac{1.84\Phi_0}{w_j^2}.$$ The London penetration depth in the (001) electrode is given by the penetration depth in the ab-planes $\lambda_{001} = \lambda_{ab}$. Instead, as a result of the London penetration depth anisotropy in YBCO $\lambda_{103}$ is given by a combination of $\lambda_{ab}$ and the c-axis penetration depth $\lambda_c$, which depends on the grain boundary angle[@Jesper][^2].
Equation \[Rosenthal1\] was used on a number of soft patterned junctions and the extracted width was compared to the nominal width measured by AFM and SEM[@Davidnanoletter]. The values were at most differing by 40%; This shows that the width of the superconducting transport channels extracted from the magnetic pattern was very close to the measured junction widths.
For one of the conventionally nanopatterned junction a $\Delta B$ of approximately 1 T was extracted from the magnetic pattern. Using equation \[Rosenthal2\] resulted in a width of 60 nm, significantly less than the nominal junction width of 300 nm. Similar results where obtained for two other junctions 300 and 500 nm wide. This in combination with the j$_C$ and $\rho_N$ values shows that a substantial part of the grain boundary, approximately 100 nm wide, on each lateral side of the junction does not feature any Josephson coupling.
The magnetic patterns of conventionally nanopatterned junctions have revealed the presence of a highly non uniform grain boundary, having a much reduced region with Josephson coupling compared to the nominal one. However, this does not give a clear image of the total area which retains the Josephson coupling along the grain boundary. In fact, it only tells us that the largest spacing between superconducting channels is significantly less than the nominal junction width. To estimate the area of both the Cooper pair and quasiparticle transport channels we analyzed the voltage noise of the junctions caused by single charge traps in the GB barrier, which will be discussed in section \[sec:Results\].
![I vs B for (a) a 10 $\mu$m wide grain boundary with many parallel channels. (b) a sample cut by FIB, which consists of only 1 or 2 parallel channels. The grey scale represents the logarithmic conductance and the darkest region corresponds to $I_C$.[]{data-label="Fig3"}](IvsB2.eps)
Noise theory for grain boundary junctions {#sec:NoiseTheory}
=========================================
Noise measurements are a helpful tool to extract information about the electrical transport through the junction and hence to obtain information about the nanostructure of the grain boundary interface. In this work we focused on the low frequency noise spectra of both the critical current fluctuations $\delta I_C$ and normal resistance fluctuations $\delta R_N$, which are related to the transport mechanisms of the cooper pairs and quasiparticles, respectively.
It is well established that at low frequencies the critical current and normal resistance fluctuations are governed by bistable charge trapping states in the junction barrier [@Rogers]. The trapping of a charge will locally increase the junction barrier making it less transparent. This process can be considered as a reduction of the total junction area $A_{j}$ by an amount which is proportional to the cross section of the localized charge trap state $A_{t}$. The fluctuating barrier transparency (or equivalently junction area) results in fluctuations of the critical current $I_C$ and normal resistance $R_N$. Each individual charge trap causes a random telegraph switching (RTS) signal between two states, with respective mean lifetimes $\tau_1$ and $\tau_2$, of both the junction normal resistance and critical current. The corresponding frequency spectrum is given by a Lorentzian [@Machlup]: $$\begin{aligned}
\label{Lorentzian}
S_R^{RTS}(f) & = & \frac{4\langle(\frac{\delta R_N}{R_N})^2\rangle \tau_{eff}}{1+(2\pi f \tau_{eff})^2}, \nonumber\\
S_I^{RTS}(f) & = & \frac{4\langle(\frac{\delta I_C}{I_C})^2\rangle \tau_{eff}}{1+(2\pi f
\tau_{eff})^2},\end{aligned}$$ where $\tau_{eff} = (\tau_1^{-1}+\tau_2^{-1})^{-1}$ is the effective lifetime of the underlying RTS signal and $f$ is the frequency. $\langle(\delta R_N /R_N)^2\rangle$ and $\langle(\delta I_C
/I_C)^2\rangle$ are the mean squared relative fluctuations of the normal resistance and critical current caused by the charge trap. For large enough junction areas many bistable charge trapping states will contribute to the total noise. Assuming a constant distribution of transition rates $1/\tau_{eff}$ the resulting noise power spectrum will have a $1/f$ shape.
The values of the relative root mean square (rms) fluctuations $\delta I_C/I_C$ and $\delta R_N/R_N$ can be determined by measuring the voltage noise across the junction at various bias current values. For a Josephson junction having a non hysteretic current voltage characteristic the total voltage fluctuations across the junction at a fixed bias current $I$ are given by[@Miklich] $$\label{SV}
S_V(f)=(V-R_dI)^2S_I(f)+V^2S_R(f)+k(V-R_dI)VS_{IR}(f),$$ where $V$ is the dc-voltage across the junction, $R_d = \partial
V/\partial I$ is the differential resistance, $S_I =|\delta
I_C/I_C|^2$, $S_R =|\delta R_N/R_N|^2$, and $S_{IR} =|\delta
I_C/I_C||\delta R_N/R_N|$ is the cross spectral density of the fluctuations. Here it is assumed that $S_I$ and $S_R$ are composed of an ensemble of RTS signals, $S_I^{RTS}$ and $S_R^{RTS}$, respectively. The value $k$ represents the correlation between the $\delta I_C$ and $\delta
R_N$ fluctuations. One has $k=-2$ and $k=2$ for perfectly antiphase and inphase correlated fluctuations, respectively. For uncorrelated fluctuations one obtains $k=0$. From equation \[SV\] it follows that at bias currents close to the critical current the voltage fluctuations are dominated by critical current fluctuations $S_I$ due to the large differential resistance. For large bias currents, where the differential resistance approaches the asymptotic normal resistance, the voltage noise is governed by resistance fluctuations $S_V = V^2S_R$. The correlation term $S_{IR}$ will only contribute to the voltage noise in the intermediate bias current regime, while it is negligible close to the critical current and for large bias currents.
The values of the relative fluctuations $\delta I_C/I_C$ and $\delta
R_N/R_N$ depend on the nature of the junction barrier. Indeed, from the ratio $q=|\delta I_C/I_C|/|\delta R_N/R_N|$ between the relative fluctuations one can extract information about the homogeneity of the junction barrier as we will discuss on the basis of the following three junction models applicable to grain boundary junctions:
For a homogenous junction barrier one can assume that the $I_CR_N$ product is a constant, independent of the critical current density $j_c$ and resistivity $\rho_n$. This is for example the case for a superconductor-insulator-superconductor (SIS) junction [@Tinkham], where cooper pairs and quasiparticles tunnel (directly) through the same parts of the junction. From the constant $I_CR_N$ product it follows directly that the relative fluctuations of the critical current and normal resistance have the same amplitude and are anticorrelated $\delta I_C/I_C = -\delta R_N/R_N$, resulting in a ratio $q=1$.
In the Intrinsic Shunted Junction (ISJ) model[@Gross; @GrossPRB], instead, where the barrier is assumed to be inhomogeneous containing a high density of localized electronlike states, the quasiparticle transport is dominated by resonant tunneling via the localized states. On the contrary, due to Coulomb repulsion cooper pairs can only tunnel directly through the barrier. Detailed calculations show that the $I_CR_N$ product is not anymore constant, instead it follows the scaling behavior $I_C R_N \propto (j_c)^p$, where $j_c$ is the critical current density of the junction and $p$ is a constant depending on the position of the localized states. For localized states sitting in the middle of the barrier the scaling power is $p=0.5$. From this $I_CR_N$ scaling behavior one obtains for the ratio of the normalized fluctuations $|\delta I_C/I_C|/|\delta
R_N/R_N|=q=1/(1-p)$[@Gross]. Typical experimental values of $q$ range between 2 and 4 [@Kawasaki1992; @Marx1995a; @Marx1995b; @Marx1998].
The channel model proposed by Micklich [*et al.*]{} [@Miklich] assumes a junction which consists of N parallel channels, where all channels have the same resistance but only one channel carries a supercurrent. For a large number of channels $N$ the fluctuations in critical current can be much higher than the fluctuations in resistance, giving a high ratio $q$. Since the supercurrent and the quasiparticles have separate channels no correlation between the critical current and resistance fluctuations is expected.
![a) Sketch of the interface geometry. The crystallographic orientations of the (001) and (103) YBCO is indicated by arrows. $\theta$ is the interface angle and is defined with respect to the \[001\] MgO direction.[]{data-label="Fig8"}](SampleAngle.eps){width="90mm"}
![(a) Voltage noise at 10 Hz as a function of bias current (open symbols) for a soft nano patterned junction (nominal interface angle = 30$^\circ$). A theoretical fit (solid line) is included to determine S$_I$ and S$_R$. The inset shows the voltage noise as a function of frequency for three different bias points. The “hump” moves to higher frequencies when the bias is increased. (b) Conductivity as a function of bias voltage of the soft nano patterned junction. The central part ($|V|<1$) mV is related to the Josephson effect.[]{data-label="Fig4"}](noise6_T.eps){width="90mm"}
![Noise spectrum measured at $I \gg I_C$ after the subtraction of the $1/f$ background. The black line is a Lorentzian fit to equation \[SV\_RTS\]. The plateau of the Lorentzian is given by $4 \tau_{eff}\langle (\delta V/V)^2 \rangle$, with $2 \pi \tau_{eff} = 302$ Hz and $\delta V/V = 0.0044$. The inset shows the respective time trace with $\Delta V/V \simeq 0.0095$.[]{data-label="Fig5"}](timetrace_Lorentzian_1.eps){width="90mm"}
![Voltage noise at 10 Hz as a function of bias current (open symbols) of a soft nano patterned junction with an interface angle of 50$^\circ$. The solid line is a theoretical fit used to determine S$_I$ and S$_R$.[]{data-label="Fig10"}](NoiseR22_50DegreeGreenPhase.eps){width="90mm"}
Results and Discussion {#sec:Results}
======================
The voltage noise spectral density was measured using a room temperature voltage preamplifier with an input noise of 4 nV/Hz$^{1/2}$ followed by a Stanford Research Dynamic Signal analyzer SR785 for a number of bias points. These measurements were done at 4 K for the soft nanostructured junctions, where the current voltage characteristic was non hysteretic. For the conventionally fabricated samples a temperature of 280 mK was needed to avoid thermal smearing of the IVC (due to the low $I_C$ of these junctions)[@Miklich].
All noise measurements on the conventionally patterned samples refer to grain boundaries obtained by patterning the STO seed layer parallel to the \[100\] direction ($\theta = 0^\circ$) of the MgO substrate[@Jesper] (see Fig. \[Fig8\]). The soft patterned junctions had nominal interface angles ($\theta$) of 30$^\circ$ and 50$^\circ$. Grain boundaries with a small interface angle $\theta$ have a micro structure close to a basal-plane like type ($45^\circ$ \[010\] tilt), see Fig. \[Fig8\], where the a-b planes of the (001) YBCO electrode meet one single a-b plane on the (103) YBCO electrode side[@Cedergren]. These grain boundaries have proven to be low dissipative [@Bauch2005; @Bauch2006; @Lombardi_PhysicaC] compared to \[001\]-tilt ones and suitable for applications in quantum circuitry. By increasing the GB angle $\theta$ the interface gradually evolves into a $45^\circ$ \[010\] twist GB at $\theta = 90^\circ$ (see Fig. \[Fig8\]).
--------------------------------------------- --------------------------------------------- -------------- ------------------------------- ------------------------- -----------------------------------
Junction GB $A_{j}$ $S_I^{1/2}$ (10 Hz) $q=(S_{I}/S_{R})^{1/2}$ $A_{j}^{1/2}S_{I}^{1/2}$ (10 Hz)
technology type ($\mu$m$^2$) $\times 10^{-4}$(Hz$^{-1/2}$) $\times 10^{-6}(\mu$m/Hz$^{1/2})$
YBCO/STO/MgO biepitaxial $45^\circ$ \[010\] tilt 0.02 $1.0\pm 0.1$ $1.0\pm 0.1$ 14
$\theta\simeq 0^\circ$ (this work) ($T=4.2$K)
YBCO/STO/MgO biepitaxial $\alpha \cdot \{45^\circ$ \[010\] tilt$\}+$ 0.052 $1.6\pm 0.1$ $1.8\pm 0.2$ 36
$\theta\simeq 50^\circ$ (this work) $\beta \cdot \{45^\circ$ \[010\] twist$\}$ ($T=4.2$K)
YBCO/NdGaO$_3$ bicrystal $2\times14^\circ$ \[100\] tilt 0.06 $0.36\pm 0.06$ $1.05\pm 0.1$ 10
(Ref. [@Liatti2006]) ($T=55$K)
YBCO/STO bicrystal $25^\circ$ \[001\] tilt 1.0 $0.32$ $2.5$ 32
(Ref. [@Kawasaki1992]) ($T=70$K)
Bi$_2$Sr$_2$CaCu$_2$O$_{8+x}$/STO bicrystal $24^\circ$ \[001\] tilt 1.6 $0.24\pm 0.06$ $1.9\pm 0.3$ 30
(Ref. [@Marx1995a]) ($T=40-70$K)
YBCO/STO bicrystal $24^\circ$ \[001\] tilt 3.8 $0.18\pm 0.01$ $3.8\pm 0.6$ 35
(Ref. [@Marx1995b; @Marx1998]) ($T=25-70$K)
YBCO/STO bicrystal $36.8^\circ$ \[001\] tilt 1.0 $0.35\pm 0.12$ $3.7\pm 0.8$ 35
(Ref. [@Marx1995b; @Marx1998]) ($T=30-70$K)
--------------------------------------------- --------------------------------------------- -------------- ------------------------------- ------------------------- -----------------------------------
Noise properties of soft patterned junctions
--------------------------------------------
In Figure \[Fig4\] (a) the voltage noise spectral density at 10 Hz is plotted as a function of the bias current $I$ for one of our soft patterned nanojunctions having a width $w\simeq 200$ nm and nominal interface angle of 30$^\circ$. However, by examining the grain boundary by AFM the actual interface angle was closer to 0$^\circ$.
As expected, the noise peaks close to the critical current when $I\simeq I_C = $4 $\mu$A. A second peak appears close to $I=$5.2 $\mu$A. This is due to a resonance feature in the IVC causing the differential resistance (R$_d$) to spike. For higher bias, where the resistance fluctuations dominate, the noise increases quadratically. The hump structure around 50 $\mu$A is caused by a single charge trap causing a RTS signal with a typical Lorentzian spectrum on top of a $1/f$ background. The occurrence of such Loretzians is typical for a limited number of charge traps in submicron sized GB junctions [@Herbstritt]. The voltage dependence of the effective lifetime of the charge trap causes the Lorentzian to move to higher frequencies for increasing bias current, which is shown in the inset of Figure \[Fig4\].
To fit the measured voltage noise spectral density at 10 Hz as a function of bias current (see solid line in Figure \[Fig4\] (a)) to equation \[SV\], which assumes a pure $1/f$ noise spectrum, we neglected the data between 20 $\mu$A and 80 $\mu$A caused by a single charge trap. From the fit we obtain $S_{I}\simeq S_R\simeq
10^{-8}/$Hz resulting in $q=|\delta I_C/I_C|/|\delta R_N/R_N|\simeq
1$, and $k\simeq -1.3$. The ratio $q\simeq 1$ indicates that our junction has a rather homogeneous barrier, where quasi particles and Cooper pairs tunnel directly through the same parts of the barrier [@Miklich]. Together with a tunnel like conductance spectrum (see Figure \[Fig4\] (b)) we can conclude that our junction barrier is very similar to that of a SIS junction, consistent with the band bending model [@Mannhart1998; @Hilgenkamp1998]. Similar results have only been found in $2\times 14^{\circ}$ \[100\]-tilt YBCO GB Josephson junctions [@Liatti2006]. Furthermore, our result is incompatible with the Intrinsically shunted junction model [@Gross; @GrossPRB; @Marx1995a; @Marx1995b] and the channel model [@Miklich], where $q$-values larger than 2 are expected. The deviation of the correlation between the critical current and resistance fluctuations from perfect anti-correlation ($k = -2$) could be caused by the limited amount of two level fluctuators in the small junction area not representing a perfect ensemble. It is important to point out that the noise properties of our nano junction close to an ideal SIS Josephson junction underline once more the pristine character of the junction barrier that can be obtained by using the soft nano-patterning method.
In Figure \[Fig10\] the voltage noise spectral density at 10 Hz is plotted for a 520 nm wide sample having a nominal interface angle of $\theta\simeq50^\circ$. This value was confirmed by the AFM inspection of the GB. From the fit we obtain $S_{I}\simeq 2
\cdot10^{-8} - 3 \cdot10^{-8} /$Hz, $S_R\simeq 8 \cdot 10^{-9}/$Hz, and $k\simeq -0.5$ resulting in $q=1.8 \pm0.2$. The $q$ value close to 2 indicates that the transport across the GB barrier cannot be described by a direct tunneling model, e.g. a homogeneous SIS tunnel junction. Instead, our result shows that for this kind of GB type (mixture of $45^\circ$ \[010\] twist and $45^\circ$ \[010\] tilt) the barrier is better described by the ISJ model, where quasiparticles tunnel resonantly via localized states. In table \[Table\_coll\] we summarize the noise data of the soft nanopatterned biepitaxial YBCO GB junctions together with results from literature on HTS Josephson junctions of various GB types. Comparing the ratios $q=(S_{I}/S_{R})^{1/2}$ between different GB types one can clearly see that only GBs where the ab-planes in at least one of the electrodes are tilted around an axis parallel to the GB interface, e.g. $45^\circ$ \[010\] tilt (this work) and $2\times 14^\circ$ \[100\] tilt [@Liatti2006], have a ratio $q\simeq 1$. All the other GB types such as \[001\] tilt [@Kawasaki1992; @Marx1995a; @Marx1995b; @Marx1998] and $\alpha \cdot
\{45^\circ$ \[010\] tilt$\}+\beta \cdot \{45^\circ$ \[010\] twist$\}$, with $0<\beta \leq 1$ exhibit ratios $q\gtrsim 2$. These facts give a strong indication that the nature of HTS GB barriers depends on how the ab-planes meet at the interface. GBs with ab-planes tilted around an axis parallel to the GB interface, such as a basal plane GB, can be described by a direct tunneling model consistent with a homogeneous SIS barrier. All other GB types deviating from a bare rotation of the ab-planes around the GB line are characterized by resonant quasiparticle tunneling via localized states (ISJ model).
From the spectral density of the critical current fluctuations and the junction area one can obtain information about the areal charge trap density $n_t$ and the cross sectional area $A_t$ of the charge traps [@Marx1997; @VanHarlingen2004]. Assuming $N$ identical and independent charge traps the spectral density of the relative critical current fluctuations scales with the junction area $A_j$ as $\langle (\delta I_C/I_C)^2 \rangle = N(A_t/A_j)^2 = n_t A_t^2/A_j$. From this equation it follows that the quantity $(S_I A_j)^{1/2}$ is proportional to the product of cross sectional area of a charge trap and the square root of the trap density $A_t n_t^{1/2}$. In table \[Table\_coll\] we show the computed product $(S_I A_j)^{1/2}$ at 10 Hz for the various GB types. Remarkably the values for GBs having ratios $q\geq2$ are close to $35\times 10^{-6} \mu$m/Hz$^{1/2}$ [^3]. Instead, the values for $(S_I A_j)^{1/2}$ in GB types with $q\simeq 1$ are roughly 3 times smaller. Assuming that the cross sectional area of a charge trap is independent of the GB type, the difference in charge trap density supports once more the different nature of the GB barriers.
In the following we will use the Lorentzian spectra sitting on top of a $1/f$ background (see inset of Fig. \[Fig4\]) to estimate the cross sectional area $A_{t}$ of a single charge trap in the barrier: A single charge trap causes the voltage across the junction to fluctuate between two bistable states with an amplitude $\Delta V$ (see inset in Fig. \[Fig5\]). For large bias currents $I \gg I_C$, when the differential resistance is asymptotically reaching the normal state resistance of the junction, we can write for the respective relative resistance change $\Delta R_N/R_N = \Delta
V/V$. Assuming that the current flow across the junction is homogeneous and the charge trap completely blocks the current flow in a small part of the junction barrier we can determine the charge trap’s cross sectional area $A_t$ from the measured voltage fluctuation amplitude $\Delta V$ $$\label{trap_area}
A_t = \frac{\Delta R_N}{R_N} A_{qp} = \frac{\Delta V}{V} A_{qp},$$ where $A_{qp}$ is the total area of quasi particle transport along the junction. Instead of extracting $\Delta V$ from a voltage time trace, one can also use the mean squared fluctuation amplitude $\langle (\delta V)^2 \rangle$ determined from a Lorentzian fit of the noise spectrum (see Fig. \[Fig5\]). The two quantities are related via [@Herbstritt] $$\label{DeltaVtime}
(\Delta V)^2 = \left( \frac{\tau_1}{\tau_2}+\frac{\tau_2}{\tau_1}+2\right) \langle (\delta V)^2 \rangle.$$ For clearly visible Lorentzians in the measured noise spectra the ratio between the two mean life times is typically in the range from 1 to 10. Hence, we can approximate the fluctuation amplitude within a factor of two by $\Delta V \simeq 2 \sqrt{\langle(\delta
V)^2\rangle} $ using the root mean squared (rms) fluctuation amplitude extracted from a Lorentzian spectrum (see Fig. \[Fig5\]): $$\label{SV_RTS}
S_V^{RTS}(f)=V^2S_R^{RTS}(f).$$
![Histogram showing the effective fluctuator area that was extracted from multiple spectra of 3 different soft patterned junctions in the high bias range. The black curve is a normal distribution with the same mean and standard deviation as the data set.[]{data-label="Fig6"}](Histogram_Afluct2.eps)
Together with equation \[Lorentzian\] we can extract $\delta
R_N/R_N$ and approximate the cross sectional area of a charge trap: $$\label{DefectArea}
A_t \simeq 2\frac{\delta R_N}{R_N} A_{qp}.$$ From the results of the previous section we can make the following considerations:
1. The $I_C$ vs $B$ measurements have shown that the modulation period corresponds to an effective width close to the nominal width of the junctions. We can therefore assume that the Cooper pair transport is along the whole grain boundary, $A_{cp}$ $\simeq$ $A_{j}$.
2. The fitting of the voltage noise spectral density has shown that $S_{I}\simeq S_R$, this tells us that the area of the superconducting channel is approximately equal to the quasiparticle one, $A_{cp} \simeq A_{qp}$.
These two facts imply that the areas of both transport channels are very close to the nominal junction area ($A_{qp} \simeq A_{cp}
\simeq A_{j}$). One can, therefore, use the nominal area, measured by AFM or SEM, in combination with the noise measurement to extract $A_{t}$. Here we also use that the junction thickness $\simeq$ film thickness (120 nm). We have made this type of analysis for 3 soft patterned junctions and fitted a total of 24 Lorentzians in the high bias range on different spectra. The extracted distribution of $A_{t}$ are plotted in Figure \[Fig6\]: We get an average area for the fluctuators of about 72 nm$^2$, which is comparable to results found in submicron \[001\]-tilt YBCO GB junctions [@Herbstritt].
In Figure \[Fig7\] the spectrum at a bias current close to $I_C$ has been fitted by 2 Lorentzians and a weak 1/f background. Since the contribution from $R_N$ fluctuations is negligible for this range of currents one can extract $\delta
I_C/I_C$ using equation \[Lorentzian\] and: $$\label{SI_RTS}
S_I^{RTS}(f)=\frac{S_V^{RTS}(f)}{(V-R_dI)^2}$$ The extracted values for $\delta I_C/I_C$ and $\delta R_N/R_N$ are fairly similar in magnitude, $\delta I_C/I_C$ being at most 3 times larger than the average of $\delta R_N/R_N$. This difference could be explained by a spread in the fluctuators area. Indeed the values of $\delta I_C/I_C$ extracted close to $I_C$ will certainly come from different two level fluctuators than those generating $\delta R_N/R_N$ fluctuations at high biases.
![ Noise spectrum (open circles) and fit of two Lorentzians with a $1/f$-background (solid line) for one of the soft nanostructured junctions measured at $I\simeq I_C$.[]{data-label="Fig7"}](Spectrum1_T2.eps)
Noise properties of conventional junctions
------------------------------------------
Identical measurements and analysis were carried out for 3 nanosized junctions fabricated by conventional nanolithography. For these samples, at bias currents slightly above $I_C$ we have observed the presence of strong Lorentzians in the low frequency spectra caused by single charge traps, see Figure \[Fig9\] (a). This circumstance makes the fitting of the data to equation \[SV\] in the low bias range impossible, therefore preventing the extraction of $S_I$ and the comparison with $S_R$. However, for these junctions we were able to fit the Lorentzian voltage noise spectra for the two different bias ranges, where they are dominated by current fluctuations (close to $I_C$), $S_I^{RTS}$, and by resistance fluctuations (far above $I_C$) , $S_R^{RTS}$, respectively, see Figure \[Fig9\]. Assuming that the average effective area of the charge traps is roughly the same as that extracted from the junctions fabricated by soft nanopatterning we can estimate A$_{cp}$ and A$_{qp}$ [^4] for the conventionally fabricated junctions. The part of the junction area manifesting Josephson coupling and quasi particle transport can be approximated by $A_{cp} \simeq A_t I_C/2\delta I_C$ and $A_{qp}
\simeq A_t R_N/2\delta R_N$, respectively. In the insets of Figure \[Fig9\] we show the spectral density of the normalized fluctuations multiplied by the frequency. The pronounced difference between the fluctuation amplitudes of the Lorentzians in the critical current and resistance noise spectra by several orders of magnitude clearly manifests the difference in area for the quasi particle and cooper pair transport channels. In table \[Table1\] we summarize the results for 3 conventionally patterned junctions. The average quasiparticle area is 25-50% less than the nominal area. However, the superconducting area varies greatly and for 2 of the junctions it is significantly less than the quasiparticle area. The A$_{qp}$ extracted from the noise measurements tells us that the grain boundaries, despite losing most of the Josephson properties, still have a quasiparticle transport channel with an average area comparable to the nominal area. The area of the quasiparticle channel does not decrease much in the fabrication process for the conventional junctions, however the resistivity of the channel seem to increase. Evaluating the critical current density based on the effective area across which cooper pair transport occurs results in $j_C^{eff} = I_C/A_{cp} \simeq 2-20$ kA/cm$^2$, where $I_C$ is the critical current of the conventionally patterned junctions. Here it is interesting to note that these values are in the same range as those found in pristine soft nano-structured GB junctions [@Davidnanoletter]. This fact clearly reflects the strong dependence of the Josephson coupling on the stoichiometry in close proximity (length scale of coherence length) of the GB. The detrimental effect of the ion etching process during the conventional nano patterning seems to locally kill the Josephson coupling rather than inducing a gradual decrease over the whole junction area. The ion beam procedure appears to be an on-off process for the Josephson coupling: the grain boundary region which survives the ion bombardment preserves the same Josephson properties as the untouched soft nano-patterned samples. The overall increase of more than one order of magnitude of the GB resistivity of conventionally fabricated nanojunctions compared to the soft nano-patterned ones can therefore be related to a reduced barrier transparency in the junction regions where the Josephson coupling has been switched off in the milling procedure.
Junction nr $A_{j}$ (nm$^2$) $A_{qp}$ (nm$^2$) $A_{cp}$ (nm$^2$)
------------- ------------------ ------------------- -------------------
Nr 1 50000 31500 1250
Nr 2 30000 14600 9060
Nr 3 30000 22300 160
: \[Table1\]Total nominal area $A_j$, area for the quasiparticle transport channels $A_{qp}$ and area for superconducting transport channels $A_{cp}$ (extracted from noise data) for 3 conventionally patterned samples.
![Noise spectrum (open circles) and fit of a Lorentzian with a $1/f$-background (solid line) for a conventionally patterned junction measured at a) $I\simeq I_C$ and b) $I \gg
I_C$. The vertical axis show the normalized fluctuations corresponding to a) $S_I$ and b) $S_R$. The insets shows normalized fluctuations multiplied by the frequency to emphasize the amplitude of the charge traps.[]{data-label="Fig9"}](ConventionalSpectrum1.eps){width="90mm"}
Summary and Conclusions
=======================
To summarize we have compared the electrical transport and noise properties for nanoscaled biepitaxial YBCO grain boundary Josephson junctions fabricated by two different methods. From electrical transport and $1/f$ voltage noise properties of soft nanopatterned Josephson junctions with a GB characterized by a rotation of the ab-planes parallel to the interface (large c-axis tunneling component) we can conclude that the GB barrier is very homogeneous and has a SIS character (direct tunneling model). Instead, the noise properties of soft patterned junctions, where the transport is dominated by tunneling parallel to the ab-planes, are in accordance with a resonant tunneling model (ISJ model). From the analysis of two level fluctuators in the barrier, on the other hand, we find that the conventional nanofabrication method severely deteriorates the Josephson properties of the GB. The junction area maintaining Josephson current can on average be much smaller than the nominal area, while the quasi particle transport area is similar to the nominal one. In this case the transport across the GB interface can be well described by the transport model proposed by Miklich [@Miklich]. The resistivity in these samples is increased compared to the soft nanopatterned GB junctions. The noise properties of our nanojunctions allows to identify two classes of experiments that one can perform by taking advantage of the specifics of the transport properties:
1\) to realize quantum bits by employing soft nanopatterned junctions. The pristine grain boundary is an ideal candidate to study the intrinsic source of dissipation in HTS by measuring relaxation and coherence times in a quantum bit.
2\) to realize devices where charging effects are dominant. The large resistivity values of the conventionally patterned junctions $\rho_N
\simeq 5 \cdot 10^{-7} - 2 \cdot 10^{-6} \Omega$cm$^2$ make these junctions good candidates for the realization of all-HTS single electron transistors, which can be used to study possible subdominant order parameters in HTS materials[@Sergey].
This work has been partially supported by EU STREP project MIDAS, the Swedish Research Council (VR) under the Linnaeus Center on Engineered Quantum Systems, the Swedish Research Council (VR) under the project Fundamental properties of HTS studied by the quantum dynamics of two level systems and the Knut and Alice Wallenberg Foundation.
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[^1]: The two formulas for $\Delta B$ are valid in the short junction limit, $w_j$<4$\lambda_j$[@Likharev], where $\lambda_j$ is the Josephson penetration depth[@VanDuzer]: $\lambda_j^2=\hbar/(2ej_C\mu_0(\lambda_{001}+\lambda_{103}+d))$. For the soft nanopatterned junctions we get $\lambda_j \simeq$ 1.2$\mu$m. All measured junctions are in the short junction limit since $w_j$<4$\lambda_j$.
[^2]: $\lambda_{ab}$ for a critical temperature (T$_c$) of 89 K is approximately 160 nm[@Zuev]. The effective London penetration depth, $\lambda_{eff}$, was obtained using $\lambda_{c}$ = 2 $\mu$m[@Homes] and geometrical considerations[@Jesper] with a dependence on the grain boundary angle.
[^3]: The constant value $(S_I A_j)^{1/2}$ is equivalent to the scaling behavior of the resistance fluctuations $S_R \propto R_N$ reported by Marx et al. in Ref. [@Marx1997].
[^4]: The $I_C$ vs B pattern of the conventional junctions do not follow a Fraunhofer like pattern. This hints that the interfaces consists of an array of superconducting channels. However, as long as the individual channels have an area greater than the effective area of the defects, equation \[DefectArea\] is still valid.
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abstract: 'It is shown that a function $f$ is a generalized Stieltjes function of order $\lambda>0$ if and only if $x^{1-\lambda}(x^{\lambda-1+k}f(x))^{(k)}$ is completely monotonic for all $k\geq 0$, thereby complementing a result due to Sokal. Furthermore, a characterization of those completely monotonic functions $f$ for which $x^{1-\lambda}(x^{\lambda-1+k}f(x))^{(k)}$ is completely monotonic for all $k\leq n$ is obtained in terms of properties of the representing measure of $f$.'
author:
- 'Stamatis Koumandos and Henrik L. Pedersen[^1]'
title: On generalized Stieltjes functions
---
[*2010 Mathematics Subject Classification: Primary: 44A10, Secondary: 26A48*]{}
[*Keywords: Laplace transform, Generalized Stieltjes function, completely monotonic function*]{}
Introduction
============
In this paper we investigate a real-variable characterization of generalized Stieltjes functions obtained by Sokal, see [@Sok].
Let $\lambda >0$ be given. A function $f:(0,\infty)\to \mathbb R$ is called a generalized Stieltjes function of order $\lambda$ if $$f(x)=\int_0^{\infty}\frac{d\mu(t)}{(x+t)^{\lambda}}+c,$$ where $\mu$ is a positive measure on $[0,\infty)$ making the integral converge for $x>0$ and $c\geq 0$.
The class of ordinary Stieltjes functions is the class of generalized Stieltjes functions of order $1$.
A $C^{\infty}$-function $f$ on $(0,\infty)$ is completely monotonic if $(-1)^nf^{(n)}(x)\geq 0$ for all $n\geq 0$ and all $x>0$. Bernstein’s theorem characterizes these functions as Laplace transforms of positive measures: $f$ is completely monotonic if and only if there exists a positive measure $\mu$ on $[0,\infty)$ such that $t\mapsto e^{-xt}$ is integrable w.r.t. $\mu$ for all $x>0$ and $$f(x)=\int_0^{\infty}e^{-xt}\, d\mu(t).$$
We remark that $f$ is a generalized Stieltjes function of order $\lambda$ if and only if $$\label{eq:sokalabsolutely}
f(x)=\int_0^{\infty}e^{-xt}t^{\lambda-1}\varphi(t)\, dt+c, \quad x>0$$ for some completely monotonic function $\varphi$, and some non-negative number $c$. See [@kp2 Lemma 2.1].
Sokal (see [@Sok]) introduced for $\lambda >0$ the operators $$T_{n,k}^{\lambda}(f)(x)\equiv (-1)^nx^{-(n+\lambda-1)}\left(x^{k+n+\lambda-1}f^{(n)}(x)\right)^{(k)}, \quad n,k\geq 0$$ and obtained the following characterization.
The following are equivalent for a $C^\infty$-function $f$ defined on $(0,\infty)$.
(a) $f$ is a generalized Stieltjes function of order $\lambda$;
(b) $T_{n,k}^{\lambda}(f)(x)\geq 0$ for all $x>0$, and $n,k\geq 0$.
Sokal’s characterization is an extension of Widder’s characterization of the class of ordinary Stieltjes functions: $f$ is a Stieltjes function if and only if the function $(x^kf(x))^{(k)}$ is completely monotonic for all $k\geq 0$. (See [@Wid1].)
In [@kp3 Theorem 1.5] an analogue of Sokal’s result where the function $\varphi$ in is absolutely monotonic is obtained. See also [@karpprilepkina Theorem 2] for a result complementing [@kp3 Theorem 1.1].
\[rem:equivalent\] Notice that, by Leibniz’ rule, $$x^{-(n+\lambda-1)}\left(x^{k+n+\lambda-1}f^{(n)}(x)\right)^{(k)}=\sum_{j=0}^{k}\binom{k}{j}\frac{\Gamma(n+k+\lambda)}{\Gamma(n+j+\lambda)}x^jf^{(n+j)}(x).$$
In this paper we first show that condition (b) in Sokal’s theorem above can be replaced by the condition that $$c_k^{\lambda}(f)(x)\equiv x^{1-\lambda}(x^{\lambda-1+k}f(x))^{(k)}$$ is completely monotonic for all $k$. There is a simple relation between $T^{\lambda}_{n,k}(f)$ and ${c_{k}^{\lambda}}(f)$:
\[prop:direct\] The relation $$T^{\lambda}_{n,k}(f)(x)=(-1)^n{c_{k}^{\lambda}}(f)^{(n)}(x)$$ holds for any $n,k\geq 0$ and $x>0$.
\[cor:sokal\] The following are equivalent for a function $f\in C^{\infty}((0,\infty))$.
1. $f$ is a generalized Stieltjes function of order $\lambda$;
2. ${c_{k}^{\lambda}}(f)$ is completely monotonic for all $k\geq 0$.
3. $T^{\lambda}_{n,k}(f)\geq 0$ for all $n\geq 0$ and all $k\geq 0$.
In [@kp2] the generalized Stieltjes functions corresponding to measures having moments of all orders were charaterized in terms of properties of remainders in asymptotic expansions. (A measure $\mu$ has moments of all orders in any polynomial is integrable w.r.t. $\mu$.) In view of the results in the present paper we notice the following corollary. The proof follows by combining Corollary \[cor:sokal\] with [@kp2 Theorem 3.1] and [@kp2 Lemma 3.1].
The following are equivalent for a function $f:(0,\infty)\to \mathbb R$.
1. $f$ is a generalized Stieltjes function corresponding to a measure $\mu$ having moments of all orders;
2. ${c_{k}^{\lambda}}(f)$ is completely monotonic for all $k\geq 0$ and the function $x^{\lambda-1}f(x)$ admits for any $n$ an asymptotic expansion $$x^{\lambda-1}f(x)=\sum_{k=0}^{n-1}\frac{\alpha_k}{x^{k+1}}+r_n(x),$$ in which $x^nr_n(x)\to 0$ as $x\to \infty$.
In the affirmative case, $\alpha_k=(-1)^k(\lambda)_ks_k(\mu)/k!$ where $s_k(\mu)$ is the $k$’th moment of $\mu$, and $r_n$ has the representation $$r_n(x)=(-1)^nx^{\lambda-1}\int_0^{\infty}e^{-xt}t^{\lambda-1}\xi_n(t)\, dt,$$ where $\xi_n$ belongs to $C^{\infty}([0,\infty))$, and satisfies $\xi_n^{(j)}(0)=0$ for $j\leq n-1$ and $0\leq \xi_n^{(n)}(t)\leq s_n(\mu)$ for $t\geq 0$. Furthermore, $${c_{n}^{\lambda}}(f)(x)=x^{1-\lambda}(x^nr_n(x))^{(n)}={c_{n}^{\lambda}}\left(\mathcal L(t^{\lambda-1}(-1)^n\xi_n(t)\right)(x).$$
Our aim is also to characterize, for any given positive integer $N$, those functions $f$ for which $c_0^{\lambda}(f),\ldots,c_N^{\lambda}(f)$ are completely monotonic. In the case where $\lambda=1$ this has been carried out in [@p], but the case of general $\lambda$ requires, as we shall see, additional insight.
We thus introduce the classes $\mathcal C_N^{\lambda}$ as $$\mathcal C_N^{\lambda}=\{f\in C^{\infty}((0,\infty))\, |\, c_k^{\lambda}(f)\ \text{is completely monotonic for }\ k=0,\ldots,N\}.$$ We shall use some distribution theory so we briefly describe our notation. The action of a distribution $u$ on a test function $\varphi$ (an infinitely often differentiable function of compact support in $(0,\infty)$) is denoted by $\langle u,\varphi\rangle$. The distribution $\partial u$ is defined via $\langle \partial u,\varphi\rangle=-\langle u,\varphi' \rangle$. A standard reference to distribution theory is [@rudin].
Our results can be formulated as follows.
\[thm:main\] Let $\lambda >0$ be given, and let $N\geq 1$. The following properties of a function $f:(0,\infty)\to \mathbb R$ are equivalent.
(a) $f\in \mathcal C_N^{\lambda}$;
(b) $f$ can be represented as $$f(x)=c+\int_0^{\infty}e^{-xs}s^{\lambda-1}\, d\mu(s),$$ where $c\geq 0$, and $\mu$ is a positive measure on $(0,\infty)$ for which $\mu_k\equiv (-1)^ks^k\partial^k\mu$, (in distributional sense) is a positive measure such that $$\begin{aligned}
&\int_0^{\infty}e^{-xs}s^{\lambda-1}\, d\mu_k(s)<\infty, \quad k=0,\ldots,N.\end{aligned}$$
In the affirmative case, $${c_{k}^{\lambda}}(f)(x)=x^{1-\lambda}\left(x^{\lambda-1+k}f(x)\right)^{(k)}=\int_0^{\infty}e^{-xs}s^{\lambda-1}\, d\mu_k(s)$$ for $k=0,\ldots,N$.
We notice the following corollary characterizing those non-negative functions $f$ for which ${c_{1}^{\lambda}}(f)$ is completely monotonic. The proof follows from Propostion \[prop:specialcase\] and Lemma \[lemma:integrability\].
\[cor:bernstein\] Let $f$ be a non-negative $C^{\infty}$-function defined on $(0,\infty)$. Then $x^{1-\lambda}\left(x^\lambda f(x)\right)'=\lambda f(x)+xf'(x)$ is completely monotonic if and only if $$f(x)=\alpha+\frac{\beta}{x^{\lambda}}+\int_0^{\infty}e^{-xs}s^{\lambda-1}\int_s^{\infty}\frac{d\mu(t)}{t^{\lambda}}\, ds,$$ for some non-negative numbers $\alpha$ and $\beta$ and some positive measure $\mu$ on $(0,\infty)$ making the integral convergent.
It is easy to see that $e^{-xs}s^{\lambda-1}\int_s^{\infty}d\mu(t)/t^{\lambda}$ is integrable on $(0,\infty)$ if and only if $s^{\lambda-1}\int_s^{\infty}d\mu(t)/t^{\lambda}$ is integrable at $0$, and that this is the case if and only if $\int_0^1\, d\mu(t)<\infty$ and $\int_1^{\infty}\, d\mu(t)/t^{\lambda}<\infty$.
Corollary \[cor:bernstein\] can be reformulated as follows. Let $g$ be a non-negative $C^{\infty}$-function on $(0,\infty)$. Then $x^{1-\lambda}g'(x)$ is completely monotonic if and only if $$\label{eq:bernstein}
g(x)=\alpha x^{\lambda}+\beta+\int_0^{\infty}\int_0^{xt}e^{-u}u^{\lambda-1}\, du\frac{d\mu(t)}{t^{\lambda}}.$$ Formulated in this way the corollary is related to the class of Bernstein functions. A Bernstein function is by definition a non-negative function $g$ on $(0,\infty)$ for which $g'$ is completely monotonic. These functions admit an integral representation (see [@SSV Theorem 3.2] or [@bergforst]), which we for the reader’s convenience state here: $g$ is a Bernstein function if and only if $$g(x)=\alpha x+\beta+\int_0^{\infty}(1-e^{-xt})\, d\nu(t),$$ where $\alpha$ and $\beta$ are non-negative numbers, and $\nu$, called the Lévy measure, is a positive measure on $(0,\infty)$ satisfying $\int_{0}^{1}t d\nu(t)<\infty$ and $\int_{1}^{\infty} d\nu(t)<\infty$.
When $\lambda=1$, we have $$\int_0^{xt}e^{-u}u^{\lambda-1}\, du=1-e^{-xt},$$ and reduces to the integral representation of a Bernstein function with the corresponding Lévy measure being $d\mu(t)/t$. Corollary \[cor:bernstein\] contains a characterization of what could be called “generalized Bernstein functions of order $\lambda$”.
Proofs
======
[*Proof of Proposition \[prop:direct\]:*]{} The key to the proof is the following relation $$\label{eq:key}
T^{\lambda}_{n,k}(f)(x)=(-1)^n\left(\sum_{j=0}^k(\lambda-1)_{k-j}\binom{k}{j}\left(x^{j}f(x)\right)^{(j)}\right)^{(n)},$$ which we verify now. A standard application of Leibniz’ formula yields $$\begin{aligned}
\left(x^{j}f(x)\right)^{(j+n)}&=\sum_{l=0}^{n+j}\binom{n+j}{l}\left( x^j\right)^{(l)}f^{(n+j-l)}(x)\\
&=\sum_{l=0}^{j}\binom{n+j}{l}\frac{j!}{(j-l)!}x^{j-l}f^{(n+j-l)}(x)\\
&=\sum_{m=0}^{j}\binom{n+j}{j-m}\frac{j!}{m!}x^{m}f^{(n+m)}(x).\end{aligned}$$ Hence, the right hand side of equals $$\begin{aligned}
\lefteqn{ (-1)^n\sum_{j=0}^k(\lambda-1)_{k-j}\binom{k}{j}\left(x^{j}f(x)\right)^{(j+n)}}\\
&= (-1)^n\sum_{j=0}^k(\lambda-1)_{k-j}\binom{k}{j}\sum_{m=0}^{j}\binom{n+j}{j-m}\frac{j!}{m!}x^{m}f^{(n+m)}(x)\\
&= (-1)^n\sum_{m=0}^k\left\{\sum_{j=m}^{k}(\lambda-1)_{k-j}\binom{k}{j}\binom{n+j}{j-m}\frac{j!}{m!}\right\}x^{m}f^{(n+m)}(x).\end{aligned}$$ The expression in the brackets can be written in another form. Indeed $$\sum_{j=m}^{k}(\lambda-1)_{k-j}\binom{k}{j}\binom{n+j}{j-m}\frac{j!}{m!}=\binom{k}{m}\frac{\Gamma(n+k+\lambda)}{\Gamma(n+m+\lambda)},$$ by a corollary to the Chu-Vandermonde identity (see [@aar p. 70]). This gives us $$\begin{aligned}
\lefteqn{(-1)^n\left(\sum_{j=0}^k(\lambda-1)_{k-j}\binom{k}{j}\left(x^{j}f(x)\right)^{(j)}\right)^{(n)}}\\
&=(-1)^n\sum_{m=0}^k\binom{k}{m}\frac{\Gamma(n+k+\lambda)}{\Gamma(n+m+\lambda)}x^{m}f^{(n+m)}(x)=T^{\lambda}_{n,k}(f)(x).\end{aligned}$$ For $n=0$ the identity reads $$\sum_{j=0}^k(\lambda-1)_{k-j}\binom{k}{j}\left(x^{j}f(x)\right)^{(j)}=T^{\lambda}_{0,k}(f)(x)={c_{k}^{\lambda}}(f)(x),$$ and the proposition is proved.$\square$
To prove Theorem \[thm:main\] we need a few preliminary results.
\[lemma:recursion\] For $f\in C^{\infty }((0,\infty))$ we have $${c_{k+1}^{\lambda}}(f)(x)=(\lambda+k){c_{k}^{\lambda}}(f)(x)+x{c_{k}^{\lambda}}(f)'(x).$$
[*Proof.*]{} This follows by computation: $$\begin{aligned}
{c_{k+1}^{\lambda}}(f)(x)&=x^{1-\lambda}(xx^{\lambda-1+k}f(x))^{(k+1)}\\
&=x^{2-\lambda}(x^{\lambda-1+k}f(x))^{(k+1)}+(k+1)x^{1-\lambda}(x^{\lambda-1+k}f(x))^{(k)}\\
&=x^{2-\lambda}(x^{\lambda-1}{c_{k}^{\lambda}}(f)(x))'+(k+1){c_{k}^{\lambda}}(f)(x)\\
&=x{c_{k}^{\lambda}}(f)'(x)+(\lambda+k){c_{k}^{\lambda}}(f)(x).
\end{aligned}$$
$\square$
\[prop:recursion\] Suppose that $f\in \mathcal C_N^{\lambda}$, and let for $k=0,\ldots,N$ $${c_{k}^{\lambda}}(f)(x)=\int_0^{\infty}e^{-xs}\, d\mu_k(s)+b_k,$$ where $\mu_k$ is a positive measure on $(0,\infty)$ and $b_k\geq 0$. Then, in the distributional sense, $$(-1)^ks^k\partial^k(s^{1-\lambda}\mu_0)=s^{1-\lambda}\mu_k.$$
[*Proof.*]{} From Lemma \[lemma:recursion\] it follows that (for $k\leq N-1$) $$\begin{aligned}
\int_0^{\infty}e^{-xs}\, d\mu_{k+1}(s)+b_{k+1}&=(\lambda+k)\int_0^{\infty}e^{-xs}\, d\mu_k(s)+(\lambda+k)b_k\\
&{}\qquad -x\int_0^{\infty}se^{-xs}\, d\mu_k(s).\end{aligned}$$ Letting $x\to \infty$ yields $b_{k+1}=(\lambda+k)b_k$ so that $$\begin{aligned}
\frac{1}{x}\int_0^{\infty}e^{-xs}\, d\mu_{k+1}(s)&=(\lambda+k)\frac{1}{x}\int_0^{\infty}e^{-xs}\, d\mu_k(s)\\
&{}\qquad -\int_0^{\infty}se^{-xs}\, d\mu_k(s).\end{aligned}$$ By the uniqueness of the Laplace transform we obtain $$s\mu_k=((\lambda+k)\mu_k-\mu_{k+1})\ast m.$$ (Here, $m$ denotes Lebesgue measure on $(0,\infty)$.) We get by differentiation (as distributions) that $$s\partial \mu_k=(\lambda+k-1)\mu_k-\mu_{k+1}.$$ We shall obtain the assertion in the proposition by induction, using this recursive relation: for $k=0$ the assertion is valid. Before verifying the induction step notice that $$s\partial (s^k\partial^k(s^{1-\lambda}\mu_0))=ks^k\partial^k(s^{1-\lambda}\mu_0)+s^{k+1}\partial^{k+1}(s^{1-\lambda}\mu_0).$$ Suppose now that the assertion holds for $k$. Then $$\begin{aligned}
s^{k+1}\partial^{k+1}(s^{1-\lambda}\mu_0)&=s\partial (s^k\partial^k(s^{1-\lambda}\mu_0))-ks^k\partial^k(s^{1-\lambda}\mu_0)\\
&=s\partial ((-1)^ks^{1-\lambda}\mu_k)-k(-1)^ks^{1-\lambda}\mu_k\\
&=(-1)^k\{s(1-\lambda)s^{-\lambda}\mu_k+s^{1-\lambda}s\partial \mu_k-ks^{1-\lambda}\mu_k\}\\
&=(-1)^ks^{1-\lambda}\{(1-\lambda)\mu_k+(\lambda+k-1)\mu_k-\mu_{k+1}-k\mu_k\}\\
&=(-1)^{k+1}s^{1-\lambda}\mu_{k+1}.\end{aligned}$$ The assertion holds also for $k+1$, and the proof follows.$\square$
[*Proof that (a) implies (b) in Theorem \[thm:main\].*]{} If $f\in \mathcal C_N^{\lambda}$ then the function ${c_{k}^{\lambda}}(f)$ is completely monotonic for $0\leq k\leq N$. In particular $$f(x)={c_{0}^{\lambda}}(f)(x)=\int_0^{\infty}e^{-xs}\,d\mu_0(s)+b_0=\int_0^{\infty}e^{-xs}s^{\lambda-1}\,d(s^{1-\lambda}\mu_0)(s)+b_0.$$ Let $\mu=s^{1-\lambda}\mu_0$ and notice that by Proposition \[prop:recursion\] $(-1)^ks^k\partial^k\mu\, (=s^{1-\lambda}\mu_k)$ is a positive measure with the property that $$\int_0^{\infty}e^{-xs}s^{\lambda-1}\, d((-1)^ks^k\partial^k\mu)(s)<\infty.$$ Thus (b) follows. $\square$
The next result is a special case of (b) implies (a) in Theorem \[thm:main\]. We state and prove it separately in order to describe the method, which will be alluded to in the following proof.
\[prop:specialcase\] Let $f$ have the representation $$f(x)=c+\int_0^{\infty}e^{-xs}s^{\lambda-1}\, d\mu(s),$$ where $c\geq 0$ and $\mu$ is a positive measure on $(0,\infty)$. If $-s\partial \mu(s)$ is a positive measure then $${c_{1}^{\lambda}}(f)(x)=\lambda c+\int_0^{\infty}e^{-xs}s^{\lambda-1}\, d(-s\partial \mu(s))$$ is completely monotonic.
[*Proof.*]{} Let $n\geq 1$ and take $\varphi_n\in C^{\infty}((0,\infty))$ such that $0\leq \varphi_n(t)\leq 1$, $$\varphi_n(t)=\left\{\begin{array}{ll}
0,&t<1/(2n)\\
1,&1/n\leq t\leq n\\
0,&t>2n
\end{array}\right.,$$ $|\varphi_n'(t)|\leq \text{Const}\, n$ for $t\in(1/(2n),1/n)$, and $|\varphi_n'(t)|\leq \text{Const}$ for $t\in(n,2n)$. By definition of the derivative in distributional sense we have $$\begin{aligned}
\lefteqn{\int_0^{\infty}e^{-xs}s^{\lambda-1}\varphi_n(s)\, d(-s\partial \mu(s))}\\
&=\langle -s\partial \mu(s),e^{-xs}s^{\lambda-1}\varphi_n(s)\rangle\\
&=\langle \mu,(e^{-xs}s^{\lambda}\varphi_n(s))'\rangle\\
&=-x\int_0^{\infty}e^{-xs}s^{\lambda}\varphi_n(s)\, d\mu(s)+\int_0^{\infty}e^{-xs}\lambda s^{\lambda-1}\varphi_n(s)\, d\mu(s)\\
&\quad +\int_0^{\infty}e^{-xs}s^{\lambda}\varphi_n'(s)\, d\mu(s).
\end{aligned}$$
Using dominated convergence it follows that the sum of first and second term on the right hand side tends to $$-x\int_0^{\infty}e^{-xs}s^{\lambda}\, d\mu(s) +\lambda \int_0^{\infty}e^{-xs}s^{\lambda-1}\, d\mu(s).$$ The third term tends to zero, again due to dominated convergence and the estimate (using $|s\varphi'(s)|\leq \text{Const}$ for $s\leq 1/n$) $$\begin{aligned}
\lefteqn{\int_0^{\infty}\left|e^{-xs}s^{\lambda}\varphi_n'(s)\right|\, d\mu(s)\leq}\\ & \text{Const}\left(\,\int_{\sfrac{1}{2n}}^{\sfrac{1}{n}}e^{-xs}s^{\lambda-1}\, d\mu(s)+\int_{n}^{2n}e^{-xs}s^{\lambda}\, d\mu(s)\right).\end{aligned}$$ Hence, letting $n$ tend to infinity, we obtain that $$\begin{aligned}
{c_{1}^{\lambda}}(f)(x)&=\lambda f(x)+xf'(x)\\
&=\lambda c+\lambda\int_0^{\infty}e^{-xs}s^{\lambda-1}\, d\mu(s)-x\int_0^{\infty}e^{-xs}s^{\lambda}\, d\mu(s)\\
&=\lambda c+\int_0^{\infty}e^{-xs}s^{\lambda-1}\, d(-s\partial \mu(s)).
\end{aligned}$$ Thus ${c_{1}^{\lambda}}(f)$ is completely monotonic, and $e^{-xs}s^{\lambda-1}$ is integrable w.r.t. the measure $-s\partial \mu(s)$.$\square$
[*Proof that (b) implies (a) in Theorem \[thm:main\].*]{} We suppose that $f$ has the representation $$f(x)=c+\int_0^{\infty}e^{-xs}s^{\lambda-1}\, d\mu(s),$$ with $c\geq 0$, and $\mu_k\equiv (-1)^ks^k\partial^k \mu$ being a positive measure for $k=0,\ldots,N$. It is easy to verify that $\mu_{k+1}=k\mu_k-s\partial \mu_k$ for $k=0,\ldots,N-1$. Proposition \[prop:specialcase\] yields that ${c_{1}^{\lambda}}(f)$ is completely monotonic and has the representation $${c_{1}^{\lambda}}(f)(x)=\lambda c+\int_0^{\infty}e^{-xs}s^{\lambda-1}\, d\mu_1(s).$$ Let us now assume that ${c_{k}^{\lambda}}(f)$ is completely monotonic and has the representation $${c_{k}^{\lambda}}(f)(x)=b+\int_0^{\infty}e^{-xs}s^{\lambda-1}\, d\mu_k(s).$$ Then $$\begin{aligned}
{c_{k+1}^{\lambda}}(f)(x)&=(\lambda+k){c_{k}^{\lambda}}(f)(x)+x{c_{k}^{\lambda}}(f)'(x)\\
&=(\lambda+k)\int_0^{\infty}e^{-xs}s^{\lambda-1}\, d\mu_k(s)-x\int_0^{\infty}e^{-xs}s^{\lambda}\, d\mu_k(s).\end{aligned}$$ Now, taking $\varphi_n$ as before it follows that $$\begin{aligned}
\lefteqn{\int_0^{\infty}e^{-xs}s^{\lambda-1}\varphi_n(s)\, d\mu_{k+1}(s)}\\
&=\int_0^{\infty}e^{-xs}s^{\lambda-1}\varphi_n(s)\, d(k\mu_k-s\partial \mu_k(s))\\
&=\langle k\mu_k-s\partial \mu_k(s),e^{-xs}s^{\lambda-1}\varphi_n(s)\rangle\\
&=\langle k\mu_k,e^{-xs}s^{\lambda-1}\varphi_n(s)\rangle+\langle \mu_k,(e^{-xs}s^{\lambda}\varphi_n(s))'\rangle\\
&=-x\int_0^{\infty}e^{-xs}s^{\lambda}\varphi_n(s)\, d\mu_k(s)+(k+\lambda)\int_0^{\infty}e^{-xs} s^{\lambda-1}\varphi_n(s)\, d\mu_k(s)\\
&\quad -\int_0^{\infty}e^{-xs}s^{\lambda}\varphi_n'(s)\, d\mu_k(s).
\end{aligned}$$ As before, letting $n$ tend to infinity, and applying dominated convergence we get that $${c_{k+1}^{\lambda}}(f)(x)=\int_0^{\infty}e^{-xs}s^{\lambda-1}\, d\mu_{k+1}(s)$$ is completely monotonic.$\square$
Additional results and comments
===============================
Suppose that ${c_{k}^{\lambda}}(f)$ is completely monotonic for some $k\geq 1$. What can be said about the functions ${c_{0}^{\lambda}}(f),\ldots,{c_{k-1}^{\lambda}}(f)$? Are they also completely monotonic? The answer is given in Proposition \[prop:question\].
\[prop:question\] Let $k\geq 1$, $f\in C^{\infty}((0,\infty))$ and suppose that the functions ${c_{0}^{\lambda}}(f),\ldots,{c_{k-1}^{\lambda}}(f)$ are non-negative. If ${c_{k}^{\lambda}}(f)$ is completely monotonic then ${c_{j}^{\lambda}}(f)$ is also completely monotonic for $j\leq k-1$.
The proof of this proposition requires some preliminary results. Define $${\gamma_{k}^{\lambda}}(f)(x)\equiv x^{-\lambda}(x^{\lambda-1+k}f(x))^{(k-1)},\quad k\geq 1.$$ Notice that ${\gamma_{k}^{\lambda}}(f)=c^{\lambda +1}_{k-1}(f)$.
\[lemma:first\] For $f\in C^{(k+1)}((0,\infty))$ we have $${\gamma_{k}^{\lambda}}(f)(x)={c_{k-1}^{\lambda}}(f)(x)+(k-1){\gamma_{k-1}^{\lambda}}(f)(x).$$
[*Proof:*]{} This follows by a direct computation.$\square$
The next lemma is an immediate consequence of Lemma \[lemma:first\].
If ${c_{j}^{\lambda}}(f)(x)\geq 0$ for all $j=0,\ldots,k$ then ${\gamma_{j}^{\lambda}}(f)(x)\geq 0$ for all $j=1,\ldots,k+1$
\[lemma:comp\] Let $k\geq 1$ be given and assume that ${c_{j}^{\lambda}}(f)(x)\geq 0$ for all $j=0,\ldots,k$. Then:
1. $(x^{\lambda-1+k}f(x))^{(j)}\geq 0$ for all $j=0,\ldots,k$;
2. $\lim_{x\to 0}(x^{\lambda-1+k}f(x))^{(j)}= 0$ for all $j=0,\ldots,k-2$;
3. $\lim_{x\to 0}(x^{\lambda-1+k}f(x))^{(k-1)}\in [0,\infty)$;
[*Proof.*]{} We use induction in $k$. For $k=1$ (i) is clearly satisfied, (ii) needs not be checked, and (iii) follows by noticing that $x^\lambda f(x)$ is non-negative and increasing. For $k=2$, $(x^{\lambda+1}f(x))''=x^{\lambda-1}{c_{2}^{\lambda}}(f)(x)\geq 0$, $(x^{\lambda+1}f(x))'=x^{\lambda}{c_{1}^{\lambda}}(f)(x)+\lambda x^\lambda {c_{0}^{\lambda}}(f)(x)\geq 0$, and thus (i) is satisfied. Property (ii) is clearly satisfied, and (iii) follows since $(x^{\lambda+1}f(x))'$ is non-negative and increasing.
Next we assume that $f$ satisfies ${c_{j}^{\lambda}}(f)\geq 0$ for all $j\leq k+1$, and aim at verifying (i), (ii), and (iii) with $k$ replaced by $k+1$. For $j=k+1$ we get $(x^{\lambda+k}f(x))^{(j)}=x^{\lambda-1}{c_{k+1}^{\lambda}}(f)(x)\geq 0$. For $1\leq j\leq k$ we use $$(x^{\lambda+k}f(x))^{(j)}=x(x^{\lambda-1+k}f(x))^{(j)}+j(x^{\lambda-1+k}f(x))^{(j-1)}\geq 0,$$ and (i) is verified. To see (ii), notice that $$(x^{\lambda+k}f(x))^{(k-1)}=x(x^{\lambda-1+k}f(x))^{(k-1)}+(k-1)(x^{\lambda-1+k}f(x))^{(k-2)},$$ The last term tends to zero by the induction hypothesis, and the first term equals $x$ times a non-negative and increasing function. Hence (ii) holds for $k+1$. Property (iii) for $k+1$ follows since $(x^{\lambda+k}f(x))^{(k)}$ is a positive and increasing function. This proves the lemma.$\square$
\[lemma:integrability\] Let $f\in C^{\infty}((0,\infty))$ and suppose that ${c_{0}^{\lambda}}(f),\ldots,{c_{k-1}^{\lambda}}(f)$ are non-negative functions. If ${c_{k}^{\lambda}}(f)$ is completely monotonic then ${\gamma_{k}^{\lambda}}(f)$ is also completely monotonic and $${\gamma_{k}^{\lambda}}(f)(x)=\frac{l_k}{x^{\lambda}}+\frac{b_k}{\lambda}+\int_0^{\infty}M_k(u)u^{\lambda-1}e^{-xu}\, du,$$ where $$M_k(u)=\int_u^{\infty} s^{-\lambda}d\mu_k(s).$$
[*Proof.*]{} By the complete monotonicity we may write $$(x^{\lambda-1+k}f(x))^{(k)}=x^{\lambda-1}{c_{k}^{\lambda}}(f)(x)=x^{\lambda-1}\int_0^{\infty}e^{-xs}\, d\mu_k(s)+b_kx^{\lambda-1},$$ where $b_k\geq 0$ and $\mu_k$ is a positive measure on $(0,\infty)$. The assumptions on non-negativity yield that the function $x^{\lambda}{\gamma_{k}^{\lambda}}(f)(x)=(x^{\lambda-1+k}f(x))^{(k-1)}$ is non-negative and increasing. Hence $$l_k\equiv \lim_{x\to 0}x^{\lambda}{\gamma_{k}^{\lambda}}(f)(x)\geq 0.$$ Furthermore, $$\begin{aligned}
x^{\lambda}{\gamma_{k}^{\lambda}}(f)(x)-l_k&=\int_0^x(t^{\lambda-1+k}f(t))^{(k)}\,dt\\
&=\int_0^xt^{\lambda-1}\left(\int_0^{\infty}e^{-ts}\, d\mu_k(s)+b_k\right)\,dt\\
&=\frac{b_k}{\lambda}x^{\lambda}+\int_0^{\infty}\int_0^xt^{\lambda-1}e^{-ts}\, dt\, d\mu_k(s)\\
&=\frac{b_k}{\lambda}x^{\lambda}+x^{\lambda}\int_0^{\infty}\int_0^su^{\lambda-1}e^{-xu}\, du\, s^{-\lambda}d\mu_k(s)\\
&=\frac{b_k}{\lambda}x^{\lambda}+x^{\lambda}\int_0^{\infty}\int_u^{\infty} s^{-\lambda}d\mu_k(s)u^{\lambda-1}e^{-xu}\, du,\end{aligned}$$ by Fubini’s theorem. Consequently, $$M_k(u)=\int_u^{\infty} s^{-\lambda}d\mu_k(s)$$ is finite and $M_k(u)u^{\lambda-1}$ is integrable at $0$.
The formulas above also show that $${\gamma_{k}^{\lambda}}(f)(x)=\frac{l_k}{x^{\lambda}}+\frac{b_k}{\lambda}+\int_0^{\infty}M_k(u)u^{\lambda-1}e^{-xu}\, du$$ is completely monotonic.$\square$
[*Proof of Proposition \[prop:question\]:*]{} From Lemma \[lemma:integrability\], $$(x^{\lambda-1+k}f(x))^{k-1}=l_k+\frac{b_k}{\lambda}x^{\lambda}+x^{\lambda}\int_0^{\infty}M_k(u)u^{\lambda-1}e^{-xu}\, du,$$ where $l_k,b_k\geq 0$ and $M_k(u)=\int_u^{\infty}s^{-\lambda}\, d\mu(s)$. Notice that $$\label{eq:limk}
u^{\lambda}M_k(u)\to 0, \quad u\to 0.$$ (To see this, rewrite as follows $$u^{\lambda}M_k(u)=\int_u^1\left(\frac{u}{s}\right)^{\lambda}\, d\mu(s)+u^{\lambda}\int_1^{\infty}\frac{d\mu(s)}{s^{\lambda}},$$ and use the dominated convergence theorem on the first term.)
Integrating this relation from $\epsilon$ to $x$, and letting $\epsilon$ tend to 0 we get, using (ii) of Lemma \[lemma:comp\], that $$\begin{aligned}
(x^{\lambda-1+k}f(x))^{(k-2)}&=l_kx+\frac{b_k}{\lambda(\lambda+1)}x^{\lambda+1}\\
&{}\qquad +x^{\lambda+1}\int_0^{\infty}M_{k-1}(u)u^{\lambda}e^{-xu}\, du,\end{aligned}$$ where $$M_{k-1}(u)=\int_u^{\infty}\frac{M_k(s)}{s^2}\, ds.$$ Continuing this process (using in each step (ii) of Lemma \[lemma:comp\]) we get $$\begin{aligned}
x^{\lambda-1+k}f(x)&=\frac{l_k}{(k-1)!}x^{k-1}+\frac{b_k}{(\lambda)_k}x^{\lambda+k-1}\nonumber\\
&{}\qquad +x^{\lambda+k-1}\int_0^{\infty}M_1(u)u^{\lambda+k-2}e^{-xu}\, du, \label{eq:above1}\end{aligned}$$ where $$M_{j}(u)=\int_u^{\infty}\frac{M_{j+1}(s)}{s^2}\, ds, \quad j=1,\ldots,k-1.$$ Division by $x^{\lambda-1+k}$ in shows that $f$ is completely monotonic, and has the representation $$\begin{aligned}
f(x)&=\frac{l_k}{(k-1)!}x^{-\lambda}+\frac{b_k}{(\lambda)_k}+\int_0^{\infty}M_1(u)u^{\lambda+k-2}e^{-xu}\, du\nonumber \\
&=\frac{b_k}{(\lambda)_k}+\int_0^{\infty}\left(M_1(u)u^{k-1}+\frac{l_k}{(k-1)!}\right)e^{-xu}u^{\lambda-1}\, du.\label{eq:conclusionquestion}\end{aligned}$$ In order to show that the functions ${c_{1}^{\lambda}}(f),\ldots,{c_{k-1}^{\lambda}}(f)$ are completely monotonic it suffices (Theorem \[thm:main\]) to verify that $(-1)^j\partial^j \left(M_1(u)u^{k-1}\right)\geq 0$ for $j=1,\ldots,k-1$. Now, $$M_1(u)u^{k-1}=u^{k-1}\int_u^{\infty}\frac{M_2(s)}{s^2}\, ds=\int_1^{\infty}\frac{M_2(ut)(ut)^{k-2}}{t^k}\, dt,$$ so it is enough to verify that $(-1)^j\partial^j\left(M_2(u)u^{k-2}\right)\geq 0$ in order to obtain that $(-1)^j\partial^j \left(M_1(u)u^{k-1}\right)\geq 0$. Repeating this argument we end up having to verify that $$\label{eq:star}
(-1)^{j}\partial^{j} \left(M_{k-j+1}(u)u^{j-1}\right)\geq 0,\quad 1\leq j\leq k-1.$$ These inequalities are verified using induction. For $j=1$ it reads $\partial M_k(u)\leq$ which is true since $M_k$ is a decreasing function. Next assuming that holds for some $j\leq k-2$ we aim at verifying it for $j+1$. We rewrite the expression $(-1)^{j+1}\partial^{j+1} \left(M_{k-j}(u)u^{j}\right)$ in two ways: $$\begin{aligned}
(-1)^{j+1}\partial^{j+1} \left(M_{k-j}(u)u^{j}\right)&=(-1)^{j+1}u\partial^{j+1} \left(M_{k-j}(u)u^{j}\right)\\
&{}\quad +(-1)^{j+1}(j+1)\partial^{j} \left(M_{k-j}(u)u^{j}\right);\\
(-1)^{j+1}\partial^{j+1} \left(M_{k-j}(u)u^{j}\right)&=(-1)^{j+1}\partial^{j} \left(-\frac{M_{k-j+1}(u)}{u^2}u^{j+1}\right)\\
&{}\quad +(-1)^{j+1}(j+1)\partial^{j} \left(M_{k-j}(u)u^{j}\right).\end{aligned}$$ Comparing these two identities we infer that $$(-1)^{j+1}u\partial^{j+1} \left(M_{k-j}(u)u^{j}\right)=(-1)^{j}\partial^{j} \left(M_{k-j+1}(u)u^{j-1}\right),$$ and thus holds for $j+1$.$\square$
Introducing the functions $N_j(u)\equiv M_j(1/u)$ for $j=1,\ldots,k$ it follows that $$N_k(u)=\int_{1/u}^{\infty}s^{-\lambda}\, d\mu(s)=\int_{0}^{u}t^{\lambda}\, d\widehat{\mu}(t),$$ where $\widehat{\mu}$ denotes the image measure $\phi(\mu)$, with $\phi(x)=1/x$. For $j\leq k-1$ the relation between $N_j$ and $N_{j+1}$ is $$N_j(u)=\int_{1/u}^{\infty}\frac{M_{j+1}(s)}{s^2}\, ds=\int_{0}^{u}N_{j+1}(t)\, dt.$$ Consequently we see that the derivatives $N_1^{(j)}(u)$ for $j\leq k-1$ are all non-negative, and take the value $0$ at $u=0$. In terms of these functions the representation can be rewritten as $$\begin{aligned}
f(x)&=\frac{b_k}{(\lambda)_k}+\frac{l_k\Gamma(\lambda)}{(k-1)!x^{\lambda}}+\int_0^{\infty}M_1(u)u^{k-1}e^{-xu}u^{\lambda-1}\, du\\
&=\frac{b_k}{(\lambda)_k}+\frac{l_k\Gamma(\lambda)}{(k-1)!x^{\lambda}}+\int_0^{\infty}N_1(s)s^{-k-\lambda}e^{-x/s}\, ds.
\end{aligned}$$
The next proposition shows that for any given $N$ the classes $\mathcal C_N^{\lambda}$ become larger as $\lambda$ increases. As remarked in [@Sok] it is not clear how to verify this even for $N=\infty$ only considering the operators $T_{n,k}^{\lambda}$.
If $\lambda_1<\lambda_2$ then $\mathcal C_{N}^{\lambda_1}\subset \mathcal C_{N}^{\lambda_2}$ for all $N$.
[*Proof.*]{} This follows from Leibniz’ formula. Assume $f\in \mathcal C_N^{\lambda_1}$ and let $\lambda_2>\lambda_1$. Then $$f(x)=c+\int_0^{\infty}e^{-xs}s^{\lambda_1 -1}\, d\mu(s)=c+\int_0^{\infty}e^{-xs}s^{\lambda_2 -1}\, s^{\lambda_1-\lambda_2}d\mu(s),$$ where $(-1)^js^{j}\partial^j\mu\geq 0$ for all $j\leq N$. Hence, for $k\leq N$, $$\begin{aligned}
&(-1)^ks^k\partial^k(s^{\lambda_1-\lambda_2}\mu)\\
&=(-1)^ks^k\sum_{j=0}^k\binom{k}{j}(-1)^{k-j}\frac{\Gamma(k-j+\lambda_2-\lambda_1)}{\Gamma(\lambda_2-\lambda_1)}s^{\lambda_1-\lambda_2+j-k}\partial^j\mu\\
&=s^{\lambda_1-\lambda_2}\sum_{j=0}^k\binom{k}{j}\frac{\Gamma(k-j+\lambda_2-\lambda_1)}{\Gamma(\lambda_2-\lambda_1)}(-1)^js^{j}\partial^j\mu\geq 0.\end{aligned}$$ $\square$
[99]{} G.E. Andrews, R. Askey, R. Roy, Special Functions, Cambridge Univ. Press, Cambridge (1999). C. Berg and G. Forst, Potential theory on locally compact abelian groups, Ergebnisse der Mathematik und ihrer Grenzgebiete 87, Springer, Berlin Heidelberg New York (1975). D.B. Karp and E.G. Prilepkina, Applications of the Stieltjes and Laplace transform representations of the hypergeometric functions, arxiv 1705.02493. S. Koumandos and H.L. Pedersen, Completely monotonic functions of positive order and asymptotic expansions of the logarithm of Barnes double gamma function and Euler’s gamma function, [*J. Math. Anal. Appl.*]{}, [**355**]{} (2009), 33–40.
S. Koumandos and H.L. Pedersen, On Asymptotic Expansions of Generalized Stieltjes Functions, *Comput. Methods Funct. Theory* **15** (2015), 93–115 (DOI: 10.1007/s40315-014-0094-7).
S. Koumandos and H.L. Pedersen, On the Laplace transform of absolutely monotonic functions, *Results Math.* (2016) (DOI:10.1007/s00025-016-0638-4).
H.L. Pedersen, Pre Stieltjes Functions, *Mediterr. J. Math.* **8** (2011), 113–122 (DOI:10.1007/s00009-010-0043-2).
W. Rudin, Functional Analysis (2nd ed.), McGraw-Hill, New York (1991).
R.L. Schilling, R. Song and Z. Vondraček, Bernstein functions. De Gruyter Studies in Mathematics **37**, De Gruyter, Berlin (2010).
A.D. Sokal, Real-variables characterization of generalized Stieltjes functions, *Expo. Math.* **28** (2010), 179–185.
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D.V. Widder, The Laplace transform. Princeton University Press, Princeton (1946).
Stamatis Koumandos\
Department of Mathematics and Statistics\
The University of Cyprus\
P. O. Box 20537\
1678 Nicosia, Cyprus\
[*email*]{}:
Henrik Laurberg Pedersen\
Department of Mathematical Sciences\
University of Copenhagen\
Universitetsparken 5\
DK-2100, Denmark\
[*email*]{}:
[^1]: Research supported by grant DFF–4181-00502 from The Danish Council for Independent Research $|$ Natural Sciences
|
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abstract: 'Exact sum rules for the longitudinal and transverse part of the vector channel spectral functions at nonzero momentum are derived in the first part of the paper. The sum rules are formulated for the finite temperature spectral functions, from which the vacuum component has been subtracted, and represent a generalization of previous work in which sum rules were derived only for the zero-momentum limit. In the second part of the paper, we demonstrate how the sum rules can be used as constraints in spectral fits to lattice data at various temperatures, with the latest dynamical lattice QCD data at zero momentum.'
author:
- Philipp Gubler
- Daisuke Satow
title: Finite temperature sum rules in the vector channel at finite momentum
---
Introduction {#sec:intro}
============
Hadrons, whose dynamics is governed by quantum chromodynamics (QCD), are deconfined at high temperature ($T$), where quarks and gluons are expected to be the fundamental degrees of freedom. Such matter is called quark-gluon plasma, and is experimentally investigated by heavy ion collision experiments. In analyzing the experimental data, electromagnetic (EM) probes such as dilepton spectra are particularly useful [@Adare:2014fwh] because once generated in the medium, they are expected to reach the detector without further QCD interaction with other particles. The electric conductivity is also an important quantity since it may increase the lifetime of magnetic fields generated in the early stage of the heavy ion collision [@McLerran:2013hla; @Tuchin:2013apa; @Tuchin:2015oka; @Hattori:2016emy]. Furthermore, how the spectrum of the vector meson is modified at finite $T$ has been discussed for a long time from the point of view of the chiral symmetry restoration [@Bernard:1988db]. The spectral function of the EM current at finite $T$, which is the focus of this paper, contains information on all the above three quantities.
There are many approaches for evaluating the EM spectral function at finite $T$, such as perturbative QCD [@Baier:1988xv], holographic QCD [@CaronHuot:2006te], model calculations [@Gale:2014dfa], sum rules [@Gubler:2015yna; @Huang:1994fs; @Kapusta:1993hq; @Zschocke:2002mn; @Kim:2017nyg], low-energy effective theory [@Chanfray; @Klingl], and lattice QCD [@Ding:2010ga; @Meyer:2011gj; @Brandt:2015aqk; @Amato:2013naa; @Aarts:2007wj]. Nevertheless, none of these approaches are perfect. Especially in the lattice QCD approach, which allows fully nonperturbative first principle QCD calculations, the spectral function cannot be analyzed directly since it is a quantity defined in real, not imaginary time. Therefore, one needs to make some assumption about the functional form of the spectral function to analyze it, or otherwise has to rely on some method to analytically continue imaginary time data to real time, such as the maximum entropy method (MEM) [@Jarrell:1996rrw; @Asakawa:2000tr; @Gubler:2010cf], the Backus-Gilbert method [@Backus:1968; @Backus:1970; @Brandt:2015sxa] or the Schlessinger point method [@Schlessinger:1967; @Tripolt:2016cya] (see Ref.[@Tripolt:2017] for a comparison of these three methods).
In our previous work of Ref.[@Gubler:2016hnf], we derived three sum rules at zero momentum, which constrain the spectral function, and used them to improve the ansatz employed in previous lattice QCD analysis. One aim of this paper is to derive similar sum rules for the small but finite spatial momentum case. At finite momentum, the EM spectral function no longer has a single independent component, but two, corresponding to the transverse and the longitudinal channels. In the longitudinal channel, a novel and robust structure, a sharp peak corresponding to the diffusion mode of the EM charge, appears.
As shown in Ref.[@Gubler:2016hnf], the sum rules can be used to improve the analysis of lattice QCD data by constraining the shape of the spectral functions and derive transport coefficients that do not appear directly in the spectral functions. In Ref.[@Gubler:2016hnf], this was demonstrated by using two sum rules (1 and 3 in this and the previous work), but the other sum rule (2) was not used. The other aim of this paper is to update the analysis such that all three sum rules can be used, and to employ the latest lattice QCD data including dynamical quarks as input for the spectral function fit.
The paper is organized as follows: In the next section, we introduce the quantities in quantum field theory that are necessary in our analysis, and explain how to derive the sum rules from the operator product expansion (OPE) and hydrodynamics [@Romatschke:2009ng]. Section \[sec:sumrule\] is devoted to the derivation of the sum rules in transverse and longitudinal channels, at small but finite spatial momentum. We also confirm that the spectral function evaluated at weak coupling and in the chiral limit satisfies these sum rules, and check to which energy region the sum rules are sensitive. We demonstrate that the sum rules can be used to improve the lattice QCD analysis for the zero-momentum case in Sec. \[sec:lattice\]. We summarize the paper and give concluding remarks in Sec. \[sec:summary\]. In the three appendixes, we evaluate the contributions to the spectral functions from the transport peak, the continuum, and the UV tail, at weak coupling and in the chiral limit.
In this paper, we recapitulate known results such as the sum rules in the transverse channel at zero momentum, Eqs.(\[eq:sumrule-1-p0\]) and (\[eq:sumrule-3-T\]). Their derivation can be found in our previous work [@Gubler:2016hnf], but we rederive them to make our paper self-contained. The recapitulation of the evaluation of the transport peak, the continuum, and the UV tails in the three appendixes is provided for the same reason.
Preliminaries
=============
In this section, we explain the method for deriving the sum rules developed in Ref. [@Romatschke:2009ng]. Only the asymptotic behaviors of the EM current correlator in the UV and IR energy regions are necessary for this purpose. We also discuss these behaviors and give their analytic form obtained from OPE and hydrodynamics in this section.
Formalism
---------
We begin by introducing quantities that will be used in the derivation of the sum rules. The retarded Green function of the EM current ($j^\mu\equiv e\sum_f q_f \overline{\psi}_f \gamma^\mu \psi_f $) is defined as $G^R_{\mu\nu}(\omega,{{\mathbf{p}}})\equiv i \int dt \int d^3{{\mathbf{x}}}e^{i\omega t-i{{\mathbf{p}}}\cdot {{\mathbf{x}}}}
\theta(t) \langle [j_\mu(t,{{\mathbf{x}}}),j_\nu(0,{{\mathbf{0}}})] \rangle$, where the average is taken over the thermal ensemble, $e$ is the coupling constant of quantum electrodynamics, $q_f$ is the charge of each quark flavor (in units of $e$), and $\psi_f$ the quark field with flavor $f$, respectively.
At finite temperature, the medium effect breaks Lorentz symmetry so that the tensor structure of the Green function has two independent components, $$\begin{aligned}
\label{eq:tensor-decomposition}
G^R_{\mu\nu}(p)&=G^T(p)P^T_{\mu\nu}(p)
+G^L(p)P^L_{\mu\nu}(p),\end{aligned}$$ where $p^\mu\equiv(\omega,{{\mathbf{p}}})$ is a shorthand notation for the energy and the spatial momentum, and $P^T_{\mu\nu}(p)\equiv g^{i}_\mu g^{j}_\nu (\delta^{ij}-{p^i p^j}/{{{\mathbf{p}}}^2})$ and $P^L_{\mu\nu}(p)\equiv P^0_{\mu\nu}(p) -P^T_{\mu\nu}(p)$ are the projection operators to the transverse and longitudinal parts with $P^0_{\mu\nu}(p)\equiv -(g_{\mu\nu}-{p_\mu p_\nu}/{p^2})$. The first (second) term in Eq. (\[eq:tensor-decomposition\]) corresponds to the transverse (longitudinal) component in three dimensions. When ${{\mathbf{p}}}$ is along the $z$-direction, they are related to the components of the Green function as $G_{T}=G^R_{11}=G^R_{22}$ and $G_L=p^2G^R_{00}/{{\mathbf{p}}}^2$.
Here we recapitulate the method of deriving sum rules at finite temperature that was developed in Ref.[@Romatschke:2009ng]. Using the residual theorem for the contour[^1] drawn in Fig. \[fig:contour\], we get $$\begin{aligned}
\label{eq:residue-theorem}
\begin{split}
&\delta G^{R}_{\mu\nu}(i\omega,{{\mathbf{p}}})-\delta G^{R}_{\mu\nu}(\infty,{{\mathbf{p}}}) \\
&~~~= \frac{1}{2\pi i}\oint_C d\omega'
\frac{\delta G^{R}_{\mu\nu}(\omega',{{\mathbf{p}}})-\delta G^{R}_{\mu\nu}(\infty,{{\mathbf{p}}})}{\omega'-i\omega},
\end{split}\end{aligned}$$ where $\delta$ stands for the subtraction of the $T=0$ part: $\delta G^R_{\mu\nu}\equiv G^R_{\mu\nu}-G^R_{\mu\nu}|_{T=0}$. Because of this subtraction of the zero temperature part and another subtraction of $\delta G^{R}_{\mu\nu}(\infty,{{\mathbf{p}}})$ done in the expression above, all the UV divergences are regularized in all the cases we consider. Thus, the integral on the contour $C$ can be safely replaced with the integral on the real axis. Moreover, in deriving Eq. (\[eq:residue-theorem\]) we have used the property that the retarded Green function is analytic in the upper $\omega'$ plane.
Now, by taking the $\omega\rightarrow 0$ limit, Eq. (\[eq:residue-theorem\]) reduces to $$\begin{aligned}
\begin{split}
&\delta G^{R}_{\mu\nu}(0,{{\mathbf{p}}})-\delta G^{R}_{\mu\nu}(\infty,{{\mathbf{p}}}) \\
&~~~= {\text P}\frac{1}{\pi i}\int^\infty_{-\infty} d\omega'
\frac{\delta G^{R}_{\mu\nu}(\omega',{{\mathbf{p}}})-\delta G^{R}_{\mu\nu}(\infty,{{\mathbf{p}}})}{\omega'},
\end{split}\end{aligned}$$ where we have used $1/(\omega'-i\omega)\rightarrow {\text P}(1/\omega')+i\pi\delta(\omega')$. We consider only the case of $\mu=\nu$ in this paper. Then, the real (imaginary) part of $\delta G^{R}_{\mu\nu}(\omega,{{\mathbf{p}}})$ is even (odd) in terms of $\omega$. This property enables us to simplify the equation above as $$\begin{aligned}
\label{eq:derivation}
\begin{split}
\delta G^{R}_{\mu\nu}(0,{{\mathbf{p}}})-\delta G^{R}_{\mu\nu}(\infty,{{\mathbf{p}}})
= \frac{2}{\pi}\int^\infty_{0} d\omega
\frac{\delta \rho_{\mu\nu}(\omega,{{\mathbf{p}}})}{\omega},
\end{split}\end{aligned}$$ where we have introduced the spectral function[^2] of the EM current, $\rho_{\mu\nu}(p)\equiv{\text{Im}}G^{R}_{\mu\nu}(p)$, and changed the label of the integration variable as $\omega'\rightarrow \omega$ for simplicity. We hence see that the asymptotic behaviors of the retarded Green function in the UV and IR regions determine the integral of the spectral function.
Let us here briefly discuss the differences between the sum rules derived here and the so-called finite energy sum rules (FESR), which are widely used in the literature (see for instance Ref.[@Ayala:2013vra]). In contrast to the procedure of this paper, one in the FESR does not subtract the zero temperature contribution as we have done in Eq.(\[eq:residue-theorem\]) and below. Instead, to avoid an ultraviolet divergence, one does not take the radius of the contour in Fig.\[fig:contour\] to infinity, but sets it to some threshold value, which however should be large enough such that the OPE is still approximately valid. Doing this, one can derive sum rules that are not exact, but practically useful, as they can constrain the spectral function below the threshold value that usually contains the most interesting physical content.
![The contour $C$, used in the integral of Eq. (\[eq:residue-theorem\]).[]{data-label="fig:contour"}](contour.eps){width="25.00000%"}
UV behavior
-----------
The behavior in the UV region can be described with the help of the OPE. At leading order in the coupling constant, the Wilson coefficients of the operators with dimension 4 read [@CaronHuot:2009ns; @Shifman:1978by] $$\begin{aligned}
\label{eq:OPE}
\begin{split}
& \delta G^{R}_{\mu\nu}(\omega,{{\mathbf{p}}}) \\
&=e^2\sum q^2_f \frac{1}{p^2} \\
&~~~\times \Bigl[\left\{2m_f \delta\left\langle\overline{\psi}_f \psi_f \right\rangle
+\frac{1}{12}\delta\left\langle \frac{\alpha_s}{\pi}G^2\right\rangle \right\}P^0_{\mu\nu}(p) \\
&~~~-2\delta\left\langle T^{\alpha\beta}_f \right\rangle A_{\mu\nu \alpha\beta}(p)
\Bigr]
+{\cal O}(\omega^{-4}),
\end{split}\end{aligned}$$ where $A_{\mu\nu \alpha\beta}(p)\equiv g_{\mu\alpha}g_{\nu\beta}+g_{\mu\beta}g_{\nu\alpha}-2(g_{\mu\alpha}p_{\nu}p_{\beta}+
g_{\nu\alpha}p_{\mu}p_{\beta}-g_{\mu\nu}p_\alpha p_\beta)/p^2$, $G^{\mu\nu}_a\equiv \partial^\mu A^\nu_a-\partial^\nu A^\mu_a -gf_{abc} A^\mu_b A^\nu_c$ is the field strength, $G^2\equiv G^a_{\mu\nu}G^{a\mu\nu}$, $T^{\alpha\beta}_{f}\equiv i{\cal {ST}} \overline{\psi}_f\gamma^\alpha D^\beta \psi_f $ is the quark component to the traceless part of the energy-momentum tensor, $D^\mu\equiv \partial^\mu +igA^\mu_a t^a$ is the covariant derivative, $A^\mu_a$ is the gluon field, $t^a$ is the generator of the $SU({N_c})$ group in the fundamental representation, $f_{abc}$ is the structure constant of the $SU({N_c})$ group, $m_f$ is the current quark mass, $g$ is the QCD coupling constant, $\alpha_s\equiv g^2/(4\pi)$, and ${N_c}$ is the number of the colors. ${\cal {ST}}$ makes a tensor symmetric and traceless, ${\cal {ST}}O^{\alpha\beta}\equiv (O^{\alpha\beta}+O^{\beta\alpha})/2-g^{\alpha\beta}O^\mu_\mu /4$.
We decompose Eq.(\[eq:OPE\]) into transverse and longitudinal components as $$\begin{aligned}
\nonumber
\delta G_{T}(\omega,{{\mathbf{p}}})
&=e^2\sum q^2_f \frac{1}{p^2} \\
\nonumber
&~~~\times \Bigl[\left\{2m_f \delta\left\langle\overline{\psi}_f \psi_f \right\rangle
+\frac{1}{12}\delta\left\langle \frac{\alpha_s}{\pi}G^2\right\rangle \right\}\\
\label{eq:OPE-wo-RG-T}
&~~~+\frac{8}{3}\frac{\omega^2+{{\mathbf{p}}}^2}{p^2}\delta\left\langle T^{00}_f \right\rangle
\Bigr]
+{\cal O}(\omega^{-4}),\\
\nonumber
\delta G^{R}_{00}(\omega,{{\mathbf{p}}})
&=e^2\sum q^2_f \frac{1}{p^2}\frac{{{\mathbf{p}}}^2}{p^2} \\
\nonumber
&~~~\times \Bigl[\left\{2m_f \delta\left\langle\overline{\psi}_f \psi_f \right\rangle
+\frac{1}{12}\delta\left\langle \frac{\alpha_s}{\pi}G^2\right\rangle \right\} \\
\label{eq:OPE-wo-RG-L}
&~~~+\frac{8}{3} \delta\left\langle T^{00}_f \right\rangle
\Bigr]
+{\cal O}(\omega^{-6}),\end{aligned}$$ where we have used the isotropy of the system and the traceless property of $T^{\alpha\beta}_f$.
Because we consider the $\omega\rightarrow \infty$ limit, we need to take into account the rescaling/mixing effect of the operators. Due to their vanishing anomalous dimensions, the chiral and gluon condensate terms remain unchanged, but the quark component of the energy-momentum tensor changes. To describe its behavior, we rewrite this operator as $$\begin{aligned}
\label{eq:Tf-decompose}
T^{00}_{f}=T'{}^{00}_{f}+\frac{1}{4{C_F}+{N_f}}\left(T^{00}+\frac{2}{{N_f}}\tilde{T}^{00}\right),\end{aligned}$$ where $$\begin{aligned}
T'{}^{00}_{f}\equiv T^{00}_{f}-\frac{1}{{N_f}}\sum_{f'} T^{00}_{f'},\\
T^{00}\equiv \sum_{f'} T^{00}_{f'}+T^{00}_g,\\
\tilde{T}^{00}\equiv 2C_F \sum_{f'} T^{00}_{f'}-\frac{{N_f}}{2} T^{00}_g .\end{aligned}$$ Here, $T^{\mu\nu}_g\equiv -G^{\mu\alpha}_{a}G^\nu{}_{\alpha a} +g^{\mu\nu}G^2/4$ is the gluon component of the traceless part of the energy-momentum tensor, $N_f$ is the flavor number, and $C_F\equiv ({N_c}^2-1)/(2{N_c})$. We note that $T^{\mu\nu}$ is the traceless part of the full energy-momentum tensor, not the energy-momentum tensor itself. A standard renormalization group (RG) analysis yields the following scaling properties [@Peskin:1995ev]: $$\begin{aligned}
\label{eq:scaling-T'-Ttilde}
\begin{split}
T'{}^{00}_{f}(\kappa)&= \left[\frac{\ln\left(\kappa^2_0/\Lambda^2_{\text{QCD}} \right)}{\ln\left(\kappa^2/\Lambda^2_{\text{QCD}} \right)}\right]^{a'}
T'{}^{00}_{f}(\kappa_0),\\
\tilde{T}^{00}(\kappa)
&= \left[\frac{\ln\left(\kappa^2_0/\Lambda^2_{\text{QCD}} \right)}{\ln\left(\kappa^2/\Lambda^2_{\text{QCD}} \right)}\right]^{\tilde{a}}
\tilde{T}^{00}(\kappa_0),
\end{split} \end{aligned}$$ while $T^{00}$ is independent of $\kappa$. Here $\kappa$ and $\kappa_0$ are renormalization scales, $\Lambda_{\text{QCD}}$ is the QCD scale parameter, $a'\equiv 8{C_F}/(3b_0)$, and $\tilde{a}\equiv 2(4{C_F}+{N_f})/(3b_0)$, where $b_0\equiv (11{N_c}-2{N_f})/3$, which appears in the expression $$\begin{aligned}
\label{eq:RG-alphas}
\alpha_s(\kappa)=\frac{4\pi}{b_0 \ln(\kappa^2/\Lambda^2_{\text{QCD}})}.\end{aligned}$$ In the $\omega\rightarrow \infty$ limit, it is natural to choose the RG scale[^3] as $\kappa^2=\omega^2$.
We see that, except for the $T^{00}$ term, all terms in Eq. (\[eq:Tf-decompose\]) are suppressed logarithmically at large $\omega$. Thus, Eqs. (\[eq:OPE-wo-RG-T\]) and (\[eq:OPE-wo-RG-L\]) become $$\begin{aligned}
\nonumber
\delta G_{T}(\omega,{{\mathbf{p}}})
&=e^2\sum q^2_f \frac{1}{p^2} \\
\nonumber
&~~~\times \Bigl[\left\{2m_f \delta\left\langle\overline{\psi}_f \psi_f \right\rangle
+\frac{1}{12}\delta\left\langle \frac{\alpha_s}{\pi}G^2\right\rangle \right\}\\
\label{eq:OPE-w-RG-T}
&~~~+\frac{8}{3}\frac{1}{4{C_F}+{N_f}}\frac{\omega^2+{{\mathbf{p}}}^2}{p^2}\delta\left\langle T^{00} \right\rangle
\Bigr] \nonumber \\
&~~~+{\cal O}(\omega^{-4}),\\
\nonumber
\delta G^{R}_{00}(\omega,{{\mathbf{p}}})
&=e^2\sum q^2_f \frac{1}{p^2}\frac{{{\mathbf{p}}}^2}{p^2} \\
\nonumber
&~~~\times \Bigl[\left\{2m_f \delta\left\langle\overline{\psi}_f \psi_f \right\rangle
+\frac{1}{12}\delta\left\langle \frac{\alpha_s}{\pi}G^2\right\rangle \right\} \\
\label{eq:OPE-w-RG-L}
&~~~+\frac{8}{3}\frac{1}{4{C_F}+{N_f}} \delta\left\langle T^{00} \right\rangle
\Bigr]
+{\cal O}(\omega^{-6}). \end{aligned}$$ We note that, in the $\omega\rightarrow \infty$ limit, which is relevant to the derivation of the sum rule, the asymptotic freedom of QCD guarantees that the above expression is exact. In other words, all higher order $\alpha_s$ corrections vanish in this limit.
IR behavior {#ssc:IR}
-----------
On the other hand, the asymptotic behavior in the IR region is described by hydrodynamics [@Kovtun:2012rj], as long as the spatial momentum is small enough. In the channel of the EM current, the basic equations consist of the conservation law and the constituent equation, $$\begin{aligned}
\label{eq:conservation-law}
\partial_0j^0&= -\nabla\cdot {{\mathbf{j}}},\\
\nonumber
{{\mathbf{j}}}&= -D\nabla j^0+\sigma{{\mathbf{E}}}-\sigma\tau_J\partial_0 {{\mathbf{E}}}+\kappa_B\nabla\times{{\mathbf{B}}}\\
\label{eq:constituent-eq}
&~~~+{\cal O}(\partial^2 {{\mathbf{E}}}, \partial^2 {{\mathbf{B}}}, \partial^2 j^0 ),\end{aligned}$$ where $D$ is the diffusion constant, $\sigma$ the electrical conductivity and $\tau_J$ and $\kappa_B$ second order transport coefficients corresponding to $\partial_0{{\mathbf{E}}}$ and $\nabla\times{{\mathbf{B}}}$, respectively. ${{\mathbf{E}}}\equiv -\nabla A^0-\partial_0 {{\mathbf{A}}}$ and ${{\mathbf{B}}}\equiv\nabla\times{{\mathbf{A}}}$ are the electric and magnetic fields, where $A^\mu$ is the vector potential.
After performing the Fourier transformation that is defined as $f(p)\equiv \int d^4x e^{ip\cdot x} f(x)$, Eqs. (\[eq:conservation-law\]) and (\[eq:constituent-eq\]) become $$\begin{aligned}
\label{eq:conservation-law-p}
\omega j^0&= {{\mathbf{p}}}\cdot {{\mathbf{j}}},\\
\nonumber
{{\mathbf{j}}}&= -Di{{\mathbf{p}}}j^0+i\sigma (-{{\mathbf{p}}}A^0+\omega {{\mathbf{A}}})\\
\nonumber
&~~~-\sigma\tau_J \omega (-{{\mathbf{p}}}A^0+\omega {{\mathbf{A}}})-\kappa_B ([{{\mathbf{p}}}\cdot{{\mathbf{A}}}]{{\mathbf{p}}}-{{\mathbf{p}}}^2{{\mathbf{A}}}) \\
\label{eq:constituent-eq-p}
&~~~+{\cal O}(p^2A, p^2j^0).\end{aligned}$$ Let us solve these equations for the transverse and the longitudinal components of the current. By introducing the transverse component, $j^i_T(p)\equiv P^{ij}_T(p) j^j(p)$, we get the solutions as follows: $$\begin{aligned}
{{\mathbf{j}}}_T(p)&= \left(i\sigma\omega-\sigma\tau_J\omega^2+\kappa_B{{\mathbf{p}}}^2
\right) {{\mathbf{A}}}_T \nonumber \\
&~~~ +{\cal O}(\omega^3{{\mathbf{A}}}_T, \omega{{\mathbf{p}}}^2{{\mathbf{A}}}_T, {{\mathbf{p}}}^4{{\mathbf{A}}}_T),\\
\nonumber
\omega j^0(p)&= -iD{{\mathbf{p}}}^2 j^0
-i{{\mathbf{p}}}^2\sigma A^0 \\
&~~~+ {\cal O}(\omega{{\mathbf{p}}}^2 A^0, {{\mathbf{p}}}^4 A^0, \omega{{\mathbf{p}}}^2 j^0, {{\mathbf{p}}}^4 j^0) \nonumber \\
&~~~+({\text {terms that are proportional to }}{{\mathbf{p}}}\cdot{{\mathbf{A}}}).\end{aligned}$$
By using the linear response theory, the induced current is written as $$\begin{aligned}
\label{eq:linear-response}
j_\mu(p)=-G^R_{\mu\nu}(p)A^\nu(p),\end{aligned}$$ from which we obtain $$\begin{aligned}
\label{eq:hydro-GR-T}
G_{T}(p) &= i\sigma\omega-\sigma\tau_J\omega^2+\kappa_B{{\mathbf{p}}}^2
+{\cal O}(\omega^3, \omega{{\mathbf{p}}}^2, {{\mathbf{p}}}^4),\\
\label{eq:hydro-GR-L}
G^R_{00}(p) &= i\sigma{{\mathbf{p}}}^2\frac{1+{\cal O}(\omega,{{\mathbf{p}}}^2)}{\omega+iD{{\mathbf{p}}}^2+{\cal O}(\omega{{\mathbf{p}}}^2,{{\mathbf{p}}}^4)}.\end{aligned}$$ We note that there is a pole at $\omega=-iD{{\mathbf{p}}}^2$ in the longitudinal channel, which we call the diffusion pole. This is a novel structure that appears only at finite ${{\mathbf{p}}}$. This pole appears as a peak in the spectral function, $$\begin{aligned}
\label{eq:hydro-rho-L}
\rho_{00}(p) &= \sigma{{\mathbf{p}}}^2
\frac{\omega}{\omega^2+(D{{\mathbf{p}}}^2)^2} ,\end{aligned}$$ while at $|{{\mathbf{p}}}|=0$, it reduces to the delta function, $$\begin{aligned}
\label{eq:hydro-rho-L-p=0}
\rho_{00}(\omega,{{\mathbf{0}}})
&= \pi\frac{\sigma}{D}\omega\delta(\omega) .\end{aligned}$$
For constructing our sum rules, we, in principle, also need to evaluate the zero temperature part, which should be subtracted later. At $T=0$, Lorentz invariance guarantees the tensor structure of the correlator, $$\begin{aligned}
G^R_{\mu\nu}(p)
&= p^2P^0_{\mu\nu}(p) \tilde{G}^R(p^2),\end{aligned}$$ where $\tilde{G}^R(0)=0$ due to the renormalization condition of the electric charge [@Peskin:1995ev]. The transverse component is given as $G_T(p)= p^2\tilde{G}^R(p^2)$. $\tilde{G}^R(p^2)$ is regular at $p^2=0$ due to the renormalization condition, so it is easy to see that there are no contributions to $\sigma$, $\tau_J$, and $\kappa_B$ at $T=0$, from Eq. (\[eq:hydro-GR-T\]). The $T=0$ contribution only enters in the higher order terms, which are neglected in Eq. (\[eq:hydro-GR-T\]).
On the other hand, the longitudinal component reads $G^R_{00}(p)={{\mathbf{p}}}^2 \tilde{G}^R(p^2)$, so $G^R_{00}(p)$ at $T=0$ and $\omega=0$ is of order ${{\mathbf{p}}}^2 \tilde{G}^R(-{{\mathbf{p}}}^2)$. The $T=0$ contribution does not affect the sum rules 2 (\[eq:sumrule-2-L\]) and 3 (\[eq:sumrule-3-L\]) because as is explained in Sec. \[ssc:sumrule-L\], the relevant quantities to the derivation of these sum rules are $\omega^2\delta G^R_{00}(p)|_{\omega\rightarrow 0}$ and $\omega^4\delta G^R_{00}(p)|_{\omega\rightarrow 0}$, and the $T=0$ contributions vanish. The only sum rule which $T=0$ terms may affect is sum rule 1 (\[eq:sumrule-1-L\]). Nevertheless, we only consider terms in $G^R_{00}(p)|_{\omega\rightarrow 0}$ up to ${{\mathbf{p}}}^2$ in this paper, as is shown in Sec. \[ssc:sumrule-L\]. Therefore, sum rule 1 is also unaffected.
We note that we have so far neglected effects of possible hydro modes, which adds a $\omega^{3/2}$ term at zero momentum to $G^T$ [@Kovtun:2003vj]. It was suggested that this approximation is justified in the large ${N_c}$ limit [@Kovtun:2003vj]. Among our sum rules, sum rule 3 (\[eq:sumrule-3-T\]) in the transverse channel may be changed at finite ${N_c}$ due to this effect. In the analysis of lattice QCD data, this effect practically can be neglected because the IR cutoff of the current lattice QCD is not small enough for such IR energy effects to be detectable.
Sum rules at finite momentum {#sec:sumrule}
============================
In this section, we derive the sum rules at finite momentum in the transverse and the longitudinal channels. For later convenience, we also give the expressions for the sum rules at zero momentum in the transverse channel, which were already obtained in Ref. [@Gubler:2016hnf]. We also confirm that the sum rules are satisfied by the expressions of the spectral function in the chiral and weak coupling limits.
Transverse channel
------------------
Before deriving the sum rules, let us discuss what kind of structure can be expected to appear in the spectral function in a perturbative analysis. First, in the low-energy region, a peak with a width of order $g^4T$ due to the collision effect is expected to appear. This peak is called the transport peak. Its derivation is recapitulated in Appendix \[app:transport-peak\]. At larger energy, $\omega\sim T$, the pair-creation process yields a continuum in the spectral function (see Appendix \[app:continuum\] for its expression). Also, at $\omega\gg T$, the OPE analysis predicts a UV tail, whose derivation is recapitulated in Appendix \[app:UVtail\]. These structures are summarized in Fig. \[fig:spectral-longpaper\], where the expressions in the chiral and weak coupling limits for ${N_c}={N_f}=3$, Eqs. (\[eq:transport-T\]), (\[eq:continuum-T\]), and (\[eq:UVtail-T\]) have been used. The following parameters are chosen for illustrative purposes: $\tau^{-1}/T=0.5$, $|{{\mathbf{p}}}|/T=0.5$, $\kappa_0/T=1$, and $\Lambda_{\text{QCD}}/T=0.67$. We note that the corrections due to the ${{\mathbf{p}}}^2$ terms are almost negligible for this case, though the $|{{\mathbf{p}}}|$ value adopted here is not very small compared to $T$. We furthermore caution that the plots for each structure are reliable only at their energy regions of applicability. Namely, the transport peak is reliable at low energy, the continuum at intermediate and high energy and the UV tail at high energy, respectively. These regions are marked by the vertical lines with attached arrows in the figure. Note that these boundaries are not exact and should only be considered as indicative. One should not take the curves seriously when they are outside of the adequate energy regions.
![ The transport peak, the continuum, and the UV tail of the EM current spectral function in the transverse channel $\delta\rho_T$ as a function of $\omega$. The energy unit is $T$. To draw the figure, the parameters are set as ${N_c}={N_f}=3$, $\tau^{-1}/T=0.5$, $\kappa_0/T=1$, and $\Lambda_{\text{QCD}}/T=0.67$. The solid (dashed) lines correspond to a spatial momentum of $|{{\mathbf{p}}}|/T = 0.5$ ($|{{\mathbf{p}}}| = 0$). The vertical lines with the attached arrows indicate the regions for which the respective analytic expressions can be trusted.[]{data-label="fig:spectral-longpaper"}](spectral-longpaper-T.ranges.eps){width="49.00000%"}
### Sum rule 1
By using the asymptotic expression of $\delta G^R_T$ in the UV and IR energy regions, Eqs. (\[eq:OPE-w-RG-T\]) and (\[eq:hydro-GR-T\]), Eq. (\[eq:derivation\]) for $\mu=\nu=1$ becomes $$\begin{aligned}
\label{eq:sumrule-1-T}
\begin{split}
\kappa_B {{\mathbf{p}}}^2
+{\cal O}({{\mathbf{p}}}^4)
= \frac{2}{\pi}\int^\infty_{0} d\omega
\frac{\delta \rho_{T}(\omega,{{\mathbf{p}}})}{\omega},
\end{split}\end{aligned}$$ where the contribution from the UV part has vanished. This is the first sum rule in the transverse channel (sum rule 1). We note that $|{{\mathbf{p}}}|$ should be small enough to trust this relation, because we have assumed that the IR region is well described by hydrodynamics, which is valid only at small momentum and energy. $\kappa_B$ has been evaluated in lattice QCD [@Brandt:2013faa] with methods that do not suffer from the problem of analytic continuation, so this sum rule can be used to constrain the spectral function. At $|{{\mathbf{p}}}|=0$, it reduces to $$\begin{aligned}
\label{eq:sumrule-1-p0}
\begin{split}
0&= \frac{2}{\pi}\int^\infty_{0} d\omega
\frac{\delta \rho_{T}(\omega,{{\mathbf{0}}})}{\omega},
\end{split}\end{aligned}$$ which was already obtained in Ref. [@Gubler:2016hnf], and also in Ref. [@Bernecker:2011gh] by using the conservation of the EM current.
To get a feeling of how this sum rule is satisfied, let us check the respective contributions from the transport peak, the continuum, and the UV tail below. This also gives us an indication about the sensitivity of the sum rule integral to these three structures. The transport peak in the spectral function at small momentum is given by Eq. (\[eq:transport-T\]), and its contribution to sum rule 1 reads $$\begin{aligned}
\label{eq:T-sumrule1-transport}
\begin{split}
\frac{2}{\pi}\int^\infty_{0} d\omega
\frac{\delta \rho_{T}(\omega,{{\mathbf{p}}})}{\omega}
&\simeq {C_{\text{em}}}{N_c}\chi \frac{2}{3} \frac{2}{\pi}
\int^\infty_{0} d\omega \frac{\tau^{-1} }{\omega^2+\tau^{-2}} \\
&~~~\times \left[1 +\frac{{{\mathbf{p}}}^2}{5} \frac{(3\omega^2-\tau^{-2})}{(\omega^2+\tau^{-2})^2}\right]\\
&= {C_{\text{em}}}{N_c}\chi \frac{2}{3},
\end{split}\end{aligned}$$ where ${C_{\text{em}}}\equiv e^2\sum_f q^2_f$, $\tau\sim (g^4T)^{-1}$ is the relaxation time introduced in the Boltzmann equation, and $\chi\equiv T^2/6$. For the leading term we have used $\int^\infty_{0} d\omega \tau^{-1}/(\omega^2+\tau^{-2})=\pi/2$, while the momentum dependent term vanishes because of $\int^\infty_{0} d\omega (3\omega^2-\tau^{-2})/(\omega^2+\tau^{-2})^3=-[\omega/(\omega^2+\tau^{-2})^2]^\infty_0=0$.
The contribution from the continuum (\[eq:continuum-T\]) to sum rule 1 is given as $$\begin{aligned}
\label{eq:T-sumrule1-cont}
\begin{split}
& \frac{2}{\pi}\int^\infty_{0} d\omega
\frac{\delta \rho_{T}(\omega,{{\mathbf{p}}})}{\omega} \\
&\simeq -C_{\text{em}}{N_c}\frac{1}{3\pi^2}
\int^\infty_{0} d\omega \omega
\Biggl[{n_F}\left(\frac{\omega}{2}\right) \\
&~~~ +{{\mathbf{p}}}^2\left\{\frac{1}{20}{n_F}''\left(\frac{\omega}{2}\right)
-\frac{1}{\omega^2}{n_F}\left(\frac{\omega}{2}\right)\right\}\Biggr] \\
&\simeq -C_{\text{em}}{N_c}\frac{1}{3}
\Biggl[\frac{T^2}{3}
-\frac{{{\mathbf{p}}}^2}{2\pi^2}\ln\frac{T}{\mu}\Biggr] ,
\end{split}\end{aligned}$$ where ${n_F}(k^0)\equiv[e^{k^0/T}+1]^{-1}$ is the Fermi distribution function. We have used $\int^\infty_{0} d\omega \omega {n_F}(\omega/2)=\pi^2T^2/3$, and introduced the IR cutoff $\mu$ for the continuum contribution because of a logarithmic IR divergence. The nonsingular parts of the ${{\mathbf{p}}}^2$ terms have been omitted. The result for the continuum (\[eq:continuum-T\]) is obtained from a one-loop calculation, and becomes unreliable when $\omega\lesssim gT$, where the hard-thermal loop resummation becomes necessary [@Moore:2006qn; @Braaten:1990wp]. Therefore, we see that $\mu\sim gT$.
The contribution from the UV tail (\[eq:UVtail-T\]) is estimated as $$\begin{aligned}
\label{eq:T-sumrule1-UVtail}
\begin{split}
\frac{2}{\pi}\int^\infty_{0} d\omega
\frac{\delta \rho_{T}(\omega,{{\mathbf{p}}})}{\omega}
&\sim {C_{\text{em}}}g^2T^2, {C_{\text{em}}}g^2{{\mathbf{p}}}^2.
\end{split} \end{aligned}$$ Here we have used the fact that the IR cutoff of the UV tail is of order $\sim T$, because the derivation of the UV tail is based on the OPE, which is valid when $\omega\gg T$.
Now let us check that sum rule 1 (\[eq:sumrule-1-T\]) is satisfied. Because $\kappa_B=0$ at weak coupling (see Appendix \[app:transport-peak\]), the integral in Eq. (\[eq:sumrule-1-T\]) needs to vanish in order to satisfy the sum rule. We first see that for the ${{\mathbf{p}}}$-independent part, the contributions from the transport peak and the continuum, which are of order ${C_{\text{em}}}T^2$, cancel while the contribution from the UV tail is of higher order ($\sim {C_{\text{em}}}g^2T^2$), by looking at Eqs.(\[eq:T-sumrule1-transport\]-\[eq:T-sumrule1-UVtail\]). Therefore, the ${{\mathbf{p}}}$-independent part was shown to satisfy sum rule 1 at leading order already in Ref.[@Gubler:2016hnf]. For the ${{\mathbf{p}}}^2$ term, we see that the transport peak does not contribute, and the continuum contribution is of order $\sim {C_{\text{em}}}{{\mathbf{p}}}^2\ln (1/g)$ while the UV tail contribution is suppressed by a factor of $g^2$. Because the continuum contribution is sensitive to the IR cutoff, we need to improve the evaluation by performing the HTL resummation [@Moore:2006qn; @Braaten:1990wp], in order to confirm that this contribution becomes negligible so that sum rule 1 is satisfied at order ${{\mathbf{p}}}^2$. It is furthermore understood that sum rule 1 is mainly sensitive to the transport peak as well as the continuum, while the contribution of the UV tail is small.
### Sum rule 2 {#sssec:sumrule2-T}
In the derivation of Eq.(\[eq:derivation\]), we used only the fact that the retarded Green function is analytic in the upper $\omega$ plane. Thus, we can derive a similar equation in which $\delta G^R(\omega)$ ($\delta \rho(\omega)$) is replaced with $\omega^2\delta G^R(\omega)$ ($\omega^2\delta \rho(\omega)$), $$\begin{aligned}
\begin{split}
&\omega^2\delta G_{T}(\omega,{{\mathbf{p}}})|_{\omega\rightarrow 0}-\omega^2\delta G_{T}(\omega,{{\mathbf{p}}})|_{\omega\rightarrow \infty} \\
&= \frac{2}{\pi}\int^\infty_{0} d\omega \omega
\delta \rho_{T}(\omega,{{\mathbf{p}}}),
\end{split}\end{aligned}$$ for the transverse component. By using Eqs. (\[eq:OPE-w-RG-T\]) and (\[eq:hydro-GR-T\]), this equation becomes $$\begin{aligned}
\label{eq:sumrule-2-T}
\begin{split}
&-e^2\sum q^2_f
\Biggl[\left\{2m_f \delta\left\langle\overline{\psi}_f \psi_f \right\rangle
+\frac{1}{12}\delta\left\langle \frac{\alpha_s}{\pi}G^2\right\rangle \right\}\\
&~~~+\frac{8}{3}\frac{1}{4{C_F}+{N_f}}\delta\left\langle T^{00} \right\rangle
\Biggr]
= \frac{2}{\pi}\int^\infty_{0} d\omega \omega
\delta \rho_{T}(\omega,{{\mathbf{p}}}).
\end{split}\end{aligned}$$ We call this equation sum rule 2 in what follows. It should be noted that there is no explicit $|{{\mathbf{p}}}|$ dependence on the left-hand side, but again we are implicitly assuming that $|{{\mathbf{p}}}|$ is small enough so that the hydrodynamics well describes the behavior of $\omega^2\delta G_{T}(\omega,{{\mathbf{p}}})|_{\omega\rightarrow 0}$. Also, we emphasize that the expectation values of the local operators on the left-hand side can be evaluated nonperturbatively by lattice QCD without suffering from the problem of analytic continuation. Therefore, sum rule 2 can be used to constrain the shape of the spectral function. As it was already discussed in the previous paper [@Gubler:2016hnf], lattice QCD results show that the left-hand side of Eq.(\[eq:sumrule-2-T\]) is found to be dominated by the $\left\langle T^{00} \right\rangle$ for almost all temperatures around and above $T_c$.
Let us evaluate the contributions to sum rule 2 from the transport peak, continuum, and UV tail. The contribution from the transport peak is found by using Eq. (\[eq:transport-T\]) as $$\begin{aligned}
\label{eq:T-sumrule2-transport}
\begin{split}
\frac{2}{\pi}\int^\infty_{0} d\omega \omega
\delta \rho_{T}(\omega,{{\mathbf{p}}})
&= {C_{\text{em}}}{N_c}\chi \frac{2}{3} \frac{2}{\pi}\int^\infty_{0} d\omega
\frac{\tau^{-1} \omega^2}{\omega^2+\tau^{-2}} \\
&~~~\times \left[1 +\frac{{{\mathbf{p}}}^2}{5} \frac{(3\omega^2-\tau^{-2})}{(\omega^2+\tau^{-2})^2}\right]\\
&= {C_{\text{em}}}{N_c}\chi \frac{2}{3}
\left[ \frac{2}{\pi}\tau^{-1}\varLambda +\frac{{{\mathbf{p}}}^2}{5} \right] ,
\end{split}\end{aligned}$$ where we have introduced the UV cutoff $\varLambda$ for the transport peak because of the linear UV divergence, and used $\int^\infty_{0} d\omega \omega^2(3\omega^2-\tau^{-2})/(\omega^2+\tau^{-2})^3=\pi \tau/2$. As the Boltzmann equation cannot be used when $\omega\gtrsim gT$ since the instantaneous scattering description becomes invalid [@Moore:2006qn], we set $\varLambda\sim gT$.
The contribution from the continuum (\[eq:continuum-T\]) reads $$\begin{aligned}
\label{eq:T-sumrule2-cont}
\begin{split}
& \frac{2}{\pi}\int^\infty_{0} d\omega \omega
\delta \rho_{T}(\omega,{{\mathbf{p}}}) \\
&= - C_{\text{em}}{N_c}\frac{1}{3\pi^2}
\int^\infty_{0} d\omega \omega^3
\Biggl[{n_F}\left(\frac{\omega}{2}\right) \\
&~~~~ +{{\mathbf{p}}}^2\left\{\frac{1}{20}{n_F}''\left(\frac{\omega}{2}\right)
-\frac{1}{\omega^2}{n_F}\left(\frac{\omega}{2}\right)\right\}\Biggr] \\
&= - C_{\text{em}}{N_c}\frac{1}{45} T^2
\Biggl[ 14\pi^2T^2 +{{\mathbf{p}}}^2 \Biggr] ,
\end{split}\end{aligned}$$ where we have used $\int^\infty_{0} d\omega \omega^3 {n_F}(\omega/2)=14\pi^4T^4/15$ and $\int^\infty_{0} d\omega \omega^3 {n_F}'' (\omega/2)=8\pi^2T^2$.
The UV tail contribution is, by using Eq. (\[eq:UVtail-T\]), found to be $$\begin{aligned}
\label{eq:T-sumrule2-UVtail}
\begin{split}
\frac{2}{\pi}\int^\infty_{0} d\omega \omega
\delta \rho_{T}(\omega,{{\mathbf{p}}})
&= \frac{2}{\pi} {C_{\text{em}}}{N_c}{C_F}\frac{4\pi^2T^4}{27} \alpha_s(\kappa_0) \\
&~~~\times \int^\infty_{X_0} dX
\left[\frac{X_0}{X}\right]^{\tilde{a}+1} \\
&~~~+{\cal O}\left({C_{\text{em}}}g^2T^2{{\mathbf{p}}}^2\right)\\
&= {C_{\text{em}}}{N_c}{C_F}\frac{4\pi^2T^4}{9} \frac{2}{4{C_F}+{N_f}} \\
&~~~+{\cal O}\left({C_{\text{em}}}g^2T^2{{\mathbf{p}}}^2\right),
\end{split}\end{aligned}$$ where we have introduced $X\equiv \ln(\omega/\Lambda_{\text{QCD}})$ and $X_0\equiv \ln(\kappa_0/\Lambda_{\text{QCD}})$, used Eq. (\[eq:RG-alphas\]) and introduced the IR cutoff of the UV tail as $\kappa_0\sim T$.
Let us check whether sum rule 2 is satisfied. From Eqs.(\[eq:T-sumrule2-transport\]-\[eq:T-sumrule2-UVtail\]), we see that for the ${{\mathbf{p}}}$-independent part, the contributions from the continuum and the UV tail have the same order of magnitude ($\sim {C_{\text{em}}}T^4$) while the contribution from the transport peak is much smaller, $\sim {C_{\text{em}}}T^2 \tau^{-1}\varLambda\sim {C_{\text{em}}}g^5 T^4$. These contributions are found to agree with the left-hand side of Eq. (\[eq:sumrule-2-T\]) by using $\delta \langle T^{00}\rangle={N_c}\pi^2T^4(8{C_F}+7{N_f})/60$, which is obtained from Eqs. (\[eq:T00f-free\]) and (\[eq:T00g-free\]). For the ${{\mathbf{p}}}^2$ term, the contributions from the transport peak and the continuum cancel, while the UV tail contribution is much smaller. Therefore, we have confirmed that sum rule 2 is satisfied up to order ${{\mathbf{p}}}^2$, in the chiral and weak coupling limits.
### Sum rule 3
The derivation of the third sum rule turns out to be somewhat more tricky. Equation (\[eq:residue-theorem\]) for $\mu=\nu=1$ can be rewritten as $$\begin{aligned}
\begin{split}
&\delta G_{T}(i\omega,{{\mathbf{p}}})-\delta G_{T}(\infty,{{\mathbf{p}}})
= \frac{1}{\pi }\int^\infty_{0} d\omega' \\
&~~~\times\frac{\omega' \delta \rho_{T}(\omega',{{\mathbf{p}}})+\omega[{\mathrm{Re}}\delta G_{T}(\omega',{{\mathbf{p}}})-\delta G_{T}(\infty,{{\mathbf{p}}})]}{\omega'{}^2+\omega^2}\\
&~~~= \frac{2}{\pi }\int^\infty_{0} d\omega'
\frac{\omega' \delta \rho_{T}(\omega',{{\mathbf{p}}})}{\omega'{}^2+\omega^2} ,
\end{split}\end{aligned}$$ where we have used the relation $$\begin{aligned}
&0 = \nonumber \\
& \int^\infty_{-\infty} d\omega'
\frac{\omega' \delta \rho_{T}(\omega',{{\mathbf{p}}})-\omega[{\mathrm{Re}}\delta G_{T}(\omega',{{\mathbf{p}}})-\delta G_{T}(\infty,{{\mathbf{p}}})]}{\omega'{}^2+\omega^2},\end{aligned}$$ in the last line, which is obtained by using the residual theorem for the integral $\oint_C d\omega'
[\delta G_{T}(\omega',{{\mathbf{p}}})-\delta G_{T}(\infty,{{\mathbf{p}}})]/(\omega'+i\omega)$. By subtracting Eq. (\[eq:derivation\]) and $i\omega \delta G_T{}'=i\omega^2\delta G_T{}'2\int^\infty_0 d\omega'/[\pi(\omega^2+\omega'{}^2)]$, which is necessary to regularize the IR singularity in the integral, we get $$\begin{aligned}
\begin{split}
&\delta G_{T}(i\omega,{{\mathbf{p}}})-\delta G_{T}(0,{{\mathbf{p}}}) -i\omega \delta G_T{}'(0,{{\mathbf{p}}}) \\
&= \frac{2}{\pi }\omega^2 \int^\infty_{0} d\omega' \frac{1}{\omega^2+\omega'{}^2}
\left[ \delta \rho_{T}(\omega',{{\mathbf{p}}})\frac{-1}{\omega'}
+\delta \rho'_T{}(0,{{\mathbf{p}}})\right],
\end{split}\end{aligned}$$ where $'$ stands for the derivative in terms of energy ($\omega$, $\omega'$). Taking the $\omega \to 0$ limit, this reduces to $$\begin{aligned}
\label{eq:derivation-sumrule3-T}
\begin{split}
\frac{1}{2}\delta G_{T}{}''(0,{{\mathbf{p}}})
= \frac{2}{\pi } \int^\infty_{0} d\omega \frac{1}{\omega^3}
\left[ \delta \rho_{T}(\omega,{{\mathbf{p}}})
-\omega \delta \rho'_T{}(0,{{\mathbf{p}}})\right].
\end{split}\end{aligned}$$ Here, we have changed the integration variable from $\omega'$ to $\omega$ for simplicity. To get an explicit form of this sum rule, one needs to evaluate $\delta G_{T}{}''(0,{{\mathbf{p}}})$ and $\delta \rho'_T{}(0,{{\mathbf{p}}})$. In the expansion of Eq.(\[eq:hydro-GR-T\]), we get only the $|{{\mathbf{p}}}|=0$ terms as $\delta G_{T}{}''(0,{{\mathbf{p}}})=-2\sigma\tau_J+{\cal O}({{\mathbf{p}}}^2)$ and $\delta \rho'_T{}(0,{{\mathbf{p}}})=\sigma+{\cal O}({{\mathbf{p}}}^2)$. Therefore, we can obtain a sum rule for the $|{{\mathbf{p}}}|=0$ case, which reads $$\begin{aligned}
\label{eq:sumrule-3-T}
-\sigma\tau_J
&= \frac{2}{\pi} \int^\infty_{0} \frac{d\omega}{\omega^3}
\left[\delta \rho_T(\omega,{{\mathbf{0}}})-\sigma\omega\right].\end{aligned}$$ We call this equation sum rule 3 for the transverse channel. We note that the transport coefficients in the left-hand side cannot be computed by lattice QCD without suffering from the problem of analytic continuation. We do not check that sum rule 3 is satisfied in the chiral and the weak coupling limits since this was already done in our previous paper [@Gubler:2016hnf]. Instead we just cite the order of magnitude of the three contributions: the transport peak contribution is of order ${C_{\text{em}}}T^2 \tau^2\sim {C_{\text{em}}}g^{-8}$ and is equal to the left-hand side of sum rule 3, while the continuum is much smaller, ${C_{\text{em}}}g^{-5}$. The UV tail contribution is the smallest and of order ${C_{\text{em}}}g^{-4}$.
Explicitly taking into account higher order terms in Eq. (\[eq:hydro-GR-T\]) and expanding $\delta\rho_T$ in $|{{\mathbf{p}}}|^2$ order by order, it should be possible to obtain a corresponding sum rule at finite momentum. We leave this task for future work.
Longitudinal channel {#ssc:sumrule-L}
--------------------
Before discussing the sum rules, let us remember that the retarded Green function in the longitudinal channel is exactly known at zero momentum [@Ding:2010ga] from the conservation law of the charge: $$\begin{aligned}
\label{eq:rho-L-p=0-exact}
\rho_{00}(\omega,{{\mathbf{0}}})
&= \pi \chi_q \omega\delta(\omega) ,\end{aligned}$$ where $\chi_q\equiv \int d^3{{\mathbf{x}}}\langle j^0({{\mathbf{x}}}) j^0({{\mathbf{0}}}) \rangle/T$ is the charge susceptibility. By matching this result with the hydro result of Eq.(\[eq:hydro-rho-L-p=0\]), we see that the hydro result is exact for all $\omega$ at zero momentum, and $\sigma/D=\chi_q$. For this reason, the sum rules in the longitudinal channel provide nontrivial information only when ${{\mathbf{p}}}$ is finite. Therefore, we consider only the finite momentum case in this subsection.
At finite momentum, the diffusion peak appears in the longitudinal spectral function in addition to the three structures that were already present in the transverse channel, as was explained in Sec. \[ssc:IR\]. To get a feeling about the possible shape of the spectral function in the longitudinal channel, we plot the diffusion peak (\[eq:hydro-rho-L\]), the transport peak (\[eq:transport-00\]), the continuum (\[eq:continuum-00\]), and the UV tail (\[eq:UVtail-00\]) in Fig. \[fig:spectral-longpaper-L\]. The parameters are the same as in Fig. \[fig:spectral-longpaper\]. Again, the approximate regions for which the above analytic descriptions are expected to be valid are indicated by the vertical lines and arrows.
Here we comment on the treatment of the diffusion peak in the traditional QCD sum rule literature: In the conventional sum rule approach, the delta function structure that is similar to the hydro result of Eq.(\[eq:hydro-rho-L-p=0\]) was suggested based on the perturbative calculation in Ref.[@Bochkarev:1985ex], and has been assumed in the subsequent works. Though the two approaches give the same form at $|{{\mathbf{p}}}|=0$ as they should follow the exact results (\[eq:rho-L-p=0-exact\]), the perturbative approach is not generally reliable at $\omega=|{{\mathbf{p}}}|=0$ even when $g$ is small, so that the hydro approach should be adopted. Actually, once we consider finite $|{{\mathbf{p}}}|$, they yield different results.
![ The transport peak, the continuum, and the UV tail of the EM current spectral function in the longitudinal channel $\delta\rho_{00}$ as a function of $\omega$. The energy unit is $T$. The parameters and the meaning of the vertical line with attached arrows are the same as in Fig. \[fig:spectral-longpaper\].[]{data-label="fig:spectral-longpaper-L"}](spectral-longpaper-00.ranges.eps){width="49.00000%"}
### Sum rule 1
The first sum rule for $\delta\rho_{00}$ is obtained from Eq. (\[eq:derivation\]) for $\mu=\nu=0$ by using Eqs. (\[eq:OPE-w-RG-L\]) and (\[eq:hydro-GR-L\]): $$\begin{aligned}
\label{eq:sumrule-1-L}
\begin{split}
\frac{\sigma}{D}
+{\cal O}({{\mathbf{p}}}^2)
= \frac{2}{\pi}\int^\infty_{0} d\omega
\frac{\delta \rho_{00}(\omega,{{\mathbf{p}}})}{\omega}.
\end{split}\end{aligned}$$ We call this sum rule 1 for the longitudinal channel. Since $\sigma/D$ agrees with the susceptibility $\chi_q$, the left-hand side can be evaluated nonperturbatively by lattice QCD without the problem of analytic continuation.
Let us evaluate the contributions from the diffusion peak, the transport peak, the continuum, and the UV tail, in the weak coupling and the chiral limits. By using Eq. (\[eq:hydro-rho-L\]), the contribution from the diffusion peak is evaluated as $$\begin{aligned}
\begin{split}
\frac{2}{\pi}\int^\infty_{0} d\omega
\frac{\delta \rho_{00}(\omega,{{\mathbf{p}}})}{\omega}
&= \frac{2}{\pi} \sigma{{\mathbf{p}}}^2
\int^\infty_{0} d\omega \frac{1}{\omega^2+(D{{\mathbf{p}}}^2)^2}\\
&= \frac{\sigma}{D},
\end{split}\end{aligned}$$ which is of order ${C_{\text{em}}}T^2$. This contribution is equal to the left-hand side of Eq.(\[eq:sumrule-1-L\]). All the other contributions \[the transport peak (\[eq:transport-00\]), continuum (\[eq:continuum-00\]), and the UV tail (\[eq:UVtail-00\])\] are found to be proportional to ${{\mathbf{p}}}^2$, so the contribution from the diffusion peak is dominant when the momentum is small, and sum rule 1 is satisfied in the limit considered here.
### Sum rule 2 {#sum-rule-2}
The second sum rule is obtained by replacing $\delta G^R(\omega)$ \[$\delta \rho(\omega)$\] with $\omega^2\delta G^R(\omega)$ \[$\omega^2\delta \rho(\omega)$\] in the derivation of sum rule 1, as in Sec. \[sssec:sumrule2-T\]. The result is $$\begin{aligned}
\label{eq:sumrule-2-L}
0= \frac{2}{\pi}\int^\infty_{0} d\omega \omega
\delta \rho_{00}(\omega,{{\mathbf{p}}}).\end{aligned}$$ This is sum rule 2 for the longitudinal channel, which constrains the spectral function. We note that this sum rule can be obtained also by using the current conservation [@Bernecker:2011gh]. Therefore, actually this sum rule is exact, and valid at any value of $|{{\mathbf{p}}}|$, not only for small $|{{\mathbf{p}}}|$.
We again evaluate the contributions to sum rule 2 in the weak coupling and the chiral limits. First we check the diffusion peak contribution. By using Eq. (\[eq:hydro-rho-L\]), we get the order estimate $$\begin{aligned}
\begin{split}
\frac{2}{\pi}\int d\omega \omega
\delta \rho_{00}(\omega,{{\mathbf{p}}})
& = \frac{2}{\pi} \sigma{{\mathbf{p}}}^2
\int^\infty_{0} d\omega \frac{\omega^2}{\omega^2+(D{{\mathbf{p}}}^2)^2} \\
&\sim {C_{\text{em}}}T^2\tau^2 {{\mathbf{p}}}^4 ,
\end{split}\end{aligned}$$ where we have introduced a UV cutoff of the diffusion peak, which is of order $D{{\mathbf{p}}}^2\sim \tau{{\mathbf{p}}}^2$.
The transport peak (\[eq:transport-00\]) contributes as $$\begin{aligned}
\begin{split}
& \frac{2}{\pi}\int^\infty_{0} d\omega \omega
\delta \rho_{00}(\omega,{{\mathbf{p}}}) \\
&= \frac{2}{\pi} \frac{2}{3}{C_{\text{em}}}{N_c}\chi {{\mathbf{p}}}^2
\int^\infty_{0} d\omega
\frac{\tau^{-1}}{\omega^2+\tau^{-2}} \\
&~~~\times \left[1
+{{\mathbf{p}}}^2 \frac{2}{5}\left(\tau^{-2}+\frac{11}{3}\omega^2\right)
\frac{1}{(\omega^2+\tau^{-2})^2}
\right] \\
&= \frac{2}{3}{C_{\text{em}}}{N_c}\chi {{\mathbf{p}}}^2
\left[1+\frac{{{\mathbf{p}}}^2 \tau^2}{3} \right] ,
\end{split}\end{aligned}$$ where we have used $\int^\infty_{0} d\omega \tau^{-1}(\tau^{-2}+11\omega^2/3)/(\omega^2+\tau^{-2})^3= \tau^2 (\pi/2)(5/6)$.
The continuum contribution is estimated by using Eq. (\[eq:continuum-00\]), $$\begin{aligned}
\begin{split}
\frac{2}{\pi}\int^\infty_{0} d\omega \omega
\delta \rho_{00}(\omega,{{\mathbf{p}}})
&= -C_{\text{em}}{N_c}\frac{{{\mathbf{p}}}^2}{3\pi^2} \int^\infty_{0} d\omega \omega \\
&~~~\times \left[{n_F}\left(\frac{\omega}{2}\right)+\frac{{{\mathbf{p}}}^2}{40}{n_F}''\left(\frac{\omega}{2}\right)\right] \\
&= -C_{\text{em}}{N_c}\frac{{{\mathbf{p}}}^2}{3\pi^2}
\left[\frac{\pi^2T^2}{3} +\frac{{{\mathbf{p}}}^2}{20}\right],
\end{split}\end{aligned}$$ where we have used $\int^\infty_{0} d\omega \omega{n_F}''\left(\omega/2\right)=2$.
The UV tail contribution is estimated to be $$\begin{aligned}
\begin{split}
\frac{2}{\pi}\int^\infty_{0} d\omega \omega
\delta \rho_{00}(\omega,{{\mathbf{p}}})
&= {\cal O}\left( C_{\text{em}} g^2T^2{{\mathbf{p}}}^2, C_{\text{em}} g^2 {{\mathbf{p}}}^4\right),
\end{split}\end{aligned}$$ by using Eq. (\[eq:UVtail-00\]).
By looking at all the contributions, we see that the contributions from the transport peak and the continuum are larger than the one from the UV tail at order ${{\mathbf{p}}}^2$. However, these two contributions are found to cancel each other, so that sum rule 2 is satisfied at leading order in ${{\mathbf{p}}}^2$. For the ${{\mathbf{p}}}^4$ terms, the contributions from the diffusion peak and the transport peak are larger than the ones from the continuum and the UV tail. These contributions are expected to cancel, but we cannot confirm this here since the former contribution was not explicitly calculated.
### Sum rule 3
The third sum rule is obtained by replacing $\delta G^R(\omega)$ \[$\delta \rho(\omega)$\] with $\omega^4\delta G^R(\omega)$ \[$\omega^4\delta \rho(\omega)$\] in the derivation of sum rule 1. It gives $$\begin{aligned}
\label{eq:sumrule-3-L}
\begin{split}
&-e^2\sum q^2_f {{\mathbf{p}}}^2
\Bigl[\left\{2m_f \delta\left\langle\overline{\psi}_f \psi_f \right\rangle
+\frac{1}{12}\delta\left\langle \frac{\alpha_s}{\pi}G^2\right\rangle \right\} \\
&~~~+\frac{8}{3}\frac{1}{4{C_F}+{N_f}} \delta\left\langle T^{00} \right\rangle
\Bigr]
= \frac{2}{\pi}\int^\infty_{0} d\omega \omega^3
\delta \rho_{00}(\omega,{{\mathbf{p}}}),
\end{split}\end{aligned}$$ which we call sum rule 3 in the longitudinal channel. We note that there is no ${{\mathbf{p}}}^4$ correction on the left-hand side, but implicitly the smallness of $|{{\mathbf{p}}}|$ is assumed so that hydrodynamics is reliable. As before, the operators on the left-hand side of this sum rule can be evaluated by lattice QCD without having the problem of analytic continuation.
Let us evaluate the contribution to this sum rule in the weak coupling and the chiral limits. The diffusion peak contribution is, by using Eq. (\[eq:hydro-rho-L\]), estimated as $$\begin{aligned}
\begin{split}
\frac{2}{\pi}\int^\infty_{0} d\omega \omega^3
\delta \rho_{00}(\omega,{{\mathbf{p}}})
&= \frac{2}{\pi}\sigma{{\mathbf{p}}}^2
\int^\infty_{0} d\omega
\frac{\omega^4}{\omega^2+(D{{\mathbf{p}}}^2)^2}\\
&\sim {C_{\text{em}}}T^2 \tau^4{{\mathbf{p}}}^8,
\end{split}\end{aligned}$$ where we have used the fact that the UV cutoff of the diffusion peak is of order $\tau {{\mathbf{p}}}^2$.
The transport peak contributes as $$\begin{aligned}
\begin{split}
& \frac{2}{\pi}\int^\infty_{0} d\omega \omega^3
\delta \rho_{00}(\omega,{{\mathbf{p}}}) \\
&= \frac{2}{\pi}\frac{2}{3}{C_{\text{em}}}{N_c}\chi {{\mathbf{p}}}^2
\int^\infty_{0} d\omega \frac{\omega^2 \tau^{-1}}{\omega^2+\tau^{-2}} \\
&~~~\times \left[1
+ \frac{2}{5}\left(\tau^{-2}+\frac{11}{3}\omega^2\right)
\frac{{{\mathbf{p}}}^2}{(\omega^2+\tau^{-2})^2} \right] \\
&= {\cal O}({C_{\text{em}}}T^2{{\mathbf{p}}}^2\tau^{-1}\varLambda)
+ {C_{\text{em}}}{N_c}\chi {{\mathbf{p}}}^4 \frac{2}{5} ,
\end{split}\end{aligned}$$ where we have used $\int^\infty_{0} d\omega \left(\tau^{-2}+11\omega^2/3\right) \omega^2 \tau^{-1}/(\omega^2+\tau^{-2})^3=(\pi/2)(3/2)$.
The contribution from the continuum (\[eq:continuum-00\]) reads $$\begin{aligned}
\begin{split}
\frac{2}{\pi}\int^\infty_{0} d\omega \omega^3
\delta \rho_{00}(\omega,{{\mathbf{p}}})
&= -C_{\text{em}}{N_c}\frac{{{\mathbf{p}}}^2}{6\pi} \frac{2}{\pi}\int^\infty_{0} d\omega \omega^3 \\
&~~~\times \left[{n_F}\left(\frac{\omega}{2}\right)+\frac{{{\mathbf{p}}}^2}{40}{n_F}''\left(\frac{\omega}{2}\right)\right] \\
&= -C_{\text{em}}{N_c}\frac{{{\mathbf{p}}}^2}{3} \\
&~~~\times \left[\frac{14\pi^2T^4}{15}+ \frac{{{\mathbf{p}}}^2}{5}T^2\right].
\end{split}\end{aligned}$$
Finally, the UV tail contribution (\[eq:UVtail-00\]) is $$\begin{aligned}
\begin{split}
\frac{2}{\pi}\int^\infty_{0} d\omega \omega^3
\delta \rho_{00}(\omega,{{\mathbf{p}}})
&= \frac{2}{\pi}{C_{\text{em}}}\alpha_s(\kappa_0) {N_c}{C_F}\frac{4\pi^2T^4}{27} {{\mathbf{p}}}^2 \\
&~~~\times \int^\infty_{X_0} dX
\left[\frac{X_0}{X}\right]^{\tilde{a}+1} \\
&~~~+{\cal O}({C_{\text{em}}}g^2T^2{{\mathbf{p}}}^4)\\
&= {C_{\text{em}}}{N_c}{{\mathbf{p}}}^2 \frac{8\pi^2T^4}{9}
\frac{{C_F}}{4{C_F}+{N_f}} \\
&~~~ +{\cal O}({C_{\text{em}}}g^2T^2{{\mathbf{p}}}^4).
\end{split}\end{aligned}$$
Let us check the sum rule order by order in ${{\mathbf{p}}}^2$. For the ${{\mathbf{p}}}^2$ terms, we see that the contributions from the continuum and the UV tail are dominating, and their sum is equal to $-C_{\text{em}}{N_c}{{\mathbf{p}}}^2\pi^2T^4 2(8{C_F}+7{N_f})/[45(4{C_F}+{N_f})]$, which agrees with the left-hand side of sum rule 3 (\[eq:sumrule-3-L\]). The order ${{\mathbf{p}}}^4$ terms are dominated by the transport peak and the continuum, but they cancel each other. Therefore, sum rule 3 is shown to be satisfied up to order ${{\mathbf{p}}}^4$.
### Sum rules for $\delta\rho_L$
Before ending this section, we note that two sum rules for $\delta\rho_L$ can be derived from our three sum rules for $\delta\rho_{00}$, (\[eq:sumrule-1-L\]), (\[eq:sumrule-2-L\]), and (\[eq:sumrule-3-L\]). Using $\rho_L(p)=p^2 \rho_{00}(p)/{{\mathbf{p}}}^2$, we get $$\begin{aligned}
& -\frac{\sigma}{D}
+{\cal O}({{\mathbf{p}}}^2)
= \frac{2}{\pi}\int^\infty_{0} d\omega
\frac{\delta \rho_{L}(\omega,{{\mathbf{p}}})}{\omega}, \\
\nonumber
& -e^2\sum q^2_f
\Bigl[\left\{ 2m_f \delta\left\langle\overline{\psi}_f \psi_f \right\rangle
+\frac{1}{12}\delta\left\langle \frac{\alpha_s}{\pi}G^2\right\rangle \right\} \\
&+\frac{8}{3}\frac{1}{4{C_F}+{N_f}} \delta\left\langle T^{00} \right\rangle
\Bigr]
= \frac{2}{\pi}\int^\infty_{0} d\omega \omega
\delta \rho_{L}(\omega,{{\mathbf{p}}})\end{aligned}$$ Naively, one would expect these to agree at $|{{\mathbf{p}}}|=0$ with the corresponding sum rules for $\delta \rho_T$, Eqs.(\[eq:sumrule-1-T\]) and (\[eq:sumrule-2-T\]). This is indeed the case for the latter sum rule, but the former one does not agree, because of the $-\sigma/D$ term on its left-hand side. This disagreement can be traced back to the singularity of $\delta \rho_L$ at the diffusion pole $\omega=-iD{{\mathbf{p}}}^2$.
Application to lattice QCD data analysis {#sec:lattice}
========================================
In this section, we demonstrate that our sum rules can be used to improve a fit of the spectral function to the latest Euclidean time lattice QCD data given in Ref.[@Brandt:2015aqk]. In doing this, we make use of the sum rules in three different ways.
1. Providing guidance in the choice of the functional forms used to parametrize the spectral function.
2. Reducing the number of fitting parameters (sum rules 1 and 2).
3. Determining transport coefficients (sum rule 3).
We note that we use all the sum rules including sum rule 2, which was missing in the previous work [@Gubler:2016hnf].
In the following discussion of lattice QCD data, we restrict ourselves to the zero-momentum case, as most currently available lattice QCD data are provided only in this limit. This simplifies the situation in the sense that for $|{{\mathbf{p}}}| = 0$ the correlator has only one independent component and some parameters drop out of the sum rules \[such as $\kappa_B$ in Eq.(\[eq:sumrule-1-T\])\]. In this section, we use the notation $\rho(\omega, T) = \rho_{T}(\omega, |{{\mathbf{p}}}|=0) = \rho_{L}(\omega, |{{\mathbf{p}}}|=0)$ for the spectral function. The application of our sum rules to nonzero momentum lattice QCD data is left for future work. We hence only use the sum rules of Eqs.(\[eq:sumrule-1-p0\]) and (\[eq:sumrule-2-T\]) at $|{{\mathbf{p}}}| = 0$ and (\[eq:sumrule-3-T\]).
Parametrization of the spectral function in vacuum and at finite temperature
----------------------------------------------------------------------------
Our goal here is to find a functional form of the spectral function that is consistent with the sum rules 1-3, that is, that does not lead to divergent results for these sum rules. Note that some terms of the finite temperature spectra used in Ref.[@Brandt:2015aqk] in fact lead to divergences for sum rules 2 and 3 and therefore violate them.
First, we start with the spectral function in vacuum, for which we can follow the parametrization of Ref.[@Brandt:2015aqk], $$\begin{aligned}
& \frac{1}{C_{\mathrm{em}}} \rho(\omega, T \simeq 0) \nonumber \\
=& \,\, \frac{\pi}{3} a_V \delta(\omega - m_V) \nonumber \\
&\,\, + (1 + k_1) \frac{1}{4 \pi} \Theta(\omega - \Omega_0)
\omega^2 \tanh \Bigl( \frac{\omega \beta_0}{4} \Bigr)
\label{eq:vac} \end{aligned}$$ This form is adapted to our notation, which differs from that of Ref.[@Brandt:2015aqk] by a factor of $1/6$ and the treatment of $C_{\mathrm{em}}$. Here $\beta_0 = 1/T_0$, where $T_0$ corresponds to the low temperature of the “vacuum" lattice ensemble of Ref.[@Brandt:2015aqk] ($T_0 \simeq 32\,\mathrm{MeV}$). The values of the parameters $a_V$, $m_V$, $k$ and $\Omega_0$ are determined by the fit. Here, the $\delta$-function peak corresponds to the $\rho$ meson, as an isospin 1 current was used in Ref.[@Brandt:2015aqk]. To check whether the above is a reasonable parametrization, we have performed a MEM analysis [@Jarrell:1996rrw; @Asakawa:2000tr; @Gubler:2010cf] of the vacuum data provided in Ref.[@Brandt:2015aqk]. In this simple analysis, we have ignored correlations between data points at different time slices. The result is shown as a red line in Fig.\[fig:MEM.fit\]. Here, we have chosen the default model, which is an input of the MEM algorithm, to match the perturbative value of the spectral function at high energy \[$\rho(\omega)/\omega^2 = 3/(2\pi)$, blue dashed line in Fig.\[fig:MEM.fit\]\].
![Result of a MEM analysis of vacuum data of Ref.[@Brandt:2015aqk] (red solid line), compared with a simple fit using Eq.(\[eq:vac\]) (black solid line). The blue dashed line shows the default model used in the MEM analysis.[]{data-label="fig:MEM.fit"}](MEM.fit.3.eps){width="49.00000%"}
It is seen in the figure that at low energy the spectral function is dominated by a single peak, while at high energy, there is an almost flat continuum. We furthermore see hints of excited states at around $a\omega \simeq 0.8$, but their effect seems to to be weak. In all, Eq.(\[eq:vac\]) turns out to be a reasonable, if somewhat rough, parametrization of the spectral function at temperatures much below $T_c$. As is also shown in Fig.\[fig:MEM.fit\] and further discussed later, the fit results for the peak position $m_V$ and the onset of the continuum $\Omega_0$ agree well with those obtained from the MEM analysis.
Next, we consider the spectral function at temperatures around and above $T_c$, where large modifications are expected. As it is already discussed in detail in this work, we most importantly expect the appearance of a transport peak at $\omega \simeq 0$ and an UV tail at high energy. We again follow the parametrization used in Ref.[@Brandt:2015aqk], but modify it such that sum rules 1-3 can be satisfied. Specifically, we use $$\begin{aligned}
& \frac{1}{C_{\mathrm{em}}} \rho(\omega, T) \nonumber \\
=&\,\, \frac{\omega A_T \Gamma_T}{3(\Gamma_T^2 + \omega^2)} \bigl[ 1 - A(\omega) \bigr]
+ \frac{\pi}{3} a_T \delta(\omega - m_V) \nonumber \\
&\,\, + A(\omega) \bigl[1 + \tilde{k}(\omega) \bigr] \frac{1}{4 \pi}
\omega^2 \tanh \Bigl( \frac{\omega \beta_T}{4} \Bigr) \nonumber \\
&\,\, + \frac{c_0}{4 \pi} \Theta(\omega - \Omega_0) \frac{1}{\omega^2} \frac{1}{\bigl[\ln(\omega/\Lambda_{\mathrm{QCD}})\bigr]^{1+ \tilde{a}}},
\label{eq:finiteT} \end{aligned}$$ with $$\begin{aligned}
A(\omega) = \tanh\Bigl( \frac{\omega^2}{\Delta^2}\Bigr)
\label{eq:A}\end{aligned}$$ and $$\begin{aligned}
\tilde{k}(\omega) = k_1 + k_2 \Biggl[1 - \tanh^2\Bigl( \frac{\omega}{\Omega_0 \eta} \Bigr) \Biggr].
\label{eq:k}\end{aligned}$$ Let us here mention the differences between our above parametrization and that of Ref.[@Brandt:2015aqk]. First, the factors $\bigl[ 1 - A(\omega) \bigr]$ and $A(\omega)$, which were introduced already in our previous work [@Gubler:2016hnf], are there for cutting off the divergence in the integral of sum rules 2 and 3. Furthermore, the factor $1/\bigl[\ln(\omega/\Lambda_{\mathrm{QCD}})\bigr]^{1+ \tilde{a}}$, which is adapted from the perturbative expression, is employed to make integral of sum rule 2 finite.
Lattice data used in the fit
----------------------------
Reference [@Brandt:2015aqk] provides correlator data of altogether five lattice ensembles with respective temperatures $32$, $169$, $203$, $254$ and $338\,\mathrm{MeV}$, which in the setting of that work corresponds to $0.15$, $0.8$, $1.0$, $1.25$ and $1.76\,T_c$. The spatial extent of the lattice is $N_s = 64$ for all ensembles, while the temporal size is $N_{\tau} = 128$, 24, 20, 16 and 12. The lattice spacing is $a = 0.0486\,\mathrm{fm}$ and the vacuum pion mass $270\,\mathrm{MeV}$, which shows that this is not yet a physical point simulation. For more details, we refer the reader to the original publication of Ref.[@Brandt:2015aqk].
The lattice data are related to the spectral function $\rho(\omega)$ through the following integral: $$\begin{aligned}
G^{\mathrm{E}}(\tau, T) = \int_{0}^{\infty} \frac{d \omega}{2 \pi} \frac{\rho(\omega,T)}{C_{\mathrm{em}}} \frac{\cosh[\omega(\tau - 1/2T)]}{\sinh(\omega/2T)}.
\label{eq:Eucl.vec.correlator}\end{aligned}$$ Here, $\tau$ is defined in the interval $0 \leq \tau < 1/T$ and is symmetric with respect to the central point $\tau = 1/(2T)$. Hence, only half of the temporal data points can be used as independent information for the fit. Furthermore, for small $\tau$ values, the data could contain lattice artifacts. We follow Ref.[@Brandt:2015aqk] and use data points $\tau/a \in [4:48]$ for the vacuum ensemble and $\tau/a \in [4:N_{\tau}/2]$ for the others.
Treatment of the parameters
---------------------------
We treat the various parameters introduced in Eqs.(\[eq:vac\])-(\[eq:k\]) as follows. We keep $m_V$, $k_1$ and $\Omega_0$ fixed for both vacuum and finite temperature. $\eta$, which only enters in the nonzero temperature parametrization, is also kept fixed for all ensembles. All other parameters, $A_T$, $\Gamma_T$, $\Delta$, $a_T$, $k_2$ and $c_0$ are allowed to depend on temperature. To reduce the numbers of unknowns to be fitted, we however make use of the sum rules 1 and 2 to constrain the parameters. To be specific, the sum rules are used to fix $A_T/\Gamma_T$ and $c_0$ at each temperature. To be able to use sum rule 2, one, in principle, needs the quark- and gluon-condensate values and the energy density as a function of temperature. For the simple trial analysis of this work, we employ the values provided in Ref.[@Bazavov:2014pvz], which we rescale to two flavors according to the ratio in the perturbative high-temperature limit. As the quark masses and other lattice parameters in Ref.[@Bazavov:2014pvz] are different from those of Ref.[@Brandt:2015aqk], this is not a completely consistent treatment and will have to be improved in a more quantitative analysis in the future.
Fit results
-----------
Performing our fit, it turns out that even if one uses both sum rules 1 and 2 as constraints, the lattice data are not sufficient to determine all parameters of Eqs.(\[eq:finiteT\])-(\[eq:k\]) with good accuracy. We found for instance that certain combinations of the ratio $A_T/\Gamma_T$, which is proportional to the electric conductivity $\sigma_{\mathrm{el}}$, and $\Delta$ lead to similar $\chi^2$ values. This means that the systematic uncertainty of these quantities is large and that more data points with smaller errors will be needed to further constrain the parameters, especially for the lattice ensembles with temperatures above $T_c$, where the number of usable data points is rather small. In the following we provide one possible fit result, which is obtained by demanding that $\sigma_{\mathrm{el}}/T$ is a monotonically increasing function of $T$ and take values in reasonable agreement with previous works [@Brandt:2015aqk; @Aarts:2007wj; @Francis:2011bt; @Ding:2010ga; @Ding:2013qw; @Ding:2014dua; @Amato:2013naa; @Brandt:2012jc]. The numerical result of the fit is given in Table\[tab:fit.values\]. These give a $\chi^2$ value of $\chi^2/\mathrm{d.o.f.} = 0.87$. We emphasize once more that this is not the only solution and should therefore be regarded as an illustrative example rather than the final result. Nevertheless, the purpose of our analysis is to demonstrate the possibility to use the sum rules to improve the lattice QCD analysis, and to this end, we believe that our fit suffices.
Fitted parameters Fixed $0.8\,T_c$ $1.0\,T_c$ $1.25\,T_c$ $1.67\,T_c$
-------------------- ---------- ------------ ------------ ------------- -------------
$m_V$ 0.206 - - - -
$a_V$ 0.000860 - - - -
$\Omega_0$ 0.319 - - - -
$k_1$ 0.0909 - - - -
$\eta$ 2.04 - - - -
$a_T/T^3$ - 4.15 2.94 2.15 1.08
$\Gamma_T/T$ - 1.36 0.54 $\infty$ $\infty$
$A_T/(\Gamma_T T)$ - 0.450 0.728 0.728 0.758
$\Delta_T/T$ - 4.06 2.97 0.79 1.23
$c_0/T^3$ - -166 -250 -186 - 95
$k_2$ - -0.0294 -0.0377 -0.0322 -0.0040
Let us now discuss the resulting spectral functions in some detail. First, we compare in Fig.\[fig:MEM.fit\] the vacuum spectral function of Eq.(\[eq:vac\]) with that obtained by MEM. It is seen that the two methods indeed give a quantitatively similar result, while the details naturally disagree due to the roughness of the parametrization of Eq.(\[eq:vac\]) and the limited resolution of MEM. Next, we consider the spectral functions for temperatures around and above $T_c$. The spectra are shown in Fig.\[fig:full.spec\].
![The fitted spectral functions for temperatures between $0.8$ and $1.67\,T_c$. To improve the visibility, we have transformed the $\rho$ meson peak, which in the fit is assumed to be a delta function, into a Gaussian with a small, but nonzero width.[]{data-label="fig:full.spec"}](full.spec.2.eps){width="49.00000%"}
It is seen in this figure that the transport peak gradually grows with increasing temperature. One furthermore observes from the behavior of the discontinuity around $a\omega\simeq 0.3$ that the UV tail, which is parametrized to appear above $\Omega_0$, grows with larger temperature. We note that the coefficient of the UV tail ($c_0$) is negative in this fit, which might indicate that $\langle \tilde{T}^{00}(\omega\sim T)\rangle $ is negative, as is suggested from Eq. (\[eq:deltarho\_T\_UVtail\]). Also note that we here have divided the spectral function by $\omega$, which means that it approaches a constant at small $\omega$, while it approaches a form proportional to $\omega$ at large energy.
To examine the behavior of the spectral function more in detail, we show the same spectral functions with the vacuum part subtracted in Fig.\[fig:subtr.spec\].
![Same as Fig.\[fig:full.spec\], but with vacuum part subtracted.[]{data-label="fig:subtr.spec"}](subtr.spec.2.eps){width="49.00000%"}
At low energy, the same features can be observed as in the previous figure. At high energy, the dominant part of the continuum gets subtracted and only the UV tail remains. Also note that because the residue of the $\rho$ meson mass is reduced at finite temperature, the residue becomes negative for the subtracted spectral function shown here.
Next, let us examine the quality of our fit by numerically integrating the spectral function as in Eq.(\[eq:Eucl.vec.correlator\]) and compare the result with the lattice data of Ref.[@Brandt:2015aqk]. This comparison is shown as red and blue data points in Fig.\[fig:comp.latt\].
{width="45.00000%"} {width="45.00000%"} {width="45.00000%"} {width="45.00000%"} {width="45.00000%"}
In these plots we show the ratio between the lattice data and the integrated spectral function, which should be 1 for a perfect fit. Overall, it can be seen that both our and the fit of Ref.[@Brandt:2015aqk] can reproduce the lattice data fairly well, especially if one considers the very small errors of most of the lattice data points. For the lowest temperature ($T = 0.15\,T_c$), both fits exhibit almost the same behavior for small $\tau/a$ values, while for $\tau/a \gtrsim 40$, the fit of Ref.[@Brandt:2015aqk] works slightly better than ours. At higher temperatures, both fits perform equally well, being consistent with the lattice results within errors for most $\tau/a$ values. This shows that the available lattice data are not sufficient to distinguish our finite temperature spectral functions from those obtained in Ref.[@Brandt:2015aqk].
Comparison with fit of Ref.[@Brandt:2015aqk]
--------------------------------------------
In Ref.[@Brandt:2015aqk] a fit similar to ours was performed with the following functional form. For the vacuum part, the same parametrization as in Eq.(\[eq:vac\]) was employed, while for nonzero temperature $$\begin{aligned}
& \frac{1}{C_{\mathrm{em}}} \rho(\omega, T) \nonumber \\
&= \,\, \frac{\omega A_T \Gamma_T}{3(\Gamma_T^2 + \omega^2)}
+ \frac{\pi}{3} a_T \delta(\omega - m_V) \nonumber \\
&~~~ + \Theta(\omega - \Omega_T) \bigl[1 + \tilde{k}(\omega) \bigr] \frac{1}{4 \pi}
\omega^2 \tanh \Bigl( \frac{\omega \beta_T}{4} \Bigr) \nonumber \\
&~~~ + \frac{c_0}{4 \pi} \Theta(\omega - \Omega_0) \frac{1}{\omega^2}
\label{eq:finiteT.Brandt} \end{aligned}$$ was used, which we have again adapted to our notation. It should be noted that this form is contradicting with the sum rule 2, as its first and fourth terms lead to divergences in that sum rule. If one however only uses sum rule 1, a fit is possible. The authors of Ref.[@Brandt:2015aqk] have tested several models, in which some combinations of parameters are set to 0 or other fixed values for certain temperatures. For details, we refer the reader to Ref.[@Brandt:2015aqk].
Here, we compare our spectral function with the results obtained for model 2c in Ref.[@Brandt:2015aqk]. The other models of that work have different features in certain regions of $\omega$, but the same overall behavior. In Fig.\[fig:full.spec.comp\] we show the full (nonsubtracted) spectral functions for four different temperatures.
{width="45.00000%"} {width="45.00000%"} {width="45.00000%"} {width="45.00000%"}
It is seen in these figures that while the details of model 2c differ from our spectral function, the general structure is the same. The biggest difference between our spectral function and those of Ref.[@Brandt:2015aqk] can be found for the $T = 0.8\,T_c$ case in the region around the $\rho$ meson peak, where all models of Ref.[@Brandt:2015aqk] are close to 0, while our spectral function is smoothly connected to the continuum. The true spectral function in this region likely lies between these two extremes, as the $\rho$ meson in reality has quite a large width and is placed on top of a smooth $\pi \pi$ continuum, but still is the dominating structure below energies of about $1$ GeV (see, for instance, Fig.1 of Ref.[@Kwon:2008vq]).
Evaluation of physical quantities
---------------------------------
From the above fit results, we can determine the electric conductivity from the numerical values of $A_{T}/\Gamma_{T}$. In our parametrization, it is given as $$\begin{aligned}
\frac{\sigma_{\mathrm{el}}}{C_{\mathrm{em}}} = \frac{1}{C_{\mathrm{em}}} \lim_{\omega \to 0} \frac{\rho(\omega,T)}{\omega} = \frac{1}{3} \frac{A_T}{\Gamma_T}.
\label{eq:elec.conduct}\end{aligned}$$ From the values of Table\[tab:fit.values\], $\sigma_{\mathrm{el}}$ is obtained as shown in the second column of Table\[tab:le.constants\] and on the upper panel of Fig.\[fig:low.en.const\].
![The electric conductivity $\sigma_{\mathrm{el}}$ and the second order transport coefficient $\tau_{J}$ as functions of the temperature $T$. Both quantities are made dimensionless by multiplying appropriate powers of $T$. For $\sigma_{\mathrm{el}}$, the values obtained in Ref.[@Brandt:2015aqk] are also shown. Here, we have for simplicity combined the various errors quoted in Ref.[@Brandt:2015aqk] into a single one for each data point.[]{data-label="fig:low.en.const"}](sigma.2.eps "fig:"){width="49.00000%"} ![The electric conductivity $\sigma_{\mathrm{el}}$ and the second order transport coefficient $\tau_{J}$ as functions of the temperature $T$. Both quantities are made dimensionless by multiplying appropriate powers of $T$. For $\sigma_{\mathrm{el}}$, the values obtained in Ref.[@Brandt:2015aqk] are also shown. Here, we have for simplicity combined the various errors quoted in Ref.[@Brandt:2015aqk] into a single one for each data point.[]{data-label="fig:low.en.const"}](tau.eps "fig:"){width="49.00000%"}
We however emphasize once again here, that other fits with similar $\chi^2$ values but rather different electric conductivities are possible and that the numbers shown here just represent one of many possible solutions. On the upper panel of Fig.\[fig:low.en.const\], we furthermore show the results obtained in Ref.[@Brandt:2015aqk] for comparison. It is seen that with the exception of the point at $T = 1.67\,T_c$, the results are not consistent, even though they show the same general tendency. This once more indicates that the systematic uncertainty in the evaluation of this quantity is still rather large.
Next, we can now use sum rule 3 to estimate the value of the second order transport coefficient $\tau_{J}$, as all other ingredients in that sum rule are known. The results of such a computation are found in the third column of Table\[tab:le.constants\] and on the lower plot of Fig.\[fig:low.en.const\].
$T$ $\sigma_{\mathrm{el}}/(C_{\mathrm{em}}T)$ $T\tau_{J}$
------------- ------------------------------------------- -------------
$0.8\,T_c$ 0.150 1.21
$1.0\,T_c$ 0.243 2.16
$1.25\,T_c$ 0.243 1.96
$1.67\,T_c$ 0.253 1.29
: The electric conductivity $\sigma_{\mathrm{el}}$ and the second order transport coefficient $\tau_{J}$ at various temperatures, as obtained from the fit to lattice QCD data.[]{data-label="tab:le.constants"}
It is seen in this figure that $\tau_{J}$ exhibits quite an interesting behavior as a function of $T$. Namely, it increases for temperatures below $T_c$, takes a maximum at $T = T_c$ and then decreases again for temperatures above $T_c$. It remains to be seen whether this behavior is an artifact of our fit and/or our parametrization of the spectral function or if it is a real physical effect.
As a last result, we give the thermal dilepton rate $dN_{l^{+}l^{-}}/d\omega d^3 {{\mathbf{p}}}$ for vanishing spatial momentum ($|{{\mathbf{p}}}| = 0$), which can be easily obtained from the relation between the spectral function and the dilepton rate: $$\begin{aligned}
\frac{dN_{l^{+}l^{-}}}{d\omega d^3 {{\mathbf{p}}}}(|{{\mathbf{p}}}|=0) = \frac{\alpha_{\mathrm{em}}^2}{\pi^3 \omega^2}
\frac{\rho(\omega,T)}{e^{\omega/T} - 1}. \end{aligned}$$ The results are shown in Fig.\[fig:dilepton.rate\], where we have adjusted the horizontal axis to physical units (MeV) and where we also show the corresponding model 2c results of Ref.[@Brandt:2015aqk]. We see that our result is larger than the one in Ref.[@Brandt:2015aqk], especially at $T=0.8T_c$. This difference can be understood from the absence of the spectrum near the vector meson peak in Ref.[@Brandt:2015aqk], which was discussed in the previous subsection.
![The thermal dilepton rate for two-flavor QCD, extracted from our fit to lattice QCD data of Ref.[@Brandt:2015aqk]. The thick solid lines show the result of our fit, while the thin dashed lines correspond to model 2c of Ref.[@Brandt:2015aqk].[]{data-label="fig:dilepton.rate"}](dilepton.mod.2c.eps){width="49.00000%"}
Summary and Concluding Remarks {#sec:summary}
==============================
In the first part of this paper, we derived and discussed five exact sum rules (two for the transverse and three for the longitudinal parts) for the vacuum-subtracted spectral functions of the vector channel at finite temperature, which are determined completely by the UV and IR behavior of the vector correlator. The UV part can be obtained from the OPE, while the IR behavior is accurately described by hydrodynamics. The sum rules are valid for nonzero momentum ${{\mathbf{p}}}$, which should however be small enough such that the hydrodynamic description of the vector correlator in the IR regime can be trusted. In the limit $|{{\mathbf{p}}}| \to 0$, transverse and longitudinal parts approach each other such that the sum rules which we have already derived in Ref.[@Gubler:2016hnf] remain.
In perturbation theory, it has been known that three (four) distinct structures emerge in the transverse (longitudinal) channel: a transport peak, an exponentially suppressed continuum, and a power suppressed UV tail in both channels, and a diffusion peak in the longitudinal channel. All these structures can, in principle, contribute to the various sum rules and therefore need to be taken into account to test their validity. Doing this, we found that the sum rules are indeed satisfied in the weak coupling regime. This exercise also gives a rough idea about how the different parts of the full spectral function can be expected to contribute in different ways to each sum rule.
In the second part of the paper, we have employed recent two-flavor dynamical lattice QCD data at almost zero and finite temperature and at zero momentum to perform a spectral fit, in which two sum rules (1 and 2) are used as constraints to reduce the number of parameters to be fitted. The third sum rule (3) in turn enables us to extract the value of the second order transport coefficient $\tau_J$. We note that the second sum rule was not used in our previous work [@Gubler:2016hnf]. It, however, needs to be emphasized here that we found the fit not to be completely stable in the sense that we confirmed the existence of several local minima with comparable $\chi^2$ values, which means that more data points with increased precision will be needed for uniquely determining the true shape of the spectral function. Nevertheless, we succeeded in demonstrating that the sum rules can be used to improve the lattice QCD data analysis.
In our fit, we employed lattice data at zero momentum and have therefore used the three sum rules already derived in Ref.[@Gubler:2016hnf]. Once lattice data at nonzero momentum are available, it would be interesting to apply the sum rules derived in this paper to their analysis. Also, once the lattice QCD analysis at the physical point and in the continuum limit becomes available, it will be possible to use the phenomenological form of the vector spectral function at $T=0$ obtained from experiment. We leave these topics for future work.
Acknowledgements {#acknowledgements .unnumbered}
================
The research of D. S. is supported by the Alexander von Humboldt Foundation. The research of P.G. is supported by the Mext-Supported Program for the Strategic Foundation at Private Universities, “Topological Science" (No. S1511006). We thank Bastian B. Brandt for fruitful discussions.
Evaluation of the transport peak at weak coupling {#app:transport-peak}
=================================================
In this appendix, we evaluate the transport peak appearing in the spectral function by using the Boltzmann equation in the relaxation time approximation for massless quarks. This appendix is in essence a recapitulation of the literature, for example Refs. [@Satow:2014lva; @Hong:2010at]. The Boltzmann equation reads $$\begin{aligned}
\begin{split}
&v\cdot\partial_X n_{\pm f}({{\mathbf{k}}}, X)
\pm eq_f\left({{\mathbf{E}}}+{{\mathbf{v}}}\times{{\mathbf{B}}}\right)(X) \cdot\nabla_{{{\mathbf{k}}}} n_{\pm f}({{\mathbf{k}}}, X)\\
&= C[n_{\pm f}],
\end{split}\end{aligned}$$ where $n_{\pm f}({{\mathbf{k}}}, X)$ is the distribution function for the quark (antiquark) with momentum ${{\mathbf{k}}}$ at point $X$, and $v^\mu\equiv (1,{{\mathbf{v}}})$ with ${{\mathbf{v}}}\equiv {{\mathbf{k}}}/|{{\mathbf{k}}}|$. $C[n_{\pm f}]$ represents the collision effect among the quarks, which is given later.
Now we consider the situation in which the system at equilibrium is disturbed by weak external EM fields, so that the distribution function slightly deviates from the equilibrium, $n_{\pm f}({{\mathbf{k}}}, X)={n_F}(|{{\mathbf{k}}}|)+\delta n_{\pm f}({{\mathbf{k}}}, X)$. By linearizing the Boltzmann equation in terms of $\delta n_{\pm f}({{\mathbf{k}}}, X)$ and EM fields, we get $$\begin{aligned}
\label{eq:EOM-1}
\begin{split}
v\cdot\partial_X \delta n_{\pm f}({{\mathbf{k}}}, X)
\pm eq_f {{\mathbf{E}}}(X) \cdot{{\mathbf{v}}}{n_F}'(|{{\mathbf{k}}}|)
&= \delta C[n_{\pm f}],
\end{split}\end{aligned}$$ where the magnetic field term disappears due to the isotropy of the system at equilibrium. $\delta C[n_{\pm f}]$ is a linearized form of $C[n_{\pm f}]$, whose expression reads $$\begin{aligned}
\delta C[n_{\pm f}]
&= -\tau^{-1} \left(\delta n_{\pm f}({{\mathbf{k}}}, X)
\mp{n_F}'(|{{\mathbf{k}}}|) \delta\mu_f(X)
\right), \end{aligned}$$ in the relaxation time approximation. Here we have introduced the relaxation time $\tau\sim(g^4T)^{-1}$, and the shift of the chemical potential ($\delta\mu_f$) caused by the EM fields. The second term in the expression above is necessary, since the deviation of the distribution created by the shift of the chemical potential does not relax. The shift of the chemical potential is determined by the conservation law of particle number, $$\begin{aligned}
0&=\int\frac{d^3{{\mathbf{k}}}}{(2\pi)^3}\sum_{s=\pm 1} s \delta C[n_{s f}] ,\end{aligned}$$ which reduces to $$\begin{aligned}
\label{eq:EOM-2}
\delta\mu_f(X)
&= -\frac{1}{\chi}
\int\frac{d^3{{\mathbf{k}}}}{(2\pi)^3}\sum_{s=\pm 1} s \delta n_{s f}({{\mathbf{k}}}, X).\end{aligned}$$ Here, $\chi\equiv T^2/6$. From Eqs. (\[eq:EOM-1\]) and (\[eq:EOM-2\]), we obtain the solutions, $$\begin{aligned}
\delta\mu_f(p)
&= ieq_f\frac{{{\mathbf{E}}}(p)\cdot \hat{{{\mathbf{p}}}}}{|{{\mathbf{p}}}|} \frac{1-(\omega+i\tau^{-1})A(p)}{1-i\tau^{-1}A(p)}, \\
\label{eq:Boltzmanneq-solution}
\delta n_{s f}({{\mathbf{k}}}, p)
&= -is{n_F}'(|{{\mathbf{k}}}|)\frac{eq_f{{\mathbf{E}}}(p)\cdot{{\mathbf{v}}}-\tau^{-1}\delta\mu_f(p)}{v\cdot p+i\tau^{-1}},\end{aligned}$$ where we have performed the Fourier transformation $(X\rightarrow p)$ and introduced $A(p)\equiv \ln[(\omega+|{{\mathbf{p}}}|+i\tau^{-1})/(\omega-|{{\mathbf{p}}}|+i\tau^{-1})]/(2|{{\mathbf{p}}}|)$.
The induced current is given by $$\begin{aligned}
j^\mu(p)
&= 2e\sum_f q_f{N_c}\int\frac{d^3{{\mathbf{k}}}}{(2\pi)^3} v^\mu
\sum_{s=\pm 1} s\delta n_{sf}({{\mathbf{k}}},p).\end{aligned}$$ By using this expression in momentum space and the linear response theory relation of Eq.(\[eq:linear-response\]), Eq. (\[eq:Boltzmanneq-solution\]) leads to the retarded Green function $G^R_{\mu\nu}(p)$, as is shown below. We note that $j^0=-2e\sum_f q_f{N_c}\chi\delta\mu_f$, which indicates that $\chi$ is essentially the susceptibility.
Transverse channel
------------------
The transverse component of $G^R$ is given by $$\begin{aligned}
\begin{split}
G_T(p)
&=4 {C_{\text{em}}}{N_c}\frac{1}{(2\pi)^3}
\int^{2\pi}_0 d\phi \int^1_{-1} d\cos\theta \int^\infty_0 d|{{\mathbf{k}}}| |{{\mathbf{k}}}|^2 \\
&~~~\times{n_F}'(|{{\mathbf{k}}}|)\frac{\omega\sin^2\theta\cos^2\phi}{\omega-|{{\mathbf{p}}}|\cos\theta+i\tau^{-1}} \\
&= -{C_{\text{em}}}{N_c}\chi\frac{\omega}{{{\mathbf{p}}}^2} \\
&~~~\times \left[\omega+i\tau^{-1}-A(p)(p^2+2i\tau^{-1}\omega+(i\tau^{-1})^2)\right].
\end{split}
\label{eq:GTfull}\end{aligned}$$ In the hydro limit, $\omega,|{{\mathbf{p}}}|\ll\tau^{-1}$, this reduces to $$\begin{aligned}
\begin{split}
G_T(p)
&\simeq {C_{\text{em}}}{N_c}\chi \frac{2}{3}\tau \omega
\left[i-\tau\omega\right],
\end{split} \end{aligned}$$ where we have used $$\begin{aligned}
\begin{split}
A(p) & \simeq \frac{1}{i\tau^{-1}}
\left[1-\frac{\omega}{i\tau^{-1}}+\frac{3\omega^2+{{\mathbf{p}}}^2}{3(i\tau^{-1})^2}
-\frac{\omega(\omega^2+{{\mathbf{p}}}^2)}{(i\tau^{-1})^3}
\right].
\end{split}\end{aligned}$$ By comparing this expression with the hydro result of Eq.(\[eq:hydro-GR-T\]), we derive the following expression for the transport coefficients: $$\begin{aligned}
\sigma &= {C_{\text{em}}}{N_c}\chi \frac{2}{3}\tau
= {C_{\text{em}}}{N_c}\frac{T^2}{9}\tau,\\
\tau_J&=\tau, \\
\kappa_B&=0.\end{aligned}$$
Next, we obtain the spectral function for small $|{{\mathbf{p}}}|$. Expanding Eq.(\[eq:GTfull\]) in terms of $|{{\mathbf{p}}}|$, we derive $$\begin{aligned}
\begin{split}
G_T(p)
&\simeq -{C_{\text{em}}}{N_c}\chi \frac{2}{3}\frac{\omega}{\omega+i\tau^{-1}}
\left[1+\frac{1}{5}\frac{{{\mathbf{p}}}^2}{(\omega+i\tau^{-1})^2}\right],
\end{split}
\label{eq:GTexpanded}\end{aligned}$$ where we have used $$\begin{aligned}
\begin{split}
& A(p) \simeq \\
& \frac{1}{\omega+i\tau^{-1}}
\left[1+\frac{1}{3}\left(\frac{|{{\mathbf{p}}}|}{\omega+i\tau^{-1}}\right)^2
+\frac{1}{5}\left(\frac{|{{\mathbf{p}}}|}{\omega+i\tau^{-1}}\right)^4\right].
\end{split}\end{aligned}$$ The imaginary part of Eq.(\[eq:GTexpanded\]) reads $$\begin{aligned}
\label{eq:transport-T}
\begin{split}
\rho_T(p)
&\simeq {C_{\text{em}}}{N_c}\chi \frac{2}{3} \frac{\tau^{-1} \omega}{\omega^2+\tau^{-2}}
\left[1 +\frac{{{\mathbf{p}}}^2}{5} \frac{(3\omega^2-\tau^{-2})}{(\omega^2+\tau^{-2})^2}\right].
\end{split} \end{aligned}$$
Longitudinal channel {#longitudinal-channel}
--------------------
The longitudinal component of $G^R$ is given by $$\begin{aligned}
\begin{split}
G^R_{00}(p)
&= 2{C_{\text{em}}}{N_c}\chi\frac{1-(\omega+i\tau^{-1})A(p)}{1-i\tau^{-1}A(p)}.
\end{split}
\label{eq:G00full}\end{aligned}$$
In the hydro limit, $\omega,|{{\mathbf{p}}}|\ll\tau^{-1}$, this reduces to $$\begin{aligned}
\label{eq:G00-hydrolim}
\begin{split}
G^R_{00}(p)
&\simeq i{{\mathbf{p}}}^2 \frac{2{C_{\text{em}}}{N_c}\chi \tau}{3}
\frac{1}
{(\omega+i\tau{{\mathbf{p}}}^2/3)},
\end{split} \end{aligned}$$ where we have retained only the leading order terms. Comparing this expression with the result of hydrodynamics (\[eq:hydro-GR-L\]), we get $$\begin{aligned}
D&= \frac{\tau}{3}, \end{aligned}$$ and also confirm the Einstein relation, $$\begin{aligned}
\sigma
&= 2{C_{\text{em}}}{N_c}\chi D,\end{aligned}$$ where the factor 2${N_c}$ originates from the spin and color degrees of freedom of the quark.
Now, we can obtain the spectral function for small $|{{\mathbf{p}}}|$. Expanding Eq.(\[eq:G00full\]) in terms of $|{{\mathbf{p}}}|$, we derive $$\begin{aligned}
\begin{split}
G^R_{00}(p)
&\simeq -\frac{2}{3}{C_{\text{em}}}{N_c}\chi \frac{|{{\mathbf{p}}}|^2}{\omega+i\tau^{-1}}
\frac{1}{\omega} \\
&~~~\times \left[1+\left(\frac{3}{5}+\frac{i\tau^{-1}}{3\omega}\right)
\left(\frac{|{{\mathbf{p}}}|}{\omega+i\tau^{-1}}\right)^2
\right].
\end{split} \end{aligned}$$ The imaginary part of the above expression reads $$\begin{aligned}
\label{eq:transport-00}
\begin{split}
\rho^{00}(p)
&\simeq
\frac{2}{3}{C_{\text{em}}}{N_c}\chi \frac{{{\mathbf{p}}}^2}{\omega} \frac{\tau^{-1}}{\omega^2+\tau^{-2}} \\
&~~~\times \left[1
+{{\mathbf{p}}}^2 \frac{2}{5}\left(\tau^{-2}+\frac{11}{3}\omega^2\right)
\frac{1}{(\omega^2+\tau^{-2})^2}
\right].
\end{split} \end{aligned}$$
Evaluation of the continuum at weak coupling {#app:continuum}
============================================
In this appendix, we evaluate the continuum in the weak coupling and massless limit. In the free limit, the Green function of the EM current can be calculated by using Wick’s theorem as [@Altherr:1989jc] $$\begin{aligned}
\begin{split}
G^R_{\mu\nu}(x)
&= i\theta(t) C_{\text{em}}{N_c}{\mathrm{Tr}}\Bigl[\gamma_\mu S^>(x)\gamma_\nu S^<(-x) \\
&~~~-\gamma_\mu S^<(x)\gamma_\nu S^>(-x)
\Bigr],
\end{split}\end{aligned}$$ where $S^>(x)\equiv \langle \psi(x)\overline{\psi}(0) \rangle$ and $S^<(x)\equiv \langle \overline{\psi}(0) \psi(x)\rangle$. Here, we have omitted the flavor indices for simplicity. By performing the Fourier transformation and taking the imaginary part, the spectral function reads $$\begin{aligned}
\begin{split}
\rho_{\mu\nu}(p)
&= \frac{1}{2} C_{\text{em}}{N_c}\int \frac{d^4k}{(2\pi)^4}
{\mathrm{Tr}}\Bigl[\gamma_\mu S^>(k)\gamma_\nu S^<(k-p) \\
&~~~-\gamma_\mu S^<(k)\gamma_\nu S^>(k-p)
\Bigr]\\
&=2 C_{\text{em}}{N_c}\int \frac{d^4k}{(2\pi)^4}
{\mathrm{Tr}}\Bigl[\gamma_\mu {\ooalign{\hfil/\hfil\crcr$k$}}\gamma_\nu ({\ooalign{\hfil/\hfil\crcr$k$}}-{\ooalign{\hfil/\hfil\crcr$p$}}) \Bigr] \\
&~~~\times \rho^0(k) \rho^0(k-p)
[{n_F}(k^0-\omega)-{n_F}(k^0)],
\end{split}
\label{eq:rhomunu1}\end{aligned}$$ where we have used $S^>(k)= {\ooalign{\hfil/\hfil\crcr$k$}}2\rho^0(k)[1-{n_F}(k^0)]$ and $S^<(k)={\ooalign{\hfil/\hfil\crcr$k$}}2\rho^0(k){n_F}(k^0)$, and introduced the free quark spectral function as $\rho^0(k)\equiv \pi{\mathrm{sgn}}(k^0)\delta(k^2)$. The two delta functions can be written as $$\begin{aligned}
\label{eq:deltafunc-1}
\delta(k^2)&=\sum_{s=\pm 1}\frac{\delta(k^0-s|{{\mathbf{k}}}|)}{2|{{\mathbf{k}}}|},\\
\nonumber
\delta([k-p]^2)&= \frac{1}{2|{{\mathbf{k}}}||{{\mathbf{p}}}|}\delta\left[\cos\theta-\frac{2k^0\omega-p^2}{2|{{\mathbf{k}}}||{{\mathbf{p}}}|}\right] \\
\label{eq:deltafunc-2}
&~~~\times\theta\left[-p^2\left\{(k^0)^2-k^0\omega+\frac{p^2}{4}\right\}\right],\end{aligned}$$ where we adopted the standard polar coordinate, by assigning ${{\mathbf{p}}}$ to point into the $z$-direction. We also note that $\theta$ in the first line is the angle between ${{\mathbf{k}}}$ and ${{\mathbf{p}}}$ while that in the second line is a step function. Therefore, the expression of Eq.(\[eq:rhomunu1\]) becomes $$\begin{aligned}
\begin{split}
\rho_{\mu\nu}(p)
&= C_{\text{em}}{N_c}\frac{1}{8\pi^2|{{\mathbf{p}}}|}
\int^{2\pi}_0 d\phi \int^\infty_0 d|{{\mathbf{k}}}| \sum_{s=\pm 1} \\
&~~~\times \Bigl[k_\mu (k-p)_\nu+k_\nu (k-p)_\mu+g_{\mu\nu} k\cdot p \Bigr] \\
&~~~\times {\mathrm{sgn}}(s|{{\mathbf{k}}}|-\omega)
s \theta\left[-\left\{(k^0)^2-k^0\omega+\frac{p^2}{4}\right\}\right]\\
&~~~\times [{n_F}(s|{{\mathbf{k}}}|-\omega)-{n_F}(s|{{\mathbf{k}}}|)],
\end{split}
\label{eq:rhomunu2}\end{aligned}$$ where the values for $k^0$ and $\cos\theta$ are determined by the delta functions of Eqs. (\[eq:deltafunc-1\]) and (\[eq:deltafunc-2\]), and we have used $p^2>0$, which is justified because we consider the case that $\omega \sim T\gg |{{\mathbf{p}}}|$. From now on, we focus on the case for which $\omega>0$. Then, only the contribution with $s=+1$ remains, and the step function restricts the range of $|{{\mathbf{k}}}|$ to $k_-<|{{\mathbf{k}}}|<k_+$, where $k_\pm\equiv (\omega\pm|{{\mathbf{p}}}|)/2$. Thus, Eq.(\[eq:rhomunu2\]) becomes $$\begin{aligned}
\label{eq:rho-continuum}
\begin{split}
\rho_{\mu\nu}(p)
&= -C_{\text{em}}{N_c}\frac{1}{8\pi^2|{{\mathbf{p}}}|}
\int^{2\pi}_0 d\phi \int^{k_+}_{k_-} d|{{\mathbf{k}}}| \\
&~~~\times \Bigl[k_\mu (k-p)_\nu+k_\nu (k-p)_\mu+g_{\mu\nu} k\cdot p \Bigr] \\
&~~~\times [1-{n_F}(-|{{\mathbf{k}}}|+\omega)-{n_F}(|{{\mathbf{k}}}|)],
\end{split}\end{aligned}$$ where we have used $\omega-|{{\mathbf{k}}}|>(\omega-|{{\mathbf{p}}}|)/2>0$. From the distribution factor $[1-{n_F}(-|{{\mathbf{k}}}|+\omega)-{n_F}(|{{\mathbf{k}}}|)]=[1-{n_F}(-|{{\mathbf{k}}}|+\omega)][1-{n_F}(|{{\mathbf{k}}}|)]-{n_F}(-|{{\mathbf{k}}}|+\omega){n_F}(|{{\mathbf{k}}}|)$, we see that the physical processes corresponding to this expression are the quark antiquark pair-creation process and its inverse.
Transverse channel
------------------
$\delta\rho^T(p)$ can be evaluated by setting $\mu=\nu=1$ and subtracting the $T=0$ part from Eq. (\[eq:rho-continuum\]): $$\begin{aligned}
\begin{split}
\delta\rho_{T}(p)
&= C_{\text{em}}{N_c}\frac{1}{8\pi^2|{{\mathbf{p}}}|}
\int^{2\pi}_0 d\phi \int^{k_+}_{k_-} d|{{\mathbf{k}}}| \\
&~~~\times \Bigl[2{{\mathbf{k}}}^2\cos^2\phi(1-\cos^2\theta)-|{{\mathbf{k}}}|\omega+|{{\mathbf{k}}}||{{\mathbf{p}}}|\cos\theta \Bigr] \\
&~~~\times [{n_F}(-|{{\mathbf{k}}}|+\omega)+{n_F}(|{{\mathbf{k}}}|)]\\
&= -C_{\text{em}}{N_c}\frac{p^2}{4\pi|{{\mathbf{p}}}|^3}
\int^{|{{\mathbf{p}}}|/2}_{-|{{\mathbf{p}}}|/2} d|{{\mathbf{k}}}|
\left(\frac{{{\mathbf{p}}}^2}{4}+{{\mathbf{k}}}^2\right) \\
&~~~\times \left[{n_F}\left(\frac{\omega}{2}-|{{\mathbf{k}}}|\right)+{n_F}\left(\frac{\omega}{2}+|{{\mathbf{k}}}|\right)\right],
\end{split}\end{aligned}$$ where in the last line, we have used $\cos\theta=(2|{{\mathbf{k}}}|\omega-p^2)(2|{{\mathbf{k}}}||{{\mathbf{p}}}|)$ and have changed the integration variable as $|{{\mathbf{k}}}|\rightarrow |{{\mathbf{k}}}|-\omega/2$.
We can safely expand this in terms of $|{{\mathbf{k}}}|/\omega$, because $|{{\mathbf{k}}}|\simeq |{{\mathbf{p}}}|\ll \omega$, which leads to $$\begin{aligned}
\label{eq:continuum-T}
\begin{split}
& \delta\rho_{T}(p) \\
&\simeq - C_{\text{em}}{N_c}\frac{\omega^2}{\pi|{{\mathbf{p}}}|^3}\left(1-\frac{{{\mathbf{p}}}^2}{\omega^2}\right)
\int^{|{{\mathbf{p}}}|/2}_{0} d|{{\mathbf{k}}}|
\left(\frac{{{\mathbf{p}}}^2}{4}+{{\mathbf{k}}}^2\right) \\
&~~~\times \left[{n_F}\left(\frac{\omega}{2}\right)+\frac{{{\mathbf{k}}}^2}{2}{n_F}''\left(\frac{\omega}{2}\right)\right]\\
&\simeq - C_{\text{em}}{N_c}\frac{\omega^2}{6\pi}\left(1-\frac{{{\mathbf{p}}}^2}{\omega^2}\right)
\left[{n_F}\left(\frac{\omega}{2}\right)+\frac{{{\mathbf{p}}}^2}{20}{n_F}''\left(\frac{\omega}{2}\right)\right]\\
&\simeq -C_{\text{em}}{N_c}\frac{\omega^2}{6\pi}
\Biggl[{n_F}\left(\frac{\omega}{2}\right) \\
&~~~ +{{\mathbf{p}}}^2\left\{\frac{1}{20}{n_F}''\left(\frac{\omega}{2}\right)
-\frac{1}{\omega^2}{n_F}\left(\frac{\omega}{2}\right)\right\}\Biggr] .
\end{split}\end{aligned}$$
Longitudinal channel {#longitudinal-channel-1}
--------------------
By setting $\mu=\nu=0$ in Eq. (\[eq:rho-continuum\]), we get $$\begin{aligned}
\begin{split}
\delta\rho_{00}(p)
&= C_{\text{em}}{N_c}\frac{1}{2\pi|{{\mathbf{p}}}|}
\int^{|{{\mathbf{p}}}|/2}_{-|{{\mathbf{p}}}|/2} d|{{\mathbf{k}}}|
\left[ {{\mathbf{k}}}^2- \frac{{{\mathbf{p}}}^2}{4} \right] \\
&~~~\times \left[{n_F}\left(\frac{\omega}{2}-|{{\mathbf{k}}}|\right)+{n_F}\left(\frac{\omega}{2}+|{{\mathbf{k}}}|\right)\right],
\end{split}\end{aligned}$$ where we have changed the integration variable as before.
Expanding the integrand in terms of $|{{\mathbf{p}}}|/\omega$, we derive $$\begin{aligned}
\label{eq:continuum-00}
\begin{split}
\delta\rho_{00}(p)
&= C_{\text{em}}{N_c}\frac{2}{\pi|{{\mathbf{p}}}|}
\int^{|{{\mathbf{p}}}|/2}_{0} d|{{\mathbf{k}}}|
\left[ {{\mathbf{k}}}^2- \frac{{{\mathbf{p}}}^2}{4} \right] \\
&~~~\times \left[{n_F}\left(\frac{\omega}{2}\right)+\frac{{{\mathbf{k}}}^2}{2}{n_F}''\left(\frac{\omega}{2}\right)\right]\\
&= -C_{\text{em}}{N_c}\frac{{{\mathbf{p}}}^2}{6\pi}
\left[{n_F}\left(\frac{\omega}{2}\right)+\frac{{{\mathbf{p}}}^2}{40}{n_F}''\left(\frac{\omega}{2}\right)\right] .
\end{split} \end{aligned}$$
Evaluation of the UV tail at weak coupling {#app:UVtail}
==========================================
In this appendix, we briefly recapitulate the derivation of the UV tail in the EM current spectral function from the OPE [@CaronHuot:2009ns]. The UV behavior of the EM current retarded correlator is described by the OPE of Eqs.(\[eq:OPE-wo-RG-T\]) and (\[eq:OPE-wo-RG-L\]). Among the three terms in these expressions, only $\langle T^{00}_f\rangle$ is not RG invariant. This operator yields imaginary parts of the retarded correlator, as can be understood as follows: The scaling relation (\[eq:scaling-T’-Ttilde\]) can be rewritten as $$\begin{aligned}
\begin{split}
T'{}^{00}_{f}(\kappa)
&\simeq T'{}^{00}_{f}(\kappa_0)
+a'\ln\left(\frac{\kappa^2_0}{\kappa^2}\right)
\frac{b_0}{4\pi}\alpha_sT'{}^{00}_{f} ,\\
\tilde{T}^{00}(\kappa)
&\simeq \tilde{T}^{00}(\kappa_0)
+\tilde{a}\ln\left(\frac{\kappa^2_0}{\kappa^2}\right)
\frac{b_0}{4\pi}\alpha_s\tilde{T}^{00}_{f},
\end{split} \end{aligned}$$ when $\kappa$ is close to $\kappa_0$. It was shown in Ref.[@CaronHuot:2009ns] that the factor $\ln(\kappa^2_0/\kappa^2)$ generates an imaginary contribution $i\pi$, due to the analytic continuation to the real time. Following this prescription, the imaginary parts of the retarded correlators (\[eq:OPE-wo-RG-T\]) and (\[eq:OPE-wo-RG-L\]) read $$\begin{aligned}
\nonumber
\delta \rho_T(p)
&=e^2\sum q^2_f \frac{8}{9}\frac{\omega^2+{{\mathbf{p}}}^2}{(p^2)^2}
\alpha_s(\omega) \\
\label{eq:deltarho_T_UVtail}
&~~~\times \left( 2{C_F}\delta\left\langle T'{}^{00}_{f}(\omega) \right\rangle
+\frac{1}{{N_f}} \delta\left\langle \tilde{T}^{00}(\omega) \right\rangle \right) ,\\
\nonumber
\delta \rho_{00}(p)
&= e^2\sum q^2_f \frac{8}{9}\frac{{{\mathbf{p}}}^2}{(p^2)^2}
\alpha_s(\omega) \\
&~~~\times \left( 2{C_F}\delta\left\langle T'{}^{00}_{f}(\omega) \right\rangle
+\frac{1}{{N_f}} \delta\left\langle \tilde{T}^{00}(\omega) \right\rangle \right) .\end{aligned}$$ We note that this expression is valid when the OPE is reliable ($\omega\gg T,\Lambda_{\text {QCD}}$).
In the chiral and weak coupling limits, the operator expectation values at the renormalization scale $\kappa_0\sim T$ read $$\begin{aligned}
\label{eq:T00f-free}
\langle T^{00}_f\rangle&= {N_c}\frac{7\pi^2T^4}{60},\\
\label{eq:T00g-free}
\langle T^{00}_g\rangle&=2{C_F}{N_c}\frac{\pi^2T^4}{15},\end{aligned}$$ which, by using the scaling relation of Eq.(\[eq:scaling-T’-Ttilde\]), leads to $$\begin{aligned}
\nonumber
\delta \rho_T(p)
&={C_{\text{em}}}\frac{1}{\omega^2}
\left(1+3\frac{{{\mathbf{p}}}^2}{\omega^2}\right)
\alpha_s(\kappa_0) {N_c}{C_F}\frac{4\pi^2T^4}{27} \\
\label{eq:UVtail-T}
&~~~\times \left[\frac{\ln\left(\kappa_0/\Lambda_{\text{QCD}} \right)}{\ln\left(\omega/\Lambda_{\text{QCD}} \right)}\right]^{\tilde{a}+1} ,\\
\nonumber
\delta \rho_{00}(p)
&={C_{\text{em}}}\frac{{{\mathbf{p}}}^2}{\omega^4}
\left(1+2\frac{{{\mathbf{p}}}^2}{\omega^2}\right)
\alpha_s(\kappa_0) {N_c}{C_F}\frac{4\pi^2T^4}{27}\\
\label{eq:UVtail-00}
&~~~\times \left[\frac{\ln\left(\kappa_0/\Lambda_{\text{QCD}} \right)}{\ln\left(\omega/\Lambda_{\text{QCD}} \right)}\right]^{\tilde{a}+1}. \end{aligned}$$ Here, we have retained terms up to next-to-leading order in the small $|{{\mathbf{p}}}|$ expansion.
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[^1]: This contour actually runs slightly above the real axis, so that it does not overlap with the singularities such as the continuum or the diffusion pole at zero momentum, which appear on it.
[^2]: We note that our convention for $\rho_{\mu\nu}$ differs from the popular $\rho_{\mu\nu}=2{\text{Im}}G^R_{\mu\nu}$ by a factor of 2.
[^3]: We could also choose $\kappa^2=p^2$, which however would not change the results of this paper.
|
---
abstract: 'We study numerically the dependence of the critical magnetic Reynolds number $\Rmc$ for the turbulent small-scale dynamo on the hydrodynamic Reynolds number $\Re$. The turbulence is statistically homogeneous, isotropic, and mirror–symmetric. We are interested in the regime of low magnetic Prandtl number $\Pm=\Rm/\Re<1$, which is relevant for stellar convective zones, protostellar disks, and laboratory liquid-metal experiments. The two asymptotic possibilities are $\Rmc\to\const$ as $\Re\to\infty$ (a small-scale dynamo exists at low $\Pm$) or $\Rmc/\Re=\Pmc\to\const$ as $\Re\to\infty$ (no small-scale dynamo exists at low $\Pm$). Results obtained in two independent sets of simulations of MHD turbulence using grid and spectral codes are brought together and found to be in quantitative agreement. We find that at currently accessible resolutions, $\Rmc$ grows with $\Re$ with no sign of approaching a constant limit. We reach the maximum values of $\Rmc\sim500$ for $\Re\sim3000$. By comparing simulations with Laplacian viscosity, fourth-, sixth-, and eighth-order hyperviscosity and Smagorinsky large-eddy viscosity, we find that $\Rmc$ is not sensitive to the particular form of the viscous cutoff. This work represents a significant extension of the studies previously published by Schekochihin et al. (2004a) and Haugen et al. (2004a) and the first detailed scan of the numerically accessible part of the stability curve $\Rmc(\Re)$.'
author:
- |
A. A. Schekochihin, N. E. L. Haugen, A. Brandenburg, S. C. Cowley, J. L. Maron,\
and J. C. McWilliams
title: 'The Onset of a Small-Scale Turbulent Dynamo at Low Magnetic Prandtl Numbers'
---
{#section .unnumbered}
-0.55cm
The magnetic Prandtl number $\Pm$, which is the ratio of the kinematic viscosity to the magnetic diffusivity, is a key parameter of MHD turbulence. In fully ionized plasmas, $\Pm \approx 2.6\times10^{-5} T^4/n$, where $T$ is the temperature in kelvins and $n$ the ion number density in units of cm$^{-3}$. In hot rarefied plasmas, such as the warm and hot phases of the interstellar medium or the intracluster medium, $\Pm\gg1$. In contrast, in the Sun’s convective zone, $\Pm\sim10^{-7}$ to $10^{-4}$, in planets, $\Pm\sim10^{-5}$, and in protostellar disks, while estimates vary, it is also believed that $\Pm\ll1$ [e.g., @Brandenburg_Subramanian]. All these astrophysical bodies have disordered fluctuating small-scale magnetic fields and, in some cases, also large-scale “mean” fields. As they also have large Reynolds numbers and large-scale sources of energy, they are expected to be in a turbulent state. It is then natural to ask if their magnetic fields are a product of turbulent dynamo.
To be precise, there are two types of dynamo. The large-scale, or [*mean-field dynamo*]{} generates magnetic fields at scales larger than the energy-containing scale of the turbulence, as is, for example, the case in helical turbulence. [*The small-scale dynamo*]{} amplifies magnetic fluctuation energy below the energy-containing scale of the turbulence. The small-scale dynamo is due to random stretching of the magnetic field by turbulent motions and does not depend on the presence of helicity. Mean-field dynamos typically predict field growth on time scales associated with the energy-containing scale (or longer), while the small-scale dynamo amplifies magnetic energy at the turbulent rate of stretching. Thus, the small-scale dynamo is usually a much faster process than the mean-field dynamo, and the large-scale field produced by the latter can be treated as approximately constant on the time scale of the small-scale dynamo. The mean-field dynamo (or, more generally, a large-scale magnetic field of any origin) also gives rise to small-scale magnetic fluctuations because of the turbulent shredding of the mean field: this leads to algebraic-in-time growth of the small-scale magnetic energy — again, a slower generation process than the exponential-in-time small-scale dynamo.
In the systems with $\Pm\gg1$, the existence of the small-scale dynamo is well established numerically and has a solid theoretical basis (see @SCTMM_stokes for an account of the relevant theoretical and numerical results and for a long list of references). The situation is much less well understood for the case of small $\Pm$. It has been largely assumed that a small-scale dynamo should be operative in this regime as well. For example, the presence of large amounts of small-scale magnetic flux in the solar photosphere [e.g., @Title_review] has been attributed to small-scale dynamo action. This appeared to be confirmed by numerical simulations of the MHD turbulence in the convective zone [@Cattaneo_solar; @Cattaneo_Emonet_Weiss; @Nordlund_book]. However, such simulations are usually done at $\Pm\ge1$ ($\Pm=5$ in Cattaneo’s simulations). Previous attempts to simulate MHD turbulence in various contexts with $\Pm<1$ found achieving dynamo in this regime much more difficult than for $\Pm\ge1$ [@Nordlund_etal; @Brandenburg_etal_structures; @Nore_etal; @Christensen_Olson_Glatzmaier; @MCM]. A systematic numerical investigation of the effect of $\Pm$ on the efficiency of the small-scale dynamo was carried out by @SCMM_lowPr, who found that the critical magnetic Reynolds number $\Rmc$ required for the small-scale dynamo to work increases sharply at $\Pm<1$. An independent numerical study by @HBD_pre confirmed this result.
What are the basic physical considerations that should guide us in interpreting this result? First of all, let us stress that all working numerical small-scale dynamos are of the large-$\Pm$ kind (the case of $\Pm=1$ is nonasymptotic, but its properties that emerge in numerical simulations suggest that it belongs to the same class). Two essential features of the large-$\Pm$ dynamos are (1) the scale of the velocity field is larger than the scale of the magnetic field, and (2) the velocity field that drives the dynamo is spatially smooth and locally looks like a random linear shear, so the dynamo is due to exponential-in-time separation of Lagrangian trajectories and the consequent exponential stretching of the magnetic field. The basic physical picture of such dynamos (@Zeldovich_etal_linear; see discussion in @SCTMM_stokes; see also a review of an alternative but complementary approach by @Ott_review) explicitly requires these two conditions to hold. The map dynamos and the dynamos in deterministic chaotic flows that were extensively studied in the 1980s–1990s [see review by @STF] are all of this kind. The numerical dynamos with $\Pm\ge1$ (the first due to @Meneguzzi_Frisch_Pouquet) are of this kind as well because they are driven by the spatially smooth viscous-scale turbulent eddies, which have the largest turnover rate.
When $\Pm\ll1$ with both $\Rm\gg1$ and $\Re\gg1$, the characteristic scale $\lB$ of the magnetic field lies in the inertial range. For Kolmogorov turbulence, a simple estimate gives $\lB\sim\Rm^{-3/4}\lf$, where $\lf$ is the energy-containing scale. As the viscous scale is $\ld\sim\Re^{-3/4}\lf$, we have $\lf\ll\lB\ll\ld$. In a rough way, one can think of the turbulent eddies at scales $l>\lB$ as stretching the field at the rate $u_l/l$ and of the eddies at scales $l<\lB$ as diffusing the field with the turbulent diffusivity $u_l l$. In Kolmogorov turbulence, $u_l\sim l^{1/3}$, so both the dominant stretching and the dominant diffusion are due to the eddies at scale $l\sim\lB$. The resulting rates of stretching and of turbulent diffusion are of the same order, so the outcome of their competition cannot be determined on this qualitative level [@Kraichnan_Nagarajan]. An important conclusion, however, can be drawn. If the bulk of the magnetic energy is at the scale $\lB$, the existence of the dynamo is entirely decided by the action of the velocities at the scale $\lB$. Then it cannot matter where in the inertial range $\lB$ lies. But $\lB/\ld\sim\Pm^{-3/4}$, so the value of $\Pm$ does not matter as long as it is asymptotically small. Therefore, there are two possibilities: either there is a dynamo at low $\Pm$ and $\Rmc\to\const$ as $\Re\to\infty$ or there is not and there exists a finite $\Pmc=\Rmc/\Re\to\const$ as $\Re\to\infty$. Strictly speaking, the third possibility is that $\Rmc\propto\Re^\alpha$, where $\alpha$ is some fractional power, but this can only happen if the intermittency of the velocity field (non–self-similarity of the inertial range) is important for the existence of the dynamo.[^1]
The two possibilities identified above are illustrated in . Several authors [@Vainshtein_lowPr; @Rogachevskii_Kleeorin; @Boldyrev_Cattaneo] showed that, given certain reasonable assumptions, the first possibility ($\Rmc\to\const$) is favored by the @Kazantsev model: the small-scale dynamo in a Gaussian white-in-time velocity field. In particular, @Boldyrev_Cattaneo found that the Kazantsev model gives $\Rmc$ that is roughly 10 times larger in the $\Pm\ll1$ regime than in the $\Pm\gg1$ regime (@Rogachevskii_Kleeorin predict $\Rmc\sim400$, which is consistent with that). This prompted them to declare the issue settled on the grounds that the failure of the dynamo in numerical experiments at current limited resolutions is compatible with such an increase in $\Rmc$. However, the $\Pm\ll1$ dynamo in the Kazantsev model is a quantitative mathematical consequence of the model, and it is not known if and how it is affected by such drastic and certainly unrealistic assumptions as the Gaussian white-noise statistics for the velocity field.[^2] The existence of a dynamo in real turbulence is also a quantitative question (see discussion above), so it cannot be decided by a model that is not a quantitative approximation of turbulence.
Thus, the issue cannot be considered settled until definitive numerical evidence is produced. This is an especially hard task because we do not know just how high a magnetic Reynolds number we must achieve in order to clearly see the distinction between $\Rmc\to\const$ and $\Rmc/\Re\to\const$. In this Letter, we have collected numerical results from two independent computational efforts: simulations using an incompressible spectral MHD code [see code description in @Maron_Goldreich; @MCM] and weakly compressible simulations using a grid-based high-order MHD code (the [Pencil Code]{}[^3]).
The equations we solved numerically (in a triply periodic cube) are \[NSEq\] \_t+ &=& -[p]{} + [()4]{} + + ,\
\_t&=& () + \^2, \[eq\_B\] where $\vu$ is the velocity and $\vB$ is the magnetic field (the [Pencil Code]{}, in fact, solves the evolution equation for the vector potential ${\bf A}$ and then computes $\vB=\vdel\times{\bf A}$). All runs reported below are in the kinematic regime, $|\vB|\ll|\vu|$, so the Lorentz force in plays no role. Turbulence is driven by a random white-in-time nonhelical body force $\vf$ concentrated at $k=\kf$, where $\kf$ is the wavenumber associated with the box size. The (hyper)viscous force is = [1]{}, \[def\_F\] where $\nu_n$ is the fluid viscosity and S\_[ij]{} = [12]{}([u\_ix\_j]{} + [u\_jx\_i]{}) - [13]{}\_[ij]{}. In the spectral simulations, the density $\rho=1$, and the incompressibility constraint $\vdel\cdot\vu=0$ is enforced exactly via the determination of the pressure $p$. The grid simulations are isothermal: $p=\cs^2\rho$ with sound speed $\cs=1$, and the density satisfies \_t+ () = 0. We stay in the weakly compressible regime of low Mach numbers $M=\usq^{1/2}/\cs\sim 10^{-1}$ and $\rho\simeq\<\rho\>=1$ (angular brackets denote volume averages). Some numerical results on the onset of dynamo action at larger Mach numbers are given in @Haugen_Brandenburg_Mee.
The dissipation in the induction equation is always Laplacian with magnetic diffusivity $\eta$ (we choose not to tamper with magnetic dissipation because we are interested in the sensitive question of field growth or decay). With regard to the viscous dissipation, we perform three kinds of simulations:
1. Simulations with Laplacian viscosity: $n=1$ in .
2. Simulations with fourth-, sixth-, and eighth-order hyperviscosities: $n=2$, $3$, and $4$, respectively, in .
3. Large-eddy simulations (LES) with the Smagorinsky effective viscosity [e.g., @Pope_book]: in , $n=1$, and $\nu_1$ is replaced by $\nu_{\rm S} = \bl(C_{\rm S}\Delta\br)^2\bl(2\vS:\vS\br)^{1/2}$, where $\Delta$ is the mesh size and $C_{\rm S}=0.2$ is an empirical coefficient.
The magnetic Reynolds number is defined by $\Rm=\usq^{1/2}/\kf\eta$, where $\kf$ is the box wavenumber (the smallest wavenumber in the problem). For the runs with Laplacian viscosity ($n=1$), the hydrodynamic Reynolds number is $\Re=\usq^{1/2}/\kf\nu_1$. For hyperviscous runs and for LES, we define $\Re$ by replacing $\nu_1$ with the effective viscosity: \[nueff\_def\] =/2: = / (the second expression is for the spectral simulations, where $\vdel\cdot\vu=0$ exactly). Here $\epsilon=\<\vf\cdot\vu\>$ is the total injected power and is equal to the total energy dissipation. As the forcing $\vf$ is white in time, $\epsilon=\const$: indeed, given $\<f^i(t,\vx)f^j(t',\vx')\> = \delta(t-t')\,\epsilon^{ij}(\vx-\vx')$, it is easy to show that $\epsilon=\frac{1}{2}\,\epsilon^{ii}(0)$. The results of all our simulations are presented in , where $\Rmc$ is plotted versus $\Re$. Each value of $\Rmc$ was computed by interpolating between least-squares–fitted growth/decay rates of a growing and a decaying run. Error bars are based on $\Rm$ and $\Re$ for these pairs of runs. The only exception is the point enclosed in a circle, which corresponds to $(\Rm,\Re)$ for a run that appeared to be marginal (in this case we could not afford the resolution necessary to achieve a decaying case). The run times in all cases were long enough for the least-squares–fitted growth rates to stop changing appreciably (typically this required about $20$ box-crossing times, but cases close to marginal needed longer running times).
We see that there is good agreement between the results for runs with different forms of viscous dissipation; this confirms the natural assumption that the field-generation properties of the turbulence at low $\Pm$ are not sensitive to the way the velocity spectrum is cut off. It is also encouraging that results from two very different codes are in quantitative agreement.
Our previous studies [@SCMM_lowPr; @HBD_pre] had the maximum value of $\Rmc\sim200$. The results reported here raise it to $\sim500$, with the corresponding values of $\Pmc$ around 0.15. While a roughly 10-fold increase with respect to $\Rmc$ for the $\Pm=1$ dynamo has now been achieved, there is thus far no sign of $\Rmc$ reaching an asymptotically constant value. This said, the current resolutions are clearly still insufficient to make a definitive judgement, although we are now very close to values of $\Rmc$ predicted by the theories based on the Kazantsev model [@Rogachevskii_Kleeorin; @Boldyrev_Cattaneo] — whether or not the model yields quantitatively correct predictions should become clear in the near future.
The numerical results reported above concerned the dependence $\Rmc(\Re)$ for the turbulent small-scale dynamo, i.e., the ability of turbulent velocity fluctuations to amplify magnetic energy at scales smaller than the energy-containing scale of the turbulence. The $\Rmc(\Re)$ dependence is also an interesting issue for other kinds of dynamo.
If the velocity field is non–mirror-symmetric, it can often drive the mean-field dynamo (MFD), which means the growth of the magnetic field at scales larger than the energy-containing scale of the turbulence [@Krause_Raedler]. This large-scale field generated by the MFD, just like a mean field imposed externally, can induce small-scale magnetic fluctuations as it is shredded by the turbulence, so the total field has both a mean (large-scale) and a fluctuating component. In many cases, the breaking of the mirror symmetry leads to a non-zero value of the net helicity $\<\vu\cdot(\vdel\times\vu)\>\neq0$ (the average is over all scales smaller than the energy-containing scale of the turbulence, [*not*]{} over the entire volume of the system). The mean-field generation is then referred to as the $\alpha$-effect. The stability curve $\Rmc(\Re)$ for the $\alpha$-effect is different than for the small-scale dynamo: it is essentially a condition for at least one unstable large-scale mode to fit into the system. In a numerical study done with the same code as the grid simulations reported above but with fully helical random forcing, @Brandenburg_alpha found much lower values of $\Rmc$ than for the small-scale dynamo and very little dependence of $\Rmc$ on $\Pm$ for $\Pm\ge0.1$.
A non-zero net helicity is not a necessary condition for the MFD [e.g., @Gilbert_Frisch_Pouquet]. In fact, it has been suggested recently by @Rogachevskii_Kleeorin_sc that the MFD can be driven simply by the presence of a constant mean velocity shear (shear-current or $\delta$-effect) — a very generic possibility of obvious relevance to systems with mean flows. Mean flows are present in many astrophysical cases and in all current laboratory dynamo experiments [@Gailitis_etal; @Mueller_Stieglitz_Horanyi; @Bourgoin_etal; @Lathrop_Shew_Sisan; @Forest_etal]. A mean flow can be a dynamo in its own right: an MFD (field growth at scales above the flow scale) and, if the flow has chaotic trajectories in three dimensions, also a small-scale dynamo (field growth at scales $\sim\Rm^{-1/2}$ times the scale of the flow; see @STF — as noted above, small-scale dynamos in deterministic chaotic flows are equivalent to the large-$\Pm$ case). When $\Re$ is large, the energy of the turbulent velocity fluctuations is comparable to the energy of the mean flow. The critical $\Rm$ required for field growth will have some dependence on $\Re$, which reflects the effect of the turbulence on the structure of the mean flow and/or on the effective value of the magnetic diffusivity [the $\beta$-effect; see @Krause_Raedler]. This dependence was the subject of two recent numerical studies: of the dynamo in a turbulence with a constant Taylor-Green forcing by @Ponty_etal, and of the Madison dynamo experiment (propeller driving in a spherical domain) by @Bayliss_Forest. The $\Re$ dependence of $\Rmc$ that emerges from such simulations is distinct from that for a pure small-scale dynamo. Indeed, Y. Ponty et al. (2005, private communication) have shown that, in the limit of large $\Re$, the value of $\Rmc$ in their simulations tends to a constant that coincides with $\Rmc$ calculated for the mean flow alone, i.e., for the velocity field with fluctuations removed by time averaging. In contrast, the subject of the present Letter has been the possibility of a small-scale dynamo driven solely by turbulent fluctuations, in the absence of a mean flow. The importance of this possibility or lack thereof is that such a dynamo, if it exists, occurs at the turbulent stretching rate associated with the resistive scale. This is much faster (by a factor of $\sim\Rm^{1/2}$; see discussion above) than the growth rate of any MFD or of a small-scale dynamo associated with the mean flow, i.e., than the stretching rate at the energy-containing scale or at the scale of the mean flow.
It is a pleasure to acknowledge discussions with participants of the program “Magnetohydrodynamics of Stellar Interiors” at the Isaac Newton Institute, Cambridge (UK), especially S. Fauve, N. Kleeorin, Y. Ponty, M. Proctor, and I. Rogachevskii. We also thank C. Forest and J.-F. Pinton for discussions of both laboratory and numerical dynamos. Simulations were done at the UKAFF (Leicester), NCSA (Illinois), Norwegian High Performance Computing Consortium (Trondheim and Bergen), and the Danish Center for Scientific Computing. This work was supported in part by NSF grant AST 00-98670 and by the US DOE Center for Mutiscale Plasma Dynamics. A.A.S. was supported by the UKAFF Fellowship. N.E.L.H. was supported in part by the David Crighton Visiting Fellowship (DAMTP, Cambridge).
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[^1]: The role of coherent structures can be prominent in quasi–two-dimensional dynamos (three-component velocity field depending on two spatial variables), where the inverse cascade characteristic of the two-dimensional turbulence gives rise to persistent large-scale vortices, which drive the dynamo [@Smith_Tobias].
[^2]: @Vainshtein_Kichatinov argue that the equations that arise from the Kazantsev model are valid for non-white velocity fields if $n$-point joint probability density functions of Lagrangian displacements satisfy Fokker-Planck equations with some diffusion tensor. They further assume (on dimensional grounds) that this diffusion tensor scales as the scale-dependent turbulent diffusion $\sim u_l l$. This is, in fact, a closure scheme that we believe to be equivalent to Kazantsev’s zero-correlation-time theory.
[^3]: See http://www.nordita.dk/software/pencil-code.
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abstract: 'The exact quantization of two models, the massive vector meson model and the Higgs model in the London limit, both describing massive photons, is presented. Even though naive arguments (based on gauge-fixing) may indicate the equivalence of these models, it is shown here that this is not true in general when we consider these theories on manifolds with boundaries. We show, in particular, that they are equivalent only for a special choice of the boundary conditions that we are allowed to impose on the fields.'
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### \#1
SU-4240-575\
August 1994
Inequivalence of the Massive Vector Meson and
Higgs Models on a Manifold with Boundary
L. Chandar and E. Ercolessi[^1]
*Department of Physics, Syracuse University,*
*Syracuse, NY 13244-1130*
\[se:1\]
It is known that gauge theories involve only massless gauge bosons unless either the gauge symmetry is spontaneously broken [@Broke] or topological terms [@CS; @MCS] are added to the action. Such models are not merely of theoretical interest. They have important applications in particle physics and in condensed matter physics. In the former they arise, for example, in models that incorporate the electroweak interactions [@Broke]. Applications in condensed matter physics arise in situations such as superconductivity [@Super] where it is known that photons acquire a mass in the superconducting regime. Such models also arise in effective theories describing the long wavelength physics [@Zee] of 2+1 dimensional systems.
It is usually believed that, under certain approximations, such theories involving massive gauge bosons are equivalent to the massive vector meson model. By the massive vector meson model is here meant the action whose gauge symmetry has been [*explicitly*]{} broken by the addition of a quadratic mass term [@Massive]. The arguments that are used [@Broke] to show the equivalence of these models depend quite crucially on gauge-fixing in some form or the other. On the other hand, we know that gauge-fixing arguments for manifolds with boundaries are always suspect because of the following two reasons.
Firstly, the Gauss law for manifolds with boundaries has to be defined by smearing it with appropriate test functions so as to ensure that they generate canonical transformations [@Gauge] on the phase space. Such a requirement restricts the allowed gauge transformations [@Gauge] which, by definition, are the canonical transformations generated by Gauss law. Since gauge-fixing arguments do not usually pay attention to this feature, they are not to be believed without further justifications. The second reason is a subtle one and is related to the fact that on a manifold with boundary, the Hamiltonian is self-adjoint and bounded from below only if the fields and their momenta satisfy suitable boundary conditions (BC’s). Most of the gauge-fixing arguments either pick a special choice of BC’s or do not talk about it at all.
Because of the above reasons, it is clear that the equivalence of the two models is proven only on manifolds without boundaries (like [**R$^{n}$**]{}). For manifolds with boundaries, a more careful analysis is warranted to display the similarities/differences between the massive vector meson model and a gauge theory with massive gauge bosons.
In Section 2, we look at the exact treatment (following Dirac) of the massive vector meson model. It is shown that the quantization here depends on a one-parameter family of BC’s. We specialize to a particular BC for which we are able to carry out the quantization completely. We are not able to do this exact quantization for the most general BC’s. In Section 3, we consider the Higgs model in the London limit (the modulus of the Higgs field is frozen to its vacuum expectation value). Here the exact treatment leads to a quadratic Hamiltonian along with a Gauss law. In this case, quantization depends on a two-parameter family of BC’s and unlike the earlier case, we are able to carry through the exact quantization for the most general BC’s. In Section 4, we compare the quantizations carried out in Sections 2 and 3. It turns out that there is a natural identification of the BC’s of sections 2 and 3 provided one of the two parameters of section 3 is set to zero. We show that the Hamiltonian of the massive vector meson and Higgs models are different in general. In this Section, we also briefly compare these two theories with one other model describing a massive photon, namely the Maxwell-Chern-Simons (MCS) [@CS; @MCS] theory.
\[se:2\] In this section we will consider the usual Maxwell action augmented by a mass term for the vector potential $A_\m$ on a space time manifold $D
\times {\bf R}^1$, the two dimensional disc $D$ representing the spatial manifold and ${\bf R}^1$ denoting time: S = \_[D \^1]{}d\^[3]{}x { - - \^2 } . \[ml\] Here $F_{\m\n} = \pa_\m A_\n - \pa_\n A_\m$ is the electromagnetic tensor whose components are the electric field $E_i = \pa_0 A_i -
\pa_i A_0$ and the magnetic field $B = \pa_1 A_2 - \pa_2 A_1 =
\frac{1}{2} \e_{ij} F_{ij}$ .
It is well known [@Massive] that this model describes a constrained system, with the second class constraints given by ¶\_0 0 \^2 A\_0 + \_i \_i 0 , \[con\] $\P_\m = (\P_0,\Pi _i)$ being the momenta conjugate to $A_\m$. $\Pi _{i}$ here is related to $E_{i}$ by $\Pi _{i} = \frac{1}{e^{2}}E_{i}$. Of course, the above constraints (\[con\]) have to be smeared with appropriate test functions so that they generate well-defined canonical transformations [@Gauge]. However, since these constraints are second class, they can be imposed strongly. Following Dirac’s procedure [@Dirac] for the system described by (\[ml\]), the Hamiltonian (up to irrelevant surface terms that depend on the test functions used to smear the above constraints) we end up with is H = \_[D]{} d\^[2]{}x{ e\^2 \_[i]{} \^[2]{} + (\_i \_[i]{})\^[2]{} + B\^2 + \^2 \^2 } , \[mh\] where the variables $A_i$ and $\Pi _i$ satisfy the usual canonical commutation relations. For the following discussion it is convenient to rewrite the Hamiltonian (\[mh\]) using the notation of differential forms . After integrating by parts the second and third terms in (\[mh\]) and neglecting the surface integrals, we get: H = - \_[D]{} { \*(e\^2 + d\*d\*) - A \*(\^2 + \*d\*d) A } . \[fh\]
We can rewrite this expression in a very compact form if on the vector space of forms $\a^{(p)}$ of degree $p$ we introduce the scalar product $<\a^{(p)},\b^{(p)}> \, := (-1)^p \int \bar{\a^{(p)}} \, *\b^{(p)}$, where the bar denotes complex conjugation. Now (\[fh\]) becomes H = <,(e\^2 + d\*d\*) > + <A,(\^2 + \*d\*d) A> . \[sh\]
In order to quantize this Hamiltonian, we need to expand the fields $A$ and $\Pi$ in a complete basis of the Hilbert space of one-forms. Since $H$ is constructed from the differential operators $d*d*$ and $*d*d$, we would like to expand the fields in a basis of eigenfunctions of such operators [@MCS]. In other words, we need to find a domain of self-adjointness for these operators. Let us first find a domain of self-adjointness [@Domain] for the operator $*d*d$. From the relation: 0 = <,\*d\*d > - <\*d\*d , > = \_[D]{} { |[\*d]{} - | \*d } ,\[sa\] where $\a,\b$ are any two one-forms, it is easy to see that the operator $*d*d$ is self-adjoint on the domain \_ł= { : \*d |\_[r=R]{} = - ł\_ |\_[r=R]{} } ł\^1 . \[do\] To go from (\[sa\]) to (\[do\]) we have required that the fields satisfy local rotationally invariant BC’s. \[By local BC’s, we mean BC’s which mix fields and their derivatives only at the same point.\] In (\[do\]), $r$ and $\q$ are polar coordinates on the disc $D$ with $r=R$ giving its boundary and $A_\q = A_i \frac{\pa x^i}{\pa \q} (r,\q)$.
$\l$ here can be any real number. But the Hamiltonian (\[sh\]) is bounded from below only if $\l \geq 0$. This can be seen by noticing that = <d ,d > - \_[D]{} | (\*d ) = <d ,d > + ł\_[D]{} |\_|\^2 R d . \[po\] Since $<d \a ,d \a > \geq 0$, $<\a ,*d*d \a >$ is nonnegative iff $\l \geq 0$. Therefore, from now on, we will only consider the domains $\cd_\l $ with $\l \geq 0$. Let us now turn to the problem of solving the eigenvalue equation d\*d A = ø\^2 A \[ep\] for the one-form $A$ satisfying the BC’s d A |\_[r=R]{} = - łA\_ |\_[r=R]{} , ł0 . \[bc\] This problem has already been examined and solved in [@MCS]. Here, we will not repeat the calculations and will only list the eigenmodes of $*d*d$, together with the corresponding eigenvalues $\o ^2$.
The solutions for $\o ^2 \neq 0$ are of the form: .
[l]{} \^[(1)]{} = \_[nm]{}\^[(1)]{} \*d\[e\^[in]{} J\_n (\^[(1)]{} r)\] \
\_[-nm]{}\^[(1)]{} = |[\^[(1)]{}]{}
} n 0 , m > 0 , \[m1\] where $J_n (x)$ is the real Bessel function of order $n$[^2] and the eigenvalues $\o = \o_{nm}^{(1)}$ are fixed by the BC’s (\[bc\]) which now read: \^[(1)]{} J\_n (\^[(1)]{} R) = ł\_[r=R]{} . \[bcm1\]
There is also a set of zero modes, solutions of (\[ep\]) with $\o =0$, given by: .
[l]{} = \_[nm]{}\^[(0)]{} d\[e\^[in]{} J\_n (\^[(0)]{} r)\] \
\_[-nm]{}\^[(0)]{} = |
} n 0 , m > 0 , \[m0\] the Bessel functions now satisfying the condition $J_n (\wn^{(0)}R) = 0$.
The functions (\[m1\]) and (\[m0\]) form a complete set of solutions for (\[ep\]),(\[bc\]) if $\l >0$. On the contrary, if $\l
=0$ there is another set of zero modes, given by the so-called harmonic forms $$\begin{aligned}
\hn = \cn_n^{(h)}\; d z^n \;\;\; ,\;\;\; \bar{\hn} = \cn_n^{(h)} \; d \bar{z}^n
&& n >0 \label{hm}\end{aligned}$$ where $z= x_1 + i x_2 = r e^{i\q}$ is the complex coordinate on $D$.
It is important to notice that the harmonic functions are eigenfunctions of the\
operator \*, $*h_n = i h_n$, and that, in addition, if $\l =0$ there exists a relationship between the nonzero modes (\[m1\]) and the zero modes (\[m0\]) $\pnm^{(1)} = *\ponm \; \mbox{ with }\; \o_{nm}^{(1)} = \o_{nm}
^{(0)}$. These two relations imply that, for $\l =0$, the complete set of eigenfunctions $(\Psi_{nm}^{(1)},\Psi_{nm}^{0},h_n )$ of the operator $*d*d$ is also a complete set of eigenfunctions of the operator $d*d*$. Indeed, to diagonalise $H$, we can expand the fields $A$ and $\Pi$ in (\[sh\]) as A &=& \^[(1)]{} \^[(1)]{} + \^[(0)]{} + q\^[(h)]{}\_n + c.c. ,\
&=& \^[(1)]{} \^[(1)]{} + \^[(0)]{} + p\^[(h)]{}\_n + c.c. ,\[exp\] where (as in the following) repeated indices are summed over. The Hamiltonian is then H &= & { + + .\
& + & . } , \[dh\] where the only non zero commutation relations satisfied by the operators $q^{(j)}$’s and $p^{(j)}$’s ($j=0,1,h$) are $
\left[ \qn^{(1)},p_{n'm'}^{(1)\dagger} \right] = i \d_{nn'} \d_{mm} = \left[
\qon,p_{n'm'\dagger}^{(0)} \right] \;\; , \;\;
\left[ q^{(h)}_n ,p^{(h)\dagger}_{n'} \right] = i \d_{nn'}$.
Here $\omega_{nm}=\omega _{nm}^{(0)}=\omega _{nm}^{(1)}$ while the commutation relations follow from those of the variables $A_{i},\Pi _{i}$. Let us define the annihilation-creation operators .
[l]{} a\_[nm]{}\^[(j)]{} = \
a\_[nm]{}\^[(j)]{} =
} j=0,1 \[ca0\]
[l]{} a\_[n]{}\^[(h)]{} =\
a\_[n]{}\^[(h)]{} =
\[ca\] , with commutators $
\left[ a_{nm}^{(j)} , a_{n'm'}^{(j)\dagger} \right] =
i \d_{nn'} \d_{mm'} \;\; (j=0,1)\;\; , \;\;
\left[ a_{n}^{(h)} , a_{n'}^{(h)\dagger} \right] =
i \d_{nn'}$, where Ø\_[nm]{}\^[(1)]{}=Ø\_[nm]{}\^[(0)]{}= , Ø\_[n]{}\^[(h)]{} = e . \[om\]
Then (\[dh\]) becomes H = Ø\_[nm]{}\^[(1)]{} \_[nm]{}\^[(1)]{} \_[nm]{}\^[(1)]{} + Ø\_[nm]{}\^[(0)]{} \_[nm]{}\^[(0)]{} \_[nm]{}\^[(0)]{} + Ø\_[n]{}\^[(h)]{} a\_n\^ a\_n . \[hd\] The spectrum of $H$ can be read off from (\[hd\]).
We remark here that the lowest energy modes are the ones corresponding to the harmonic functions and are all degenerate, having an energy $\O_{n}^{(h)}$ that depends only on the mass $\ma$ of the vector potential, for all $n$.
We conclude this section by looking briefly at the $\l >0$ case. Now it is no longer true that the operator $d*d*$ is diagonal in the basis $(\Psi_{nm},\Psi_{nm}^{(0)})$ of eigenfunctions of $*d*d$. If we continue to use a field expansion similar to (\[exp\]), with the harmonic modes now missing, we can write (\[sh\]) in the non-diagonal form H & = & { . +\
& & . } \[ndh\] where the overlap coefficients $f_{mm'}:= <\Psi_{nm}^{(1)} , *\Psi_{nm'}^{(1)}>
$ are different from zero for every $m,m'$. Thus, each of the $\pn^{(0)}$ mode of the momentum field is coupled to an infinite number of other such modes. We do not know how to diagonalize (\[ndh\]).
\[se:3\]
As before, we will work on the space-time manifold $D
\times {\bf R}^1$. Consider then a $U(1)$ Higgs model with the modulus $\r$ of the Higgs field $\f = \r e^{iq\j}$ frozen to its vacuum value. In this limit (the London limit), in addition to the vector potential $A_\m$, the only other degree of freedom is the real phase $\j$. The action in these variables is then: S = \_[D \^1]{} d\^[3]{}x{ - F\^- (\_- ) (\^- ) } \[hl\] where $m{H}e = q \r$ is the mass of the vector meson $A_\m$.
The Hamiltonian corresponding to (\[hl\]) is H = \_[D]{} d\^[2]{}x{ \_[i]{} \^2 + B\^2 + ¶\^2 + (\_i - )\^2 } \[hh\] where $\Pi _{i}=\frac{E_{i}}{e^{2}}$ is the momentum conjugate to $A_{i}$ and $\Pi$ the momentum conjugate to $\psi$. This Hamiltonian has to supplemented by the Gauss law $\P - \pa_i \Pi _{i} \approx 0$. The non-zero Poisson Brackets are { (x), ¶(y) } = \^2 (x-y) , { A\_i (x) , \_j (y) } = \_[ij]{} \^2 (x-y) . \[hc\] As explained in [@Gauge] and after equation (\[con\]), the correct way of reading the Gauss law is by smearing it with a test function $\L^{(0)}$: (Ł\^[(0)]{}) = \_[D]{} d\^[2]{}xŁ\^[(0)]{} ( ¶- \_i \_[i]{} ) = 0 , \[sg\] where $\L^{(0)}$ is zero on the boundary of the disc, $\L^{(0)}|_{\pa D} = 0$.
Let us rewrite both (\[hh\]) and (\[sg\]) using the form notation. To do so, let us introduce the space of vectors $(\j ,A)$ where $\j$ is a zero-form and $A$ is a one-form, with the scalar product $<(\j ,A),(\j' ,A')> = <\j,\j'>_0 + <A,A'>_1$ where $< \cdot , \cdot >_0$ , $< \cdot , \cdot >_1$ are the scalar products respectively on zero and one forms, as previously defined. In addition, to simplify the notation, we set $f := m_{H} \j$, $P := \P /m_{H}$, ${\cal A}_{i}:= A_{i}/e$ and ${\cal E}_{i}:=
e\Pi _{i}$.
After integrating (\[hh\]) by parts and neglecting surface terms, the Hamiltonian becomes H &=& ( <(f,[A]{}), \_0 (f,[A]{})>+<(P,[P]{}),(P,[E]{})>) \[ch\]\
\_0 &=& while the Gauss law (\[sg\]) reads (Ł\^[(0)]{}) = <Ł\^[(0)]{},P + \*d\* [E]{}> = 0 . \[cg\]
Our analysis will now proceed as in the massive vector meson case. We have first to look for suitable BC’s on the fields $(f,{\cal A})$ that make the Hamiltonian $H$ diagonalizable. Let $f,g$ be zero-forms and ${\cal A},{\cal
B}$ be one-forms. Then, from &&<(g,[B]{}),\_0 (f,[A]{})> - <\_0 (g,[B]{}),(f,[A]{})> =\
&& = \_[D]{} { |[\*d [B]{}]{} [A]{} - |[[B]{}]{} \*d [A]{} } - \_[D]{} { |[g]{} \*(df - m\_[H]{}e[A]{}) - |[\*(dg-m\_[H]{}e[B]{})]{} f }, \[st\] we see that $\hat{H}_{0}$ is self-adjoint on the domain \_[ł]{} = { (f,[A]{}) : \*d[A]{} |\_[r=R]{} = - ł[A]{}\_ |\_[r=R]{} f |\_[r=R]{} = -\_ |\_[r=R]{} } ł,\^1 . \[hdo\] We note here that $\lambda$ has the same role as the $\lambda$ that appeared in section 2, while $\mu$ is a new parameter that did not exist in the previous case. We have as before imposed locality and rotational invariance to obtain (\[hdo\]).
On this domain, the Hamiltonian can be rewritten as H &=& { <[E]{},[E]{}>\_1 + <P,P>\_0 + <d[A]{},d[A]{}>\_1 + <df-m\_[H]{}e[A]{},df-m\_[H]{}e[A]{}> } +\
&+& \_[D]{} R d{ ł|[A]{}\_|\^2 + f\^[2]{} }\_[r=R]{} , \[hpd\] so that it is nonnegative iff $\l ,\m \geq 0$. Thus from now on we will consider only domains $\cd_{\l \m}$ with $\l \geq 0 \; ,
\mu \geq 0$.
Our task then is to solve the eigenvalue problem \_0 (
[c]{} f\
[A]{}
) = ø\^2 (
[c]{} f\
[A]{}
) \[eph\] with the BC’s \*d[A]{} |\_[r=R]{} &=& - ł[A]{}\_ |\_[r=R]{} \[bch1\]\
f |\_[r=R]{} &=& -\_ |\_[r=R]{} . \[bch2\] The eigenmodes of (\[eph\]) subject to the above BC’s can be found by an analysis very similar to that used to solve (\[ep\]).
If $\o^2 \neq 0$, this system of equations can be decoupled into one differential equation for ${\cal A}$ and one equation defining $f$ as a function of ${\cal A}$: (d\*d\* + \*d\*d) [A]{} &=& (ø\^2 -m\^2\_[H]{} e\^2 ) [A]{} ,\[ea\]\
f &=& - \*d\*[A]{} . \[ef\]
Let us first look at the modes of ${\cal A}$ obtained from equation (\[ea\]) when $\o^2 = m^2_{H} e^{2}$. In this case the harmonic one forms (\[hm\]) satisfy (\[ea\]) and the BC (\[bch1\]) if $\l = 0$. In addition, from (\[ef\]) and (\[hm\]) it follows that $f \equiv 0$, so that (\[bch2\]) is satisfied only for $\m =0$. This means that the Hamiltonian (\[ch\]) admits harmonic modes for $\l =0$ iff $\m$ is also zero. Since we are interested in comparing the Higgs model in the broken phase with the massive vector meson model and since the latter does admit harmonic modes for $\l =0$, from now on we will set $\m \equiv 0$.
Thus, if $\l =0$, we have a set of solutions $(f,{\cal A})$ of (\[eph\]) corresponding to the eigenvalue $\o ^2 = m^2_{H} e^{2}$ given by $$\begin{aligned}
\Hn = \cm_n^{(h)}\; \left( 0,d z^n \right) \;\;\; ,\;\;\; \bar{\Hn} =
\cm_n^{(h)}\; \left( 0,d \bar{z}^n \right)&&
n >0 \; . \label{hhm}\end{aligned}$$
If $\o ^2 > m^2_{H} e^{2}$, (\[ea\]) and (\[ef\]) admit the following two sets of solutions for $\l \geq 0$: .
[l]{} \^[()]{} = \_[nm]{}\^[()]{} ( - e\^[in]{} J\_n(\_[nm]{} r) , d\[e\^[in]{} J\_n (\_[nm]{} r)\] ) \
\_[-nm]{}\^[()]{} = |[\^[()]{}]{}
} n 0 , m > 0 , \[saa\] .
[l]{} \^[()]{} = \_[nm]{}\^[()]{} ( 0 , \*d\[e\^[in]{} J\_n (\_[nm]{} r)\] ) \
\_[-nm]{}\^[()]{} = |[\^[()]{}]{}
} n 0 , m > 0 . \[sab\] corresponding to the eigenvalues $\o^{(\a )2}_{nm} = \a_{nm}^2 +
m^2_{H}e^{2}$ and $\o^{(\b)2}_{nm}=\b_{nm}^2 +m^2 _{H}e^{2}$ respectively, where $\a_{nm}$ and $\b_{nm}$ are determined by the BC’s (\[bch1\]) and (\[bch2\]) with $\m =0$: J\_n (\_[nm]{}R) &=& 0 ,\[bc11\]\
\_[nm]{} J\_n (\_[nm]{}R) &=& ł\_[r=R]{} .\[bc12\] (Notice that the $\a _{nm}$ and $\b _{nm}$ above are respectively identical to the $\omega _{nm}^{(0)}$ and $\omega _{nm}^{(1)}$ of Section 2.)
Finally we obtain a set of solutions to (\[eph\]) when $\o^2 =0$, which are given by: .
[l]{} \_[nm]{}\^[(0)]{} = \_[nm]{}\^[(0)]{} ( e\^[in]{} J\_n (\_[nm]{}r), d\[e\^[in]{} J\_n (\_[nm]{}r)\] ) \
\_[-nm]{}\^[(0)]{}=|[\_[nm]{}\^[(0)]{}]{}
} n 0 , m > 0 , \[sao\] with the $\g_{nm}$’s fixed by the condition $J_n (\g_{nm} R) = 0$ (so that $\g_{nm}=\a_{nm}$).
In conclusion, for $\l =0$, (\[hhm\],\[saa\],\[sab\],\[sao\]) form a complete set of eigenfunctions that allow us to expand the fields $(f,{\cal A})$ and $(P,{\cal E})$ as (f,[A]{}) &=& \^[()]{} \^[()]{} + \^[()]{} \^[()]{} ++ q\^[(h)]{}\_n + c.c.\
(P,[E]{}) &=& \^[()]{} \^[()]{} + \^[()]{} \^[()]{} ++ p\^[(h)]{}\_n + c.c. \[hexp\] where, by virtue of (\[hc\]), the only nonzero commutation relations are $\left[ q_{nm}^{(j)} , p_{n'm'}^{(j)} \right] = i \d_{nn'} \d_{mm'}
\; \; , \; \;
\left[ q_{n}^{(h)} , p_{n'}^{(h)} \right] = i \d_{nn'}$, where $j=\alpha ,\beta$ or $0$.
If $\l >0$, the field expansions look very similar to (\[hexp\]), except for the fact that in this case the harmonic modes $H_n$ are absent. We now will turn our attention to the Gauss law (\[cg\]). Since in (\[cg\]) the test function $\L^{(0)}$ vanishes on the boundary $\pa
D$, it can be chosen in particular to be $e^{in\q} J_n (\g_{nm} r)$ with $J_n (\g_{nm} R)=0$. It is then immediate to verify that Gauss law simply implies $\pn^{(0)} \approx 0$.
We can also introduce creation-annihilation operators as in (\[ca0\],\[ca\]), where now $j=\a ,\b$, and Ø\_[nm]{}\^[()]{}= , Ø\_[nm]{}\^[()]{}= , Ø\_[n]{}\^[(h)]{} = m\_[H]{}e . \[omm\]
Therefore, the Hamiltonian (\[ch\]) acting on the physical states becomes: H = Ø\_[nm]{}\^[()]{} a\_[nm]{}\^[()]{} a\_[nm]{}\^[()]{} + Ø\_[nm]{}\^[()]{} a\_[nm]{}\^[()]{} a\_[nm]{}\^[()]{} + Ø\_[n]{}\^[(h)]{} a\_n\^[(h)]{} a\_n\^[(h)]{} , \[hdd\]
Notice that the Hamiltonian (\[hdd\]) has been explicitly derived from (\[ch\]) for $\l =0$. In this case, $\Omega _{nm}^{(\alpha )}=\Omega
_{nm}^{(\beta )}=\Omega _{nm}$ just as in the massive vector meson case for $\lambda =0$. For $\l >0$, (\[ch\]) assumes a form which is almost identical to (\[hdd\]), the only difference being that the harmonic modes are no longer solutions of (\[ea\]) and hence do not appear in the Hamiltonian.
\[se:4\]
Let us now compare the vector meson model with the Higgs model in the London limit. From (\[exp\]) and (\[hexp\]) it is clear that there is a one-to-one correspondence between the modes of the fields for these two theories (if we take into account the Gauss law $p_{nm}^{(0)}\approx 0$ which kills one set of modes for the Higgs theory). But what about the Hamiltonians?
It is known [@Broke] that on an infinite plane these two models are indeed equivalent, both describing a massive electromagnetic potential $A_\m$. We see this also in our approach, by noting that the only BC’s that are suitable to a plane geometry require that all the fields vanish at infinity and hence force both $\l$ and $\m$ to be zero. In the latter case, the Hamiltonian for the massive vector meson model (\[hd\]) and the one for the Higgs model (\[hdd\]) are exactly the same, once we identify $m_A$ with $m_H$. Therefore, both on an infinite plane and on a disc with BC’s $\l = \m =0$, these two models coincide.
This is not the case if we confine the theory on a disc and impose BC’s with $\l >0$ (but still $\m =0$). The two Hamiltonians are then different: while for the Higgs model, (\[hdd\]) is diagonal, in the massive vector meson model it has the form (\[ndh\]), in which every mode of the electric field is coupled to infinitely many others. Thus the massive vector meson model and the Higgs model are equivalent on a disc only if we choose boundary conditions for the fields characterized by the value zero for the parameter(s) that appear in (\[bc\]) and (\[bch1\],\[bch2\]).
We would like to end this paper by briefly comparing the models under consideration to yet another model describing a massive vector meson, namely the Maxwell-Chern-Simons (MCS) theory. The action for this model reads: S = \_[D\^1 ]{}d\^[3]{}x { - F\_ F\^ + \^ A\_ \_ A\_ } . \[mcs\]
This Lagrangian has been studied in detail in [@MCS], where it has been shown that the fields have to satisfy BC’s characterized by a nonnegative parameter $\l$, exactly like in (\[bc\]) or (\[bch1\]). As in section 2, one can show that the Hamiltonian of this system is diagonalized by the modes of the operator $*d*d$ only if $\l =0$ and that, as soon as $\l$ deviates from this value, the Hamiltonian couples an infinite number of such modes. In fact, the Hamiltonian for the MCS theory derived in [@MCS] and the one for the massive vector meson model, (\[ndh\]), do coincide if we make the identifiction $m_A = \frac{ek}{2\p}$. In particular, both these Hamiltonians concide with the Higgs Hamiltonian (\[hdd\]) when $\l$ is chosen to be zero. In this latter case, the MCS theory admits an additional set of observables (“edge” observables) which commute with the Hamiltonian and are completely localized at the boundary of the spatial manifold $D$.
[**Acknowledgements**]{} We thank A.P. Balachandran for many useful suggestions that he gave us throughout this work. We also thank P. Teotonio and A. Momen for stimulating discussions. This work was supported by the DOE, U.S.A. under contract number DE-FG02-85ER40231.
[99]{} C. Itzykson and J.B. Zuber, [*Quantum Field Theory*]{}, McGraw-Hill (1980). J.F. Schonfeld, [*Nucl. Phys.*]{} [**B185**]{} (1981) 157; S. Deser, R. Jackiw and S. Templeton, [*Phys. Rev. Lett.*]{} [**48**]{} (1982) 975, [*Ann. Phys.*]{} [**140**]{} (1982) 372; T.J. Allen, M. Bowick and A. Lahiri, [*Mod. Phys. Lett.*]{} [**A6**]{} (1991) 559; J.A. Minahan and R.C. Warner, Florida Preprint UFIFT-HEP-89-15 (1989); A.P. Balachandran and P. Teotonio-Sobrinho, [*Int. Jour. Mod. Phys.*]{} [**A8**]{} (1993) 723, [*Int. Jour. Mod. Phys.*]{} [**A9**]{} (1994) 1569. A.P. Balachandran, L. Chandar, E. Ercolessi, T.R. Govindarajan and R. Shankar, [*Int. Jour. Mod. Phys.*]{} [**A9**]{} (1994) 3417; A.P.Balachandran, L.Chandar and E.Ercolessi, Syracuse University preprint SU-4240-574, hep-th/9411164 (1994). J.R. Schrieffer, [*Theory of Superconductors*]{}, Benjamin (1964); L.D. Landau and E.M. Lifshitz [*Statistical Physics*]{} Part 2, Pergamon Press (1984). A. Zee in Proceedings of the 1991 Kyoto Conference on [*Low Dimensional Field Theories and Condensed Matter Physics*]{}. P.A.M. Dirac, [*Can. Jour. Math.*]{} [*3*]{} (1951) 1; K. Sundermeyer, [*Constrained Dynamics*]{}, Lectures Notes in Physics 169, Springer Verlag (1982). A.P. Balachandran, G. Bimonte, K.S. Gupta and A. Stern, [*Int. Jour. Mod. Phys.*]{} [**A7**]{} (1992) 4655, 5855; A.P. Balachandran, [*Gauge Symmetries, Topology and Quantization*]{} in the Proceedings of the Summer Course on [*Low Dimensional Quantum Field Theories for Condensed matter Phycists*]{}, ICTP, Trieste 24 Aug.-4 Sept. 1992, Syracuse University preprint SU-4240-506 (1992) and World Scientific (in press). P.A.M. Dirac, [*Can. Jour. Math.*]{} [**2**]{} (1950) 129, [*Proc. Roy. Soc.*]{} [**A246**]{} (1958) 326; [*Lectures on Quantum Mechanics*]{}, Yeshiva University, Academic Press (1967). R. Abraham, J.E. Marsden and T. Ratiu, [*Manifolds, Tensor Analysis and Applications*]{}, Springer Verlag (1988). M. Reed and B. Simon, [*Methods of Modern Mathematical Physics*]{} Vol. I, Academic Press (1972).
[^1]: Present address: Dipartimento di Fisica, Universit$\bar{\mbox{a}}$ di Bologna,
[^2]: Here and in the following, we adopt the convention that the normalization contants, such as $\cn_{nm}^{(1)}$ in (\[m1\]), are fixed by the conditions $<\Psi_{nm},\Psi_{nm}>=1$ and $\cn_{nm}^{(1)} > 0$.
|
---
abstract: 'In this paper, we discuss the collection of a corpus associated to tropical storm Harvey, as well as its analysis from both spatial and topical perspectives. From the spatial perspective, our goal here is to get a first estimation of the quality and precision of the geographical information featured in the collected corpus. From a topical perspective, we discuss the representation of Twitter posts, and strategies to process an initially unlabeled corpus of tweets.'
author:
- |
Etienne Brangbour, Pierrick Bruneau,\
Stéphane Marchand-Maillet, Renaud Hostache,\
Patrick Matgen, Marco Chini, Thomas Tamisier
bibliography:
- 'refs.bib'
title: Extracting localized information from a Twitter corpus for flood prevention
---
Introduction
============
Twitter is a popular micro-blogging service based on short textual posts. Its users interact by sharing their opinions, experiences or moods. Its availability on mobile platforms lets users react in real time to catastrophic events such as forest fires, earthquakes or floods. Modelling and predicting flood outbreak and spread are essential in order to mitigate the ecological and economical impact of such events. The classical approach for prediction relies on simulations of the flow and saturation resulting from rainwater [@bates00]. More recently, assimilation models were proposed as means to incorporate external information, such as multispectral satellite imagery, to ensembles of physical models [@hostache15]. The Publimape project explores the extension of this approach to social data, as posted on platforms such as Twitter or Instagram. The tropical storm Harvey, occurred in 2017 and widely discussed then[^1], is used as a validation use case in the course of the project.
Using social data to perform event detection has been considered from two perspectives in the literature. From the one hand, *Volunteered Geographical Information* explicitly asks users to go out on the field, and capture information linked to the event of interest [@griesbaum_direct_2017]. This point of view somehow relates to a crowdsourcing campaign. On the other hand, *Participatory Sensing* views Twitter users as units of a sensor network [@burke_participatory_2006; @crooks_earthquake:_2013]. Relevant information is then collected passively. The latter perspective is considered in the context of Publimape.
In this paper, we discuss the collection of a corpus associated to tropical storm Harvey, as well as its analysis from both spatial and topical perspectives. From the spatial perspective, our goal here is to get a first estimation of the quality and precision of the geographical information featured in the collected corpus. From a topical perspective, we discuss the representation of Twitter posts, and strategies to process an initially unlabeled corpus of tweets.
In section \[sec:sota\], we first review methodologies and constraints inherent to Twitter content collection in the context of event detection. The design choices for our use case is then described in section \[sec:harvey\]. We also showcase insights derived from geographical information featured in Twitter content in section \[sec:experiments\]. Adapted textual representations are discussed in section \[sec:textual\], and their categorization in section \[sec:classification\].
Collecting Tweets for Event Detection {#sec:sota}
=====================================
Twitter exposes an API that enables the real-time collection of posted messages[^2]. It allows to filter the Twitter stream w.r.t. hashtags, keywords, or GPS bounding boxes. It is also possible to obtain a fully random sample of the stream, limited to 1% of the complete stream rate of flow [@cheng_event_2014]. It is also possible to query content for a past period, but this is a paying feature (i.e. Twitter *Enterprise* API[^3]).
Tweets obtained via one of these APIs are collected as JSON objects, i.e. dictionary objects where fields may be literals (numbers or strings), or themselves dictionary objects, implementing a hierarchical data structure.
Among the root fields of a tweet object (approx. 30), in the context of this paper we will pay special attention to the following fields:
- *coordinates*: the geotag (i.e. GPS coordinates) of the tweet,
- *place.full\_name*: the place attached to the tweet,
- *place.bounding\_box*: the bounding box of the above-mentioned place (*bbox* in the remainder of the document),
- *user.location*: the place featured in the profile of the user that posted the tweet. Specifically, this is a free form text field.
- *text*: the actual tweet textual content. Whenever the *truncated* field is true, *text* is a truncated version, and the full version has to be fetched from *extended\_tweet.full\_text*.
Sharing tweet databases is forbidden *a priori*. However, sharing collections of tweet IDs is allowed, delegating the charge of *hydrating* them to full objects to the recipient. In the context of non-commercial research projects, it is hence possible to share ID collections of unlimited size[^4].
For collecting tweets in the context of event detection, the usage of keywords has been contrasted to that of geographical bounding boxes [@ozdikis_survey_2017]. Filtering w.r.t. keywords or hashtags has been commonly used [@starbird_chatter_2010], for example in the context of a 2011 earthquake [@crooks_earthquake:_2013]. Spatio-temporal distributions of the filtered results may then be estimated [@sriram_short_2010; @helwig_analyzing_2015].
In [@cheng_event_2014], the authors claim that collecting according to keywords neglects diffusion effects, i.e. when users copy and paste tweet text without explicitly using the retweet mechanism. Even if experiments in [@sakaki_tweet_2013] show a moderate impact of this phenomenon, collecting according to a geographical filter completely alleviates this bias [@ozdikis_survey_2017]. Geographical filters are also used in [@gao_mapping_2018], excluding retweets and non-English language afterwards. Let us note here that the free streaming Twitter API is limited w.r.t. the Entreprise API in that it is impossible to query tweets posted by user having their profile in the Region of Interest (RoI). Only content explicitly tied to the RoI (i.e. coordinates or place fields) is retrieved. Our usage of geographical filters is described in section \[sec:harvey\].
Initiatives have recently emerged in favor of persistently sharing tweet collections. Harvard Dataverse[^5] stores such sets as an incentive for reproducible research. Actually, a set of 35M tweet IDs associated to hurricanes Harvey and Irma can be found there [@littman_hurricanes_2017]. Those were collected using keyword and hashtag filters: it is hence somehow complementary to the corpus introduced in section \[sec:harvey\]. The former corpus served as basis for a MediaEval task recently [@bischke_multimedia_nodate], where task managers selected a subset of 10K tweets featuring a picture, and had those pictures annotated w.r.t. the presence of a passable or flooded street using the Figure Eight platform[^6].
Creation of the Harvey corpus {#sec:harvey}
=============================
With hydrologist partners in the project, we defined the RoI as indicated in figure \[fig:collection\]a, for the period spanning from the 19$^{\text{th}}$ of August to the 21$^{\text{st}}$ of September. In figure \[fig:collection\]b, we see this period is linked to a peak of queries featuring the term *harvey*.
Using such an ensemble of rectangular bounding boxes is required when using Twitter APIs. Using the Enterprise API, we query tweets with any of *coordinates*, *places*, or *user.location* fields overlapping the RoI, for the period mentioned above. This lead to the collection of 7.5M tweets.
![*a)* RoI used for the corpus collection. *b)* Popularity of the *harvey* query through the period of interest according to *Google Trends*.[]{data-label="fig:collection"}](region_interest.png){width="80.00000%"}
We store this data set as a document collection in a NoSQL database. In order to facilitate further spatial analysis, annotations of these documents are stored in a separate collection. An annotation is made of a tweet ID, and a type (*geotag* or *bbox*). For annotations of the *geotag* type, GPS coordinates are attached directly as a field. We also created a specific *place* collection, that stores the detailed information for *bbox* annotations (including the bounding box coordinates themselves). *bbox* annotations refer to IDs from the *place* collection as foreign keys.
Tweets obtained via the *Enterprise* API have an extended user field (*user.derived*), featuring normalized place names, but no associated bounding boxes, merely the *geotag* a geographical centroid. As means to encode annotations of *bbox* type, we used the Nominatim[^7] geocoding service to recover the bounding boxes. The risk of false recovery is limited by the fact that derived place names are obtained from Twitter’s own geocoding service. We use the first retrieved bounding box that contains the geographical centroid. We distinguish bounding boxes obtained from user profiles by creating annotations with type *pbbox* (i.e. *profile bounding box*). All types considered, we stored approximately 8.3M annotations, referring to 8434 distinct places.
Tweets were collected whether their *coordinates*, *place* or *user* field overlaps the RoI defined in figure \[fig:collection\]a. Let us note that tweets posted by a user profile matching the RoI, but with *coordinates* or *place* field out of the RoI is unlikely to be relevant for our use case (e.g. Houston inhabitant in vacation in Europe). Thus we post-process the annotation set, by excluding situations highlighted above. This led to exclude approximately 170K annotations and 4700 places, so approx. 2% of annotations and more than 50% of places. These numbers are consistent with the discussion above, and removes noise *a priori* from the analysis presented in the next section.
Geographical analysis of the corpus {#sec:experiments}
===================================
One of the objectives of the project introduced in this paper is to map content posted on Twitter with as much spatial precision as possible. We initially assume that *geotag* fields are exact. They make up for approximately 1% of the annotation collection, as indicated in figure \[fig:sunburst\], and reported previously in the literature [@middleton_real-time_2014].
![Scatterplots of the places frequency in the corpus as a function of their surface, for *bbox* (*a*) and *pbbox* (*b*) annotations.[]{data-label="fig:scatterplots"}](scatterplots.png){width="95.00000%"}
Hence we focus on the surfaces of the places identified in the corpus, specifically on the link between the surface of a bounding box and its frequency in the corpus: is a place with a large surface more frequent? Scatterplots in figure \[fig:scatterplots\] display this link. A log-log scale is used for better legibility, as surfaces and frequencies, taken independently, are both exponentially distributed. Graphics in figure \[fig:scatterplots\] were obtained from an interactive application where hovering over glyphs reveals the place name, its surface and its frequency, as means to facilitate exploration. *bbox* and *pbbox* annotations are represented separately.
For the *bbox* case, a significant correlation exists according to Pearson’s and Kendall’s tests ($p < 10^{-10}$), with an estimation of $0.73$ for the Pearson correlation coefficient. Inspecting figure \[fig:scatterplots\]a, we see that highly specific places like *Cypress Park High School* are mentioned only 3 times, when *Houston, TX* and *Texas, USA* appear in respectively 3.9M and 34K tweets. For reliable social mapping, the latter places are not specific enough. In practice, we use figure \[fig:scatterplots\]a to establish a specificity threshold beyond which a bounding box is not relevant for the use case of interest. We qualitatively set this threshold to 350 km$^2$, excluding points on the r.h.s. of \[fig:scatterplots\]a. This value separates *Lake Houston* from *The Woodlands* (see figure \[fig:threshold\]): this surface range is considerable, practically limiting the risk of excluding useful information from our study.
We transferred this threshold to the *pbbox* annotations on figure \[fig:scatterplots\]b. In the latter case, Pearson and Kendall are small ($0.19$ in both cases) and weakly significant ($p = 0.01$ for Pearson’s test). Users often indicate a town as their profile location, which are generally less specific than locations given as examples before. However the chosen threshold retains most of them.
![Bounding boxes for *Lake Houston* (*a*) and *The Woodlands* (*b*).[]{data-label="fig:threshold"}](350threshold.png){width="95.00000%"}
We define sub-categories to *bbox* and *pbbox* according to our threshold (*s* for *small* and *l* for *large*). The cross distribution between annotation types is displayed in figure \[fig:sunburst\]. There we see that only 17.4% of the given geographical information is usable in our applicative context.
![*Sunburst* view of the annotation type cross distribution. The *source* field is also reported.[]{data-label="fig:sunburst"}](sunburst-en.png){width="95.00000%"}
On figure \[fig:sunburst\], the *source* field, indicating the originating application of the tweet, is also reported. The vast majority of tweets is emitted from iPhone, Android and web Twitter clients ($76\%$ for these 3 cumulated categories). Next most popular sources are other social platforms ($3\%$ for Facebook and $2\%$ for Instagram) and secondary Twitter clients ($1\%$ for both iPad and TweetDeck). Content posted by bots has minor presence in our corpus: the two first are SocialOomph and IFTTT with $1\%$ each.
Thanks to the interactive version of the view in figure \[fig:sunburst\], we focus on geographical information relevant to us (i.e. type *geotag*, and the *small* sub-types). Instagram is the origin of most tweets annotated by a *geotag* ($63\%$). Focusing on *geotag* annotations also reveals several minority resources relevant for our use case, e.g. meteorological (*CWIS Twitter Feed*) and traffic (*TTN HOU Traffic*) reports. However we still have to evaluate the reliability of geotags emitted by such automated resources.
A large majority of annotations in the *s\_bbox* sub-category is sent from official Twitter clients. The latter also make up for 3 quarters of the *s\_pbbox* sub-category. The last quarter gathers less popular clients mentioned above with many minor sources (e.g. most of IFTTT). Hence it is reasonable to first focus on *geotag* and *s\_pbbox* annotations.
![Density map of *geotag* and *s\_bbox* types among tweets featuring *flood* or *harvey* keywords (*a*), zoomed on the Wharthon area (*b*).[]{data-label="fig:heatmap"}](heatmap-en.png){width="95.00000%"}
In order to facilitate discussions with hydrologist partners, among the *geotag* and *s\_bbox* categories, we filtered out tweets featuring *flood* or *harvey* in their text. We display the resulting 19K tweets on a density map (see figure \[fig:heatmap\]a). Specifically, hydrologists wanted to evaluate whether it was *a priori* possible to extract social map information for the Wharthon area. Zooming the map in this region, we identify 81 potentially interesting tweets (see figure \[fig:heatmap\]b). Removing the textual filter yields approximately 3600 tweets instead. This area therefore represent an almost negligible part of our corpus. Intuitively, the tweet density is strongly tied to the population density. However, the retrieved set is small enough to allow a manual inspection, which can be useful for qualitative tests.
Textual content {#sec:textual}
===============
Representation space
--------------------
In order to categorize tweet texts, an adequate representation space has to be chosen. In the literature on event detection and sentiment analysis, the usage of chosen keywords is reported [@sakaki2010earthquake], as well as classical representations such as Tf-Idf or Bag-of-Words [@gao_mapping_2018; @kouloumpis2011twitter; @batoolprecise]. These methods are based on word frequencies, and work well with structured and curated text (e.g. Wikipedia pages). However, text posted on social networks contain many abbreviations, slang and typos. Pre-processing such text has been suggested in the literature [@lampos_nowcasting_2012], but this may lead to information loss or distortion.
Alternatively, we chose to use Tweet2Vec, a character-based representation space for social media content [@dhingra_tweet2vec:_2016]. The motivation for character-based embeddings is to be more robust to short and informal text than word-based embeddings. This method relies on extracting the final hidden state of a recurrent neural network trained to predict hashtags contained in tweets. Besides open-sourcing an implementation of their model[^8], the authors also released pre-trained model parameters, taking 2829 characters as potential input (e.g. characters, digits, punctuation, emoticons), and returning 500-dimensional embedding vectors. This pre-trained model was used for our preliminary experiments. We collected our own corpus of similar size by sampling the Twitter stream in English language for future experiments.
Named entity extraction
-----------------------
Extracting named entities, specifically place names, can be critical for event detection. In case of natural disasters, there is an important amount of tweets mentioning affected tiers [@middleton_real-time_2014] (e.g. *“My sister flooded Lumberton, Texas. Walmart on 69.. need help she is stranded and the family from Houston can not get to her”* in our corpus). In such situation, tweet geotags can be amended by the detected named entities. This detection can be performed by combining named entity detection libraries[^9] to Open Street Maps-based geocoding systems such as Nominatim[^10].
In the end, as seen in section \[sec:experiments\], only a small portion of our corpus can be localized in a satisfactory way in the context of the targetted application. In future work we will also try to benefit from named entities present in tweet texts, e.g. *“The Intersection of Asford Pkwy and Dairy Ashford Rd is singificantly higher than yesterday”*. Such an extraction has been considered in the literature about event detection in Twitter corpora [@middleton_real-time_2014]. Other work has exploited tags and laguage geographicity [@kordopatis-zilos_placing_nodate], but such approaches would be difficult to transfer to our regional scale *a priori*.
Content classification {#sec:classification}
======================
Active Learning Experiments
---------------------------
We define 3 target categories for tweets, already considered in the literature in a similar context [@bischke_multimedia_nodate]:
1. *Non Relevant*: the text is not related to the flood event
2. *Positive Indication*: the text is a report by a person directly affected by the flood
3. *Negative Indication*: the text is a report of a person safe from the flood
Owing to the size of the collected corpus, and the cost of manual work that prevents exhaustive annotations, we consider active learning [@dawodactive] as means to reduce the corpus annotation cost. We perform initial experiments anyway, with a sample of 421 tweets annotated by hand (316 for training and 105 for test). To favor class balance, we used the *flood* keyword in our sampling process.
We then assessed how several active learning strategies behaved in the context of our classification task and representation space. These experiments were coded in Python using the *libact* library [@yang2017libact] that implements the most common strategies. We compared the *Uncertainty sampling* to *Hierarchical Sampling*,with a random sampling strategy as a baseline. We monitored the precision of an SVM classifier during the execution of the strategy. The results were not seen as conclusive, as both strategies do not perform better than random. This may be due to the imbalance between the small sample size and the embedding dimensionality, thus calling for a smart way of acquiring labels.
Crowdsourcing
-------------
In order to train a quality classifier, we need to label our corpus as reliably and exhaustively as possible. In this view, we propose to combine active learning to the usage of a crowdsourcing platform such as *Mechanical Turk* [^11] or Figure Eight [^12]. Our intuition is to use active learning to select the items that will be pushed to the crowdsourcing platform.
Literature shows that crowdsourcing raises reliability issues that need to be addressed [@nowak_how_2010]. Related work has been made in the context of image concepts annotation [@nowak_how_2010; @loni_getting_2014], social media analysis [@kamel_boulos_crowdsourcing_2011] and named entity identification [@finin2010annotating]. For our task, particular attention will have to be put in the way questions are asked to workers, in order to better guide them and limit the risk of errors.
Conclusion
==========
The final objective of the project is to aggregate relevant content w.r.t. a 2D spatial grid. Spatio-temporal classification methods were proposed in the literature [@helwig_analyzing_2015; @anantharam_extracting_2015; @tamura_density-based_2013], sometimes accounting for the out of event local tweet emission rate [@gao_mapping_2018] and trending topic detection [@atefeh_survey_2015; @cordeiro_online_2016]. The present paper let us quantify the quantity and reliability of geographical information in our corpus. Besides adapting filtering methods mentionned above, we will evalute the improvement yielded by named entity extraction.
Practically, as discussed in section \[sec:classification\], we will explore the potential of active learning and crowdsourcing in order to identify relevant content more finely than using keywords (e.g. as made in section \[sec:experiments\]), and harness the multimodality of the content under study (i.e. text, GPS coordinates, time, images) while accounting for potential missing values [@brangbour_extracting_2018].
Acknowledgements
================
This work was performed in the context of the Publimape project, funded by the CORE programme of the Luxembourgish National Research Fund (FNR).
[^1]: <https://en.wikipedia.org/wiki/Hurricane_Harvey>
[^2]: <https://developer.twitter.com/en/docs/tweets/filter-realtime/overview>
[^3]: <https://developer.twitter.com/en/enterprise>
[^4]: <https://developer.twitter.com/en/developer-terms/agreement-and-policy.html>
[^5]: <https://dataverse.harvard.edu/>
[^6]: <https://www.figure-eight.com>
[^7]: <https://nominatim.openstreetmap.org/>
[^8]: https://github.com/bdhingra/tweet2vec
[^9]: https://github.com/ushahidi/geograpy
[^10]: https://nominatim.openstreetmap.org/
[^11]: https://www.mturk.com/
[^12]: https://www.figure-eight.com/
|
---
abstract: 'Bounded rational decision-makers transform sensory input into motor output under limited computational resources. Mathematically, such decision-makers can be modeled as information-theoretic channels with limited transmission rate. Here, we apply this formalism for the first time to multilayer feedforward neural networks. We derive synaptic weight update rules for two scenarios, where either each neuron is considered as a bounded rational decision-maker or the network as a whole. In the update rules, bounded rationality translates into information-theoretically motivated types of regularization in weight space. In experiments on the MNIST benchmark classification task for handwritten digits, we show that such information-theoretic regularization successfully prevents overfitting across different architectures and attains results that are competitive with other recent techniques like dropout, dropconnect and Bayes by backprop, for both ordinary and convolutional neural networks.'
author:
- |
Felix Leibfried$^{1,2,3}$ [and]{.nodecor} Daniel A. Braun$^{1,2}$\
$^1$Max Planck Institute for Intelligent Systems, Tübingen, Germany\
$^2$Max Planck Institute for Biological Cybernetics, Tübingen, Germany\
$^3$Graduate Training Center of Neuroscience, Tübingen, Germany\
bibliography:
- 'refs.bib'
title: 'Bounded Rational Decision-Making in Feedforward Neural Networks'
---
INTRODUCTION
============
Intelligent systems in biology excel through their ability to flexibly adapt their behavior to changing environments so as to maximize their (expected) benefit. In order to understand such biological intelligence and to design artificial intelligent systems, a central goal is to analyze adaptive behavior from a theoretical point of view. A formal framework to achieve this goal is decision theory. An important idea, originating from the foundations of decision theory, is the principle of maximum expected utility [@Neumann1944]. According to the principle of maximum expected utility, an intelligent agent is formalized as a decision-maker that chooses optimal actions that maximize the expected benefit of an outcome, where the agent’s benefit is quantified by a utility function.
A fundamental problem of the maximum expected utility principle is that it does not take into account computational resources that are necessary to identify optimal actions—it is for example computationally prohibitive to compute an optimal chess move because of the vast amount of potential board configurations. One way of taking computational resources into account is to study optimal decision-making under information-processing constraints [@Gersham2015; @Simon1972]. In this study, we use an information-theoretic model of bounded rational decision-making [@Genewein2015; @Ortega2015; @Ortega2013] that has precursors in the economic literature [@Sims2011; @Wolpert2004; @Mattsson2002] and that is closely related to recent advances harnessing information theory for machine learning and perception-action systems [@Blundell2015; @Still2014; @SanchezGiraldo2013; @Kappen2012; @Rawlik2012; @Rubin2012; @Tishby2011; @Friston2010; @Peters2010; @Still2009; @Todorov2009].
Previously, this information-theoretic bounded rationality model was applied to derive a synaptic weight update rule for a single reward-maximizing spiking neuron [@Leibfried2015]. It was shown that such a neuron tries to keep its firing rate close to its average firing rate, which ultimately leads to economizing of synaptic weights. Mathematically, such economizing is equivalent to a regularization that prevents synaptic weights from growing without bounds. The bounded rational weight update rule furthermore generalizes the synaptic weight update rule for an ordinary reward-maximizing spiking neuron as presented for example in [@Xie2004]. In our current work, we extend the framework of information-theoretic bounded rationality to networks of neurons, but restrict ourselves for a start to deterministic settings. In particular, we investigate two scenarios, where either each single neuron is considered as a bounded rational decision-maker or the network as a whole.
The remainder of this manuscript is organized as follows. In Section \[sec:background\], we explain the information-theoretic bounded rationality model that we use. In Section \[sec:neural\_networks\], we apply this model to derive bounded rational synaptic weight update rules for single neurons and networks of neurons. In Section \[sec:experimental\_results\], we demonstrate the regularizing effect of these bounded rational weight update rules on the MNIST benchmark classification task. In Section \[sec:conclusion\], we conclude.
BACKGROUND ON BOUNDED RATIONAL DECISION-MAKING {#sec:background}
==============================================
A FREE ENERGY PRINCIPLE FOR BOUNDED RATIONALITY
-----------------------------------------------
A decision-maker is faced with the task to choose an optimal action out of a set of actions. Each action $y$ is associated with a given task-specific utility value $U(y)$. A fully rational decision-maker picks the action $y^*$ that globally maximizes the utility function, where $y^* = \operatorname*{arg\,max}_y U(y)$, assuming for notational simplicity that the global maximum is unique. Under limited computational resources however, the decision-maker may not be able to identify the globally optimal action $y^*$ which leads to the question of how limited computational resources should be quantified. In general, the decision-maker’s behavior can be expressed as a probability distribution over actions $p(y)$. The basic idea of information-theoretic bounded rationality is that changes in such probability distributions are costly and necessitate computational resources. More precisely, computational resources are quantified as informational cost evoked by changing from a prior probabilistic strategy $p_0(y)$ to a posterior probabilistic strategy $p(y)$ during the deliberation process preceding the choice. Mathematically, this informational cost is given by the Kullback-Leibler divergence $D_{KL}(p(y)||p_0(y)) \leq B$ between prior and posterior strategy, where computational resources are modeled as an upper bound $B \geq 0$ [@Blundell2015; @Ortega2015; @Ortega2013; @Kappen2012; @Rawlik2012; @Rubin2012; @Friston2010; @Peters2010; @Still2009; @Todorov2009; @Wolpert2004; @Mattsson2002]. Accordingly, bounded rational decision-making can be formalized by the following free energy objective $$\label{eq:free_energy_objective}
\begin{split}
& p^*(y) \\
& = \operatorname*{arg\,max}_{p(y)} ~ (1-\beta) \left\langle U(y) \right\rangle_{p(y)} - \beta D_{KL}(p(y)||p_0(y)) \\
& = \operatorname*{arg\,max}_{p(y)} ~ \left\langle (1-\beta) U(y) - \beta \ln \frac{p(y)}{p_0(y)} \right\rangle_{p(y)} ,
\end{split}$$ where $\beta \in (0;1)$ controls the trade-off between expected utility and informational cost. Note that the upper bound $B$ imposed on the Kullback-Leibler divergence determines the value of $\beta$. Choosing the value of $B$ is hence equivalent to choosing the value of $\beta$.
The free energy objective in Equation is concave with respect to $p(y)$ and the optimal solution $p^*(y)$ can be expressed in closed analytic form: $$\label{eq:solution_free_energy_objective}
p^*(y) = \frac{p_0(y)\exp(\frac{1-\beta}{\beta} U(y))}{\sum_{y'} p_0(y')\exp(\frac{1-\beta}{\beta} U(y'))} .$$ In the limit cases of none ($\beta \rightarrow 1$) and infinite ($\beta \rightarrow 0$) resources, the optimal strategy from Equation becomes $$\begin{aligned}
\label{eq:limit_solution_free_energy_objective_1}
\lim_{\beta \rightarrow 1} p^*(y) = & p_0(y) ,\\
\label{eq:limit_solution_free_energy_objective_2}
\lim_{\beta \rightarrow 0} p^*(y) = & \delta_{y y^*},\end{aligned}$$ respectively, where $y^* = \operatorname*{arg\,max}_y U(y)$ represents an action that globally maximizes the utility function. A decision-maker without any computational resources ($\beta \rightarrow 1$) sticks to its prior strategy $p_0(y)$, whereas a decision-maker that can access an arbitrarily large amount of resources ($\beta \rightarrow 0$) always picks a globally optimal action and recovers thus the fully rational decision-maker.
A RATE DISTORTION PRINCIPLE FOR CONTEXT-DEPENDENT DECISION-MAKING
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In the face of multiple contexts, fully rational decision-making requires to find an optimal action $y$ for each environment $x$, where optimality is defined through a utility function $U(x,y)$. Bounded rational decision-making in multiple contexts means to compute multiple strategies, expressed as conditional probability distributions $p(y|x)$, under limited computational resources. Limited computational resources are modeled through an upper bound $B \geq 0$ on the expected Kullback-Leibler divergence $\left\langle D_{KL}(p(y|x)||p_0(y)) \right\rangle_{p(x)} \leq B$ between the strategies $p(y|x)$ and a common prior $p_0(y)$, averaged over all possible environments described by the distribution $p(x)$ [@Genewein2015; @Rubin2012]. The resulting optimization problem may be formalized as $$\label{eq:avg_free_energy_objective}
\begin{split}
p^*(y|x) & = \operatorname*{arg\,max}_{p(y|x)} ~ (1-\beta) \left\langle U(x,y) \right\rangle_{p(x,y)} \\
& - \beta \left\langle D_{KL}\left(p(y|x)||p_0(y)\right)\right\rangle _{p(x)} ,
\end{split}$$ where $\beta \in (0;1)$ governs the trade-off between expected utility and informational cost. It can be shown that the most economic prior $p_0(y)$ is given by the marginal distribution $p_0(y) = p(y) = \sum_x p(y|x)p(x)$, because the marginal distribution minimizes the expected Kullback-Leibler divergence for a given set of conditional distributions $p(y|x)$—see [@Tishby1999]. In this case, the expected Kullback-Leibler divergence becomes identical to the mutual information $I(x,y)$ between the environment $x$ and the action $y$ [@Genewein2015; @Leibfried2015; @Still2014; @SanchezGiraldo2013; @Sims2011; @Tishby2011]. Accordingly, bounded rational decision-making can be formalized through the following objective $$\label{eq:rate_distortion_objective}
\begin{split}
& p^*(y|x) \\
& = \operatorname*{arg\,max}_{p(y|x)} ~ (1-\beta) \left\langle U(x,y) \right\rangle _{p(x,y)} - \beta I(x,y) \\
& = \operatorname*{arg\,max}_{p(y|x)} ~ \left\langle (1-\beta) U(x,y) - \beta \ln \frac{p(y|x)}{p(y)} \right\rangle_{p(x,y)} ,
\end{split}$$ which is mathematically equivalent to the rate distortion problem from information theory [@Shannon1959].
The rate distortion objective in Equation is concave with respect to $p(y|x)$ but there is unfortunately no closed analytic form solution. It is however possible to express the optimal solution as a set of self-consistent equations: $$\begin{aligned}
\label{eq:solution_rate_distortion_objective_1}
p^*(y|x) & = & \frac{p(y)\exp(\frac{1-\beta}{\beta} U(x,y) )}{\sum_{y'}p(y')\exp(\frac{1-\beta}{\beta} U(x,y') )} , \\
\label{eq:solution_rate_distortion_objective_2}
p(y) & = & \sum_x p^*(y|x)p(x).\end{aligned}$$ These self-consistent equations are solved by replacing $p(y)$ with an initial arbitrary distribution $q(y)$ and iterating through Equations and in an alternating fashion. This procedure is known as Blahut-Arimoto algorithm [@Arimoto1972; @Blahut1972] and is guaranteed to converge to a global optimum [@Csiszar1974] presupposed that $q(y)$ does not assign zero probability mass to any $y$.
In the limit cases of none ($\beta \rightarrow 1$) and infinite ($\beta \rightarrow 0$) resources, the optimal strategy from Equations and may be expressed in closed analytic form $$\begin{aligned}
\label{eq:limit_solution_rate_distortion_objective_1}
\lim_{\beta \rightarrow 1} p^*(y|x) = & p(y) = \delta_{y y^*} , \\
\label{eq:limit_solution_rate_distortion_objective_2}
\lim_{\beta \rightarrow 0} p^*(y|x) = & \delta_{y y^*_x},\end{aligned}$$ where $y^* = \operatorname*{arg\,max}_y \left\langle U(x,y) \right\rangle_{p(x)}$ refers to an action that globally maximizes the expected utility averaged over all possible environments, and $y_x^*$ refers to an action that globally maximizes the utility for one particular environment $x$—assuming for notational simplicity that global maxima are unique in both cases. In the absence of any computational resources ($\beta \rightarrow 1$), the decision-maker chooses the same strategy no matter which environment is encountered in order to minimize the deviation between the conditional strategies $p(y|x)$ and the average strategy $p(y)$. The decision-maker chooses however a strategy that maximizes the average expected utility. In case of access to an arbitrarily large amount of computational resources ($\beta \rightarrow 0$), the decision-maker picks the best action for each environment and recovers thus the fully rational decision-maker.
THEORETICAL RESULTS: SYNAPTIC WEIGHT UPDATE RULES {#sec:neural_networks}
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PARAMETERIZED STRATEGIES AND ONLINE RULES {#sec:parameterized_strategies}
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Computing the optimal solution to the rate distortion problem in Equation with help of Equations and through the Blahut-Arimoto algorithm has two severe drawbacks. First, it requires to compute and store the conditional strategies $p(y|x)$ and the marginal strategy $p(y)$ explicitly, which is prohibitive for large environment and action spaces. And second, it requires that the decision-maker is able to evaluate the utility function for arbitrary environment-action pairs $(x,y)$, which is a plausible assumption in planning, but not in reinforcement learning where samples from the utility function can only be obtained from interactions with the environment.
We therefore assume a parameterized form of the strategy $p_w(y|x)$, from which the decision-maker can draw samples $y$ for a given sample of the environment $x$, and optimize the rate distortion objective from Equation with help of gradient ascent [@Leibfried2015]—also referred to as policy gradient in the reinforcement learning literature [@Xie2004]. Gradient ascent requires to compute the derivative of the objective function $L(w)$ with respect to the strategy parameters $w$ and to update the parameters according to the rule $w \leftarrow w + \alpha \cdot \frac{\partial}{\partial w} L(w)$ in each time step, where $\alpha > 0$ denotes the learning rate and $\frac{\partial}{\partial w} L(w)$ is defined as $$\label{eq:derivative_rate_distortion_objective}
\begin{split}
& \frac{\partial}{\partial w} L(w) = \\
& \left\langle \left( \frac{\partial}{\partial w} \ln p_w(y|x) \right) (1-\beta) U(x,y) \right\rangle_{p_w(x,y)} \\
& - \left\langle \left( \frac{\partial}{\partial w} \ln p_w(y|x) \right) \beta \ln \frac{p_w(y|x)}{p_w(y)} \right\rangle_{p_w(x,y)} .
\end{split}$$ Note that the update rule from Equation requires the computation of an expected value over $p_w(x,y)$. This expected value can be approximated through environment-action samples $(x,y)$ in either a batch or an online manner. For the rest of this paper, we assume an online update rule where the agent adapts its behavior instantaneously after each interaction with the environment in response to an immediate reward signal $U(x,y)$ as is typical for reinforcement learning.
Informally, the rate distortion model for bounded rational decision-making translates into a specific form of regularization that penalizes deviations of the decision-maker’s instantaneous strategy $p_w(y|x)$, given the current environment $x$, from the decision-maker’s mean strategy $p_w(y) = \sum_x p_w(y|x) p(x)$, averaged over all possible environments. Previously, Equation was applied to a single spiking neuron that was stochastic [@Leibfried2015]. Here, we generalize this approach to deterministic networks of neurons that have neural input (environmental context $x$), neural output (action $y$) and a reward signal (utility $U$). We derive parameter update rules in the style of Equation that allow to adjust synaptic weights in an online fashion. In particular, we investigate two scenarios where either each single neuron is considered as a bounded rational decision-maker or the network as a whole.
A STOCHASTIC NEURON AS A BOUNDED RATIONAL DECISION-MAKER
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A stochastic neuron may be considered as a bounded rational decision-maker [@Leibfried2015]: the neuron’s presynaptic input is interpreted as environmental context and the neuron’s output is interpreted as action variable. The neuron’s parameterized strategy corresponds to its firing behavior and is given by $$\label{stochastic_neuron_strategy}
p_{\mathbf{w}}(y|\mathbf{x}) = y \cdot \rho(\mathbf{w}^\top\mathbf{x}) + (1-y) \cdot (1-\rho(\mathbf{w}^\top\mathbf{x})),$$ where $y \in \{0,1\}$ is a binary variable reflecting the neuron’s current firing state, $\mathbf{x}$ is a binary column vector representing the neuron’s current presynaptic input and $\mathbf{w}$ is a real-valued column vector representing the strength of presynaptic weights. $\rho \in (0;1)$ is a monotonically increasing function denoting the neuron’s current firing probability. In a similar way, the neuron’s mean firing behavior can be expressed as: $$\label{stochastic_neuron_mean_strategy}
p_{\mathbf{w}}(y) = y \cdot \bar{\rho}(\mathbf{w}) + (1-y) \cdot (1-\bar{\rho}(\mathbf{w})),$$ where $\bar{\rho}(\mathbf{w}) = \sum_{\mathbf{x}} \rho(\mathbf{w}^\top\mathbf{x}) p(\mathbf{x})$ denotes the neuron’s mean firing probability averaged over all possible inputs $\mathbf{x}$. The mean firing probability $\bar{\rho}(\mathbf{w})$ can be easily estimated with help of an exponential window in an online manner according to $$\bar{\rho}(\mathbf{w}) \leftarrow (1-\frac{1}{\tau})\bar{\rho}(\mathbf{w}) + \frac{1}{\tau} \rho(\mathbf{w}^\top\mathbf{x}) ,$$ where $\tau$ is a constant defining the time horizon [@Leibfried2015].
Assuming a task-specific utility function $U(\mathbf{x},y)$ determining the neuron’s instantaneous reward and assuming furthermore that the neuron’s output $y$ does not impact the presynaptic input $\mathbf{x}$ of the next time step, the bounded rational neuron may be thought of as optimizing a rate distortion objective according to Equation with gradient ascent as outlined in Section \[sec:parameterized\_strategies\] [@Leibfried2015]. Equation is then applicable by using the quantities $$\label{eq:derivative_log_probability}
\begin{split}
& \frac{\partial}{\partial w_i} \ln p_{\mathbf{w}}(y|\mathbf{x}) = \\
& x_i \rho'(\mathbf{w}^\top\mathbf{x}) \left( \frac{y}{\rho(\mathbf{w}^\top\mathbf{x}) } - \frac{1-y}{1-\rho(\mathbf{w}^\top\mathbf{x}) } \right),
\end{split}$$ and $$\label{eq:mi_term}
\begin{split}
& \ln \frac{p_{\mathbf{w}}(y|\mathbf{x})}{p_{\mathbf{w}}(y)} = \\
& y \ln \frac{\rho(\mathbf{w}^\top\mathbf{x})}{\bar{\rho}(\mathbf{w})} + (1-y) \ln \frac{1-\rho(\mathbf{w}^\top\mathbf{x}) }{1-\bar{\rho}(\mathbf{w}) }.
\end{split}$$ By averaging over the binary quantity $y$, a more concise weight update rule is derived [@Leibfried2015]: $$\label{eq:derivative_rate_distortion_objective_stochastic neuron}
\begin{split}
& \frac{\partial}{\partial w_i} L(\mathbf{w}) = \\
& \left\langle x_i \rho'(\mathbf{w}^\top\mathbf{x}) (1-\beta) \Delta U(\mathbf{x}) \right\rangle_{p(\mathbf{x})} \\
& - \left\langle x_i \rho'(\mathbf{w}^\top\mathbf{x}) \beta \ln \frac{\rho(\mathbf{w}^\top\mathbf{x}) (1-\bar{\rho}(\mathbf{w}))}{\bar{\rho}(\mathbf{w}) (1-\rho(\mathbf{w}^\top\mathbf{x}))} \right\rangle_{p(\mathbf{x})} ,
\end{split}$$ where $\Delta U(\mathbf{x}) = U(\mathbf{x},y=1) - U(\mathbf{x},y=0)$ denotes the difference in utility between firing ($y=1$) and not firing ($y=0$) for a given $\mathbf{x}$. If the conditional and marginal strategies are initialized to be roughly equal $p_{\mathbf{w}_0}(y) \approx p_{\mathbf{w}_0}(y|\mathbf{x})$, where $\mathbf{w}_0 \approx \mathbf{0}$ refers to the initial value of $\mathbf{w}$, the hyperparameter $\beta$ determines how fast the decision-maker’s strategy converges. A high value of $\beta$ implies little computational resources and quick convergence due to the fact that conditional and marginal strategies are initially almost equal. On the opposite, a low value of $\beta$ indicating vast computational resources allows the decision-maker to find an optimal strategy for each environment where conditional and marginal strategies may deviate substantially.
A DETERMINISTIC NEURON AS A BOUNDED RATIONAL DECISION-MAKER
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In a deterministic setup, the neuron’s parameterized firing behavior in a small time window $\Delta t$ may be expressed through its firing rate $\phi(\mathbf{w}^\top \boldsymbol\xi)$ as: $$\label{det_firing_behavior}
p_{\mathbf{w}}(y|\boldsymbol\xi) = y \cdot \phi(\mathbf{w}^\top\boldsymbol\xi) \Delta t + (1-y) \cdot (1-\phi(\mathbf{w}^\top\boldsymbol\xi) \Delta t),$$ where $\boldsymbol\xi$ is a real-valued column vector indicating the presynaptic firing rates and $\phi > 0$ is a monotonically increasing function. In a similar fashion, the neuron’s mean firing behavior is given by $$\label{mean_firing_behavior}
p_{\mathbf{w}}(y) = y \cdot \bar{\phi}(\mathbf{w}) \Delta t + (1-y) \cdot (1-\bar{\phi}(\mathbf{w}) \Delta t),$$ where $\bar{\phi}(\mathbf{w}) = \sum_{\boldsymbol \xi} \phi(\mathbf{w}^\top \boldsymbol\xi) p(\boldsymbol \xi)$ refers to the neuron’s mean firing rate averaged over all possible presynaptic firing rates $\boldsymbol \xi$. In accordance with the previous section, the mean firing rate $\bar{\phi}(\mathbf{w})$ can be conveniently approximated in an online manner through an exponential window with a time constant $\tau$ as: $$\bar{\phi}(\mathbf{w}) \leftarrow (1-\frac{1}{\tau})\bar{\phi}(\mathbf{w}) + \frac{1}{\tau} \phi(\mathbf{w}^\top \boldsymbol \xi) .$$ Using the quantities introduced above, we can define a mutual information rate between the presynaptic firing rates $\boldsymbol\xi$ and the instantaneous firing state of the neuron $y \in \{0;1\}$: $$\label{eq:mutual_information_rate}
\begin{split}
& \lim_{\Delta t \rightarrow 0} \frac{1}{\Delta t} I(\boldsymbol\xi,y) \\
& = \lim_{\Delta t \rightarrow 0} \frac{1}{\Delta t} \left\langle \sum_y p_{\mathbf{w}}(y|\boldsymbol\xi) \ln \frac{p_{\mathbf{w}}(y|\boldsymbol\xi)}{p_{\mathbf{w}}(y)} \right\rangle_{p(\boldsymbol\xi)} \\
& = \left\langle \phi(\mathbf{w}^\top \boldsymbol\xi) \ln \frac{\phi(\mathbf{w}^\top \boldsymbol\xi)}{\bar{\phi}(\mathbf{w})} \right\rangle_{p(\boldsymbol\xi)} .
\end{split}$$ A derivation of Equation can be found in Section \[sec:mutual\_info\_rate\_det\_neu\]. Assuming a rate-dependent utility function $U(\boldsymbol\xi,\phi(\mathbf{w}^\top \boldsymbol\xi))$, a deterministic neuron can be interpreted as a bounded rational decision-maker similar to Equation with the following rate distortion objective $$\label{eq:deterministic_rate_distortion_objective}
\begin{split}
\mathbf{w}^* & = \operatorname*{arg\,max}_{\mathbf{w}} ~ (1-\beta) \left\langle U(\boldsymbol\xi,\phi(\mathbf{w}^\top \boldsymbol\xi)) \right\rangle _{p(\boldsymbol\xi)} \\
& - \beta \lim_{\Delta t \rightarrow 0} \frac{1}{\Delta t} I(\boldsymbol\xi,y) \\
& = \operatorname*{arg\,max}_{\mathbf{w}} ~ \left\langle (1-\beta) U(\boldsymbol\xi,\phi(\mathbf{w}^\top \boldsymbol\xi)) \right\rangle_{p(\boldsymbol\xi)} \\
& - \left\langle \beta \phi(\mathbf{w}^\top \boldsymbol\xi) \ln \frac{\phi(\mathbf{w}^\top \boldsymbol\xi)}{\bar{\phi}(\mathbf{w})} \right\rangle_{p(\boldsymbol\xi)} .
\end{split}$$
Optimizing the neuron’s weights with gradient ascent, a similar weight update rule as in Equation is derived for the deterministic case: $$\label{eq:deterministic_derivative_rate_distortion_objective}
\begin{split}
& \frac{\partial}{\partial w_i} L(\mathbf{w}) = \\
& \left\langle \xi_i \phi'(\mathbf{w}^\top \boldsymbol\xi) (1-\beta) \frac{\partial}{\partial \phi}U(\boldsymbol\xi,\phi(\mathbf{w}^\top \boldsymbol\xi)) \right\rangle_{p(\boldsymbol\xi)}\\
& - \left\langle \xi_i \phi'(\mathbf{w}^\top \boldsymbol\xi) \beta \ln \frac{\phi(\mathbf{w}^\top \boldsymbol\xi)}{\bar{\phi}(\mathbf{w}) } \right\rangle_{p(\boldsymbol\xi)} ,
\end{split}$$ where $\frac{\partial}{\partial \phi}U(\boldsymbol\xi,\phi(\mathbf{w}^\top \boldsymbol\xi))$ denotes the derivative of the utility function with respect to the neuron’s firing rate. The solution in Equation requires the derivative of two terms with respect to $w_i$. The derivative of the expected utility $\left\langle U(\boldsymbol\xi,\phi(\mathbf{w}^\top \boldsymbol\xi)) \right\rangle_{p(\boldsymbol\xi)}$ is straightforward, whereas the derivative of the mutual information rate $\lim_{\Delta t \rightarrow 0} \frac{1}{\Delta t} I(\boldsymbol\xi,y)$ is not so trivial and explained in more detail in Section \[sec:derivative\_mutual\_info\_rate\].
A NEURAL NETWORK OF BOUNDED RATIONAL DETERMINISTIC NEURONS {#sec:consortium}
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Here, we consider a feedforward multilayer perceptron that can be imagined to consist of individual bounded rational deterministic neurons as described in the previous section. Assuming that all neurons aim at maximizing a global utility function while at the same time minimizing their local mutual information rate, each neuron $n$ may be interpreted as solving a deterministic rate distortion objective where the utility function is shared among all neurons but the mutual information cost is neuron-specific: $$\label{eq:global_deterministic_rate_distortion_objective}
\begin{split}
{\mathbf{w}^n}^* & = \operatorname*{arg\,max}_{\mathbf{w}^n} ~ (1-\beta) \left\langle U(\boldsymbol\xi^{\text{in}},\mathbf{f}(\mathcal{W},\boldsymbol\xi^{\text{in}})) \right\rangle _{p(\boldsymbol\xi^{\text{in}})} \\
& - \beta \lim_{\Delta t \rightarrow 0} \frac{1}{\Delta t} I(\boldsymbol\xi^n,y^n) ,
\end{split}$$ where ${\mathbf{w}^n}$, $\boldsymbol\xi^n$ and $y^n$ refer to the presynaptic weight vector, the presynaptic firing rates and the current firing state of neuron $n$ respectively and where $\mathcal{W}$ denotes the entirety of all weights in the whole neural network. The global utility $U(\boldsymbol\xi^{\text{in}},\mathbf{f}(\mathcal{W},\boldsymbol\xi^{\text{in}}))$ is expressed as a function of the network’s input rates $\boldsymbol\xi^{\text{in}}$ and the network’s output rates $\mathbf{f}(\mathcal{W},\boldsymbol\xi^{\text{in}})$.
The corresponding synaptic weight update rule for gradient ascent is similar to Equation and given by $$\label{eq:global_deterministic_derivative_rate_distortion_objective}
\begin{split}
& \frac{\partial}{\partial w^n_i} L^n(\mathcal{W}) = \\
& \left\langle (1-\beta) \frac{\partial}{\partial w^n_i} U(\boldsymbol\xi^{\text{in}},\mathbf{f}(\mathcal{W},\boldsymbol\xi^{\text{in}})) \right\rangle_{p(\boldsymbol\xi^{\text{in}})} \\
& - \left\langle \beta \xi^n_i \phi'({\mathbf{w}^n}^\top \boldsymbol\xi^n) \ln \frac{\phi({\mathbf{w}^n}^\top \boldsymbol\xi^n)}{\bar{\phi}(\mathbf{w}^n) } \right\rangle_{p(\boldsymbol\xi^{\text{in}})},
\end{split}$$ where $L^n(\mathcal{W})$ refers to the rate distortion objective of neuron $n$. The derivative of the utility function with respect to the weight $\frac{\partial}{\partial w^n_i} U(\boldsymbol\xi^{\text{in}},\mathbf{f}(\mathcal{W},\boldsymbol\xi^{\text{in}}))$ can be straightforwardly derived via ordinary backpropagation [@LeCun1998].
A DETERMINISTIC NEURAL NETWORK AS A BOUNDED RATIONAL DECISION-MAKER {#sec:global_decision_maker}
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While focusing on individual neurons as bounded rational decision-makers in the previous section, it is also possible to interpret an entire feedforward multilayer perceptron as one bounded rational decision-maker. To allow for this interpretation, we consider in the following the network’s output rates $f_j(\mathcal{W},\boldsymbol \xi) \in (0;1)$ as the event probabilities of a categorical distribution (for example, by using a softmax activation function in the last layer). Importantly, the categorical distribution is considered as a bounded rational strategy $$p_\mathcal{W}(\mathbf{y}|\boldsymbol \xi) = \sum_j y_j f_j(\mathcal{W},\boldsymbol \xi) ,$$ that generates a binary unit output vector $\mathbf{y}$ given the input rates $\boldsymbol \xi$ and the set of all weights in the entire network denoted by $\mathcal{W}$. The average bounded rational strategy is then given by $$p_\mathcal{W}(\mathbf{y}) = \sum_j y_j \bar{f}_j(\mathcal{W}) ,$$ where $\bar{f}_j(\mathcal{W})$ is the mean rate of output unit $j$ that can again be approximated in an online manner according to $$\bar{f}_j(\mathcal{W}) \leftarrow (1-\frac{1}{\tau})\bar{f}_j(\mathcal{W}) + \frac{1}{\tau} f_j(\mathcal{W},\boldsymbol \xi) ,$$ by use of an exponential window with a time constant $\tau$ in line with previous sections.
Accordingly, the informational cost can be quantified by the mutual information between $\boldsymbol \xi$ and $\mathbf{y}$: $$\label{eq:mutual_information_global}
\begin{split}
I(\boldsymbol \xi,\mathbf{y}) & = \left\langle \sum_{\mathbf{y}} p_\mathcal{W}(\mathbf{y}|\boldsymbol \xi) \ln \frac{p_\mathcal{W}(\mathbf{y}|\boldsymbol \xi)}{p_\mathcal{W}(\mathbf{y})} \right\rangle_{p(\boldsymbol \xi)} \\
& = \left\langle \sum_{j} f_j(\mathcal{W},\boldsymbol \xi) \ln \frac{f_j(\mathcal{W},\boldsymbol \xi)}{\bar{f}_j(\mathcal{W})} \right\rangle_{p(\boldsymbol \xi)} .
\end{split}$$ Presupposing again a rate dependent utility function $U(\boldsymbol\xi,\mathbf{f}(\mathcal{W},\boldsymbol\xi))$, the entire deterministic network may be interpreted to solve the subsequent rate distortion objective $$\label{eq:global_network_deterministic_rate_distortion_objective}
\begin{split}
\mathcal{W}^* & = \operatorname*{arg\,max}_{\mathcal{W}} ~ (1-\beta) \left\langle U(\boldsymbol\xi,\mathbf{f}(\mathcal{W},\boldsymbol\xi)) \right\rangle _{p(\boldsymbol\xi)} \\ & - \beta I(\boldsymbol\xi,\mathbf{y}) \\
& = \operatorname*{arg\,max}_{\mathcal{W}} \ \left\langle (1-\beta) U(\boldsymbol\xi,\mathbf{f}(\mathcal{W},\boldsymbol\xi)) \right\rangle_{p(\boldsymbol\xi)} \\
& - \left\langle \beta \sum_{j} f_j(\mathcal{W},\boldsymbol\xi) \ln \frac{f_j(\mathcal{W},\boldsymbol\xi)}{\bar{f}_j(\mathcal{W})} \right\rangle_{p(\boldsymbol\xi)} ,
\end{split}$$
Assuming that synaptic weights are updated via gradient ascent, the following weight update rule can be derived $$\label{eq:global_network_deterministic_derivative_rate_distortion_objective}
\begin{split}
& \frac{\partial}{\partial w^n_i} L(\mathcal{W}) = \\
& \left\langle (1-\beta) \sum_j \left( \frac{\partial}{\partial w^n_i} f_j(\mathcal{W},\boldsymbol\xi) \right) \left( \frac{\partial}{\partial f_j} U(\boldsymbol\xi,\mathbf{f}(\mathcal{W},\boldsymbol\xi)) \right) \right\rangle_{p(\boldsymbol\xi)} \\
& - \left\langle \beta \sum_j \left( \frac{\partial}{\partial w^n_i} f_j(\mathcal{W},\boldsymbol\xi) \right) \ln \frac{f_j(\mathcal{W},\boldsymbol\xi)}{\bar{f}_j(\mathcal{W})} \right\rangle_{p(\boldsymbol\xi)} ,
\end{split}$$ where $ \frac{\partial}{\partial w^n_i}$ denotes the derivative with respect to the $i$th weight of neuron $n$, and $\frac{\partial}{\partial f_j} U(\boldsymbol\xi,\mathbf{f}(\mathcal{W},\boldsymbol\xi))$ denotes the derivative of the utility function with respect to the firing rate of the $j$th output neuron. Equation requires to differentiate two terms with respect to $w^n_i$. The derivative of the expected utility is straightforward while the derivative of the mutual information is explained in Section \[sec:derivative\_mutual\_info\_global\].
Note that the derivative of the rate distortion objective $\frac{\partial}{\partial w^n_i} L(\mathcal{W})$ takes a convenient form which can be easily computed by extending ordinary backpropagation [@LeCun1998]. In ordinary backpropagation, the quantity $\frac{\partial}{\partial f_j} U(\boldsymbol\xi,\mathbf{f}(\mathcal{W},\boldsymbol\xi))$ is propagated backwards through the network. The core algorithm of ordinary backpropagation can be employed for computing $\frac{\partial}{\partial w^n_i} L(\mathcal{W})$ by simply replacing the derivative of the utility function $\frac{\partial}{\partial f_j} U(\boldsymbol\xi,\mathbf{f}(\mathcal{W},\boldsymbol\xi))$ with the more general quantity $(1-\beta)\frac{\partial}{\partial f_j} U(\boldsymbol\xi,\mathbf{f}(\mathcal{W},\boldsymbol\xi)) - \beta \ln \frac{f_j(\mathcal{W},\boldsymbol\xi)}{\bar{f}_j(\mathcal{W})}$.
EXPERIMENTAL RESULTS: MNIST CLASSIFICATION {#sec:experimental_results}
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In our simulations, we applied both types of rate distortion regularization (the local type from Section \[sec:consortium\] and the global type from Section \[sec:global\_decision\_maker\]) on the MNIST benchmark classification task. In particular, we investigated in how far this information-theoretically motivated regularization subserves generalization. To this end, we trained classification on the MNIST training set, consisting of $60,000$ grayscale images of handwritten digits, and tested generalization on the MNIST test set, consisting of $10,000$ examples. For all our simulations, we used a network with two hidden layers of rectified linear units [@Glorot2011] and a top layer of 10 softmax units implemented in Lua with Torch [@Collobert2011]. We chose as optimization criterion the negative cross entropy between the class labels and the network output [@Simard2003] $$\label{negative_cross_entropy}
U(\boldsymbol\xi,\mathbf{f}(\mathcal{W},\boldsymbol\xi)) = \sum_j \delta_{j l(\boldsymbol\xi)} \ln f_j(\mathcal{W},\boldsymbol\xi),$$ where $\delta$ denotes the Kronecker delta and $\boldsymbol\xi$ the vectorized input image—note that pixels were normalized to lie in the range $[0;1]$. The variable $j \in ~ \{1, 10\}$ is an index over the network’s output units and $l(\boldsymbol\xi) \in ~ \{1, 10\}$ denotes the label of image $\boldsymbol\xi$.
In order to assess the robustness of our regularizers, we performed our experiments with networks of different architectures. In particular, we used network architectures with two hidden layers and varied the number of neurons $\#_{\text{neu}} \in \{529,1024,2025,4096\}$ per hidden layer. We performed gradient ascent with a learning rate $\alpha = 0.01$ updating weights online after each training example. We trained the networks for 70 epochs where one epoch corresponded to one sweep through the entire training set. After each epoch, the learning rate decayed according to $\alpha \leftarrow \frac{\alpha}{1 + t \cdot \eta}$ where $t$ denotes the current epoch and $\eta = 0.002$ is a decay parameter. Weights were updated by use of a momentum $\gamma=0.9$ according to $\Delta w_i^n \leftarrow \gamma \Delta w_i^n + (1-\gamma) \frac{\partial}{\partial w_i^n} L(\mathcal{W})$ and were randomly initialized in the range $(-(\#_\text{in}(n))^{-0.5}; (\#_\text{in}(n))^{-0.5})$ with help of a uniform distribution at the beginning of the simulation where $\#_\text{in}(n)$ denotes the number of inputs to neuron $n$. Each non-input neuron had an additional bias weight that was initialized in the same way as the presynaptic weights of that neuron. Rate distortion regularization required furthermore to compute the mean firing rate $\bar{\phi}(\mathbf{w}^n)$ of individual neurons $n$ through an exponential window in an online fashion with a time constant $\tau = 1000$. In order to ensure numerical stability when using rate distortion regularization, terms of the form $\ln \frac{\phi({\mathbf{w}^n}^\top \boldsymbol\xi^n)}{\bar{\phi}(\mathbf{w}^n)}$ in the weight update rules were computed according to $\ln \max\{\phi({\mathbf{w}^n}^\top \boldsymbol\xi^n), \varepsilon\} - \ln \max\{\bar{\phi}(\mathbf{w}^n),\varepsilon\}$ with $\varepsilon = 2.22 \cdot 10^{-16}$.
{width="1\linewidth"}
To find optimal values for the rate distortion trade-off parameter $\beta$, we conducted pilot studies with small networks comprising 529 neurons per hidden layer that were trained for only 50 epochs on the MNIST training set according to the aforementioned training scheme and subsequently evaluated on the MNIST test set. While this might induce overfitting of $\beta$ on the test set in the small networks, we used the same $\beta$-values as a heuristic for all larger architectures and did not tune the hyperparameter any further. In global rate distortion regularization (Grdi), the best test error was achieved around $\beta=0.2$ although Grdi seems to behave rather robust in the range $\beta \in [0;0.8]$—see middle left panel in Figure \[fig:results\]. In local rate distortion regularization (Lrdi), the best test error was achieved for $\beta=0$ with ordinary utility maximization without regularization—see middle right panel. However, when measuring the performance in terms of expected utility on the test set, Lrdi achieved a significant performance increase compared to ordinary utility maximization in the range $\beta \in [10^{-5},5 \cdot 10^{-4}]$—see upper right panel. In our final studies, we could furthermore ascertain that Lrdi performs reasonably well on larger architectures as it achieved a test error of 1.26% compared to 1.43% in ordinary utility maximization when increasing the number of units per hidden layer to 4096.
[Method]{} [$\#_{\text{neu}}$]{} [$\text{Error [\%]}$]{}
----------------------------------- ----------------------- -------------------------
Bayes by backprop [@Blundell2015] 1200 1.32
Dropout [@Wan2013] 800 1.28
Dropconnect [@Wan2013] 800 1.20
Dropout [@Srivastava2014] 4096 **1.01**
529 1.36
Local rate distortion (Lrdi) 1024 1.34
$\beta=10^{-5}$ 2025 1.28
4096 **1.26**
529 1.23
Global rate distortion (Grdi) 1024 1.17
$\beta=0.2$ 2025 1.14
4096 **1.11**
: Classification Errors on the MNIST Test Set in the Permutation Invariant Setup[]{data-label="tab:best_results_non-conv"}
The results of our final studies where we trained networks for 70 epochs are illustrated in Table \[tab:best\_results\_non-conv\] which compares rate distortion regularization to other techniques from the literature for different network architectures comprising two hidden layers. It can be seen that both local and global rate distortion regularization (Lrdi and Grdi respectively) attain results in the permutation invariant setting (Lrdi: 1,26%, Grdi: 1.11%) that are competitive with other recent techniques like dropout (1.01% [@Srivastava2014] and 1.28% [@Wan2013]), dropconnect (1.20% [@Wan2013]) and Bayes by backprop (1.32% [@Blundell2015]). It is furthermore shown that both rate distortion regularizers lead to a decreasing generalization error when increasing the number of neurons in hidden layers which demonstrates successful prevention of overfitting. Successful prevention of overfitting is additionally demonstrated by applying global rate distortion regularization (Grid, $\beta=0.2$) to a convolutional neural network with an architecture according to [@Wan2013]—see Section B.2 in [@Wan2013]—attaining an error of 0.61% without tuning any hyperparameters (see Table \[tab:best\_results\_conv\]). This result is also competitive with other recent techniques in the permutation non-invariant setting—compare to dropout (0.59% [@Wan2013]) and dropconnect (0.63% [@Wan2013]). In line with [@Srivastava2014], we preprocessed the input with ZCA whitening and added a max-norm regularizer to limit the size of presynaptic weight vectors to at most $3.5$.
[Method]{} [$\text{Error [\%]}$]{}
----------------------------------- -------------------------
Conv net + Dropconnect [@Wan2013] 0.63
Conv net + Grdi ($\beta=0.2$) 0.61
Conv net + Dropout [@Wan2013] **0.59**
: Classification Errors on the MNIST Test Set in the Permutation Non-Invariant Setup[]{data-label="tab:best_results_conv"}
The lower panels of Figure \[fig:results\] show the development of the test set error over epochs for both rate distortion regularizers (red) compared to ordinary utility maximization without regularization (Umax, black) for the different network architectures that we used in the permutation invariant setting. It can be seen that the global variant of our regularizer (Grdi with $\beta=0.2$, see lower left panel in Figure \[fig:results\]) leads to a significant increase in performance across different architectures as demonstrated by the two separate clusters of trajectories. In addition, Grdi also leads to faster learning as the red trajectories in the lower left panel of Figure \[fig:results\] decrease significantly faster then the black trajectories during the first ten epochs of training. For the local variant of our regularizer (Lrdi with $\beta=10^{-5}$, see lower right panel in Figure \[fig:results\]), the performance improvements are less prominent when compared to the global variant.
CONCLUSION {#sec:conclusion}
==========
Previously, a synaptic weight update rule for a single reward-maximizing spiking neuron was devised, where the neuron was interpreted as a bounded rational decision-maker under limited computational resources with help of rate distortion theory [@Leibfried2015]. It was shown that such a bounded rational weight update rule leads to an efficient regularization by preventing synaptic weights from growing without bounds. In our current work, we extend these results to deterministic neurons and neural networks. On the MNIST benchmark classification task, we have demonstrated the regularizing effect of our approach as networks were successfully prevented from overfitting. These results are robust as we conducted experiments with different network architectures achieving performance competitive with other recent techniques like dropout [@Srivastava2014], dropconnect [@Wan2013] and Bayes by backprop [@Blundell2015] for both ordinary and convolutional networks. The strength of rate distortion regularization is that it is a more principled approach than for example dropout and dropconnect as it may be applied to general artificial agents with parameterized policies and not only to neural networks. Parameterized policies that optimize the rate distortion objective have been previously applied to unsupervised density estimation tasks with autoencoder networks [@SanchezGiraldo2013]. Our current work extends this kind of approach to the theory of reinforcement and supervised learning with feedforward neural networks, and also provides evidence that this approach scales well on large data sets.
APPENDIX {#sec:appendix}
========
MUTUAL INFORMATION RATE OF A DETERMINISTIC NEURON {#sec:mutual_info_rate_det_neu}
-------------------------------------------------
$$\label{eq:derivation_mutual_information_rate}
\begin{split}
& \lim_{\Delta t \rightarrow 0} \frac{1}{\Delta t} I(\boldsymbol\xi,y) \\
& = \lim_{\Delta t \rightarrow 0} \frac{1}{\Delta t} \left\langle \sum_y p_{\mathbf{w}}(y|\boldsymbol\xi) \ln \frac{p_{\mathbf{w}}(y|\boldsymbol\xi)}{p_{\mathbf{w}}(y)} \right\rangle_{p(\boldsymbol\xi)} \\
& = \lim_{\Delta t \rightarrow 0} \frac{1}{\Delta t} \left\langle \phi(\mathbf{w}^\top \boldsymbol\xi) \Delta t \ln \frac{\phi(\mathbf{w}^\top \boldsymbol\xi)}{\bar{\phi}(\mathbf{w})} \right\rangle_{p(\boldsymbol\xi)} \\
& + \lim_{\Delta t \rightarrow 0} \frac{1}{\Delta t} \left\langle (1-\phi(\mathbf{w}^\top \boldsymbol\xi) \Delta t) \underbrace{\ln \frac{1-\phi(\mathbf{w}^\top \boldsymbol\xi) \Delta t}{1-\bar{\phi}(\mathbf{w})\Delta t}}_{\rightarrow 0} \right\rangle_{p(\boldsymbol\xi)} \\
& = \left\langle \phi(\mathbf{w}^\top \boldsymbol\xi) \ln \frac{\phi(\mathbf{w}^\top \boldsymbol\xi)}{\bar{\phi}(\mathbf{w})} \right\rangle_{p(\boldsymbol\xi)} .
\end{split}$$
DERIVATIVE OF THE MUTUAL INFORMATION RATE {#sec:derivative_mutual_info_rate}
-----------------------------------------
$$\label{eq:derivative_mutual_information_rate}
\begin{split}
& \frac{\partial}{\partial w_i} \lim_{\Delta t \rightarrow 0} \frac{1}{\Delta t} I(\boldsymbol\xi,y) \\
& = \lim_{\Delta t \rightarrow 0} \frac{1}{\Delta t} \frac{\partial}{\partial w_i} \left\langle \sum_y p_{\mathbf{w}}(y|\boldsymbol\xi) \ln \frac{p_{\mathbf{w}}(y|\boldsymbol\xi)}{p_{\mathbf{w}}(y)} \right\rangle_{p(\boldsymbol\xi)} \\
& = \lim_{\Delta t \rightarrow 0} \frac{1}{\Delta t} \left\langle \sum_y \left( \frac{\partial}{\partial w_i} p_{\mathbf{w}}(y|\boldsymbol\xi) \right) \ln \frac{p_{\mathbf{w}}(y|\boldsymbol\xi)}{p_{\mathbf{w}}(y)} \right\rangle_{p(\boldsymbol\xi)} \\
& + \lim_{\Delta t \rightarrow 0} \frac{1}{\Delta t} \underbrace{ \left\langle \sum_y p_{\mathbf{w}}(y|\boldsymbol\xi) \left( \frac{\partial}{\partial w_i} \ln p_{\mathbf{w}}(y|\boldsymbol\xi) \right) \right\rangle_{p(\boldsymbol\xi)} }_{= \left\langle \sum_y \frac{\partial}{\partial w_i} p_{\mathbf{w}}(y|\boldsymbol\xi) \right\rangle_{p(\boldsymbol\xi)} = \frac{\partial}{\partial w_i} 1 = 0} \\
& - \lim_{\Delta t \rightarrow 0} \frac{1}{\Delta t} \underbrace{\left\langle \sum_y p_{\mathbf{w}}(y|\boldsymbol\xi) \left( \frac{\partial}{\partial w_i} \ln p_{\mathbf{w}}(y) \right) \right\rangle_{p(\boldsymbol\xi)} }_{ = \sum_y \frac{\partial}{ \partial w_i} p_{\mathbf{w}}(y) = \frac{\partial}{ \partial w_i} 1 = 0} \\
& = \lim_{\Delta t \rightarrow 0} \frac{1}{\Delta t} \left\langle \xi_i \phi'(\mathbf{w}^\top \boldsymbol\xi) \Delta t \ln \frac{\phi(\mathbf{w}^\top \boldsymbol\xi)}{\bar{\phi}(\mathbf{w})} \right\rangle_{p(\boldsymbol\xi)} \\
& - \lim_{\Delta t \rightarrow 0} \frac{1}{\Delta t} \left\langle \xi_i \phi'(\mathbf{w}^\top \boldsymbol\xi) \Delta t \underbrace{\ln \frac{1-\phi(\mathbf{w}^\top \boldsymbol\xi) \Delta t}{1-\bar{\phi}(\mathbf{w})\Delta t}}_{\rightarrow 0} \right\rangle_{p(\boldsymbol\xi)} \\
& = \left\langle \xi_i \phi'(\mathbf{w}^\top \boldsymbol\xi) \ln \frac{\phi(\mathbf{w}^\top \boldsymbol\xi)}{\bar{\phi}(\mathbf{w})} \right\rangle_{p(\boldsymbol\xi)} .
\end{split}$$
DERIVATIVE OF THE GLOBAL MUTUAL INFORMATION {#sec:derivative_mutual_info_global}
-------------------------------------------
$$\label{eq:derivative_mutual_information_global}
\begin{split}
& \frac{\partial}{\partial w^n_i} I(\boldsymbol\xi,\mathbf{y}) \\
& = \frac{\partial}{\partial w^n_i} \left\langle \sum_\mathbf{y} p_{\mathcal{W}}(\mathbf{y}|\boldsymbol\xi) \ln \frac{p_{\mathcal{W}}(\mathbf{y}|\boldsymbol\xi)}{p_{\mathcal{W}}(\mathbf{y})} \right\rangle_{p(\boldsymbol\xi)} \\
& = \left\langle \sum_\mathbf{y} \left( \frac{\partial}{\partial w^n_i} p_{\mathcal{W}}(\mathbf{y}|\boldsymbol\xi) \right) \ln \frac{p_{\mathcal{W}}(\mathbf{y}|\boldsymbol\xi)}{p_{\mathcal{W}}(\mathbf{y})} \right\rangle_{p(\boldsymbol\xi)} \\
& + \underbrace{ \left\langle \sum_\mathbf{y} p_{\mathcal{W}}(\mathbf{y}|\boldsymbol\xi) \left( \frac{\partial}{\partial w^n_i} \ln p_{\mathcal{W}}(\mathbf{y}|\boldsymbol\xi) \right) \right\rangle_{p(\boldsymbol\xi)}}_{= \left\langle \sum_{\mathbf{y}} \frac{\partial}{\partial w^n_i} p_{\mathcal{W}}(\mathbf{y}|\boldsymbol\xi) \right\rangle_{p(\boldsymbol\xi)} = \frac{\partial}{\partial w^n_i} 1 = 0 } \\
& - \underbrace{ \left\langle \sum_\mathbf{y} p_{\mathcal{W}}(\mathbf{y}|\boldsymbol\xi) \left( \frac{\partial}{\partial w^n_i} \ln p_{\mathcal{W}}(\mathbf{y}) \right) \right\rangle_{p(\boldsymbol\xi)}}_{ = \sum_{\mathbf{y}} \frac{\partial}{\partial w^n_i} p_{\mathcal{W}}(\mathbf{y}) = \frac{\partial}{\partial w^n_i} 1 = 0} \\
& = \left\langle \sum_j \left( \frac{\partial}{\partial w^n_i} f_j(\mathcal{W},\boldsymbol\xi) \right) \ln \frac{f_j(\mathcal{W},\boldsymbol\xi)}{\bar{f}_j(\mathcal{W})} \right\rangle_{p(\boldsymbol\xi)} .
\end{split}$$
### Acknowledgements {#acknowledgements .unnumbered}
This study was supported by the DFG, Emmy Noether grant BR4164/1-1.
### References {#references .unnumbered}
|
---
abstract: |
We investigate the distribution of the gas (ionized, neutral atomic and molecular), and interstellar dust in the complex star forming region NGC6357 with the goal of studying the interplay between the massive stars in the open cluster Pis24 and the surrounding interstellar matter.
Our study of the distribution of the ionized gas is based on narrow-band , , and images obtained with the Curtis-Schmidt Camera at CTIO, Chile, and on radio continuum observations at 1465 MHz taken with the VLA with a synthesized beam of 40. The distribution of the molecular gas is analyzed using $^{12}$CO(1-0) data obtained with the Nanten radiotelescope, Chile (angular resolution = 27). The interstellar dust distribution was studied using mid-infrared data from the GLIMPSE survey and far-infrared observations from IRAS.
NGC6357 consists of a large ionized shell and a number of smaller optical nebulosities. The optical, radio continuum, and near- and mid-IR images delineate the distributions of the ionized gas and interstellar dust in the regions and in previously unknown wind blown bubbles linked to the massive stars in Pis24 revealing surrounding photodissociation regions. The CO line observations allowed us to identify the molecular counterparts of the ionized structures in the complex and to confirm the presence of photodissociation regions. The action of the WR star HD157504 on the surrounding gas was also investigated. The molecular mass in the complex is estimated to be (4$\pm$2)$\times$10$^5$ . Mean electron densities derived from the radio data suggest electron densities $>$200 , indicating that NGC6357 is a complex formed in a region of high ambient density. The known massive stars in Pis24 and a number of newly inferred massive stars are mainly responsible for the excitation and photodissociation of the parental molecular cloud.
author:
- |
C.E. Cappa$^{1,2}$[^1], R. Barbá$^{3,4}$, N.U. Duronea$^{1}$, J. Vasquez$^{1,2}$, E.M. Arnal$^{1,2}$, W.M. Goss$^{5}$, and E. Fernández Lajús$^{2,6}$\
$^{1}$Instituto Argentino de Radioastronomía (CCT-La Plata, CONICET), C.C. No. 5, 1894 Villa Elisa, Argentina\
$^{2}$Facultad de Ciencias Astronómicas y Geofísicas, Universidad Nacional de La Plata, Paseo del Bosque s/n, 1900 La Plata, Argentina\
$^{3}$Instituto de Ciencias Astronómicas, de la Tierra y del Espacio (ICATE-CONICET), Av. España Sur 1512, J5402DSP San Juan, Argentina\
$^4$Departamento de Física, Universidad de La Serena, Cisternas 1200 Norte, La Serena, Chile\
$^{5}$National Radio Astronomy Observatory, P.O. Box 0, Socorro, NM 87801, USA\
$^{6}$Instituto de Astrofísica de La Plata (IALP-CONICET), Paseo del Bosque s/n, 1900, La Plata, Argentina
date: 'Accepted 2000 \*\*\*\*\*. Received 2010 \*\*\*\*'
title: ' A multifrequency study of the active star forming complex NGC6357. I. Interstellar structures linked to the open cluster Pis24'
---
\[firstpage\]
ISM: regions – ISM: individual objects: NGC6357 – clusters: open clusters – open clusters: individual: Pis24
Introduction
============
NGC 6357 ($\equiv$ W22 $\equiv$ RCW131 $\equiv$ Sh2-11) is a large complex located in the Sagittarius spiral arm. The complex consists of an incomplete large shell of about 60 in diameter, many other bright optical nebulosities in different evolutionary stages, many OB stars belonging to the open cluster Pis24, and bright infrared (IR) sources, some of them young stellar object (YSO) candidates [@lortet84; @felli90; @bohigas04; @wang07; @russ10].
@lortet84 showed that the large shell was a low-excitation and ionization bounded region. This shell, which can also be identified in images in the mid-IR [@wang07], opens to the north. The shell shows no evidence of expansion motions in optical lines [@lortet84] and has been interpreted as an ionized gas bubble created by the strong winds of the current massive stars in Pis24 or by a previous generation of stars [@lortet84; @bohigas04; @wang07].
The two brightest regions in the complex are G353.2+0.9 and G353.1+0.6. G353.2+0.9 is the brightest region at optical and radio wavelengths [e.g., @felli90]. Both extended radio sources were detected at several frequencies [e.g., @shaver70; @haynes78]. Detailed studies of these two bright regions were performed using VLA observations at 5 GHz with an angular resolution of 100 [@felli90]. Optical images of the whole NGC 6357 region show a number of additional ionized regions, with no previous optical and radio studies.
The distribution of the molecular gas associated with selected regions in the complex was investigated by @mcbreen83 and @massi97. These last authors found that the bulk of the molecular gas related to the complex has velocities in the range \[-14,+4\] .
The massive open cluster Pis24, which lies in the central cavity of NGC6357 and has an age of about 210$^6$ yr, is considered to be the only ionization source of the complex because of the huge number of UV photons emitted by the massive stars [@massey01]. The cluster contains at least a dozen O-type stars [@wang07], including two of the most luminous stars known in the Galaxy [@bohigas04]. These stars were classified O3.5If (Pis24-1) and O3.5III(f\*) (Pis24-17) by @walborn02. Recent studies by @maizapellaniz07 showed that Pis24-1 is a multiple system. The Wolf-Rayet star WR93 ($\equiv$ HD157406, WC7 + O7-9, @vanderhucht01) is also projected onto this region. Massey et al. (2001) consider this star to be a probable member of the cluster. However, the large uncertainties in reddening and distance determinations cast doubts on its relation to the cluster. In addition to the visible stars, massive OB stars hidden in areas with high reddening may also contribute to the ionization and photodissociation of the gas [@felli90].
The study by @persi86 of this region focussed on stellar formation activity. This study, along with more recent searches for YSOs in the region, resulted in the discovery of a large number of infrared and X-ray source candidates to YSOs and massive stars with high reddening [e.g., @bohigas04; @wang07], showing that the whole complex is an active area of recent and on-going star formation.
Distance determinations to the ionized complex span the range 1.7 to 2.6 kpc. @neckel78 and @lortet84 derived a distance of 1.7 kpc from UBV and H$\beta$ data, and from the nebular reddening, respectively. @massey01, based on recent spectral classification and absolute magnitude calibration, redetermined the distance to Pismis 24, adopting $(m-M)_o$ = 12.0 mag, or $d$ = 2.5 kpc. Following @massey01, we adopt a distance $d$ = 2.5$\pm$0.5 kpc for the complex.
With the aim of investigating the interplay between the massive stars belonging to Pis24 and the different components of the neighbouring interstellar material, we analyzed , and images obtained at CTIO, VLA radio continuum observations at 1.465 GHz, and CO(1-0) observations obtained using the NANTEN telescope, as well as IRAS archive images (IRAS-HIRES) at 60 and 100 $\mu$m, and GLIMPSE IRAC archive images in the near- and mid-IR. Optical line ratios are extremely useful in investigating the excitation conditions in different areas of the complex. High resolution radio images lead to the distribution of ionized gas in highly obscured regions, and in the derivation of the physical parameters of ionized regions. Images in the mid and far infrared lead to characterization of the interstellar dust in the complex. As a substantial amount of molecular gas is expected to be present in star forming regions, we also observed the whole complex in the CO(1-0) line at 115 GHz.
This is the first of a series of papers dealing with this complex. This paper focusses on some of the most interesting structures in the complex, excited by the massive stars in the open cluster Pis24. The bright regions G353.24+0.64 and G353.1+0.6 will be analyzed in a subsequent paper.
Our study provides new information about the interstellar dust, and the ionized and neutral gas distributions in the complex and the dust and gas masses. Interstellar bubbles, regions, and photodissociation regions (PDRs) excited by known members of the open cluster Pis24 and newly inferred massive stars in the complex are revealed.
Observations
============
Optical images
--------------
Narrow-band , , and , and broad-band $V$ and $R$ CCD images were obtained on May 1999, using the Curtis-Schmidt Camera at Cerro Tololo Inter-American Observatory (CTIO), Chile. The camera was equipped with a Site2K $2048\times2048$ array and has a pixel scale of 2.3 arcsec pixel$^{-1}$. The seeing during observations was typically about 1 arcsec giving a pronounced undersampling for point sources. The narrow-band images were obtained with filters[^2] centred at 5027Å, 6567Å, and 6744Å, with a FWHM of about 50Å, 68Å, and 50Å, respectively. The filter also includes some contamination of neighbouring \[NII\] nebular emission lines at 6548Å and 6584Å.
The individual images in each filter were corrected by bias level and flat-field, and then combined into a single mosaic using IRAF routines[^3]. These mosaics were registered by using hundreds of stars in the overlaping region. For each final mosaic an astrometric solution was found using stellar positions derived from the Guide Star Catalog 2.0. The absolute coordinate accuracy for each mosaic is better than 0.4 arcsec, although the typical relative uncertainty in the registration between images has been reduced to less than 0.1 pixels.
The surface brightness calibration process was performed using narrow-band images of Pis24 obtained with the Advanced Camera for Surveys (ACS) and the Wide Field and Planetary Camera 2 (WFPC2) on board the Hubble Space Telescope (HST). The datasets correspond to those obtained by Observing Programs No 9857 (PI: O. De Marco) and No 9091 (PI: J. Hester) for ACS and WFPC2 instruments, repectively. The ACS/WFC filter F658N ($\lambda_c \sim6584$, FWHM$\sim$73Å) has similar characteristics as the filter used in the Curtis-Schmidt camera. The WFPC2 filters F673N ($\lambda_c\sim6732$, FWHM$\sim$65Å) and F502N ($\lambda_c\sim5012$, FWHM$\sim$37Å) have also similar characteristics as the and filters used in the Curtis-Schmidt Camera. The relative surface brightness zero points were obtained using two areas of about 30 arcsec in common with ACS and WFPC2 images. Uncertainties in the relative calibrations for the Curtis-Schmidt filters are below 20%. This figure is derived from the comparison of measurements in adyacent areas.
Radio images
------------
### Radio continuum at 1.46 GHz
The distribution of the ionized gas was also investigated using radio continuum data. The surveyed field, which corresponds to the central region of the complex, was observed at 1.465 MHz (20cm) using the Very Large Array in the DnC configuration on 2000 July 3 and 5 as part of the AC555 observing program. The sources 1328+307 ($\equiv$ 3C286l, S$_{1.46GHz}$ = 14.9 Jy) and 1748-253 (S$_{1.46GHz}$ = 1.3 Jy) were used as primary and secondary flux density calibrators, respectively. The bandwidth was 50 MHz and the total integration time 2 hours. The coordinates of the field center are 17$^h$25$^m$856,–34111304 (J2000). The synthesized beam is 439343 at a position angle of +41.
The data were edited, calibrated, self-calibrated, and imaged using AIPS tasks. The rms in the central part of the image is 20 .
### CO line data
Intermediate angular resolution and medium sensitivity [CO]{} data were obtained with the 4-m [NANTEN]{} millimetre-wave telescope of Nagoya University. At the time of the observations this telescope was installed at the Las Campanas Observatory, Chile. The half-power beamwidth and the system temperature, including the atmospheric contribution towards the zenith, were 26 and $\sim$220K (SSB) at 115 GHz.
The data were gathered using the position switching mode. Observations of points devoid of [CO]{} emission were interspersed among the program positions. The coordinates of these points were retrieved from a database that was kindly made available to us by the [NANTEN]{} staff. The spectrometer used was an acusto-optical with 2048 channels providing a velocity resolution of $\sim$ 0.055 . For intensity calibrations, a room-temperature chopper wheel was employed @penzias73. An absolute intensity calibration was achieved by observing Orion [KL]{} (RA,Dec. \[J2000\] = 5$^h$40$^m$145, $-$5$^\circ$22493) and $\rho$ Oph East (RA,Dec. \[J2000\] = 16$^h$32$^m$228, –24$^\circ$28331). The absolute radiation temperature, T$_R^\ast$, of Orion [KL]{} and $\rho$ Oph East were assumed to be 65 K and 15 K, respectively [@ulich76; @kutner81].
The [CO]{} observations were carried out in April 2001 and the surveyed area is defined by 35251$\leq$ [*l*]{} $\leq$35395 and [^4]. An inner area defined by 35296 $\leq$ [*l*]{} $\leq$35350 and 059$\leq$ [*b*]{} $\leq$ 113[^5] was sampled at one beam width intervals, while the remaining area was sampled every 54 (two beamwidths). The total number of observed pointings is 349. The integration time per pointing was 16s resulting in a typical rms noise of $\Delta$T$_{rms}$$\sim$0.34 K.
Infrared data
-------------
The [**warm**]{} dust distribution was analyzed using high-resolution (HIRES) IRAS data obtained through [*IPAC*]{}[^6]. The IR data in the [*IRAS*]{} bands at 60 and 100 $\mu$m have angular resolutions of 15 and 17.
High angular resolution IRAC images from the Galactic Legacy Infrared Mid-Plane Survey Extraordinaire (GLIMPSE; @benjamin03) at 3.6 $\mu$m (with angular resolution $\phi$ = 15), 4.5 $\mu$m ($\phi$ = 17), 5.8 $\mu$m ($\phi$ = 18), and 8.0 $\mu$m ($\phi$ = 194) obtained with SPITZER were also used. Additional information on IRAC images is available from @fazio04 and from the Spitzer Science Center Observer Support Website[^7].
General characteristics of the complex
======================================
In this section we describe the general distribution of the ionized and neutral gas, while a detailed analysis of some of the most interesting regions will be discussed in the next sections.
Figure 1 shows a composite image of NGC 6357. is shown in green, in blue and in red. The filamentary structure of the ionized gas is evident. In addition to the large shell, clearly detected in and \[SII\] emissions, the complex shows numerous shell-like features, dust lanes and globules. In Fig. 2 we indicate the location of the most prominent features to facilitate their identification. The known regions G353.2+0.9, and G353.1+0.6, the large shell, and a number of other interesting optical structures, like G353.12+0.86 and G353.24+0.64 are indicated in Fig. 2. An elongated structure at RA,Dec.(J2000) = 172430,–3425 (referred to as Structure 1), an optical filament probably linked to WR 93, and the positions of WR93 and Pis24 are also indicated in the figure.
In this paper we investigate the characteristics of the region G353.2+0.9 and those of the ionized shell G353.12+0.86 and the large shell including Structure 1. The prominent regions G353.1+0.6 and G353.24+0.64 will be analyzed in a subsequent paper.
Figures 3 and 4 show the / and / line ratios. Based on the different excitation conditions, these images provide the evidence to distinguish whether the sources are HII regions, interstellar bubbles or PDRs. Low / and / line ratios are indicative of low excitation conditions associated with regions such as the large shell; on the other hand, higher / line ratios can be associated with interstellar bubbles. In this context, G353.12+0.86 is a stellar wind bubble.
Figure 5 displays the VLA image at 1.46 GHz, which covers most of the complex. A comparison with the optical images reveals the radio continuum counterparts of most of the optical features indicated in Fig. 2. In particular, the brightest regions G353.2+0.9 and G353.1+0.6, which were studied by @felli90 at 5.0 GHz, are easily identified. On the contrary, G353.2+0.7 (centered at 172535.52, –3416546), which was also investigated by @felli90, can not be identified in the radio continuum image as a distinct physical structure. The western border of this feature, as delineated by @felli90, is part of G353.12+0.86, while the eastern section may be part of G353.24+0.64 or of the large shell, as pointed out by @felli90. The radio counterpart of the shell-like structure G353.12+0.86 is first identified. The radio emitting region at 1725, –341220, located slightly to the southeast of the region G353.2+0.9, lacks an optical counterpart.
The upper panel of Fig. \[IR-1\] shows the distribution of the dust colour temperature T$_c$ (in contours and grayscale), while an overlay of the H${\alpha}$ emission distribution (in grayscale) and T$_c$ (in contours) is displayed in the lower panel. The dust colour temperature distribution was obtained from the $HIRES$ images at 60 and 100 $\mu$m, following the procedure described by @cichowolski01. Derived temperatures correspond to $n$ = 1.5. The parameter $n$ is related to the dust absorption efficiency ($\kappa_\nu\ \propto\ \nu^n$). We adopted $\kappa_\nu$ = 40 cm$^2$ g$^{-1}$, a value derived from the expressions by @hildebrand83. Dust temperatures range from $\sim$25 to $\sim$50 K, with the highest values close to G353.2+0.9, in agreement with previous results from @persi86. The region of low dust temperatures at 172515, –3418, encircled by regions with higher temperatures, coincides with the eastern section of the cavity of G353.12+0.86. Dust linked to the W arm of the large shell has a dust temperature of 30 K, while values as high as 50 K were derived for the environs of G353.2+0.9. Regions with high T$_c$ far from Pis24 (for example G353.1+0.6) are suggestive of the existence of unknown excitation sources [[@damke06]]{}.
The derived dust temperatures are typical for regions. A comparison with the optical image shows that areas with high dust temperature coincide with bright H$\alpha$ or radio continuum emitting regions. This behaviour is compatible with the fact that the stellar UV radiation field of the massive stars in the region is responsible for the heating of the interstellar dust.
Fig. \[IR-2\] displays a composite image of the brightest areas of NGC6357 using the IRAC images: 3.6 $\mu$m is in blue, 4.5 $\mu$m in green, and 8 $\mu$m in red. Emission at 3.6 $\mu$m originates in a faint and diffuse PAH feature at 3.3 $\mu$m and in dispersed stellar light; the 4.5 $\mu$m band shows emission from Br$\alpha$ and Pf$\beta$ and vibrational H$_2$ lines and roto-vibrational CO lines, typical from shocked gas [e.g., @churchwell06; @watson08]. Emission at 8 $\mu$m includes strong features from polycyclic aromatic hydrocarbons (PAHs), which are prominent. PAH emission and ionized gas emission (shown by the H$\alpha$ and 1.46 GHz images) have different spatial distributions, with the ionized gas located in the inner area of the structures, closer to the excitation sources than PAH emission. PAHs cannot survive inside regions [@cesarsky96], but they do on neutral PDRs at the interface between ionized and molecular gas. Thus, the emission at 8 $\mu$m shows that PDRs are widespread in this complex, a characteristic also indicative of the presence of molecular gas. PDRs are clearly delineated in this image (areas in magenta and white). The detection of CI radio recombination lines from several areas in NGC6357 is also indicative of the presence of PDRs (Quireza et al. 2006). The emission at 4.5 $\mu$m, coincident with the PDRs (detected at 8 $\mu$m) probably indicates the presence of shocked gas in these last regions. The brightest region in all IRAC bands is G353.2+0.9.
Elephant trunks are present throughout the region. Clear examples are located at 172500, –3414 and at 172450, –341050 (this last one first identified by @bohigas04).
Figure 8 displays a series of images showing the emission distribution within the velocity interval in steps of 2.5 covering the whole complex. The images show that the bulk of the molecular gas related to the complex has velocities in the range \[–10.0,0.0\] , coincident with previous findings by @massi97. The most outstanding feature is a CO depresion near the centre of the images, encircled by a ring-like structure of strong CO emission easily identified in the range \[$-$7.5,0.0\] (indicated as Shell A in Fig. 8). G353.12+0.86 coincides with the central cavity of Shell A.
In addition to this structure, other molecular clouds are identified in the CO images. These are labelled clouds B to H and appear to be linked to different nebulosities in the complex. They will be analyzed in Sect. 4 to 7.
The large scale CO emission distribution coincides with areas of high extinction and with the massive dense cores identified by @russ10.
After this general description, we analyze in the next sections the shell G353.12+0.86, the region G353.2+0.9, and the large shell.
The shell-like feature G353.12+0.86
===================================
The central region of NGC6357
-----------------------------
Two optical shells are apparent near the centre of Fig. 1. They are centred at 172435, and at 172513, , respectively. The central cavities and surrounding envelopes are clearly identifiable in the composite images displayed in Figs. 1, 2, 5, and 7. Figures 1 and 7 reveal that the intense emission that delineates the shells is almost completely encircled by emission at 8 $\mu$m. A comparison of Figs. 1 and 5 shows that both optical shells have radio counterparts.
The optical shells are separated by a dust cloud clearly discernible in Fig. 1 These images suggest that this dust cloud is a foreground object. In this scenario, both shells would be a unique structure with the most massive stars inside.
The CO emission distribution shown in Fig. 8 reveals a region without molecular emission centred at 172445,$-$3416, coincident with the position of the two optical cavities. As pointed out before, the CO hole is almost completely encircled by molecular gas with velocities in the range \[–7.5,0.0\] (indicated as Shell A in Fig. 8). A comparison with the optical image clearly shows that these two optical shells are surrounded by molecular material. Thus, the CO emission distribution confirms the presence of PDRs at the interface between the ionized and molecular gases.
A comparison of the distribution of the massive dense cores found by @russ10 with Shell A shows that 65% out of the $\sim$ 70 massive cores are projected onto the CO shell. Velocities of the most massive cores are in the range –2.0 to –4.3 , in agreement with the velocity of Shell A. This indicates that the molecular counterpart of G353.12+0.86 is a region of active star formation that has probably been triggered by the expansion of G353.12+0.86 (see for example @hes05).
The dust column that separates the optical shells can be identified as an elephant trunk pointing towards the centre of G353.12+0.86. The dust cloud coincides in position with a CO cloud having velocities in the range \[$-$12.5,$-$7.5\] (named cloud B in Fig. 8). However, the existence of molecular gas linked to this dust cloud with velocities can not be ruled out because of the poor angular resolution of the CO data in comparison with the optical and radio images. Our results are compatible with those by Massi et al. (1997), who found molecular gas linked to this dust cloud with velocities in the range \[$-$11.5,$-$6.5\] (the South-Eastern Complex, S.E.C). At least four high density molecular cores listed by @russ10 are projected onto this dust column. For the two most massive cores (\#112 and \#115 in their table 2) these authors find velocities of –5.19 and , compatible with the velocity of cloud B. The fact that the borders of the dust cloud are ionized, the presence of PAH emission at 8 $\mu$m adjacent to the ionized gas [**(see Fig. 7)**]{}, and the existence of molecular gas in the region, clearly indicates that molecular gas is beeing photodissociated by the UV photons of Pis24.
As pointed out by @massi97, the fact that the molecular gas linked to the dust cloud has velocities more negative than the CO associated with Shell A gives additional support to the suggestion that the dust cloud (and Cloud B) is in front of the shells. Consequently, both ionized shells may be one structure ionized by the UV photons of some of the massive stars in Pis24 and swept-up by the strong stellar winds. From here on, this structure of 20$\times$9 in size (14.5$\times$6.6 pc at 2.5 kpc), will be referred to as G353.12+0.86, being Shell A its molecular counterpart.
The molecular hydrogen column density $N_{H2}$ and the molecular mass associated with Shell A and cloud B were estimated from the $^{12}$CO data, making use of the empirical relation between the integrated emission $W_{CO}$ ($\equiv \ \int T dv)$ and $N_{H2}$. We adopted $N_{H2}$ = (1.9$\pm$0.3) W$_{CO}$ $\times$ 10$^{20}$ cm$^{-2}$ (K )$^{-1}$, obtained by @murphy91. The amount of molecular gas in Shell A between $-$7.5 and 0 is (1.2 $\pm$ 0.6) $\times$10$^5$ , while for Cloud B, we derive (3.4 $\pm$ 1.7) $\times$10$^3$ taking into account the emission in the range \[$-$12.5,$-$7.5\] . Velocity intervals and H$_2$ masses of the clouds identified in NGC6357 are listed in Table 2.
In the following paragraphs we analyze in some detail the characteristics of this structure.
Morphology and characteristics of G353.1+0.86
---------------------------------------------
The central cavity of G353.12+0.86 shows diffuse emission revealing that it is filled by hot gas at T $\approx$ 10$^4$ K. The / ratio derived for the cavity (in the range $\approx$ 0.09-0.12, Fig. 4) indicates a region with high excitation conditions. Pis24 is the main excitation source of this shell, since most of its massive stars are seen in projection onto the northern part of the cavity.
The northwestern section of the cavity is sharply bounded by the bright region G353.2+0.9 (this region is analyzed in Sect.5).
The western boundary of G353.12+0.86 (near 172416, –341138) is thick, with an intricate net of filaments detected in optical lines. The filaments are located in the inner part of the wall (Fig. 1). The radio continuum image (Fig. 5) displays faint and diffuse emission coincident in position with the area where the and filaments are distributed. Emission at 8 $\mu$m reveals a complex network of filaments, mixed with those observed at (Figs. 1 and 7). The emission distribution at different wavelengths in this region suggests that PDRs and optical filaments at different distances along the western wall of the shell are observed.
The southweastern rim of the cavity, near 172430,$-$3419, is sharp and bright in optical lines, in the radio continuum, and at 8 $\mu$m (Figs. 1, 5, and 7). A close inspection of Fig. 1 shows that the emission in the different optical bands does not coincide in position, being the bright emission closer to the excitation sources than the emission, while the strong emission is located slightly to the south of the emission. The and the radio continuum emissions are closely coincident. The emission distribution at different wavelengths indicates the existence of an excitation gradient, with the high excitation regions closer to the massive stars in Pis24 than the low excitation ones. The emission at 8 $\mu$m reveals a filament lying $\approx$12 south of the radio emission, indicating that a PDR has developed at the surface of Shell A. Emission at 3.6 and 4.5 $\mu$m is also present in these regions (Fig. 9), with the emission at 4.5 $\mu$m almost coincident with the emission. The location of the emission farther away from the excitation sources than the emission is typical in PDRs. A similar stratified ionization structure is present at the interface between the ionized gas and the molecular material in the pillars of M16, where PDRs have developed [@hester96].
East of 1725, the shell displays strong emission encircling the H$\alpha$ emitting region. No OB stars (see @massey01) are known to be projected onto this section of the cavity. Inspection of table 5 by @wang07 resulted in the identification of a dozen stars projected onto the cavity. Based on 2MASS colour-colour and colour-magnitude diagrams, these stars are A-type or later.
In summary, the southern, eastern, and western edges of G353.12+0.86 show a clear stratification, with the high excitation ionized gas closer to the massive stars in Pis24 that the low excitation ionized gas and the PAHs, pinpointing the last ones the interface between the ionized and molecular gas.
The emission distribution in all bands displayed in Figs. 1, 5 and 7 reveal that the shell is an interstellar bubble blown by the massive stars in the cluster Pis24. The interstellar gas has been swept-up and compressed onto the molecular wall by the stellar winds of the massive stars in the stellar cluster. The emission distribution resembles some of the IR bubbles described by @watson08 [@watson09].
The physical parameters of the ionized gas were estimated from the image at 1.46 GHz. These values were derived from the expressions by @mezger67. The ionized mass was multiplied by 1.27 to take into account the contribution of He singly ionized (10% He abundance). We adopted a distance of 2.5 kpc and an electron temperature of 8000 $\pm$ 1000 K, in agreement with estimates from radio recombination lines by @wilson70 and @quireza06. We have taken into account that the plasma is distributed in a shell of 1.4 width and covers 20% of the area of the shell. Derived values are listed in Table 1, where we include the flux density at 1.46 GHz, the emission measure EM, the size of the structure in arcmin and pc, the rms electron density and the mass and the ionized mass (including singly ionized He). Errors in electron densities and mases take into account an uncertainty of 20$\%$ in distance are about 50$\%$.
G353.2+0.9 and its close environs
=================================
Figure 5 reveals that the radio continuum emission in this region is concentrated in two bright clumps: the strongest one coincides with the optically bright region G353.2+0.9 (172447, –3410), while the other one is centered at (named G353.19+0.84 from hereon). A small blister linked to the O6.5V ((f)) star N36 ($\equiv$ Pis 24+3, @massey01) was detected by @bohigas04 inside G353.2+0.9. In the following subsections, we analyze the region G353.2+0.9, the small blister linked to N36, and G353.19+0.84.
G353.2+0.9
----------
A comparison between Fig. 1 and 5 suggests that and emissions correlate with the radio emission. The sharp southern boundary detected both at optical lines and in the radio continuum suggests that the ionized gas is being pushed by the massive stars in Pis24, located close to the southern border of the bright region [@bohigas04]. The emission distribution at 8 $\mu$m shows PAH emission in the IRAC 8 $\mu$m band projected onto the bright region (white area in Fig. 7).
Figure [**9**]{} displays an enlargement of this region in the IRAC bands at 3.6 (in blue), 4.5 (in green), and 8 $\mu$m (in red). The southern border of G353.2+0.9 is very bright at 4.5 and 8 $\mu$m (area in orange). Emission at 4.5 $\mu$m, detected in the southern area may originate in shocked gas. The elephant trunk detected at 172445, –341055 in optical data by Bohigas et al. (2004), which points directly towards the open cluster, radiates in the PAH features. The southern border of G353.2+0.9 and the pillar were analyzed in detail by @west10, who found kinematical evidence of strong interaction between Pis24 and the gas in the pillar. Figure 6 shows that the dust colour temperature in this region (near 172545,–3412) is higher than expected for an region, which is compatible with a region where shock and ionization fronts are present.
Figure 8 shows the existence of molecular gas probably related to this region with velocities in the range \[–7.5,+2.5\] . This material partially overlaps the region, extending to the north (Cloud C in Fig. 8). CO gas at negative velocities is clearly interacting with the nebula, while material at positive velocities seems to be located behind the ionized gas, in agreement with the detection of PAH emission superimposed to the ionized region. These results coincide with those by @massi97, who found molecular gas with peak velocities at –6 to the north and northeast of the nebula, and at –2 located behind the ionized gas, and with @russ10, who detected dense molecular cores with similar velocities (–3.9 and –5.6 ). The fact that the ionized region is very bright in optical lines is compatible with most of the dense molecular gas being to the north or behind the nebula, in agreement with previous findings by @bohigas04 and @wang07. The molecular gas distribution confirms the previous suggestion by Bohigas et al. that the region is ionization bounded.
The amount of molecular gas in the components peaking at $-$6 and $-$2 (which can not be separated in our data set and correspond to Cloud C), as estimated from our CO data, is (2.4$\pm$1.2)$\times$10$^4$ .
The physical parameters of the ionized gas, including electron densities and ionized masses derived from our radio continuum image, are summarized in Table 1. We have adopted a background emission of 0.5 to perform these estimates, and an electron temperature of 9500 K [@bohigas04]. The ionized mass is similar to that estimated by @bohigas04. Electron densities obtained by those authors from line ratios are about 2000 , higher than our estimates (410 ). An estimate of the filling factor $f$ can be obtained as $f$ = $\sqrt{n_{radio}/n_{opt}}$, where $n_{radio}$ is the r.m.s. electron density derived from radio data, and $n_{opt}$ is the local electron density estimated from line ratios, which is more sensitive to higher density regions. This parameter indicates the volume of the region that is really occupied by plasma. The derived value is $f \approx$ 0.5 and suggests that ionized gas is present in most of the volume of this region.
The blister related to N36
--------------------------
The presence of a small cavity created by the O6.5V ((f)) star N36 (17244568,–349399, with an optical extintion of about 6.5 mag) inside the brightest section of G353.2+0.9, at 172446, –34950, is evident in the and images, and in the ratio images, mainly in the / image (Fig. 4).
The cavity was already detected by @bohigas04 and @wang07. The value of the / ratio changes sharply from about 0.1 in the periphery to about 0.25 in the center of the blister. The / line ratio, which is almost uniform accross the whole region, suggests that the change in the / ratio is mainly due to local extinction.
Figure 9 reveals a complex net of filaments at 3.6 and 8 $\mu$m (areas in magenta), which also depicts diffuse emission at 4.5 $\mu$m. Most of these concentric filaments seems to have a common centre, located close to the position of N36. Small dust regions are also detected in connection with these filaments.
The cavity around N36 can be also identified in the 5GHz VLA image obtained with a synthesized beam of 35 by @felli90. Our VLA image is consistent with these findings, since the area of highest radio emission is projected towards the south of the blister.
G353.19+0.84
------------
The emission at 1.46 GHz extends to the southeast of G353.2+0.9, where an ionized clump whose brightest section is centered at 172500, –341220 can be identified (Fig. 5). This clump was barely detected in the image at 5 GHz obtained by @felli90 as a small region of low emission. It can not be identified at optical lines (Fig. 1).
This ionized region is behind an area of strong visual absorption described in Sect. 4. The ionized clump is projected onto a ring of PAH emission evident both in Figs. 4 and 9. The emission in the far IR at 60 $\mu$m (not shown here) is relatively strong in all the area. These facts suggest that cold and large interstellar grains are mixed with the ionized gas, where PAHs are destroyed.
Part of Cloud B, detected in the range , may be either linked to this ionized clump or located in front of it.
A striking elephant trunk pointing towards Pis24 appears projected onto the southern border of this radio source (at 172459.8, –341250). The IR point source located at the top of the pillar (172459.4,–3412498) might be an evaporating green globule (EGG), as the ones described by @hester96.
The existence of a PDR bordering the ionized region can not be ruled out.
The x-ray source at 172455.85, –3412340 (number 654 in the list by @wang07), which has a bright stellar counterpart, lie close to the borders of the strong radio source. The MSX sources G353.1998+00.8506 and G353.2021+00.8313, also projected near the borders of the ionized region, can be classified as MYSOs and CHII, respectively, and might be the excitation sources of this region.
The main physical parameters of the ionized gas in this region are summarized in Table 1. A background emission of 0.3 and T$_e$ = 9500 K were adopted.
Finally, parameters for the whole G353.2+0.9 region including G353.19+0.84, and the low level emission region that surrounds these sources are also included in Table 1. An electron temperature of 9500 K was assumed.
---------------- --------------- -------------------- ----------------- ----------------- ------- ------------ ------------ -----------------
$S_{1.46GHz}$ $EM$ 2R 2R $n_e$ $M_{HII}$ $M_i$ log N$_{Ly-c}$
Jy 10$^5$pc cm$^{-6}$ pc M$_{\sun}$ M$_{\sun}$
G353.12+0.86 34.5 0.7 20$\times$9 14.5$\times$6.6 200 240 330 50.0
G353.2+0.9 41.0 7.8 5.4$\times$3.3 3.9$\times$2.4 410 160 200 49.3
G353.19+0.84 5.0 1.5 4.2$\times$2.9 3.1$\times$2.1 190 40 50 48.5
Whole area$^1$ 93.6 3.0 14.6$\times$5.4 10.6$\times$3.9 200 710 900 49.7
---------------- --------------- -------------------- ----------------- ----------------- ------- ------------ ------------ -----------------
[1 Includes G353.2+0.9, G353.19+0.84, and lower emission areas]{}
A ring nebula related to WR93?
==============================
The bright WR star HD157406 (RA,Dec.(J2000) = 17250888,–3411128) is a probable member of Pis24 [@massey01]. Spectroscopic studies estimate a mass loss rate of 2.5$\times$10$^{-5}$ and a wind terminal velocity of 2290 for the star [@prinja90; @vanderhucht01]. With these parameters, it is expected that the massive star will deeply perturb the surrounding gas.
The star is projected close to a bright filament seen in optical lines and in the radio continuum at 1.46 GHz, extending from 17256, –34625 to 172515, –3412 (the filament is indicated in Figs. 2 and 5). The filament is particularly bright in and was interpreted as part of a ring nebula related to the WR star
[@marston94].
The emission at 8.3 $\mu$m delineates the eastern border of the filament, reinforcing the suggestion that it is being excited by massive stars located to the W of the filament, where the WR star is located.
The complexity of the gas distribution in NGC6357, the large number of shells, filaments and dust patches (in particular near the WR star) insure that the identification of an interstellar bubble created by the strong stellar winds of this star is unlikely.
The large shell
===============
The thick E and W arms of the large shell are clearly identified in the composite image of Fig. 1, with the W arm bending to the east near Dec.(J2000) = –3427. The ionized filament detected from 172330, –3427 to approximately 1725, –3424 (named Structure 1 in Figs. 2 and 5) is about 13020 in size. The large shell, of about 60 in size (or 44 pc at 2.5 kpc) opened to the north, is mainly detected in emission. Most of the filaments are also easily identified in the faint emission (e.g. near RA,Dec.(J2000) = 172556,–332510). The emission is an order of magnitude lower than the emission. The high / line ratios ($>$0.17, Fig. 4) of the large shell and the lack of emission confirm the low excitation conditions, as previously suggested by [@lortet84]{}.
A comparison of the CO emission distribution at different velocities (Fig. 6) with the emission shows molecular material probably related to the large shell to the east, north and south of the nebula. The E arm of the large shell appears bounded by CO emission located near 172640, –345 with velocities in the range (Cloud D in Fig. 8). This cloud, of about 25 in length, consists of at least three bright clumps and is projected onto a faint emission region. Many dense cores are projected onto two of these clumps [@russ10]. We note that part of the material of the CO clumps may belong to Shell A. The coincidence of the molecular emission region with a region of low optical emission sugests that the molecular gas and the associated interstellar dust are in front of the optical filaments, in agreement with the existence of high extinction regions [@russ10]. Due to the relatively small field of view, our radio continuum image does not include the E arm and, consequently, it is not clear if optical emission related to this arm is present behind Cloud D.
Towards the north, strong CO emission is also present between –5 to 0 , at –3345, extending from 1723 to 172630 (Cloud E in Fig. 8), and near 1723,–3356 in the range \[–12.5,–7.5\] (Cloud F in Fig. 8). This material is placed to the north of the optical filaments of the nebula. Cloud F probably make expansion of the ionized gas towards the north difficult.
Structure 1 is detected at 1.46 GHz (Fig. 5) and in and lines, being brighter the easthern extreme (at 172505, –3424). The lack of emission is compatible with low excitation conditions. Emission in the IRAC band at 8 $\mu$m borders the southern part of Structure 1 (Fig. 7), extending to the east up to 172525, –342320, behind a dust cloud. The strong emission at 8 $\mu$m to the south of the ionized gas reveals the existence of a PDR at the interface between the ionized and molecular gas. Structure 1 is projected onto molecular material detected in the range \[–2.5,0\] (Cloud G in Fig. 8), at 172430, –3430. CO velocities coincide with the velocity of the ionized gas ($\approx$–7 ), as shown by RRL observations obtained by Quireza et al. (2006) towards this area.
Strong CO emission with velocities in the range is present at 1725, –3425 (Cloud H). Part of this material is most probably connected to G353.1+0.6.
The amount of molecular gas associated with the large shell was estimated by integrating the CO emission in the velocity interval \[–12.5,0\] . A molecular mass of (5.7$\pm$2.8)$\times$10$^4$ was derived for Cloud D, (4.5$\pm$2.2)$\times$10$^4$ for Cloud E, (9.5$\pm$4.7)$\times$10$^3$ for Cloud F, (2.6$\pm$1.3)$\times$10$^4$ for Cloud G, and (3.6$\pm$1.8)$\times$10$^3$ for Cloud H. The total amount of molecular gas connected to the outer shell is 1.4$\times$10$^5$ .
It is worth mentioning that the angular resolution of the CO data is 7 times larger than that of the radio image, thus making it difficult the association of CO structures with both optical and radio features.
The origin of the large shell was discussed by Wang et al. (2007). This shell may have originated in the massive stars of Pis24. We can not discard the fact that the massive progenitor of the WR star have contributed to the shaping of the outer shell (e.g. Wang et al. 2007).
------------------- --------------- -------------
(v$_1$,v$_2$) M$_{H2}$
10$^3$
Shell A –7.5,0.0 120$\pm$60
Cloud B –12.5,–7.5 3.4$\pm$1.7
Cloud C –7.5,+2.5 24$\pm$12
[*Outer shell*]{}
Cloud D –7.5,0.0 57$\pm$28
Cloud E –5.0,0.0 45$\pm$22
Cloud F –12.5,–7.5 9.5$\pm$4.2
Cloud G –2.5,0 9$\pm$4
Cloud H –12.5,–7.5 3.6$\pm$1.8
------------------- --------------- -------------
: Parameters of the molecular gas
Masses, densities, and excitation sources
=========================================
The UV photons necessary to keep the gas of the different regions in the complex ionized can be estimated from the radio continuum emission. These values were derived using N$_{Ly-C }$(10$^{48}$s$^{-1}$) = 3.5110$^{-5}$n$_e^2$(cm$^{-3}$)R$^3$(pc). These results are listed in the last column of Table 1. Taking into account that at 25-50% of the UV photons produced by massive stars are absorbed by interstellar dust in regions [@inoue], a photon flux of about (3-8)$\times$10$^{50}$ s$^{-1}$ is necessary to maintain G353.12+0.86, G353.2+0.9, and G353.19+0.84 ionized. Ionized gas linked to other regions in the complex was not taken into account.
Bearing in mind photon flux estimates by @martins02 and Vacca et al. (1996), the massive stars in Pis24 can supply a UV photon flux of (1.4-3.3)$\times$ 10$^{50}$ s$^{-1}$. Although the massive stars in Pis24 are major contributors to the ionization of the nebula, in agreement with @massi97 and @bohigas04, additional massive stars should be identified in NGC6357 to explain the ionization of the gas in the whole complex.
An estimate of the total molecular hydrogen mass involved in the complex can be derived by integrating the CO emission in the range \[–12.5,+5.0\] , within the region displayed in Fig. 9. This value turns out to be (4$\pm$2)$\times$10$^5$ . Our estimate also includes molecular gas linked to G353.1+0.6 and G353.24+0.64
.
Summary
=======
In this paper we have investigated the distribution of the ionized, neutral gas, and interstellar dust towards NGC6357. Our goal was to study the interplay between the massive stars in the open cluster Pis24 and the surrounding interstellar matter.
The distribution of the ionized gas was analyzed using narrow-band , , and images obtained with the Curtis-Schmidt Camera at CTIO (Chile), and radio continuum observations at 1465 MHz taken with the VLA with a synthesized beam of 40. The distribution of the molecular gas and of the interstellar dust were studied using data obtained with the Nanten radiotelescope, Chile, and near-and mid-IR data from the GLIMPSE and IRAS surveys, respectively.
NGC6357 consists of a large ionized shell and numerous smaller shell-like features, dust lanes, globules and elephant trunks. / and / line ratios provide the evidence to distinguish among regions, interstellar bubbles, and PDRs. Thus, this study revealed new interstellar bubbles surrounded by photodissociation regions in the complex. Molecular observations allowed us to identify the molecular counterparts of the ionized structures in the complex and to confirm the presence of photodissociation regions.
The shell G353.12+0.86 is located near the centre of the complex. It is 15.06.8 pc in size at the adopted distance of 2.5 kpc. The shell is detected in and emission, as well as in the radio continuum at 1.46 GHz. The emission reveals that it is filled by hot gas at 10$^4$ K. PAH emission surrounds the ionized gas emission, indicating the presence of PDRs. A shell of molecular gas was identified in the CO emission distribution, confirming the presence of PDRs. A number of dense cores coincide with the CO shell, indicating that the shell is an active region of star formation, probably triggered by the expansion of G353.12+0.86.
The dust column that appears to separate the optical shell in two independient structures is most probably a foreground object. The difference in velocity between the molecular gas associated with the dust column (in the range \[–12.5,–7.5\] ) and the molecular gas linked to G353.12+0.86 (in the range \[–7.5,0.0\] strongly reinforces this interpretation. In this scenario, G353.12+0.86 is a unique structure with the massive stars of the open cluster Pis24 inside. The emission distribution in the optical, IR, and radio bands show that this structure is an interstellar bubble blown by the massive stars of Pis24. The interstellar gas has been swept-up and compressed onto the molecular wall.
G353.2+0.9 is the brightest region at optical, IR, and radio wavelengths. The fact that G353.2+0.9 is very bright both in radio continuum and in optical lines is compatible with the location of most of the dense molecular gas to the north or behind the nebula, confirming the previous suggestion by Bohigas et al. (2004) that the region is ionization bounded. Electron densities derived from our radio continuum image show that this is the region with the highest rms electron density in the complex. An estimate of the filling factor suggests that ionized gas occupies most of the volume of this region.
The synthesized beam of the radio continuum image allowed us to detect the ionized clump G353.19+0.84, 3.12.1 pc in size, located slightly to the southeast of G353.2+0.9. This ionized clump is behind an area of strong visual absorption and can not be detected in optical lines. It can be identified in the far IR emission and is partially projected onto a ring of PAH emission. These characteristics suggest that large interstellar grains are mixed with the ionizad gas, where PAHs are destroyed. Two MSX point sources classified as massive young stellar object and compact region might be the excitation sources of this region. A striking elephant trunk pointing towards Pis24 appears projected onto the southern border of this radio source.
The action of the WR star HD157504 on the surrounding gas was also investigated. The star may be linked to a bright filament seen in optical lines and in the radio continuum emission. The complexity of the gas distribution near the position insures that the identification of an interstellar bubble created by the strong stellar winds the WR star is unlikely.
The large shell, with a diameter of 44 pc at 2.5 kpc, opens to the north. It is detected in and emissions. The high / line ratios and the lack of emission confirm the low excitation conditions. Molecular gas having velocities in the range \[–12.5,+5.0\] appears related to the easthern, northern, and southern sections of the shell. The total amount of molecular gas connected to the large shell was estimated as 1.4$\times$10$^5$ . The massive progenitor of the WR star have probably contributed to the shaping of the large shell.
Mean electron densities derived from the radio data suggest electron densities excess 200 , indicating that NGC6357 is a complex formed in a region of high ambient density. The total molecular hydrogen mass involved in the complex is estimated as (4$\pm$2)$\times$10$^5$ .
Estimates of the UV photon flux emitted by the massive stars of Pis24 indicate that they are the main contributors to the ionization of the nebula. However, additional massive stars should be identified in NGC6357 to explain the ionization of the gas.
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank the anonymous referee for many helpful comments and suggestions, which helped to improve the presentation of this paper. C.E.C. acknowledge the kind hospitality during her stays at Universidad de La Serena, Chile. We acknowledge Jesús Maiz Apellaniz for allow us to use JMAPLOT software. The VLA is operated by the National Radio Astronomy Observatory. The NRAO is a facility of the National Science Foundation operated under a cooperative agreement by the associated Universities, Inc. This project was partially financed by the Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET) of Argentina under projects PIP 112-200801-02488 and PIP 112-200801-01299, Universidad Nacional de La Plata (UNLP) under project 11/G093, and Agencia Nacional de Promoción Científica y Tecnológica (ANPCYT) under project PICT 2007-00902.
This research has made use of the NASA/IPAC Infrared Science Archive, which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. The MSX mission is sponsored by the Ballistic Missile Defense Organization (BMDO). We acknowledge the use of NASA’s SkyView facility (http://skyview.gsfc.nasa.gov) located at NASA Goddard Space Flight Center. This research has made use of the SIMBAD database and ALADIN software, operated at CDS, Strasbourg, France.
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[^1]: Email:ccappa@fcaglp.fcaglp.unlp.edu.ar
[^2]: Passbands are plotted in\
http://www.ctio.noao.edu/instruments/filters/filters$_{34}$.html \[www.ctio.noao.edu\]
[^3]: IRAF is distributed by the National Optical Observatories which is operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation.
[^4]: In equatorial coordinates, the four vertixes of this region are RA,Dec. (J2000) = (17$^h$25$^m$5821, $-$35$^\circ$1044), (17$^h$20$^m$1126, $-$34$^\circ$2198), (17$^h$24$^m$694, $-$33$^\circ$1018), and (17$^h$29$^m$5135, $-$33$^\circ$58107).
[^5]: The four vertixes of this region are RA,Dec. (J2000) = (17$^h$25$^m$229, –34$^\circ$32320), (17$^h$23$^m$1309, –34$^\circ$14177), (17$^h$24$^m$4119, –33$^\circ$47321), and (17$^h$26$^m$5064, –34$^\circ$5404).
[^6]: [*IPAC*]{} is funded by NASA as part of the [*IRAS*]{} extended mission under contract to Jet Propulsion Laboratory (JPL) and California Institute of Technology (Caltech).
[^7]: Available at http://ssc.spitzer.caltech.edu/ost
|
---
abstract: 'A precise understanding of the radio emission from extensive air showers is of fundamental importance for the design of cosmic ray radio detectors as well as the analysis and interpretation of their data. In recent years, tremendous progress has been made in the understanding of the emission physics both in macroscopic and microscopic frameworks. A consistent picture has emerged: the emission stems mainly from time-varying transverse currents and a time-varying charge excess; in addition, Cherenkov-like compression of the emission due to the refractive index gradient in the atmosphere can lead to time-compression of the emitted pulses and thus high-frequency contributions in the signal. In this article, I discuss the evolution of the modelling in recent years, present the emission physics as it is understood today, and conclude with a description and comparison of the models currently being actively developed.'
author:
- |
T. Huege\
IKP, Karlsruhe Institute of Technology (KIT)
title: Theory and simulations of air shower radio emission
---
Introduction
============
Radio emission from cosmic ray air showers has been measured for the first time in the 1960s [@JelleyFruinPorter1965]. This started a flurry of activities both on the experimental and theoretical side; for a review of this early history, please see [@Allan1971]. However, difficulties in the interpretation of the measurements and limitations of the analogue detection technique led to a complete cease of activities in the 1970s.
The field was revived in the early 2000s by the LOPES [@FalckeNature2005] and CODALEMA [@ArdouinBelletoileCharrier2005] experiments. It was clear from their start that a sound understanding of the radio emission physics was imperative for the interpretation of measurements and the optimization of the experimental designs. Therefore, in parallel to the revival of experimental activities, also theorists started new efforts for the modelling of air shower radio emission.
Here, I give an overview of these “modern” efforts. I shortly review how the models have evolved and describe the approaches currently employed. Afterwards, I give a summary of the current understanding of the radio emission physics and conclude by presenting some results gathered by a task force aimed at comparing the existing models in detail.
The evolution of modern radio emission modelling
================================================
A driver for the modern efforts in radio detection of cosmic ray air showers has been the paper by Falcke & Gorham [@FalckeGorham2003]. It predicted that air shower radio emission should be measurable with digital radio detectors, and presented a first estimate of the field strengths to be expected on the basis of the “geosynchrotron” approach.
Obsolete approaches
-------------------
As a direct follow-up to this paper, the geosynchrotron concept was followed up with semi-analytic calculations in the frequency domain [@HuegeFalcke2003a] and with Monte Carlo calculations in the time domain [@SuprunGorhamRosner2003]. In parallel, full-fledged Monte Carlo simulation codes were developed, in particular REAS1 [@HuegeFalcke2005a; @HuegeFalcke2005b] and ReAires [@DuVernoisIcrc2005]. REAS1 made the transition to REAS2 [@HuegeUlrichEngel2007a] by replacing the parameterized air shower model with one based on detailed CORSIKA-derived histograms. Other semi-analytical calculations followed [@MeyerVernetLecacheuxArdouin2008; @ChauvinRiviereMontanet2010]. For a review of the evolution of these models in the time from 2003 to 2009, please see [@HuegeArena2008].
However, it turned out that all of these models were employing an incomplete modelling of the radio emission physics. They did not take the radiation associated with the varying number of charges in air showers into account correctly, and therefore gave inconsistent results. A particular signature of this inconsistency was that the models always predicted unipolar time-domain pulses, whereas other approaches predicted bipolar pulses. For a detailed discussion of the cause for and solution of these discrepancies, I kindly refer the reader to [@HuegeLudwigScholtenARENA2010]. Readers should be aware that the models listed in this section should be considered obsolete and should thus no longer be actively used. Still, many of the predictions that were made by these models, e.g. on coherence effects [@HuegeFalcke2003a], air shower geometry [@HuegeFalcke2005b], and energy and depth of shower maximum sensitivity [@HuegeUlrichEngel2008] were qualitatively correct.
Current approaches {#sec:currentmodels}
------------------
With the understanding of the inconsistencies in the models, the field has made tremendous progress. Nowadays, many different approaches exist, and they all agree in their qualitative description of the radio emission features. The approaches can be grouped in the two categories of *microscopic* and *macroscopic* models. The microscopic models follow individual particles (electrons and positrons) in an air shower and calculate the associated radio emission. Macroscopic models rather describe the radio emission physics on the basis of macroscopic quantities such as currents and net charge. Both have their virtues and limitations.
### Microscopic approaches
A number of microscopic models exist today. The formalisms used to calculate the electromagnetic radiation differ, but they have in common that no assumptions are made on the actual “radio emission mechanism”. The electromagnetic radiation is calculated on the basis of classical electrodynamics applied in a specific formalism. The radiation that is emitted is thus directly governed by the movement, and in particular acceleration, of the individual particles. (In this sense, it might be more appropriate to call these approaches “simulations” rather than “models”.) The following is a list of existing and actively maintained microscopic models with a short description for each of them. (Other microscopic models exist but are no longer actively maintained and thus not listed here.)
- [REAS3.1 [@LudwigHuege2010] uses the endpoint formalism [@JamesFalckeHuege2012] to calculate the radio emission on the basis of histogrammed CORSIKA showers (the same as in REAS2) in the time-domain. ]{}
- [SELFAS2 [@MarinRevenu2012] calculates the radio emission on the basis of a formalism separating the electric field into components for the static contribution, charge variation and current variation in the time-domain. Its air shower model is based on the universality of histogrammed CORSIKA simulations [@LafebreEngelFalcke2009], so no air shower simulation is needed for a radio simulation.]{}
- [CoREAS [@HuegeARENA2012a] employs the endpoint formalism as in REAS3.1, but calculates the radio emission directly in CORSIKA on a per-particle level, without any intermediate histogramming step. It also works in the time-domain.]{}
- [ZHAireS [@AlvarezMunizCarvalhoZas2012] employs the ZHS algorithm [@ZasHalzenStanev1992] on a per-particle level in AIRES in either the time- or the frequency-domain.]{}
All of these models include a realistic treatment of the atmospheric refractive index. The two “full Monte Carlo” simulations CoREAS and ZHAireS describe the underlying air shower with the highest degree of complexity.
### Macroscopic approaches
In contrast to the microscopic models, which calculate the emission from individual air shower particles and in principle make no assumptions on the emission “mechanism”, the macroscopic models calculate the emission as emanating from the “bulk features” of the particle distributions, such as currents, net charge or dipole moments. Their advantage is that they are faster and give useful insights in the emission physics. On the other hand, in some cases free parameters have to be set, and it is sometimes difficult to separate the different contributions [@JamesFalckeHuege2012].
- [MGMR [@ScholtenWernerRusydi2008] calculates the radio emission on the basis of time-varying transverse currents, time varying net charge and a time-varying dipole moment in the time-domain. The air shower model is based on parameterizations, some important parameters such as the drift velocity of the particles and the shower-disk thickness need to be chosen.]{}
- [EVA [@WernerDeVriesScholten2012] employs the same time-domain emission calculation as MGMR, but parameterizes the air shower particle distributions on the basis of individual CONEX simulations. It thus has no free parameters to set. The parameterizations are currently one-dimensional, but could be generalized to multiple dimensions.]{}
- [The semi-analytical model developed by Dave Seckel [@SeckelARENA2012] employs a similar approach as MGMR, but works in the frequency domain.]{}
Of these models, EVA also includes a realistic treatment of the atmospheric refractive index.
The emission physics
====================
Having shortly reviewed the models available and actively maintained today, it is instructive to discuss our current understanding of the radio emission physics.
Identified contributions
------------------------
While disentangling different “mechanisms” for the radio emission from extensive air showers (or any complex radiating system) is difficult and sometimes even misleading [@JamesFalckeHuege2012], it is still often instructive to try. For air shower radio emission, the following contributions could be identified:
- [The dominant contribution to the radio emission is of geomagnetic origin. The emission is linearly polarized with the electric field vector aligned in the direction of the Lorentz force ($\vec{v} \times \vec{B}$), independent of the observer location as shown in Figure \[fig:polpatterns\]a). The radiation can be understood as emanating from a time-varying transverse current. The current is a consequence of the equilibrium between acceleration of electrons and positrons in the geomagnetic field and deceleration by interactions with atmospheric molecules. The time-variation arises from the growth and decline of the number of charges during the air shower evolution. This is the mechanism that was already proposed in the 1960s by Kahn & Lerche [@KahnLerche1966].]{}
- [A second contribution to the radio signal arises from the time-variation of the net charge present in an air shower. This is the same mechanism responsible for radio emission of showers in dense media. In air, however, the contribution is only sub-dominant with respect to the geomagnetic emission. The emission is linearly polarized with the electric field vectors oriented radially with respect to the shower axis. Consequently, the polarization angle varies with observer location as presented in Figure \[fig:polpatterns\]b). The net charge in an air shower arises mainly because electrons are knocked out of atmospheric molecules and are swept along with the shower disk. It is time-varying also because of the growth and decline of the number of charges in the air shower. This is the emission mechanism proposed by Askaryan [@Askaryan1962a; @Askaryan1965].]{}
- [To which extent the originally proposed “geosynchrotron” emission, i.e. the radiation associated directly with the acceleration of electrons and particles in the geomagnetic field, contributes to the radio signal is not yet fully resolved; it is conceptually difficult to disentangle completely from the geomagnetic effects leading to the transverse currents. It might play a role at very high frequencies, where recent simulations have predicted polarization characteristics similar to what has been predicted in the earlier “geosynchrotron” models [@HuegeARENA2012a].]{}
- [It has been known for a long time that atmospheric electric fields can influence the radio emission from air showers, and the effects have also been studied theoretically [@BuitinkHuegeFalcke2010]. In recent modelling efforts, it could be shown that, depending on the relative orientation of electric field and shower axis, the transverse current and charge excess contributions can be amplified or dampened by atmospheric electric fields, see Figure \[fig:efieldeffects\].]{}
- [An effect first noticed by Dave Seckel in REAS3 simulations is a secondary pulse arising at later times for suitable air shower geometries. This arises from the ground impact of the particles, which are stopped instantaneously when reaching the observation level in the REAS3 simulation. A more realistic treatment should take into account the electric properties of the soil and model the “transition radiation” arising from this ground impact. This could well be relevant at kHZ frequencies.]{}
- [Some macroscopic models include other contributions such as a time-varying dipole or the ions which are left behind in the wake of the air shower. This might be seen as an illustration of the complexity of “summing up mechanisms”.]{}
![Alignment of the electric field vectors as a function of observer position (shower core at the center) for a vertical air shower. Left: transverse current emission, right: charge-excess emission. Both contributions are linearly polarized.[]{data-label="fig:polpatterns"}](patterns.eps){width="70.00000%"}
![North-south component of the electric field for an observer 250 m north of the core of a 30$^{\circ}$ inclined 10$^{16}$ eV air shower simulated at the LOPES site with CoREAS 1.0 in the presence of various vertical atmospheric electric fields.[]{data-label="fig:efieldeffects"}](efieldeffects.eps){width="70.00000%"}
In summary, the main mechanisms for the radio emission are the time-varying transverse currents and the time-varying net charge present in the air shower. The superposition of these differently polarized contributions directly leads to prominent asymmetries in the radio emission footprint (in particular an east-west asymmetry for vertical air showers), as is obvious from Figure \[fig:polpatterns\].
Refractive index effects
------------------------
Another effect is important for the radio emission from extensive air showers. The refractive index of the atmosphere is not unity but has a value of $\approx 1.000292$ at sea level and decreases with height as the atmospheric density decreases. The fact that the index is larger than unity means that for suitable geometries radiation emitted at different times and locations can reach a given observer at the same time. In such situations, the pulses are thus compressed in time, and consequently the frequency spectra of the emitted radiation can extend up to very high frequencies well in the GHz regime [@AlvarezMunizCarvalhoZas2012; @DeVriesBergScholten2011; @WernerDeVriesScholten2012]. Many such high-frequency results have recently been presented [@HuegeARENA2012a; @CarvalhoARENA2012; @ScholtenARENA2012], and they all agree qualitatively in their prediction of a “Cherenkov ring” appearing at high frequencies. The diameter of this ring is related to the geometrical distance between source and observer, and thus carries information on the air shower evolution. While it is adequate to label these refractive index effects “Cherenkov effects”, this should not be mixed up with what is usually referred to as classical “Cherenkov radiation” [@JamesFalckeHuege2012].
Where we stand today
====================
The modelling of radio emission from extensive air showers has come a long way, and with the resolution of the discrepancies present in the first “modern” approaches, a consistent picture has emerged. The goal of the modelers should be to understand the radio emission on a level on par with the systematic uncertainties of existing and future experiments, which means on a $\approx 10\%$ level.
Model-model comparisons
-----------------------
One way of judging how close the models are to reaching this goal is to compare them systematically. A task force involving authors of all models discussed in section \[sec:currentmodels\] has started to undertake this effort, initiated at a workshop at Ohio State University in early 2012. In preparation for the ARENA conference, these comparisons were updated once more, and I have the honor of presenting a small excerpt of these on behalf of the task force in Figures \[fig:comppulsesn1\], \[fig:compspectran1\] and \[fig:comppulsesnr\].
In these figures, it is striking that the results from CoREAS and ZHAireS are very similar. These two codes model the interaction and emission physics with the highest complexity, i.e. on a microscopic per-particle level, and are independent in their technical implementations of both the air shower simulation (CORSIKA vs. AIRES) and the radio emission calculation (endpoint formalism vs. ZHS algorithm). Their correspondence is thus a very promising indication that the models are indeed converging. REAS and SELFAS show slight deviations from the full Monte Carlos, which might be related to the fact that they are based on information from histogrammed particle distributions rather than individual particles. Another striking observation is that the macroscopic MGMR and EVA models predict significantly higher amplitudes, especially for observers close to the shower axis. These differences to the microscopic models are not yet understood.
![West-component of the electric field pulses for a vertical $10^{17}$ eV proton-induced air shower at the site of the Pierre Auger Observatory as simulated with the indicated models. The refractive index of the atmosphere was set to unity.[]{data-label="fig:comppulsesn1"}](pulsesn1.eps){width="\textwidth"}
![Absolute amplitudes of the frequency spectra for a vertical $10^{17}$ eV proton-induced air shower at the site of the Pierre Auger Observatory as simulated with the indicated models. The refractive index of the atmosphere was set to unity.[]{data-label="fig:compspectran1"}](spectran1.eps){width="\textwidth"}
![West-component of the electric field pulses for a vertical $10^{17}$ eV proton-induced air shower at the site of the Pierre Auger Observatory as simulated with the indicated models. The refractive index of the atmosphere was modelled according to atmospheric density.[]{data-label="fig:comppulsesnr"}](pulsesnr.eps){width="\textwidth"}
Model-data comparisons
----------------------
While comparisons between different models are important and studying the reasons for deviations will be imperative to further improving the models, it is comparisons with data which matter in the end. Unfortunately, currently no data set is publicly available for the modelers to compare their models against. It would be of great use if experiments could publish a data set with all necessary details (including not only air shower parameters such as energy and angles but also the exact core positions and relative antenna locations) so that modelers can use them to benchmark their predictions. Until then, comparisons carried out by the experimental collaborations [@LudwigARENA2012] will be the only way to judge the models.
Open questions
--------------
One issue that bothered modelers over the past year was the question whether the established calculations worked correctly very near the Cherenkov angle of individual particle tracks. A number of contributions to these proceedings have addressed this issue and the preliminary conclusion is that for the geometries relevant for air showers, the treatments seem to be valid [@JamesARENA2012; @BelovARENA2012]. Other aspects that should be studied in more detail than they have before include the transition radiation arising from the particle ground impact, the influence of realistically modelled atmospheric electric fields and possible reflection or scattering of radio waves on the particle plasma in the shower disk itself.
Conclusions
===========
The modelling of radio emission from cosmic ray air showers has made tremendous progress in the past few years. A consistent picture has emerged, and several microscopic and macroscopic models are available and agree in their qualitative predictions. In the next few years, the models have to be benchmarked against each other and against measured data, with the goal to reach and prove a modelling accuracy within the systematic uncertainties of experiments.
Acknowledgments {#acknowledgments .unnumbered}
===============
I would like to thank very much all colleagues who participated in the coordinated comparison of models of which I had the honor of presenting a small excerpt at the ARENA conference, in particular: J. Alvarez-Muñiz, W. Carvalho Jr., M. Ludwig, V. Marin, B. Revenu, D. Seckel and K.D. de Vries. Also, I would like to thank M. Gelb for his work on modelling electric field effects.
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---
abstract: 'A finitely generated group is lacunary hyperbolic if one of its asymptotic cones is an $\mathbb{R}$-tree. In this article we give a necessary and sufficient condition on lacunary hyperbolic groups in order to be stable under free product by giving a dynamical characterization of lacunary hyperbolic groups. Also we studied limits of elementary subgroups as subgroups of lacunary hperbolic groups and characterized them. Given any countable collection of increasing union of elementary groups we show that there exists a lacunary hyperbolic group whose set of all maximal subgroups is the given collection. As a consequence we construct a finitely generated divisible group. First such example was constructed by V. Guba in [@Gu86]. In section \[ripssection\] we show that given any finitely generated group $Q$ and a non elementary hyperbolic group $H$, there exists a short exact sequence $1\rightarrow N\rightarrow G\rightarrow Q\rightarrow 1$, where $G$ is a lacunary hyperbolic group and $N$ is a non elementary quotient of $H$. Our method allows to recover [@AS14 Theorem 3]. In section \[maximalvonneumann\], we extend the class of groups $\mathcal{R}ip_{\mathcal{T}}(Q)$ considered in [@CDK19] and hence give more new examples of property $(T)$ von Neumann algebras which have maximal von Neumann subalgebras without property $(T)$.'
author:
- Krishnendu Khan
title: Subgroups of Lacunary Hyperbolic Groups and Free Products
---
\[section\] \[theorem\][Lemma ]{} \[theorem\][Assumption ]{} \[theorem\][Proposition ]{} \[theorem\][Corollary ]{} \[theorem\][Definition ]{} \[theorem\][Definition ]{} \[theorem\][Notation ]{} \[theorem\][Example ]{}
Introduction
============
The interplay between general group theory and combinatiorial investigation of specific groups having some geometric structure has been exploited by Gromov by introducing and developing the notion of hyperbolic groups in his celebrated article [@Gr87]. In this influencing work Gromov proposed wide range research program.
Von Neumann introduced the notion of amenable groups in [@vN29] around 1929 and conjectured that a group $G$ is non amenable if and only if $G$ contains a free subgroup of rank $2$ after showing that no amenable group contains a free subgroup of rank $2$. This conjecture is shown to be false by A. Olshanskii in in [@Ol79]. Later in [@Ol93], A. Olshanskii developed small cancellation theory over hyperbolic groups and showed that one can construct non abelian groups without free subgroups by taking quotient of non elementary hyperbolic groups, which can be realized as a direct limit of hyperbolic groups. One can infact get non amenable groups in the limit by starting with property (T) hyperbolic group.
Intuitive idea behind an asymptotic cone of a metric space is to study “large scale” geometry of a metric space by viewing it from “infinite distance”. The term “asymptotic cone” was coined by Gromov in [@Gr81] to prove that a group of polynomial growth is virtually nilpotent. Later the notion of asymptotic cone was formally developed for arbitrary finitely generated groups by L. van den Dries, A.J. Wilkie in [@vdDW84]. Using asymptotic cones, one can characterize several important classes of groups. For example, groups of polynomial growth are precisely groups with all asymptotic cones locally compact \[[@Gr81; @Pau89; @D02]\].In [@Gr87] M. Gromov characterized hyperbolic groups in terms of asymptotic cones.
A finitely generated group is hyperbolic if and only if all of its asymptotic cones are $\mathbb{R}$-tree.
It was shown in [@DP99] that asymptotic cones of non-elementary hyperbolic groups are all isometric to the complete homogeneous $\mathbb{R}$-tree of valence continuum. Asymptotic cones of quasi-isometric spaces are bi-Lipschitz equivalent and hence the topology of an asymptotic cone of a finitely generated group does not depend on the choice of the generating set. For a survey of results on asymptotic cones and quasi isometric rigidity results see [@D02; @DS05]. In the paper [@OOS07], the authors have considered lacunary hyperbolic groups (see definition, sec. \[lacunarydef\]) and showed that there exists a finitely generated group, one of whose asymptotic cone $\mathcal{C}$ has non trivial countable fundamental group ($\pi(\mathcal{C})\equiv \mathbb{Z}$). We studied those limit groups in this article via hyperbolicity function (see section 2,3) and characterized lacunary hyperblic groups by hyperbolicity functions which measures the thinness of triangles in a geodesic metric spaces. We show that the thinness of triangles can grow sublinearly with respect to the size of perimeter of triangles in hyperbolic metric space and thinness of triangle grows sublinearly with respect to an increasing sequence of the size of perimeter of triangles in lacunary hyperbolic metric space.
Let G be a finitely generated group with generating set $S$. Then $G$ is lacunary hyperbolic if and only if the hyperbolicity function $f_{G}$ of the corresponding Cayley graph $\Gamma(G,S)$ satisfies $\liminf_{t\rightarrow\infty}\frac{f_{G}(t)}{t}=0$.
One advantage of having a dynamical characterization is that one can get a necessary and sufficient condition when lacunary hyperbolicity is preserved under free product. In general lacunary hyperbolicity is not stable under free product [@OOS07 Example 3.16]. See Definition \[synchrodef\] for the definition of synchronized LH.
Let $G_1=G* H$. Then $G_1$ is lacunary hyperbolic if and only if $G$ and $H$ are synchronized LH.
Maximal elementary subgroup $E(g)$ of a non elementary hyperbolic group $G$, containing an element $g$ of infinite order, is characterized by $E(g)=\{x\in G| \ x^{-1}g^nx=g^{\pm n} \ for \ some \ n=n(x)\in\mathbb{N}-\{0 \}\}$. $E(g)$ in a hyperbolic group. In section \[section4\] of this article we take this algebraic definition (denote by $E^{\mathcal{L}}(g)$) and characterize for lacunary hyperbolic groups.
Let $G$ be a lacunary hyperbolic group and $g\in G$ be an infinite order element. Then $E^{\mathcal{L}}(g)$ has a locally finite normal subgroup $N\triangleleft G$ such that:\
Either $E^{\mathcal{L}}(g)/N$ is an abelian group of Rank 1(i.e. $E^{\mathcal{L}}(g)/N<(\mathbb{Q},+,\cdot )$) or $E^{\mathcal{L}}(g)/N$ is an extension of a rank one group by involutive automorphism(i.e, $a\rightarrow a^{-1}$).
Conversely, for any rank one abelian group $E$, there exists a lacunary hyperbolic group $G$ with an infinite order element $g\in G$ such that $E=E^{\mathcal{L}}(g)$. More generally we obtained the following,
\[rankoneintro\] For any torsion free non elementary hyperbolic group $G$ and a countable family $\mathscr{F}:=\{Q^i_m\}_{i\in\mathbb{N}}$ of subgroups of $(\mathbb Q,+)$, there exists a non elementary, torsion free, non abelian lacunary hyperbolic quotient $G^{\mathscr{F}}$ of $G$ such that the set of all maximal subgroups of $G^{\mathscr{F}}$ is equal up to isomorphism to $\{Q^i_m\}_{i\in\mathbb{N}}$ i.e, every maximal subgroup of $G^{\mathscr{F}}$ is isomorphic to $Q^i_m$ for some $i\in\mathbb{N}$ and for every $i$ there exists a maximal subgroup of $G^{\mathscr{F}}$ that is isomorphic to $Q^i_m$.
Taking $\mathscr{F}=\{\mathbb{Q}\}$ in previous theorem we recover the following theorem by V.S. Guba,
[@Gu86 Theorem 1] There exists a non trivial finitely generated torsion free divisible group.
By taking $\mathscr{F}=\{\mathbb{Z}\}$ we get following,
[@Ol93 Corollary 1] Every non cyclic torsion free hyperbolic group $G$ has a non cyclic quotient $\bar{G}$ such that every proper non trivial subgroups of $\bar{G}$ is infinite cyclic.
A group $G$ has unique product property whenever for all pairs of non empty finite subsets $A$ and $B$ of $G$ the set of products $AB$ has an element $g\in G$ with a unique representation of the form $g=ab$ with $a\in A$ and $b\in B$. In [@RS87], Rips and Segev gave the first examples of torsion-free groups without the unique product property. Other examples of torsion free groups without unique product can be found in [@Pas77; @Pro88; @Car14; @Al91]. As a corollary of Theorem \[rankoneintro\] we get following,
For every rank one abelian group $Q_m$, there exists a non elementary, torsion free, property $(T)$, lacunary hyperbolic group without the unique product property $G^{Q_m}$ such that any maximal subgroup of $G^{Q_m}$ is isomorphic to $Q_m$.
In particular, there exist a non trivial property $(T)$ torsion free divisible lacunary hyperbolic group without the unique product property.
In subsection \[rankonebyfinitesection\] we discuss locally finite by rank one abelian subgroups of lacunary hyperbolic groups. We denote the class of increasing union of elementary subgroups as ${}_{rk-1}\mathscr{E}_{F}$ (see Notation \[notationelementary\]). Even though Theorem \[rankoneintro\] can be thought of as a corollary of the following theorem, we choose to discuss those in two different subsections in order to make a clear exposition.
Let $G$ be a torsion free non elementary hyperbolic group and $\mathscr{C}:=\{E^j \}_{j\in\mathbb{N}}$ be a countable collection of groups with $E^j\in {}_{rk-1}\mathscr{E}_{F}$ for all $j\geq 1$. Then there exists a non elementary lacunary hyperbolic quotient $G^{\mathscr{C}}$ of $G$ such that $\{E^{{\mathcal{L}}}(h)\ | \ h\in (G^{\mathscr{C}})^0 \}=\mathscr{C}$.
Moreover $\mathscr{C}$ is the set of all maximal proper subgroups of the group $G^{\mathscr{C}}$.
We also investigate Rips type construction in this article in order to construct lacunary hyperbolic groups from a given finitely generated group by closely following the proof strategy of E. Rips for hyperbolic group along with small cancellation condition developed by A. Olshanskii in [@Ol93], later in [@OOS07] by A.O’lshanskii, D. Osin and M. Sapir, and in [@Os'10] by D. Osin.
\[ripsintro\] Let $Q$ be a finitely generated group and $H$ be a non elementary hyperbolic group. Then there exist groups $G$ and $N$, for which we get a short exact sequence; $$1\rightarrow N\rightarrow G\rightarrow Q\rightarrow 1$$ such that;
1. $G$ is a lacunary hyperbolic group.
2. $N$ is a $2$ generated non elementary quotient of $H$.
3. If $H$ is torsion free then so are $G$ and $N$.
$G$ cannot be hyperbolic if $Q$ has no finite presentation.
Note that if one takes $H$ to be a torsion free, property $(T)$, hyperbolic group without the unique product property in the Theorem \[ripsintro\], then one can recover the following theorem (see, Section 5 for details),
Let $Q$ be a finitely generated group. Then there exists a short exact sequence $$\begin{aligned}
1\rightarrow N\rightarrow G\rightarrow Q\rightarrow 1\nonumber
\end{aligned}$$
such that;
- $G$ is a torsion-free group without the unique product property which is a direct limit of Gromov hyperbolic groups.
- $N$ is a subgroup of $G$ with Kazhdan’s Property $(T)$ and without the unique product property.
The group von Neumann algebra ${\mathcal{L}}G$ corresponding to a countable discrete group $G$ is the weak operator topology closer of the group algebra $\mathbb{C}G$ inside $\mathscr{B}(\ell^2 G)$, the space of bounded linear operators on the Hilbert space $\ell^2G$. The group constructed in Theorem \[rankoneintro\] can be used to construct interesting concrete examples of a group such that the corresponding group von Neumann algebra has property $(T)$ while all of its maximal subalgebras does not have property $(T)$. Theorem 4.4 in [@CDK19] shows how the Belegradek-Osin’s group Rips construction techniques and Ol’shanski’s type monster groups can be used in conjunction with Galois correspondence results for II$_1$ factors $\grave{a}$ $la$ Choda [@Ch78] to produce many maximal von Neumann subalgebras arising from group. In particular, many examples of II${_1}$ factors are constructed in [@CDK19] with property (T) that have maximal von Neumann subalgebras without property (T). In section \[maximalvonneumann\], we extend the class of groups $\mathcal{R}ip_{T}(Q)$ considered in [@CDK19] by generalizing [@CDK19 Theorem 3.10] and hence provide new examples of such groups (see section \[maximalvonneumann\] for details). For groups with property $(T)$ considered in Theorem \[rankoneintro\] there exists a property $(T)$ group $N$ such that $Q\hookrightarrow Out(N)$ and $[Out(N):Q]<\infty$ by [@BO06]. Denote $N\rtimes Q$, the groups getting from [@BO06] as described above, by the $\mathcal{R}ip_{T}(Q)$ for fixed $Q$. Note that $N\rtimes Q$ has property $(T)$.
$\mathcal M^i_m$ is a maximal von Neumann algebra of $\mathcal M$ for every $i$. In particular, when $N\rtimes Q\in \mathcal Rip_{\mathcal T}(Q)$ then for every $i$, $\mathcal M^i_m$ is a non-property (T) maximal von Neumann subalgebra of a property (T) von Neumann algebra $\mathcal M$.
Hyperbolic spaces
=================
Hyperbolic metric spaces
------------------------
I this section we shall mention equivalent definitions for hyperbolic spaces and their relations. In later sections we shall use whichever definition is convenient for the purpose.
Let $\delta\geq 0$. A geodesic triangle in a metric space is said to be $\delta$-slim if each of it’s side is contained in $\delta $ neighborhood of the union of other two sides. A geodesic metric space $X$ is said to be $\delta$-hyperbolic if all geodesic triangles in $X$ are $\delta$ hyperbolic. When a space is said to be hyperbolic it means that it is $\delta$-hyperbolic for some $\delta>0$.
Given a geodesic triangle (formed by joining three points $x,y,z\in X$ by geodesics, denote by $\triangle(x,y,z)$) in a metric space, the triangle equality tells us that there exists unique non-negative numbers $a,b,c$ such that $d(x,y)=a+b$, $d(x,z)=a+c$ and $d(y,z)=b+c$. Consider the metric tree $T_{\triangle}:=T(a,b,c)$, which consists of $3$ vertices with valency $1$ and $1$ vertices with valency 3, $3$ edges of length $a,b$ and $c$. There exists an isometry from $\{x,y,z\}$ to a subset of vertices of $T_{\triangle}$ (vertices with valency 1). Call these vertices $\{v_x,v_y,v_z\}$. This map $\{x,y,z\}\rightarrow \{v_x,v_y,v_z\}$ extends uniquely to a map $\chi_{\triangle}:\triangle \rightarrow T_{\triangle}$ such that restriction to each side of the triangle is an isometry. Denote $o_{\triangle}$ to be the central vertex of $T_{\triangle}$.
Let $\triangle$ be a geodesic triangle in a metric space $(X,d)$ and consider the map $\chi_{\triangle}:\triangle \rightarrow T_{\triangle}$ as defined above. Let $\delta\geq 0$. $\triangle$ is said to be $\delta$-thin if $p,q\in (\chi_{\triangle})^{-1}(t)$ implies $d(p,q)\leq \delta$ for all $t\in T_{\triangle}$. The diameter of $\chi_{\triangle}^{-1}(o_{\triangle})$ is denoted by insize $\triangle$.\
A geodesic metric space is called $\delta $-hyperbolic if there exists a $\delta\geq 0$ such that all geodesic triangles are $\delta$-thin. When a space is said to be hyperbolic it means that it is $\delta$-hyperbolic for some $\delta$.
These two definitions are equivalent but hyperbolicity constant may be different for two definitions. In that case one can take hyperbolicity as the maximum of hyperbolicity constants arising from different definition. For proofs we refer [@BH99], chapter $III$.\
Next property of hyperbolic space can be deduce from cutting hyperbolic $n$-gon into triangles.
[@Ol93]\[hyperbolicngon\] For $n\geq 3$, any side of a geodesic $n$-gon in a $\delta$-hyperbolic space belongs to the $(1+\log_2(n-1))\delta$ neighborhood of the other $(n-1)$ sides.
Ultrafilter and asymptotic cone
-------------------------------
In a metric space a segment is defined to be a subset isometric to an interval of the set of real numbers $\mathbb{R}$. A ray is defined to be a segment isometric to $[0,\infty)$.
An $\mathbb{R}$-tree is a complete metric space $T$ such that, for all points $x,y$ in $T$, there is a unique compact arc with endpoints $x$ and $y$, such that this arc is a segment.
A non principle ultrafilter $\omega$ is a finitely additive measure on all subsets $\mathcal{S}$ of natural numbers $\mathbb{N}$ such that , $\omega(\mathcal{S})\in \{0,1 \}, \omega(\mathbb{N})=1$ and $\omega(\mathcal{S})=0$ for all finite subset of $\mathbb{N}$.
For a bounded sequence of numbers $\{x_n \}_{n\in \mathbb{N}}$ the ultralimit $\lim^{\omega}_i$ with respect to $\omega$ is the unique number $a$ such that ; $$\omega(\{i\in \mathbb{N}| \ |a-x_i|<\epsilon \})=1, \ \ for \ all \ \epsilon>0$$ Similarly $\lim^{\omega}_i x_i=\infty$ if; $$\omega(\{i\in \mathbb{N}| \ x_i>M \})=1, \ \ for \ all \ M<\infty$$ For two infinite series of real numbers $\{a_n\}$, $\{b_n\}$ we write $a_n=o_{\omega}(b_n)$ if ${\lim^{\omega}}_i \frac{a_i}{b_i}=0$ and $a_n=\Theta_{\omega}(b_n)$ (respectively $a_n=\mathcal{O}_{\omega}(b_n)$) if $0 < {\lim^{\omega}}_i (\frac{a_i}{b_i})<\infty$ (respectively ${\lim_i^{\omega}} (\frac{a_i}{b_i})<\infty$).
Let $(X_n,d_n)$ be a metric space for every $n\in \mathbb{N}$. Let $\{e_n \}=:e$ be a sequence of points such that $e_n\in X_n$. Consider the set $\mathcal{F}$ of sequences $g=\{g_n \}$ where $g_n\in X_n$ such that $d_n(e_n,g_n)\leq C(g)$ $(C$ is a constant depending only on $g)$. Define an equivalence relation between elements of $\mathcal{F}$ as follows;
$f=\{f_n\}$ is equivalent to $h=\{h_n\}$ in $\mathcal{F}$ if and only if $\lim^{\omega}_id_i(f_i,h_i)=0$
The equivalence class of an element $\{g_n\}$ of $\mathcal{F}$ is denoted by $\{g_n\}^{\omega}$. The $\omega$-limit $\lim^{\omega}(X_n)_e$ is the quotient space of $\mathcal{F}$ by this equivalence relation and distance between two elements is defined as follows;
$Dist(f,g)=\lim^{\omega}d_i(f_i,g_i)$ for $f=\{f_i\}^{\omega}$, $g=\{g_i\}^{\omega}$ in $\lim^{\omega}(X_n,d_n)_e$.
An asymptotic cone $Con^{\omega}(X,e,l)$ for a metric space $(X,D)$, is an $\omega$-limit of metric space $X_n$ as above, where $X_n=X$ for all $n\in \mathbb{N}$, $d_n=\frac{D}{l_n}$ for a given non decreasing infinite sequence $l=\{l_n\}$ of positive real numbers and for a given sequence of points $e=\{e_n\}$ in $X$.
If $\{Y_n\}$ is a sequence of subsets of $X$ endowed with the induced metric, define $\lim^{\omega}(Y_n)_e$ to be the subset of $Con^{\omega}(X,e,l)$ consisting of $x\in Con^{\omega}(X,e,l)$ that can be represented by a sequence $\{x_n\}$, where $x_n\in Y_n$.
The asymptotic cone is a complete metric space. Moreover asymptotic cone $Con^{\omega}(X,e,l)$ is a geodesic metric space if $X$ is a geodesic metric space [@DS05; @Gr96].
The asymptotic cone of a finitely generated group $G$ with a word metric is asymptotic cone of it’s Cayley graph. Asymptotic cone of a finitely generated group with respect to two generating sets are bi-Lipscitz.\
A geodesic path $p$ in $Con^{\omega}(X,e,l)$ is called limit geodesic if $p=\lim^{\omega}(p_n([0,\lambda_n]))_e$ where $p_n:[0,\lambda_n]\rightarrow X$ is geodesic for all $n\in \mathbb{N}$.
[@DS05] Let $X$ be a complete geodesic metric space. Let $\mathcal{P}$ be a collection of closed geodesic non-empty subsets (called $''pieces''$). If the following properties are satisfied:\
$(T_1)$ Every two different pieces have at-most one common point.\
$(T_2)$ Every nontrivial simple geodesic triangle (A simple loop consists with 3 geodesics) in $X$ is contained in one piece.\
Then we say $X$ is tree-graded space with respect to $\mathcal{P}$.
$(T_2')$ Every non trivial simple loop in $X$ is contained in one piece.\
Here one can replace condition ($T_2$) by ($T_2'$).\
Here by loop we mean a path with same initial point and terminal point.
A loop $\gamma:[a,b]\rightarrow X$ is simple if $\gamma(t_1)\neq \gamma(t_2)$ for $t_1\neq t_2$ in $[a,b)$.
\[corollary 4.18, [@DS05b]\]\[limitgeodesictrg\] Assume that in an asymptotic cone $Con^\omega(X,e,d)$, a collection of closed subsets $\mathcal{P}$ satisfying ($T_1$) and every non trivial simple geodesic triangle in $Con^\omega(X,e,d)$ whose sides are limit geodesic is contained in a subset from the collection $\mathcal{P}$. Then $\mathcal{P}$ satisfy ($T_2$) i.e, $Con^\omega(X,e,d)$ is tree-graded with respect to $\mathcal{P}$.
\[asymptoticgromov\] A geodesic metric space is hyperbolic if and only if all of its asymptotic cones are $\mathbb{R}$-tree.
Hyperbolic group and hyperbolicity function
-------------------------------------------
Let $(X,d)$ be a geodesic metric space. For every positive real number $t$, define the following set: $$S_t:=\{ set \ of \ all \ geodesic \ triangles \ with \ perimeter \ \leq t \}$$ For a geodesic triangle triangle $\triangle$ with sides $A,B,C$ in $X$ define, $$\delta_{\triangle}:= \underset{p\in A\cup B\cup C}{\sup}d(p,union \ of \ two\ sides\ of\ \triangle \ not \ containing \ p)$$ Define the hyperbolicity function $f_X:[0,\infty)\rightarrow [0,\infty)$ by $f_X(t):=\underset{\triangle \in S_t}{\sup}\delta_{\triangle}$.
One can reformulate lemma $\color{blue}{\ref*{hyperbolicngon}}$ for a geodesic metric space in terms of it’s hyperbolicity function as follows,
\[hyperbolicgeodesicngon1\] For $n\geq 3$, any side of a geodesic $n$-gon $T_n$, in a geodesic metric space $X$, belongs to the closed $(1+\log_2(n-1))f_X(|T_n|)$ neighborhood of other $(n-1)$ sides, where $f_X$ is the hyperbolicity function of the geodesic metric space $X$ and $|T_n|$ is the perimeter of the $n$-gon $T_n$.
A function $f:[0,\infty)\rightarrow [0,\infty)$ is called sub-linear if $\lim_{t\rightarrow \infty }\frac{f(t)}{t}=0$.
\[hyptriangle\] A geodesic metric space $(X,d)$ is hyperbolic if and only if the hyperbolicity function $f_X$ is sub-linear.
If $X$ is $\delta$ hyperbolic space for some $\delta>0$, then by the definition of hyperbolicity function $f_X$ is bounded by $\delta$ and hence sub-linear.
Suppose the hyperbolicity function $f_X$ of $(X,d)$ is sub-linear. Lets assume that $(X,d)$ is not hyperbolic. By ${\color{blue}\ref*{asymptoticgromov}}$, there exists an asymptotic cone which is not $\mathbb{R}$-tree (Say, $Con^{\omega}(X,e,d=\{d_n\})$). Hence by definition of $\mathbb{R}$-tree, there exists a non-trivial simple loop in $Con^{\omega}(X,e,d)$. As in the definition of tree graded space ($T_1$) and ($T_2'$) are equivalent, we can assume that there exists a non-trivial simple geodesic triangle in $Con^{\omega}(X,e,d)$. According to lemma $\color{blue}{\ref*{limitgeodesictrg}}$ applied to the collection $\mathcal{P}$ of all one-element subsets of $Con^{\omega}(X,e,d)$, to show that $Con^{\omega}(X,e,d)$ is a tree it suffices
(1,0) – (5,0); (7,2) parabola (5,0); (-1,2) parabola (1,0); (7,2) parabola (4,5); (1,4) parabola (4,5); (-1,2) parabola (1,4); at (0.15,0.75)[$P^1_n=o_{\omega}(d_n)$]{}; at (5.85,0.75)[$P^2_n=o_{\omega}(d_n)$]{}; at (3,4.75)[$P^3_n=o_{\omega}(d_n)$]{}; (4.90,3.5) parabola (3,0); at (4.25,0.5)[$\leq \alpha_n$]{}; at (0,5)[$\mathbf{H_n}$]{}; at (1,0)[$a_n^{\prime}$]{}; at (5,0)[$b_n$]{}; at (7,2)[$b_n^{\prime}$]{}; at (-1,2)[$a_n$]{}; at (4,5)[$c_n$]{}; at (1,4)[$c_n^{\prime}$]{}; at (3,-0.25)[${A_n}$]{}; at (5,4)[${B_n}$]{}; at (0,3)[${C_n}$]{};
to prove that it contains no simple non-trivial limit geodesic triangles[^1].
Let $\triangle_{\infty}$ be a simple non-trivial limit geodesic triangle in $Con^{\omega}(X,e,d)$ with sides A, B and C with,
$\lim^{\omega}{A_n}=A,
\lim^{\omega}{B_n}=B,
\lim^{\omega}{C_n}=C$
where $A_n,B_n,C_n$ are geodesics in $X$ with endpoints $a_n',b_n$ for $A_n$ and $b_n',c_n$ for $B_n$ and $c_n',a_n$ for $C_n$, for all $n\in \mathbb{N}$. Let ${\triangle}_{\infty}$ is approached by hexagons $H_n$, formed by vertices $a_n{a_n}'b_n{b_n}'c_n{c_n}'$ in $X$ with $\ell (P^i_n)$[^2]$=o_{\omega}(d_n)$ for $i=1,2,3$, where, $ P^1_n$ is a geodesic joining $a_n,a_n'$ in $X$ and respectively $P^2_n$ is a geodesic joining $b_n,b_n'$ in $X$ and $P^3_n$ is a geodesic joining $c_n,c_n'$ in $X$.
Denote the perimeter of the hexagon $H_n$ by $|H_n|$. Let $A$ be a nontrivial side of $\triangle_{\infty}$. By lemma $\color{blue}{\ref*{hyperbolicgeodesicngon1}}$, $A_n$ belongs to closed $4f_X(|H_n|)$ neighborhood of other $5$ sides in $H_n$. In particular $A_n$ is contained in closed $\alpha_n$ neighborhood of $B_n\cup C_n $, where $$\begin{aligned}
\label{A}
\alpha_n=4f_X(|H_n|) + \max_{i=1,2,3}\{\ell(P^i_n) \}.
\end{aligned}$$ We have, $$\begin{aligned}
\label{B}
\ell(P^i_n)=o_{\omega}(d_n)\\
\ell(A_n) = \Theta_{\omega}(d_n)\nonumber
\end{aligned}$$ Hence we get $|H_n|=\Theta_{\omega}(d_n)$. As the function $f_X$ is sub-linear, we also have, $$\begin{aligned}
\label{C}
f_X(|H_n|)=o_{\omega}(|H_n|)=o_{\omega}(d_n).
\end{aligned}$$ Finally we get that $A$ is contained in the union of other two sides in $Con^{\omega}(X,e,d)$, as $\alpha_n=o_{\omega}(d_n)$ (combining $\color{blue}{(\ref*{A}),(\ref*{B}),(\ref*{C})}$). This contradicts out assumption that the triangle $\triangle_{\infty}$ is simple. Hence $Con^{\omega}(X,e,d)$ is an $\mathbb{R}$-tree. That contradicts our assumption that $X$ is not hyperbolic.
Lacunary hyperbolic groups {#lacunarydef}
==========================
A metric space $X$ is called lacunary hyperbolic if one of the asymptotic cones of $X$ is an $\mathbb{R}$-tree. A finitely generated group is called lacunary hyperbolic if it has a finite generating set and the corresponding Cayley graph is lacunary hyperbolic.
Note that lacunary hyperbolicity does not depend on finite generating set. It follows from Theorem \[OOS\] below.
Let $\alpha: G\rightarrow H$ be a group homomorphism and $G=\langle A\rangle ,H=\langle B \rangle$. The injectivity radius $r_A(\alpha)$ is the radius of largest ball centered at identity of $G$ in the Cayley graph of $G$ with respect to $A$ on which the restriction of $\alpha$ is injective.
In the above setting, $r_A(\alpha)$ can be $\infty$, for example when $\alpha$ is injective.
[@OOS07 Theorem 1.1 ]\[OOS\] Suppose $G$ be a finitely generated group. Then following are equivalent:\
$\mathbf{a.)}$ $G$ is lacunary hyperbolic group.\
$\mathbf{b.)}$ There exists a scaling sequence $d=(d_n)$, such that $Con^{\omega}(G,d)$ is an $\mathbb{R}$-tree for all non-principal ultrafilter $\omega$.\
$\mathbf{c.)}$ $G$ is the direct limit of a sequence of finitely generated hyperboolic groups and epimorphisms;\
$$\begin{aligned}
\label{lacunarylim}
G_1\overset{\alpha_1}{\longrightarrow} G_{2}\overset{\alpha_{2}}{\longrightarrow}\cdots G_i\overset{\alpha_i}{\longrightarrow} G_{i+1}\overset{\alpha_{i+1}}{\longrightarrow}G_{i+2}\overset{\alpha_{i+2}}{\longrightarrow}\cdots
\end{aligned}$$ where $G_i$ is generated by a finite set $\langle S_i\rangle $ and $\alpha_i(S_i)=S_{i+1}$. Also $G_i$’s are $\delta_i$ hyperbolic where $\delta_i$=$o(r_{S_i}(\alpha_i))$ (where $r_{S_i}(\alpha_i)$=injective radius of $\alpha_i$ w.r.t. $S_i$ ).
The following theorem gives us a characterization of lacunary hyperbolic group in terms of the hyperbolicity function of the corresponding Cayley graph.
\[lhgdef\] Let G be a finitely generated group with generating set $S$. Then $G$ is lacunary hyperbolic if and only if the hyperbolicity function $f_{G}$ of the corresponding Cayley graph $\Gamma(G,S)$ satisfies $\liminf_{t\rightarrow\infty}\frac{f_{G}(t)}{t}=0$.
Suppose $G$ be lacunary hyperbolic group. Consider any geodesic triangle of perimeter less than or equal to $r_{S_i}(\alpha_i)$, then we use part $c.)$ of Theorem ${\color{blue}\ref*{OOS}}$ and get that $f_G(r_{S_i}(\alpha_i))=\mathcal{O}(\delta_i)=o(r_{S_i}(\alpha_i))$, as $\delta_i=o(r_{S_i}(\alpha_i))$. Hence, $$\lim_{n\rightarrow\infty}\frac{f_G(r_{S_i}(\alpha_i))}{r_{S_i}(\alpha_i)}=0\Rightarrow \liminf_{n\rightarrow\infty}\frac{f_G(t)}{t}=0.$$\
\
Suppose the hyperbolicity function $f_G$ satisfies the following condition: $$\liminf_{t\rightarrow\infty}\frac{f_G(t)}{t}=0$$ Hence there exists a non-decreasing scaling sequence, say $k=(k_n)$ such that: $$\begin{aligned}
\label{S}
\lim_{n\rightarrow\infty}\frac{f_G(d_n)}{d_n}=0.
\end{aligned}$$ Fix the sequence $d=\{d_n\}$. Then we have, $$\begin{aligned}
\label{lac}
f_G(d_n)=o(d_n).
\end{aligned}$$ Let $\omega$ be an arbitrary non principle ultrafilter. According to lemma $\color{blue}{\ref*{limitgeodesictrg}}$ applied to the collection $\mathcal{P}$ of all one-element subsets of $Con^{\omega}(G,d)$ to show that $Con^{\omega}(G,d)$ is an $\mathbb{R}$-tree it is sufficient to prove that it contains no non-trivial simple limit geodesic triangles.\
Let $\triangle_{\infty}$ be a simple non-trivial limit geodesic triangle in $Con^{\omega}(G,d)$ with sides A, B and C with,
$\lim^{\omega}{A_n}=A,
\lim^{\omega}{B_n}=B,
\lim^{\omega}{C_n}=C$
where $A_n,B_n,C_n$ are geodesics in $X:=\Gamma(G,S)$ with endpoints $a_n',b_n$ for $A_n$ and $b_n',c_n$ for $B_n$ and $c_n',a_n$ for $C_n$, for all $n\in \mathbb{N}$. Let ${\triangle}_{\infty}$ is approached by hexagons $H_n$, formed by vertices $a_n{a_n}'b_n{b_n}'c_n{c_n}'$ in $X$ with $\ell (P^i_n)$[^3]$=o_{\omega}(d_n)$ for $i=1,2,3$, where, $ P^1_n$ is a geodesic joining $a_n,a_n'$ in $X$ and respectively $P^2_n$ is a geodesic joining $b_n,b_n'$ in $X$ and $P^3_n$ is a geodesic joining $c_n,c_n'$ in $X$.\
Denote the perimeter of the hexagon $H_n$ by $|H_n|$. Let $A$ be a nontrivial side of $\triangle_{\infty}$. By lemma $\color{blue}{\ref*{hyperbolicgeodesicngon1}}$, $A_n$ belongs to closed $4f_G(|H_n|)$ neighborhood of other $5$ sides in $H_n$. In particular $A_n$ is contained in closed $\beta_n$ neighborhood of $B_n\cup C_n $, where $$\begin{aligned}
\label{A1}
\beta_n=4f_G(|H_n|) + \max_{i=1,2,3}\{\ell(P^i_n) \}.
\end{aligned}$$ We have, $$\begin{aligned}
\label{B1}
\ell(P^i_n)=o_{\omega}(d_n)\\
\ell(A_n) = \Theta_{\omega}(d_n)\nonumber\\
\ell(B_n),\ell(C_n)=\mathcal{O}_{\omega}(d_n)\nonumber
\end{aligned}$$ Hence we get $|H_n|=\Theta_{\omega}(d_n)$. By $\color{blue}{\ref*{S}}$, we also have, $$\begin{aligned}
\label{C1}
f_G(|H_n|)=o_{\omega}(|H_n|)=o_{\omega}(d_n).
\end{aligned}$$ Finally we get that $A$ is contained in the union of other two sides in $Con^{\omega}(X,e,d)$, as $\beta_n=o_{\omega}(d_n)$ (combining $\color{blue}{(\ref*{A1}),(\ref*{B1}),(\ref*{C1})}$). This contradicts out assumption that the triangle $\triangle_{\infty}$ is simple. Hence $Con^{\omega}(X,e,d)$ is an $\mathbb{R}$-tree. Which implies that the group $G$ is lacunary hyperbolic.
One can use similar proof to characterize lacunary hyperbolic spaces i.e, a geodesic metric space $(X,d)$ is lacunary hyperbolic if and only if the corresponding hyperbolicity function $f_X$ satisfies $\liminf_{t\rightarrow \infty}\frac{f_X(t)}{t}=0$.
Let $(X,d_X),(Y,d_Y)$ be metric spaces and $f:X\rightarrow Y$ be a map. The map $f$ is called a $(\lambda,c)$ quasi isometric embedding if for all $x,y\in X$,
$\frac{d_X(x,y)}{\lambda}-c < d_Y(f(x),f(y))< \lambda d_X(x,y)+c$
A $(\lambda,c)$ quasi isometric embedding is a $(\lambda,c)$ quasi-isometry if there exists a constant $D\geq0 $ such that for all $y\in Y$ there exists $x\in X$ with $d_Y(f(x),Y)<D$.
We say $X$ is quasi isometric to $Y$ is there exists a quasi isometry from $X$ to $Y$ for some $\lambda$ and $c$.
[@BH99] Quasi isometry is an equivalence relation.
\[quasigeodesic\] A $(\lambda,c)$quasi-geodesic in a metric space $(X,d_X)$ is a $(\lambda,c)$ quasi-isometric embedding of $\mathbb{R}$ into $(X,d_X)$. More precisely a map $q:\mathbb{R}\rightarrow (X,d_X)$ such that there exists $\lambda,c>0$ so that for all $s,t\in \mathbb{R}$, $$\frac{|s-t|}{\lambda}-c < d_X(f(x),f(y))< \lambda|s-t|+c$$ is called a $(\lambda,c)$ quasi geodesic.
[@OOS07 Theorem 3.18 ]\[aa\] Let $G$ be a lacunary hyperbolic group. Then, every finitely generated undistorted (i.e, quasi-isometrically embedded) subgroup of $G$ is lacunary hyperbolic.
So it is natural to expect the “same" result to be true for lacunary hyperbolic space.
\[qisubspace\] Let $X$ be a lacunary hyperbolic space and $q: Y\rightarrow X$ is a quasi isometric embedding. Then $Y$ is also lacunary hyperbolic space.
Proof follows from similar argument from Theorem ([\[aa\]]{}).
Suppose $X,Y$ are two quasi-isometric geodesic metric space. Then $X$ is lacunary hyperbolic if and only if $Y$ is lacunary hyperbolic.
\[synchrodef\] Let $G_i$ be LHG (lacunary hyperbolic group) for $i=1,2,\cdots ,k$ with corresponding hyperbolicity function $f_{G_i}$ for $G_i$ for all $i=1,2,\cdots,k$. Then we call $G_i$’s are synchronized LH if there exists an increasing infinite sequence $\{x_j\}_{j\in \mathbb{N}}$ of real numbers and finite generating sets $X_i$ of $G_i$ for all $i$ such that the hyperbolicity function with respect to generating sets $X_i$, $f_{G_i}$ satisfies $\lim_{j\rightarrow\infty}\frac{f_{G_i}(x_j)}{x_j}=0$ for all $i=1,2,\cdots,k$ (equivalently $\liminf_{t\rightarrow \infty} \frac{\sum_{i=1}^{k}f_{G_i}(t)}{t}=0$).
Free product of two lacunary hyperbolic group is not necessarily a lacunary hyperbolic (Example 3.16, $\cite{OOS07}$).
\[synchro\] Let $G=G_0* G_1$ with $G_i$’s LHG. Then $G$ is LHG if and only if $G_0$ and $G_1$ are synchronized LH.
Let $G_1,G_2$ are synchronized LH. Let $\Delta=T_1T_2T_3$ be a geodesic triangle in $G:=G_1*G_2$ with geodesic sides $T_1,T_2,T_3$, where $G_i$s are hyperbolic groups with finite presentation $G_i={\langle}X_i|R_i{\rangle}$ for $i=1,2$. Since $\Delta=1$ in $G$, consider the normal form $\Delta=\prod_{i=1}^{k}g_i^{-1}r_i^{\pm 1}g_i$ in the free group $\mathcal{F}(X_1\cup X_2)$ for some positive integer $k$ where $g_i\in (X_1\cup X_2)^{\pm 1}$ and $r_i\in R_1\cup R_2$. We are going to induct on the sum of lengths of the normal form of $\Delta$. Suppose the sum of length of normal form is $\leq 3$. Then the triangle $\Delta$ belongs to one or the free factors either $G_1$ or $G_2$ since the product is $1$. Let us assume that the sum of lengths of normal form is $n\geq 4$. Let we have a cancellation in $T_1T_2=T_1'xyT_2'$ where $xy=1$ in $G_1$ (respectively in $G_2$). Then the triangle $\Delta $ can be written as union of triangle $\Delta_1=xy$ (which is a bigon with non trivial side $x$ and $y$) and the triangle $\Delta_2$ with sides $T_1',T_2'$ and $T_3$. Notice that sum of lengths normal form of $\Delta_1,\Delta_2$ say $n_1,n_2$ respectively, are strictly less than $n$. Hence we get $$f_{G}(n=n_1+n_2) \leq \max\{f_{G_1}(n_1),f_{G_1}(n_2), f_{G_2}(n_1), f_{G_2}(n_2)\}
\leq \max\{ f_{G_1}(n),f_{G_2}(n)\}.\nonumber$$ Last inequality is coming from the fact that hyperbolicity function is non decreasing. When there is a cancellation $xyz=1$ in $G_1$ (or in $G_2$) then we are in the following setting, $T_1=T_1'x, T_2=y$ and $T_3=zT_2'$, where $xyz=1$ in $G_1$ (or in $G_2$). In this case we can also write the triangle $\Delta$ as union of triangle $\Delta$ formed by $x,y,z$ and the triangle $\Delta_2$ (bigon) formed by $T_1',T_2'$ with lengths of normal forms $n_1,n_2$. Then we get $$f_{G}(n=n_1+n_2) \leq \max\{ f_{G_1}(n_1),f_{G_1}(n_2), f_{G_2}(n_1), f_{G_2}(n_2)\}
\leq \max\{ f_{G_1}(n),f_{G_2}(n)\}.\nonumber$$ Hence by induction we can see that $f_G(t)\leq \max\{ f_{G_1}(t),f_{G_2}(t)\}$ for all $t$. Which implies that, $$\begin{aligned}
\liminf_{t\rightarrow \infty}\frac{f_G(t)}{t}\leq \liminf_{t\rightarrow\infty}\frac{f_{G_1}(t)+f_{G_2}(t)}{t} \rightarrow 0\nonumber. \end{aligned}$$ Hence $G$ is a lacunary hyperbolic group.
For the other direction let $G=G_1* G_2$ is LHG. Note that we have $$\begin{aligned}
f_{G_i}(t)\leq f_G(t)\ \textit{for all } t> 0\ \textit{and for all }i=1,2.
\end{aligned}$$ since every geodesic triangles in $G_i$ is also a geodesic triangle in $G$ for all $i=1,2$ as the embedding $G_i\rightarrow G_1*G_2$ is a $(1,0)$ quasi isometric embedding for all $i=1,2$. There exists a sequence, say $\{ y_i\}$ such that, $\lim_{i\rightarrow\infty }\frac{f_G(y_i)}{y_i}=0$. So we have $\lim_{i\rightarrow\infty }\frac{f_{G_j}(y_i)}{y_i}=0$ for all $j=1,2$. Hence by definition $G_1,G_2$ are synchronized LH.
\[freeprodlacuremark\] Above proposition is in fact true for free product of finite number of LHGs i.e, for a finite number of lacunary hyperbolic groups $\{G_i\}_{i=1}^{k}$, $G=*^{k}_{i=1}G_i$ is LHG if and only if $\{G_i\}$ are synchronized LH. In particular $*_{i=1}^kG$ is a lacunary hyperbolic group whenever $G$ is a lacunary hyperbolic group for any positive integer $k$.
Algebraic properties of lacunary hyperbolic groups {#section4}
==================================================
Small cancellation theory
-------------------------
### van Kampen diagrams
Given a word $W$ in alphabets $\mathcal{S}$, we denote its length by $\|W\|$. We also write $W\equiv V$ to express the letter-for-letter equality for words $U,V$.
Let $G$ be a group generated by alphabets $\mathcal{S}$. A van Kampen diagram $\triangle$ over a presentation $$\begin{aligned}
\label{vankampen}
G=\langle S|\mathcal{R}\rangle\end{aligned}$$ where $\mathcal{R}$ is cyclically reduced words over alphabets $S$, is a finite, oriented, connected, planar 2-complex endowed with a labeling function $Lab:E(\triangle)\rightarrow \mathcal{S}^{\pm 1}$, where $E(\triangle)$ denotes the set of oriented edges of $\triangle$, such that $Lab(e^{-1})\equiv (Lab(e))^{-1}$. Given a cell $\Pi$ of $\triangle$, $\partial \Pi$ denotes its boundary; similarly $\partial \triangle$ denote boundary of $\triangle$. The labels of $\partial \triangle$ and $\partial \Pi$ are defined up to cyclic permutations. Also one additional requirment is that the label for any cell $\Pi$ of $\triangle$ is equal to (up to a cyclic permutation) $R^{\pm 1}$, where $R\in\ \mathcal{R}$.
By van Kampen lemma, a word $W$ over alphabets $\mathcal{S}$ represents the identity element in the group given by () if and only if there exists a van Kampen diagram $\triangle$ over () such that, $Lab(\partial\triangle)\equiv W$. \[Ch.5, Theorem 1.1\][@LS77].
### Small cancellation over hyperbolic groups
Let $G=\langle X\rangle$ be a finitely generated group.
A word $W$ in the alphabet $X^{\pm 1}$ is called $(\lambda,c)$-quasi geodesic (respectively geodesic) in $G$ if any path in the Cayley graph $\Gamma(G,X)$ labeled by $W$ is $(\lambda,c)$-quasi geodesic (respectively geodesic).
Let $G=\langle X\rangle$ be a finitely generated group, and let $\mathcal{R}$ be a symmetric set of words (i.e. it is closed under operations of taking cyclic shifts and inverse of words and all the words are cyclically reduces) from $X^*$.
[@LS77] Let $$\begin{aligned}
G=\langle X\rangle
\end{aligned}$$ be a group generated by $X$, and $\mathcal{R}$ be a symmetrized set of reduced words in a finite set of alphabets $X^{\pm 1}$. A common initial sub-word of any two distinct words in $\mathcal{R}$ is called a piece. We say that $\mathcal{R}$ satisfies $C'(\mu)$ condition if any piece contained (as a sub-word) in a word $R\in\mathcal{R}$ has length less than $\mu\|R\|$.
Let $G$ be a group generated by a set $X$. A subword $U$ of a word $R\in \mathcal{R}$ is called an $\epsilon$-piece (reference [@Ol93]) for $\epsilon\geq 0$ if there exists a word $R'\in \mathcal{R}$ such that
(a) $R\equiv UV$ and $R^{\prime}\equiv U^{\prime}V^{\prime}$ for some $U^{\prime},V^{\prime}\in \mathcal{R}$.
(b) $U^{\prime}= YUZ$ in $G$ for some $Y,Z\in X^*$ where $\| Y\|,\| Z\| \leq \epsilon$.
(c) $YRY^{-1}{\neq} R^{\prime}$ in the group $G$.
It is said that the system satisfies the $C(\lambda,c,\epsilon,\mu,\rho)$-condition for some $\lambda\geq 1,c\geq 0,\epsilon\geq0,\mu>0,\rho>0$ if,
a) $\| R\| \geq \rho$ for any $R\in \mathcal{R}$.
b) any word $R\in \mathcal{R}$ is a $(\lambda,c)$-quasi geodesic.
c) for any $\epsilon$-$piece$ of any word $R\in \mathcal{R}$, the inequalities $\| U\|,\| U^{\prime } \| <\mu \| R\|$ holds.
Let $U^{\pm 1}$ be a subword of $R\in \mathcal{R}$ and we have,
1. $R\equiv UVU^{\prime} V^{\prime}$ for some $V,U^{\prime} ,V^{\prime}\in X^*$,
2. $U^{\prime}=YU^{\pm 1}Z$ in the group $G$ for some words $Y,Z\in X^*$ where $\| Y\| ,\| Z\| \leq \epsilon$.
Then we call $U$ an $\epsilon^{\prime}$-$piece$ of the word $R$. If $\mathcal{R}$ satisfies the $C(\lambda,c,\epsilon,\mu,\rho)$-condition and in addition for all $R\in \mathcal{R}$, the above decomposition implies $\| U\| , \| U^{\prime} \| < \mu \| R\|$, we say that $\mathcal{R}$ satisfies $C'(\lambda,c,\epsilon,\mu,\rho)$-condition.
Let $\epsilon\geq 0,\mu \in (0,1),$ and $\rho>0$. We say that a symmetrized set $\mathcal{R}$ of words over the alphabet $X^{\pm 1}$ satisfies condition $C(\epsilon,\mu,\rho)$ for the group $G$, if
1. All words from $\mathcal{R}$ are geodesic in $G$.
2. $\|R\|\geq \rho$ for all $R\in \mathcal{R}$.
3. The length of any $\epsilon$-piece contained in any word $R\in \mathcal{R}$ is smaller than $\mu\|R\|$.
Suppose now that $G$ be a group defined by $$\begin{aligned}
\label{hyppresent}
G=\langle X|\mathcal{O}\rangle\end{aligned}$$ where $\mathcal{O}$ is the set of all relators (not only defining) of $G$. Given a symmetrized set of words $\mathcal{R}$, we consider the quotient group , $$\begin{aligned}
\label{representation}
H=\langle G|\mathcal{R}\rangle =\langle X|\mathcal{O}\cup \mathcal{R}\rangle.\end{aligned}$$ A cell over a van Kampen diagram over ($\color{blue}{\ref*{representation}}$) is called an $\mathcal{R}$-cell (respectively, an $\mathcal{O}$-cell) if it’s boundary label is a word from $\mathcal{R}$ (respectively, $\mathcal{O}$). We always consider van Kampen diagram over $(\color{blue}{\ref*{representation}})$ up to some elementary transformations. For examples we do not distinguish diagrams if one can be obtained from other by joining two distinct $\mathcal{O}$-cells having a common edge or by inverse transformation, etc (ref, [@Ol93 Section 5 ]).
Let $\triangle$ be a van Kampen diagram over ($\color{blue}{\ref*{representation}}$), $q$ be a sub-path of it’s boundary $\partial \triangle$, $\Pi,\Pi'$ some $\mathcal{R}$-cells of $\triangle$. Suppose $p=s_1q_1s_2q_2$ be a simple closed path in $\triangle$, where $q_1$ (respectively $q_2$) is a sub-path of the boundary $\partial \Pi$ (respectively $q$ or $\partial \Pi'$) with $\max\{\|s_1\|,\|s_2\| \}\leq \epsilon $ for some constant $\epsilon$. Then denote $\Gamma$ to be the sub-diagram of $\triangle$ bounded by $p$. We call $\Gamma$ is an $\epsilon$-contiguity sub-diagram of $\Pi$ to the part $q$ of $\partial \triangle$ (or $\Pi'$ respectively) if $\Gamma$ contains no $\mathcal{R}$-cells. The sub-paths $q_1,q_2$ are called contiguity arcs of $\Gamma$ and the ratio $\|q_1\|/\|\partial \triangle\|$ is called contiguity degree of $\Pi$ to $\partial \triangle$ (or to $\Pi'$ respectively). Contiguity degree is denoted by $(\Pi,\Gamma,\partial\triangle)$ or $(\Pi,\Gamma,\Pi')$.\
We call a (disc) van Kampen diagram over $(\color{blue}{\ref*{representation}})$ minimal if it has minimal number of $\mathcal{R}$-cells among all disc diagrams with the same boundary label.
Let $G$ be a $\delta$ hyperbolic group having presentation $\langle X|\mathcal{O}\rangle$ as $(\color{blue}{\ref*{hyppresent}})$, $\epsilon \geq 2\delta$, $0<\mu\leq 0.01$, and $\rho$ is large enough (it suffices to choose $\rho > 10^6\frac{\epsilon}{\mu}$). Let $H$ is given by, $$\begin{aligned}
H=\langle G|\mathcal{R}\rangle =\langle X|\mathcal{O}\cup \mathcal{R}\rangle.
\end{aligned}$$ as in ($\color{blue}{\ref*{representation}}$) where $\mathcal{R}$ is a symmetrized set of words in $X^{\pm 1}$ satisfying the $C(\epsilon,\mu,\rho)$-condition. Then the following holds,
1\. Let $\triangle$ be a minimal disc diagram over ($\color{blue}{\ref*{representation}}$). Suppose that $\partial\triangle=q^1q^2\cdots q^t$, where the labels of $q^1,q^2,\cdots ,q^t$ are geodesic in $G$ and $t\leq 12$. Then provided $\triangle $ has an $\mathcal{R}$-cell, there exists an $\mathcal{R}$-cell $\Pi$ in $\triangle$ and disjoint $\epsilon$-contiguity sub-diagrams $\Gamma_1,\Gamma_2,\cdots,\Gamma_t$ (some of them may be absent) of $\Pi$ to $q^1,\cdots,q^t$ respectively such that, $$(\Pi,\Gamma_1,q^1)+\cdots +(\Pi,\Gamma_t,q^t)>1-23\mu.$$
2\. $H$ is a $\delta_1$ hyperbolic group with $\delta_1\leq 4L$, where $L=\max\{\|R\| | R\in \mathcal{R} \}$.
Elementary subgroups of hyperbolic groups
-----------------------------------------
A group $E$ is called elementary if it is virtually cyclic. We now state an elementary properties of elementary group.
If $E$ is a torsion free elementary group then $E$ is cyclic.
Let $E$ be an infinite elementary group. Then it contains normal subgroups $T\leq E^{+}\leq E$ such that $|E:E^{+}|\leq 2$ , $T$ is finite and $E^{+}/T\simeq \mathbb{Z}$. If $E\neq E^+$ then $E/T\simeq D_{\infty}$(infinite dihedral group).
Let $G$ be a hyperbolic group and $g\in G$ be an infinite order element. Then the elementary subgroup containing $g$ is equal to the following set,
$E(g):=\{x\in G| \ x^{-1}g^nx=g^{\pm n} \ for\ some \ n=n(x)\in \mathbb{N}-\{0\} \}$
For hyperbolic group $E(g)$ is unique maximal elementary subgroup of $G$ containing the infinite order element $g\in G$ (see [@Ol93 Lemma 1.16]). Geometrically $E(g)$ is the kernel of the natural action of non elementary hyperbolic group $G$ on its hyperbolic boundary.
Necessary lemmas and theorems
-----------------------------
[@BH99]\[Hau\] Let $G=\langle X\rangle $ be a $\delta$ hyperbolic group. Then there exists a constant $R_{\lambda,c}$ depending on $\lambda,c$ such that any $(\lambda,c)$ quasi geodesic in $\Gamma(G,X)$ is $R_{\lambda,c}$ Hausdorff distance away from a geodesic.
The following theorem can also be found as a corollary of the combination theorem in [@BF93].
[@MO98 Corollary 7]\[olmik\] Let $G$ and $H$ be hyperbolic groups, $A$ and $B$ be infinite elementary subgroups of $G$, $H$ respectively. Then the free product of the groups $G$ and $H$ with amalgamated subgroups $A$ and $B$ is hyperbolic if and only if either $A$ is a maximal elementary subgroup of $G$ or $B$ is a maximal elementary subgroup of $H$.
[@Ol93 Lemma 6.7, 7.5]\[ol93lemma\] Let $H_1,H_2,\cdots,H_k$ be non elementary subgroups of a hyperbolic group $G$ and $\lambda>0$. Then there exists $\mu>0$ such that for any $c\geq 0$, there is $\epsilon\geq 0$ such that for any $N\geq 0$ there is $\rho>0$ with the following property:
Let the symmetrized presentation in (\[representation\]) satisfies $C'(\lambda,c,\epsilon,\mu,\rho)$ condition. Then the quotient $H$ is a hyperbolic group. Moreover, $W=_H1$ if and only if $W=_G1$ for every word $W$ with $\|W\|\leq N$ and the images of $H_1,H_2,\cdots,H_k$ are non elementary subgroups in the quotient group $H$.
We are going to state a simple version of a lemma that we will be using later
\[properpower\] Let $H={\langle}X{\rangle}$ be a torsion free non elementary hyperbolic group and $G=H/{\langle}{\langle}\mathcal{R}{\rangle}{\rangle}$ be a quotient group where $\mathcal{R}$ satisfies $C'(\lambda,c,\epsilon,\mu,\rho)$ condition for spars enough parameters $\lambda,c,\epsilon,\mu,\rho$ with $1>1-122\lambda\mu>0$. Suppose $U,W\in \mathcal{F}(X)$ such that $U=_GW^k$ for some $k\geq 2$ and $$\|U\| <\frac{\mu\rho-c}{\lambda}-\epsilon.$$ Then $U=_HW^k$.
Words with small cancellations
------------------------------
Let $G={\langle}X{\rangle}$ be a non elementary torsion free $\delta$-hyperbolic group for some $\delta>0$. Let us consider the set $\mathcal{R}$ of words consisting of the form, $$\begin{aligned}
\label{systemwords}
R_i=z_iU^{m_{i,1}}VU^{m_{i,2}}V\cdots VU^{m_{i,j_i}} , \ i=1,2,\cdots,k\end{aligned}$$ and their cyclic shifts, where $k\in\mathbb{N},U,V,z_1,z_2,\cdots,z_k\in \mathcal{F}(X)$ are geodesic words in $G$, $U,V\neq_G 1$ and $m_{i,t}\in\mathbb{N}$ (also assume that $m_{i,j}\neq m_{i',j'}$ if $(i,j)\neq (i',j')$) for $1\leq i\leq k,1\leq t\leq j_i$. Denote $Z=\{ z_1,z_2,\cdots ,z_k\}$. Let $L:=\max \{\|U\|,\|V\|,\|z_1\|,\|z_2\|,\cdots,\|z_k\| \}$, $\underline{m}:=\min\{m_{i,t} |1\leq i\leq k,1\leq t\leq j_i \}$ and $\overline{m}_i:=\max\{m_{i,t}|1\leq t\leq j_i \}$.
We also specify $m_{i,t}$ as follows: choose $m_{1,1}$ and for all $1\leq i\leq k, \ m_{i,1}=2^{i-1}m_{1,1}, j_i=m_{i,1}-1$ and for all $1\leq t\leq j_i, \ m_{i,t}=m_{i,1}+(t-1)$.
[@Dar17 Lemma 5.1]\[smallwords\] For the set of words $\mathcal{R}$ suppose that $V\notin E(U)$, $z_i\notin E(U)$ for $1 \leq i \leq k$. Then there exist constants $\lambda = c = \tilde{K} \in\mathbb{N}$ depending on $G, U, V$ and $Z$, such that the words of the system (\[systemwords\]) are $(\lambda, c)$-quasi-geodesic in $\Gamma(G, X)$, provided that $\underline{m} \geq \tilde{K}$ .
Let $\lambda,c$ is defined from Lemma \[smallwords\], $\mathcal{R}$ is defined as above. Since all the words in the system of words $\mathcal{R}$ is $(\lambda,c)$-quasi geodesic by Lemma \[smallwords\], there exists a constant $R_{\lambda,c}$ such that these words are $R_{\lambda,c}$ Hausdorff distance away from a geodesic by Lemma \[Hau\]. Also assume that with respect to given constants $\epsilon\geq 0,\mu>0,\rho>0$, following conditions are satisfied. $$\begin{aligned}
\label{allconditions}
\|R\|\geq \rho, \ for \ all\ R\in\mathcal{R}\nonumber\\
\underline{m}\geq\tilde{K}\\
\mu\rho\geq 6L(\overline{m}_i+1) \ for \ 1\leq i\leq k\nonumber \\
\underline{m}\geq \frac{2\epsilon'}{\|U\|}\cdot 12\lambda,\ where\ \epsilon'=\epsilon +2\|c\|+5(2R_{\lambda,c}+182\delta +\frac{\|c\|}{2})\nonumber \end{aligned}$$
[@Dar17 Lemma 5.2]\[lemma5\] Using the setting of the previous lemma and assuming that the above described conditions take place, let us consider the system of words $\mathcal{R}$ given by (\[systemwords\]). Let $\lambda,c$ be defined by the Lemma \[smallwords\]. Then, if for the given constants $\epsilon \geq 0,
\mu > 0, \rho > 0$, the conditions in (\[allconditions\]) are satisfied, then the system $\mathcal{R}$ satisfies the $C'(\lambda,c,\epsilon,\mu,\rho)$-condition.
For any given constants $\epsilon \geq 0,\mu > 0, \rho > 0$, we denote a system of words $\mathcal{R}$ as described above, by $\mathcal{R}(Z,U,V,\lambda,c,\epsilon,\mu,\rho)$.
Elementary subgroups of lacunary hyperbolic groups
--------------------------------------------------
Let $G$ be a lacunary hyperbolic group and $g\in G$ be an infinite order element. Then define $E^{\mathcal{L}}(g)=\{x\in G |xg^nx^{-1}=g^{\pm n}$, for some $n=n(x)\in \mathbb{N}-\{0\}\}$.
\[elementarygroup\]\[elementarytheorem\] Let $G$ be a lacunary hyperbolic group and $g\in G$ be an infinite order element. Then $E^{\mathcal{L}}(g)$ has a locally finite normal subgroup $N\triangleleft G$ such that:\
Either $E^{\mathcal{L}}(g)/N$ is an abelian group of Rank 1(i.e. $E^{\mathcal{L}}(g)/N \ embeds \ in \ (\mathbf{Q},+)$) or $E^{\mathcal{L}}(g)/N$ is an extension of a rank one group by involutive automorphism (i.e, $a\rightarrow a^{-1}$).\
By Theorem \[OOS\], there exist hyperbolic groups $G_i$ and epimorphism $\alpha_i$:\
$$G_i\overset{\alpha_i}{\longrightarrow}G_{i+1}\overset{\alpha_{i+1}}{\longrightarrow}G_{i+2}\cdots$$\
where $G_i$ is generated by a finite set ${\langle}S_i{\rangle}$ and $\alpha_i(S_i)=S_{i+1}$. Also $G_i$’s are $\delta_i$ hyperbolic where $\delta_i$=$o(r_{S_i}(\alpha_i))$ (where $r_{S_i}(\alpha_i)$=injective radius of $\alpha_i$ w.r.t. $S_i$ ) and $$G=\underrightarrow{\lim} \ G_i$$ Take a representative[^4] $\{g_i\}_{i\in \mathbb{N}}$ of the element $g$ in the direct limit. Then $g_i$ has infinite order in $G_i$ for all $i\in \mathbb{N}$ as $g\in G$ has infinite order. So for all $i\in \mathbb{N}$, there exists a unique maximal elementary subgroup of $G_i$ containing $g_i$ (as $G_i$ is hyperbolic), namely $E(g_i)$.
\[lemma1\] Suppose $H$ and $K$ are two hyperbolic groups and $\alpha :H\longrightarrow K$ be an epimorphism. If $h\in H$ and $\alpha(h)\in K$ has infinite order then $$\alpha(E(h))\subset E(\alpha(h))$$ where $E(h)$ and $E(\alpha(h))$ are unique maximal elementary subgroup containing $h$ and $\alpha(h)$ respectively.
From a characterization of the unique maximal elementary subgroup of hyperbolic group we can write, $E(h)=\{x\in H|xh^nx^{-1}=h^{\pm n}$, for some $n=n(x)\in \mathbb{N}-\{0\}\}$ and similarly for $E(\alpha(h))=\{x\in K|x\alpha(h)^mx^{-1}=\alpha(h)^{\pm m}$,for some $m=m(x)\in \mathbb{N}-\{0\}\}$. So the lemma clearly follows for this.
By previous lemma we can see that $\alpha_i(E(g_i))\subset E(g_{i+1})$. Define $\overline{E}(g)=\underrightarrow{\lim}E(g_i)$. Next lemma shows that this definition is independent of choice of that representative $\{g_i\}_{i\in \mathbb{N}}$.
\[lemma2\] $(a)$ $\overline{E}(g)$ is independent of choice of $\{g_i\}_{i\in \mathbb{N}}$.\
$(b)$ $\overline{E}(g)=E(g)$.
$(a):$ Suppose $\{g'_j\}_{j\in \mathbb{N}}$ be another representative of $g$ in the direct limit. Then define $\overline{E'}(g)=\underrightarrow{\lim}E(g'_i)$. Let $x\in \overline{E'}(g)$ and $\{x_j\}$ be a representative of $x$ in direct limit. Hence $x_{j_0}\in E(g'_{j_0})$ for some $j_0$ hence for all $j\geq j_0$. But $\{g_i\}_{i\in \mathbb{N}}$ and $\{g'_j\}_{j\in \mathbb{N}}$ are both representative of the element $g$, So there exists a number $k\in \mathbb{N}$ such that $g'_k=g_k$. Hence $x_t\in E(g'_t)=E(g_t)$ for all $t\geq k$. Which implies $x\in \overline{E}(g)$. So we have $\overline{E'}(g)\subset \overline{E}(g)$. Now by doing the same method we can show that $\overline{E}(g)\subset \overline{E'}(g)$. That proves $(a)$.\
\
$(b):$ From the definition of $E^{\mathcal{L}}(g)$ and $\overline{E}(g)$ it is clear that $\overline{E}(g)\subset E_G(g)$. Now to prove the other inclusion let assume that $x\in E(g)$. Then there exists representative $\{x_i\}_{i\in \mathbb{N}}$ for $x$ and $\{g_i\}_{i \in \mathbb{N}}$ for $g$ and natural numbers $s,n$ such that: $$x_sg_{s}^nx_s^{-1}=g_s^{\pm n}$$ Hence $x_s\in E(g_s)<G_s $ for large enough $s(\geq j_0)$ by the characterization we mentioned in the proof of lemma $\color{blue}{\ref*{lemma1}}$. Hence $x\in \underrightarrow{\lim}E(g_i)$ (But this direct limit is independent of choice of representative of $g$ by part $(a)$). So $x\in \overline{E}(g)$. Hence we have $\overline{E}(g)=E(g)$.
Now define $E^{+}(g)=\{x\in G|xg^nx^{-1}=g^n$ for some $n=n(x)\in \mathbb{N}-\{0\}\}$. Then clearly $E^+(g)<E(g)$.\
Take a representative $\{g_i\}_{i\in \mathbb{N}}$ of the element $g$ and define $E^+(g_i)=\{x\in G_i|xg_i^mx^{-1}=g_i^m$, for some positive integer $m=m(x)\}$. Clearly $E^+(g_i)<E(g_i)$ and $|E(g_i):E^+(g_i)|\leq 2$ for every $i\in\mathbb{N}$.\
Now from lemma $\color{blue}{\ref*{lemma1}}$ and lemma $\color{blue}{\ref*{lemma2}}$ it follows that $E^+(g)$ is independent of choice of representative of $g$ and $E^+(g)=\underrightarrow{\lim}E^+(g_i)$.\
Hence it follows from the definition that $|E(g):E^+(g)|\leq 2$. As we can see that $E(g)$ and $E^+(g)$ is independent of choice of representative of the element $g$, we can fix a representative $\{g_i\}_{i\in \mathbb{N}}$ of $g$. Define $T_i<E^+(g_i)$ be the set of all torsion elements of $E^+(g_i)$. Now $T_i$ is finite normal subgroup of $E(g_i)$ for every $i\in \mathbb{N}$.\
In particular for every $i\in \mathbb{N}$ we get a series of normal subgroups of $G_i$:\
$$1\leq T_i\leq E^+(g_i)\leq E(g_i),$$ where $|E^+(g_i):E(g_i)|\leq 2$ and $E^+(g_i)/T_i$ is infinite cyclic and $T_i$ is finite.\
\
Now we can see that $\alpha_i(T_i)\subset T_{i+1}$.(Follows from the definition that $T_i$ is torsion subgroup of $E^+(g_i)$ and $\alpha_i(E^+(g_i))\subset E^+(g_{i+1})$) So we define $T=\underrightarrow{\lim}T_i$, the set of torsion elements of $E^{\mathcal{L}}(g)$. And this definition of $T$ is also independent of choice of representative of the element $g$.
$T$ is locally finite normal subgroup of $E(g)$.
A direct limit of finite groups is always locally finite and a limit of normal subgroups is always normal.
$E^+(g)/T$ is an abelian group of rank 1.
Let $[x],[y]\in E^+(g)/T$ and $x,y\in G$ with representative $\{x_i\}$ and $\{y_i\}$ respectively. Let $\{g_i\}$ be a representative of $g$. Then there exists $i$ such that $x_i,y_i\in E^+(g_i)$ in $G_i$, i.e. $x_i^{-1}y_i^{-1}x_iy_i\in T_i$. Hence $[x][y]=[y][x]$ in $E^+(g)/T$ in $G$, i.e. $E^+(g)/T$ is abelian.\
Let $[x],[y]\in E^+(g)/T$. As this group is abelian we get $x_iy_i=y_ix_i$ (mod T) in $G_i$ for some $i\in \mathbb{N}$, where $\{x_i\},\{y_i\}$ are representatives of $x,y$ respectively. Hence if $y_i\notin T_i$ we have $x_i\in E(y_i)$. So there exists integers $m,n\neq 0$ such that $x_i^my_i^n=1_{G_i}$. Hence $x^ny^m=1_G$. This proves the lemma hence the proposition.
Let $g$ be an infinite order element in a lacunary hyperbolic group $G$. Every finitely generated subgroup of $E^{\mathcal{L}}(g)$ is elementary.
Let $H={\langle}a_1,a_2,\cdots ,a_k{\rangle}$ be a finitely generated subgroup of $E^{\mathcal{L}}(g)$. Also let $G={\langle}x_1,\cdots ,x_n{\rangle}$. By definition of $E^{\mathcal{L}}(g)$ there exists natural numbers $\{l_i\}_{i=1}^{k}$ such that $$\begin{aligned}
\label{elementaryrel}
a_ig^{l_i}a^{-1}_i=g^{\pm l_i} \ for\ all \ i=1,2,\cdots ,k
\end{aligned}$$ So there exists a natural number $n_0$ such that (\[elementaryrel\]) is satisfied in $G_j$ for all $j\geq n_0$ (where $G_i$’s are from \[lacunarylim\] of Theorem\[OOS\]). One can see that $H\leq E(g)$ in $G_{n_0}$. Hence $H$ is virtually cyclic.
Converse of the Theorem \[elementarytheorem\], is also true in general. We are going to see torsion free version of the converse now with some interesting corollary.
\[rank1\] Given any non trivial rank one abelian group $L$ and a non elementary hyperbolic group $G_0$, there exists a lacunary hyperbolic quotient $G$ of $G_0$ and an infinite order element $g\in G$ such that $E^{\mathcal{L}}(g)$ is isomorphic to $L$.
To construct such group we are going to use the following method (section 11,12 of [@Dar17]). We are going to construct the following chain;\
$$\begin{aligned}
G_0\overset{\beta_0}{\hookrightarrow} H_1\overset{\gamma_1}{\twoheadrightarrow} G_1\overset{\beta_1}{\hookrightarrow} H_2 \overset{\gamma_1}{\twoheadrightarrow} G_2 \cdots
\end{aligned}$$ where $H_i,G_i$ are hyperbolic for all $i$ and $\gamma_i\circ \beta_{i-1}$ is surjective for all $i\geq 1$.
Being a rank $1$ abelian group $L$ can be written as $L=\cup_{i=0}^{i=\infty}L_i$, where $L_i=\langle g_i\rangle_{\infty}$ and $g_i=(g_{i+1})^{m_{i+1}}$ for some $m_{i+1}\in \mathbb{N}$.
Let $G=\langle X|\mathcal{O}\rangle$ be a torsion free non elementary hyperbolic group with $a,b\in X$ and $h\in G\setminus\{e\} $, not a proper power in $G$ such that $a,b\notin E(h),b\notin E(a)$. We define, $$\begin{aligned}
\label{amalgamprod}
H:=G*_{h=(g')^m}\langle g'\rangle_{\infty}
\end{aligned}$$ where $\langle g'\rangle_{\infty}$ is an infinite cyclic group and $m\in\mathbb{N}$.
Then we shall construct a non elementary hyperbolic factor group $G'$ such that , $$\begin{aligned}
G\overset{\beta}{\hookrightarrow} H\overset{\gamma}{\twoheadrightarrow} G'
\end{aligned}$$ Where $\beta$ is an embedding induced from $Id:X\rightarrow X$ and $\gamma\circ \beta$ is a surjective homomorphism.
By Theorem \[olmik\], the group $H$ in (\[amalgamprod\]) is hyperbolic as $\langle g'\rangle_{\infty}$ is maximal elementary subgroup of $G*\langle g'\rangle_{\infty}$.
Let $N$ be a positive number. For any $\epsilon\geq 0,\mu>0,\rho>0$ with $\lambda,c$ given by Lemma \[smallwords\] and $\lambda,\mu,\epsilon,\rho,N,c$ satisfies condition for Theorem \[ol93lemma\] for the hyperbolic group $G$. Let $\mathcal{R}(Z,U,V,\lambda,c,\epsilon,\mu,\rho)$ be a set of words as described in Section 4.4, with $Z=\{z \}$ such that $z,U,V$ are geodesic words in $\mathcal{F}(X)$ representing the elements $h,a,b$ respectively i.e, $z=_Gh, U=_Ga,V=_Gb$. Combining the fact that $\mathcal{R}(Z,U,V,\lambda,c,\epsilon,\mu,\rho)$ satisfies $C'(\lambda,c,\epsilon,\mu,\rho)$ and Theorem \[ol93lemma\] with $H_1:=gp\{a,b\},H_2:=gp\{a,h\},H_3:=gp\{b,h\}$, we get that the factor group $G':=\langle H|\mathcal{R}(Z,U,V,\lambda,c,\epsilon,\mu,\rho)\rangle $ is hyperbolic and the injective radius of the factor homomorphism is $\geq N$. Note that we can choose $N$ as large as we want. Also we get that $\gamma \circ \beta$ is surjective homomorphism. Moreover we get that for $i=1,2,3$, images of $H_i$ are non elementary in the factor group $G'$, i.e. $a,b\notin E(h),a\notin E(b)$ in $G'$. Note that by Lemma \[properpower\], $g'$ is not a proper power in $G'$ for spars enough parameters $\lambda,c,\epsilon,\mu,\rho$. Also note that this fact can be deduced directly from [@Ol93 Theorem 2, property (5)].
We start with the given non elementary hyperbolic group $G_0$ and let $F(a,b,h)$ be a free subgroup of $G_0$ with three generator over the alphabets $\{a,b,h\}$. Note that it was shown in [@Gr87 page 157] that every non elementary hyperbolic group contains a copy of free group with two generator hence one can always choose a free subgroup of any rank as a subgroup of a non elementary hyperbolic group. $a,b\notin E(h),b\notin E(a)$ in $G_0$. Now we perform the step described in beginning of the proof with $H:=G_0*_{h=(g_1)^{m_1}}\langle g_1\rangle_{\infty}$ to get the factor group $G_1$ with $a,b\notin E(h),b\notin E(a)$ in $G_1$ and $g_1$ is not a proper power in the factor group. Hence one can now use induction to get $G_i$ from $G_{i-1}$ for all $i\geq 1$. Define $G:=\underset{\rightarrow}{\lim} G_i$. $G$ is lacunary hyperbolic group as injectivity radius of factor homomorphism can be as large as we want in every step. Note that we have $E_i(h)\equiv L_i$ for every $i$ by construction, where $E_i(h)$ is the maximal elementary subgroup containing the infinite order element $h$ in $G_i$. One can see that $E^{\mathcal{L}}(h)=\underset{\rightarrow}{\lim}E_i(h)=\cup_{i=1}^{\infty}L_i$ and hence $E^{\mathcal{L}}(h)\cong L$.
There exists lacunary hyperbolic group $G$ and an infinite order element $g\in G$ such that there is no maximal elementary subgroup of $G$ containing the element $g$.
For hyperbolic group G we know that for two infinite order element $a,b\in G$ , $E(a)\cap E(b)$ is finite if $E(a)\neq E(b)$ (ref. [@Ol93]). In the case of lacunary hyperbolic group one can obtain the following,
Let $G=\underset{\rightarrow}{lim} G_i$ be a lacunary hyperbolic group with two infinite order element $a,b\in G$. If $E^{\mathcal{L}}(a)\neq E^{\mathcal{L}}(b)$ then $E^{\mathcal{L}}(a)\cap E^{\mathcal{L}}(b)$ is a locally finite group.
Let $\{a_i\},\{b_i\} $ be representatives of $a,b$ respectively. Note that for any non trivial element $g\in G$, $E^{{\mathcal{L}}}(g)=\underset{\rightarrow}{\lim}E_i(g)$, where $E_i(g)$ is the maximal elementary subgroup in $G_i$ containing the element $g$ when viewed as an element of $G_i$ for every $i$. We get $E^{\mathcal{L}}(a)\cap E^{\mathcal{L}}(b)= \underset{\rightarrow}{\lim}E_i(a_i)\cap E_i(b_i) $, where $E_i(a),E_i(b)$ are the maximal elementary subgroups of the hyperbolic group $G_i$ containing the elements $a_i,b_i$ respectively. $E_i(a_i)\cap E_i(b_i) $ is finite (see [@Ol93 Lemma 1.6]). Being a limit of finite group $E^{\mathcal{L}}(a)\cap E^{\mathcal{L}}(b)$ is locally finite.
Rank one abelian subgroups of lacunary hyperbolic groups {#rank1monstergroup}
--------------------------------------------------------
In this section we are going to describe rank one abelian subgroups of lacunary hyperbolic groups and maximal subgroups.
Two element $g,h$ of infinite order in a (hyperbolic) group is said to be commensurable if $g^k=ah^la^{-1}$ for some non zero integers $k,l$ and some element $a\in G$.
By [@Ol93 Lemma 1.6], $g$ and $h$ are commensurable if and only if the maximal elementary subgroups $E(g),E(h)$ containing two elements $g$ and $h$ respectively, are conjugate.
Let $G_0={\langle}X{\rangle}$ be a torsion free $\delta$-hyperbolic group with respect to $X$, where $X=\{x_1,x_2,\cdots,x_n \}$ is a finite generating set. Without loss of generality we assume that $E(x_i)\cap E(x_j)=\{e\}$ for $i\neq j$. Let $X$ be linearly ordered such that $x_i^{-1}<x_j^{-1}<x_i<x_j$ if $i<j$. Let $F'(X)$ denote the set of non empty reduced words on $X$, and $F'(X)=\{w_1,w_2,\cdots \}$ be an enumeration with $w_i<w_j$ for $i<j$ according to the lexicographic order induced from the order on $X$. Note that $w_1=(x_1)^{-1}$ and $w_2=(x_2)^{-1}$. We now order the set $\mathcal S:= F'(X)\times F'(X)$ lexicographically and enumerate them as, $$\begin{aligned}
\label{u_kwords}
\mathcal S=\{(u_1,v_1),(u_2,v_2),\cdots \}\end{aligned}$$ where for $i<j$ we have $(u_i,v_i)<(u_j,v_j)$.
We are going to construct the following chain;\
$$\begin{aligned}
G_0\overset{\beta_0}{\hookrightarrow} H_1\overset{\alpha_1}{\twoheadrightarrow}G'_1\overset{\gamma_1}{\twoheadrightarrow} G_1\overset{\beta_1}{\hookrightarrow} H_2 \overset{\alpha_2}{\twoheadrightarrow}G'_2\overset{\gamma_1}{\twoheadrightarrow} G_2 \cdots\end{aligned}$$ where $H_i,G_i,G'_i$ are hyperbolic for all $i$ and $\gamma_i\circ \alpha_{i-1}\circ \beta_{i-1}$ is surjective for all $i\geq 1$ and takes generating set to generating set. In particular we are going to show that the following chain $$\begin{aligned}
G_0\overset{\gamma_1\circ \alpha_{1}\circ \beta_{0}}{\twoheadrightarrow}G_1\overset{\gamma_2\circ \alpha_{2}\circ \beta_{1}}{\twoheadrightarrow} G_2\overset{\gamma_3\circ \alpha_{3}\circ \beta_{2}}{\twoheadrightarrow}\cdots G_{i-1}\overset{\gamma_i\circ \alpha_{i}\circ \beta_{i-1}}{\twoheadrightarrow}G_i\rightarrow \cdots\end{aligned}$$ satisfies part c. of Theorem \[OOS\].
Let $\mathscr{C}=\{Q^p\}_{p\in\mathbb{N}}$ be a countable family of rank one abelian groups. Then for every $p\in\mathbb{N}$, $Q^j$ can be written as $Q^p=\cup_{i=0}^{i=\infty}Q^p_i$, where $Q^p_i=\langle g^p_i\rangle_{\infty}$ and $g^p_i=(g^p_{i+1})^{m^p_{i+1}}$ for some $m^p_{i+1}\in \mathbb{N}$.
Then there exists a smallest index $j_i\geq i$ such that $v_{j_i}\notin E(u_{j_i})$. For $m\in \mathbb N$, define $$\begin{aligned}
\label{amalgum}
H^k_{i+1}:=H^{k-1}_{i+1}\underset{u_k=(g^k_{(k,i+1)})^{m^k_{i+1}}}{*}{\langle}g^k_{(k,i+1)} {\rangle}_{\infty}\ where\ H^0_{i+1}=G_i\ and \ g^k_{(k,i+1)}=g^k_{i+1} \ for\ k=1,2,\cdots ,j_i.\end{aligned}$$ For $i \geq 0$ let $H_{i+1}$ to be $H^{j_i}_{i+1}$. Note that $H_{i+1}$ is hyperbolic as $H^{k}_{i+1}$ is hyperbolic for all $k$ by [@MO98 Theorem 3]. By construction there is a natural embedding $\beta_i:G_i\hookrightarrow H_{i+1}$. Take $c_i,c_i'\in G$ such that $c_i',c_i\notin E(u_k)$ for all $1 \leq k\leq j_i$ and $c_i',c_i\notin E(v_{j_i}) $. Such $c_i$ and $c_i'$ exists as there are infinitely many elements in a non elementary hyperbolic group which are pairwise non commensurable by [@Ol93 Lemma 3.2]. Now we define the following set $$\begin{aligned}
Y_i:=\{g_{(k,i+1)}|1\leq k\leq j_i \}\end{aligned}$$ Let us consider $\mathcal{\tilde{R}}_{i+1}:=\mathcal{R}(Y_i,c_i,c_i',\lambda,c,\epsilon,\mu,\rho)$ as defined in the section 4.4.
Consider the natural quotient map $\alpha_{i+1}:H_{i+1}\twoheadrightarrow G'_{i+1}$ to the quotient $G'_{i+1}:={\langle}H_{i+1}|\tilde{\mathcal{R}}_{i+1}{\rangle}$. Also the factor group $G'_{i+1}$ is hyperbolic by [@Ol93]. it is easy to see that $\alpha_{i+1}\circ \beta_{i}$ is a surjective map that takes generators to generators.
Consider the following set $$Z_i:=\{x\in X|x\notin E(u_{j_i}) \}.$$ Let $G_{i+1}:= G'_{i+1}/{\langle}{\langle}\mathcal R (Z_i,u_{j_i},v_{j_i},\lambda,c,\epsilon,\mu,\rho){\rangle}{\rangle}$ and let $\gamma_{i+1}:G'_{i+1}\twoheadrightarrow G_{i+1}$ be the quotient map. Hence we get that the group $G_{i+1}$ is hyperbolic by Theorem \[ol93lemma\] since $\mathcal R (Z_i,u_{j_i},v_{j_i},\lambda,c,\epsilon,\mu,\rho)$ satisfies $C'(\lambda,c,\epsilon,\mu,\rho)$ small cancellation condition as discussed in section 4.4 and the map $\gamma_{i+1}$ takes generating set to generating set. In particular $\eta_{i+1}:=\gamma_{i+1}\circ \alpha_{i+1}\circ \beta_{i}$ is surjective homomorphism which takes the generating set of $G_i$ to the generating set of $G_{i+1}$. Let $G^{\mathscr{C}}:=\underset{\rightarrow}{\lim}G_i$. We summarize the above discussion in the following statement.
\[limitmax\] The above construction satisfies the following properties:
1. $G_i$ is non elementary hyperbolic group for all $i$;
2. Either $u_i\in E(v_i)$ or the group genarated by $\{u_i,v_i \}$ is equal to all of $G_i$;
3. For every $k\in\mathbb{N}$ we have $E(u_k)\simeq Q^k_l$ in $G_l$ for all $l\geq k$, where the element $u_k$ is defined in $\color{blue}(\ref{u_kwords})\color{black}$.
Part 1. follows from [@Ol93 Lemma 7.2]. To see part 2. notice that by definition if $j_i>i$ then $v_i\in E(u_i)$ in $G_i$. Otherwise if $j_i=i$ then $v_i\notin E(u_i)$ in $G_i$ and $G_i=gp\{u_i,v_i\}$. For part 3. note that we have $u_k=(g^{k}_{l})^{m^{k}_l}$ in $H_l$ for all $l\geq k$ by (\[u\_kwords\]) hence $E(u_k)\simeq Q^k_l$ in $H_l$. By part $(5)$ of [@Ol93 Theorem 2] we get $E(u_k)\simeq Q^{k}_l$ since centralizer of image of an element inside the injectivity radius in the quotient group is same as image of centralizer of that element. For the same reason we have $E(u_k)\simeq Q^{k}_l$ in the quotient group $G_l$.
\[maxele\] Note that every element $g\in G$ is equal to $u_k$ for some $k\in\mathbb{N}$ and hence $E^{{\mathcal{L}}}(g)=\underset{\rightarrow}{\lim} E_l(u_k)$ where $E_l(u_k)$ is $E(u_k)$ in $G_l$. By part 3. of lemma \[limitmax\] we get that $E_l(u_k)\simeq Q^k_l$ for every $l\geq k$. Hence $E^{{\mathcal{L}}}(g)=\underset{\rightarrow}{\lim} E_l(u_k)=\underset{\rightarrow}{\lim}Q^k_l\simeq Q^k$
We now give the main theorem of this section.
\[monster\] For any torsion free non elementary hyperbolic group $G$ and a countable family $\mathscr{F}:=\{Q^i_m\}_{i\in\mathbb{N}}$ of subgroups of $(\mathbb Q,+)$, there exists a non elementary, torsion free, non abelian lacunary hyperbolic quotient $G^{\mathcal{C}}$ of $G$ such that the set of all maximal subgroups of $G^{\mathscr{F}}$ is equal up to isomorphism to $\{Q^i_m\}_{i\in\mathbb{N}}$ i.e, every maximal subgroup of $G^{\mathscr{F}}$ is isomorphic to $Q^i_m$ for some $i\in\mathbb{N}$ and for every $i$ there exists a maximal subgroup of $G^{\mathscr{F}}$ that is isomorphic to $Q^i_m$.
By letting $\mathscr{F}=\mathscr{C}$ in the above construction we get $G^{\mathscr{C}}=G^{\mathscr{F}}$, where $Q^i_j=\langle g^i_j\rangle_{\infty}$ and $g^i_j=({g^i_{j+1}})^{m^i_{j+1}}$ for some $m^i_{j+1}\in \mathbb{N}$ and $Q^i_m=\cup_{j=1}^{\infty}Q^i_j$. One can choose sparse enough parameters to satisfy the injectivity radius condition in Definition \[lacunarylim\] which in turn will ensure that $G$ is lacunary hyperbolic. By remark \[maxele\] we get that for any $g\in G$, $E^{{\mathcal{L}}}(g)=Q^i$ for some $i$ depending of $g$. Suppose $P\nleqslant G$ is a maximal subgroup of $G$. As $P$ is a proper subgroup, $P$ is abelian by part $2.$ of Lemma \[limitmax\]. Now let $e\neq h\in G$. Note that being abelian $P$ is contained in the centralizer of $h$. Now from Definition of $E^{{\mathcal{L}}}(g)$, it follows that $g\in P\leq E^{{\mathcal{L}}}(g)(\cong Q_m)\nleqslant G$. By maximality of $P$ we get that $P\cong Q_m$. Thus, all maximal subgroups of $G$ are isomorphic to $Q^i_m$ for some $i$ and hence the theorem is proved.
As a corollary of this theorem we can recover the seminal result of monster groups constructed by Ol’shanskii in $\cite{Ol93}$.
For every non-cyclic torsion free hyperbolic group $G$, there exists a non abelian torsion free quotient $\overline{G}$ such that all proper subgroups of $\overline{G}$ are infinite cyclic.
Take $Q^i_m=\mathbb{Z}$ for all $i\in\mathbb{N}$ in Theorem \[monster\].
A group $G$ is called divisible if for any element $g$ of $G$ and any non zero integer $n$ the equation $x^n=g$ has a solution in $G$. The first example of non trivial finitely generated divisible group was constructed by V. S. Guba in [@Gu86]. Later Ol’shanskii and Mikhajlovskii proved following;
[@MO98 Corollary 3] For every non cyclic torsion free hyperbolic group $G$ there exists a non abelian torsion free divisible quotient $H$ of $G$.
Take $\mathcal{C}=\{\mathbb{Q} \}$ in Theorem \[monster\].
[@Gu86 Theorem 1] There exists a non trivial finitely generated torsion free divisible group.
If one start with non trivial property $(T)$ torsion free hyperbolic group in Theorem \[monster\] (same for Corollary 3,[@MO98]), then one gets finitely generated non amenable divisible group with property $(T)$, since property $(T)$ is preserved under taking quotients.
A group $G$ has the unique product property (or said to be unique product group) whenever for all pairs of non empty finite subsets $A$ and $B$ of $G$ the set of products $AB$ has an element $g\in G$ with a unique representation of the form $g=ab$ with $a\in A$ and $b\in B$.
Note that unique product groups are torsion free. The first examples of torsion-free groups without the unique product property was given by Rips and Segev in [@RS87]. Existence of property $(T)$ hyperbolic group without the unique product property has been shown in [@AS14]. By starting with property $(T)$ hyperbolic group without unique product property in Theorem \[monster\], we obtain following:
\[monsteruniqueproduct\] For every rank one abelian group $Q_m$, there exists a non elementary, torsion free, property $(T)$, lacunary hyperbolic group without the unique product group $G^{Q_m}$ such that any maximal subgroup of $G^{Q_m}$ is isomorphic to $Q_m$.
In particular, there exist a non trivial property $(T)$ torsion free divisible lacunary hyperbolic group without the unique product property.
Note that when we are adding relations to the starting group we need to start with a relation of sufficiently large length (bigger that the size of the sets $A,B$ and $AB$ for with the starting group does not have unique product property).
Locally finite by rank one abelian subgroups of lacunary hyperbolic groups {#rankonebyfinitesection}
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In this section we are going to prove the full converse of Theorem \[elementarygroup\]. First we are going to recall some definitions and theorems.
Given a subgroup $H\leq G$, $H^0$ denotes the set of all infinite order elements of $H$.
Let $H$ be a non elementary subgroup of a hyperbolic group $G$. Then $E_G(H):=\underset{x\in H^0}{\bigcap}E(x)$ is the unique maximal finite subgroup of $G$, normalized by the subgroup $H$, where $E(x)$ is the unique maximal elementary subgroup of $G$ containing the infinite order element $x$.
Now we are going to state a well known fact about elementary subgroups of hyperbolic groups.
For every infinite order element $g$ in a hyperbolic group $G$ there exists a series of normal subgroups; $$\begin{aligned}
1\leq T(g)\leq E^{+}(g)\leq E(g)
\end{aligned}$$ where $T(g)$ is the set of torsion elements of the unique maximal elementary subgroup $E(g)$ of the hyperbolic group $G$, $|E(g):E^+(g)|\leq 2$, $E^+(g)/T$ is infinite cyclic. Note that $T(g)$ is finite.
A subgroup $H\leq G$ is called suitable, if there exist two non commensurable elements $g,h$ in $H^0$ such that $E(g)\cap E(h)=\{1\}$.
Note that a subgroup $H$ of a hyperbolic group $G$ is same as $H$ is non elementary and $E_G(H)=\{1\}$. We would like to state a simple version of a very beautiful theorem by D. Osin in our context,
\[osin\] Let $H$ be a suitable subgroup of a hyperbolic group $G$ and $T:=\{t_1,t_1,\cdots,t_n\}\subset G$. Then there exists $\{w_1,w_2,\cdots,w_n\}\subset H$ such that the quotient group $\overline{G}:=G/{\langle}{\langle}t_1w_1,t_2w_2,\cdots,t_nw_n{\rangle}{\rangle}$ satisfies following;
1. $\overline{G}$ is hyperbolic.
2. Image of $H$ in $\overline{G}$ is a suitable subgroup of $\overline{G}$.
3. If $G$ is torsion free then so is $\overline{G}$.
\[osinlacunar\] Note that one can choose lengths of the elements $w_1,w_2,\cdots,w_n$ to be as large as one wants.
\[rankonefinite\] Let $G$ be a non elementary hyperbolic group, $\{E_i\}_{i\geq 1}$ be a collection of elementary groups with $E_i\subset E_{i+1}$ for all $i\geq 1$ and denote $E:=\underset{i\geq 1}{\bigcup}E_i$. Then there exists a lacunary hyperbolic quotient $\overline{G}$ of $G$ with an infinite order element $g\in \overline{G}$, such that $E^{{\mathcal{L}}}(g)\simeq E$.
Let $G_0={\langle}S{\rangle}, \ |S|<\infty $ be a non elementary hyperbolic group, $E_i$ be elementary groups with ${\langle}g_i{\rangle}$ be maximal cyclic subgroup of $E_i$ and $E_i={\langle}g_i{\rangle}\cup x_1{\langle}g_i{\rangle}\cup \cdots \cup x_{n_i}{\langle}g{\rangle}$ for $i=1,2$ and $n_1\leq n_2$. Note that $x_1,x_2,\cdots,x_{n_2}$ are elements of finite order and $g_1=(g_2)^{m_2}$ for some $m_2\in \mathbb{N}$. Let $H_0$ be a suitable subgroup of $G_0$. One can choose $H_1$ since any non elementary hyperbolic group contains a copy of free group with countably many generators.
Consider the group, $$\begin{aligned}
G'_0:=(G_0*E_1)\underset{E_1}{*}E_2.
\end{aligned}$$ Note that $G'_0$ is a non elementary hyperbolic group by Theorem \[olmik\] as $E_1$ is maximal elementary in the hyperbolic group $G_0*E_1$. Note that $H_0$ is a suitable subgroup of $G'_0$. Let $N\in\mathbb{N}$. Now we apply Lemma \[osin\] with $T=\{x_1,x_2,\cdots,x_{n_2} \}, \ G=G'_0$ and $H=H_0$. We choose $w_1,w_2,\cdots,w_n$ such that the injectivity radius of the the quotient map $\phi:G'_0\twoheadrightarrow \overline{G'_0}:=G'_0/{\langle}{\langle}x_1w_1,\cdots,x_{n_2}w_{n_2}{\rangle}{\rangle}$ is greater that $N$ (by Remark \[osinlacunar\]). We record the fact that the group $\overline{G_1}$ is generated by $X\cup \{g_2 \}$ and image of the suitable group $H_0$, say $H'_0$ is also suitable in the quotient group $\overline{G'_0}$. Now we consider the set $\mathcal{R}_0$ of cyclic shifts of following set of words $$\begin{aligned}
R:=g_2U^{l_1}VU^{l_{2}}V\cdots U^{l_{n_2}}
\end{aligned}$$ where $U,V$ are geodesic representative of two non commensurable elements $h_1,h_2$ in the suitable group $H'_0$ such that $E(h_1)\cap E(h_2)=\{1\}$ and $E(h_i)={\langle}h_i{\rangle}_{\infty}$ for $i=1,2$. Then there exists $\lambda,c,N$ such that for all $\epsilon,\mu,\rho$ and $N<l_1<l_2<\cdots<l_{n_2}$, $\mathcal{R}_0$ satisfies $C'(\epsilon,\mu,\lambda,c,\rho)$ condition over the group $\overline{G'_0}$.
The quotient group $G_1:={\langle}\overline{G'_0}|\mathcal{R}_0{\rangle}$ enjoys following properties:
1. $G_1$ is hyperbolic.
2. Injectivity radius of the quotient map $\epsilon: G_0\twoheadrightarrow G_1$ is $\geq N$.
3. $\epsilon(g_2)$ is not a proper power in $G_1$.
4. $\epsilon(H_0)$ is again suitable in $G_1$.
5. $E(\epsilon(g_2))\simeq E_2$.
$1.$ and $2.$ and $4.$ follows from Theorem \[osin\] and Lemma \[ol93lemma\].
By property $(5)$ of [@Ol93 Theorem 2] we get that the centralizer $C_{G_1}(\epsilon(a))$ for every $a$ in the injectivity ball of $\epsilon$ is the image of centralizer $\epsilon(C_{G_0}(a))$. Suppose $\epsilon(g_2)=\epsilon(y)^m=\epsilon(y^m)$ for some $y\in G_0$. Then $\epsilon(y)\in C_{G_1}(\epsilon(g_2))=\epsilon(C_{G_0}(g_2))$. This implies that $y\in C_{G_0}(g_2)\ \Rightarrow y\in {\langle}g_2{\rangle}$, as $y$ has infinite order. Hence $m=1$ and $\epsilon(g_2)$ is not a proper power in $G_1$.
$5.$ follows from the construction and above properties.
We start with a given non elementary hyperbolic group $G$. By [@Ol93 Proposition 1] Every non–elementary hyperbolic group $G$ contains a unique maximal normal finite subgroup $K \leq G$, in fact $K$ is precisely the kernel of the $G$-action on the boundary of $G$. Thus passing to the quotient $G/K$ if necessary we may assume that $G$ has no nontrivial finite normal subgroups. Take $G=G_0$ in the above process to obtain a quotient hyperbolic group $G_1$ of $G$ with an element $g_2$ such that $E(g_2)\simeq E_2$ with $N_1$ the injectivity radius of $\epsilon_1:G\twoheadrightarrow G_1$. Now one can repeat above procedure with a slide modification: take $G'_0=G_1*_{E(g_2)}E_{3}$. Then called the quotient group $G_3$ and the quotient map $\epsilon_2:G_1\twoheadrightarrow G_3$ and the injectivity radius of the map $\epsilon_2$ is $N_3$. We get $E(\epsilon_2(g_2))\simeq E_3$. Using induction we get, $$\begin{aligned}
G=G_0\overset{\epsilon_1}{\twoheadrightarrow} G_1\overset{\epsilon_2}{\twoheadrightarrow} G_3\overset{\epsilon_3}{\twoheadrightarrow} G_4\overset{\epsilon_4}{\twoheadrightarrow}\cdots \end{aligned}$$ Define $\overline{G}:=\underset{\rightarrow}{\lim}G_i$. Note that one can take the numbers $N_i$ to be as large as one wants in order to make $\overline{G}$ a lacunary hyperbolic group. Observe that $E^{{\mathcal{L}}}(g_2)=\underset{i\geq 1}{\bigcup} E_i=E$. Note that we are viewing $g_2$ as an element of the limit group.
\[notationelementary\] We denote the class of increasing union of elementary subgroups as ${}_{rk-1}\mathscr{E}_{F}$. Note that increasing union of elementary groups and also one can write those as rank one by finite groups.
By using the fact that $G$ is countable and by doing the process described in Theorem \[rankonefinite\] we can get following,
\[rk1byfinite\] Let $G$ be a torsion free non elementary hyperbolic group and $\mathscr{C}:=\{E^j \}_{j\in\mathbb{N}}$ be a countable collection of groups with $E^j\in {}_{rk-1}\mathscr{E}_{F}$ for all $j\geq 1$. Then there exists a non elementary lacunary hyperbolic quotient $G^{\mathscr{C}}$ of $G$ such that $\{E^{{\mathcal{L}}}(h)\ | \ h\in (G^{\mathscr{C}})^0 \}=\mathscr{C}$.
Moreover $\mathscr{C}$ is the set of all maximal proper subgroups of the group $G^{\mathscr{C}}$.
Proof of this theorem is basically a suitable combination of Theorem \[rankonefinite\] and the construction in in Section \[rank1monstergroup\]. Let $G={\langle}X{\rangle}$ be the given non elementary torsion free $\delta$-hyperbolic group with respect to $X$, where $X=\{x_1,x_2,\cdots,x_n \}$ is a finite generating set. Without loss of generality we assume that $E(x_i)\cap E(x_j)=\{e\}$ for $i\neq j$. Let $X$ be linearly ordered such that $x_i^{-1}<x_j^{-1}<x_i<x_j$ if $i<j$. Let $F'(X)$ denote the set of non empty reduced words on $X$, and $F'(X)=\{w_1,w_2,\cdots \}$ be an enumeration with $w_i<w_j$ for $i<j$ according to the lexicographic order induced from the order on $X$. Note that $w_1=(x_1)^{-1}$ and $w_2=(x_2)^{-1}$. We now order the set $\mathcal S:= F'(X)\times F'(X)$ lexicographically and enumerate them as, $$\mathcal S=\{(u_1,v_1),(u_2,v_2),\cdots \}$$ where for $i<j$ we have $(u_i,v_i)<(u_j,v_j)$.
We are going to construct the following chain;\
$$\begin{aligned}
G_0\overset{\beta_0}{\hookrightarrow} H_1\overset{\alpha_1}{\twoheadrightarrow}G'_1\overset{\gamma_1}{\twoheadrightarrow} G_1\overset{\beta_1}{\hookrightarrow} H_2 \overset{\alpha_2}{\twoheadrightarrow}G'_2\overset{\gamma_1}{\twoheadrightarrow} G_2 \cdots\end{aligned}$$ where $H_i,G_i,G'_i$ are hyperbolic for all $i$ and $\gamma_i\circ \alpha_{i-1}\circ \beta_{i-1}$ is surjective for all $i\geq 1$ and takes generating set to generating set. In particular we are going to show that the following chain $$\begin{aligned}
G_0\overset{\gamma_1\circ \alpha_{1}\circ \beta_{0}}{\twoheadrightarrow}G_1\overset{\gamma_2\circ \alpha_{2}\circ \beta_{1}}{\twoheadrightarrow} G_2\overset{\gamma_3\circ \alpha_{3}\circ \beta_{2}}{\twoheadrightarrow}\cdots G_{i-1}\overset{\gamma_i\circ \alpha_{i}\circ \beta_{i-1}}{\twoheadrightarrow}G_i\rightarrow \cdots\end{aligned}$$ satisfies part c. of Theorem \[OOS\].
We have $\mathscr{C}=\{E^p\}_{p\in\mathbb{N}}$, a countable family of groups from ${}_{rk-1}\mathscr{E}_{F}$ . Then for every $p\in\mathbb{N}$, $E^j$ can be written as $E^p=\cup_{i=0}^{i=\infty}E^p_i$, where $E^p_i=\langle g^p_i\rangle_{\infty}\cup a^p_1\langle g^p_i\rangle_{\infty}\cup a^p_2\langle g^p_i\rangle_{\infty}\cup\cdots\cup a^p_{n_i}\langle g^p_i\rangle_{\infty}$ with order of $a^p_i$ is finite, $n_i\leq n_{i+1}$ for every $i\geq 1$ and $g^p_i=(g^p_{i+1})^{m^p_{i+1}}$ for some $m^p_{i+1}\in \mathbb{N}$.
Then there exists a smallest index $j_i\geq i$ such that $v_{j_i}\notin E(u_{j_i})$. For $m\in \mathbb N$, define $$\begin{aligned}
\label{amalgumrankonebyfinite}
H^k_{i+1}:=H^{k-1}_{i+1}\underset{u_k=(g^k_{(k,i+1)})^{m^k_{i+1}}}{*} E^k_i \ ,\ where\ H^0_{i+1}=G_i\ and \ g^k_{(k,i+1)}=g_{i+1} \ for\ k=1,2,\cdots ,j_i.\end{aligned}$$ For $i \geq 0$ let $H_{i+1}$ to be $H^{j_i}_{i+1}$. Note that $H_{i+1}$ is hyperbolic as $H^{k}_{i+1}$ is hyperbolic for all $k$ by [@MO98 Theorem 3]. By construction there is a natural embedding $\beta_i:G_i\hookrightarrow H_{i+1}$. Take $c_i,c_i'\in G$ such that $c_i',c_i\notin E(u_k)$ for all $1 \leq k\leq j_i$ and $c_i',c_i\notin E(v_{j_i}) $. Such $c_i$ and $c_i'$ exists as there are infinitely many elements in a non elementary hyperbolic group which are pairwise non commensurable by [@Ol93 Lemma 3.2]. Now we define the following set $$\begin{aligned}
Y_i:=\{g_{(k,i+1)}|1\leq k\leq j_i \}\end{aligned}$$ Let us consider $\mathcal{R}(Y_i,c_i,c_i',\lambda,c,\epsilon,\mu,\rho)$ as defined in the section 4.4 and apply Theorem \[osin\] with $G:=H_{i+1}/\mathcal{R}(Y_i,c_i,c_i',\lambda,c,\epsilon,\mu,\rho)$, $T=\{a^s_{l}|1\leq s\leq j_i, 1\leq l\leq n_{j_i} \}$ and suitable subgroup $H_{i}$. Let $\tilde{\mathcal{R}}_{i+1}= \mathcal{R}(Y_i,c_i,c_i',\lambda,c,\epsilon,\mu,\rho)\cup \{a^s_{l}w^s_{l}|1\leq s\leq j_i, 1\leq l\leq n_{j_i} \}$
Consider the natural quotient map $\alpha_{i+1}:H_{i+1}\twoheadrightarrow G'_{i+1}$ to the quotient $G'_{i+1}:={\langle}H_{i+1}|\tilde{\mathcal{R}}_{i+1}{\rangle}$. Also the factor group $G'_{i+1}$ is hyperbolic by [@Ol93]. it is easy to see that $\alpha_{i+1}\circ \beta_{i}$ is a surjective map that takes generators to generators.
Consider the following set $$Z_i:=\{x\in X|x\notin E(u_{j_i}) \}.$$ Let $G_{i+1}:= G'_{i+1}/{\langle}{\langle}\mathcal R (Z_i,u_{j_i},v_{j_i},\lambda,c,\epsilon,\mu,\rho){\rangle}{\rangle}$ and let $\gamma_{i+1}:G'_{i+1}\twoheadrightarrow G_{i+1}$ be the quotient map. Hence we get that the group $G_{i+1}$ is hyperbolic by Theorem \[ol93lemma\] since $\mathcal R (Z_i,u_{j_i},v_{j_i},\lambda,c,\epsilon,\mu,\rho)$ satisfies $C'(\lambda,c,\epsilon,\mu,\rho)$ small cancellation condition as discussed in section 4.4 and the map $\gamma_{i+1}$ takes generating set to generating set. In particular $\eta_{i+1}:=\gamma_{i+1}\circ \alpha_{i+1}\circ \beta_{i}$ is surjective homomorphism which takes the generating set of $G_i$ to the generating set of $G_{i+1}$. Let $G^{\mathscr{C}}:=\underset{\rightarrow}{\lim}G_i$. We summarize the above discussion in the following statement.
\[limitmaxrankonebyfinite\] The above construction satisfies the following properties:
1. $G_i$ is non elementary hyperbolic group for all $i$;
2. Either $u_i\in E(v_i)$ or the group genarated by $\{u_i,v_i \}$ is equal to all of $G_i$;
3. For every infinite order element $u_k$, $E(u_k)\simeq E^p_{n_0}$ in $G_{n_0}$ for some $ n_0$ and $p$.
Part 1. follows from [@Ol93 Lemma 7.2]. To see part 2. notice that by definition if $j_i>i$ then $v_i\in E(u_i)$ in $G_i$. Otherwise if $j_i=i$ then $v_i\notin E(u_i)$ in $G_i$ and $G_i=gp\{u_i,v_i\}$.
For every infinite order element $u_k$ note that $E(u_k)\simeq E^p_{n_0}$ for some $p$ and $n_0$ in $H_{n_0}$ for some $n_0$ by (\[amalgumrankonebyfinite\]). Then by property $(5)$ of [@Ol93 Theorem 2] and Theorem \[osin\] $E(u_k)\simeq E^p_{n_0}$ in $G_{n_0}$.
\[lemma4.41\]
1. By part 3. of Lemma we get that $E(u_k)\simeq E^p_{n_0}$ in $G_{n_0}$ for some $p$ and $n_0$. Hence by (\[amalgumrankonebyfinite\]) we have $E(u_k)\simeq E^p_{n_0+1}$ in $H_{n_0+1}$. Then by property $(5)$ of [@Ol93 Theorem 2] and Theorem \[osin\] $E(u_k)\simeq E^p_{n_0+1}$ in $G_{n_0+1}$. By applying same process we get that $E(u_k)\simeq E^p_{n_0+i}$ in $G_{n_0+i}$ for every $i\geq 0$. Hence $E^{{\mathcal{L}}}(u_k)=\underset{\rightarrow}{\lim}E_l(u_k)=\underset{\rightarrow}{\lim} E^p_{n_0+i}=E^p$, where $E_l(u_k)$ is the maximal elementary subgroup containing the infinite order element $u_k$.
2. Note that by part (7) of [@Ol93 Theorem 2] and Theorem \[osin\] we get that if $x\in G_k$ has finite order then $x\in E^p_i$ for some $p,i$ in $G_k$.
By above construction it is clear that $\{E^{{\mathcal{L}}}(g)|\ g\in(G^{\mathscr{C}})^0 \}=\mathscr{C}$. Suppose that $P$ be a proper maximal subgroup of the group $G^{\mathscr{C}}$. Let $x\in P$ is of finite order. Then by part 2. of remark \[lemma4.41\] $x\in E^p$ for some $p\in\mathbb{N}$. If $y\in P\setminus E^p$ then $yg^p_1$ is of infinite order and not in $E^p$. So by construction we get $G=gp\{E^p,yg^p \}\leq P\leq G$. Hence $P=E^p$. Similarly for $x\in P$ has infinite order one can show $P=E^p$ for some $p$ and hence $\mathscr{C}$ is the class of maximal proper subgroups of $G^{\mathscr{C}}$.
The group $G^{\mathscr{C}}$ in Theorem \[rk1byfinite\] also has following properties:
1. Every finitely generated subgroup of the group $G^{\mathscr{C}}$ is either $G^{\mathscr{C}}$ or finite or elementary.
2. $G^{\mathscr{C}}$ has no finite presentation.
3. Every finitely presented subgroup of $G^{\mathscr{C}}$ is hyperbolic (being a finitely presented subgroup of a lacunary hyperbolic group) and hence is either finite or elementary. Therefore every finitely presented subgroup of $G^{\mathscr{C}}$ is amenable.
4. If one starts with a hyperbolic group without the unique product property then it is possible to make $G^{\mathscr{C}}$ a lacunary hyperbolic group without the unique product property.
\[rankbyfinitemonster\] Let $E\in {}_{rk-1}\mathscr{E}_{F}$ be a group and $G$ be a torsion free non elementary hyperbolic group. Then there exists a lacunary hyperbolic quotient $\overline{G}$ of $G$ such that every proper maximal subgroup of $G$ is isomorphic to $E$.
Rips type construction {#ripssection}
======================
In this section we explore the construction of small cancellation group from any finitely presented group developed by E. Rips [@Rip82].
Suppose $Q$ is a finitely presented group. Then there exists groups $G$ and $K$ , for which we get a short exact sequence; $$1\rightarrow K\rightarrow G\rightarrow Q\rightarrow 1$$ such that;
1. $G$ is a hyperbolic group.
2. $K$ is a $2$ generated group.
Various types of Rips constructions have been studied in order to construct powerful pathological examples in geometric group theory, see [@AS14; @BO06; @OW04; @W03]. General theme of Rips construction is to study exotic properties of normal subgroups of hyperbolic groups by allowing to construct groups with certain group theoretic/geometric properties from a countable group.
\[Rips\]
Let $Q$ is a finitely generated group and $H$ be a non elementary hyperbolic group. Then there exists groups $G$ and $N$ , and a short exact sequence; $$\begin{aligned}
1\rightarrow N\rightarrow G\rightarrow Q\rightarrow 1\nonumber
\end{aligned}$$ such that;
1. $G$ is a torsion-free lacunary hyperbolic group.
2. $N$ is a $2$ generated non elementary quotient of $H$.
3. If $H$ is torsion free then so are $G$ and $N$.
Note that $G$ cannot be hyperbolic if $Q$ has no finite presentation.
If $H$ is not $2$ generated then consider a finite generating set $H={\langle}X{\rangle}=\{x_1,x_2,\cdots,x_l\}$ with $l\geq 3$ such that $x_i$ has infinite order for all $l\geq i\geq 1$. Note that one can choose such generating set since for every torsion element $s$ in a non elementary hyperbolic group $H$, there exists an infinite order element $t$ such that $st^m$ has infinite order for some $m\in\mathbb{N}$. Without loss of generality assume that $x_1,x_2$ are non commensurable and $E(x_1)\cap E(x_2)=\{1\}$. Note that the subgroup $gp\{x_1,x_2\}$, generated by $x_1,x_2$, is a suitable subgroup of $H$. Applying Theorem \[osin\] with non elementary hyperbolic group $H$, suitable subgroup $gp\{x_1,x_2\}$ and $T=\{x_3,x_4,\cdots,x_l \}$ we get that there exists elements $v_3,v_4,\cdots,v_l\in gp\{x_1,x_2\}$ such that the quotient group $\overline{H}:=H/{\langle}{\langle}x_3v_3,x_4v_4,\cdots,x_lv_l{\rangle}{\rangle}$ is non elementary hyperbolic. Note that the quotient map is surjective on $gp\{x_1,x_2\}$ by choice of the set $T$. Hence if $H$ is not $2$ generated we replace $H$ by the quotient $gp\{x_1,x_2\}$ as described above which is a $2$ generated non elementary hyperbolic quotient of the group $H$. Without loss of generality we now assume that $H$ is $2$ generated i.e, $H={\langle}X=\{x,y\}{\rangle}, \ |X|=2 $ and $x\notin E(y)$.
Consider a presentation of $Q$. $$\begin{aligned}
Q={\langle}g_1,g_2,\cdots,g_n|r_1,r_2,r_3,\cdots{\rangle}\nonumber
\end{aligned}$$ We assume that $\| r_i\|_{\mathcal{F}(X)} \geq \| r_{i-1}\|_{\mathcal{F}(X)} \ \ \forall \ i\in \mathbb{N}$. where $X=\{g_1,g_2,\cdots,g_n \}$ and $\mathcal{F}(X)$ is free group generated by $X$. Let $$\begin{aligned}
Q_i:=\langle g_1,g_2,\cdots,g_n|r_1,r_2,\cdots,r_i\rangle\nonumber
\end{aligned}$$
First consider $G_0:=H*\mathcal{F}(g_1,g_2,\cdots,g_n)$, the free product of non elementary hyperbolic group $H$ with the free group generated by generators of $Q$. Note that $G_0$ is a hyperbolic group. Since $H$ is non elementary hyperbolic group, $H$ is a suitable subgroup of $G_0$. Let $T=\{{g_j}^{-1}x{g_j},g_jx{g_j}^{-1}\ | \ 1\leq j\leq n, \ x\in X \}$. Now by Theorem \[osin\] there are elements $\{w_1,w_2,\cdots \}\subset H$ such that the quotient $\overline{G_0}$ is hyperbolic and image of $H$, say $H_0$ is suitable in $\overline{G_0}$. Hence there exists two elements $a,b\in H_0$ such that $E(a)\cap E(b)=\{1\}$.
Define following set of words as defined in Section 4.4, $$\begin{aligned}
R_1:=z_1U^{m_{1,1}}VU^{m_{1,2}}V\cdots VU^{m_{1,j_1}}
\end{aligned}$$ where $U,V$ are geodesic representatives of $a,b$ respectively in the group $G_0$ and $z_1$ is a geodesic representative of $r_1$ in $\overline{G_0}$. Let $\mathcal{R}_1$ be the set of all cyclic shifts of $R_1$. Let $N_1$ be a positive number. For any $\epsilon\geq 0,\mu>0,\rho>0$ with $\lambda,c$ given by Lemma \[smallwords\] and $\lambda,\mu,\epsilon,\rho,N_1,c$ satisfies condition for Theorem \[ol93lemma\] for the hyperbolic group $\overline{G_0}$. Combining the fact that $\mathcal{R}_1$ satisfies $C'(\lambda,c,\epsilon,\mu,\rho)$ and Theorem \[ol93lemma\] with $H_0:=gp\{a,b\},H_1:=gp\{a,z_1\}$ and $H_2:=\{b,z_1 \}$, we get that the factor group $G_1:=\langle H|\mathcal{R}_1\rangle $ is hyperbolic and the injective radius of the factor homomorphism is $\geq N_1$. Moreover we get that image of $H_0,H_1,H_2$ are non elementary in the factor group $G_1$, i.e. $a,b\notin E(z_i),a\notin E(b)$ in $G_1$.
Define $K_1$ to be the images of $H_0$ in $G_1$. We continue by starting with $G_1$ instead of $G_0$ and add relations of the form $$\begin{aligned}
R_i:=z_iU^{m_{i,1}}VU^{m_{i,2}}V\cdots VU^{m_{i,j_i}}
\end{aligned}$$ where $z_i=r_i$ for $i\geq 2$, in the hyperbolic group $G_{i-1}$. Hence by induction we get the required group $G$ as a limit of hyperbolic groups $G_i$ and by choosing large enough $N_i$ in every step one can ensure that $G$ is lacunary hyperbolic. Also we get $N$ as inductive limit of groups $K_i$.
Let $Q$ has no finite presentation and $G$ is hyperbolic. Then $G$ has a finite presentation and the kernel is finitely generated which implies that $Q$ has a finite presentation. This contradicts the hypothesis that $Q$ has no finite presentation. Hence $G$ is not hyperbolic when $Q$ has no finite presentation. Note that being a non elementary image of a $2$ generated group, $N$ is $2$ generated.
As a corollary of Theorem \[Rips\] together with the existence of hyperbolic property $(T)$ group without unique product property, one can get following corollary:
\[uniqueproduct\] Let $Q$ be a finitely generated group. Then there exists a short exact sequence $$\begin{aligned}
1\rightarrow N\rightarrow G\rightarrow Q\rightarrow 1\nonumber
\end{aligned}$$
such that;
- $G$ is a torsion-free group without the unique product property which is a direct limit of Gromov hyperbolic groups.
- $N$ is a subgroup of $G$ with Kazhdan’s Property $(T)$ and without the unique product property.
In Theorem \[Rips\], let $H$ to be a hyperbolic property $(T)$ group without unique product property and in the proof one can choose $N_1$ and the sizes of $w_1,w_2,\cdots$ to be greater than the size of the finite sets $A,B$ and $AB$ for which unique product does not hold in the group $H$. Note that one can choose sizes of $w_1,w_2,\cdots$ to be arbitrary large by Remark \[osinlacunar\] and also one can choose $N_1$ to be as large as one wants.
Note that the group $G$ in Corollary \[uniqueproduct\] is a lacunary hyperbolic group which is a special class of direct limits of Gromov hyperbolic groups.
Applications to von Neumann algebras {#maximalvonneumann}
====================================
If $\mathcal{M}$ is a von Neumann algebra then a von Neumann subalgebra $\mathcal{N}\subset\mathcal{M}$ is called *maximal* if there is no intermediate von Neumann subalgebra $\mathcal{N}\nsubseteq\mathcal{P}\subsetneq\mathcal{M}$. Theorem \[monster\] generalizes Theorem 3.10 of [@CDK19] and hence we enlarge the class of property $(T)$ groups introduced in [@CDK19] that give rise to property $(T)$ von Neumann algebras which have maximal von Neumann subalgebras without property $(T)$. For readers’ convenience, we provides short details and notations that were used in [@CDK19 Section 3 & 4].
First we are going to state one of the main ingredient for our purpose, the Rips construction due to I. Belegradek and D. Osin.
[@BO06]\[beleosinripsconstr\] Let $H$ be a non-elementary hyperbolic group, $Q$ be a finitely generated group and $S$ a subgroup of $Q$. Suppose $Q$ is finitely presented with respect to $S$. Then there exists a short exact sequence $$1\rightarrow N\rightarrow G\overset{\epsilon}{\rightarrow} Q\rightarrow 1,$$ and an embedding $\iota:Q\rightarrow G$ such that
1. $N$ is isomorphic to a quotient of $H$.
2. $G$ is hyperbolic relative to the proper subgroup $\iota(S)$.
3. $\iota\circ \epsilon=Id$.
4. If $H$ and $Q$ are torsion free then so is $G$.
5. The canonical map $\phi :Q\hookrightarrow Out(N)$ is injective and $[Out(N):\phi(Q)]<\infty$.
In our setting $H$ is torsion free and has property (T) and $Q=S$ and it is torsion free. In this situation Theorem \[beleosinripsconstr\] implies that $G$ is admits a semidirect product decomposition $G= N\rtimes Q$ and it is hyperbolic relative to $\{Q\}$. We now state another key lemma
\[trivrel\] Let $N$ be an icc group and let $Q$ be a group together with an outer action $Q {\curvearrowright}^{\sigma} N$. Then $\mathcal L(N)' \cap \mathcal L(N \rtimes_{\sigma}Q)= \mathbb{C}$.
\[ripsvn\] Consider the lacunary hyperbolic groups $Q$ from Theorem \[monster\] together with the collection of maximal rank one subgroups $\mathscr{F}:=\{Q^i_m< Q\}_i$. Also let $N\rtimes Q\in \mathcal Rip (Q)$ be the semidirect product obtained via the Rips construction together with the subgroups $N\rtimes Q^i_m<N\rtimes Q$. Throughout this section we will consider the corresponding von Neumann algebras $\mathcal M^i_m :=\mathcal L(N\rtimes Q^i_m)\subset \mathcal L(N\rtimes Q):=\mathcal M$.
Assuming Notation \[ripsvn\], we now show following:
\[maximalvN\] $\mathcal M^i_m$ is a maximal von Neumann algebra of $\mathcal M$ for every $i$. In particular, when $N\rtimes Q\in \mathcal Rip_{\mathcal T}(Q)$ then for every $i$, $\mathcal M^i_m$ is a non-property (T) maximal von Neumann subalgebra of a property (T) von Neumann algebra $\mathcal M$.
The proof goes along the same line of the proof of [@CDK19 Theorem 4.4] Fix $\mathcal P$ be any intermediate subalgebra $\mathcal M^i_m\subseteq \mathcal P\subseteq \mathcal M$ for some $i$. Since $\mathcal M^i_m \subset \mathcal M$ is spatially isomorphic to the crossed product inclusion $\mathcal L(N)\rtimes Q^i_m \subset \mathcal L(N)\rtimes Q$ we have $\mathcal L(N)\rtimes Q^i_m \subseteq \mathcal P \subseteq \mathcal L(N)\rtimes Q$. By Lemma \[trivrel\] we have that $(\mathcal L(N)\rtimes Q^i_m)' \cap (\mathcal L(N)\rtimes Q ) \subseteq \mathcal L(N)' \cap ( \mathcal L (N)\rtimes Q)=\mathbb{C}$. In particular, $\mathcal P$ is a factor. Moreover, by the Galois correspondence theorem [@Ch78] (see also [@CD19 Corollary 3.8]) there is a subgroup $Q^i_m\leqslant K\leqslant Q$ so that $\mathcal P= \mathcal L(N)\rtimes K$. However since by construction, $Q^i_m$ is a maximal subgroup of $Q$ for every $i$, we must have that $K=Q^i_m$ or $Q$. Thus we get that $\mathcal P = \mathcal M^i_m$ or $\mathcal M$ and the conclusion follows.
For the remaining part note by [@CJ] that $\mathcal M$ has property (T). Also, since $N \rtimes Q^i_m$ surjects onto an infinite abelian group then it does not have property (T). Thus by [@CJ] again $\mathcal M^i_m =\mathcal L(N\rtimes Q^i_m)$ does not have property (T) either.
Acknowledgements {#acknowledgements .unnumbered}
================
The author would like to thank Prof. Alexander Ol’shanskii and Prof. Jesse Peterson for fruitful conversations and their constant support. The author is grateful to Prof. Ionut Chifan and Sayan Das for having stimulating conversation regarding this project.
[banchod]{}
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<span style="font-variant:small-caps;">Department of Mathematics, Vanderbilt University, 1326 Stevenson Center, Nashville, TN 37240, U.S.A.</span>\
[Email: ]{}[krishnendu.khan@vanderbilt.edu]{}\
[^1]: by limit geodesic triangles we mean a triangle in $Con^{\omega}(X,e,d)$ whose sides are limit geodesics
[^2]: $\ell(P^i_n)$ denotes the length of the geodesic $P^i_n$ for $i=1,2,3$ and for all $n\in \mathbb{N}$
[^3]: $\ell(P^i_n)$ denotes the length of the geodesic $P^i_n$ for $i=1,2,3$ and for all $n\in \mathbb{N}$
[^4]: representative of an element $g\in G$ we mean $\{g_i\}$, $g_i\in G_i$ with $\alpha_i(g_i)=g_{i+1}$ for all $i$
|
---
abstract: |
The structure and growth of thin films of titanium on $\alpha-\makebox{Al}_2\makebox{O}_3$ at room temperature were investigated though [*in situ*]{} RHEED observations. Two different structures coexists at low coverage. One corresponds to the $\makebox{Ti}(0001)
\parallel \makebox{Al}_2\makebox{O}_3\makebox{(0001)} $, $\makebox{Ti} [ 1 \overline{1}00]
\parallel \makebox{Al}_{2}\makebox{O}_{3}[2\overline{1}\overline{1}
0] $ and $\makebox{Ti}[10\overline{1}0]\parallel
\makebox{Al}_2\makebox{O}_3[1\overline{1}00]$ epitaxy of the $\alpha$ phase of titanium reported before for thick films prepared at high temperature. The other structure can be explained by co-existence of $\alpha$ and $\omega$ Ti in thin films. It was shown, with the use of tight-binding total energetic calculations that, the $\omega$ phase could actually be stabilized by the substrate. In addition, it was demonstrated that the presence of this extra structure has a dramatic effect on the epitaxial growth of the Ag overlayers on the system. This can be the origin of the non-trivial buffer effect of titanium previously observed.
address:
- 'Laboratoire CNRS-Saint Gobain, “Surface du Verre et Interfaces”, 93303 Aubervilliers, France.'
- 'DSM/DRECAM/SPCSI, CEA Saclay, F-91 191 Gif sur Yvette, France'
- 'Groupe de Physique des Solides, Universités Paris 6 et 7, 140 rue de Lourmel, 75015 Paris, France'
author:
- 'E. S[ø]{}nderg[å]{}rd'
- 'O. Kerjan'
- 'C. Barreteau'
- 'J.Jupille'
title: 'Epitaxy and growth of titanium buffer layers on $Al_2O_3(0001)$'
---
,
,
and
\[sec:Intro\]Introduction
=========================
Finding surface treatments which can force layer-by-layer growth is among the key issues in heteroepitaxy [@Bauer]. Indeed, many attempts have been made to promote the wetting of materials of different and non interacting nature such as the metal on oxide interface [@evans; @Ti2]. The challenge is not only to grow durable interfaces but also to control the morphology of the metal film as it affects the physical properties of the system. The subject is of increasing importance as many industrial applications rely on the mastering of interfaces such as copper interconnects on oxides for microelectronics and silver on oxides for optical purposes. As a trend, adhesion energy decreases from early transition metals to noble metals [@campbell; @JJ], due to the increased filling of the 3d band [@Johnson; @Alemany]. These differences have major consequences regarding the wetting properties of metals on oxides and lead to a large variety of growth situations. The thermodynamic condition for two-dimensional growth of a metal film on a substrate requires the adhesion energy $\beta$ to be equal to or higher than twice the surface energy of the metal $\sigma_M$. The way $\beta$ compares to $\sigma_M$ is of central importance for the growth of the film. While the condition $ \beta < \sigma_M$ implies $\theta
> 90^\circ$ and a growth in a three-dimensional fashion up to very high coverage, $\beta >
\sigma_M$ implies $\theta < 90^\circ$ and leads to a quickly percolating film. Clearly, noble metals and late transition metals on wide band gap oxides belong to the first category [@D.; @Chatain] and early transition metals to the second [@Pedden]. A method often used to improve the wetting of noble metals on oxide surfaces is pre-deposition of early transition metals [@Köstlmeier]. This so-called buffer effect involves both electronic and geometric contributions [@Köstlmeier].
In the present work, attention is focused on the effect of titanium buffers for noble metal growth on alumina, a subject receiving attention because of its application in microelectronic devices [@Ti2; @Köstlmeier], and references therein. Suzuki et al. [@Ti1] found by ion scattering that there is no interdiffusion at the interface between the titanium overlayer and the Al-terminated $\alpha -
\makebox{Al}_2\makebox{O}_3\makebox{(0001)}$ substrate. By performing surface diffraction experiments on a 20 monolayer thick titanium film annealed to 1100 K, they showed that the epitaxial relationship was $\makebox{Ti}(0001)
\parallel \makebox{Al}_2\makebox{O}_3\makebox{(0001)} $, $\makebox{Ti} [1 \overline{1}00]
\parallel \makebox{Al}_{2}\makebox{O}_{3}[2\overline{1}\overline{1}
0]$ and $\makebox{Ti}[10\overline{1}0]\parallel
\makebox{Al}_2\makebox{O}_3[1\overline{1}00]$. This means that the triangular lattice of Ti(111) is rotated by $30^\circ$ with respect to the triangular lattice formed by the outer Al atoms of $\makebox{Al}_2\makebox{O}_3\makebox{(0001)}$, see figure \[fig:Al2O3\]. Dehm et al. [@Ti2] came to similar conclusions through combined transmission electron microscopy (TEM) and reflection high energy electron diffraction (RHEED) observations of a 10-100 nm thick titanium film on $\makebox{Al}_2\makebox{O}_3\makebox{(0001)}$. These authors did not investigate in details the crystal structure of the interface Ti/$\makebox{Al}_2\makebox{O}_3$. Recently, a first principle calculation of Ti on $\makebox{Al}_2\makebox{O}_3\makebox{(0001)}$ predicted a tendency to intermixing for aluminium and titanium at the interface such that the most stable site for the titanium adatoms is above or in replacement of the surface aluminium atoms [@verdozzi]. These results have still not been experimentally verified.
Concerning the buffer effect it was reported in Ref. [@Ti2] that copper grew in a epitaxial manner on thick titanium films (10-100 nm) on $\makebox{Al}_2\makebox{O}_3\makebox{(0001)}$. The buffer effect of titanium was indirectly demonstrated on silver for much thinner films through optical reflection [*in situ*]{} measurements of the silver plasmon [@SGR]. It was found that the use of titanium buffers had a significant influence on the morphology of silver films, although the effect of the titanium film was shown to depend highly on its thickness. Very thin titanium layers ($< 0.3$ nm) hardly produce any change in the three dimensional growth of silver with respect to the bare $\makebox{Al}_2\makebox{O}_3$ surface. In contrast, a titanium layer of 0.6 nm leads to a better wetting of silver between silver and the substrate. However, for higher titanium coverage, the quality of the wetting decreases, although it remains better than on the bare surface. Similar results where obtained by surface diffraction experiments [@ISSPICJPD]. The very intriguing complexity of this scenario can hardly be explained by the surface energy of titanium alone. In the present work the structure of the titanium buffer is investigated as a fonction of its thickness. The corresponding behavior of a silver overlayer is further discussed.
Experimental
============
Single crystalline $\makebox{Al}_2\makebox{O}_3\makebox{(0001)}$ substrates from Mateck Gmbh with a $\leq 0.5^{\circ}$ miscut were prepared using acetone plus ultrasonic bath followed by a cleaning in a 10 percent dilution of a standard buffered basic soap with PH=10.6. After washing in deionized water and drying in isopropanol vapor, an [*ex situ*]{} annealing at 1320 K was performed under an atmospheric pressure of oxygen to obtain a good crystallinity. Afterwards, samples were inserted in a MECA2000 molecular beam epitaxy apparatus, annealed under a partial pressure of $5 \cdot 10^{-5}$ torr $\makebox{O}_2$ to get rid of carbon contamination, first at 820 K over a couple of hours and then at 1070 K for one hour. Samples were cooled under oxygen prior to transfer to the evaporation chamber where the base pressure was $1 \cdot 10^{-10}$ torr. In a similar vacuum chamber equipped with a photoemission analyser, is was checked that this treatment leads to clean $\makebox{Al}_2\makebox{O}_3\makebox{(0001)}$ surfaces [@SGR].
Titanium and silver films were evaporated on the samples using a Telemark electron gun. The flux was between 0.08-0.1 nm per min and the samples were held at 400 K. As titanium is a very reactive material special care was taken to keep the chamber pressure within the $10^{-10}$ torr range during evaporation. The thickness was measured with a quartz microbalance calibrated using [*ex situ*]{} electron microprobe measurements to obtain the number of atoms deposited. To eliminate any distorsions of the diffraction pattern due to stray fields from the electron guns, special care was taken to align the optical center of the RHEED with the electron guns in operation condition. The evolution of the growth was followed by a Staib RHEED instrument and the diffraction patterns were on a CCD camera. During the deposition any physical movement of the samples used for the lattice measurements were avoided to keep the substrate reference for the diffraction patterns. When analysing short lattice spacings, for which the diffraction patterns would be outside the screen, the scattered beams were deflected on the screen by the means of an electrostatic device. It was verified several times that this method does not introduce any errors on the positions of the diffraction peaks on the bare sapphire substrate.
Results
=======
The following section first presents the results and the discussion related to the growth of titanium on $\makebox{Al}_2
\makebox{O}_3$. The influence of titanium on the growth of silver is then briefly considered. Directions along which RHEED analysis is performed are always given with respect to the crystallographic axes in figure \[fig:Al2O3\]. Distances in real and reciprocal space will be labelled $l$ and $g$ respectively.
Growth of the titanium film
---------------------------
Figure \[fig:axe1\] shows the typical evolution of the RHEED pattern of a titanium layer collected along the $\makebox{Al}_2\makebox{O}_3 [ \overline{1}100 ]$ zone axis which means that the observed surface streaks correspond to the $\makebox{Al}_2\makebox{O}_3 [ 2\overline{11}0 ]$ axis. The film starts growing with a reciprocal lattice parameter close to the substrate lattice, $g_{s2}$. Up to 1 nm, the broad and not very intense diffraction rods indicate a low degree of order. Upon increasing the titanium coverage, the diffraction pattern globally gains intensity. Between approximately 1.8- 6 nm, two separated lattice parameters $g_{\omega 2}$ and $g_{\alpha 1}$ coexist. At lower coverage, the external rod $g_{\omega 2}$ is the most intense (figure \[fig:axe1\], 0.3-3.6 nm.). For increasing coverage the internal rod $g_{\alpha 1 }$ increases in intensity to finally dominate the pattern (figure \[fig:axe1\], 5.6 nm). Note that the thickness where the switch in intensity between $g_{\omega 2}$ and $g_{\alpha 1}$ occurs depends on the sample. For some crystals it is impossible to get a correct diffraction pattern below 2 nm of Ti - for others the switching between the lattice spacings $g_{\omega 2}$ and $g_{\alpha 1}$ is already observed at a lower coverage. But, in all cases, in-plane rotations of $360^{\circ}$ show that the symmetry of the structure is 6 over the whole thickness range.
To collect the peak corresponding to Ti$[ 10\overline{1}0 ]$, the RHEED pattern of the growing titanium was also recorded along the $\makebox{Al}_2\makebox{O}_3 [2\overline{11}0 ]$ azimuth. This was delicate because the working distance in the RHEED setup is mainly aimed at analysing large lattice spacings. Small lattice spacings can only be observed by deflection of the diffraction pattern on the CCD camera. Therefore, the diffraction patterns were collected by halfs. To get a complete image the two halfs were recombined using the (00) streaks as a reference (figure \[fig:axe2\]). The corresponding lattice parameters were obtained as the mean value found from the half images. The diffraction patterns in the $\makebox{Al}_2\makebox{O}_3 [ 2\overline{11}0 ]$ zone axis were in general of low intensity. Interestingly, the patterns obtained around 2 nm of titanium are close to those of the clean substrate (figure \[fig:axe2\]). This pattern disappears as the streaks corresponding to the Ti$[ 10\overline{1}0]$ appear. A full rotation of the sample reveals a six fold symmetry of the pattern.
The profiles of the diffraction features were extracted to analyse the pattern in the $\makebox{Al}_2\makebox{O}_3 [ 1 \overline{1}00
]$ zone axis. In order to smooth the noise of the data, three photos recorded for each film thickness were summed and intensities were cumulated along the diffraction peaks. To avoid zones where the diffraction streaks can be curved only the part of the diffraction pattern close to the shadow edge of the sample was used. Intensity profiles are showed in figure \[fig:peaks\] (upper right corner). The analysis of this data is complicated by the presence of two overlapping and broad peaks. Nevertheless, since in a wide coverage range, the streaks are split into two lines separated by a valley, it has been chosen to decompose the profiles into two components over the whole titanium coverage range. Diffractions peaks are Lorenzian functions but deviations from this are expected when working with a CCD camera. Therefore several types of functions were tested to obtain the best possible model over as large a range of coverage as possible. It proved to be a Voigt function with a 80% ratio of the Lorenzian to Gaussian. All the diffraction peaks could be independently fitted by means of two Voigt functions of equal width (figure \[fig:peaks\]). Due to the high background level and the presence of Kikuchi lines, profiles associated with low titanium coverage were difficult to decompose. In particular for coverage below 2 nm, the existence of the $g_{\alpha 1}$ line could be questioned. But at higher coverage of titanium the fits are unambiguous. The fact that the in-plane atomic distances are constant over a wide range of coverage (figure \[fig:lat\]) while the residue of the decomposition is very small (figure \[fig:peaks\]) proves that the decomposition of the diffraction line in two peaks is physically sound.
Figure \[fig:lat\] a) shows the evolution of the area of the fitted peaks. In the 1-3 nm range, the $g_{\alpha 1}$ and $g_{\omega 2}$ peak areas are about constant with $Ag_{\omega 2} >
Ag_{\alpha 1}$, while, above 3 nm, $g_{\alpha 1}$ increases and $g_{\omega 2}$ progressively vanishes. Due to the existence of multiple diffraction effects in RHEED these intensities are only a qualitative indications of the presence of the structures corresponding to the observed peaks in the film. The relative positions of the fitted peaks $g_{\alpha 1}$ and $g_{\omega 2}$ converted into the real space lattice constants $l_{\alpha}$ and $l_{\omega}$ are presented in figure \[fig:lat\] b). Titanium first grows with a lattice constant close to that of the substrate $l_s$ and then undergoes a contraction of $\approx 4\%$ to a lattice constant $l_{\omega}$ and finally it expands to the value $l_{\alpha}$ close to the interatomic distance of bulk Ti$[1\overline{1}00]$ which is $7.3\%$ larger than $l_s$, see ref. [@crystals1]. The intermediate contraction of the film is very surprising as the titanium lattice is already contracted on the substrate, there should be no gain in energy by a further contraction.
An analysis along the $\makebox{Al}_2\makebox{O}_3[2\overline{11}0]$ zone axis also reveals a change in lattice spacing of the titanium film even if the diffraction patterns are more blurred than along the $\makebox{Al}_2\makebox{O}_3[1\overline{1}00]$ axis (figure \[fig:axe2\]). Around 2 nm, $g_{\omega 1}$ streaks are observed in a position which corresponds to a compression by $4\%$ with respect to the substrate. These streaks disappear while the $g_{\alpha 2 }$ streaks become visible.
Titanium as a buffer layer for silver growth
--------------------------------------------
To elucidate the role of the thickness of the titanium buffer effect, 2 nm thick silver films were grown with titanium buffers of different thickness. Figure \[fig:silver\] shows typical RHEED patterns of the silver films with the beam along the $\makebox{Al}_2\makebox{O}_3[1\overline{1}00]$ axis. On the bare $\makebox{Al}_2\makebox{O}_3\makebox{(0001)}$ substrate, between 300-600 K, silver grows without any epitaxial relation and develops the ring structure typical for polycrystalline films constituted of three dimensional domains (figure \[fig:silver\] a). For a titanium layer of 0.5-1.5 nm, silver deposition leads to streaks indicating a better spread of the film (figure \[fig:silver\] b.). A full rotation in the plane shows that the film has a sixfold symmetry corresponding to an epitaxy of a Ag(111) plane. However, at $\pm 15^{\circ}$ around the principal axes of the substrate, the diffraction pattern does not change considerably and only decreases in intensity meaning that the epitaxy is loose. On thicker titanium films (2-4 nm), when the two streaks $g_{\omega 2}$ and $g_{\alpha 1}$ are present, the silver layer exhibits a peculiar structure (figure \[fig:silver\] c). The diffraction pattern is composed of dots which suggest a bad wetting of the silver on substrate. The pattern seen in figure \[fig:silver\] c) presents spots which are characteristic of a long range order, but do not correspond to any low index direction of silver. Finally, when silver is deposited on a titanium film thicker than 4 nm and when the $g_{\alpha 1}$ streak dominates, the diffraction streaks are unperturbed by the silver deposition, which means that the silver wets the substrate and grows epitaxially (figure \[fig:silver\]d). Therefore, the behavior of the silver film unambiguously shows that an important change occurs in the titanium buffer between 4 and 5 nm.
Discussion
==========
Growth of titanium
------------------
The data connected to the growth of the titanium film will first be discussed. At high titanium coverage, the epitaxial relationship is dominated by $\makebox{Ti}(0001)
\parallel \makebox{Al}_2\makebox{O}_3\makebox{(0001)} $, $\makebox{Ti} [1 \overline{1}00]
\parallel \makebox{Al}_{2}\makebox{O}_{3}[2\overline{1}\overline{1}
0]$ and $\makebox{Ti}[10\overline{1}0]\parallel
\makebox{Al}_2\makebox{O}_3[1\overline{1}00]$ in agreement with the epitaxy found by other groups for thicker films at much higher temperature [@Ti2; @Ti1]. But at lower coverage, extra structures appear in the diffraction diagram in both zone axes in the surface plane (figures \[fig:axe1\] and \[fig:axe2\]). As the $g_{\omega 1}$ and the $g_{\omega 2}$ streaks show up in the same coverage range and behave similarly, it is reasonable to attach them to the same structure. The extra streaks can either be due to multiple scattering between the principal streaks and an overlying domaine structure or the existence of another epitaxial relation between the substrate and the titanium film. Both structures exhibit the same 6 fold symmetry as the substrate and they are the only visible structures during a full rotation of the sample. The dramatic change in the growth mode of the silver overlayers is the indication of a significant difference between the underlying structures of the titanium. This and the absence of extra streaks around the zero order peak and the asymmetry of the $g_{\omega 1}$ and the $g_{\omega 2}$ streaks make the existence of a domaine structure unlikely.
If we assume that the observed pattern is the diffraction from a crystal plane then the ratio $g_{\omega 2}/g_{\omega 2}= \sqrt{3}$ suggests a hexagonal structure with a lattice parameter $a \approx
\makebox{0.455}$ nm. The orientation of the hexagon is the same as the hexagon formed by the outer Al atoms on the $\makebox{Al}_2\makebox{O}_3(0001)$ surface. This plane does not correspond to any hexagonal plane of the $\alpha$-Ti but it compares with that of the (001) plane of the high pressure hexagonal $\omega$ phase of titanium [@omegaphase]. Therefore, the double peak observed in figure \[fig:axe1\] can be explained by the coexistence between two epitaxial hexagonal configurations, one is $\alpha$ phase and the other is considered to be the $\omega$ phase of titanium .
The $\omega$-Ti phase is a distorted bcc and one of the first steps in the high pressure hcp to bcc phase transition exhibited in the $IV$ metals [@omegahcp]. This phase is frequently encountered in studies of the mechanical properties of titanium. The different high pressure phases of titanium were recently extensively studied both experimentally in a diamond anvil cell [@omegaphase; @omegaphase2] and theoretically [@omegaphase3; @omegaphase4]. Under hydrostatic pressure the $\omega$ phase coexists with the $\alpha$ phase from around 2 GPa to 9 GPa where the $\omega$ phase dominates. The phase is stable up to 116 GPa. Once the $\omega$ phase is obtained it can be kept at the ambient pressure although it is metastable. The $\omega$-Ti(001) plane has been schematized ( figure \[fig:omega\]) using the lattice parameters given in Ref. [@omegaphase3] and the structure description in Ref. [@NRLsite]. The $\omega$ phase can be seen as the successive stacking of a hexagonal lattice and a graphite like sheet, the hexagon being the origin of the periodicity of the lattice. The parameter of this hexagon a=0.4598nm [@NRLsite] is indeed close to what was observed herein.
A few substrate-induced phase transitions have been observed in metal on metal epitaxial systems. One of most studied exemples is the fcc - bcc martensitic transition of iron on metallic substrates like Cu and $Cu_3Au(001)$. A very elaborated exemple is the STM and RHEED study of the Bains transition of iron on $Cu_3Au(001)$, [@Fe0; @Fe1; @Fe2]. As in the present case, the authors obtained a critical thickness for the fcc phase between 1 and 10 ML depending on the substrate preparation and the films exhibited a high amount of in-plane disorder.
To evaluate the energetic balance between the $\alpha$ and $\omega$ phases under various compressive and extensive conditions a serie of total energy calculations was undertaken. The very small energy difference between the $\alpha$ and $\omega$ phase makes such calculations difficult. As a matter of fact, this phase transition is not properly described by first-principles methods by which the equilibrium $\omega$-Ti phase is found to be slightly lower in energy (about 7 meV) than $\alpha$-Ti [@omegaphase3]. Therefore, a total energy calculations was performed using the set of tight-binding parameters recently developed by Papaconstantopoulos and co-workers [@omegaphase3; @code] to reproduce as accurately as possible the $\alpha$ to $\omega$ phase transition. The calculated equilibrium lattice constants of the $\alpha$-Ti and $\omega$-Ti phases are $a_{\alpha}=0.294$ nm ([*i.e*]{}, $l_{\alpha}=\sqrt{3}a_{\alpha}=0.51$ nm) and $a_{\omega}=0.459$ nm respectively, showing a lattice mismatch with respect to the substrate of $+7.4\%$ and $-3.4\%$. The total energy per atom of $\alpha$-Ti (resp. $\omega$-Ti) has been calculated for hexagonal lattice parameters ranging from perfect matching with the substrate, up (resp. down) to the equilibrium lattice constants (figure \[fig:energ\_hcp\_omega\]). Each point of the curve is obtained by fixing the hexagonal lattice parameter $a$ and minimizing the total energy with respect to the parameter $c$. It clearly appears that, at the lattice constant giving the perfect matching with the substrate, $\omega$-Ti is favored with respect to $\alpha$-Ti, whereas it is the reverse at the equilibrium lattice constant. The small energy difference between the two states makes the coexistence between the two phases over a range of deposited thicknesses very plausible. The presence of defects on the substrate surface can also modify the energetic balance between the two phases. The RHEED data do not give any indication about the arrangement of the two phases which are assumed to coexists within the titanium layer nor does it give insight about the burried part of for the titanium. However, data show clearly that above a certain thickness the surface of the titanium film is exclusively made of the $\alpha$- Ti with the $\makebox{Ti}(0001)
\parallel \makebox{Al}_2\makebox{O}_3\makebox{(0001)} $, $\makebox{Ti} [1 \overline{1}00]
\parallel \makebox{Al}_{2}\makebox{O}_{3}[2\overline{1}\overline{1}
0]$ and $\makebox{Ti}[10\overline{1}0]\parallel
\makebox{Al}_2\makebox{O}_3[1\overline{1}00]$ epitaxy.
Influence of the cristallography of titanium on the wetting of silver
---------------------------------------------------------------------
Concerning the growth of silver on titanium the strength of the metallic bond should clearly favor the good spreading of silver on titanium. Furthermore the lattice spacing the of two most compact hexagonal planes of titanium and silver are the Ti(0001) with $a =
0.295$ nm and the Ag(111) with $a = 0.2889$ nm [@crystals1]. Therefore, it can be anticipated that the Ag/Ti epitaxy is favored over the Ag/$\makebox{Al}_2\makebox{O}_3$ epitaxy because it corresponds to a lower mismatch (-2% versus 5.34%). Thus, both the surface energetic and the lattice constants seems to be in favor of a better wetting of silver on titanium on $\makebox{Al}_2\makebox{O}_3$ at least when bulk values are considered.
The data presented in figure \[fig:silver\] present a more complex picture and suggest that a part of the buffer effect of titanium on silver is linked to the structure and the stress of the interface. On the bare alumina substrate silver grows as three dimensional polycrystals. Silver does not wet the alumina which transfers little order to the metal overlayer. However, the presence of very thin titanium films improves the wetting of alumina by silver ($\approx$ 1 nm, see Ref. [@SGR], the silver film showing a texture around Ag(111) orientation meaning that the titanium film transfers a part of the substrate order. Taking into account the very diffuse diffraction patterns for thin Ti layers in this region it is possible that the silver film only replicates the underlying Ti structure. At higher coverage, for titanium layers where $g_{\omega 2}$ dominates and the $g_{\alpha 1}$ just starts to become visible, the wetting of the substrate by silver becomes poor, figure \[fig:silver\]c. Finally, when the titanium film is thicker and has a nearly relaxed lattice parameter, silver grows epitaxially in a laminar way. The $\mbox{Ag}(111) \|
\mbox{Ti}(0001), \mbox{Ag}[110] \| \mbox{Ti}[2\overline{11}0]$ epitaxy corresponds to expectation and it is similar to that found for the Cu/Ti/$\mbox{Al}_2\mbox{O}_3$ system [@Ti2]. Such a structure is consistent with the observation of a pure $\alpha$ Ti phase by electron diffraction. The bad wetting of silver on the intermediate titanium coverage is an evidence that a titanium structure drastically different from $\alpha$ Ti is present. Indeed, a like explanation is the presence of the $\omega$ Ti at the surface of the titanium film. The change in the interfacial energy is due the large mismatch (9.9%) between $\omega$ Ti(0001) and Ag(111) and prevents the spreading of silver.
The present data show that the silver layers exhibit a remarkable amount of different textures. Laminar growth is only obtained when the lattice constant of the titanium film is very close to the silver bulk value. Therefore, the buffer effect of titanium for noble metals is not only the result of a change in surface and interface energies. It is also very dependent on the crystallography of the buffer and therefore on the elastic contributions to the interfacial energy.
Conclusion
==========
The room temperature growth of thin titanium films on $\alpha -
\makebox{Al}_2\makebox{O}_3(0001)$ has been shown be more complex than hereto reported. For films above 4 nm the $\alpha$-Ti phase is obtained in an epitaxial relation already observed and described in literature for the high temperature growth of thick titanium films on sapphire. For thin films an additional structure is present. This structure is assigned to the (0001) plane of the $\omega$ phase where the hexagon is aligned with the one of an Al-terminated $\alpha - \makebox{Al}_2\makebox{O}_3(0001)$. The two phases coexist at the beginning of the growth of the titanium film in the range of coverage ($\approx$ 1-4 nm). The substrate-induced existence of an $\omega$ phase is plausible when taking simple lattice considerations into account, but the details behind the growth and stabilisation of this phase still need to be explained.
The structure of the titanium film has a dramatic influence on the so-called buffer effect, which has been tested herein by depositing silver on a $\alpha -
\makebox{Al}_2\makebox{O}_3\makebox{(0001)}$ surface pre-covered by titanium at various coverage. Indeed, the titanium film structure and strain was proved to strongly affect, not only the structure, but also the wetting of the silver film. The silver film, which clearly grows in a 3D manner when the film structure contains the structure we assign to the $\omega$-Ti phase, spreads out on the $\alpha$-Ti phase. Therefore, the growth mode and spreading of noble metals deposited on titanium-covered alumina arises not only from thermodynamics but from a subtle interplay between the structure and the surface energy of the titanium film.
Acknowledgements
================
We wish to thank the group of Dr. Claude Fermon at the CEA-France for the use of their RHEED/UHV facility. Furthermore Dr. A. Marty gave helpful comments during the redaction of this paper. Dr. C. Barreteau wish to thank the Drs. D.A. Papaconstantopoulos and N. Bernstein at the Naval Research Center for the access to their tight binding code and the stay at NRL founded by an ONR grant. The group of Dr. Patrice Lehuede, Saint Gobain Recherche, helped with the electron microprobe measurements.
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![Schematic Top view of an Al terminated $\alpha$ Al$_2$O$_3$(0001) surface. The hexagon indicates the position of the outer Al- layer.[]{data-label="fig:Al2O3"}](Al2O3_last.eps){width="15cm"}
![ RHEED images showing the growth of the Ti film with the beam along the $Al_2O3[1\overline{1}00]$ axis.[]{data-label="fig:axe1"}](sondfig1.eps){width="19cm"}
![RHEED images showing the growth of the Ti film with the beam along $Al_2O_3[2\overline{11}0]$. The images presented were obtained as the recombination of two deflected images in order to observe the diffraction peak due to the $Ti[10\overline{1}0]$.[]{data-label="fig:axe2"}](sondfig2.eps){width="12cm"}
![The intensity profiles of the diffraction streaks recorded along $Al_2O_3[1\overline{1}00]$ axis (figure \[fig:axe1\]) for titanium films of thickness ranging from 0.9 to 6 nm and their corresponding fits with Voigt functions. The gradual shift in intensity from $g_{\omega 2}$ to $g_{\alpha 1}$ positions is seen on the inserted figure.[]{data-label="fig:peaks"}](sondfig4.eps){width="13cm"}
![a) The area of the diffraction peaks of figure \[fig:axe1\] given by the fitting procedure. b) The fitted lattice spacings, $l_{\alpha}$ and $l_{\omega}$, relative to the substrate, $l_s$.[]{data-label="fig:lat"}](sondfig6A.eps "fig:"){width="13cm"} ![a) The area of the diffraction peaks of figure \[fig:axe1\] given by the fitting procedure. b) The fitted lattice spacings, $l_{\alpha}$ and $l_{\omega}$, relative to the substrate, $l_s$.[]{data-label="fig:lat"}](sondfig6B.eps "fig:"){width="13cm"}
![The diffraction patterns of silver on various Ti/$Al_2O_3$ films. Observed with the beam along $Al_2O_3[1\overline{1}00]$.[]{data-label="fig:silver"}](sondfig3.eps){width="15cm"}
![Schematic top view of the $\omega$(0001) surface. The structure can be seen as the successive stacking a triangular lattice and a graphite like sheet. The indicated hexagon with an edge of 0.455 nm corresponds to the underlying lattice.[]{data-label="fig:omega"}](sondfig6.eps){width="16cm"}
![ Calculated total energy per atom of $\alpha$-Ti (full line) (resp. $\omega$-Ti (dashed line)), for lattice parameters ranging from the one giving the perfect lattice matching with the substrate ([*i.e.*]{} $a_{\alpha}=0.475/\sqrt{3}$ nm for $\alpha$-Ti and $a_{\omega}=0.459$Åfor $\omega$-Ti), up (down) to the equilibrium lattice constant. Note that the total energy of $\alpha$-Ti is represented as a function of $l_{\alpha}=\sqrt{3}
a_{\alpha}$ whereas the total energy of $\omega$-Ti is represented as a function of $l_{\omega}= a_{\omega}$.[]{data-label="fig:energ_hcp_omega"}](sondfig7.eps){width="15cm"}
|
---
abstract: 'We present a three dimensional velocity analysis of Milky Way disk kinematics using LAMOST K giant stars and the GPS1 proper motion catalogue. We find that Galactic disk stars near the anticenter direction (in the range of Galactocentric distance between $R=8$ and 13 kpc and vertical position between $Z=-$2 and 2 kpc) exhibit asymmetrical motions in the Galactocentric radial, azimuthal, and vertical components. Radial motions are not zero, thus departing from circularity in the orbits; they increase outwards within $R\lesssim 12$ kpc, show some oscillation in the northern ($0 < Z < 2$ kpc) stars, and have north-south asymmetry in the region corresponding to a well-known nearby northern structure in the velocity field. There is a clear vertical gradient in azimuthal velocity, and also an asymmetry that shifts from a larger azimuthal velocity above the plane near the solar radius to faster rotation below the plane at radii of 11-12 kpc. Stars both above and below the plane at $R\gtrsim 9$ kpc exhibit net upward vertical motions. We discuss some possible mechanisms that might create the asymmetrical motions, such as external perturbations due to dwarf galaxy minor mergers or dark matter sub-halos, warp dynamics, internal processes due to spiral arms or the Galactic bar, and (most likely) a combination of some or all of these components.'
author:
- |
Haifeng Wang,$^{1,2}$[^1] Martín López-Corredoira,$^{3,4}$ Jeffrey L. Carlin,$^{5}$ Licai Deng$^{1,2}$\
$^{1}$Key Lab of Optical Astronomy, National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100012, P.R.China\
$^{2}$University of the Chinese Academy of Sciences, Beijing, 100049, China, P.R.China\
$^{3}$Instituto de Astrofísica de Canarias, E-38205 La Laguna, Tenerife, Spain\
$^{4}$Departamento de Astrofísica, Universidad de La Laguna, E-38206 La Laguna, Tenerife, Spain\
$^{5}$LSST, 950 North Cherry Avenue, Tucson, AZ 85719, USA\
date: 'Accepted 2018 March 16. Received 2018 March 11; in original form 2018 February 7'
title: 3D Asymmetrical motions of the Galactic outer disk with LAMOST K giant stars
---
\[firstpage\]
Galaxy: kinematics and dynamics $-$ Galaxy: disk $-$ Galaxy: structure
Introduction
============
We know that there are non-axisymmetries in the Milky Way disk, since many velocity substructures, streams, and moving groups such as Coma Berenices, Sirius, Hyades, Pleiades, HD 1614, Arcturus were discovered. One of the most famous is the Hercules stream [@Denhen98; @Fux01; @Antoja12; @Xia15], which might be caused by the Galactic bar’s outer Lindblad resonance (OLR). Other asymmetric structures can be created by internal perturbations due to the bar or spiral arms [@Denhen00; @Fux01; @Quillen05], by external minor mergers such as the Sagittarius dwarf galaxy passing by, or by interaction with the Magellanic Clouds [@Gomez121; @Gomez122; @Minchev09; @Minchev10].
Radial asymmetrical motions have been found in RAVE data for stars within 1 kpc from the Sun. @Siebert11 discovered that the mean radial velocity is larger than 10 km s$^{-1}$ from the Sun toward the Galactic centre, while it becomes smaller than $-$10 km s$^{-1}$ from the Sun toward the Galactic anti-centre, which might be caused by the two-armed spiral perturbation in which the Sun is again close to the inner ultra-harmonic 4:1 resonance [@Siebert12]. @Liu171 found that the younger F-type main-sequence stars have a quite different in-plane velocity field than the older sub-giant branch stars in the solar neighbourhood local spiral arm. Beyond the solar neighbourhood, @Lop16 and @Tian172 used red clump populations to show that the mean radial velocity is negative within R $\sim$ 9 kpc and positive beyond. This is likely because of the perturbation induced by the rotating bar. The zero-crossing radius of the velocity, R $\sim$ 9 kpc, essentially indicates the rough location of the Outer Lindblad Resonance. Non-circular streaming motions and radial streaming motions of gas have also been discovered in external galaxies [@Adler96; @Tra08; @Sellwood10].
Vertical asymmetrical motions have also been identified in recent years. For example, the Galactic disk has been found to be oscillating in both stellar density and kinematics in the solar neighbourhood [@Widrow12]; the use of these “ripples” in the disk to identify the perturber has come to be known as Galactoseismology. @Williams13 detected a rarefaction-compression behaviour in the vertical velocity pattern at $R \lesssim$ 9 kpc based on red clump stars from RAVE, which they identified as a “breathing” mode with oscillations in $<Z^{2}>$, $<Zv>$ and $<v^{2}>$ [@Banik17]. This mode has odd parity in the $V_Z$ distribution and even parity in the density distribution. This is in contrast to what is known as a “bending” mode, with oscillations in $<Z>$ and $<v>$ [@Widrow14; @Banik17], and even parity in $V_Z$ with odd parity in the density distribution. With kinematics of F-type stars from the LAMOST survey, similar vertical asymmetrical substructures were also found by @Carlin13. The wave-like pattern in the mean vertical density of stars as mapped in detail with SDSS data by @Xu15 shows more stars in the north (i.e., above the Galactic plane) at distances of about 2 kpc (North near) from the Sun, more stars in the south (below the plane) at 4$-$6 kpc (South middle) from the Sun, then excess stars in the north at distances of 8$-$10 kpc from the Sun, and an overdensity in the south at distances of 12$-$16 kpc from the Sun. The residual of the density profiles after subtracting the best fit models show different oscillating patterns in almost all radii that have been probed [@Wang17].
There is not yet consensus regarding the physical scenario for producing $V_Z$ and $V_R$ asymmetries. @Tian15 showed that asymmetric motion may be related to the age of stars with different dynamical relaxation times. The nearby asymmetric radial motions can be explained by the perturbation due to either the spiral arms or the bar [@Siebert12; @Gomez13; @Faure14; @Monari16]. However, the asymmetric radial motions in outer disk are likely mainly contributed by the bar dynamics with a given pattern speed [@Grand15; @Monari15; @Tian172; @Liu18]. The explanation must be more complex for vertical asymmetrical structure, including scenarios such as minor mergers due to the passing of the Sagittarius dwarf galaxy proposed by @Gomez13 or the LMC [@Laporte18], or the disk response to bombardment by merging lower-mass satellites [@Donghia16]. The effects of even lower-mass dark matter subhalos have also been invoked as a possible explanation [@Widrow14]. The stellar disk has a clear warp [@Lop02]. According to simple analyses of vertical velocities [@roskar10; @Lop141], the kinematical signature of the Galactic warp’s line-of-nodes is located close to the Galactic anti-center. This is also a possible explanation of the observed vertical bulk motions.
Some possible mechanisms contributing to $V_R$ are also probably influencing the azimuthal velocity. Recently, @Tian172 found an intriguing U-shape along the Galactic anti-center direction, and pointed out that the reshaping of the orbits at different radius may lead to non-zero and oscillating mean in-plane velocities in the disk. @Lop142 also used red clump stars to find an almost flat rotation curve with a slight fall-off, including a puzzling low average rotation speed at the outermost and most off-plane regions which might invite us to reconsider the dark matter distribution. Both of these works show that we need more data and higher precision proper motions to map the disk kinematical details.
In this work, we will further investigate radial, azimuthal and vertical velocities in the Galactic disk. We are aspiring to figure out whether there are velocity asymmetrical motions as far away as 5 kpc from the solar location at low Galactic latitudes. The LAMOST spectroscopic survey [@Cui12; @Zhao12] is used to select a larger sample of K giant stars with line-of-sight velocities and metallicities than was previously available. After these K-giants are combined with the GPS1 proper motion catalogue, which has unprecedented precision before Gaia DR2 [@Tian171], we can decipher the outer disk 3-dimensional kinematical structure and asymmetries in detail.
The paper is organised as follows. Section 2 describes our K giant star sample and coordinate transformations. Section 3 shows the 3-dimensional (3D) velocity distribution projected onto 2D maps. Then we show the results of 1D velocity distributions in Section 4. In Section 5, discussions about the asymmetrical mechanisms are included. Finally, brief conclusions are given in Section 6.
Sample, distance and velocity
=============================
LAMOST catalogue and distance
-----------------------------
The Large Aperture Multi-Object Fiber Spectroscopic Telescope (LAMOST, also called the Guo Shou Jing Telescope), is a quasi$-$meridian reflecting Schmidt telescope with an effective aperture of about 4 meters. A total of 4000 fibers, capable of obtaining low resolution spectra (R $\sim$ 1800) covering the range from 380 to 900 nm simultaneously, are installed on its $5^{\circ}$ focal plane [@Cui12; @Zhao12]. In the 5-year survey plan, LAMOST will obtain a few million stellar spectra in about 20000 square degrees in the northern sky [@Deng12]. The LAMOST DR3 catalogue contains 5756075 spectra; among these, the LAMOST pipeline has provided stellar astrophysical parameters (effective temperature, surface gravity, and metallicity) as well as line-of-sight velocities for about 3.2 million stars. The sample is mainly distributed in the Galactic Anti-Center direction due to special conditions at the site [@Yao12].
The K giant stars are selected according to the descriptions from @Liu142. Distances are estimated from a Bayesian approach [@Car15] with uncertainties of about 20%. Interstellar extinction is derived using the Rayleigh-Jeans Colour Excess (RJCE) method [@Zasowski13] with 2MASS H band [@Skru06] and WISE W2-band photometry [@Wright10]. The difference between our derived extinction corrections and the values from @Green14 [@Green15] is less than 0.2 mag, and the uncertainty in extinction likely contributes less than 10% to the uncertainty in distance [@Liu173]. Fig. \[XYZR\] shows spatial distribution of the 65000 K giant stars used in this work. The top panel is looking down on the Galactic plane at Galactic $X$ and $Y$ coordinates, and shows that our sample is mainly distributed in the anti-center direction. The bottom panel of Fig. \[XYZR\] is the distribution in $R$ and $Z$ Galactic cylindrical coordinates, color coded by star counts on a log scale. We have adopted a Galactic coordinate system with $X$ increasing outward from the Galactic centre, $Y$ in the direction of rotation, and $Z$ positive towards the North Galactic Pole (NGP) [@Williams13].
![Spatial distribution of the 65000 K giant stars used in this work. The top panel is the $X$ and $Y$ in Galactic coordinates, and the bottom panel shows $R$ and $Z$ in cylindrical coordinates, color coded by star counts on a log scale. The majority of stars in our sample are outside the Solar radius in the direction of the Galactic Anticenter.[]{data-label="XYZR"}](XYZR_new.pdf){width="48.00000%"}
Proper motion and velocity transformation
-----------------------------------------
The proper motion catalog known as GPS1 (Gaia$-$PS1$-$SDSS) [@Tian171] has a precision of $\sim$ 1.5$-$2 mas yr$^{-1}$ with systematic error 0.3 mas yr$^{-1}$. For this work, we match our LAMOST-selected K giants to GPS1, then calculate the 3D Galactocentric cylindrical coordinates for the K giant stars by adopting a location of the Sun of $R_{\odot}$ = 8.34 kpc [@Reid14] and $Z_{\odot}$ = 27 pc [@Chen01]. The heliocentric rectangular components of the Galactic space velocity $U, V$, and $W$ are determined by the right-handed coordinate system based on @Johnson87, with $U$ positive towards the Galactic centre, $V$ positive in the direction of Galactic rotation and $W$ positive towards the north Galactic pole. Cylindrical velocities $V_R$, $V_\theta$, and $V_Z$ are defined as positive with increasing $R$, $\theta$, and $Z$, with the latter towards the NGP. For the solar motion we use @Tian15 value, \[$U_{\odot}$ $V_{\odot}$ $W_{\odot}$\] = \[9.58, 10.52, 7.01\] km s$^{-1} $. The circular speed of the LSR is adopted as 238 km s$^{-1} $ [@Schonrich12]. We also correct the radial velocities of the samples by adding 4.4 km s$^{-1}$ in this work which is smaller than @Tian15 5.7 km s$^{-1}$ due to the updating of the software of the data process compared with APOGEE data [@Blanton17].
The behavior of proper motion measurements for K giant stars within the range of $Z = [-2, 2]$ kpc is shown in Figures \[RZPMRA\] and \[RZPMDEC\], with each figure showing a color-coded map of the median proper motion (top panel) and its associated bootstrap error (bottom) for the right ascension (Fig. \[RZPMRA\]) and declination (Fig. \[RZPMDEC\]) proper motion components in the $R$ (radial distance from the Galactic centre), $Z$ (vertical distance from the midplane) plane. We only use the bins of which the median number of stars per pixel is greater than 50. The choice of bin size is 0.33 kpc for $R$ and 0.2 kpc for $Z$. Bootstrap errors are determined by resampling (with replacement) 100 times for each bin, and the uncertainties of the estimates are determined using 15% and 85% percentiles of the bootstrap samples. For both components, most of the bootstrap errors are less than 1.2 mas yr$^{-1}$. The GPS1 catalogue systematic errors are nearly an order of magnitude better than those in PPMXL [@Roeser10] and UCAC4 [@Zacharias10], and their random errors are roughly a four-fold improvement relative to PPMXL and UCAC4, making this the ideal catalogue for our purposes.
![Behavior of proper motion measurements for K giant stars in the range of $Z=[-2, 2]$ kpc. The color-coded map shows the distribution of the median (top) and bootstrap error (bottom) for the right ascension proper motion components on the $R, Z$ plane. The median number of stars per pixel is larger than 50.[]{data-label="RZPMRA"}](RZ_PMra.pdf){width="48.00000%"}
![Similar to Fig. \[RZPMRA\], but for proper motion in the declination direction.[]{data-label="RZPMDEC"}](RZ_PMdec.pdf){width="48.00000%"}
For our sample during this paper, the stars located inside $|Z|$ $<$ 2 kpc and 6 $<$ R $<$ 13 kpc are selected to map the lower disk kinematics in a region in which we have small random and systematic errors. The stars with LAMOST spectroscopic SNR $<$ 20 are not included, and we also exclude stars with \[Fe/H\] $<$ $-$1.0 dex, so we can focus on Galactic disk populations. We also set some criteria in velocity to remove fast-moving halo stars: $V_R$=\[-150, 150\] km s$^{-1}$, $V_{\theta}$=\[-50, 350\] km s$^{-1}$, and $V_Z$=\[-150, 150\] km s$^{-1}$; this sacrifices a small fraction of stars, yielding a final sample of around 65000 disk K-giant stars.
2$-$D velocity asymmetric structure
===================================
In this section, we focus on describing 2D asymmetrical motions in the Galactic disk. The top panel of Fig. \[RZVR\] shows the variation of Galactocentric radial velocity $V_R$ in the $R, Z$ plane. The errors in the bottom panel are computed by a bootstrap method. The error is mainly contributed by the distance uncertainty, proper motion random errors and statistical errors. The most prominent feature displayed in the upper panel of Fig. \[RZVR\] is that there is a large oscillating structure on the northern ($0 < Z < 2$ kpc) side. Inside the solar radius (from $6 < R \lesssim 8$ kpc), $V_R$ has a positive (mean) value, then it becomes negative from $R \sim$ 8.4 to 9.8 kpc (on average), then it becomes positive when $R \gtrsim$ 10 kpc. On the southern side, most of the bins are moving inward with negative value, mixed with a few positive bins. It is intriguing that there seems to be a negative-velocity (blue colors in Fig. \[RZVR\]) C-shaped feature extending from $(R,Z)$ = (9, 1) to (8, 0) to (12, -1) kpc, suggesting a connection between velocity features above and below the plane. We also note that there is a north-south asymmetry at $R \sim$ 10-11 kpc, $Z \sim$ 0.5 kpc (denoted by the two ellipses in the figure), corresponding to mean positive values in the north and negative $V_{\rm R}$ in the south. The average $V_R$ error in most regions is around 2 km s$^{-1}$.
![Average radial Galactocentric velocity in spatial bins; the upper panel shows the median $V_R$ value in each bin, and the lower panel their corresponding bootstrap errors. In this figure, the line-of-sight velocity is from LAMOST and the proper motion is from GPS1. Notice that there is an oscillating structure with radius at $0 < Z < 2$ kpc, and there is a north-south asymmetry at $R \sim$ 10-11 kpc, $Z \sim$ 0.5 kpc. The two ellipses highlight the regions where we see a north/south asymmetry.[]{data-label="RZVR"}](RZ_VR.pdf){width="48.00000%"}
The distribution of $V_{\theta}$ (Figure \[RZVPHI\]) shows different trends from $V_R$, exhibiting a clear gradient with $|Z|$ on both sides of the disk that represents the transition from thin to thick disk. The in-plane kinematical features of $V_{\theta}$ in the Galactic outer disk shown in @Tian172 are not seen in this panel. There are clear asymmetrical rotational motions: when $R \geq$ 10 kpc and $|Z| \geq$ 0.5 kpc, stars in the south are rotating faster than those at symmetrical locations in the north by $\sim$ 10-40 km s$^{-1}$. At $R \approx $ 8 $-$ 9 kpc and $Z \lesssim$ 0.5 kpc, there is a similar trend but reversed; here, the northern stars are faster than those in the south, but the feature is not as clear as in the outer region. We note that the asymmetric motions we are discussing in relation to Figure \[RZVPHI\] are in addition to the well-known “asymmetric drift” that is responsible for the vertical gradient in $V_\theta$. The asymmetrical motions we refer to are differences between the mean velocities at symmetrically located positions on either side of the Galactic plane. We show ellipses in Figure \[RZVPHI\] to help the reader identify the features we are discussing. We will discuss these further in the next section (see, e.g., Fig. \[1DVPHI\_test1\]).
![Similar to Fig. \[RZVR\], but for azimuthal velocity. There is a clear gradient with $Z$ representing the transition from thin to thick disk. The four ellipses highlight regions exhibiting north $-$ south asymmetries in their $V_\theta$ velocities.[]{data-label="RZVPHI"}](RZ_Vphi.pdf){width="48.00000%"}
Figure \[RZVZ\] shows the average $V_{Z}$ with position in the $R, Z$ plane. There is clear evidence of a compression motion inside the solar radius: above the midplane, stars are moving downward, and below the plane, stars are moving upward. This is very similar to substructure found by @Carlin13. Some works describe this as the “breathing” mode [@Widrow14; @Carrillo17], but in this work we will simply call it asymmetrical motion. Outside the solar location, we can see in the figure that stars are on average moving upward at almost all radii for $|Z|\le $2 kpc, which is partly different from @Carlin13. This illustrates what is often called a “bending” mode. This is comparable to the vertical bulk motion maps in Fig. 4 of @Lop141, but the accuracy of their results is low due to the large PPMXL proper motion errors, and they mentioned they only reported a tentative detection of vertical motion with low significance. We discuss some possible mechanisms to create the bending modes observed in Figure \[RZVZ\] in the discussion.
![Similar to Fig. \[RZVR\], but for vertical velocity, $V_Z$. Notice that there is compression motion inside the solar circle, and significant bulk motions at all radii and heights outside the Sun’s radius, extending until $R \sim$ 13 kpc.[]{data-label="RZVZ"}](RZ_VZ.pdf){width="48.00000%"}
1$-$D velocity asymmetric structure
===================================
Velocity profiles in the vertical direction
-------------------------------------------
In this section, we continue to describe the trends in median velocity as a function of $(R, Z)$ position in the Galaxy for K giant stars with GPS1 catalogue proper motions. Error bars give the measured bootstrap errors. We set a bin size of 0.2 kpc, and keep only bins with star counts larger than 50. When we have low density of stars, in order to keep at least 50 stars per bin, the $R$ bin is set to 2 kpc, which is relatively larger than other works [@Liu18; @Tian172]. It will not affect us when discussing low latitude outer disk asymmetrical kinematics. In the figures of this section, we have used stars with \[Fe/H\] $>$ $-$0.4 dex and we also tested to consider other solar motions [@Schonrich10]. The default value we set for the circular velocity of the local standard of rest (LSR) is 220 km s$^{-1}$, with a solar radius of 8 kpc, but we have used other values and we have seen that the overall trends are stable in this work by changing those parameters.
![The trends in median velocity as a function of ($R, Z$) position in the Galaxy for K giant stars with GPS1 catalogue proper motions. Error bars give the measured bootstrap errors. The bin size is 0.2 kpc with star counts required to be larger than 50 per bin. Different colors represent median values of $V_R$ as a function of $Z$ for different $R$ ranges, with \[Fe/H\] $>$ $-$1.0 dex. Notice that at $R \sim$ 10-11 kpc, $Z \sim$ 0.5 kpc, there is an asymmetrical velocity substructure, with inward motions below the plane, and outward $V_R$ above the plane. Velocities for each $R$ slice have been shifted by a constant value; the dashed line represents $V_R = 0$ km s$^{-1}$.[]{data-label="1DVR"}](VR_oscillations.pdf){width="48.00000%"}
![Difference between median $V_{R}$ velocities in North-South symmetric regions as a function of $|Z|$ in different distance slices. At $R\approx$ 11 kpc, $Z\approx$ 0.5 kpc there is a clear velocity substructure corresponding to the North-near structure of @Xu15.[]{data-label="1DVR_test"}](VR_oscillations_NS.pdf){width="46.00000%"}
The median $V_R$ vs. $Z$ at $R=8-12$ kpc is shown in Fig. \[1DVR\]. Different colors represent median values of $V_R$ as a function of $Z$ for different $R$ ranges. We can see that at $R \sim$ 10-11 kpc, $Z \sim$ 0.5 kpc, there is a north-south asymmetrical velocity substructure which is consistent with the Fig. \[RZVR\] substructure mentioned before, and there is an almost symmetric outward motion at $R=12$ kpc. For other locations, there is an almost flat $V_R$ curve and no significant asymmetric motion. In order to quantitatively describe this behaviour, in Fig. \[1DVR\_test\] we show the North - South (N - S) difference in median $V_R$ with $|Z|$ at different $R$. There is clear substructure around $R \sim$ 11 kpc, $Z \sim$ 0.5 kpc; at 10 or 12 kpc, there is a similar trend but weaker.
The median $V_{\theta}$ vs. $Z$ at $R=8-12$ kpc is shown in Fig. \[1DVPHI\]. The curves at $R \sim$ 8-9 kpc are almost symmetrical about $Z=0$ kpc. These transition to an asymmetrical distribution for distances larger than 10 kpc, where the median $V_\theta$ in the south is typically larger than that in the north (on average) at a given radius. As we did for $V_R$, we compare the median $V_\theta$ for northern stars to southern stars in Fig. \[1DVPHI\_test1\]. For almost all $|Z|$ bins the north-south velocity residuals are negative at $R=11, 12$ kpc; this implies that the stars in the south are rotating faster than those in the north. At R $ \approx $ 8 $-$ 9 kpc and $|Z| \lesssim$ 0.5 kpc, there is a similar trend but reversed pattern showing northern stars are rotating faster than those in the south. We speculate it is likely a real effect in the discussion.
![Similar to Fig. \[1DVR\], but for azimuthal velocity. At distances larger than 10 kpc, there is an asymmetry when comparing regions at similar heights above and below the plane. These asymmetries vary in magnitude and sign as a function of $R$. []{data-label="1DVPHI"}](VPhi_oscillations.pdf){width="48.00000%"}
![Similar to Fig. \[1DVR\_test\], but for azimuthal velocity. All stars in the south are rotating faster than those in the north for $R=11$ and 12 kpc (corresponding to negative values of $V_{\theta, north} - V_{\theta, south}$), although some points have large error bars.[]{data-label="1DVPHI_test1"}](VPhi_oscillations_NS.pdf){width="46.00000%"}
Median $V_{Z}$ vs. $Z$ at $R=8-12$ kpc is shown in Fig. \[1DVZ\]. The median $V_Z$ velocity for $R=8$ kpc oscillates from a positive value of $\sim$ 8 km s$^{-1}$ to negative ($\sim -5$ km s$^{-1}$) between $Z = -2$ kpc and $Z = 2$ kpc. The behaviour is similar for $R=9$ kpc, but the trend at 9 kpc is weaker. For $R >$ 9 kpc, there are significant upward motions from $V_Z \sim $ 3 km s$^{-1}$ to 15 km s$^{-1}$, especially between $-1 < Z < 1$ kpc. The trends at $R = 8$ kpc are similar to what is typically known as a breathing mode, while those at $R>9$ kpc may represent a bending mode. In this work, we will simply refer to these as vertical asymmetrical motions in general. Some possible mechanisms to excite these motions are discussed in the next section.
![Similar to Fig. \[1DVR\], but for vertical velocity. At $R=8$ kpc there is a clear oscillation from positive velocities below the plane to negative $V_Z$ above the plane. For $R=9$ kpc, the trend is similar to that at 8 kpc, but the pattern is weaker. From 10 kpc outward, there are significant upward bulk motions in the range $-$2 kpc $<$ Z $<$ 2 kpc.[]{data-label="1DVZ"}](VZ_oscillations.pdf){width="48.00000%"}
{width="98.00000%"}
Velocity profiles in the radial direction
-----------------------------------------
The radial profiles of the binned median $V_R$, $V_{\theta}$, and $V_{Z}$ for stars with \[Fe/H\] $>$ $-$1.0 dex in the range of $Z$ = \[$-$2, $+$2\] kpc and $R$=\[6, 13\] kpc are shown in Fig. \[1DRVRVPHIVZ\]. In order to see more details and avoid the influence of moving groups or streams on our discussion, we divide our samples into the lower region stars ($Z$ = \[-1, 1\] kpc) and higher region stars ($|Z|$ = \[1, 2\] kpc).
The top panel of Fig. \[1DRVRVPHIVZ\] shows the variation of median $V_R$ for K giant stars. The red vertical solid and dashed lines in all panels mark the locations of spiral arms [@Reid14]. The solid black line with error bars represents stars in the range of $Z$ = \[-1, 1\] kpc, and the magenta dashed line and error bars represent stars in the range $Z$ = \[-2, -1\] kpc and $Z$ = \[1, 2\] kpc. The top panel shows that lower region stars have larger average $V_R$ than the higher region stars in the ranges of R $\sim$ 6 to 7.2 kpc and 10 to 12 kpc. At other radii, there are only small differences between $V_R$ of the lower/higher samples. The middle panel displays the variation of median $V_{\theta}$ with $R$. The value ranges from $\sim$198 km s$^{-1} $ to 220 km s$ ^{-1}$ for the higher region stars, while the lower region stars have small variations from an average of $\sim220$ km s$ ^{-1}$. There are clear differences between the lower-$Z$ stars and higher stars which reflect the rotational velocity gradient with height from the midplane. The bottom panel shows the vertical velocity $V_{Z}$ along $R$. There is a clear trend with radius from negative (downward) to positive (upward) bulk motions. Furthermore, the vertical velocities of the lower region stars are larger than those in the higher region at all radii except $R \sim 10.8$ to 11.4 kpc. Some possible scenarios to explain the trends noted here are discussed in the next section. We note that we only concentrate on stars at $R<13$ kpc. At distances larger than $\sim13$ kpc, we do not have enough stars in our sample, and the errors become too large to see small-amplitude velocity trends.
Discussions
===========
Possible implications and comparisons
-------------------------------------
Radial velocity oscillations might be produced by the spiral structures that are always spatially correlated with spiral arms [@Siebert12; @Faure14; @Grand15]. However, the top panel of Fig. \[1DRVRVPHIVZ\] does not show such a correlation with any spiral arm in the outer disk. We speculate that the spiral arms are not the main contributor to, or at least have only a small effect on, the observed asymmetrical velocity structure. The overall trends we see are similar to those found by @Tian172 among old red clump stars. Minor mergers (i.e., accreting dwarf galaxies such as the Sgr dwarf) may raise vertical waves [@Gomez13], but these perturbations may not intensively affect the in-plane velocity [@Tian172]. Therefore, we speculate that the radial oscillation we see may be mainly induced by bar dynamics, the Galactic dark matter halo, a disk warp, or other mechanisms such that the mean orbit of the disk stars is intrinsically elliptical or there is a net secular expansion of the disk [@Lop16]. From the top panel in Fig. \[1DRVRVPHIVZ\], we can also see that these possible effects have larger dynamical effects on the lower disk than stars further from the midplane.
Notice that there is clear asymmetrical structure in the Fig. \[1DVR\] around $Z\approx$ 0.5 kpc and $R\approx$ 10$-$11 kpc. We have found that this corresponds to the @Wang17 overdensity, which is called the north near structure in @Xu15. The location is around 2 kpc from the Sun. So we find both a density and velocity signature of the north near structure simultaneously in this series of works.
In @Tian172, an interesting U-shaped profile is found showing different trends in $V_{\theta}$ with Galactocentric radius. In the region of $R \lesssim$ 10.5 kpc, the mean azimuthal velocities for both the young and old red clump stars mildly decrease by $\sim$10 km s$^{-1}$. Beyond R $\sim$ 10.5 kpc, the mean azimuthal velocities for both populations increase to 240 $-$ 245 km s$^{-1}$ at R $\sim$13 kpc. We do not detect similar structure in our K giant sample (see middle panel of Fig. \[1DRVRVPHIVZ\]) possibly due to sampling different stellar populations, or using a different deprojection method. Our general trend among lower region stars is similar to @Lop142 for $R$ less than 13 kpc. As mentioned, we find that there is an asymmetry at $R \approx $ 8 $-$ 9 kpc, at least for $Z \lesssim$ 0.5 kpc and $R \sim$ 10-11 kpc. We do a simple test for the azimuthal velocity distribution by limiting the sample to \[Fe/H\]$>$ -0.4 dex and $Z$ = \[-1, 1\] kpc to see more details of the midplane and test whether there are still clear disk asymmetries and variations. We can see in Fig. \[1DVPHI\_test2\], at $R=$ 11-12 kpc, that stars on the south side are still faster than north side when $|Z|$ $ >$ 0.7 kpc, and notice that it looks like it moves gradually from asymmetry with $V_{\theta}$ larger above the plane at $R \approx$ 8 $-$ 9 kpc, to $V_{\theta}$ larger below the plane at $R$ = 11 $-$ 12 kpc. Recently, a southern rising trend of $V_{\theta}$ in the range $Z$ = \[-2, 2\] kpc, $R$ = \[8, 10\] kpc is also found by @Pearl17 using LAMOST DR3 F-type stars. This may be a real effect that merits future validation with higher precision proper motions and a larger sample. Some similar possible mechanisms have been introduced in the radial velocity section, whose theoretical details on the mechanisms are a task for a future work.
![Similar to Fig. \[1DVPHI\], but for azimuthal velocity of stars with \[Fe/H\]$>$ -0.4 dex and $Z$=\[-1, 1\] kpc. The variation of azimuthal velocity at radii 8-12 kpc still exists: the stars in the south are rotating faster than those in the north when $|Z|$ $>$ 0.7 kpc for the 11 and 12 kpc slices. Notice that it looks like it moves gradually from asymmetry with $V_{\theta}$ larger above the plane at $R \approx$ 8 $-$ 9 kpc, to $V_{\theta}$ larger below the plane at $R \approx$ 11 $-$ 12 kpc.[]{data-label="1DVPHI_test2"}](VPhi_oscillations_midplane.pdf){width="48.00000%"}
The bottom panel of Fig. \[1DRVRVPHIVZ\] shows that the median $V_{Z}$ varies from negative to positive with radius from inside to outside $R_\odot$. Recently, Fig. 14 of @Carrillo17 displayed a clear negative gradient above the plane inside the solar location. The positive gradient outside the solar radius is similar to that found by @Liu172 in the range of $R$ less than 12 kpc, which showed a significant bulk motion for lower old red clump stars. The authors suggested that it is hard to explain their results by the perturbation due to a merging event [@Gomez13] or spiral arms [@Debattista99], because their measured vertical velocity is too large compared to perturbations seen in the models. @Liu172 suggested that the kinematics result from warp line-of-node kinematics. But our measured velocity (see the bottom panel of Fig. \[1DRVRVPHIVZ\]) is similar to that predicted by the minor merger simulation of @Gomez13. Their results displayed variations of vertical motions less than $\sim10$ km s$ ^{-1} $. They also predicted radial and azimuthal variations of the mean vertical velocity, correlating with the spatial structure as mentioned previously. We think that spiral arms are not likely important for vertical motions in the outer disk. @Monari15 pointed out the Galactic bar is unlikely to induce mean vertical motions greater than $\sim$ 0.5 km s$^{-1}$ in the outer disk; our bulk motions are larger than 0.5 km s$^{-1} $, so we do not favor a bar/spiral arm explanation for the vertical asymmetric motions. A minor merger or other external perturbation might be the most reasonable explanation. However, we can not exclude a warp contribution, as suggested by @Lop141, or that from the dark matter halo.
In summary, by combining our measured asymmetrical distributions and comparisons with other works, we conclude that planar asymmetrical motions are not likely mainly contributed by the spiral arms or minor mergers, and vertical asymmetrical motions are not likely mainly contributed by the bar or spiral arms. Other mechanisms such as the warps, the dark matter halo, etc. are needed to be considered. In the future, we will use Gaia DR2 and test particle simulations to validate these results.
Reconstructing previous LAMOST work
-----------------------------------
Figure \[RZVRVZ\_Carlin\] shows the binned median radial and vertical velocities $V_{R}$ and $V_{Z}$ in the $R, Z$ plane in the range $R$ = 7.8-9.8 kpc, $Z$ = -2 to 2 kpc. Although we do not have as many K giant stars with proper motion as the 400000 F-stars used by @Carlin13 (hereafter, Carlin13), and the spatial coverage is worse, we directly compare with Carlin13 for corresponding regions that are well sampled. The upper panel of Fig. \[RZVRVZ\_Carlin\] shows that the $V_{R}$ trend has some differences in the north with respect to the Carlin13 work. We measure median $V_{R}$ in the range $R\approx $ 8.6 to 9.2 kpc to be negative (on average), while Carlin13 found $V_R$ in this region to be around zero, and furthermore did not see the oscillation that we find in Fig. \[RZVR\]. The difference might be caused by the lower precision of the PPMXL proper motions used in that work, as well as differences in the distance determination methods. @Carrillo17 showed that proper motion uncertainty and distance error can lead to different or reversed asymmetrical motions in the solar neighborhood by reconsidering the bending or breathing mode with Radial Velocity Experiment (RAVE) latest sample [@Kunder17] and the Tycho-Gaia astrometric solution catalogue (TGAS) [@Gaia161]. Other factors such as the solar motion, solar location, selection criteria of different tracers, and the adopted local standard of rest velocity can also produce some differences. In the south, for $V_{R}$, the overall trends are similar in our work to those of Carlin13; many bins are less than or around 0 km s$^{-1}$ with some positive bins mixed in. There is little difference between the $V_{Z}$ trends inside the solar radius; above the plane the velocity is moving downward on average, while below the plane, the velocities are moving upward. However, outside the Sun, we find upward bulk vertical motions, while Carlin13 found that the compression-like motions continued to beyond the solar radius.
The Carlin13 work has also been revisited by @Pearl17 with new corrections to the PPMXL zero points; their results are very similar to what we have found. They also do not find substantially positive $V_{R}$ at high $Z$ due to elimination of data with larger errors, and they do not observe the north region stars moving down toward the plane similar to our work. @Pearl17 suggest that the difference arises because the Carlin13 sample contained an overwhelming amount of data in the northern third quadrant. These differences are clearer for us to see between Fig. \[RZVRVZ\_Carlin\] in this work and Fig. 2 in @Carlin13.
![Similar to the top panel of Fig. \[RZVR\], but for vertical and radial velocity on the $R, Z$ plane. The sample is limited to $Z$=\[-2, 2\] kpc and $R$=\[7.8, 9.8\] kpc, in order to reconstruct and compare with earlier LAMOST work by @Carlin13.](RZ_VRVZ_Carlin.pdf){width="52.00000%"}
. \[RZVRVZ\_Carlin\]
Conclusions
===========
In this work, we investigate the kinematics of K giant stars in the Galactic disk between $R$ = 6 to 13 kpc, $Z$ = -2 to 2 kpc with the GPS1 proper motion catalogue. From asymmetrical variations of the velocities with Galactocentric radius and height for 65000 K giant or RGB stars, we conclude the following:
The median radial velocity profile $V_R$ has a large northern oscillating structure from $R=$6 kpc to 13 kpc. There is velocity substructure located at $Z\sim$ 0.5 kpc and $R\sim$ 10$-$11 kpc for the $V_R$ distribution corresponding to the north $-$ near overdensity we have confirmed previously.
We discover asymmetrical rotational motions when comparing stars above and below the plane at $R \geq$ 10 kpc and $|Z| \geq$ 0.5 kpc. Stars in the south are, on average, rotating faster than those in the north. At $R \approx $ 8 $-$ 9 kpc, $|Z| \lesssim$ 0.5 kpc, we show that the northern stars are rotating faster than their counterparts in the south.
We also find that there is a compression motion at radii inside the solar location, and upward asymmetrical bulk motions outside the solar radius until $\sim$ 13 kpc. The radial profile of median $V_{Z}$ displays a trend from downward to upward bulk motions in the range of 6 to 13 kpc, with all regions in our study beyond $R \gtrsim 9$ kpc moving upward on average.
With the help of previous works, we discuss that in$-$plane asymmetries are not mainly contributed by the spiral arms or minor merger, and because we see vertical asymmetries, we can rule out that the vertical features are mainly caused by spiral arms or the bar. We cannot exclude contributions such as the warps, the dark matter halo, or other mechanisms. It will require more detailed mapping of disk velocity substructure(s) to decipher the mechanisms creating these structures in the future.
This paper can be thought of as a pilot paper for outer disk kinematics. ESA’s mission Gaia [@Gaia161] will soon release highly accurate parallaxes and proper motions for over a billion sources, which will provide an opportunity to map the outer disk in more detail and hopefully answer some of puzzles presented here. However, fully exploiting the proper motions from Gaia for disk kinematics requires a large spectroscopic sample such as LAMOST to provide the radial velocities and stellar parameters.
Acknowledgements {#acknowledgements .unnumbered}
================
We would like to thank the anonymous referee for his/her helpful comments. We also thank Liu C., Lawrence M. Widrow, Francesca Figueras, Alice C. Quillen, Ismael E. Carrillo, and Ivan Minchev for helpful discussions and comments. This work was supported by the National Key Research and Development Program of China through grant 2017YFA0402702. MLC was supported by grant AYA2015-66506-P of the Spanish Ministry of Economy and Competitiveness. Guoshoujing Telescope (the Large Sky Area Multi-Object Fiber Spectroscopic Telescope LAMOST) is a National Major Scientific Project built by the Chinese Academy of Sciences. Funding for the project has been provided by the National Development and Reform Commission. LAMOST is operated and managed by the National Astronomical Observatories, Chinese Academy of Sciences.
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[^1]: E-mail: hfwang@bao.ac.cn
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abstract: 'Let $p$ be a prime number and $f$ an overconvergent $p$-adic automorphic form on a definite unitary group which is split at $p$. Assume that $f$ is of “classical weight” and that its Galois representation is crystalline at $p$, then $f$ is conjectured to be a classical automorphic form. We prove new cases of this conjecture in arbitrary dimensions by making crucial use of the patched eigenvariety constructed in [@BHS].'
address:
- |
Christophe Breuil\
C.N.R.S.\
Laboratoire de Mathématiques d’Orsay\
Université Paris-Sud\
Université Paris-Saclay\
F-91405 Orsay\
France\
Christophe.Breuil@math.u-psud.fr
- |
Eugen Hellmann\
Mathematisches Institut\
Universität Bonn\
Endenicher Allee 60\
D-53115 Bonn\
Germany\
hellmann@math.uni-bonn.de
- |
Benjamin Schraen\
C.N.R.S.\
Laboratoire de Mathématiques\
Université de Versailles St. Quentin\
45 Av. des États-Unis\
F-78035 Versailles\
France\
benjamin.schraen@uvsq.fr
author:
- 'Christophe Breuil, Eugen Hellmann and Benjamin Schraen'
title: Smoothness and classicality on Eigenvarieties
---
Introduction
============
Let $p$ be a prime number. In this paper we are concerned with classicality of $p$-adic automorphic forms on some unitary groups, i.e. we are looking for criteria that decide whether a given $p$-adic automorphic form is classical or not. More precisely we work with $p$-adic forms of finite slope, that is, in the context of *eigenvarieties*.
Let $F^+$ be a totally real number field and $F$ be an imaginary quadratic extension of $F^+$. We fix a unitary group $G$ in $n$ variables over $F^+$ which splits over $F$ and over all $p$-adic places of $F^+$, and which is compact at all infinite places of $F^+$. Associated to such a group $G$ (and the choice of a tame level, i.e. a compact open subgroup of $G({\mathbb{A}}_{F^+}^{p\infty})$) there is a nice Hecke eigenvariety which is an equidimensional rigid analytic space of dimension $n[F^+:{\mathbb{Q}}]$, see e.g. [@Cheeig], [@BelChe] or [@Emerton]. One may view a [*$p$-adic overconvergent eigenform of finite slope*]{}, or simply overconvergent form, as a point $x$ of such an eigenvariety and one can associate to each overconvergent form a continuous semi-simple representation $\rho_x:{\rm Gal}(\overline F/F)\rightarrow {\mathrm{GL}}_n(\overline {\mathbb{Q}}_p)$ which is unramified outside a finite set of places of $F$ and which is trianguline in the sense of [@Co] at all places of $F$ dividing $p$ ([@KPX]).
A natural expectation deduced from the Langlands and Fontaine-Mazur conjectures is that, if $\rho_x$ is de Rham (in the sense of Fontaine) at places of $F$ dividing $p$, then $x$ is a [*classical*]{} automorphic form (see Definition \[defclass\] and Proposition \[compaclas\] for the precise definition). However, the naive version of this statement fails for two reasons: (1) a classical automorphic form for $G({\mathbb{A}}_{F^+})$ can only give Galois representations which have distinct Hodge-Tate weights (in each direction $F\hookrightarrow \overline{\mathbb{Q}}_p$) and (2) the phenomenon of [*companion*]{} forms shows that there can exist classical and non-classical forms giving the same Galois representation. However, we can resolve (1) by requiring $\rho_x$ to have distinct Hodge-Tate weights and (2) by requiring $x$ to be of “classical” (or dominant) weight. In fact, since the Hodge-Tate weights of $\rho_x$ are related to the weight of $x$, requiring the latter automatically implies the former, once $\rho_x$ is assumed to be de Rham. As a conclusion, it seems reasonable to expect that any overconvergent form $x$ of classical weight such that $\rho_x$ is de Rham at places of $F$ dividing $p$ is a classical automorphic form (see Conjecture $\ref{classiconj}$ and Remark \[extra\]).
Such a classicality theorem is due to Kisin ([@Kisinoverconvergent]) in the context of Coleman-Mazur’s eigencurve, i.e. in the slightly different setting of ${\rm GL}_2/{\mathbb{Q}}$. Note that, at the time of [@Kisinoverconvergent], the notion of a trianguline representation was not available, and in fact [@Kisinoverconvergent] inspired Colmez to define trianguline representations ([@Co]).
In the present paper we prove new cases of this classicality conjecture (in the above unitary setting). In particular we are able to deal with cases where the overconvergent form $x$ is [*critical*]{}. Throughout, we assume that $\rho_x$ is crystalline at $p$-adic places. Essentially the same proof should work if $\rho_x$ is only assumed crystabelline, but the crystalline assumption significantly simplifies the notation.
To state our main results, we fix an overconvergent form $x$ of classical weight such that $\rho_x$ is crystalline at all places dividing $p$. Such an overconvergent form can be described by a pair $(\rho_x,\delta_x)$, where $\rho_x$ is as above and $\delta_x=(\delta_{x,v})_{v\in S_p}$ is a locally ${\mathbb{Q}}_p$-analytic character of the diagonal torus of $G(F^+\otimes_{{\mathbb{Q}}}{\mathbb{Q}}_p)\cong \prod_{v\in S_p}{\mathrm{GL}}_n(F^+_v).$ Here $S_p$ denotes the set of places of $F^+$ dividing $p$. There are nontrivial relations between $\rho_{x,v}:=\rho\vert_{{\rm Gal}(\overline {F^+_v}/F^+_v)}$ and $\delta_{x,v}$, in particular the character $\delta_{x,v}$ defines an ordering of the eigenvalues of the crystalline Frobenius on $D_{\rm cris}(\rho_{x,v})$. If we assume that these Frobenius eigenvalues are pairwise distinct, then this ordering defines a Frobenius stable flag in $D_{\rm cris}(\rho_{x,v})$. We can therefore associate to $x$ for each $v\in S_p$ a permutation $w_{x,v}$ that gives the relative position of this flag with respect to the Hodge filtration on $D_{\rm cris}(\rho_{x,v})$, see §\[weyl\] (where we rather use another equivalent definition of $w_{x,v}$ in terms of triangulations). Following [@BelChe §2.4.3] we say that $x$ is [*noncritical*]{} if, for each $v$, the permutation $w_{x,v}$ is trivial. The invariant $(w_{x,v})_{v\in S_p}$ can thus be seen as “measuring” the criticality of $x$.
In the statement of our main theorem, we need to assume a certain number of Working Hypotheses (basically the combined hypotheses of all the papers we use). We denote by ${\overline{\rho}}_x$ the mod $p$ semi-simplification of $\rho_x$. These Working Hypotheses are:
1. The field $F$ is unramified over $F^+$ and $G$ is quasi-split at all finite places of $F^+$;
2. the tame level of $x$ is hyperspecial at all finite places of $F^+$ inert in $F$;
3. ${\overline{\rho}}_x({\rm Gal}(\overline F/F(\zeta_p))$ is adequate ([@Thorne]) and $\zeta_p\notin \overline F^{\ker({\rm ad}{\overline{\rho}}_x)}$;
4. the eigenvalues of $\varphi$ on $D_{\rm cris}(\rho_{x,v})$ are sufficiently generic for any $v\in S_p$ (Definition \[veryreg\]).
Our main theorem is:
\[mainintro\] Let $p>2$ and assume that the group $G$ and the tame level satisfy (i) and (ii). Let $x$ be an overconvergent form of classical weight such that $\rho_x$ is crystalline and satisfies (iii) and (iv). If $w_{x,v}$ is a product of distinct simple reflections for all places $v$ of $F^+$ dividing $p$, then $x$ is classical.
Note that the assumption on the $w_{x,v}$ in Theorem \[mainintro\] is empty when $n=2$, and already this $n=2$ case was not previously known (to the knowledge of the authors). The noncritical case of Theorem \[mainintro\], i.e. the special case where all the $w_{x,v}$ are trivial, is already known and due to Chenevier ([@Chenevierfern Prop.4.2]). Thus the main novelty, and difficulty, in Theorem \[mainintro\] is that it deals with possibly critical (though not too critical) points.
In fact we give a more general classicality criterion and prove that it is satisfied under the assumptions of Theorem $\ref{mainintro}$. This criterion is formulated in terms of the rigid analytic space of trianguline representations $X_{\rm tri}^\square({\overline{\rho}}_{x,v})$ defined in [@HellmannFS] and [@BHS §2.2]. For every $v\in S_p$ there is a canonical morphism from the eigenvariety to $X_{\rm tri}^\square({\overline{\rho}}_{x,v})$.
\[mainintro2\] Let $p>2$ and assume that the group $G$ and the tame level satisfy (i) and (ii). Let $x$ be an overconvergent form of classical weight such that $\rho_x$ is crystalline and satisfies (iii) and (iv). If for any $v\in S_p$ the image $x_v$ of $x$ in $X_{\rm tri}^\square({\overline{\rho}}_{x,v})$ is contained in a unique irreducible component of $X_{\rm tri}^\square({\overline{\rho}}_{x,v})$, then $x$ is classical.
According to this theorem we need to understand the local geometry of the space $X_{\rm tri}^\square({\overline{\rho}}_{x,v})$ at $x_v$. It turns out that much of this local geometry is controlled by the Weyl group element $w_{x,v}$ associated to $x$ which only depends on the image $x_v$ of $x$ in $X_{\rm tri}^\square({\overline{\rho}}_{x,v})$. For $v\in S_p$ denote by $\lg(w_{x,v})$ the length of the permutation $w_{x,v}$ and by $d_{x,v}$ the rank of the ${\mathbb{Z}}$-module generated by $w_{x,v}(\alpha)-\alpha$, as $\alpha$ ranges over the roots of $({\mathrm{Res}}_{F_v^+/{\mathbb{Q}}_p}{\mathrm{GL}}_n)\times_{{\mathbb{Q}}_p} \overline {\mathbb{Q}}_p\cong \prod_{\tau:\, F_v^+\hookrightarrow \overline{\mathbb{Q}}_p}{\mathrm{GL}}_n$. Then $d_{x,v}\leq \lg(w_{x,v})$, with equality if and only if $w_{x,v}$ is a product of distinct simple reflections (Lemma \[coxeter\]).
\[mainlocal\] Let $v\in S_p$ and let $X\subseteq X_{\rm tri}^\square(\overline \rho_{x,v})$ be a union of irreducible components that contain $x_v$ and satisfy the [*accumulation property*]{} of Definition \[accu\] at $x_v$. Then $$\dim T_{X,x_v}\leq \dim X+ \lg(w_{x,v})-d_{x,v}=\dim X_{\rm tri}^\square(\overline \rho_{x,v}) + \lg(w_{x,v})-d_{x,v},$$ where $T_{X,x_v}$ is the tangent space to $X$ at $x_v$. In particular $X$ is smooth at $x_v$ when $w_{x,v}$ is a product of distinct simple reflections.
The accumulation condition in Theorem \[mainlocal\] actually prevents us from directly applying it to $X=X_{\rm tri}^\square(\overline \rho_{x,v})$ and thus directly deducing Theorem \[mainintro\] from Theorem \[mainintro2\]. Hence we have to sharpen Theorem \[mainintro2\], see Theorem \[classicalitycrit\].
Assuming the classical modularity lifting conjectures for ${\overline{\rho}}_x$ (in all weights with trivial inertial type), there is a certain union $\widetilde X_{\rm tri}^\square({\overline{\rho}}_{x,v})$ of irreducible components of $X_{\rm tri}^\square({\overline{\rho}}_{x,v})$ such that $\prod_{v\in S_p}\widetilde X_{\rm tri}^\square({\overline{\rho}}_{x,v})$ is (essentially) described by the patched eigenvariety $X_p({\overline{\rho}}_x)$ defined in [@BHS] (see Remark \[conjvariant\]). In the last section of the paper (§\[modularity\]), we prove (assuming modularity lifting conjectures) that the inequality in Theorem \[mainlocal\] for $X=\widetilde X_{\rm tri}^\square({\overline{\rho}}_{x,v})$ is an [*equality*]{} for all $v\in S_p$, $$\label{mainequality}
\dim T_{\widetilde X_{\rm tri}^\square({\overline{\rho}}_{x,v}),x_v}=\dim X_{\rm tri}^\square(\overline \rho_{x,v}) + \lg(w_{x,v})-d_{x,v}\ \ \ \ \ \ {\rm (assuming\ modularity),}$$ see Corollary \[bonnedim\]. The precise computation (\[mainequality\]) of the dimension of the tangent space is intimately related to (and uses in its proof) the existence of many *companion points* on the patched eigenvariety $X_p({\overline{\rho}}_x)$. These companion points are provided by the following unconditional theorem, which is of independent interest.
\[companionptsintro\] Let $y=((\rho_v)_{v\in S_p},\epsilon)$ be a point on $X_p({\overline{\rho}}_x)$. Let $T$ be the diagonal torus in ${\mathrm{GL}}_n$ and let $\delta$ be a locally ${\mathbb{Q}}_p$-analytic character of $T(F^+\otimes_{{\mathbb{Q}}}{\mathbb{Q}}_p)$ such that $\epsilon\delta^{-1}$ is an algebraic character of $T(F^+\otimes_{{\mathbb{Q}}}{\mathbb{Q}}_p)$ and such that $\epsilon$ is strongly linked to $\delta$ in the sense of [@HumBGG §5.1] (as modules over the Lie algebra of $T(F^+\otimes_{{\mathbb{Q}}}{\mathbb{Q}}_p)$). Then $((\rho_v)_{v\in S_p},\delta)$ is also a point on $X_p({\overline{\rho}}_x)$.
We also prove that the equality (\[mainequality\]) for all $v\in S_p$ (and thus the modularity lifting conjectures) imply that the initial Hecke eigenvariety is itself [*singular*]{} at $x$ as soon as the Weyl element $w_{x,v}$ is [*not*]{} a product of distinct simple reflections for some $v\in S_p$, see Corollary \[singhecke\].
Let us now outline the strategy of the proofs of Theorems \[mainintro2\] and \[mainlocal\].
The proof of Theorem \[mainlocal\] crucially uses results of Bergdall ([@Bergdall]), together with a fine analysis of the various conditions on the infinitesimal deformations of $\rho_{x,v}$ carried by vectors in $T_{X,x_v}$, see §\[wedgesection\] and §\[endofproof\]. Very recently, Bergdall proved an analogous bound for the dimension of the tangent space of the initial Hecke eigenvariety at $x$ assuming standard vanishing conjectures on certain Selmer groups ([@Bergdraft]).
The proof of Theorem \[mainintro2\] makes use of the patched eigenvariety $X_p({\overline{\rho}}_x)$ constructed in [@BHS] by applying Emerton’s construction of eigenvarieties [@Emerton] to the locally analytic vectors of the patched Banach $G(F^+\otimes_{{\mathbb{Q}}}{\mathbb{Q}}_p)$-representation $\Pi_\infty$ of [@CEGGPS]. As usual with the patching philosophy, the space $X_p({\overline{\rho}}_x)$ can be related to another geometric object which has a much more local flavour, namely the space $X_{\rm tri}^\square(\overline \rho_{x,p}):=\prod_{v\in S_p}X_{\rm tri}^\square(\overline \rho_{x,v})$ of trianguline representations. More precisely, by [@BHS Th.3.20] there is a Zariski closed embedding: $$\label{maininclusion}
X_p(\overline\rho_x)\hookrightarrow {\mathfrak{X}}_{{\overline{\rho}}_x^p}\times {\mathbb{U}}^g \times X_{\rm tri}^\square(\overline \rho_{x,p}),$$ identifying the source with a union of irreducible components of the target. Here ${\mathbb{U}}^g$ is an open polydisc (related to the patching variables) and ${\mathfrak{X}}_{{\overline{\rho}}_x^p}$ is the rigid analytic generic fiber of the framed deformation space of ${\overline{\rho}}_x$ at all the “bad” places prime to $p$. Moreover the Hecke eigenvariety containing $x$ can be embedded into the patched eigenvariety $X_p(\overline\rho_x)$ (see [@BHS Th.4.2]). As previously, we denote by $x_v$ the image of $x$ in $X_{\rm tri}^\square(\overline \rho_{x,v})$ via (\[maininclusion\]).
For $v\in S_p$ let us write ${\bf k}_{v}$ for the set of labelled Hodge-Tate weights of $\rho_{x,v}$, and $R_{\overline \rho_{x,v}}^{\square,{\bf k}_v{\rm -cr}}$ for the quotient defined in [@Kisindef] of the framed deformation ring of $\overline \rho_{x,v}$ parametrizing crystalline deformations of $\overline \rho_{x,v}$ of Hodge-Tate weight ${\bf k}_v$, and ${\mathfrak{X}}_{\overline \rho_{x,v}}^{\square,{\bf k}_v{\rm -cr}}$ for the rigid space $({\mathrm{Spf}}\, R_{\overline \rho_{x,v}}^{\square,{\bf k}_v{\rm -cr}})^{\rm rig}$. We relate ${\mathfrak{X}}_{\overline \rho_{x,v}}^{\square,{\bf k}_v{\rm -cr}}$ to $X_{\rm tri}^\square(\overline \rho_{x,v})$ by introducing a third rigid analytic space $\widetilde {\mathfrak{X}}_{\overline \rho_{x,v}}^{\square,{\bf k}_v{\rm -cr}}$ finite over ${\mathfrak{X}}_{\overline \rho_{x,v}}^{\square,{\bf k}_v{\rm -cr}}$ parametrizing crystalline deformations $\rho_v$ of $\overline \rho_{x,v}$ of Hodge-Tate weights ${\bf k}_v$ [*together with an ordering*]{} of the Frobenius eigenvalues on $D_{\rm cris}(\rho_{v})$, see §\[variant\] for a precise definition. The space $\widetilde {\mathfrak{X}}_{\overline \rho_{x,v}}^{\square,{\bf k}_v{\rm -cr}}$ naturally embeds into $X_{\rm tri}^\square(\overline \rho_{x,v})$ and contains the point $x_v$ (and is smooth at $x_v$). We prove that there is a unique irreducible component $Z_{{\rm tri}}(x_v)$ of $X_{\rm tri}^\square(\overline \rho_{x,v})$ containing the unique irreducible component of $\widetilde {\mathfrak{X}}_{\overline \rho_{x,v}}^{\square,{\bf k}_v{\rm -cr}}$ passing through $x_v$ (Corollary \[defnZtri(x)\]). Let $Z_{\rm tri}(x):=\prod_{v\in S_p}Z_{\rm tri}(x_v)$, which is thus an irreducible component of $X_{\rm tri}^\square(\overline \rho_{x,p})$ containing $x$. Then Theorem \[mainintro2\] easily follows from the following theorem (see (i) of Remark \[uniqueirredcompo\]):
\[mainclasscrit\] Assume that ${\mathfrak{X}}_{{\overline{\rho}}_x^p}\times{\mathbb{U}}^g\times Z_{\rm tri}(x)\subseteq X_p(\overline\rho_x)$ via (\[maininclusion\]). Then the point $x$ is classical.
Let us finally sketch the proof of Theorem \[mainclasscrit\] (in fact, for the same reason as above, we have to sharpen Theorem \[mainclasscrit\], see Theorem \[classicalitycrit\]). Let $R_\infty$ be the usual patched deformation ring of ${\overline{\rho}}_x$, there is a canonical morphism of rigid spaces $X_p(\overline\rho_x)\longrightarrow {\mathfrak X}_\infty:=({\mathrm{Spf}}\, R_\infty)^{\rm rig}$. Let $L(\lambda)$ be the finite dimensional algebraic representation of $G(F^+\otimes_{{\mathbb{Q}}}{\mathbb{Q}}_p)$ associated (via the usual shift) to the Hodge-Tate weights $({\bf k}_v)_{v\in S_p}$. Proving classicality of $x$ turns out to be equivalent to proving that the image of $x$ in ${\mathfrak X}_\infty$ is in the support of the $R_\infty$-module $\Pi_\infty(\lambda)'$ which is the continuous dual of: $$\Pi_\infty(\lambda):={\mathrm{Hom}}_{{\prod}_{v\in S_p}{\mathrm{GL}}_n(\mathcal{O}_{F_{\tilde{v}}})}\big(L(\lambda),\Pi_\infty\big).$$ By [@CEGGPS Lem.4.17], the $R_\infty$-module $\Pi_\infty(\lambda)'$ is essentially a Taylor-Wiles-Kisin “usual” patched module for the trivial inertial type and the Hodge-Tate weights $({\bf k}_v)_{v\in S_p}$. Forgetting the factors ${\mathfrak{X}}_{{\overline{\rho}}_x^p}$ and ${\mathbb{U}}^g$ which appear in ${\mathfrak X}_\infty$, its support is a union of irreducible components of the smooth rigid space $\prod_{v\in S_p}{\mathfrak{X}}_{\overline \rho_{x,v}}^{\square,{\bf k}_v{\rm -cr}}$. It is thus enough to prove that the unique irreducible component $Z_{\rm cris}(\rho_x)$ of $\prod_{v\in S_p}{\mathfrak{X}}_{\overline \rho_{x,v}}^{\square,{\bf k}_v{\rm -cr}}$ passing through $(\rho_{x,v})_{v\in S_p}$ contains a point which is in the support of $\Pi_\infty(\lambda)'$. But it is easy to find a point $y$ in $Z_{\rm tri}(x)$ sufficiently close to $x$ such that $(\rho_{y,v})_{v\in S_p}\in Z_{\rm cris}(\rho_x)$ (in particular $\rho_{y,v}$ is crystalline of the same Hodge-Tate weights as $\rho_{x,v}$) [*and*]{} moreover $\rho_{y,v}$ is [*generic*]{} in the sense of [@BHS Def.2.8] for all $v\in S_p$. The assumption in Theorem \[mainclasscrit\] implies $y\in X_p(\overline\rho_x)$ and it is now not difficult to prove that such a generic crystalline point of $X_p(\overline\rho_x)$ is always classical, i.e. is in the support of $\Pi_\infty(\lambda)'$.
We end this introduction with the main notation of the paper.
If $K$ is a finite extension of ${\mathbb{Q}}_p$ we denote by $\mathcal{G}_K$ the absolute Galois group $\mathrm{Gal}(\overline{K}/K)$ and by $\Gamma_K$ the Galois group $\mathrm{Gal}(K(\zeta_{p^n},n\geq 1)/K)$ where $(\zeta_{p^n})_{n\geq 1}$ is a compatible system of primitive $p^n$-th roots of $1$ in $\overline K$. We normalize the reciprocity map ${\mathrm{rec}}_K:\, K^\times\rightarrow \mathcal{G}_K^{\rm ab}$ of local class field theory so that the image of a uniformizer of $K$ is a geometric Frobenius element. We denote by $\varepsilon$ the $p$-adic cyclotomic character and recall that its Hodge-Tate weight is $1$.
For $a\in L^\times$ (where $L$ is any finite extension of $K$) we denote by ${\rm unr}(a)$ the unramified character of $\mathcal{G}_K$, or equivalently of $\mathcal{G}_K^{\rm ab}$ or $K^\times$, sending to $a$ (the image by ${\mathrm{rec}}_K$ of) a uniformizer of $K$. For $z\in L$, we let $\vert z\vert_K:=p^{-[K:{\mathbb{Q}}_p]{{\mathrm{val}}}(z)}$ where ${{\mathrm{val}}}(p)=1$. We let $K_0\subseteq K$ be the maximal unramified subfield (we thus have $(\vert \ \vert_K)\vert_{K^\times}={\rm unr}(p^{-[K_0:{\mathbb{Q}}_p]})={\rm unr}(q^{-1})$ where $q$ is the cardinality of the residue field of $K$).
If $X={\mathrm{Sp}}\,A$ is an affinoid space, we write $\mathcal{R}_{A,K}$ for the Robba ring associated to $K$ with $A$-coefficients (see [@KPX Def.6.2.1] though our notation is slightly different). Given a continuous character $\delta: K^\times \rightarrow A^\times$ we write ${\mathcal{R}}_{A,K}(\delta)$ for the rank one $(\varphi,\Gamma_K)$-module on ${\mathrm{Sp}}\,A$ defined by $\delta$, see [@KPX Construction 6.2.4]. If $X$ is a rigid analytic space over $L$ (a finite extension of ${\mathbb{Q}}_p$) and $x$ is a point on $X$, we denote by $k(x)$ the residue field of $x$ (a finite extension of $L$), so that we have $x\in X(k(x))$. If $X$ and $Y$ are two rigid analytic spaces over $L$, we often write $X\times Y$ instead of $X\times_{{\mathrm{Sp}}\,L}Y$.
If $X$ is a “geometric object over ${\mathbb{Q}}_p$” (i.e. a rigid space, a scheme, an algebraic group, etc.), we denote by $X_K$ its base change to $K$ (for instance if $X$ is the algebraic group ${\mathrm{GL}}_n$ we write ${\mathrm{GL}}_{n,K}$). If $H$ is an abelian $p$-adic Lie group, we let $\widehat{H}$ be the rigid analytic space over ${\mathbb{Q}}_p$ which represents the functor mapping an affinoid space $X={\mathrm{Sp}}\,A$ to the group ${\mathrm{Hom}}_{\rm cont}(H,A^\times)$ of continous group homomorphisms (or equivalently locally ${\mathbb{Q}}_p$-analytic group homomorphisms) $H\rightarrow A^\times$. Finally, if $M$ is an $R$-module and $I\subseteq R$ an ideal, we denote by $M[I]\subseteq M$ the submodule of elements killed by $I$, and if $S$ is any finite set, we denote by $\vert S\vert$ its cardinality.
Crystalline points on the trianguline variety {#localpart1}
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We give several important definitions and results, including the key local statement bounding the dimension of some tangent spaces on the trianguline variety (Theorem \[upperbound\]).
Recollections {#begin}
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We review some notation and definitions related to the trianguline variety.
We fix two finite extensions $K$ and $L$ of ${\mathbb{Q}}_p$ such that: $$\vert{\mathrm{Hom}}(K,L)\vert=[K:{\mathbb{Q}}_p]$$ and denote by $\mathcal{O}_K$, ${\mathcal O}_L$ their respective rings of integers. We fix a uniformizer $\varpi_K\in \mathcal{O}_K$ and denote by $k_L$ the residue field of ${\mathcal O}_L$. We let $\mathcal{T}:=\widehat{K^\times}$ and $\mathcal{W}:=\widehat{\mathcal{O}_K^\times}$. The restriction of characters to $\mathcal{O}_K^\times$ induces projections $\mathcal{T}\twoheadrightarrow \mathcal{W}$ and $\mathcal{T}_L\twoheadrightarrow \mathcal{W}_L$. If ${\bf k}:=(k_\tau)_{\tau:\, K\hookrightarrow L}\in{\mathbb{Z}}^{{\mathrm{Hom}}(K,L)}$, we denote by $z^{\bf k}\in \mathcal{T}(L)$ the character: $$\begin{aligned}
\label{charc}
z\longmapsto\prod_{\tau\in{\mathrm{Hom}}(K,L)}\tau(z)^{k_\tau}\end{aligned}$$ where $z \in K^\times$. For ${\bf k}=(k_{\tau,i})_{1\leq i\leq n,\tau:\, K\hookrightarrow L}\in(\mathbb{Z}^n)^{{\mathrm{Hom}}(K,L)}$, we denote by $\delta_{\bf k}\in \mathcal{T}^n(L)$ the character: $$(z_1,\dots,z_n)\longmapsto\prod_{\stackrel{1\leq i\leq n}{\tau:\, K\hookrightarrow L}}\tau(z_i)^{k_{\tau,i}}$$ where $(z_1,\dots,z_n)\in (K^\times)^n$. We also denote by $\delta_{\bf k}$ its image in $\mathcal{W}^n(L)$ (i.e. its restriction to $({\mathcal O}_K^\times)^n$). We say that a point $\delta\in \mathcal{W}^n_L$ is [*algebraic*]{} if $\delta=\delta_{\bf k}$ for some ${\bf k}=(k_{\tau,i})_{1\leq i\leq n,\tau:\, K\hookrightarrow L}\in(\mathbb{Z}^n)^{{\mathrm{Hom}}(K,L)}$. We say that an algebraic $\delta=\delta_{\bf k}$ is [*dominant*]{} (resp. [*strictly dominant*]{}) if moreover $k_{\tau,i}\geq k_{\tau,i+1}$ (resp. $k_{\tau,i}> k_{\tau,i+1}$) for $i\in \{1,\dots,n-1\}$ and $\tau\in {\mathrm{Hom}}(K,L)$.
We write ${\mathcal{T}}_{\rm reg}\subset {\mathcal{T}}_L$ for the Zariski-open complement of the $L$-valued points $z^{-\bf k}, \vert z\vert_K z^{{\bf k}+{\bf 1}}$, with ${\bf k}=(k_\tau)_{\tau:K\hookrightarrow L}\in {\mathbb{Z}}_{\geq 0}^{{\mathrm{Hom}}(K,L)}$. We write ${\mathcal{T}}_{\rm reg}^n$ for the Zariski-open subset of characters $(\delta_1,\dots,\delta_n)$ such that $\delta_i/\delta_j\in {\mathcal{T}}_{\rm reg}$ for $i\neq j$.
We fix a continuous representation ${\overline{r}}:\mathcal{G}_K\rightarrow {\mathrm{GL}}_n(k_L)$ and let $R_{{\overline{r}}}^{\square}$ be the framed local deformation ring of ${\overline{r}}$ (a local complete noetherian $\mathcal{O}_L$-algebra of residue field $k_L$). We write $\mathfrak{X}_{{\overline{r}}}^\square:=({\mathrm{Spf}}\, R_{{\overline{r}}}^\square)^{{\mathrm{rig}}}$ for the rigid analytic space over $L$ associated to the formal scheme ${\mathrm{Spf}}\, R_{{\overline{r}}}^\square$. Recall that a representation $r$ of ${\mathcal{G}}_K$ on a finite dimension $L$-vector space is called *trianguline of parameter* $\delta=(\delta_1,\dots,\delta_n)$ if the $(\varphi,\Gamma_K)$-module $D_{{\mathrm{rig}}}(r)$ over ${\mathcal{R}}_{L,K}$ associated to $r$ admits an increasing filtration ${\mathrm{Fil}}_\bullet$ by sub-$(\varphi,\Gamma_K)$-modules over ${\mathcal{R}}_{L,K}$ such that the graded piece ${\mathrm{Fil}}_i/{\mathrm{Fil}}_{i-1}$ is isomorphic to ${\mathcal{R}}_{L,K}(\delta_i)$. We let $X_{\rm tri}^\square({\overline{r}})$ be the associated framed trianguline variety, see [@BHS §2.2] and [@HellmannFS]. Recall that $X_{\rm tri}^\square({\overline{r}})$ is the reduced rigid analytic space over $L$ which is the Zariski closure in $\mathfrak{X}_{{\overline{r}}}^\square\times \mathcal{T}^n_L$ of: $$\label{ucris}
U_{\rm tri}^\square({\overline{r}}):=\{{\rm points}\ (r,\delta)\ {\rm in}\ {\mathfrak{X}}^\square_{{\overline{r}}}\times{\mathcal{T}}^n_{\rm reg}\ {\rm such\ that}\ r\ \text{is trianguline of parameter}\ \delta\}$$ (the space $U_{\rm tri}^\square({\overline{r}})$ is denoted $U_{\rm tri}^\square({\overline{r}})^{\rm reg}$ in [@BHS §2.2]). The rigid space $X_{\rm tri}^\square({\overline{r}})$ is reduced equidimensional of dimension $n^2+[K:{\mathbb{Q}}_p]\frac{n(n+1)}{2}$ and its subset $U_{\rm tri}^\square({\overline{r}})\subset X_{\rm tri}^\square({\overline{r}})$ turns out to be Zariski-open, see [@BHS Th.2.6]. Moreover by [*loc. cit.*]{} the rigid variety $U_{\rm tri}^\square({\overline{r}})$ is smooth over $L$ and equidimensional, hence there is a bijection between the set of connected components of $U_{\rm tri}^\square({\overline{r}})$ and the set of irreducible components of $X_{\rm tri}^\square({\overline{r}})$.
We denote by $\omega:X_{\rm tri}^\square({\overline{r}})\rightarrow \mathcal{W}^n_L$ the composition $X_{\rm tri}^\square({\overline{r}})\hookrightarrow \mathfrak{X}_{{\overline{r}}}^\square\times \mathcal{T}^n_L\twoheadrightarrow \mathcal{T}^n_L \twoheadrightarrow \mathcal{W}^n_L$. If $x$ is a point of $X_{\rm tri}^\square({\overline{r}})$, we write $x=(r,\delta)$ where $r\in \mathfrak{X}_{{\overline{r}}}^\square$ and $\delta=(\delta_1,\dots,\delta_n)\in \mathcal{T}_L^n$. We say that a point $x=(r,\delta)\in X_{\rm tri}^\square({\overline{r}})$ is *crystalline* if $r$ is a crystalline representation of $\mathcal{G}_K$.
\[paramofcrystpt\] Let $x=(r,\delta)\in X_{\rm tri}^\square({\overline{r}})$ be a crystalline point. Then for $i\in \{1,\dots,n\}$ there exist ${\bf k}_i=(k_{\tau,i})_{\tau:\, K\hookrightarrow L}\in\mathbb{Z}^{{\mathrm{Hom}}(K,L)}$ and $\varphi_i\in k(x)^\times$ such that: $$\delta_i=z^{{\bf k}_i}{\rm unr}(\varphi_i).$$ Moreover the $(k_{\tau,i})_{i,\tau}$ are the labelled Hodge-Tate weights of $r$ and the $\varphi_i$ are the eigenvalues of the geometric Frobenius on the (unramified) Weil-Deligne representation ${\rm WD}(r)$ associated to $r$ (cf. [@Fo]).
The fact that the $(k_{\tau,i})_{i,\tau}$ are the Hodge-Tate weights of $r$ follows for instance from [@BHS Prop.2.9]. By [@KPX Th.6.3.13] there exist for each $i$ a continuous character $\delta'_i:K^\times \rightarrow k(x)^\times$ such that $r$ is trianguline of parameter $\delta':=(\delta'_1,\dots,\delta'_n)$ and such that $\delta_i/\delta'_i$ is an algebraic character of $K^\times$ (i.e. of the form $z^{{\bf k}}$ for some ${\bf k}\in{\mathbb{Z}}^{{\mathrm{Hom}}(K,L)}$). It thus suffices to prove that each $\delta'_i$ is of the form $z^{{\bf k}'_i}{\rm unr}(\varphi_i)$ for some ${\bf k}'_i\in{\mathbb{Z}}^{{\mathrm{Hom}}(K,L)}$ where the $\varphi_i\in k(x)^\times$ are the eigenvalues of the geometric Frobenius on ${\rm WD}(r)$, or equivalently (using the definition of ${\rm WD}(r)$) are the eigenvalues of the linearized Frobenius $\varphi^{[K_0:{\mathbb{Q}}_p]}$ on the $K_0\otimes_{{\mathbb{Q}}_p}k(x)$-module $D_{\rm cris}(r):=(B_{\rm cris}\otimes_{{\mathbb{Q}}_p}r)^{\mathcal{G}_K}$. By [@Berger Th.3.6] there is an isomorphism (recall $t$ is “Fontaine’s $2i\pi$”): $$\begin{aligned}
\label{berger}
D_{\rm cris}(r)\cong D_{{\mathrm{rig}}}(r)[\tfrac{1}{t}]^{\Gamma_K},\end{aligned}$$ and a triangulation ${\mathrm{Fil}}_\bullet$ of $D_{{\mathrm{rig}}}(r)$ with graded pieces giving the parameter $\delta'$ induces a complete $\varphi$-stable filtration ${\mathcal{F}}_\bullet$ on $D_{\rm cris}(r)$ such that ${\mathcal{F}}_i/{\mathcal{F}}_{i-1}$ is the filtered $\varphi$-module associated to ${\mathcal{R}}_{L,K}(\delta'_i)={\mathrm{Fil}}_i/{\mathrm{Fil}}_{i-1}$ by the same recipee as (\[berger\]). It follows from this and from [@KPX Example 6.2.6(3)] that $\delta'_i$ is of the form $z^{{\bf k}'_i}{\rm unr}(a)$ where $a\in k(x)^\times$ is the unique element such that $\varphi^{[K_0:{\mathbb{Q}}_p]}$ acts on the underlying $\varphi$-module of ${\mathcal{F}}_i/{\mathcal{F}}_{i-1}$ by multiplication by $1\otimes a\in K_0\otimes_{{\mathbb{Q}}_p} k(x)$. This finishes the proof.
Note that Lemma \[paramofcrystpt\] implies that if $x=(r,\delta)\in X_{\rm tri}^\square({\overline{r}})$ is a crystalline point, then $\omega(x)$ is algebraic ($=\delta_{\bf k}$ for ${\bf k}:=(k_{\tau,i})_{1\leq i\leq n,\tau:\, K\hookrightarrow L}$ where the $k_{\tau,i}$ are as in Lemma \[paramofcrystpt\]). We say that a point $x=(r,\delta)\in X_{\rm tri}^\square({\overline{r}})$ such that $\omega(x)$ is algebraic is [*dominant*]{} (resp. [*strictly dominant*]{}) if $\omega(x)$ is dominant (resp. strictly dominant).
A variant of the crystalline deformation space {#variant}
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We define a certain irreducible component $Z_{{\rm tri},U}(x)$ of a sufficiently small open neighbouhood $U\subseteq X_{\rm tri}^\square({\overline{r}})$ containing a crystalline strictly dominant point $x$ (Corollary \[defnZtri(x)\]).
We fix ${\bf k}=(k_{\tau,i})_{1\leq i\leq n,\tau:\, K\hookrightarrow L}\in ({\mathbb{Z}}^n)^{{\mathrm{Hom}}(K,L)}$ such that $k_{\tau,i}> k_{\tau,i+1}$ for all $i,\tau$ and write $R_{\overline r}^{\square,{\bf k}{\rm -cr}}$ for the crystalline deformation ring of $\overline r$ with Hodge-Tate weights ${\bf k}$, i.e. the reduced and ${\mathbb{Z}}_p$-flat quotient of $R^\square_{\overline r}$ such that, for any finite extension $L'$ of $L$, a morphism $x:{\mathrm{Spec}}\, L'\rightarrow {\mathrm{Spec}}\, R_{{\overline{r}}}^\square$ factors through ${\mathrm{Spec}}\, R_{\overline r}^{\square, {\bf k}{\rm -cr}}$ if and only if the representation ${\mathcal{G}}_K\rightarrow {\mathrm{GL}}_n(L')$ defined by $x$ is crystalline with labelled Hodge-Tate weights $(k_{\tau,i})_{1\leq i\leq n,\tau:\, K\hookrightarrow L}$. That this ring exists is the main result of [@Kisindef]. We write ${\mathfrak{X}}_{\overline r}^{\square,{\bf k}{\rm -cr}}$ for the rigid analytic space associated to ${\mathrm{Spf}}\, R_{\overline r}^{\square,{\bf k}{\rm -cr}}$. By [@Kisindef], it is smooth over $L$.
Let $\tilde r:{\mathcal G}_K\rightarrow {\mathrm{GL}}_n(R_{\overline r}^{\square,{\bf k}{\rm -cr}})$ be the corresponding universal deformation. By [@Kisindef Th.2.5.5] or [@BergerColmez Cor.6.3.3] there is a coherent $K_0\otimes_{{\mathbb{Q}}_p}{\mathcal{O}}_{\!{\mathfrak{X}}_{\overline r}^{\square,{\bf k}{\rm -cr}}}$-module ${\mathcal{D}}$ that is locally on ${\mathfrak{X}}_{\overline r}^{\square,{\bf k}{\rm -cr}}$ free over $K_0\otimes_{{\mathbb{Q}}_p}{\mathcal{O}}_{\!{\mathfrak{X}}_{\overline r}^{\square,{\bf k}{\rm -cr}}}$ together with a $\varphi\otimes {\mathrm{id}}$-linear automorphism $\Phi_{\rm cris}$ such that: $$({\mathcal{D}},\Phi_{\rm cris})\otimes_{{\mathcal{O}}_{\!{\mathfrak{X}}_{\overline r}^{\square,{\bf k}{\rm -cr}}}} k(x)\cong D_{\rm cris}\big(\tilde r\otimes_{R_{\overline r}^{\square,{\bf k}{\rm -cr}}} k(x)\big)$$ for all $x\in {\mathfrak{X}}_{\overline r}^{\square,{\bf k}{\rm -cr}}$. Fixing an embedding $\tau_0:K_0\hookrightarrow L$ we can define the associated family of Weil-Deligne representations: $$({\rm WD}(\tilde r),\Phi):=\big({\mathcal{D}}\otimes_{K_0\otimes_{{\mathbb{Q}}_p}{\mathcal{O}}_{\!{\mathfrak{X}}_{\overline r}^{\square,{\bf k}{\rm -cr}}},\tau_0\otimes{\mathrm{id}}}{\mathcal{O}}_{\!{\mathfrak{X}}_{\overline r}^{\square,{\bf k}{\rm -cr}}},\Phi_{\rm cris}^{[K_0:{\mathbb{Q}}_p]}\otimes{\mathrm{id}}\big)$$ on ${\mathfrak{X}}_{\overline r}^{\square,{\bf k}{\rm -cr}}$ whose isomorphism class does not depend on the choice of the embedding $\tau_0$.
Let $T^{\mathrm{rig}}\cong ({\mathbb G}_{{\rm m}}^{\mathrm{rig}})^{n}$ be the rigid analytic space over ${\mathbb{Q}}_p$ associated to the diagonal torus $T\subset {\mathrm{GL}}_{n}$ and let ${\mathcal{S}}_n$ be the Weyl group of $({\mathrm{GL}}_{n},T)$ acting on $T$, and thus on $T^{\mathrm{rig}}$, in the usual way. Recall that the map: $${\mathrm{diag}}(\varphi_1,\varphi_2,\dots,\varphi_n)\mapsto {\rm coefficients\ of}\ (X-\varphi_1)(X-\varphi_2)\dots (X-\varphi_n)$$ induces an isomorphism of schemes over ${\mathbb{Q}}_p$: $$T/{\mathcal{S}}_n\buildrel\sim\over\longrightarrow {\mathbb G}_{{\rm a}}^{n-1}\times_{{\mathrm{Spec}}\,{\mathbb{Q}}_p} {\mathbb G}_{{\rm m}}$$ and also of the associated rigid analytic spaces. We deduce that the coefficients of the characteristic polynomial of the Frobenius $\Phi$ on ${\rm WD}(\tilde r)$ determine a morphism of rigid analytic spaces over $L$: $${\mathfrak{X}}_{\overline r}^{\square,{\bf k}{\rm -cr}}\longrightarrow T_L^{{\mathrm{rig}}}/{\mathcal{S}}_n.$$ Let us define: $$\widetilde{\mathfrak{X}}_{\overline r}^{\square,{\bf k}\rm -cr}:={\mathfrak{X}}_{\overline r}^{\square,{\bf k}\rm -cr}\times_{T_L^{\mathrm{rig}}/{\mathcal{S}}_n}T_L^{\mathrm{rig}}.$$ Concretely $\widetilde{\mathfrak{X}}_{{\overline{r}}}^{\square,{\bf k}\rm-cr}$ parametrizes crystalline framed ${\mathcal{G}}_K$-deformations $r$ of $\overline r$ of labelled Hodge-Tate weights ${\bf k}$ together with an ordering $(\varphi_1,\dots,\varphi_n)$ of the eigenvalues of the geometric Frobenius on ${\rm WD}(r)$.
\[reduced\] The rigid analytic space $\widetilde{\mathfrak{X}}_{\overline r}^{\square,{\bf k}\rm -cr}$ is reduced.
It is sufficient to prove this result locally. Let $\mathrm{Sp}\,C$ be an admissible irreducible affinoid open subspace of $\mathfrak{X}_{\overline{r}}^{\square,{\rm {\bf k}-cr}}$ whose image in $T_L^{\rm rig}/\mathcal{S}_n$ is contained in an admissible affinoid open irreducible subspace $\mathrm{Sp}\,A$ of $T_L^{\rm rig}/\mathcal{S}_n$. As both $\mathfrak{X}_{\overline{r}}^{\square,{\rm {\bf k}-cr}}$ and $T_L^{\rm rig}/\mathcal{S}_n$ are smooth over $L$ we can find an admissible open affinoid covering of $\mathfrak{X}_{\overline{r}}^{\square,{\rm {\bf k}-cr}}$ by such $\mathrm{Sp}\,C$. The map $T_L^{\rm rig}\rightarrow T_L^{\rm rig}/\mathcal{S}_n$ is finite flat being the rigidification of a map of affine schemes $T_L\rightarrow T_L/\mathcal{S}_n$ which is finite flat. Consequently the inverse image of $\mathrm{Sp}\,A$ in $T_L^{\rm rig}$ is an admissible affinoid open subspace $\mathrm{Sp}\,B$ with $B$ an affinoid algebra which is finite flat over $A$. As $B$ is a finite $A$-algebra, we have an isomorphism $C\widehat{\otimes}_AB\simeq C\otimes_AB$. It follows, by definition of the fiber product of rigid analytic spaces, that the rigid analytic spaces of the form $\mathrm{Sp}(C\otimes_AB)$ form an admissible open covering of $\widetilde{\mathfrak{X}}_{\overline r}^{\square,{\bf k}\rm -cr}=\mathfrak{X}_{\overline{r}}^{\square,{\rm {\bf k}-cr}}\times_{T_L^{\rm rig}/\mathcal{S}_n}T_L^{\rm rig}$. It is sufficient to prove that rings $C\otimes_AB$ as above are reduced. From Lemma \[red\] below it is sufficient to prove that $C\otimes_AB$ is a finite flat generically étale $C$-algebra. As $B$ is finite flat over $A$, the $C$-algebra $C\otimes_AB$ is clearly finite flat. It is sufficient to prove that it is a generically étale $C$-algebra. As $B$ is generically étale over $A$, it is sufficient to prove that the map $\mathrm{Spec}\,C\rightarrow\mathrm{Spec}\,A$ is dominant. It is thus sufficient to prove that the map of rigid analytic spaces $\mathfrak{X}_{\overline{r}}^{\square,{\rm {\bf k}-cr}}\rightarrow T_L^{\rm rig}/\mathcal{S}_n$ is open. This follows from the fact that it has, locally on $\mathfrak{X}_{\overline{r}}^{\square,{\rm {\bf k}-cr}}$, a factorization: $$\mathfrak{X}_{\overline{r}}^{\square,{\rm {\bf k}-cr}}\longrightarrow ({\mathrm{Res}}_{K_0/{\mathbb{Q}}_p}{\mathrm{GL}}_{n,K_0} \times_{{\mathrm{Sp}}\,{\mathbb{Q}}_p} {\rm Flag})^{\mathrm{rig}}\times_{{\mathrm{Sp}}\,{\mathbb{Q}}_p}{\mathrm{Sp}}\,L\longrightarrow T_L^{\rm rig}/\mathcal{S}_n$$ where the first map is the smooth map in the proof of Lemma \[genericopensinXcris\] below, and the second is the projection on $(\mathrm{Res}_{K_0/\mathbb{Q}_p}\mathrm{GL}_{n,K_0})_L^{\rm rig}$ followed by the base change to $L$ of the rigidification of the morphism $\mathrm{Res}_{K_0/\mathbb{Q}_p}\mathrm{GL}_{n,K_0}\rightarrow T/\mathcal{S}_n$ defined in [@HartlHellmann (9.1)]. The first map being smooth is flat and thus open by [@Bo Cor.9.4.2], and the last two are easily seen to be open.
The following (well-known) lemma was used in the proof of Lemma \[reduced\].
\[red\] Let $A$ a commutative noetherian domain and $B$ a finite flat $A$-algebra. Then the ring $B$ has no embedded component, i.e. all its associated ideals are minimal prime ideals. Moreover if $B$ is generically étale over $A$, i.e. $\mathrm{Frac}(A)\otimes_A B$ is a finite étale $\mathrm{Frac}(A)$-algebra, then the ring $B$ is reduced.
As $B$ is flat over $A$, the map $\mathrm{Spec}\,B\rightarrow\mathrm{Spec}\,A$ has an open image, and $A$ being a domain it contains the unique generic point of $\mathrm{Spec}\,A$, which implies that the natural map $A\rightarrow B$ is injective. Moreover $B$ being finite over $A$, the image of $\mathrm{Spec}\,B\rightarrow\mathrm{Spec}\,A$ is closed, hence it is $\mathrm{Spec}\,A$ since $\mathrm{Spec}\,A$ is connected. In particular $B$ is a faithfully flat $A$-algebra. As $B$ is a flat $A$-module, it follows from [@BourbakiAC §IV.2.6 Lem.1] applied with $E=A$ and $F=B$ that $\mathfrak{p}\in\mathrm{Ass}(B)$ implies $\mathfrak{p}\cap A=0$ ($A$ is a domain, so $\mathrm{Ass}(A)=\{0\}$). It then follows from [@BourbakiAC §V.2.1 Cor.1] that if $\mathfrak{p}\in\mathrm{Ass}(B)$, then $\mathfrak{p}$ is a minimal prime of $B$. Indeed, $A$ being noetherian and $B$ a finite $A$-module, $B$ is an integral extension of $A$. We can apply [*loc. cit.*]{} to the inclusion $\mathfrak{q}\subseteq\mathfrak{p}$ where $\mathfrak{q}$ is a minimal prime ideal of $B$ (both ideals $\mathfrak{q}$ and $\mathfrak{p}$ being above the prime ideal $(0)$ of $A$ since $\mathfrak{p}\cap A=\mathfrak{q}\cap A=0$).
Let $\mathfrak{d}_{B/A}$ be the discriminant of $B/A$ (its existence comes from the fact that $B$ is a finite faithfully flat $A$-algebra, hence a finite projective $A$-module). As the extension is generically étale, we can find $f\in\mathfrak{d}_{B/A}$ such that $B_f$ is étale over $A_f$. As $A_f$ is a domain, $B_f$ is then reduced. Thus the nilradical $\mathfrak{n}$ of $B_f$ is killed by some power of $f$. Replacing $f$ by this power, we can assume that the vanishing ideal of $\mathfrak{n}$ contains $f$. Assume that $\mathfrak{n}$ is nonzero and let $\mathfrak{p}$ be a prime ideal of $B$ minimal among prime ideals containing $\mathrm{Ann}_B(\mathfrak{n})$. It follows from [@BourbakiAC §IV.1.3 Cor.1] that $\mathfrak{p}$ is an associated prime of the $B$-module $\mathfrak{n}$ and consequently of $B$. But we have $f\in\mathfrak{p}$ which contradicts the fact that $\mathfrak{p}\cap A=0$.
We now embed this “refined” crystalline deformation space $\widetilde{\mathfrak{X}}_{\overline r}^{\square,{\bf k}\rm -cr}$ into the space $X_{\rm tri}^\square({\overline{r}})$ as follows. We define a morphism of rigid spaces over $L$: $$\begin{aligned}
\label{cristotriang}
{\mathfrak{X}}_{\overline r}^{\square,{\bf k}\rm -cr}\times_{{\mathrm{Sp}}\,L} T_L^{\mathrm{rig}}&\longrightarrow &{\mathfrak{X}}_{\overline r}^\square\times_{{\mathrm{Sp}}\,L}{\mathcal{T}}^n_L\\
\nonumber (r,\varphi_1,\dots,\varphi_n)&\longmapsto & (r, z^{{\bf k}_1}{\rm unr}(\varphi_1),\dots, z^{{\bf k}_n}{\rm unr}(\varphi_n)).\end{aligned}$$ This morphism is a closed embedding of reduced rigid spaces as both maps $r\mapsto r$ and $(\varphi_1,\dots,\varphi_n)\mapsto (z^{{\bf k}_1}{\rm unr}(\varphi_1),\dots, z^{{\bf k}_n}{\rm unr}(\varphi_n))$ respectively define closed embeddings ${\mathfrak{X}}_{\overline r}^{\square,{\bf k}\rm-cr}\!\!\hookrightarrow {\mathfrak{X}}_{\overline r}^\square$ and $T_L^{\mathrm{rig}}\hookrightarrow {\mathcal{T}}^n_L$. We claim that the restriction of the morphism $(\ref{cristotriang})$ to: $$\label{cristotriangres}
\widetilde{\mathfrak{X}}_{\overline r}^{\square,{\bf k}\rm -cr}\hookrightarrow{\mathfrak{X}}_{\overline r}^{\square,{\bf k}\rm -cr}\times_{{\mathrm{Sp}}\,L}T_L^{\mathrm{rig}}$$ factors through $X_{\rm tri}^\square(\overline r)\subset {\mathfrak{X}}_{\overline r}^\square\times_{{\mathrm{Sp}}\,L}{\mathcal{T}}^n_L$. As the source of this restriction is reduced by Lemma \[reduced\], it is enough to check it on a Zariski-dense set of points of $\widetilde{\mathfrak{X}}_{\overline r}^{\square,{\bf k}\rm -cr}$.
Let $r$ be an $n$-dimensional crystalline representation of ${\mathcal G}_K$ over a finite extension $L'$ of $L$ of Hodge-Tate weights ${\bf k}$ and let $\varphi_1,\dots,\varphi_n$ be an ordering of the eigenvalues of a geometric Frobenius on ${\rm WD}(r)$, equivalently of the eigenvalues of $\varphi^{[K_0:{\mathbb{Q}}_p]}$ on $D_{\rm cris}(r)$ (that are assumed to be in $L'^\times$). Assuming moreover that the $\varphi_i$ are pairwise distinct, this datum gives rise to a unique complete $\varphi$-stable flag of free $K_0\otimes_{{\mathbb{Q}}_p}L'$-modules: $$0={\mathcal{F}}_0\subset {\mathcal{F}}_1\subset \dots\subset {\mathcal{F}}_n=D_{\rm cris}(r)$$ on $D_{\rm cris}(r)$ such that $\varphi^{[K_0:{\mathbb{Q}}_p]}$ acts on ${\mathcal{F}}_i/{\mathcal{F}}_{i-1}$ by multiplication by $\varphi_i$ (this is a *refinement* in the sense of [@BelChe Def.2.4.1]). By the same argument as in the proof of Lemma \[paramofcrystpt\] using Berger’s dictionary between crystalline $(\varphi,\Gamma_K)$-modules and filtered $\varphi$-modules (see e.g. (\[berger\])), the filtration ${\mathcal{F}}_\bullet$ induces a triangulation ${\mathrm{Fil}}_{\bullet}$ on $D_{{\mathrm{rig}}}(r)$. If we assume that ${\mathcal{F}}_\bullet$ is [*noncritical*]{} in the sense of [@BelChe Def.2.4.5], i.e. the filtration ${\mathcal{F}}_\bullet$ is in general position with respect to the Hodge filtration ${\mathrm{Fil}}^\bullet D_{\rm dR}(r)$ on $D_{\rm dR}(r)$, that is, for all embeddings $\tau:K\hookrightarrow L$ and all $i=1,\dots,n-1$ we have: $$\begin{gathered}
\label{generalpos}
\big({\mathcal{F}}_{i}\otimes_{K_0\otimes_{{\mathbb{Q}}_p}L',\tau\otimes {\mathrm{id}}}L'\big)\oplus \big({\mathrm{Fil}}^{-k_{\tau,i+1}}D_{\rm dR}(r)\otimes_{K\otimes_{{\mathbb{Q}}_p}L',\tau\otimes {\mathrm{id}}}L'\big)=D_{\rm cris}(r)\otimes_{K_0\otimes_{{\mathbb{Q}}_p}L,\tau\otimes {\mathrm{id}}}L'\\
=D_{\rm dR}(r)\otimes_{K\otimes_{{\mathbb{Q}}_p}L',\tau\otimes {\mathrm{id}}}L',\end{gathered}$$ then ${\mathcal{F}}_{i}/{\mathcal{F}}_{i-1}$ is a filtered $\varphi$-module of Hodge-Tate weights ${\bf k}_i$, or equivalently ${\mathrm{Fil}}_i/{\mathrm{Fil}}_{i-1}\cong {\mathcal{R}}_{L',K}(\delta_i)$ with $\delta_i=z^{{\bf k}_i}{\rm unr}(\varphi_i)$.
\[genericopensinXcris\] There are smooth (over $L$) Zariski-open and Zariski-dense subsets in $\widetilde{\mathfrak{X}}_{\overline r}^{\square,{\bf k}\rm -cr}$: $$\widetilde V_{\overline r}^{\square,{\bf k}\rm -cr} \subset \widetilde U_{\overline r}^{\square,{\bf k}\rm -cr}\subset\widetilde{\mathfrak{X}}_{\overline r}^{\square,{\bf k}\rm -cr}$$ such that:
1. a point $(r,\varphi_1,\dots,\varphi_n)\in \widetilde {\mathfrak{X}}_{\overline r}^{\square,{\bf k}\rm -cr}$ lies in $\widetilde U_{\overline r}^{\square,{\bf k}\rm -cr}$ if and only if the $\varphi_i$ are pairwise distinct;\
2. a point $(r,\varphi_1,\dots,\varphi_n)\in \widetilde U_{\overline r}^{\square,{\bf k}\rm -cr}$ lies in $\widetilde V_{\overline r}^{\square,{\bf k}\rm -cr}$ if and only if it satisfies assumption $(\ref{generalpos})$ above and $z^{{\bf k}_i-{\bf k}_j}{\rm unr}(\varphi_i\varphi_j^{-1})\in{\mathcal{T}}_{\rm reg}$ for $i\ne j$.
Moreover the image of $\widetilde V_{\overline r}^{\square,{\bf k}\rm -cr}$ via (\[cristotriang\]) composed with (\[cristotriangres\]) lies in $U_{\rm tri}^\square(\overline r)$.
The idea of the proof is the same as that of [@Chenevier Lem.4.4]. It is enough to show that all the statements are true locally on $\widetilde{\mathfrak{X}}_{\overline r}^{\square,{\bf k}\rm -cr}$. Let us (locally) fix a basis of the coherent locally free $K_0\otimes_{{\mathbb{Q}}_p}{\mathcal{O}}_{\!{\mathfrak{X}}_{\overline r}^{\square,{\bf k}{\rm -cr}}}$-module ${\mathcal{D}}$ on ${\mathfrak{X}}_{{\overline{r}}}^{\square,{\bf k}\rm -cr}$. By the choice of such a basis, the matrix of the crystalline Frobenius $\Phi_{\rm cris}$ and the Hodge filtration define (locally) a morphism: $${\mathfrak{X}}_{{\overline{r}}}^{\square,{\bf k}\rm -cr}\longrightarrow ({\mathrm{Res}}_{K_0/{\mathbb{Q}}_p}{\mathrm{GL}}_{n,K_0} \times_{{\mathrm{Sp}}\,{\mathbb{Q}}_p} {\rm Flag})^{\mathrm{rig}}\times_{{\mathrm{Sp}}\,{\mathbb{Q}}_p}{\mathrm{Sp}}\,L$$ where ${\rm Flag}:=({\mathrm{Res}}_{K/{\mathbb{Q}}_p}{\mathrm{GL}}_{n,K})/({\mathrm{Res}}_{K/{\mathbb{Q}}_p} B)$ (compare [@HartlHellmann §8]). By [@HartlHellmann Prop.8.12] and the discussion preceding it, it follows that this morphism is smooth, hence so is the morphism: $$\label{morphismbasechange}
\widetilde {\mathfrak{X}}_{{\overline{r}}}^{\square,{\bf k}\rm -cr}\longrightarrow \big(({\mathrm{Res}}_{K_0/{\mathbb{Q}}_p}{\mathrm{GL}}_{n,K_0})^{\mathrm{rig}}_L\times_{T_L^{\mathrm{rig}}/{\mathcal{S}}_n}T_L^{\mathrm{rig}}\big)\times_{{\mathrm{Sp}}\,L} {\rm Flag}_L^{\mathrm{rig}}$$ where ${\mathrm{Res}}_{K_0/{\mathbb{Q}}_p}{\mathrm{GL}}_{n,K_0}\rightarrow T/{\mathcal{S}}_n$ is the morphism defined in [@HartlHellmann (9.1)]. On the other hand, using that the morphism $T\rightarrow T/{\mathcal{S}}_n$ is obviously smooth in the neighbourhood of a point $(\varphi_1,\dots,\varphi_n)\in T$ where the $\varphi_i$ are pairwise distinct, we see that the conditions of (i), resp. (ii), in the statement cut out smooth (over $L$) Zariski-open and Zariski-dense subspaces of: $$\label{RHS}
\big(({\mathrm{Res}}_{K_0/{\mathbb{Q}}_p}{\mathrm{GL}}_{n,K_0})^{\mathrm{rig}}_L\times_{T_L^{\mathrm{rig}}/{\mathcal{S}}_n}T_L^{\mathrm{rig}}\big)\times_{{\mathrm{Sp}}\,L} {\rm Flag}_L^{\mathrm{rig}}.$$ Their inverse images in $\widetilde {\mathfrak{X}}_{\overline r}^{\square,{\bf k}\rm -cr}$ via (\[morphismbasechange\]) are thus smooth over $L$ and Zariski-open in $\widetilde {\mathfrak{X}}_{\overline r}^{\square,{\bf k}\rm -cr}$. Let us prove that these inverse images are also Zariski-dense in $\widetilde{\mathfrak{X}}_{\overline r}^{\square,{\bf k}\rm -cr}$. It is enough to prove that they intersect nontrivially every irreducible component of $\widetilde {\mathfrak{X}}_{\overline r}^{\square,{\bf k}\rm -cr}$. Let ${\mathrm{Sp}}\,A$ be any affinoid open subset of $\widetilde {\mathfrak{X}}_{\overline r}^{\square,{\bf k}\rm -cr}$, it follows from [@Bo Cor.9.4.2] that the image of ${\mathrm{Sp}}\,A$ by the smooth, hence flat, morphism (\[morphismbasechange\]) is admissible open in (\[RHS\]). In particular its intersection with one of the above Zariski-open and Zariski-dense subspaces of (\[RHS\]) can’t be empty, which proves the statement. The final claim of the lemma follows from (ii), the discussion preceding Lemma \[genericopensinXcris\] and the definition (\[ucris\]) of $U_{\rm tri}^\square(\overline r)$ .
Note that $\widetilde{\mathfrak{X}}_{\overline r}^{\square,{\bf k}\rm -cr}$ is equidimensional of the same dimension as ${\mathfrak{X}}_{\overline r}^{\square,{\bf k}\rm -cr}$. Indeed, by Lemma \[genericopensinXcris\] it is enough to prove the same statement for $\widetilde U_{\overline r}^{\square,{\bf k}\rm -cr}$. But this is clear since the map $\widetilde U_{\overline r}^{\square,{\bf k}\rm -cr}\rightarrow {\mathfrak{X}}_{\overline r}^{\square,{\bf k}\rm -cr}$ is smooth of relative dimension $0$, hence étale, and since ${\mathfrak{X}}_{\overline r}^{\square,{\bf k}\rm -cr}$ is equidimensional ([@Kisindef]). Lemma \[genericopensinXcris\] also implies that $(\ref{cristotriang})$ induces (as claimed above) a morphism: $$\label{embeddingcristri}
\iota_{\bf k}:\widetilde{\mathfrak{X}}_{\overline r}^{\square,{\bf k}\rm -cr}\longrightarrow X_{\rm tri}^\square(\overline r)$$ which is obviously a closed immersion as $(\ref{cristotriang})$ is.
\[defnZtri(x)\] Let $x=(r,\delta)\in X_{\rm tri}^\square(\overline r)$ be a crystalline strictly dominant point such that $\omega(x)=\delta_{\bf k}$ and the Frobenius eigenvalues $(\varphi_1,\dots,\varphi_n)$ (cf. Lemma \[paramofcrystpt\]) are pairwise distinct and let $U$ be an open subset of $X_{\rm tri}^\square(\overline r)$ containing $x$.\
(i) The point $x$ belongs to $\iota_{\bf k}(\widetilde U_{\overline r}^{\square,{\bf k}{\rm -cr}})$ and there is a unique irreducible component $\widetilde Z_{\rm cris}(x)$ of $\widetilde{\mathfrak{X}}_{\overline r}^{\square,{\bf k}{\rm -cr}}$ containing $\iota_{\bf k}^{-1}(x)$.\
(ii) If $U$ is small enough there is a unique irreducible component $Z_{{\rm tri},U}(x)$ of $U$ containing $\iota_{\bf k}(\widetilde Z_{\rm cris}(x))\cap U$, and it is such that $Z_{{\rm tri},U}(x)\cap U'=Z_{{\rm tri},U'}(x)$ for any open $U'\subseteq U$ containing $x$.
\(i) The assumptions and Lemma \[paramofcrystpt\] imply that $x$ is in the image of the map $\iota_{\bf k}$ in (\[embeddingcristri\]) and the fact that the $\varphi_i$ are pairwise distinct implies that $x\in \iota_{\bf k}(\widetilde U_{\overline r}^{\square,{\bf k}{\rm -cr}})$. In particular $\widetilde {\mathfrak{X}}_{\overline r}^{\square,{\bf k}{\rm -cr}}$ is smooth at $\iota_{\bf k}^{-1}(x)$ by Lemma \[genericopensinXcris\] and thus $\iota_{\bf k}^{-1}(x)$ belongs to a unique irreducible component $\widetilde Z_{\rm cris}(x)$ of $\widetilde{\mathfrak{X}}_{\overline r}^{\square,{\bf k}{\rm -cr}}$.
\(ii) We have that $\iota_{\bf k}(\widetilde Z_{\rm cris}(x))\cap U$ is a Zariski-closed subset of $U$, and it is easy to see that it is still irreducible if $U$ is small enough since $\iota_{\bf k}(\widetilde Z_{\rm cris}(x))$ is smooth at $x$. Hence there exists at least one irreducible component of $U$ containing the irreducible Zariski-closed subset $\iota_{\bf k}(\widetilde Z_{\rm cris}(x))\cap U$. If there are two such irreducible components, then in particular [*any*]{} point of $\iota_{\bf k}(\widetilde Z_{\rm cris}(x))\cap U$ is singular in $U$, hence in $X_{\rm tri}^\square(\overline r)$. But Lemma \[genericopensinXcris\] implies $\widetilde Z_{\rm cris}(x)\cap \widetilde V_{\overline r}^{\square,{\bf k}\rm -cr}\ne \emptyset$ is Zariski-open and Zariski-dense in $\widetilde Z_{\rm cris}(x)$, hence: $$\big(\widetilde Z_{\rm cris}(x)\cap \widetilde V_{\overline r}^{\square,{\bf k}\rm -cr}\big)\cap \big(\widetilde Z_{\rm cris}(x)\cap \iota_{\bf k}^{-1}(U)\big) \ne \emptyset$$ from which we get $\iota_{\bf k}(\widetilde Z_{\rm cris}(x)\cap \widetilde V_{\overline r}^{\square,{\bf k}\rm -cr})\cap U\ne \emptyset$. The last statement of Lemma \[genericopensinXcris\] also implies $\iota_{\bf k}(\widetilde Z_{\rm cris}(x)\cap \widetilde V_{\overline r}^{\square,{\bf k}\rm -cr})\cap U\subseteq U_{\rm tri}^\square(\overline r)$, which is then a contradiction since $U_{\rm tri}^\square(\overline r)$ is smooth over $L$.
Finally, shrinking $U$ again if necessary, we can assume that, for any open subset $U'\subseteq U$ containing $x$, the map $Z\mapsto Z\cap U'$ induces a bijection between the irreducible components of $U$ containing $x$ and the irreducible components of $U'$ containing $x$. It then follows from the definition of $Z_{{\rm tri},U}(x)$ that $Z_{{\rm tri},U}(x)\cap U'=Z_{{\rm tri},U'}(x)$.
\[down\] [(i) Since the map $\widetilde {\mathfrak{X}}_{\overline r}^{\square,{\bf k}{\rm -cr}}\rightarrow {\mathfrak{X}}_{\overline r}^{\square,{\bf k}{\rm -cr}}$ is finite, hence closed, and since $\widetilde {\mathfrak{X}}_{\overline r}^{\square,{\bf k}{\rm -cr}}$, ${\mathfrak{X}}_{\overline r}^{\square,{\bf k}{\rm -cr}}$ are both equidimensional (of the same dimension), the image of any irreducible component of $\widetilde {\mathfrak{X}}_{\overline r}^{\square,{\bf k}{\rm -cr}}$ is an irreducible component of ${\mathfrak{X}}_{\overline r}^{\square,{\bf k}{\rm -cr}}$. In particular the image of $\widetilde Z_{\rm cris}(x)$ in (i) of Corollary \[defnZtri(x)\] is the unique irreducible component of ${\mathfrak{X}}_{\overline r}^{\square,{\bf k}{\rm -cr}}$ containing $r$.\
(ii) Either by the same proof as that for (ii) of Corollary \[defnZtri(x)\] or as a consequence of (ii) of Corollary \[defnZtri(x)\], we see that there is also a unique irreducible component $Z_{{\rm tri}}(x)$ of the whole $X_{\rm tri}^\square(\overline r)$ which contains the irreducible closed subset $\iota_{\bf k}(\widetilde Z_{\rm cris}(x))$.]{}
The Weyl group element associated to a crystalline point {#weyl}
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We review the definition of the Weyl group element associated to certain crystalline points on $X_{\rm tri}^\square(\overline r)$ (measuring their “criticality”) and state our main local results (Theorem \[upperbound\], Corollary \[Xtrismooth\]).
We keep the notation of §\[variant\]. We let $W\cong \prod_{\tau:\, K\hookrightarrow L}{\mathcal S}_n$ be the Weyl group of the algebraic group: $$({\mathrm{Res}}_{K/{\mathbb{Q}}_p}{\mathrm{GL}}_{n,K})\times_{{\mathrm{Spec}}\,{\mathbb{Q}}_p}{\mathrm{Spec}}\,L\cong \prod_{\tau:\, K\hookrightarrow L}{{\mathrm{GL}}_{n,L}}$$ and $X^\ast(({\mathrm{Res}}_{K/{\mathbb{Q}}_p}T_K)\times_{{\mathrm{Spec}}\,{\mathbb{Q}}_p}{\mathrm{Spec}}\,L)\cong \prod_{\tau:\, K\hookrightarrow L}X^\ast(T_L)$ be the ${\mathbb{Z}}$-module of algebraic characters of $({\mathrm{Res}}_{K/{\mathbb{Q}}_p}T_K)\times_{{\mathrm{Spec}}\,{\mathbb{Q}}_p}{\mathrm{Spec}}\,L$ (recall $T$ is the diagonal torus in ${\mathrm{GL}}_n$ and $T_K$, $T_L$ its base change to $K$, $L$). We write ${\rm lg}(w)$ for the length of $w$ in the Coxeter group $W$ (for the set of simple reflections associated to the simple roots of the upper triangular matrices).
Let $x=(r,\delta)=(r,\delta_1,\dots,\delta_n)$ be a crystalline strictly dominant point on $X_{\rm tri}^\square({\overline{r}})$. Then by Lemma $\ref{paramofcrystpt}$ the characters $\delta_i$ are of the form $\delta_i=z^{{\bf k}_i}{\rm unr} (\varphi_i)$ where ${\bf k}_i=(k_{\tau,i})_{\tau:\, K\hookrightarrow L}$ and the $\varphi_i\in k(x)^\times$ are the eigenvalues of the geometric Frobenius on ${\rm WD}(r)$. Assume that the $\varphi_i$ are pairwise distinct, then as in §\[variant\] the ordering $(\varphi_1,\dots,\varphi_n)$ defines a complete $\varphi$-stable flag of free $K_0\otimes_{{\mathbb{Q}}_p}k(x)$-modules $0={\mathcal{F}}_0\subset {\mathcal{F}}_1\subset \dots\subset {\mathcal{F}}_n=D_{\rm cris}(r)$ on $D_{\rm cris}(r)$ such that $\varphi^{[K_0:{\mathbb{Q}}_p]}$ acts on ${\mathcal{F}}_i/{\mathcal{F}}_{i-1}$ by multiplication by $\varphi_i$. We view ${\mathcal{F}}_i$ as a filtered $\varphi$-module with the induced Hodge filtration. If we write $(k'_{\tau,i})_{\tau:K\hookrightarrow L}$ for the Hodge-Tate weights of ${\mathcal{F}}_i/{\mathcal{F}}_{i-1}$, we find that there is a unique $w_x=(w_{x,\tau})_{\tau:\, K\hookrightarrow L}\in W=\prod_{\tau: K\hookrightarrow L}{\mathcal{S}}_n$ such that: $$\label{wxdef}
k'_{\tau, i}=k_{\tau, w_{x,\tau}^{-1}(i)}$$ for all $i\in \{1,\dots, n\}$ and each $\tau: K\hookrightarrow L$. We call $w_x$ the *Weyl group element associated to* $x$. Note that ${\mathcal{F}}_\bullet$ is noncritical (see §\[variant\]) if and only if $w_{x,\tau}=1$ for all $\tau:K\hookrightarrow L$, in which case we say that the crystalline strictly dominant point $x=(r,\delta)$ is [*noncritical*]{}.
For $w\in W$ we denote by $d_w\in {\mathbb{Z}}_{\geq 0}$ the rank of the ${\mathbb{Z}}$-submodule of $X^\ast(({\mathrm{Res}}_{K/{\mathbb{Q}}_p}T_K)\times_{{\mathrm{Spec}}\,{\mathbb{Q}}_p}{\mathrm{Spec}}\,L)$ generated by the $w(\alpha)-\alpha$ where $\alpha$ runs among the roots of $({\mathrm{Res}}_{K/{\mathbb{Q}}_p}{\mathrm{GL}}_{n,K})\times_{{\mathrm{Spec}}\,{\mathbb{Q}}_p}{\mathrm{Spec}}\,L$.
\[coxeter\] With the above notations we have: $$d_w\leq \lg(w)\leq [K:{\mathbb{Q}}_p]\frac{n(n-1)}{2}$$ and $\lg(w)= d_w$ if and only if $w$ is a product of distinct simple reflections.
Note first that the right hand side inequality is obvious. Let us write in this proof $X:=X^\ast(({\rm Res}_{K/{\mathbb{Q}}_p}T_K)\times_{{\mathrm{Spec}}\,{\mathbb{Q}}_p}{\mathrm{Spec}}\,L)$, $X_{{\mathbb{Q}}}:=X\otimes_{{\mathbb{Z}}}{\mathbb{Q}}$, and let us denote by $S$ the subset of simple reflections in $W$ (thus $\dim_{{\mathbb{Q}}}(X_{{\mathbb{Q}}})=[K:{\mathbb{Q}}_p]n$ and $\vert S\vert=[K:{\mathbb{Q}}_p](n-1)$). The rank of the subgroup of $X$ generated by the $w(\alpha)-\alpha$ for $\alpha$ as above, or equivalently by the $w(\alpha)-\alpha$ for $\alpha\in X$, is equal to the dimension of the ${\mathbb{Q}}$-vector space $(w-{\mathrm{id}})X_{{\mathbb{Q}}}$ which, by the rank formula, is equal to $\dim_{{\mathbb{Q}}}(X_{{\mathbb{Q}}})-\dim_{{\mathbb{Q}}}(\ker(w-{\mathrm{id}}))$. Let $I$ be the set of simple reflections appearing in $w$, we have $|I|\leq \lg(w)$ and $|I|= \lg(w)$ if and only if $w$ is a product of distinct simple reflections. It is thus enough to prove $\dim_{{\mathbb{Q}}}(\ker(w-{\mathrm{id}}))\geq \dim_{{\mathbb{Q}}}(X_{{\mathbb{Q}}})-|I|$ with equality when $w$ is a product of distinct simple reflections. Note that $\ker(w-{\mathrm{id}})$ obviously contains the ${\mathbb{Q}}$-subvector space of $X_{\mathbb{Q}}$ of fixed points by the subgroup $W_I$ of $W$ generated by the elements of $I$, and it follows from [@Hum Th.1.12(c)] that, when $w$ is a product of distinct simple reflections, then $\ker(w-{\mathrm{id}})$ is exactly this ${\mathbb{Q}}$-subvector space. It is thus enough to prove that this ${\mathbb{Q}}$-subvector space of $X_{\mathbb{Q}}$, which is just the intersection of the hyperplanes $\ker(s-{\mathrm{id}})$ for $s\in I$, has dimension $\dim_{{\mathbb{Q}}}(X_{\mathbb{Q}})-|I|$. However we know that for any ${\mathbb{Q}}$-subvector space $V\subset X_{{\mathbb{Q}}}$ and any reflection $s$ of $X_{{\mathbb{Q}}}$, we have $\dim_{{\mathbb{Q}}}(V\cap\ker(s-{\mathrm{id}}))\geq \dim_{{\mathbb{Q}}}(V)-1$ and thus by induction: $$\dim_{{\mathbb{Q}}}\big(V\bigcap\big(\bigcap_{s\in S}\ker(s-{\mathrm{id}})\big)\big)\geq \dim_{{\mathbb{Q}}}(V)-|S|$$ with equality if and only if $\dim_{{\mathbb{Q}}}\big(V\bigcap\big(\bigcap_{s\in J}\ker(s-{\mathrm{id}})\big)\big)= \dim_{{\mathbb{Q}}}(V)-|J|$ for all $J\subseteq S$. As the ${\mathbb{Q}}$-subvector space $X_{{\mathbb{Q}}}^W$ of fixed points by $W$ has dimension $[K:{\mathbb{Q}}_p]$ (it is generated by the characters $\tau\circ\det$ for $\tau:K\hookrightarrow L$), we have: $$\dim_{{\mathbb{Q}}}(X_{{\mathbb{Q}}}^W)=\bigcap_{s\in S}\ker(s-{\mathrm{id}})=[K:{\mathbb{Q}}_p]=\dim_{{\mathbb{Q}}}(X_{{\mathbb{Q}}})-|S|.$$ Consequently we deduce (taking $V=X_{\mathbb{Q}}$): $$\dim_{{\mathbb{Q}}}\big(\bigcap_{s\in I}\ker(s-{\mathrm{id}})\big)= \dim_{{\mathbb{Q}}}(X_{\mathbb{Q}})-|I|$$ which is the desired formula.
Recall that, if $X$ is a rigid analytic variety over $L$ and $x\in X$, the tangent space to $X$ at $x$ is the $k(x)$-vector space: $$\begin{aligned}
\label{tangent}
T_{X,x}:={\mathrm{Hom}}_{k(x)}\big({\mathfrak m}_{X,x}/{\mathfrak m}^2_{X,x},k(x)\big)={\mathrm{Hom}}_{k(x)-{\rm alg}}\big({\mathcal O}_{X,x},k(x)[\varepsilon]/(\varepsilon^2)\big)\end{aligned}$$ where ${\mathfrak m}_{X,x}$ is the maximal ideal of the local ring ${\mathcal O}_{X,x}$ at $x$ to $X$. If $X$ is equidimensional, recall also that $\dim_{k(x)}T_{X,x}\geq \dim X$ and that $X$ is smooth at $x$ if and only if $\dim_{k(x)}T_{X,x}= \dim X$.
We let $\widetilde X_{\rm tri}^\square({\overline{r}})\subseteq X_{\rm tri}^\square({\overline{r}})$ be the union of the irreducible components $C$ of $X_{\rm tri}^\square({\overline{r}})$ such that $C\cap U_{\rm tri}^\square({\overline{r}})$ contains a crystalline point. For instance it follows from Lemma \[genericopensinXcris\] that all the closed embeddings (\[embeddingcristri\]) factor as closed embeddings $\iota_{\bf k}:\widetilde{\mathfrak{X}}_{\overline r}^{\square,{\bf k}\rm -cr}\hookrightarrow \widetilde X_{\rm tri}^\square(\overline r)\subseteq X_{\rm tri}^\square(\overline r)$. In particular any point $x\in X_{\rm tri}^\square({\overline{r}})$ which is crystalline strictly dominant is in $\widetilde X_{\rm tri}^\square({\overline{r}})$.
The following statement is our main local conjecture.
\[mainconj\] Let $x\in X_{\rm tri}^\square({\overline{r}})$ be a crystalline strictly dominant point such that the Frobenius eigenvalues $(\varphi_1,\dots,\varphi_n)$ (cf. Lemma \[paramofcrystpt\]) are pairwise distinct, let $w_x$ be the Weyl group element associated to $x$ (cf. (\[wxdef\])) and let $d_x:=d_{w_x}$. Then: $$\dim_{k(x)}T_{\widetilde X_{\rm tri}^\square({\overline{r}}),x}=\lg(w_x)-d_{x}+\dim X_{\rm tri}^\square({\overline{r}})=\lg(w_x)-d_{x}+n^2+[K:{\mathbb{Q}}_p]\frac{n(n+1)}{2}.$$
In particular, since $\dim \widetilde X_{\rm tri}^\square({\overline{r}})=\dim X_{\rm tri}^\square({\overline{r}})=n^2+[K:{\mathbb{Q}}_p]\frac{n(n+1)}{2}$, we see by Lemma \[coxeter\] that $\widetilde X_{\rm tri}^\square({\overline{r}})$ should be smooth at $x$ if and only if $w_x$ is a product of distinct simple reflections.
[The reader can wonder why we don’t state Conjecture \[mainconj\] with $X_{\rm tri}^\square({\overline{r}})$ instead of $\widetilde X_{\rm tri}^\square({\overline{r}})$. The reason is that Conjecture \[mainconj\] with $\widetilde X_{\rm tri}^\square({\overline{r}})$ is actually implied by other conjectures, see §\[modularity\], and we don’t know if this is the case with $X_{\rm tri}^\square({\overline{r}})$.]{}
In order to state our main result in the direction of (a weaker version of) Conjecture \[mainconj\], we need the following two definitions.
\[veryreg\] A crystalline strictly dominant point $x=(r,\delta)\in X_{\rm tri}^\square({\overline{r}})$ is very regular if it satisfies the following conditions (where the $(\varphi_i)_{1\leq i\leq n}$ are the geometric Frobenius eigenvalues on ${\rm WD}(r)$):
1. $\varphi_i\varphi_j^{-1}\notin \{1,q\}$ for $1\leq i\ne j\leq n$;\
2. $\varphi_1\varphi_2\dots\varphi_i$ is a simple eigenvalue of the geometric Frobenius acting on $\bigwedge_{k(x)}^i\!\!{\rm WD}(r)$ for $1\leq i\leq n$.
\[varphi\] [If $x=(r,\delta)$ is crystalline strictly dominant, it easily follows from the dominance property that (i) of Definition \[veryreg\] is equivalent to $\delta_i\delta_j^{-1}\notin \{z^{-\bf h},\vert z\vert_K z^{\bf h},\ \vert z\vert_K^{-1} z^{\bf h},\ {\bf h}\in {\mathbb{Z}}_{\geq 0}^{{\mathrm{Hom}}(K,L)}\}$ for $1\leq i\ne j\leq n$. In particular it implies $\delta\in {\mathcal{T}}_{\rm reg}^n$, whence the terminology (compare also [@Bergdall §6.1]).]{}
\[accu\] Let $X$ be a union of irreducible components of an open subset of $X_{\rm tri}^\square({\overline{r}})$ (over $L$) and let $x\in X_{\rm tri}^\square({\overline{r}})$ such that $\omega(x)$ is algebraic. Then $X$ satisfies the [accumulation property at]{} $x$ if $x\in X$ and if, for any positive real number $C>0$, the set of crystalline strictly dominant points $x'=(r',\delta')$ such that:
1. the eigenvalues of $\varphi^{[K_0:{\mathbb{Q}}_p]}$ on $D_{\rm cris}(r')$ are pairwise distinct;\
2. $x'$ is noncritical;\
3. $\omega(x')=\delta'\vert_{({\mathcal O}_K^\times)^n}=\delta_{\bf k'}$ with $k'_{\tau,i}-k'_{\tau,i+1}>C$ for $i\in \{1,\dots,n-1\}$, $\tau\in {\mathrm{Hom}}(K,L);$
accumulate at $x$ in $X$ in the sense of [@BelChe §3.3.1].
It easily follows from Definition \[accu\] that $X$ satisfies the accumulation property at $x$ if and only if each irreducible component of $X$ containing $x$ satisfies the accumulation property at $x$. In particular, if $x$ belongs to each irreducible component of $X$, we see that for every $C>0$ the set of points $x'$ in the statement of Definition \[accu\] is also Zariski-dense in $X$. Since $U_{\rm tri}^\square({\overline{r}})\cap X$ is Zariski-open and Zariski-dense in $X$, we also see that each irreducible component of $X$ containing $x$ also contains points $x'$ as in Definition \[accu\] which are in $U_{\rm tri}^\square({\overline{r}})$, hence each irreducible component of $X$ containing $x$ is in $\widetilde X_{\rm tri}^\square({\overline{r}})$.
In §\[phiGammacohomology\] below we will prove the theorem that follows.
\[upperbound\] Let $x\in X_{\rm tri}^\square({\overline{r}})$ be a crystalline strictly dominant very regular point and let $X\subseteq X_{\rm tri}^\square({\overline{r}})$ be a union of irreducible components of an open subset of $X_{\rm tri}^\square({\overline{r}})$ such that $X$ satisfies the accumulation property at $x$. Then we have: $$\dim_{k(x)}T_{X,x}\leq \lg(w_x)-d_x+\dim X_{\rm tri}^\square({\overline{r}})=\lg(w_x)-d_x+n^2+[K:{\mathbb{Q}}_p]\frac{n(n+1)}{2}.$$
By Lemma \[coxeter\] we thus deduce the following important corollary.
\[Xtrismooth\] Let $x\in X_{\rm tri}^\square({\overline{r}})$ be a crystalline strictly dominant very regular point and let $X\subseteq X_{\rm tri}^\square({\overline{r}})$ be a union of irreducible components of an open subset of $X_{\rm tri}^\square({\overline{r}})$ such that $X$ satisfies the accumulation property at $x$. Assume that $w_x$ is a product of distinct simple reflections. Then $X$ is smooth at $x$.
[Note that for $X$, $x$ as above we only have $\dim_{k(x)}T_{X,x}\leq \dim_{k(x)}T_{\widetilde X_{\rm tri}^\square({\overline{r}}),x}$, thus Theorem \[upperbound\] doesn’t give an upper bound on $\dim_{k(x)}T_{\widetilde X_{\rm tri}^\square({\overline{r}}),x}$ (but Conjecture \[mainconj\] implies Theorem \[upperbound\]). However Theorem \[upperbound\] and Corollary \[Xtrismooth\] will be enough for our purpose.]{}
Crystalline points on the patched eigenvariety {#globalpart}
==============================================
We state the classicality conjecture (Conjecture \[classiconj\]) and prove new cases of it (Corollary \[mainclassic\]).
The classicality conjecture {#classic}
---------------------------
We review the definition of classicality (Definition \[class\], Proposition \[compaclas\]) and state the classicality conjecture (Conjecture \[classiconj\]).
We first recall the global setting, basically the same as [@BHS §2.4]. We fix a totally real field $F^+$, we write $q_v$ for the cardinality of the residue field of $F^+$ at a finite place $v$ and we denote by $S_p$ the set of places of $F^+$ dividing $p$ . We fix a totally imaginary quadratic extension $F$ of $F^+$ that splits at all places of $S_p$ and let ${\mathcal{G}}_F:={\rm Gal}(\overline F/F)$. We fix a unitary group $G$ in $n$ variables over $F^+$ (with $n\geq 2$) such that $G\times_{F^+}F\cong {\mathrm{GL}}_{n,F}$ and $G(F^+\otimes_{\mathbb{Q}}\mathbb{R})$ is compact. We fix an isomorphism $i:G\times_{F^+}F\buildrel \sim\over\rightarrow {\mathrm{GL}}_{n,F}$ and, for each $v\in S_p$, a place $\tilde v$ of $F$ dividing $v$. The isomorphisms $F_v^+\buildrel\sim\over\rightarrow F_{\tilde v}$ and $i$ induce an isomorphism $i_{\tilde v}:\,G(F_v^+)\xrightarrow{\sim}{\mathrm{GL}}_n(F_{\tilde v})$ for $v\in S_p$. We let $G_v:=G(F^+_v)\cong {\mathrm{GL}}_n(F_{\tilde v})$ and $G_p:=\prod_{v\in S_p}G(F^+_v)\cong \prod_{v\in S_p}{\mathrm{GL}}_n(F_{\tilde v})$. We denote by $K_v$ (resp. $B_v$, resp. $\overline B_v$, resp. $T_v$) the inverse image of ${\mathrm{GL}}_n(\mathcal{O}_{F_{\tilde{v}}})$ (resp. of the subgroup of upper triangular matrices of ${\mathrm{GL}}_n(F_{\tilde v})$, resp. of the subgroup of lower triangular matrices of ${\mathrm{GL}}_n(F_{\tilde v})$, resp. of the subgroup of diagonal matrices of ${\mathrm{GL}}_n(F_{\tilde v})$) in $G_v$ under $i_{\tilde v}$ and we let $K_p:=\prod_{v\in S_p}K_v$ (resp. $B_p:=\prod_{v\in S_p}B_v$, resp. $\overline B_p:=\prod_{v\in S_p}\overline B_v$, resp. $T_p:=\prod_{v\in S_p}T_v$). We let $T_p^0:=T_p\cap K_p=\prod_{v\in S_p}(T_v\cap K_v)$.
We fix a finite extension $L$ of ${\mathbb{Q}}_p$ that is assumed to be large enough so that $|{\mathrm{Hom}}(F^+_v,L)|=[F^+_v:{\mathbb{Q}}_p]$ for $v\in S_p$. We let $\widehat{T}_{p,\rm reg}\subset \widehat{T}_{p,L}$ the open subspace of characters $\delta=(\delta_v)_{v\in S_p}=(\delta_{v,1},\dots,\delta_{v,n})_{v\in S_p}$ such that $\delta_{v,i}/\delta_{v,j}\in {\mathcal T}_{v,\rm reg}$ for all $v\in S_p$ and all $i\ne j$, where ${\mathcal T}_{v,\rm reg}$ is defined as ${\mathcal T}_{\rm reg}$ of §\[begin\] but with $F^+_v=F_{\tilde v}$ instead of $K$.
We fix a tame level $U^p=\prod_{v}U_v\subset G({\mathbb{A}}_{F^+}^{p\infty})$ where $U_v$ is a compact open subgroup of $G(F_v^+)$ and we denote by $\widehat S(U^p,L)$ the associated space of $p$-adic automorphic forms on $G({\mathbb{A}}_{F^+})$ of tame level $U^p$ with coefficients in $L$, that is, the $L$-vector space of continuous functions $f:G(F^+)\backslash G({\mathbb{A}}_{F^+}^{\infty})/U^p\longrightarrow L$. Since $G(F^+)\backslash G({\mathbb{A}}_{F^+}^{\infty})/U^p$ is compact, it is a $p$-adic Banach space (for the sup norm) endowed with the linear continuous unitary action of $G_p$ by right translation on functions. In particular a unit ball is given by the ${\mathcal{O}}_{L}$-submodule $\widehat S(U^p,{\mathcal{O}}_{L})$ of continuous functions $f:G(F^+)\backslash G({\mathbb{A}}_{F^+}^{\infty})/U^p\longrightarrow {\mathcal{O}}_{L}$ and the corresponding residual representation is the $k_{L}$-vector space $S(U^p,k_{L})$ of locally constant functions $f:G(F^+)\backslash G({\mathbb{A}}_{F^+}^{\infty})/U^p\longrightarrow k_{L}$ (a smooth admissible representation of $G_p$). Note that $S(U^p,k_L)=\varinjlim_{U_p}S(U^pU_p,k_L)$ where the inductive limit is taken over compact open subgroups $U_p$ of $G_p$ and where $S(U^pU_p,k_L)$ is the finite dimensional $k_L$-vector space of functions $f:G(F^+)\backslash G({\mathbb{A}}_{F^+}^{\infty})/U^pU_p\longrightarrow k_L$. We also denote by $\widehat S(U^p,L)^{\rm an}\subset \widehat S(U^p,L)$ the $L$-subvector space of locally ${\mathbb{Q}}_p$-analytic vectors for the action of $G_p$ ([@STdist §7]). This is a strongly admissible locally ${\mathbb{Q}}_p$-analytic representation of $G_p$.
We fix $S$ a finite set of finite places of $F^+$ that split in $F$ containing $S_p$ and the set of finite places $v\nmid p$ (that split in $F$) such that $U_v$ is not maximal. We consider the commutative spherical Hecke algebra: $$\mathbb{T}^S:=\varinjlim_I\big(\bigotimes_{v\in I}{\mathcal{O}}_L[U_v\backslash G(F^+_v)/U_v]\big),$$ the inductive limit being taken over finite sets $I$ of finite places of $F^+$ that split in $F$ such that $I\cap S=\emptyset$. This Hecke algebra naturally acts on the spaces $\widehat S(U^p,L)$, $\widehat S(U^p,L)^{\rm an}$, $\widehat S(U^p,{\mathcal{O}}_L)$, $S(U^p,k_L)$ and $S(U^pU_p,k_L)$ (for any compact open subgroup $U_p$). If $C$ is a field, $\theta:\mathbb{T}^S\rightarrow C$ a ring homomorphism and $\rho: {\mathcal{G}}_F\rightarrow {\mathrm{GL}}_n(C)$ a group homomorphism which is unramified at each finite place of $F$ above a place of $F^+$ which splits in $F$ and is not in $S$, we refer to [@BHS §2.4] for what it means for $\rho$ to be [*associated to*]{} $\theta$.
Though we could state a more general classicality conjecture, it is convenient for us to assume right now the following two extra hypothesis: $p>2$ and $G$ quasi-split at each finite place of $F^+$ (these assumptions will be needed anyway for our partial results, note however that they imply that $4$ divides $n[F^+:{\mathbb{Q}}]$ which rules out the case $n=2$, $F^+={\mathbb{Q}}$). We fix $\mathfrak{m}^S$ a maximal ideal of $\mathbb{T}^S$ of residue field $k_L$ (increasing $L$ if necessary) such that $\widehat S(U^p,L)_{\mathfrak{m}^S}\neq 0$, or equivalently $\widehat S(U^p,{\mathcal{O}}_L)_{\mathfrak{m}^S}\neq 0$, or $S(U^p,k_L)_{\mathfrak{m}^S}=\varinjlim_{U_p}S(U^pU_p,k_L)_{\mathfrak{m}^S}\neq 0$, or $S(U^pU_p,k_L)_{\mathfrak{m}^S}\ne 0$ for some $U_p$ (note that $\widehat S(U^p,L)_{\mathfrak{m}^S}$ is then a closed subspace of $\widehat S(U^p,L)$ preserved by $G_p$). We denote by ${\overline{\rho}}={\overline{\rho}}_{\mathfrak{m}^S}:{\mathcal{G}}_F\rightarrow {\mathrm{GL}}_n(k_L)$ the unique absolutely semi-simple Galois representation associated to $\mathfrak{m}^S$ (see [@Thorne Prop.6.6] and note that the running assumption $F/F^+$ unramified in [*loc. cit.*]{} is useless at this point). We assume $\mathfrak{m}^S$ [*non-Eisenstein*]{}, that is, ${\overline{\rho}}$ absolutely irreducible. Then it follows from [@Thorne Prop.6.7] (with the same remark as above) that the spaces $\widehat S(U^p,L)_{\mathfrak{m}^S}$, $\widehat S(U^p,{\mathcal{O}}_L)_{\mathfrak{m}^S}$ and $S(U^p,k_L)_{\mathfrak{m}^S}$ become modules over $R_{{\overline{\rho}},S}$, the complete local noetherian ${\mathcal{O}}_L$-algebra of residue field $k_L$ pro-representing the functor of deformations $\rho$ of ${\overline{\rho}}$ that are unramified outside $S$ and such that $\rho'\circ c\cong \rho\otimes \varepsilon^{n-1}$ (where $\rho'$ is the dual of $\rho$ and $c\in {\rm Gal}(F/F^+)$ is the complex conjugation).
The continuous dual $(\widehat S(U^p,L)_{\mathfrak{m}^S}^{\rm an})'$ of $\widehat S(U^p,L)_{\mathfrak{m}^S}^{\rm an}:=(\widehat S(U^p,L)_{\mathfrak{m}^S})^{\rm an}=(\widehat S(U^p,L)^{\rm an})_{\mathfrak{m}^S}$ becomes a module over the global sections $\Gamma({\mathfrak{X}}_{{\overline{\rho}},S},{\mathcal{O}}_{{\mathfrak{X}}_{{\overline{\rho}},S}})$ where ${\mathfrak{X}}_{{\overline{\rho}},S}:=({\mathrm{Spf}}\, R_{{\overline{\rho}},S})^{{\mathrm{rig}}}$ (see for instance [@BHS §3.1]). We denote by $Y(U^p,{\overline{\rho}})$ the *eigenvariety of tame level $U^p$* (over $L$) defined in [@Emerton] (see also [@BHS §4.1]) associated to $\widehat S(U^p,L)_{\mathfrak{m}^S}^{\rm an}$, that is, the support of the coherent ${\mathcal{O}}_{{\mathfrak{X}}_{{\overline{\rho}},S}\times \widehat T_{p,L}}$-module $(J_{B_p}(\widehat S(U^p,L)^{\rm an}_{\mathfrak{m}^S}))'$ on ${\mathfrak{X}}_{{\overline{\rho}},S}\times \widehat T_{p,L}$ where $J_{B_p}$ is Emerton’s locally ${\mathbb{Q}}_p$-analytic Jacquet functor with respect to the Borel $B_p$ and $(\cdot)'$ means the continuous dual. This is a reduced closed analytic subset of ${\mathfrak{X}}_{{\overline{\rho}},S}\times \widehat T_{p,L}$ of dimension $n[F^+:{\mathbb{Q}}]$ whose points are: $$\label{points}
\left\{x=(\rho,\delta)\in {\mathfrak{X}}_{{\overline{\rho}},S}\times \widehat T_{p,L}\ {\rm such\ that}\ {\mathrm{Hom}}_{T_p}\big(\delta,J_{B_p}(\widehat S(U^p,L)^{\rm an}_{\mathfrak{m}^S}[\mathfrak{p}_\rho]\otimes_{k(\mathfrak{p}_\rho)}k(x))\big)\ne 0\right\}$$ where $\mathfrak{p}_\rho\subset R_{{\overline{\rho}},S}$ denotes the prime ideal corresponding to the point $\rho\in {\mathfrak{X}}_{{\overline{\rho}},S}$ under the identification of the sets underlying ${\mathfrak{X}}_{{\overline{\rho}},S}=({\mathrm{Spf}}\, R_{{\overline{\rho}},S})^{{\mathrm{rig}}}$ and ${\rm Spm}\,R_{{\overline{\rho}},S}[1/p]$ ([@deJong Lem.7.1.9]) and where $k(\mathfrak{p}_\rho)$ is its residue field. We denote by $\omega:Y(U^p,{\overline{\rho}})\rightarrow \widehat T^0_{p,L}$ the composition $Y(U^p,{\overline{\rho}})\hookrightarrow {\mathfrak{X}}_{{\overline{\rho}},S}\times \widehat T_{p,L}\twoheadrightarrow \widehat T_{p,L} \twoheadrightarrow \widehat T^0_{p,L}$.
\[changeu\] [If ${U'}^p\subseteq {U}^p$ (and $S$ contains $S_p$ and the set of finite places $v\nmid p$ that split in $F$ such that $U'_v$ is not maximal), then a point $x=(\rho,\delta)$ of $Y(U^p,{\overline{\rho}})$ is also in $Y({U'}^p,{\overline{\rho}})$ since $\widehat S(U^p,L)^{\rm an}_{\mathfrak{m}^S}[\mathfrak{p}_\rho]\subseteq \widehat S({U'}^p,L)^{\rm an}_{\mathfrak{m}^S}[\mathfrak{p}_\rho]$ and $J_{B_p}$ is left exact.]{}
We let $X_{\rm tri}^\square({\overline{\rho}}_p)$ be the product rigid analytic variety $\prod_{v\in S_p}X_{\rm tri}^\square({\overline{\rho}}_{\tilde{v}})$ (over $L$) where ${\overline{\rho}}_{\tilde v}$ is the restriction of $\rho$ to the decomposition subgroup of ${\mathcal G}_F$ at $\tilde v$ (that we identify with ${\mathcal{G}}_{F_{\tilde v}}={\rm Gal}(\overline F_{\tilde v}/F_{\tilde v})$) and where $X_{\rm tri}^\square({\overline{\rho}}_{\tilde{v}})$ is as in §\[begin\]. This is a reduced closed analytic subvariety of $({\mathrm{Spf}}\, R^\square_{{\overline{\rho}}_p})^{{\mathrm{rig}}}\times\widehat{T}_{p,L}$ where $R^\square_{{\overline{\rho}}_p}:=\widehat{\bigotimes}_{v\in S_p}R^\square_{{\overline{\rho}}_{\tilde{v}}}$. Identifying $B_v$ (resp. $T_v$) with the upper triangular (resp. diagonal) matrices of ${\mathrm{GL}}_n(F_{\tilde v})$ via $i_{\tilde v}$, we let $\delta_{B_v}:=|\cdot|_{F_{\tilde v}}^{n-1}\otimes |\cdot|_{F_{\tilde v}}^{n-3}\otimes \cdots \otimes |\cdot|_{F_{\tilde v}}^{1-n}$ be the modulus character of $B_v$ and define as in [@BHS §2.3] an automorphism $\imath_v:\widehat T_v\buildrel\sim\over\rightarrow \widehat T_v$ by: $$\imath_v(\delta_1,\dots,\delta_n):=\delta_{B_v}\cdot(\delta_1,\dots,\delta_i\cdot(\varepsilon\circ{\mathrm{rec}}_{F_{\tilde v}})^{i-1},\dots,\delta_n\cdot(\varepsilon\circ{\mathrm{rec}}_{F_{\tilde v}})^{n-1})$$ (the twist by $\delta_{B_v}$ here ultimately comes from the same twist appearing in the definition of $J_{B_v}$). It then follows from [@BHS Th.4.2] that the morphism of rigid spaces: $$\begin{aligned}
({\mathrm{Spf}}\,R_{{\overline{\rho}},S})^{{\mathrm{rig}}}\times \widehat T_{p,L}&\longrightarrow &({\mathrm{Spf}}\, R^\square_{{\overline{\rho}}_p})^{{\mathrm{rig}}}\times\widehat{T}_{p,L}\\
\nonumber \big(\rho,(\delta_v)_{v\in S_p}\big)=\big(\rho,(\delta_{v,1},\dots,\delta_{v,n})_{v\in S_p}\big)&\longmapsto & \big((\rho|_{{\mathcal{G}}_{F_{\tilde v}}})_{v\in S_p},(\imath_v^{-1}(\delta_{v,1},\dots,\delta_{v,n}))_{v\in S_p}\big)\end{aligned}$$ induces a morphism of reduced rigid spaces over $L$: $$\label{eigenvartotrianguline}
Y(U^p,{\overline{\rho}})\longrightarrow X_{\rm tri}^\square({\overline{\rho}}_p)=\prod_{v\in S_p}X_{\rm tri}^\square({\overline{\rho}}_{\tilde v})$$ (note that (\[eigenvartotrianguline\]) is thus [*not*]{} compatible with the weight maps $\omega$ on both sides). We say that a point $x=(\rho,\delta)=(\rho,(\delta_v)_{v\in S_p})\in Y(U^p,{\overline{\rho}})$ is crystalline (resp. dominant, resp. strictly dominant, resp. crystalline strictly dominant very regular etc.) if for each $v\in S_p$ its image in $X_{\rm tri}^\square({\overline{\rho}}_{\tilde v})$ via (\[eigenvartotrianguline\]) is (see §\[begin\] and Definition \[veryreg\]). Due to the twist $\imath_v$ beware that $x=(\rho,\delta)\in Y(U^p,{\overline{\rho}})$ is strictly dominant if and only if $\delta\vert_{T_v\cap K_v}$ is (algebraic) dominant for each $v\in S_p$.
Let $\delta\in \widehat T_{p,L}$ be any locally algebraic character. Then we can write $\delta=\delta_{\lambda}\delta_{\rm sm}$ in $\widehat T_{p,L}$ where $\lambda=(\lambda_v)_{v\in S_p}\in \prod_{v\in S_p}({\mathbb{Z}}^n)^{{\mathrm{Hom}}(F_{\tilde v},L)}$, $\delta_\lambda:=\prod_{v\in S_p}\delta_{\lambda_v}$ (see §\[begin\] for $\delta_{\lambda_v}\in \widehat T_{v,L}$) and $\delta_{\rm sm}$ is a smooth character of $T_{p}$ with values in $k(\delta)$ (the residue field of the point $\delta\in \widehat T_{p,L}$). Using the theory of Orlik and Strauch ([@OrlikStrauch]), we define as in [@BHS (3.7)] the following strongly admissible locally ${\mathbb{Q}}_p$-analytic representation of $G_p$ over $k(\delta)$: $${\mathcal{F}}_{\overline B_p}^{G_p}(\delta):={\mathcal{F}}_{\overline B_p}^{G_p}\big((U(\mathfrak{g}_L)\otimes_{U({\overline {\mathfrak b}}_L)}(-\lambda))^\vee, \delta_{\rm sm}\delta_{B_p}^{-1}\big)$$ where $\delta_{B_p}:=\prod_{v\in S_p}\delta_{B_v}$ and where we refer to [@BHS §3.5] for the details and notation. Recall that ${\mathcal{F}}_{\overline B_p}^{G_p}(\delta)$ has the same constituents as the locally ${\mathbb{Q}}_p$-analytic principal series $({\rm Ind}_{\overline B_p}^{G_p}\delta_{\lambda}\delta_{\rm sm}\delta_{B_p}^{-1})^{\rm an}=({\rm Ind}_{\overline B_p}^{G_p}\delta\delta_{B_p}^{-1})^{\rm an}$ but in the “reverse order” (at least generically). If $\lambda$ is dominant (that is $\lambda_v$ is dominant for each $v$ in the sense of §\[begin\]), we denote by ${\rm LA}(\delta)$ the locally algebraic representation: $$\begin{gathered}
\label{ladelta}
{\rm LA}(\delta):={\mathcal{F}}_{\overline B_p}^{G_p}(L(\lambda)',\delta_{\rm sm}\delta_{B_p}^{-1})={\mathcal{F}}_{G_p}^{G_p}\big(L(\lambda)',({\rm Ind}_{\overline B_p}^{G_p}\delta_{\rm sm}\delta_{B_p}^{-1})^{\rm sm}\big)\\
=L(\lambda)\otimes_L \big({\rm Ind}_{\overline B_p}^{G_p}\delta_{\rm sm}\delta_{B_p}^{-1}\big)^{\rm sm}\end{gathered}$$ where $L(\lambda)$ is the simple $U(\mathfrak{g}_L)$-module over $L$ of highest weight $\lambda$ relative to the Lie algebra of $B_p$ (which is finite dimensional over $L$ since $\lambda$ is dominant) that we see as an irreducible algebraic representation of $G_p$ over $L$, where $L(\lambda)'$ is its dual, and where $(-)^{\rm sm}$ denotes the smooth Borel induction over $k(\delta)$ (the second equality in (\[ladelta\]) following from [@OrlikStrauch Prop.4.9(b)]). Arguing as in [@OrlikStrauch §6] (note that $L(\lambda)'$ is the unique irreducible subobject of $(U(\mathfrak{g}_L)\otimes_{U({\overline {\mathfrak b}}_L)}(-\lambda))^\vee$), it easily follows from [@OrlikStrauch Th.5.8] (see also [@BreuilAnalytiqueI Th.2.3(iii)]) and [@HumBGG §5.1] that ${\rm LA}(\delta)$ is identified with the maximal locally ${\mathbb{Q}}_p$-algebraic quotient of ${\mathcal{F}}_{\overline B_p}^{G_p}(\delta)$ (or the maximal locally algebraic subobject of $({\rm Ind}_{\overline B_p}^{G_p}\delta\delta_{B_p}^{-1})^{\rm an}$).
It follows from (\[points\]) together with [@BreuilAnalytiqueII Th.4.3] that a point $x=(\rho,\delta)\in {\mathfrak{X}}_{{\overline{\rho}},S}\times\widehat T_{p,L}$ lies in $Y(U^p,{\overline{\rho}})$ if and only if: $$\label{adj}
{\mathrm{Hom}}_{T_p}\big(\delta,J_{B_p}(\widehat S(U^p,L)^{\rm an}_{\mathfrak{m}^S}[\mathfrak{p}_\rho]\otimes_{k(\mathfrak{p}_\rho)}k(x))\big)\cong{\mathrm{Hom}}_{G_p}\big({\mathcal{F}}_{\overline B_p}^{G_p}(\delta),\widehat S(U^p,L)_{\mathfrak{m}^S}^{\rm an}[\mathfrak{p}_{\rho}]\otimes_{k(\mathfrak{p}_\rho)}k(x)\big)\ne 0.$$
\[defclass\] A point $x=(\rho,\delta)\in Y(U^p,{\overline{\rho}})$ is called *classical* if there exists a nonzero continuous $G_p$-equivariant morphism: $${\mathcal{F}}_{\overline B_p}^{G_p}(\delta)\longrightarrow \widehat S(U^p,L)_{\mathfrak{m}^S}^{\rm an}[\mathfrak{p}_{\rho}]\otimes_{k(\mathfrak{p}_\rho)}k(x)$$ that factors through the locally ${\mathbb{Q}}_p$-algebraic quotient ${\rm LA}(\delta)$ of ${\mathcal{F}}_{\overline B_p}^{G_p}(\delta)$ (equivalently $(\rho,\delta)$ is classical if ${\mathrm{Hom}}_{G_p}({\rm LA}(\delta),\widehat S(U^p,L)_{\mathfrak{m}^S}[\mathfrak{p}_{\rho}]\otimes_{k(\mathfrak{p}_\rho)}k(x))\ne 0$).
[(i) This definition is [@BHS Def.3.14].\
(ii) It seems reasonnable to expect that if $x=(\rho,\delta)\in Y(U^p,{\overline{\rho}})$ is classical, then in fact [*any*]{} continuous $G_p$-equivariant morphism ${\mathcal{F}}_{\overline B_p}^{G_p}(\delta)\longrightarrow \widehat S(U^p,L)_{\mathfrak{m}^S}^{\rm an}[\mathfrak{p}_{\rho}]\otimes_{k(\mathfrak{p}_\rho)}k(x)$ factors through ${\rm LA}(\delta)$. See the last statement of Corollary \[mainclassic\] below for a partial result in that direction.]{}
We fix an algebraic closure $\overline{\mathbb{Q}}_p$ of $L$ and embeddings $j_\infty:\overline{\mathbb{Q}}\hookrightarrow {\mathbb{C}}$, $j_p:\overline{\mathbb{Q}}\hookrightarrow \overline{\mathbb{Q}}_p$. Recall that, if $\pi=\pi_{\infty}\otimes_{{\mathbb{C}}}\pi_f$ is an automorphic representation of $G({\mathbb{A}}_{F^+})$ over ${\mathbb{C}}$ where $\pi_{\infty}$ (resp. $\pi_f$) is a representation of $G(F^+\otimes_{\mathbb{Q}}\mathbb{R})$ (resp. of $G({\mathbb{A}}_{F^+}^\infty)$), then due to the compactness of $G(F^+\otimes_{\mathbb{Q}}\mathbb{R})$, we have that $\pi_\infty$ is a finite dimensional irreducible representation that comes from an algebraic representation of ${\mathrm{Res}}_{F^+/{\mathbb{Q}}}G$ over ${\mathbb{C}}$ (argue as in [@BelChe §§6.2.3,6.7]). Moreover, arguing again as in [*loc. cit.*]{}, $\pi_\infty$ (resp. $\pi_f$) has a ${\overline{\mathbb{Q}}}$-structure given by $j_\infty$ which is stable under the action of $({\mathrm{Res}}_{F^+/{\mathbb{Q}}}G)({\overline{\mathbb{Q}}})$ (resp. of $G({\mathbb{A}}_{F^+}^\infty)$). Hence the scalar extension of the ${\overline{\mathbb{Q}}}$-structure of $\pi_\infty$ to $\overline {\mathbb{Q}}_p$ via $j_p$ is endowed with an action of $({\mathrm{Res}}_{F^+/{\mathbb{Q}}}G)(\overline {\mathbb{Q}}_p)$, thus in particular of $({\mathrm{Res}}_{F^+/{\mathbb{Q}}}G)({\mathbb{Q}}_p)=G(F^+\otimes_{{\mathbb{Q}}}{\mathbb{Q}}_p)=G_p$. This latter representation of $G_p$ is easily seen to be defined over $L$ and of the form $L(\lambda)$ for a dominant $\lambda$ as above. We say that $\pi_\infty$ [*is of weight $\lambda$*]{} if the resulting representation of $G_p$ is $L(\lambda)$.
For the sake of completeness, we recall the following proposition showing that Definition \[defclass\] coincides with the usual classicality definition.
\[compaclas\] A strictly dominant point $x=(\rho,\delta)\in Y(U^p,{\overline{\rho}})$, that is such that $\omega(x)=\delta_{\lambda}$ for some dominant $\lambda\in \prod_{v\in S_p}({\mathbb{Z}}^n)^{{\mathrm{Hom}}(F_{\tilde v},L)}$, is classical if and only if there exists an automorphic representation $\pi=\pi_{\infty}\otimes_{{\mathbb{C}}}\pi_f^p\otimes_{{\mathbb{C}}}\pi_p$ of $G({\mathbb{A}}_{F^+})$ over ${\mathbb{C}}$ such that the following conditions hold:
1. the $G(F^+\otimes_{\mathbb{Q}}\mathbb{R})$-representation $\pi_\infty$ is of weight $\lambda$ in the above sense;\
2. the ${\mathcal{G}}_F$-representation $\rho$ is the Galois representation associated to $\pi$ (see proof below);\
3. the invariant subspace $(\pi_f^p)^{U^p}$ is nonzero;\
4. the $G_p$-representation $\pi_p$ is a quotient of $\big({\rm Ind}_{\overline B_p}^{G_p} \delta\delta_{\lambda}^{-1}\delta_{B_p}^{-1}\big)^{\rm sm}\otimes_{k(\delta)}\overline {\mathbb{Q}}_p$.
If moreover $F$ is unramified over $F^+$ and $U_v$ is hyperspecial when $v$ is inert in $F$, then such a $\pi$ is unique and appears with multiplicity $1$ in $L^2(G(F^+)\backslash G({\mathbb{A}}_{F^+}),{\mathbb{C}})$.
Let $W$ be any linear representation of $G_p$ over an $L$-vector space and $U$ any compact open subgroup of $G({\mathbb{A}}_{F^+}^{\infty})$, we define $S(U,W)$ to be the $L$-vector space of functions $f:G(F^+)\backslash G({\mathbb{A}}_{F^+}^{\infty})\longrightarrow W$ such that $f(gu)=u_p^{-1}(f(g))$ for $g\in G({\mathbb{A}}_{F^+}^{\infty})$ and $u\in U$, where $u_p$ is the projection of $u$ in $G_p$. Fixing $U^p$ as previously, we define $S(U^p,W):=\varinjlim_{U_p}S(U^pU_p,W)$ (inductive limit taken over compact open subgroups $U_p$ of $G_p$) endowed with the linear left action of $G_p$ given by $(h_p\cdot f)(g):=h_p(f(gh_p))$ ($h_p\in G_p$, $g\in G({\mathbb{A}}_{F^+}^{\infty})$) where the second $h_p$ is seen in $G({\mathbb{A}}_{F^+}^{\infty})$ in the obvious way. Note that $\mathbb{T}^S$ also naturally acts on $S(U^p,W)$ (the representation $W$ here playing no role since this action is “outside $p$”). Then it follows from [@EGH §7.1.4] that there is an isomorphism of smooth representations of $G_p$ over $\overline{\mathbb{Q}}_p$: $$\label{class}
S(U^p,L(\lambda)')\otimes_{L}\overline {\mathbb{Q}}_p\cong \bigoplus_{\pi}\big[\big((\pi_f^p)^{U^p}\otimes_{\overline {\mathbb{Q}}}\pi_p\big)\otimes_{\overline{\mathbb{Q}},j_p}\overline {\mathbb{Q}}_p\big]^{\oplus m(\pi)}$$ where the direct sum is over the automorphic representations $\pi=\pi_\infty\otimes_{{\mathbb{C}}}\pi_p$ of $G({\mathbb{A}}_{F^+})$ such that $\pi_\infty$ is of weight $\lambda$ and $(\pi_f^p)^{U^p}\ne 0$ (we take the ${\overline{\mathbb{Q}}}$-structures) and where $m(\pi)$ is the multiplicity of $\pi$ in $L^2(G(F^+)\backslash G({\mathbb{A}}_{F^+}),{\mathbb{C}})$. We then say that a point $\rho\in {\mathfrak{X}}_{{\overline{\rho}},S}$ is the Galois representation associated to $\pi$ (with $\pi_\infty$ of weight $\lambda$) if we have: $$\big[\big((\pi_f^p)^{U^p}\otimes_{\overline {\mathbb{Q}}}\pi_p\big)\otimes_{\overline{\mathbb{Q}},j_p}\overline {\mathbb{Q}}_p\big]^{\oplus m(\pi)}\subseteq S(U^p,L(\lambda)')_{\mathfrak{m}^S}[\mathfrak{p}_\rho]\otimes_{k(\mathfrak{p}_\rho)}\overline {\mathbb{Q}}_p$$ where $\mathfrak{p}_\rho$ is as in (\[points\]) (and $R_{{\overline{\rho}},S}$ acts on $S(U^p,L(\lambda)')_{\mathfrak{m}^S}$ again using [@Thorne Prop.6.7]). Note that $S(U^p,L(\lambda)')_{\mathfrak{m}^S}[\mathfrak{p}_\rho]\otimes_{k(\mathfrak{p}_\rho)}\overline {\mathbb{Q}}_p\ne 0$ (equivalently $S(U^p,L(\lambda)')_{\mathfrak{m}^S}[\mathfrak{p}_\rho]\ne 0$) if and only if there exists an automorphic representation $\pi$ such that $\pi_\infty$ is of weight $\lambda$, $(\pi_f^p)^{U^p}\ne 0$ and $\rho$ is the Galois representation associated to $\pi$.
Let $\widehat S(U^p,L)^{\lambda-\rm la}\subset \widehat S(U^p,L)^{\rm an}$ be the closed $G_p$-subrepresentation of locally $L(\lambda)$-algebraic vectors, that is the $L$-subvector space of $\widehat S(U^p,L)^{\rm an}$ (or equivalently of $\widehat S(U^p,L)$) of vectors $v$ such that there exists a compact open subgroup $U_p$ of $G_p$ such that the $U_p$-subrepresentation generated by $v$ in $\widehat S(U^p,L)\vert_{U_p}$ is isomorphic to $(L(\lambda)\vert_{U_p})^{\oplus d}$ for some positive integer $d$. Note that the subspace $\widehat S(U^p,L)^{\lambda-\rm la}$ is preserved under the action of $\mathbb{T}^S$ (since the latter commutes with $G_p$). Then it follows from [@Emerton Prop.3.2.4] and its proof that there is an isomorphism of locally ${\mathbb{Q}}_p$-algebraic representations of $G_p$ over $L$ which is $\mathbb{T}^S$-equivariant (with the action of $\mathbb{T}^S$ on the right hand side given by its action on $S(U^p,L(\lambda)')$): $$\widehat S(U^p,L)^{\lambda-\rm la}\cong L(\lambda)\otimes_L S(U^p,L(\lambda)').$$ We then deduce a $G_p$-equivariant isomorphism of $R_{{\overline{\rho}},S}$-modules: $$\label{lambdaalgm}
\widehat S(U^p,L)^{\lambda-\rm la}_{\mathfrak{m}^S}\cong L(\lambda)\otimes_L S(U^p,L(\lambda)')_{\mathfrak{m}^S}$$ where $\widehat S(U^p,L)^{\lambda-\rm la}_{\mathfrak{m}^S}:=(\widehat S(U^p,L)^{\lambda-\rm la})_{\mathfrak{m}^S}=(\widehat S(U^p,L)_{\mathfrak{m}^S})^{\lambda-\rm la}$.
Now let $x=(\rho,\delta)\in Y(U^p,{\overline{\rho}})$ with $\omega(x)=\delta_{\lambda}$ for $\lambda$ dominant and define $\mathfrak{p}_\rho$ as in (\[points\]). From Definition \[defclass\] and the definition of $\widehat S(U^p,L)^{\lambda-\rm la}$, we get that the point $x$ is classical if and only if there exists a nonzero $G_p$-equivariant morphism: $$L(\lambda)\otimes_L \big({\rm Ind}_{\overline B_p}^{G_p} \delta\delta_{\lambda}^{-1}\delta_{B_p}^{-1}\big)^{\rm sm} \longrightarrow \widehat S(U^p,L)_{\mathfrak{m}^S}^{\lambda-\rm la}[\mathfrak{p}_\rho]\otimes_{k(\mathfrak{p}_\rho)}k(x)$$ if and only if by (\[lambdaalgm\]) there exists a nonzero $G_p$-equivariant morphism: $$\big({\rm Ind}_{\overline B_p}^{G_p} \delta\delta_{\lambda}^{-1}\delta_{B_p}^{-1}\big)^{\rm sm}\longrightarrow S(U^p,L(\lambda)')_{\mathfrak{m}^S}[\mathfrak{p}_\rho]\otimes_{k(\mathfrak{p}_\rho)}k(x)$$ if and only if by (\[class\]) there exists an automorphic representation $\pi=\pi_\infty\otimes_{{\mathbb{C}}}\pi_f^p\otimes_{{\mathbb{C}}}\pi_p$ of $G({\mathbb{A}}_{F^+})$ such that $\pi_\infty$ is of weight $\lambda$, $(\pi_f^p)^{U^p}\ne 0$, $\rho$ is the Galois representation associated to $\pi$ and $\pi_p$ is a quotient of $({\rm Ind}_{\overline B_p}^{G_p} \delta\delta_{\lambda}^{-1}\delta_{B_p}^{-1})^{\rm sm}\otimes_{k(\delta)}\overline {\mathbb{Q}}_p$.
Now let us prove the last assertion. According to [@Labesse Cor.5.3], there exists an isobaric representation $\Pi=\Pi_1\boxplus\cdots\boxplus\Pi_r$ where $n=m_1+\cdots+m_r$ and $\Pi_i$ nonzero automorphic representations of $\mathrm{GL}_{m_i}(\mathbb{A}_F)$ occuring in the discrete spectrum such that $\Pi$ is a weak base change of $\pi$ in the sense of [@Labesse §4.10]. Since $\overline{\rho}$, hence $\rho$, are absolutely irreducible, we have $r=1$ and $\Pi=\Pi_1$ cuspidal. The equality $m(\pi)=1$ then follows from [@Labesse Th.5.4] (which uses the extra assumption $F/F^+$ unramified). The uniqueness of $\pi$ is a consequence of the strong base change theorem [@Labesse Th.5.9] together with the fact that $\pi_v$ is unramified at finite places $v$ of $F^+$ which are inert in $F$ (which uses the extra assumption $U_v$ hyperspecial for $v$ inert) and the fact that the $L$-packets at finite places of $F^+$ which are split in $F$ are singletons.
[With the notation of Proposition \[compaclas\], write $\delta\delta_{\lambda}^{-1}=(\delta_{{\rm sm},v,1},\dots,\delta_{{\rm sm},v,n})_{v\in S_p}$, if moreover $\delta_{{\rm sm},v,i}/\delta_{{\rm sm},v,j}\notin \{1,\vert\cdot\vert_{F_{\tilde v}}^{-2}\}$ for $1\leq i\ne j\leq n$ and $v\in S_p$, then we see from (iv) of Proposition \[compaclas\] that $\pi_p\cong ({\rm Ind}_{B_p}^{G_p} \delta\delta_{\lambda}^{-1})^{\rm sm}\otimes_{k(\delta)}\overline {\mathbb{Q}}_p$.]{}
We then have the following conjecture, which by Proposition \[compaclas\] is essentially a consequence of the Fontaine-Mazur conjecture and the Langlands philosophy, and which is the natural generalization in the context of definite unitary groups of the main result of [@Kisinoverconvergent] for ${\mathrm{GL}}_2/{\mathbb{Q}}$ (in the crystalline case).
\[classiconj\] Let $x=(\rho,\delta)\in Y(U^p,{\overline{\rho}})$ be a crystalline strictly dominant point. Then $x$ is classical.
\[extra\] [We didn’t seek to state the most general classicality conjecture. Obviously, the assumptions that $p>2$ and $G$ is quasi-split at each finite place of $F^+$ shouldn’t be crucial, and one could replace crystalline by de Rham.]{}
Proof of the main classicality result {#firstclassical}
-------------------------------------
We prove a criterion for classicality (Theorem \[classicalitycrit\]) in terms of the patched eigenvariety of [@BHS], which itself builds on the construction in [@CEGGPS] of a “big” patching module $M_\infty$. We use it to prove our main classicality result (Corollary \[mainclassic\]).
We keep the notation of §\[classic\] and make the following extra assumptions (which are required for the construction of $M_\infty$): $F$ is unramified over $F^+$, $U_v$ is hyperspecial if $v$ is inert in $F$, ${\overline{\rho}}(\mathcal{G}_{F(\zeta_p)})$ is adequate in the sense of [@Thorne Def.2.3] and $\zeta_p\notin \overline F^{\ker({\overline{\rho}}\otimes {\overline{\rho}}')}$. For instance if $p>2n+1$ and ${\overline{\rho}}\vert_{\mathcal{G}_{F(\zeta_p)}}$ is (still) absolutely irreducible, then ${\overline{\rho}}(\mathcal{G}_{F(\zeta_p)})$ is automatically adequate ([@GHTT Th.9]). We first briefly recall some notation, definitions and statements and refer to [@BHS §3.2] for more details on what follows.
We let $R_{{\overline{\rho}}_{\tilde v}}^{\overline\square}$ be the maximal reduced and ${\mathbb{Z}}_p$-flat quotient of the framed local deformation ring $R_{{\overline{\rho}}_{\tilde v}}^\square$ and set: $$R^{\rm loc}:=\widehat{\bigotimes}_{v\in S} R_{{\overline{\rho}}_{\tilde v}}^{\overline\square},\ \ R_{{\overline{\rho}}^p}:=\widehat{\bigotimes}_{v\in S\backslash S_p}R_{{\overline{\rho}}_{\tilde v}}^{\overline\square},\ \ R_{{\overline{\rho}}_p}:=\widehat{\bigotimes}_{v\in S_p}R_{{\overline{\rho}}_{\tilde v}}^{\overline\square},\ \ R_\infty:=R^{\rm loc}{{\mathchoice{\mbox{\rm [\hspace{-0.15em}[}}
{\mbox{\rm [\hspace{-0.15em}[}}
{\mbox{\scriptsize\rm [\hspace{-0.15em}[}}
{\mbox{\tiny\rm [\hspace{-0.15em}[}}}}x_1\dots,x_g{{\mathchoice{\mbox{\rm ]\hspace{-0.15em}]}}
{\mbox{\rm ]\hspace{-0.15em}]}}
{\mbox{\scriptsize\rm ]\hspace{-0.15em}]}}
{\mbox{\tiny\rm ]\hspace{-0.15em}]}}}}$$ where $g\geq 1$ is some integer which will be fixed below. We let ${\mathfrak{X}}_{{\overline{\rho}}^p}:=({\mathrm{Spf}}\, R_{{\overline{\rho}}^p})^{{\mathrm{rig}}}$, ${\mathfrak{X}}_{{\overline{\rho}}_p}:=({\mathrm{Spf}}R_{{\overline{\rho}}_p})^{{\mathrm{rig}}}$ and ${\mathfrak{X}}_\infty:=({\mathrm{Spf}}\, R_\infty)^{{\mathrm{rig}}}$ so that: $$\label{product}
{\mathfrak{X}}_\infty={\mathfrak{X}}_{{\overline{\rho}}^p}\times {\mathfrak{X}}_{{\overline{\rho}}_p}\times {\mathbb{U}}^g$$ where ${\mathbb{U}}:=({\mathrm{Spf}}\,{\mathcal{O}}_L{{\mathchoice{\mbox{\rm [\hspace{-0.15em}[}}
{\mbox{\rm [\hspace{-0.15em}[}}
{\mbox{\scriptsize\rm [\hspace{-0.15em}[}}
{\mbox{\tiny\rm [\hspace{-0.15em}[}}}}y {{\mathchoice{\mbox{\rm ]\hspace{-0.15em}]}}
{\mbox{\rm ]\hspace{-0.15em}]}}
{\mbox{\scriptsize\rm ]\hspace{-0.15em}]}}
{\mbox{\tiny\rm ]\hspace{-0.15em}]}}}})^{{\mathrm{rig}}}$ is the open unit disc over $L$. We also define $S_\infty:={\mathcal{O}}_L{{\mathchoice{\mbox{\rm [\hspace{-0.15em}[}}
{\mbox{\rm [\hspace{-0.15em}[}}
{\mbox{\scriptsize\rm [\hspace{-0.15em}[}}
{\mbox{\tiny\rm [\hspace{-0.15em}[}}}}y_1,\dots,y_t{{\mathchoice{\mbox{\rm ]\hspace{-0.15em}]}}
{\mbox{\rm ]\hspace{-0.15em}]}}
{\mbox{\scriptsize\rm ]\hspace{-0.15em}]}}
{\mbox{\tiny\rm ]\hspace{-0.15em}]}}}}$ where $t:=g+[F^+:{\mathbb{Q}}]\tfrac{n(n-1)}{2}+|S|n^2$ and $\mathfrak{a}:=(y_1,\dots,y_t)$ (an ideal of $S_\infty$).
Thanks to Remark \[changeu\] and Proposition \[compaclas\] we can (and do) assume that the tame level $U^p$ is small enough so that we have: $$\label{net}
G(F)\cap (hU^pK_ph^{-1})=\{1\}\ \ {\rm for\ all\ }h\in G({\mathbb{A}}^\infty_{F^+})$$ (indeed, let $w\nmid p$ be a finite place of $F^+$ that splits in $F$ such that $U_w$ is maximal, replace $U^p$ by $U'^p:=U'_w\prod_{v\ne w}U_v$ where $U'_w$ is small enough so that $U'^p$ satisfies (\[net\]), and use Proposition \[compaclas\] and local-global compatibility at $w$ to deduce classicality in level $U^p$ from classicality in level $U'^p$). Then there is a quotient $R_{{\overline{\rho}},S}\twoheadrightarrow R_{{\overline{\rho}}, {\mathcal{S}}}$ such that the action of $R_{{\overline{\rho}},S}$ on $\widehat S(U^p,L)_{\mathfrak{m}^S}$ factors through $R_{{\overline{\rho}},{\mathcal{S}}}$, an integer $g\geq 1$ and:
1. a continuous $R_\infty$-admissible (see [@BHS Def.3.1]) unitary representation $\Pi_\infty$ of $G_p$ over $L$ together with a $G_p$-stable and $R_\infty$-stable unit ball $\Pi_\infty^\circ\subset \Pi_\infty$;\
2. a morphism of local ${\mathcal{O}}_L$-algebras $S_\infty\rightarrow R_\infty$ such that $M_\infty:={\mathrm{Hom}}_{{\mathcal{O}}_L}(\Pi_\infty^\circ,{\mathcal{O}}_L)$ is finite projective as an $S_\infty{{\mathchoice{\mbox{\rm [\hspace{-0.15em}[}}
{\mbox{\rm [\hspace{-0.15em}[}}
{\mbox{\scriptsize\rm [\hspace{-0.15em}[}}
{\mbox{\tiny\rm [\hspace{-0.15em}[}}}}K_p{{\mathchoice{\mbox{\rm ]\hspace{-0.15em}]}}
{\mbox{\rm ]\hspace{-0.15em}]}}
{\mbox{\scriptsize\rm ]\hspace{-0.15em}]}}
{\mbox{\tiny\rm ]\hspace{-0.15em}]}}}}$-module;\
3. compatible isomorphisms $R_\infty/\mathfrak{a}R_\infty\cong R_{{\overline{\rho}},{\mathcal{S}}}$ and $\Pi_\infty[\mathfrak{a}]\cong \widehat S(U^p,L)_{\mathfrak{m}^S}$ where the latter is $G_p$-equivariant.
We then define the *patched eigenvariety* $X_p({\overline{\rho}})$ as the support of the coherent ${\mathcal{O}}_{{\mathfrak{X}}_\infty\times \widehat T_{p,L}}$-module $\mathcal{M}_\infty=(J_{B_p}(\Pi_\infty^{R_\infty-{\rm an}}))'$ on ${\mathfrak{X}}_\infty\times \widehat T_{p,L}$ (see [@BHS Def.3.2] for $\Pi_\infty^{R_\infty-{\rm an}}$; strictly speaking $(J_{B_p}(\Pi_\infty^{R_\infty-{\rm an}}))'$ is the global sections of the sheaf $\mathcal{M}_\infty$). This is a reduced closed analytic subset of ${\mathfrak{X}}_\infty\times \widehat T_{p,L}$ ([@BHS Cor.3.19]) whose points are ([@BHS Prop3.7]): $$\label{pointpatching}
\left\{x=(y,\delta)\in {\mathfrak{X}}_\infty\times \widehat T_{p,L}\ {\rm such\ that}\ {\mathrm{Hom}}_{T_p}\big(\delta,J_{B_p}(\Pi_\infty^{R_\infty-\rm an}[\mathfrak{p}_y]\otimes_{k(\mathfrak{p}_y)}k(x))\big)\ne 0\right\}$$ where $\mathfrak{p}_y\subset R_\infty$ denotes the prime ideal corresponding to the point $y\in {\mathfrak{X}}_\infty$ (under the identification of the sets underlying ${\mathfrak{X}}_\infty$ and ${\rm Spm}\,R_\infty[1/p]$) and ${k(\mathfrak{p}_y)}$ is the residue field of $\mathfrak{p}_y$. It follows from the proof of [@BHS Th.4.2] that we can recover the eigenvariety $Y(U^p,{\overline{\rho}})$ as the reduced Zariski-closed subspace of $X_p({\overline{\rho}})$ underlying the vanishing locus of the ideal $\mathfrak{a}\Gamma({\mathfrak{X}}_\infty,{\mathcal{O}}_{{\mathfrak{X}}_\infty})$.
\[CM\] The coherent sheaf $\mathcal{M}_\infty$ is Cohen-Macaulay over $X_p(\overline{\rho})$.
From the proof of [@BHS Prop.3.10] (to which we refer the reader for more details) we deduce that there exists an admissible affinoid covering $(U_i)_i$ of $X_p(\overline{\rho})$ such that $\Gamma(U_i,\mathcal{M}_\infty)$ is a finite projective module over a ring $\mathcal{O}_{\mathcal{W}_\infty}(W_i)$ whose action on $\Gamma(U_i,\mathcal{M}_\infty)$ factors through a ring homomorphism $\mathcal{O}_{\mathcal{W}_\infty}(W_i)\rightarrow\mathcal{O}_{X_p(\overline{\rho})}(U_i)$. Consequently we can deduce from [@EGAIV1 Prop.16.5.3] that $\Gamma(U_i,\mathcal{M}_\infty)$ is a Cohen-Macaulay $\mathcal{O}_{X_p(\overline{\rho})}(U_i)$-module.
It follows from [@BHS Th.3.20] that the isomorphism of rigid spaces: $$\begin{aligned}
{\mathfrak{X}}_\infty\times \widehat T_{p,L}&\buildrel\sim\over\longrightarrow &{\mathfrak{X}}_\infty\times\widehat{T}_{p,L}\\
\nonumber \big(x,(\delta_v)_{v\in S_p}\big)=\big(x,(\delta_{v,1},\dots,\delta_{v,n})_{v\in S_p}\big)&\longmapsto & \big(x,(\imath_v^{-1}(\delta_{v,1},\dots,\delta_{v,n}))_{v\in S_p}\big)\end{aligned}$$ induces via (\[product\]) a morphism of reduced rigid spaces over $L$: $$\label{patchedeigenvartoXtri}
X_p({\overline{\rho}})\longrightarrow {\mathfrak{X}}_{{\overline{\rho}}^p}\times X_{\rm tri}^\square({\overline{\rho}}_p)\times {\mathbb{U}}^g$$ which identifies the source with a union of irreducible components of the target. Note that the composition: $$Y(U^p,{\overline{\rho}})\hookrightarrow X_p({\overline{\rho}})\buildrel{(\ref{patchedeigenvartoXtri})}\over\longrightarrow{\mathfrak{X}}_{{\overline{\rho}}^p}\times X_{\rm tri}^\square({\overline{\rho}}_p)\times {\mathbb{U}}^g\twoheadrightarrow X_{\rm tri}^\square({\overline{\rho}}_p)$$ is the map (\[eigenvartotrianguline\]). An irreducible component of the right hand side of (\[patchedeigenvartoXtri\]) is of the form ${\mathfrak{X}}^p\times Z\times {\mathbb{U}}^g$ where ${\mathfrak{X}}^p$ (resp. $Z$) is an irreducible component of ${\mathfrak{X}}^p$ (resp. $X_{\rm tri}^\square({\overline{\rho}}_p)$). Given an irreducible component ${\mathfrak{X}}^p\subseteq {\mathfrak{X}}_{{\overline{\rho}}^p}$, we denote by $X_{\rm tri}^{{\mathfrak{X}}^p \rm-aut}({\overline{\rho}}_p)\subseteq X_{\rm tri}^\square({\overline{\rho}}_p)$ the union (possibly empty) of those irreducible components $Z\subseteq X_{\rm tri}^\square({\overline{\rho}}_p)$ such that ${\mathfrak{X}}^p\times Z\times {\mathbb{U}}^g$ is an irreducible component of $X_p({\overline{\rho}})$ via (\[patchedeigenvartoXtri\]). The morphism (\[patchedeigenvartoXtri\]) thus induces an isomorphism: $$\label{union}
X_p({\overline{\rho}})\buildrel\sim\over\longrightarrow \bigcup_{{\mathfrak{X}}^p} \big({\mathfrak{X}}^p\times X_{\rm tri}^{{\mathfrak{X}}^p \rm-aut}({\overline{\rho}}_p)\times {\mathbb{U}}^g\big)$$ the union (inside ${\mathfrak{X}}_{{\overline{\rho}}^p}\times X_{\rm tri}^\square({\overline{\rho}}_p)\times {\mathbb{U}}^g$) being over the irreducible components ${\mathfrak{X}}^p$ of ${\mathfrak{X}}_{{\overline{\rho}}^p}$.
We now state and prove the main result of this section, which gives a criterion for classicality on $Y(U^p,{\overline{\rho}})$. Recall that, given a crystalline strictly dominant point $x_v=(r_v,\delta_v)\in X_{\rm tri}^\square({\overline{\rho}}_{\tilde v})$ such that the geometric Frobenius eigenvalues on ${\rm WD}(r_v)$ are pairwise distinct and $V_v\subseteq X_{\rm tri}^\square({\overline{\rho}}_{\tilde v})$ a sufficiently small open neighbourhood of $x_v$, we have constructed in Corollary $\ref{defnZtri(x)}$ an irreducible component $Z_{{\rm tri},V_v}(x_v)$ of $V_v$ containing $x_v$.
\[classicalitycrit\] Let $x=(\rho,\delta)\in Y(U^p,{\overline{\rho}})$ be a crystalline strictly dominant point such that the eigenvalues $\varphi_{\tilde v,1},\dots,\varphi_{\tilde v,n}$ of the geometric Frobenius on the (unramified) Weil-Deligne representation ${\rm WD}(\rho\vert_{{\mathcal{G}}_{F_{\tilde v}}})$ satisfy $\varphi_{\tilde v,i}\varphi_{\tilde v,j}^{-1}\notin \{1,q_v\}$ for all $i\neq j$ and all $v\in S_p$. Let ${\mathfrak{X}}^p\subset {\mathfrak{X}}_{{\overline{\rho}}^p}$ be an irreducible component such that $x\in {\mathfrak{X}}^p\times X_{\rm tri}^{{\mathfrak{X}}^p \rm-aut}({\overline{\rho}}_p)\times {\mathbb{U}}^g\subseteq X_p({\overline{\rho}})$ via (\[union\]), let $x_v\in X_{\rm tri}^\square({\overline{\rho}}_{\tilde v})$ (for $v\in S_p$) be the image of $x$ via: $${\mathfrak{X}}^p\times X_{\rm tri}^{{\mathfrak{X}}^p \rm-aut}({\overline{\rho}}_p)\times {\mathbb{U}}^g\twoheadrightarrow X_{\rm tri}^{{\mathfrak{X}}^p \rm-aut}({\overline{\rho}}_p)\hookrightarrow X_{\rm tri}^\square({\overline{\rho}}_p)\twoheadrightarrow X_{\rm tri}^\square({\overline{\rho}}_{\tilde v})$$ and let $V_v\subseteq X_{\rm tri}^\square({\overline{\rho}}_{\tilde v})$ (for $v\in S_p$) be a sufficiently small open neighbourhood of $x_v$ so that $Z_{{\rm tri},V_v}(x_v)\subseteq V_v$ is defined. If we have: $$\prod_{v\in S_p} Z_{{\rm tri},V_v}(x_v)\subseteq X_{\rm tri}^{{\mathfrak{X}}^p\rm-aut}({\overline{\rho}}_p)$$ then the point $x$ is classical.
Let us write $\mathfrak{p}_y\subset R_\infty$ for the prime ideal corresponding to the image $y$ of $x$ in ${\mathfrak{X}}_\infty$ via $Y(U^p,{\overline{\rho}})\hookrightarrow X_p({\overline{\rho}})\hookrightarrow {\mathfrak{X}}_\infty\times \widehat{T}_{p,L}\twoheadrightarrow {\mathfrak{X}}_\infty$ and $\mathfrak{p}_\rho\subset R_{{\overline{\rho}},S}$ for the prime ideal corresponding to the global representation $\rho$. Then it follows from property (iii) above that we have $\mathfrak{a}R_\infty\subseteq \mathfrak{p}_{\rho}$ and $\widehat S(U^p,L)_{\mathfrak{m}^S}[\mathfrak{p}_{\rho}]=\Pi_{\infty}[\mathfrak{p}_y]$. From Definition \[defclass\] we thus need to show that ${\mathrm{Hom}}_{G_p}({\rm LA}(\delta),\Pi_{\infty}[\mathfrak{p}_y]\otimes_{k(\mathfrak{p}_y)}k(x))\ne 0$.
As in §\[classic\] let us write $\delta=\delta_{\lambda}\delta_{\rm sm}$ with $\lambda=(\lambda_v)_{v\in S_p}$ and: $$\lambda_v:=(\lambda_{v,\tau,i})_{1\leq i\leq n,\tau\in{\mathrm{Hom}}(F_{\tilde v},L)}\in {\mathbb{Z}}^{{\mathrm{Hom}}(F_{\tilde v},L)}$$ (recall that each $\lambda_v$ is dominant with respect to $B_v$). Consider the usual induction with compact support ${\rm ind}_{K_p}^{G_p}(L(\lambda)\vert_{K_p})$ (resp. ${\rm ind}_{K_v}^{G_v}(L(\lambda_v)\vert_{K_v})$) where $L(\lambda_p)$ (resp. $L(\lambda_v)$) is the irreducible algebraic representation of $G_p$ (resp. $G_v$) over $L$ of highest weight $\lambda$ (resp. $\lambda_v$) with respect to $B_p$ (resp. $B_v$). Let $\mathcal{H}(\lambda):={\mathrm{End}}_{G_p}({\rm ind}_{K_p}^{G_p}L(\lambda))$ and $\mathcal{H}(\lambda_v):={\mathrm{End}}_{G_v}({\rm ind}_{K_v}^{G_v}L(\lambda_v))$ be the respective convolution algebras (which are commutative $L$-algebras), we have $\mathcal{H}(\lambda)\cong \prod_{v\in S_p}\mathcal{H}(\lambda_v)$. Moreover by Frobenius reciprocity: $$\Pi_\infty(\lambda):={\mathrm{Hom}}_{K_p}(L(\lambda),\Pi_\infty)\cong {\mathrm{Hom}}_{G_p}\big({\rm ind}_{K_p}^{G_p}L(\lambda),\Pi_\infty\big)$$ carries an action of $\mathcal{H}(\lambda)$. By a slight extension of [@CEGGPS Lem.4.16(1)] (see the proof of [@BHS Prop.3.15]), the action of $R_{{\overline{\rho}}_{\tilde v}}^\square$ on $\Pi_\infty(\lambda)$ via $R_{{\overline{\rho}}_{\tilde v}}^\square\rightarrow R^{\rm loc}\hookrightarrow R_\infty$ factors through its quotient $R_{{\overline{\rho}}_{\tilde v}}^{\square,{\bf k}_v\rm -cr}$ where, for $v\in S_p$, ${\bf k}_v:=(k_{v,\tau,i})_{1\leq i\leq n,\tau\in{\mathrm{Hom}}(F_{\tilde v},L)}$ with $k_{v,\tau,i}:=\lambda_{v,\tau,i}-(i-1)$ (note that $\omega(x_v)=\delta_{{\bf k}_v}$ and that $R_{{\overline{\rho}}_{\tilde v}}^{\square,{\bf k}_v\rm -cr}$ is also a quotient of $R_{{\overline{\rho}}_{\tilde v}}^{\overline\square}$).
These two actions of $\mathcal{H}(\lambda_v)$ and $R_{{\overline{\rho}}_{\tilde v}}^{\square,{\bf k}_v\rm -cr}$ on the $L$-vector space $\Pi_\infty(\lambda)$ are related. By [@CEGGPS Th.4.1] and a slight extension of [@CEGGPS Lem.4.16(2)] (see the proof of [@BHS Prop.3.15]), there is a unique $L$-algebra homomorphism $\eta_v:\mathcal{H}(\lambda_v)\rightarrow R_{{\overline{\rho}}_{\tilde v}}^{\square,{\bf k}_v\rm -cr}[1/p]$ which interpolates the local Langlands correspondence (in a sense given in [@CEGGPS Th.4.1]) and such that the above action of $\mathcal{H}(\lambda_v)$ on $\Pi_\infty(\lambda)$ agrees with the action induced by that of $R_{{\overline{\rho}}_{\tilde v}}^{\square,{\bf k}_v\rm -cr}[1/p]$ composed with the morphism $\eta_v$.
In order to show that ${\rm LA}(\delta)$ admits a nonzero $G_p$-equivariant morphism to $\Pi_{\infty}[\mathfrak{p}_y]\otimes_{k(\mathfrak{p}_y)}k(x)$, we claim it is enough to show that $\Pi_\infty(\lambda)[\mathfrak{p}_y]\cong {\mathrm{Hom}}_{G_p}({\rm ind}_{K_p}^{G_p}L(\lambda),\Pi_\infty[\mathfrak{p}_y])$ is nonzero. Indeed, by what we just saw, any nonzero $G_p$-equivariant morphism ${\rm ind}_{K_p}^{G_p}L(\lambda)\rightarrow \Pi_\infty[\mathfrak{p}_y]$ induces a nonzero $G_p$-equivariant morphism: $${\rm ind}_{K_p}^{G_p}L(\lambda)\otimes_Lk(x)\longrightarrow \Pi_\infty[\mathfrak{p}_y]\otimes_{k(\mathfrak{p}_y)}k(x)$$ which factors through ${\rm ind}_{K_p}^{G_p}L(\lambda)\otimes_{\mathcal{H}(\lambda)}\theta_{\mathfrak{p}_y}$ where $\theta_{\mathfrak{p}_y}$ is the character: $$\xymatrix{
\theta_{\mathfrak{p}_y}:\mathcal{H}(\lambda)\ar[r]^<<<<{\otimes_{v\in S_p}\eta_v}&\widehat\bigotimes_{v\in S_p} R_{{\overline{\rho}}_{\tilde v}}^{\square,{\bf k}_v{\rm -cr}}[1/p]\ar[r] & k(\mathfrak{p}_y)\subseteq k(x),
}$$ the last morphism being the canonical projection to the residue field $k(\mathfrak{p}_y)$ at $\mathfrak{p}_y$ (the map $R_{{\overline{\rho}}_p}\hookrightarrow R_\infty\twoheadrightarrow R_\infty/\mathfrak{p}_y$ factoring through $\widehat\bigotimes_{v\in S_p} R_{{\overline{\rho}}_{\tilde v}}^{\square,{\bf k}_v{\rm -cr}}$ by the assumption on $\rho$). But by the compatibility with the local Langlands correspondence in [@CEGGPS Th.4.1] together with the assumption $\varphi_{\tilde v,i}/\varphi_{\tilde v,j}\neq q_v$ for $1\leq i,j\leq n$ and $v\in S_p$, we have ${\rm ind}_{K_p}^{G_p}L(\lambda)\otimes_{\mathcal{H}(\lambda)}\theta_{\mathfrak{p}_y}\cong {\rm LA}(\delta)\otimes_{k(\delta)}k(x)$.
By the same proof as that of [@CEGGPS Lem.4.17(2)], the $R_\infty \otimes_{R_{{\overline{\rho}}_p}}\widehat\bigotimes_{v\in S_p} R_{{\overline{\rho}}_{\tilde v}}^{\square,{\bf k}_v{\rm -cr}}$-module $\Pi_{\infty}(\lambda)'$ is supported on a union of irreducible components of: $${\mathfrak{X}}_{{\overline{\rho}}^p}\times \prod_{v\in S_p} {\mathfrak{X}}_{{\overline{\rho}}_{\tilde v}}^{\square,{\bf k}_v\rm -cr}\times{\mathbb{U}}^g$$ and we have to prove that $y$ is a point on one of these irreducible components. Since $y\in {\mathfrak{X}}^p\times \prod_{v\in S_p} Z_{\rm cris}(\rho_{\tilde v})\times{\mathbb{U}}^g$ where $Z_{\rm cris}(\rho_{\tilde v})$ is the unique irreducible component of ${\mathfrak{X}}_{{\overline{\rho}}_{\tilde v}}^{\square,{\bf k}_v\rm -cr}$ containing $\rho_{\tilde v}:=\rho\vert_{{\mathcal{G}}_{F_{\tilde v}}}$ (recall ${\mathfrak{X}}_{{\overline{\rho}}_{\tilde v}}^{\square,{\bf k}_v\rm -cr}$ is smooth over $L$ by [@Kisindef]), it is enough to prove that ${\mathfrak{X}}^p\times \prod_{v\in S_p} Z_{\rm cris}(\rho_{\tilde v})\times{\mathbb{U}}^g$ is one of the irreducible components in the support of $\Pi_{\infty}(\lambda)'$, or equivalently that ${\mathfrak{X}}^p\times \prod_{v\in S_p} Z_{\rm cris}(\rho_{\tilde v})\times{\mathbb{U}}^g$ contains at least one point which is in the support of $\Pi_{\infty}(\lambda)'$.
For each $v\in S_p$ let $x'_v=(r'_v,\delta'_v)$ be any point in $\iota_{{\bf k}_v}(\widetilde Z_{\rm cris}(x_v))\cap V_v\subseteq V_v\subseteq X_{\rm tri}^\square({\overline{\rho}}_{\tilde v})$ where $\widetilde Z_{\rm cris}(x_v)$ is as in (i) of Corollary \[defnZtri(x)\] (so in particular $x'_v$ is crystalline strictly dominant of Hodge-Tate weights ${\bf k}_v$ and $r'_v$ lies on $Z_{\rm cris}(\rho_{\tilde v})$ by (i) of Remark \[down\]). Then we have $x'_v\in Z_{{\rm tri},V_v}(x_v)$ for $v\in S_p$ by (ii) of Corollary \[defnZtri(x)\]. From the assumption: $$\prod_{v\in S_p}Z_{{\rm tri},V_v}(x_v)\subset X_{\rm tri}^{{\mathfrak{X}}^p\rm -aut}({\overline{\rho}}_p)$$ it then follows that there exists: $$x'=(y',\epsilon')\in {\mathfrak{X}}^p\times\prod_{v\in S_p}Z_{{\rm tri},V_v}(x_v)\times {\mathbb{U}}^g\subseteq {\mathfrak{X}}^p\times X_{\rm tri}^{{\mathfrak{X}}^p \rm-aut}({\overline{\rho}}_p)\times {\mathbb{U}}^g\buildrel (\ref{union})\over \subseteq X_p({\overline{\rho}})\subset {\mathfrak{X}}_\infty\times \widehat T_{p,L}$$ (with $y'\in {\mathfrak{X}}_\infty$, $\epsilon'\in \widehat T_{p,L}$) mapping to $x'_v$ via ${\mathfrak{X}}^p\times X_{\rm tri}^{{\mathfrak{X}}^p \rm-aut}({\overline{\rho}}_p)\times {\mathbb{U}}^g\twoheadrightarrow X_{\rm tri}^{{\mathfrak{X}}^p \rm-aut}({\overline{\rho}}_p)\hookrightarrow X_{\rm tri}^\square({\overline{\rho}}_p)\twoheadrightarrow X_{\rm tri}^\square({\overline{\rho}}_{\tilde v})$ (so $\epsilon'_v=\imath_v^{-1}(\delta'_v)$) and where $y'$ still belongs to ${\mathfrak{X}}^p\times \prod_{v\in S_p} Z_{\rm cris}(\rho_{\tilde v})\times{\mathbb{U}}^g$. It is thus enough to prove that $y'$ is in the support of $\Pi_{\infty}(\lambda)'$, i.e. that $\Pi_\infty(\lambda)[\mathfrak{p}_{y'}]\cong {\mathrm{Hom}}_{K_p}(L(\lambda),\Pi_\infty[\mathfrak{p}_{y'}])$ is nonzero.
We conclude by a similar argument as in the proof of [@BHS Prop.3.27]. By (the proof of) [@Chenevier Lem.4.4] and the same argument as at the end of the proof of Lemma \[genericopensinXcris\] (using the smoothness, hence flatness, of $\widetilde U_{{\overline{\rho}}_{\tilde v}}^{\square,{\bf k}_v\rm -cr}\rightarrow {\mathfrak{X}}_{{\overline{\rho}}_{\tilde v}}^{\square,{\bf k}_v\rm -cr}$), we may choose $x_v'\in \iota_{{\bf k}_v}(\widetilde Z_{\rm cris}(x_v))\cap V_v$ such that the crystalline Galois representation $r'_v$ is generic in the sense of [@BHS Def.2.8]. Then we claim that the nonzero $G_p$-equivariant morphism ${\mathcal{F}}_{\overline B_p}^{G_p}(\epsilon')\rightarrow \Pi_\infty^{R_\infty-\rm an}[\mathfrak{p}_{y'}]\otimes_{k(\mathfrak{p}_{y'})}k(x')$ corresponding by [@BreuilAnalytiqueII Th.4.3] to the nonzero $T_p$-equivariant morphism $\epsilon'\rightarrow J_{B_p}(\Pi_\infty^{R_\infty-\rm an}[\mathfrak{p}_{y'}]\otimes_{k(\mathfrak{p}_{y'})}k(x'))$ given by the point $x'$ factors through its locally ${\mathbb{Q}}_p$-algebraic quotient ${\rm LA}(\epsilon')$ (which provides a nonzero $K_p$-equivariant morphism $L(\lambda)\rightarrow \Pi_\infty[\mathfrak{p}_{y'}]$). Indeed, if it doesn’t, then the computation of the Jordan-Hölder factors of ${\mathcal{F}}_{\overline B_p}^{G_p}(\epsilon')$ ([@BreuilAnalytiqueII Cor.4.6]) together with [@BreuilAnalytiqueI Cor.3.4] show that there exits a point $x''=(y',\epsilon'')\in X_{p}({\overline{\rho}})$ such that $\epsilon''$ is locally algebraic of [*nondominant*]{} weight. In particular there is some $v\in S_p$ such that the image of $x''$ in $X_{\rm tri}^\square({\overline{\rho}}_{\tilde v})$ is of the form $(r'_v, \imath_v^{-1}(\epsilon''_v))$ with $\imath_v^{-1}(\epsilon''_v)$ locally algebraic [*not*]{} strictly dominant. This contradicts [@BHS Lem.2.11].
Let $x=(\rho,\delta)\in Y(U^p,{\overline{\rho}})$ be a crystalline strictly dominant point such that for all $v\in S_p$ the eigenvalues of the geometric Frobenius on ${\rm WD}(\rho|_{{\mathcal{G}}_{F_{\tilde v}}})$ are pairwise distinct. Recall that we have associated in §\[weyl\] a Weyl group element $w_{x_v}$ to the image $x_v$ of $x$ in $X_{\rm tri}^\square({\overline{\rho}}_{\tilde v})$ via (\[eigenvartotrianguline\]). We write: $$\label{wx}
w_x:=(w_{x_v})_{v\in S_p}\in \prod_{v\in S_p}\Big(\prod_{F_{\tilde v}\hookrightarrow L}{\mathcal{S}}_n\Big)$$ for the corresponding element of the Weyl group of $({\mathrm{Res}}_{F^+/{\mathbb{Q}}}G)_L\buildrel\sim\over\rightarrow \prod_{v\in S_p} ({\mathrm{Res}}_{F_{\tilde v}/{\mathbb{Q}}_p}{\mathrm{GL}}_{n,F_{\tilde v}})_L$. We then obtain the following corollary, which is our main result.
\[mainclassic\] Let $x=(\rho,\delta)\in Y(U^p,{\overline{\rho}})$ be a crystalline strictly dominant very regular point. Assume that the Weyl group element $w_{x}$ in (\[wx\]) is a product of pairwise distinct simple reflections. Then $x$ is classical. Moreover all eigenvectors associated to $x$ are classical, that is we have (see the proof of Proposition \[compaclas\] for $\widehat S(U^p,L)_{\mathfrak{m}^S}^{\lambda-{\rm la}}$): $$\mathrm{Hom}_{T_p}\big(\delta,J_{B_p}(S(U^p,L)^{\lambda-{\rm la}}_{\mathfrak{m}^S}[\mathfrak{p}_\rho]\otimes_{k(\mathfrak{p}_\rho)}k(x))\big)\buildrel\sim\over\longrightarrow\mathrm{Hom}_{T_p}\big(\delta,J_{B_p}(\widehat{S}(U^p,L)^{\rm an}_{\mathfrak{m}^S}[\mathfrak{p}_\rho]\otimes_{k(\mathfrak{p}_\rho)}k(x))\big).$$
Keep the notation of Theorem \[classicalitycrit\]. By Proposition \[acconZtri\] below, for each $v\in S_p$ there is a sufficiently small open neighbourhood $V_v$ of $x_v$ in $X_{\rm tri}^\square({\overline{\rho}}_{\tilde v})$ such that the irreducible component $Z_{{\rm tri},V_v}(x_v)$ of $V_v$ in (ii) of Corollary \[defnZtri(x)\] is defined and satisfies the accumulation property at $x_v$ (Definition \[accu\]).
Seeing $x$ in $X_p({\overline{\rho}})$ via the closed embedding $Y(U^p,{\overline{\rho}})\hookrightarrow X_p({\overline{\rho}})$, by (\[union\]) there exist irreducible components ${\mathfrak{X}}^p$ of ${\mathfrak{X}}_{{\overline{\rho}}^p}$ and $Z=\prod_{v\in S_p}Z_v$ of $X_{\rm tri}^\square({\overline{\rho}}_p)=\prod_{v\in S_p}X_{\rm tri}^\square({\overline{\rho}}_{\tilde v})$ such that: $$x\in {\mathfrak{X}}^p\times Z\times {\mathbb{U}}^g\subseteq {\mathfrak{X}}^p\times X_{\rm tri}^{{\mathfrak{X}}^p \rm-aut}({\overline{\rho}}_p)\times{\mathbb{U}}^g\buildrel (\ref{union}) \over\subseteq X_p({\overline{\rho}}).$$ Then it follows from Proposition \[acconZaut\] and Remark \[obvious\] below that $Z_v$ satisfies the accumulation property at $x_v$ for all $v\in S_p$. Let $Y_v\subseteq Z_v\cap V_v$ be a nonempty union of irreducible components of $V_v$, then $X_v:=Y_v\cup Z_{{\rm tri},V_v}(x_v)$ satisfies the accumulation property at $x_v$ since both $Y_v$ and $Z_{{\rm tri},V_v}(x_v)$ do. But $X_v$ is smooth at $x_v$ by the assumption on $w_{x_v}$ and Corollary \[Xtrismooth\] applied with $(X,x)=(X_v,x_v)$, hence there can only be one irreducible component of $X_v$ passing through $x_v$. We deduce in particular $Z_{{\rm tri},V_v}(x_v)\subseteq Y_v\subseteq Z_v$, hence $\prod_{v\in S_p}Z_{{\rm tri},V_v}(x_v)\subseteq \prod_{v\in S_p}Z_v\subseteq X_{\rm tri}^{{\mathfrak{X}}^p \rm-aut}({\overline{\rho}}_p)$ and $x$ is classical by Theorem \[classicalitycrit\]. We also deduce that the only possible $Z=\prod_{v\in S_p}Z_v$ passing through $(x_v)_{v\in S_p}$ is smooth at $(x_v)_{v\in S_p}$, hence that $X_{\rm tri}^{{\mathfrak{X}}^p \rm-aut}({\overline{\rho}}_p)$ is smooth at $(x_v)_{v\in S_p}$.
Let us now prove the last statement. From the injection: $$\mathrm{Hom}_{T_p}\big(\delta,J_{B_p}(S(U^p,L)^{\lambda-{\rm la}}_{\mathfrak{m}^S}[\mathfrak{p}_\rho]\otimes_{k(\mathfrak{p}_\rho)}k(x))\big)\hookrightarrow\mathrm{Hom}_{T_p}\big(\delta,J_{B_p}(\widehat{S}(U^p,L)^{\rm an}_{\mathfrak{m}^S}[\mathfrak{p}_\rho]\otimes_{k(\mathfrak{p}_\rho)}k(x))\big)$$ it is enough to prove that these two $k(x)$-vector spaces have the same (finite) dimension. Recall from [@BHS §3.2] that for any $x'=(y',\delta')\in X_p({\overline{\rho}})$ we have an isomorphism of $k(x')$-vector spaces: $$\label{iso1}
\mathrm{Hom}_{T_p}\big(\delta',J_{B_p}(\Pi_\infty^{R_\infty-{\rm an}}[\mathfrak{p}_{y'}]\otimes_{k(\mathfrak{p}_{y'})}k(x'))\big)\cong \mathcal{M}_\infty\otimes_{\mathcal{O}_{X_p(\overline{\rho})}} k(x').$$ If moreover $x'=(\rho',\delta')\in Y(U^p,{\overline{\rho}})\hookrightarrow X_p({\overline{\rho}})$ we have $\widehat S(U^p,L)_{\mathfrak{m}^S}[\mathfrak{p}_{\rho'}]=\Pi_{\infty}[\mathfrak{p}_{y'}]$, hence an isomorphism of $k(x')$-vector spaces: $$\label{iso2}
\mathrm{Hom}_{T_p}\big(\delta',J_{B_p}(\widehat{S}(U^p,L)^{\rm an}_{\mathfrak{m}^S}[\mathfrak{p}_{\rho'}]\otimes_{k(\mathfrak{p}_{\rho'})}k(x'))\big)\simeq\mathrm{Hom}_{T_p}\big(\delta',J_{B_p}(\Pi_\infty^{R_\infty-{\rm an}}[\mathfrak{p}_{y'}]\otimes_{k(\mathfrak{p}_{y'})}k(x'))\big).$$ We first claim that $x$ is a smooth point of $X_p(\overline{\rho})$. Indeed, by what we proved above, it is enough to show that its component $y^p=(y_v)_{v\in S\backslash S_p}$ in $\mathfrak{X}_{\overline{\rho}^p}$ is a smooth point. As $x$ is classical, by Proposition \[compaclas\] (in particular the end of the proof) it corresponds to an automorphic representation $\pi$ of $G(\mathbb{A}_{F^+})$ with cuspidal strong base change $\Pi$ to $\mathrm{GL}_n(\mathbb{A}_F)$. It then follows from [@Caraiani Th.1.2] that $\Pi$ is tempered, in particular generic, at all finite places of $F$. Then [@BLGGT Lem.1.3.2(1)] implies that $y_v$ for $v\in S\backslash S_p$ is a smooth point of $({\mathrm{Spf}}\, R_{{\overline{\rho}}_{\tilde v}}^{\overline\square})^{{\mathrm{rig}}}$. As $\mathcal{M}_\infty$ is Cohen-Macaulay (Lemma \[CM\]) and $x$ is smooth on $X_p(\overline{\rho})$, we conclude from [@EGAIV1 Cor.17.3.5(i)] that $\mathcal{M}_\infty$ is actually locally free at $x$. Consequently there exists an open affinoid neighbourhood of $x$ in $X_p(\overline{\rho})$ on which the dimension of the fibers of $\mathcal{M}_\infty$ is constant. Intersecting this neighbourhood with $Y(U^p,\overline{\rho})$ and using (\[iso1\]) and (\[iso2\]), we obtain an open affinoid neighbourhood $V_x$ of $x$ in $Y(U^p,\overline{\rho})$ on which $\dim_{k(x')}\mathrm{Hom}_{T_p}(\delta',J_{B_p}(\widehat{S}(U^p,L)^{\rm an}_{\mathfrak{m}^S}[\mathfrak{p}_{\rho'}]\otimes_{k(\mathfrak{p}_{\rho'})}k(x')))$ is constant for $x'=(\rho',\delta')\in V_x$.
Now let $x'\in V_x$ be a very classical point in the sense of [@BHS Def.3.16] and write $\omega(x')=\delta_{\lambda'}$ with dominant $\lambda'\in \prod_{v\in S_p}({\mathbb{Z}}^n)^{{\mathrm{Hom}}(F_{\tilde v},L)}$. It follows from [*loc. cit.*]{} and [@BreuilAnalytiqueII Th.4.3] that we have: $$\begin{aligned}
\label{adjla}
{\scriptstyle {\mathrm{Hom}}_{T_p}\big(\delta',\ J_{B_p}(\widehat S(U^p,L)^{\rm an}_{\mathfrak{m}^S}[\mathfrak{p}_{\rho'}]\otimes_{k(\mathfrak{p}_{\rho'})}k(x'))\big)}&\cong&{\scriptstyle {\mathrm{Hom}}_{G_p}\big({\rm LA}(\delta'),\ \widehat S(U^p,L)_{\mathfrak{m}^S}^{\rm an}[\mathfrak{p}_{\rho'}]\otimes_{k(\mathfrak{p}_{\rho'})}k(x')\big)}\\
&\cong& {\scriptstyle {\mathrm{Hom}}_{G_p}\big({\rm LA}(\delta'),\ \widehat S(U^p,L)_{\mathfrak{m}^S}^{\lambda'-{\rm la}}[\mathfrak{p}_{\rho'}]\otimes_{k(\mathfrak{p}_{\rho'})}k(x')\big)}\\
&\cong &{\scriptstyle {\mathrm{Hom}}_{T_p}\big(\delta',\ J_{B_p}(\widehat S(U^p,L)^{\lambda'-{\rm la}}_{\mathfrak{m}^S}[\mathfrak{p}_{\rho'}]\otimes_{k(\mathfrak{p}_{\rho'})}k(x'))\big)}.\end{aligned}$$ From what is proved above, it is thus enough to find a very classical point $x'$ in $V_x$ such that: $$\begin{gathered}
\label{dimequal}
\dim_{k(x')}{\mathrm{Hom}}_{T_p}\big(\delta',J_{B_p}(\widehat S(U^p,L)^{\lambda'-{\rm la}}_{\mathfrak{m}^S}[\mathfrak{p}_{\rho'}]\otimes_{k(\mathfrak{p}_{\rho'})}k(x'))\big)=\\
\dim_{k(x)}{\mathrm{Hom}}_{T_p}\big(\delta,J_{B_p}(\widehat S(U^p,L)^{\lambda-{\rm la}}_{\mathfrak{m}^S}[\mathfrak{p}_{\rho}]\otimes_{k(\mathfrak{p}_{\rho})}k(x))\big).\end{gathered}$$ Let $x''=(\rho'',\delta'')\in Y(U^p,{\overline{\rho}})$ be any classical crystalline strictly dominant point and let $\omega(x'')=\delta_{\lambda''}$. By Proposition \[compaclas\] it corresponds to a unique automorphic representation $\pi''$ which moreover has multiplicity $1$, hence we have (with the notation of the proof of Proposition \[compaclas\]): $$J_{B_p}\big(S(U^p,L)_{\mathfrak{m}^S}^{\lambda''-{\rm la}}[\mathfrak{p_{\rho''}}]\otimes_{k(\mathfrak{p_{\rho''}})}\overline{\mathbb{Q}}_p\big)\simeq J_{B_p}\big(L(\lambda'')\otimes_L\bigotimes_{v\in S_p}\pi''_{v}\big)\otimes_{{\overline{\mathbb{Q}}}}({\pi_f''}^{p})^{U^p}\otimes_{{\overline{\mathbb{Q}}},j_p}\overline{\mathbb{Q}}_p.$$ From the definition of $S$ together with [@EmertonJacquetI Prop.4.3.6] and property (iv) in Proposition \[compaclas\], it then easily follows that: $$\label{calcdim}
\dim_{k(x'')}{\mathrm{Hom}}_{T_p}\big(\delta'',J_{B_p}(\widehat S(U^p,L)^{\lambda''-{\rm la}}_{\mathfrak{m}^S}[\mathfrak{p}_{\rho''}]\otimes_{k(\mathfrak{p}_{\rho''})}k(x''))\big)=\dim_{{\overline{\mathbb{Q}}}}\Big(\bigotimes_{v\in S\backslash S_p} {\pi_v''}^{U_v}\Big).$$ Let $Z$ be the union of $x$ and of the very classical points in $V_x$, by [@BHS Thm.3.18] this set $Z$ accumulates at $x$. By [@Caraiani Th.1.2], we can apply [@Chenevierfern Lem.4.5(ii)] to the intersection of $Z$ with one irreducible component of $V_x$, and obtain that, for $v\nmid p$, the value $\dim_{{\overline{\mathbb{Q}}}}{\pi''_{v}}^{U_v}$ is constant on this intersection. In particular $\dim_{{\overline{\mathbb{Q}}}}(\bigotimes_{v\in S\backslash S_p} {\pi_v''}^{U_v})$ is also constant on this intersection, which finishes the proof by (\[calcdim\]) and (\[dimequal\]).
\[uniqueirredcompo\] [(i) Keeping the notation of Theorem \[classicalitycrit\], if there is a [*unique*]{} irreducible component $Z$ of $X_{\rm tri}^\square({\overline{\rho}}_p)$ passing through the image of $x$ in $X_{\rm tri}^\square({\overline{\rho}}_p)$, or equivalently if for each $v\in S_p$ there is a unique irreducible component of $X_{\rm tri}^\square({\overline{\rho}}_{\tilde v})$ passing through $x_v$, then $x$ is classical. Indeed, in that case there is an irreducible component ${\mathfrak{X}}^p$ of ${\mathfrak{X}}_{{\overline{\rho}}^p}$ such that $x\in {\mathfrak{X}}^p\times Z\times {\mathbb{U}}^g={\mathfrak{X}}^p\times X_{\rm tri}^{{\mathfrak{X}}^p \rm-aut}({\overline{\rho}}_p)\times{\mathbb{U}}^g$. In particular, for a sufficiently small open neighbourhood $V_v$ of $x_v$ in $X_{\rm tri}^\square({\overline{\rho}}_{\tilde v})$, we have $\prod_{v\in S_p} V_v \subseteq Z=X_{\rm tri}^{{\mathfrak{X}}^p\rm-aut}({\overline{\rho}}_p)$ and we see that the assumption in Theorem \[classicalitycrit\] is [*a fortiori*]{} satisfied.\
(ii) Let us recall the various global hypothesis underlying the statements of Theorem \[classicalitycrit\] and Corollary \[mainclassic\]: $p>2$, $G$ is quasi-split at all finite places of $F^+$, $F/F^+$ is unramified, $U_v$ is hyperspecial if $v$ is inert in $F$, ${\overline{\rho}}(\mathcal{G}_{F(\zeta_p)})$ is adequate and $\zeta_p\notin \overline F^{\ker({\overline{\rho}}\otimes {\overline{\rho}}')}$.]{}
Accumulation properties
-----------------------
We prove some accumulation properties (as in Definition \[accu\]) that are used in the proof of Corollary \[mainclassic\] in order to apply Corollary \[Xtrismooth\].
We first go back to the purely local set up of §\[localpart1\]. We call a point $x=(r,\delta_1,\dots,\delta_n)\in X_{\rm tri}^\square(\overline r)$ *saturated* if there exists a triangulation of the $(\varphi,\Gamma_K)$-module $D_{{\mathrm{rig}}}(r)$ with parameter $(\delta_1,\dots,\delta_n)$ (cf. §\[begin\]). Note that, if $x$ is crystalline strictly dominant with pairwise distinct Frobenius eigenvalues, then $x$ is saturated if and only if $x$ is noncritical (cf. §§\[variant\], \[weyl\]). Recall also from §\[begin\] that if $x$ is saturated and if $(\delta_1,\dots,\delta_n)\in {\mathcal{T}}_{\rm reg}^n$ then $x\in U_{\rm tri}^\square(\overline r)$.
\[numnoncrit\] Let $x=(r,\delta_1,...,\delta_n)\in X_{\rm tri}^\square(\overline r)$ with $\omega(x)=\delta_{\bf k}$ for some ${\bf k}\!=\!(k_{\tau,i})_{1\leq i\leq n,\tau:\, K\hookrightarrow L}\!\in(\mathbb{Z}^n)^{{\mathrm{Hom}}(K,L)}$. Assume that: $$\label{eqnnumnoncrit}
k_{\tau,i}-k_{\tau,i+1}> [K:K_0] {\mathrm{val}}\big(\delta_1(\varpi_K)\cdots \delta_i(\varpi_K)\big)$$ for $i\in \{1,\dots,n-1\}$, $\tau\in {\mathrm{Hom}}(K,L)$. Then $x$ is saturated and $r$ is semi-stable. If moreover $(\delta_1,\dots,\delta_n)\in {\mathcal{T}}^n_{\rm reg}$, then $r$ is crystalline strictly dominant noncritical.
By [@KPX Th.6.3.13] and [@BHS Prop.2.9] the representation $r$ is trianguline with parameter $(\delta'_1,\dots, \delta'_n)$ where $\delta'_i=\delta_i z^{{\bf k}_{w^{-1}(i)}-{\bf k}_i}$ for some $w=(w_\tau)_{\tau:\, K\hookrightarrow L}\in W=\prod_{\tau:K\hookrightarrow L} {\mathcal{S}}_n$. As $D_{{\mathrm{rig}}}(r)$, and hence $\wedge_{\mathcal{R}_{k(x),K}}^i D_{{\mathrm{rig}}}(r)$, are $\varphi$-modules over $\mathcal{R}_{k(x),K}$ which are pure of slope zero (being étale $(\varphi,\Gamma_K)$-modules), it follows that for all $i$: $$1\leq \big|\delta'_1(\varpi_K)\cdots\delta'_i(\varpi_K)\big|_K.$$ Since $\delta'_1(\varpi_K)\cdots\delta'_i(\varpi_K)=\delta_1(\varpi_K)\cdots\delta_i(\varpi_K)\cdot \prod_{j=1}^i\prod_\tau (\tau(\varpi_K)^{k_{\tau,w_{\tau}^{-1}(j)}-k_{\tau,j}})$ we obtain: $$\label{slope}
{\mathrm{val}}\big(\delta_1(\varpi_K)\cdots \delta_i(\varpi_K)\big) \geq \tfrac{1}{[K:K_0]}\sum_{j=1}^i\sum_{\tau}(k_{\tau,j}-k_{\tau,w_\tau^{-1}(j)}).$$ We now prove by induction on $i$ that $w_\tau^{-1}(i)=i$ for all $\tau$. The inequality (\[slope\]) for $i=1$ gives ${\mathrm{val}}(\delta_1(\varpi_K))\geq \tfrac{1}{[K:K_0]}\sum_\tau (k_{\tau,1}-k_{\tau,w_\tau^{-1}(1)})$. But assumption $(\ref{eqnnumnoncrit})$ with $i=1$ implies ${\mathrm{val}}(\delta_1(\varpi_K))< \tfrac{1}{[K:K_0]}\sum_\tau (k_{\tau,1}-k_{\tau,j})$ for $j\in \{2,\dots,n\}$ which forces $w_\tau^{-1}(1)=1$ for all $\tau$. Assume by induction that $w_\tau^{-1}(j)=j$ for all $j\leq i-1$ and all $\tau$. Then (\[slope\]) gives: $${\mathrm{val}}(\delta_1(\varpi_K)\cdots\delta_i(\varpi_K))\geq \tfrac{1}{[K:K_0]}\sum_\tau (k_{\tau,i}-k_{\tau,w_\tau^{-1}(i)})$$ and again $(\ref{eqnnumnoncrit})$ implies ${\mathrm{val}}(\delta_1(\varpi_K)\cdots\delta_i(\varpi_K))< \tfrac{1}{[K:K_0]}\sum_\tau (k_{i,\tau}-k_{j,\tau})$ for $j\in \{i,\dots,n\}$ which forces $w_\tau^{-1}(i)=i$ for all $\tau$. We thus have $(\delta_1,\dots, \delta_n)=(\delta'_1,\dots, \delta'_n)$ which implies that the point $x=(r,\delta_1,\dots, \delta_n)$ is saturated. Since $\delta$ is strictly dominant, we obtain that $r$ is semi-stable by the argument in the proof of [@Chenevier Th.3.14] (see also the proof of [@HellmSchrDensity Cor.2.7(i)]). By a slight generalisation of the proof of Lemma \[paramofcrystpt\] (that we leave to the reader), we have $\delta_i=z^{{\bf k}_i}{\rm unr}(\varphi_i)$ where the $\varphi_i$ are the eigenvalues of the linearized Frobenius $\varphi^{[K_0:{\mathbb{Q}}_p]}$ on the $K_0\otimes_{{\mathbb{Q}}_p}k(x)$-module $D_{\rm st}(r):=(B_{\rm st}\otimes_{{\mathbb{Q}}_p}r)^{\mathcal{G}_K}$. If in addition $(\delta_1,\dots,\delta_n)\in {\mathcal{T}}^n_{\rm reg}$, then it follows from Remark \[varphi\] that $\varphi_i\varphi_j^{-1}\ne p^{-[K_0:{\mathbb{Q}}_p]}$ for $1\leq i\leq j\leq n$ and the argument of [@Chenevier Th.3.14], [@HellmSchrDensity Cor.2.7(i)] then shows that the monodromy operator $N$ on $D_{\rm st}(r)$ must be zero, i.e. that $r$ is crystalline. This finishes the proof.
\[acconZtri\] Let $x=(r,\delta)\in X_{\rm tri}^\square(\overline r)$ be a crystalline strictly dominant point such that the eigenvalues of the geometric Frobenius on ${\rm WD}(r)$ are pairwise distinct. Then there exists a sufficiently small open neighbourhood $U$ of $x$ in $X_{\rm tri}^\square(\overline r)$ such that the irreducible component $Z_{{\rm tri},U}(x)$ of $U$ in (ii) of Corollary \[defnZtri(x)\] is defined and satisfies the accumulation property at $x$.
Recall that we have to prove that, for any positive real number $C$, the set of points $x'=(r',\delta')\in Z_{{\rm tri},U}(x)$ such that $r'$ is crystalline with pairwise distinct geometric Frobenius eigenvalues on ${\rm WD}(r')$ and $x'$ is noncritical with $\omega(x')=\delta_{\bf k'}$ strictly dominant satisfying: $$\label{accumulation}
k'_{\tau,i}-k'_{\tau,i+1}>C$$ for all $i=1,\dots,n-1$, $\tau:K\hookrightarrow L$ accumulates at $x$.
Let $U$ be an open subset of $x$ in $X_{\rm tri}^\square(\overline r)$ as in (iii) of Corollary \[defnZtri(x)\], i.e. such that for any open $U'\subseteq U$ containing $x$ we have $Z_{{\rm tri},U}(x)\cap U'=Z_{{\rm tri},U'}(x)$. Let $\widetilde Z_{\rm cris}(x)$ as in (i) of Corollary \[defnZtri(x)\], by Lemma $\ref{genericopensinXcris}$, the space $V:=\iota_{\bf k}(\widetilde Z_{\rm cris}(x)\cap \widetilde V_{\overline r}^{\square,{\bf k}{\rm -cr}})$ is Zariski-open and Zariski-dense in $\iota_{\bf k}(\widetilde Z_{\rm cris}(x))$, hence accumulates in $\iota_{\bf k}(\widetilde Z_{\rm cris}(x))$ at any point of $\iota_{\bf k}(\widetilde Z_{\rm cris}(x))$, in particular at $x$. We claim that it is enough to prove that the points $x'\in U$ as above accumulate in $U$ at every point of $V\cap U$. Indeed, if $U'\subseteq U$ is an open neighbourhood containing $x$, then $U'$ also contains a point $v\in V$. By the accumulation statement at $v\in V\cap U$, the Zariski closure in $U'$ of the points $x'$ contains a small neighbourhood around $v$, hence contains an irreducible component of $U'$ containing $v$. But since $v$ is a smooth point of $U'$ (over $L$) as $v\in U_{\rm tri}^\square(\overline r)$ by the last statement of Lemma \[genericopensinXcris\], there is only one such irreducible component, and since $v\in \iota_{\bf k}(\widetilde Z_{\rm cris}(x))\cap U'\subseteq Z_{{\rm tri},U'}(x)$, we see that this irreducible component must be $Z_{{\rm tri},U'}(x)$. Thus the Zariski closure in $U'$ of the points $x'$ always contains $Z_{{\rm tri},U'}(x)$. This easily implies the proposition since $Z_{{\rm tri},U'}(x)=Z_{{\rm tri},U}(x)\cap U'$.
Since $U_{\rm tri}^\square(\overline r)$ is open in $X_{\rm tri}^\square(\overline r)$, it is enough to prove that the crystalline points $x'$ in $U\cap U_{\rm tri}^\square(\overline r)$ satisfying the conditions in the first paragraph of this proof accumulate at any crystalline strictly dominant point $x$ of $U\cap U_{\rm tri}^\square(\overline r)$. The condition on their Frobenius eigenvalues is then in fact automatic by Remark \[varphi\]. Shrinking $U$ further if necessary, we can take $U$ to be contained in some quasi-compact open neighbourhood of $x$ in $X_{\rm tri}^\square(\overline r)$, and thus we may assume that for $i\in \{1,\dots,n\}$ the functions $y=(r_y,(\delta_{y,1},\dots,\delta_{y,n}))\mapsto \delta_{y,i}(\varpi_K)$ are uniformly bounded on $U$. Hence by Lemma $\ref{numnoncrit}$ we may assume that $C$ is sufficiently large so that the points $x'\in U\cap U_{\rm tri}^\square(\overline r)$ with $\omega(x)=\delta_{\bf k'}$ algebraic satisfying $(\ref{accumulation})$ are in fact also automatically crystalline noncritical. Changing notation, we see that it is finally enough to prove that the points $x'\in U_{\rm tri}^\square(\overline r)$ satisfying (\[accumulation\]) for $C$ big enough accumulate at any crystalline strictly dominant point $x$ of $U_{\rm tri}^\square(\overline r)$.
We now consider the rigid analytic spaces ${\mathcal{S}}_n$, ${\mathcal{S}}^\square(\overline r)$ appearing in the proof of [@BHS Th.2.6] (to which we refer the reader for more details; do not confuse here ${\mathcal{S}}_n$ with the permutation group!). In [*loc. cit.*]{} there is a diagram of rigid spaces over ${\mathcal{T}}_L^n$: $$\begin{xy}\xymatrix{&{\mathcal{S}}^\square(\overline r)\ar[dl]_{\pi_{\overline r}}\ar[dr]^g&\\ U_{\rm tri}^\square(\overline r) & & {\mathcal{S}}_n}
\end{xy}$$ where $\pi_{\overline r}$ is a ${\mathbb{G}}_m^n$-torsor and $g$ is a composition ${\mathcal{S}}^\square(\overline r)\hookrightarrow {\mathcal{S}}_n^{\square,{\rm adm}}\rightarrow {\mathcal{S}}_n^{{\rm adm}}\hookrightarrow {\mathcal{S}}_n$ where the first and last maps are open embeddings and the middle one is a ${\mathrm{GL}}_n$-torsor.
Let us choose a point $\tilde x\in \pi_{\overline r}^{-1}(x)$. As $\pi_{\overline r}$ is a ${\mathbb{G}}_m^n$-torsor, it is enough to prove that the points in ${\mathcal{S}}^\square(\overline r)$ satisfying $(\ref{accumulation})$ accumulate at $\tilde x$. The same argument shows that it is enough to prove that the points of ${\mathcal{S}}_n$ satisfying $(\ref{accumulation})$ accumulate at $g(\tilde x)$. But the morphism ${\mathcal{S}}_n\rightarrow {\mathcal{T}}^n_L$ is a composition of open embeddings and structure morphisms of geometric vector bundles (compare the proof of [@HellmSchrDensity Th.2.4]). It follows that $g(\tilde x)$ has a basis of neighbourhoods $(U_i)_{i\in I}$ in ${\mathcal{S}}_n$ such that $V_i:=\omega(U_i)$ is a basis of neighbourhoods of $\omega( x)$ in ${\mathcal{W}}_L^n$ and such that the rigid space $U_i$ is isomorphic to a product $V_i\times{\mathbb{B}}$ of rigid spaces over $L$ where ${\mathbb{B}}$ is some closed polydisc (compare [@Chenevier Cor.3.5] and [@HellmSchrDensity Lem.2.18]). Write $\omega( x)=\delta_{\bf k}$, it is thus enough to prove that the algebraic weights $\delta_{\bf k'}\in {\mathcal{W}}_L^n$ satisfying $(\ref{accumulation})$ accumulate at $\delta_{\bf k}$ in ${\mathcal{W}}_L^n$, which is obvious.
We are now back to the global setting of §\[firstclassical\]. Similarly to Definition \[accu\], we say that a union $X$ of irreducible components of an open subset of $X_{\rm tri}^\square({\overline{\rho}}_p)=\prod_{v\in S_p}X_{\rm tri}^\square({\overline{\rho}}_{\tilde v})$ satisfies the accumulation property at a point $x\in X$ if, for any positive real number $C>0$, $X$ contains crystalline strictly dominant points $x'=(x'_v)_{v\in S_p}$ with pairwise distinct Frobenius eigenvalues, which are noncritical, such that $\omega(x'_v)=\delta_{{\bf k}'_v}$ with $k'_{v,\tau,i}-k'_{v,\tau,i+1}>C$ for $v\in S_p$, $i\in \{1,\dots,n-1\}$, $\tau\in {\mathrm{Hom}}(F_{\tilde v},L)$ and that accumulate at $x$ in $X$.
\[acconZaut\] Let ${\mathfrak{X}}^p\subset {\mathfrak{X}}_{{\overline{\rho}}^p}$ be an irreducible component and $x\in X_{\rm tri}^{{\mathfrak{X}}^p\rm- aut}({\overline{\rho}}_p)$ be a crystalline strictly dominant point. Then $X_{\rm tri}^{{\mathfrak{X}}^p\rm-aut}({\overline{\rho}}_p)$ satisfies the accumulation property at $x$.
It is enough to show that, for $C$ large enough, the points of ${\mathfrak{X}}^p\times X_{\rm tri}^{{\mathfrak{X}}^p \rm-aut}({\overline{\rho}}_p)\times {\mathbb{U}}^g$ such that their projection to $X_{\rm tri}^{{\mathfrak{X}}^p \rm-aut}({\overline{\rho}}_p)$ is a point $x'=(x'_v)_{v\in S_p}$ of the same form as above accumulate at any point of ${\mathfrak{X}}^p\times X_{\rm tri}^{{\mathfrak{X}}^p \rm-aut}({\overline{\rho}}_p)\times {\mathbb{U}}^g$ mapping to $x$ in $X_{\rm tri}^{{\mathfrak{X}}^p \rm-aut}({\overline{\rho}}_p)$. Using (\[union\]) this claim is contained in the proof of [@BHS Th.3.18] (which itself is a consequence of [@BHS Prop.3.10]).
\[obvious\] [It is obvious from the definition that if a union $X$ of irreducible components of $X_{\rm tri}^\square({\overline{\rho}}_p)=\prod_{v\in S_p}X_{\rm tri}^\square({\overline{\rho}}_{\tilde v})$ satisfies the accumulation property at some point $x\in X$, then for each $v\in S_p$ the image of $X$ in $X_{\rm tri}^\square({\overline{\rho}}_{\tilde v})$ (which is a union of irreducible components of $X_{\rm tri}^\square({\overline{\rho}}_{\tilde v})$) satisfies the accumulation property at the image of $x$ in $X_{\rm tri}^\square({\overline{\rho}}_{\tilde v})$.]{}
On the local geometry of the trianguline variety {#phiGammacohomology}
================================================
This section is entirely local and devoted to the proof of Theorem \[upperbound\] above giving an upper bound on some local tangent spaces. We use the notation of §\[localpart1\].
Tangent spaces {#start}
--------------
We start with easy preliminary lemmas on some tangent spaces.
If $x=(r,\delta)\in X_{\rm tri}^\square({\overline{r}})$, we denote by ${\rm Ext}^1_{\mathcal{G}_K}(r,r)$ the usual $k(x)$-vector space of $\mathcal{G}_K$-extensions $0\rightarrow r\rightarrow *\rightarrow r\rightarrow 0$.
\[ext1\] Let $x=(r,\delta)\in X_{\rm tri}^\square({\overline{r}})$ be any point, then there is an exact sequence of $k(x)$-vector spaces $0\rightarrow K(r)\rightarrow T_{\mathfrak{X}_{{\overline{r}}}^\square,r}\rightarrow {\rm Ext}^1_{\mathcal{G}_K}(r,r)\rightarrow 0$ where $K(r)$ is a $k(x)$-subvector space of $T_{\mathfrak{X}_{{\overline{r}}}^\square,r}$ of dimension $\dim_{k(x)}{\mathrm{End}}_{k(x)}(r)-\dim_{k(x)}{\mathrm{End}}_{\mathcal{G}_K}(r)=n^2-\dim_{k(x)}{\mathrm{End}}_{\mathcal{G}_K}(r)$.
It easily follows from [@KisinModularity Lem.2.3.3 & Prop.2.3.5] that there is a topological isomorphism $\widehat{\mathcal{O}}_{\mathfrak{X}^\square_{{\overline{r}}},r}\cong R^\square_{r}$ where the former is the completed local ring at $r$ to the rigid analytic variety $\mathfrak{X}^\square_{{\overline{r}}}$ and the latter is the framed local deformation ring of $r$ in equal characteristic $0$. In particular from (\[tangent\]) we have $T_{\mathfrak{X}_{{\overline{r}}}^\square,r}\cong {\mathrm{Hom}}_{k(x)}\big(R^\square_{r},k(x)[\varepsilon]/(\varepsilon^2)\big)$. Then the result follows by the same argument as in [@KisinModularity §2.3.4], seeing an element of ${\rm Ext}^1_{\mathcal{G}_K}(r,r)$ as a deformation of $r$ with values in $k(x)[\varepsilon]/(\varepsilon^2)$.
\[dim\] Let $x=(r,\delta)\in X_{\rm tri}^\square({\overline{r}})$ be a point such that $H^2(\mathcal{G}_K,r\otimes r')=0$ ($r'$ being the dual of $r$), then $\dim_{k(x)}{\rm Ext}^1_{\mathcal{G}_K}(r,r)=\dim_{k(x)}{\mathrm{End}}_{\mathcal{G}_K}(r)+n^2[K:{\mathbb{Q}}_p]$.
This follows by the usual argument computing $\dim_{k(x)}H^1(\mathcal{G}_K,r\otimes r')$ from the Euler characteristic formula of Galois cohomology using $\dim_{k(x)}H^0(\mathcal{G}_K,r\otimes r')=\dim_{k(x)}{\mathrm{End}}_{\mathcal{G}_K}(r)$ and $\dim_{k(x)}H^2(\mathcal{G}_K,r\otimes r')=0$.
\[cascr\] [Lemma \[dim\] in particular holds if $x$ is crystalline and the Frobenius eigenvalues $(\varphi_i)_{1\leq i\leq n}$ (see Lemma \[paramofcrystpt\]) satisfy $\varphi_i\varphi_j^{-1}\ne q$ for $1\leq i,j\leq n$. In particular it holds if $x$ is crystalline strictly dominant very regular (cf. Definition \[veryreg\]).]{}
We now fix a point $x=(r,\delta)\in X_{\rm tri}^\square({\overline{r}})$ which is crystalline strictly dominant very regular and a union $X$ of irreducible components of an open subset of $X_{\rm tri}^\square({\overline{r}})$ such that $X$ satisfies the accumulation property at $x$ (Definition \[accu\]). It obviously doesn’t change the tangent space $T_{X,x}$ of $X$ at $x$ if we replace $X$ by the union of its irreducible components that contain $x$, hence we may (and do) assume that $x$ belongs to each irreducible component of $X$.
\[inj\] There is an injection of $k(x)$-vector spaces $T_{X,x}\hookrightarrow T_{\mathfrak{X}_{{\overline{r}}}^\square,r}$.
The embedding $X\hookrightarrow X_{\rm tri}^\square({\overline{r}})\hookrightarrow \mathfrak{X}_{{\overline{r}}}^\square\times \mathcal{T}^n_L$ induces an injection on tangent spaces (with obvious notation): $$T_{X,x}\hookrightarrow T_{\mathfrak{X}_{{\overline{r}}}^\square,r}\oplus T_{\mathcal{T}^n_L,\delta}.$$ We thus have to show that the composition with the projection $T_{\mathfrak{X}_{{\overline{r}}}^\square,r}\oplus T_{\mathcal{T}^n_L,\delta}\twoheadrightarrow T_{\mathfrak{X}_{{\overline{r}}}^\square,r}$ remains injective. Let $\vec{v}\in T_{X,x}$ which maps to $0\in T_{\mathfrak{X}_{{\overline{r}}}^\square,r}$, and thus [*a fortiori*]{} to $0$ in ${\rm Ext}^1_{\mathcal{G}_K}(r,r)$ via the surjection in Lemma \[ext1\]. We have to show that the image of $\vec{v}$ in $T_{\mathcal{T}^n_L,\delta}$ is also $0$. We know that the image of $\vec{v}$ in $T_{\mathcal{W}^n_L,\omega(x)}$ is zero since the Hodge-Tate weights don’t vary (that is, the $d_{\tau,i,\vec{v}}$ below are all zero, see the beginning of §\[wedgesection\]). To conclude that the image in $T_{\mathcal{T}^n_L,\delta}$ is also $0$, we can for instance use Bergdall’s Theorem \[bergsplit\] below (which uses the accumulation property of $X$ at $x$) together with an obvious induction on $i$.
\[boundtx\] Assume that the $k(x)$-vector space image of $T_{X,x}$ in ${\rm Ext}^1_{\mathcal{G}_K}(r,r)$ has dimension smaller or equal than: $$\dim_{k(x)}{\rm Ext}^1_{\mathcal{G}_K}(r,r)-d_x-\big([K:{\mathbb{Q}}_p]\frac{n(n-1)}{2}-\lg(w_x)\big).$$ Then Theorem \[upperbound\] is true.
From Lemma \[inj\] and Lemma \[ext1\] we obtain a short exact sequence: $$\begin{aligned}
\label{exactseq}
0\longrightarrow K(r)\cap T_{X,x}\longrightarrow T_{X,x}\longrightarrow {\rm Ext}^1_{\mathcal{G}_K}(r,r).\end{aligned}$$ Hence the assumption implies: $$\dim_{k(x)}T_{X,x}\leq \dim_{k(x)}K(r)+\lg(w_x)-d_x+\dim_{k(x)}{\rm Ext}^1_{\mathcal{G}_K}(r,r)-[K:{\mathbb{Q}}_p]\frac{n(n-1)}{2}.$$ But from Lemma \[ext1\], Lemma \[dim\] and Remark \[cascr\] we have: $$\begin{gathered}
\dim_{k(x}K(r)+\lg(w_x)-d_x+\dim_{k(x)}{\rm Ext}^1_{\mathcal{G}_K}(r,r)-[K:{\mathbb{Q}}_p]\frac{n(n-1)}{2}=\\
\lg(w_x)-d_x+n^2+[K:{\mathbb{Q}}_p]\frac{n(n+1)}{2}\end{gathered}$$ which gives Theorem \[upperbound\].
We will see below that $d_x$ correspond to the “weight conditions” and $[K:{\mathbb{Q}}_p]\frac{n(n-1)}{2}-\lg(w_x)$ correspond to the “splitting conditions”.
Tangent spaces and local triangulations {#wedgesection}
---------------------------------------
We recall some of the results of [@Bergdall] that we use to prove a technical statement on the image of $T_{X,x}$ in ${\rm Ext}^1_{\mathcal{G}_K}(r,r)$ (Corollary \[inclv\]).
We keep the notation of §\[start\], in particular $x=(r,\delta)=(r,\delta_1,\dots,\delta_n)\in X_{\rm tri}^\square({\overline{r}})$ is a crystalline strictly dominant very regular point and $X$ is a union of irreducible components of an open subset of $X_{\rm tri}^\square({\overline{r}})$, each component in $X$ satisfying the accumulation property at $x$. Taking a look at [@Bergdall §§5.1,6.1], it is easy to see from the properties of $X_{\rm tri}^\square({\overline{r}})$ and from Definition \[accu\] (together with the discussion that follows) that one can apply all the results of [@Bergdall §7] at $X$ and the point $x$ (called the “center” and denoted by $x_0$ in [*loc. cit.*]{}). We let $w_x=(w_{x,\tau})_{\tau:\, K\hookrightarrow L}\in \prod_{\tau: K\hookrightarrow L}{\mathcal{S}}_n$ be the Weyl group element associated to $x$ (§\[weyl\]).
Recall that $D_{{\mathrm{rig}}}(r)$ is the étale $(\varphi,\Gamma_K)$-module over $\mathcal{R}_{k(x),K}=k(x)\otimes_{{\mathbb{Q}}_p}\mathcal{R}_{K}$ associated to $r$. Note that ${\rm Ext}^1_{\mathcal{G}_K}(r,r)\cong {\rm Ext}^1_{(\varphi,\Gamma_K)}(D_{{\mathrm{rig}}}(r),D_{{\mathrm{rig}}}(r))$ where the right hand side denotes the extension in the category of $(\varphi,\Gamma_K)$-module over $\mathcal{R}_{k(x),K}$ (see [@BelChe Prop.5.2.6] for $K={\mathbb{Q}}_p$, the proof for any $K$ is analogous). We write $\omega(x)=\delta_{\bf k}$ for ${\bf k}=(k_{\tau,i})_{1\leq i\leq n,\tau:\, K\hookrightarrow L}\in(\mathbb{Z}^n)^{{\mathrm{Hom}}(K,L)}$. Let $\vec{v}\in T_{X,x}$, seeing the image of $\vec{v}$ in ${\rm Ext}^1_{\mathcal{G}_K}(r,r)$ as a $k(x)[\varepsilon]/(\varepsilon^2)$-valued representation of $\mathcal{G}_K$, we can write its Sen weights as $(k_{\tau,i}+\varepsilon d_{\tau,i,\vec{v}})_{1\leq i\leq n,\tau:\, K\hookrightarrow L}$ for some $d_{\tau,i,\vec{v}}\in k(x)$. The tangent space $T_{{\mathcal W}_L^n,\omega(x)}$ to ${\mathcal W}_L^n$ at $\omega(x)$ is isomorphic to $k(x)^{[K:{\mathbb{Q}}_p]n}$ and the $k(x)$-linear map of tangent spaces $d\omega : T_{X,x}\longrightarrow T_{{\mathcal W}_L,\omega(x)}$ induced by the weight map $\omega\vert_X$ sends $\vec{v}$ to the tuple $(d_{\tau,i,\vec{v}})_{1\leq i\leq n,\tau:\, K\hookrightarrow L}$. The following theorem is a direct application of [@Bergdall Th.7.1].
\[bergweight\] For any $\vec{v}\in T_{X,x}$, we have $d_{\tau,i,\vec{v}}=d_{\tau,w_{x,\tau}^{-1}(i),\vec{v}}$ for $1\leq i\leq n$ and $\tau:\, K\hookrightarrow L$.
Let $\vec{v}\in T_{X,x}$, we can see $\vec{v}$ as a $k(x)[\varepsilon]/(\varepsilon^2)$-valued point of $X$, and the composition: $${\mathrm{Sp}}\,k(x)[\varepsilon]/(\varepsilon^2)\buildrel \vec{v}\over \longrightarrow X\hookrightarrow X_{\rm tri}^\square({\overline{r}})\longrightarrow \mathcal{T}^n_L$$ gives rise to continuous characters $\delta_{i,\vec{v}}:K^\times \rightarrow (k(x)[\varepsilon]/(\varepsilon^2))^\times$ for $1\leq i\leq n$. The following theorem again follows from an examination of the proof of [@Bergdall Th.7.1].
\[bergsplit\] For any $\vec{v}\in T_{X,x}$ and $1\leq i\leq n$ we have an injection of $(\varphi,\Gamma_K)$-modules over $\mathcal{R}_{k(x)[\varepsilon]/(\varepsilon^2),K}=k(x)[\varepsilon]/(\varepsilon^2)\otimes_{{\mathbb{Q}}_p}\mathcal{R}_{K}$: $$\mathcal{R}_{k(x)[\varepsilon]/(\varepsilon^2),K}\big(\delta_{1,\vec{v}}\delta_{2,\vec{v}}\cdots \delta_{i,\vec{v}}\big)\hookrightarrow D_{{\mathrm{rig}}}(\wedge_{k(x)[\varepsilon]/(\varepsilon^2)}^ir_{\vec{v}})\cong \wedge_{\mathcal{R}_{k(x)[\varepsilon]/(\varepsilon^2),K}}^iD_{{\mathrm{rig}}}(r_{\vec{v}})$$ where the left hand side is the rank one $(\varphi,\Gamma_K)$-module defined by the character $\delta_{1,\vec{v}}\delta_{2,\vec{v}}\cdots \delta_{i,\vec{v}}$ ([@KPX Cons.6.2.4]) and where $r_{\vec{v}}$ is the $k(x)[\varepsilon]/(\varepsilon^2)$-valued representation of $\mathcal{G}_K$ associated to $\vec{v}$.
From [@BelChe Prop.2.4.1] (which readily extends to $K\ne {\mathbb{Q}}_p$) or arguing as in §\[variant\], the $(\varphi,\Gamma_K)$-module $D_{{\mathrm{rig}}}(r)$ has a triangulation ${\mathrm{Fil}}_{\bullet}$ for $\bullet\in \{1,\cdots,n\}$, the graded pieces being: $$\begin{aligned}
\label{triang}
\mathcal{R}_{k(x),K}\big(z^{{\bf k}_{w_x^{-1}(1)}}{\rm unr}(\varphi_1)\big),\dots,\mathcal{R}_{k(x),K}\big(z^{{\bf k}_{w_x^{-1}(n)}}{\rm unr}(\varphi_n)\big)\end{aligned}$$ where ${\bf k}_{w_x^{-1}(i)}:=(k_{\tau,w_{x,\tau}^{-1}(i)})_{\tau:\, K\hookrightarrow L}$ (see (\[charc\]) for $z^{{\bf k}_{j}}$). Note that we have: $$\label{dexat}
\delta_{i}(z)=z^{{\bf k}_{i}-{\bf k}_{w_x^{-1}(i)}}(z^{{\bf k}_{w_x^{-1}(i)}}{\rm unr}(\varphi_i)).$$ For $1\leq i\leq n$ we let $D_{{\mathrm{rig}}}(r)^{\leq i}:={\mathrm{Fil}}_{i}\subseteq D_{{\mathrm{rig}}}(r)$, and we set $D_{{\mathrm{rig}}}(r)^{\leq 0}:=0$. We thus have for $1\leq i\leq n$: $${\rm gr}_iD_{{\mathrm{rig}}}(r):=D_{{\mathrm{rig}}}(r)^{\leq i}/D_{{\mathrm{rig}}}(r)^{\leq i-1}=\mathcal{R}_{k(x),K}\big(z^{{\bf k}_{w_x^{-1}(i)}}{\rm unr}(\varphi_i)\big).$$ For $\tau:\, K\hookrightarrow L$ we fix a Lubin-Tate element $t_\tau\in \mathcal{R}_{L,K}$ as in [@KPX Not.6.2.7] (recall that the ideal $t_\tau\mathcal{R}_{L,K}$ is uniquely determined). If ${\bf k}:=(k_\tau)_{\tau:\, K\hookrightarrow L}\in{\mathbb{Z}}_{\geq 0}^{{\mathrm{Hom}}(K,L)}$, we let $t^{\bf k}:=\prod_{\tau:\, K\hookrightarrow L}t_\tau^{k_{\tau}}$. We set for $1\leq i\leq n$: $$\Sigma_i({\bf k},w_x):=\sum_{j=1}^i({\bf k}_{j}-{\bf k}_{w_x^{-1}(j)})\in {\mathbb{Z}}_{\geq 0}^{{\mathrm{Hom}}(K,L)}$$ (where nonnegativity comes from $k_{\tau,i}\geq k_{\tau,i+1}$ for every $i,\tau$) and we can thus define $t^{\Sigma_i({\bf k},w_x)}\in \mathcal{R}_{L,K}$. In particular we deduce from (\[dexat\]) (and the properties of the $t_\tau$): $$\begin{aligned}
\label{wedgedelta}
\mathcal{R}_{k(x),K}(\delta_{1}\cdots \delta_{i})\cong t^{\Sigma_i({\bf k},w_x)}\wedge^i_{\mathcal{R}_{k(x),K}}D_{{\mathrm{rig}}}(r)^{\leq i}\hookrightarrow \wedge^i_{\mathcal{R}_{k(x),K}}D_{{\mathrm{rig}}}(r).\end{aligned}$$
We consider for $1\leq i\leq n$ the cartesian square (which defines $V_i$): $$\xymatrix{{\rm Ext}^1_{(\varphi,\Gamma_K)}\big(D_{{\mathrm{rig}}}(r),D_{{\mathrm{rig}}}(r)\big)\ar[r]&{\rm Ext}^1_{(\varphi,\Gamma_K)}\big(t^{\Sigma_i({\bf k},w_x)}D_{{\mathrm{rig}}}(r)^{\leq i},D_{{\mathrm{rig}}}(r)\big)\\
V_i \ar[r]\ar@{^{(}->}[u]&
{\rm Ext}^1_{(\varphi,\Gamma_K)}\big(t^{\Sigma_i({\bf k},w_x)}D_{{\mathrm{rig}}}(r)^{\leq i},D_{{\mathrm{rig}}}(r)^{\leq i}\big)\ar@{^{(}->}[u]}$$ where the first horizontal map is the restriction map and where the injection on the right easily follows from the very regularity assumption (Definition \[veryreg\]). Equivalently we have: $$\begin{gathered}
\label{viker}
V_i\cong \ker\!\Big({\rm Ext}^1_{(\varphi,\Gamma_K)}\big(D_{{\mathrm{rig}}}(r),D_{{\mathrm{rig}}}(r)\big)\longrightarrow
{\rm Ext}^1_{(\varphi,\Gamma_K)}\big(t^{\Sigma_i({\bf k},w_x)}D_{{\mathrm{rig}}}(r)^{\leq i},D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq i}\big)\Big)\end{gathered}$$ where the map is defined by pushforward along $D_{{\mathrm{rig}}}(r)\twoheadrightarrow D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq i}$ and pullback along $t^{\Sigma_{i}({\bf k},w_x)}D_{{\mathrm{rig}}}(r)^{\leq i}\hookrightarrow D_{{\mathrm{rig}}}(r)$.
\[inclv\] The image of any $\vec{v}\in T_{X,x}$ in ${\rm Ext}^1_{\mathcal{G}_K}(r,r)\cong {\rm Ext}^1_{(\varphi,\Gamma_K)}(D_{{\mathrm{rig}}}(r),D_{{\mathrm{rig}}}(r))$ is in $V_1\cap \cdots \cap V_{n-1}$ (where the intersection is within ${\rm Ext}^1_{(\varphi,\Gamma_K)}(D_{{\mathrm{rig}}}(r),D_{{\mathrm{rig}}}(r))$).
Note that $V_1\cap V_2\cap \cdots \cap V_{n}=V_1\cap V_2\cap \cdots \cap V_{n-1}$. Let $\vec{v}\in T_{X,x}$, $r_{\vec{v}}$ the associated $k(x)[\varepsilon]/(\varepsilon^2)$-deformation and see $D_{{\mathrm{rig}}}(r_{\vec{v}})$ as an element of ${\rm Ext}^1_{(\varphi,\Gamma_K)}(D_{{\mathrm{rig}}}(r),D_{{\mathrm{rig}}}(r))$. We have to prove that the image of $D_{{\mathrm{rig}}}(r_{\vec{v}})$ in ${\rm Ext}^1_{(\varphi,\Gamma_K)}(t^{\Sigma_i({\bf k},w_x)}D_{{\mathrm{rig}}}(r)^{\leq i},D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq i})$ is zero for any $1\leq i\leq n$ (see (\[viker\])). The proof is by induction on $i\geq 1$. The case $i=1$ follows immediately from Theorem \[bergsplit\] and (\[wedgedelta\]) (together with Definition \[veryreg\]). We prove that the statement for $i-1$ implies the statement for $i$.
So, assume $i\geq 2$ and that the image of $D_{{\mathrm{rig}}}(r_{\vec{v}})$ in: $${\rm Ext}^1_{(\varphi,\Gamma_K)}\big(t^{\Sigma_{i-1}({\bf k},w_x)}D_{{\mathrm{rig}}}(r)^{\leq i-1},D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq i-1}\big)$$ is zero. Then by Corollary \[zero2\] the image of $D_{{\mathrm{rig}}}(r_{\vec{v}})$ in: $${\rm Ext}^1_{(\varphi,\Gamma_K)}\big(t^{\Sigma_{i}({\bf k},w_x)}D_{{\mathrm{rig}}}(r)^{\leq i-1},D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq i}\big)$$ is also zero. From the exact sequence: $$\begin{gathered}
0\rightarrow {\rm Ext}^1_{(\varphi,\Gamma_K)}\big(t^{\Sigma_{i}({\bf k},w_x)}{\rm gr}_iD_{{\mathrm{rig}}}(r),D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq i}\big)\rightarrow \\
{\rm Ext}^1_{(\varphi,\Gamma_K)}\big(t^{\Sigma_{i}({\bf k},w_x)}D_{{\mathrm{rig}}}(r)^{\leq i},D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq i}\big)\rightarrow \\
{\rm Ext}^1_{(\varphi,\Gamma_K)}\big(t^{\Sigma_{i}({\bf k},w_x)}D_{{\mathrm{rig}}}(r)^{\leq i-1},D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq i}\big)\end{gathered}$$ (where the injectivity on the left follows from Definition \[veryreg\]), we see that the image of $D_{{\mathrm{rig}}}(r_{\vec{v}})$ in ${\rm Ext}^1_{(\varphi,\Gamma_K)}\big(t^{\Sigma_{i}({\bf k},w_x)}D_{{\mathrm{rig}}}(r)^{\leq i},D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq i}\big)$ comes from a unique extension: $$D_{{\mathrm{rig}}}(r_{\vec{v}})^{(i)}\in {\rm Ext}^1_{(\varphi,\Gamma_K)}\big(t^{\Sigma_{i}({\bf k},w_x)}{\rm gr}_iD_{{\mathrm{rig}}}(r),D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq i}\big).$$ We thus have to prove that $D_{{\mathrm{rig}}}(r_{\vec{v}})^{(i)}=0$.
The twist by the rank one $(\varphi,\Gamma_K)$-module $\wedge_{\mathcal{R}_{k(x),K}}^{i-1}D_{{\mathrm{rig}}}(r)^{\leq i-1}$ is easily seen (by elementary linear algebra) to induce an isomorphism: $$\begin{gathered}
\label{wedgetwist}
{\rm Ext}^1_{(\varphi,\Gamma_K)}\big(t^{\Sigma_{i}({\bf k},w_x)}{\rm gr}_iD_{{\mathrm{rig}}}(r),D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq i}\big)\buildrel\sim\over\longrightarrow \\
{\rm Ext}^1_{(\varphi,\Gamma_K)}\big(t^{\Sigma_{i}({\bf k},w_x)}\wedge^iD_{{\mathrm{rig}}}(r)^{\leq i},(D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq i})\wedge (\wedge^{i-1}D_{{\mathrm{rig}}}(r)^{\leq i-1})\big)\end{gathered}$$ where we write $\wedge^\cdot D_{{\mathrm{rig}}}(r)$ for $\wedge_{\mathcal{R}_{k(x),K}}^\cdot D_{{\mathrm{rig}}}(r)$ and where $(D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq i})\wedge (\wedge^{i-1}D_{{\mathrm{rig}}}(r)^{\leq i-1})$ stands for the quotient: $$\begin{gathered}
\big(D_{{\mathrm{rig}}}(r)\wedge (\wedge^{i-1}D_{{\mathrm{rig}}}(r)^{\leq i-1})\big)/\big(D_{{\mathrm{rig}}}(r)^{\leq i}\wedge (\wedge^{i-1}D_{{\mathrm{rig}}}(r)^{\leq i-1})\big)\cong\\
\big(D_{{\mathrm{rig}}}(r)\wedge (\wedge^{i-1}D_{{\mathrm{rig}}}(r)^{\leq i-1})\big)/\wedge^{i}D_{{\mathrm{rig}}}(r)^{\leq i}.\end{gathered}$$ (here, $D_{{\mathrm{rig}}}(r)\wedge (\wedge^{i-1}D_{{\mathrm{rig}}}(r)^{\leq i-1})$ and $D_{{\mathrm{rig}}}(r)^{\leq i}\wedge (\wedge^{i-1}D_{{\mathrm{rig}}}(r)^{\leq i-1})$ are seen inside $\wedge^{i}D_{{\mathrm{rig}}}(r)$). Moreover the injective map $\wedge^{i-1}D_{{\mathrm{rig}}}(r)^{\leq i-1}\hookrightarrow \wedge^{i-1}D_{{\mathrm{rig}}}(r)^{\leq i}$ still induces an injection (using Definition \[veryreg\]): $$\begin{gathered}
{\rm Ext}^1_{(\varphi,\Gamma_K)}\big(t^{\Sigma_{i}({\bf k},w_x)}\wedge^iD_{{\mathrm{rig}}}(r)^{\leq i},(D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq i})\wedge (\wedge^{i-1}D_{{\mathrm{rig}}}(r)^{\leq i-1})\big)\hookrightarrow\\
{\rm Ext}^1_{(\varphi,\Gamma_K)}\big(t^{\Sigma_{i}({\bf k},w_x)}\wedge^iD_{{\mathrm{rig}}}(r)^{\leq i},(D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq i})\wedge (\wedge^{i-1}D_{{\mathrm{rig}}}(r)^{\leq i})\big).\end{gathered}$$ Denote by: $$\label{rtilde}
\widetilde D_{{\mathrm{rig}}}(r_{\vec{v}})^{(i)}\ \in \ {\rm Ext}^1_{(\varphi,\Gamma_K)}\big(t^{\Sigma_{i}({\bf k},w_x)}\wedge^iD_{{\mathrm{rig}}}(r)^{\leq i},(D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq i})\wedge (\wedge^{i-1}D_{{\mathrm{rig}}}(r)^{\leq i})\big)$$ the image of $D_{{\mathrm{rig}}}(r_{\vec{v}})^{(i)}$ (using the isomorphism (\[wedgetwist\])). It is thus equivalent to prove that $\widetilde D_{{\mathrm{rig}}}(r_{\vec{v}})^{(i)}=0$. Note that: $$\begin{gathered}
\label{wedgequot}
(D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq i})\wedge (\wedge^{i-1}D_{{\mathrm{rig}}}(r)^{\leq i})\cong \big(D_{{\mathrm{rig}}}(r)\wedge (\wedge^{i-1}D_{{\mathrm{rig}}}(r)^{\leq i})\big)/\wedge^{i}D_{{\mathrm{rig}}}(r)^{\leq i}.\end{gathered}$$
For $1\leq i\leq n$, we have a $k(x)$-linear map ${\rm Ext}^1_{\mathcal{G}_K}(r,r)\rightarrow {\rm Ext}^1_{\mathcal{G}_K}(\wedge_{k(x)}^ir,\wedge_{k(x)}^ir)$ defined by mapping a $k(x)[\varepsilon]/(\varepsilon^2)$-valued representation of $\mathcal{G}_K$ to its $i$-th exterior power over $k(x)[\varepsilon]/(\varepsilon^2)$. This induces an $\mathcal{R}_{k(x)[\varepsilon]/(\varepsilon^2),K}$-linear map: $${\rm Ext}^1_{(\varphi,\Gamma_K)}(D_{{\mathrm{rig}}}(r),D_{{\mathrm{rig}}}(r))\longrightarrow {\rm Ext}^1_{(\varphi,\Gamma_K)}\big(\wedge^i\!D_{{\mathrm{rig}}}(r),\wedge^iD_{{\mathrm{rig}}}(r)\big).$$ Let $D_{{\mathrm{rig}}}(\wedge^ir_{\vec{v}})\in {\rm Ext}^1_{(\varphi,\Gamma_K)}(\wedge^i\!D_{{\mathrm{rig}}}(r),\wedge^iD_{{\mathrm{rig}}}(r))$ be the image of $D_{{\mathrm{rig}}}(r_{\vec{v}})$. The pull-back along $\wedge^iD_{{\mathrm{rig}}}(r)^{\leq i}\hookrightarrow \wedge^iD_{{\mathrm{rig}}}(r)$ sends $D_{{\mathrm{rig}}}(\wedge^ir_{\vec{v}})$ to an element in ${\rm Ext}^1_{(\varphi,\Gamma_K)}(\wedge^iD_{{\mathrm{rig}}}(r)^{\leq i},\wedge^{i}D_{{\mathrm{rig}}}(r))$. Elementary linear algebra (recall $\varepsilon^2=0$!) shows this element in fact belongs to: $${\rm Ext}^1_{(\varphi,\Gamma_K)}\big(\wedge^iD_{{\mathrm{rig}}}(r)^{\leq i},D_{{\mathrm{rig}}}(r)\wedge (\wedge^{i-1}D_{{\mathrm{rig}}}(r)^{\leq i})\big)$$ (which embeds into ${\rm Ext}^1_{(\varphi,\Gamma_K)}(\wedge^iD_{{\mathrm{rig}}}(r)^{\leq i},\wedge^{i}D_{{\mathrm{rig}}}(r))$ again by Definition \[veryreg\]). The pushforward along: $$D_{{\mathrm{rig}}}(r)\wedge (\wedge^{i-1}D_{{\mathrm{rig}}}(r)^{\leq i})\twoheadrightarrow (D_{{\mathrm{rig}}}(r)\wedge (\wedge^{i-1}D_{{\mathrm{rig}}}(r)^{\leq i}))/\wedge^{i}D_{{\mathrm{rig}}}(r)^{\leq i}$$ now gives by (\[wedgequot\]) an element in: $${\rm Ext}^1_{(\varphi,\Gamma_K)}\big(\wedge^iD_{{\mathrm{rig}}}(r)^{\leq i},(D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq i})\wedge (\wedge^{i-1}D_{{\mathrm{rig}}}(r)^{\leq i})\big)$$ and further pull-back along $t^{\Sigma_{i}({\bf k},w_x)}\wedge^iD_{{\mathrm{rig}}}(r)^{\leq i}\hookrightarrow \wedge^iD_{{\mathrm{rig}}}(r)^{\leq i}$ finally gives an element: $$\label{wedgetilde}
\widetilde D_{{\mathrm{rig}}}(\wedge^ir_{\vec{v}})\ \in \ {\rm Ext}^1_{(\varphi,\Gamma_K)}\big(t^{\Sigma_{i}({\bf k},w_x)}\wedge^iD_{{\mathrm{rig}}}(r)^{\leq i},(D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq i})\wedge (\wedge^{i-1}D_{{\mathrm{rig}}}(r)^{\leq i})\big).$$ Now, again manipulations of elementary linear algebra show we recover the element $\widetilde D_{\rm rig}(r_{\vec{v}})^{(i)}$ of (\[rtilde\]), that is, we have $\widetilde D_{{\mathrm{rig}}}(\wedge^ir_{\vec{v}})=\widetilde D_{\rm rig}(r_{\vec{v}})^{(i)}$.
But we know from Theorem \[bergsplit\] (using (\[wedgedelta\]) and Definition \[veryreg\]) that the image of $D_{{\mathrm{rig}}}(\wedge^ir_{\vec{v}})$ (by pullback) in ${\rm Ext}^1_{(\varphi,\Gamma_K)}\big(t^{\Sigma_{i}({\bf k},w_x)}\wedge^iD_{{\mathrm{rig}}}(r)^{\leq i},\wedge^iD_{{\mathrm{rig}}}(r)\big)$ actually sits in: $${\rm Ext}^1_{(\varphi,\Gamma_K)}\big(t^{\Sigma_{i}({\bf k},w_x)}\wedge^iD_{{\mathrm{rig}}}(r)^{\leq i},\wedge^iD_{{\mathrm{rig}}}(r)^{\leq i}\big)$$ (in fact even in the image of ${\rm Ext}^1_{(\varphi,\Gamma_K)}(t^{\Sigma_{i}({\bf k},w_x)}\wedge^iD_{{\mathrm{rig}}}(r)^{\leq i},t^{\Sigma_{i}({\bf k},w_x)}\wedge^iD_{{\mathrm{rig}}}(r)^{\leq i})$). In particular its image $\widetilde D_{{\mathrm{rig}}}(\wedge^ir_{\vec{v}})$ in: $${\rm Ext}^1_{(\varphi,\Gamma_K)}\big(t^{\Sigma_{i}({\bf k},w_x)}\wedge^iD_{{\mathrm{rig}}}(r)^{\leq i},(D_{{\mathrm{rig}}}(r)\wedge (\wedge^{i-1}D_{{\mathrm{rig}}}(r)^{\leq i}))/\wedge^{i}D_{{\mathrm{rig}}}(r)^{\leq i}\big)$$ must be zero. Since $\widetilde D_{{\mathrm{rig}}}(\wedge^ir_{\vec{v}})=\widetilde D_{\rm rig}(r_{\vec{v}})^{(i)}$, we obtain $\widetilde D_{\rm rig}(r_{\vec{v}})^{(i)}=0$.
Proof of the main local theorem {#endofproof}
-------------------------------
We compute various dimensions and finish the proof of Theorem \[upperbound\].
Seeing an element of ${\rm Ext}^1_{\mathcal{G}_K}(r,r)$ as a $k(x)[\varepsilon]/(\varepsilon^2)$-valued representation of $\mathcal{G}_K$, we can write its Sen weights as $(k_{\tau,i}+\varepsilon d_{\tau,i})_{1\leq i\leq n,\tau:\, K\hookrightarrow L}$ for some $d_{\tau,i}\in k(x)$. We let $V$ be the $k(x)$-subvector space of ${\rm Ext}^1_{\mathcal{G}_K}(r,r)$ (or of ${\rm Ext}^1_{(\varphi,\Gamma_K)}(D_{{\mathrm{rig}}}(r),D_{{\mathrm{rig}}}(r))$) of extensions such that $d_{\tau,i}=d_{\tau,w_{x,\tau}^{-1}(i)}$ for $1\leq i\leq n$ and $\tau:\, K\hookrightarrow L$.
\[condweight\] We have $\dim_{k(x)}V =\dim_{k(x)}{\rm Ext}^1_{\mathcal{G}_K}(r,r)-d_x$.
The Sen map ${\rm Ext}^1_{\mathcal{G}_K}(r,r)\cong {\rm Ext}^1_{(\varphi,\Gamma_K)}(D_{{\mathrm{rig}}}(r),D_{{\mathrm{rig}}}(r))\longrightarrow k(x)^{[K:{\mathbb{Q}}_p]n}$ sending an extension to $(d_{\tau,i})_{1\leq i\leq n,\tau:\, K\hookrightarrow L}$ is easily checked to be surjective (by a dévissage argument using Définition \[veryreg\], we are reduced to the rank one case where it is obvious). The $k(x)$-subvector space of $k(x)^{[K:{\mathbb{Q}}_p]n}$ of tuples $(d_{\tau,i})_{1\leq i\leq n,\tau:\, K\hookrightarrow L}$ such that $d_{\tau,i}=d_{\tau,w_{x,\tau}^{-1}(i)}$ for $1\leq i\leq n$ and $\tau:\, K\hookrightarrow L$ has dimension $[K:{\mathbb{Q}}_p]n-d_x$ (argue as in the beginning of the proof of Lemma \[coxeter\]). The result follows.
Recall that a $(\varphi,\Gamma_K)$-module $D$ over $\mathcal{R}_{k(x),K}$ is called [*crystalline*]{} if $D[1/\prod_{\tau:\, K\hookrightarrow L}t_\tau]^{\Gamma_K}$ is free over $K_0\otimes_{{\mathbb{Q}}_p}k(x)$ of the same rank as $D$. If $D,D'$ are two crystalline $(\varphi,\Gamma_K)$-module over $\mathcal{R}_{k(x),K}$, one can define the $k(x)$-subvector space of [*crystalline extensions*]{} ${\rm Ext}^1_{\rm cris}(D,D')\subseteq {\rm Ext}^1_{(\varphi,\Gamma_K)}(D,D')$. Note that ${\rm Ext}^1_{\rm cris}(\cdot,\cdot)$ respects surjectivities on the right entry (resp. sends injectivities to surjectivities on the left entry) as there is no ${\rm Ext}^2_{\rm cris}$, see [@Benois Cor.1.4.6].
\[dim1\] For $1\leq i\leq \ell\leq n$ we have: $$\begin{gathered}
\dim_{k(x)}{\rm Ext}^1_{(\varphi,\Gamma_K)}\big({\rm gr}_iD_{{\mathrm{rig}}}(r)/(t^{\Sigma_{\ell}({\bf k},w_x)}),D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq \ell}\big)=\\
\sum_{\tau:\, K\hookrightarrow L}\big\vert\{i_1\in \{\ell+1,\dots,n\}, w_{x,\tau}^{-1}(i_1)<w_{x,\tau}^{-1}(i)\}\big\vert.\end{gathered}$$
It follows from Proposition \[cris\] below (applied with $(i,\ell)=(i,i)$ and $(i,\ell)=(i-1,i)$) together with the two exact sequences: $$\begin{gathered}
0\rightarrow {\rm Ext}^1_{(\varphi,\Gamma_K)}\big({\rm gr}_iD_{{\mathrm{rig}}}(r)/(t^{\Sigma_{i}({\bf k},w_x)}),D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq i}\big)\rightarrow \\
{\rm Ext}^1_{(\varphi,\Gamma_K)}\big(D_{{\mathrm{rig}}}^{\leq i}(r)/(t^{\Sigma_{i}({\bf k},w_x)}),D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq i}\big)\rightarrow \\
{\rm Ext}^1_{(\varphi,\Gamma_K)}\big(D_{{\mathrm{rig}}}(r)^{\leq i-1}/(t^{\Sigma_{i}({\bf k},w_x)}),D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq i}\big),\end{gathered}$$ $$\begin{gathered}
0\rightarrow {\rm Ext}^1_{\rm cris}\big({\rm gr}_iD_{{\mathrm{rig}}}(r),D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq i}\big)\rightarrow {\rm Ext}^1_{\rm cris}\big(D_{{\mathrm{rig}}}^{\leq i},D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq i}\big)
\rightarrow \\
{\rm Ext}^1_{\rm cris}\big(D_{{\mathrm{rig}}}(r)^{\leq i-1},D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq i}\big)\end{gathered}$$ (injectivity on the left following again from Definition \[veryreg\]), that we have: $$\begin{gathered}
\label{dimcrisi}
{\rm Ext}^1_{(\varphi,\Gamma_K)}\big({\rm gr}_iD_{{\mathrm{rig}}}(r)/(t^{\Sigma_{\ell}({\bf k},w_x)}),D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq \ell}\big)\cong \\
{\rm Ext}^1_{\rm cris}\big({\rm gr}_iD_{{\mathrm{rig}}}(r),D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq \ell}\big).\end{gathered}$$ By dévissage on $D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq \ell}$ using that ${\rm Ext}^1_{\rm cris}$ respects here short exact sequences (by Definition \[veryreg\] and the discussion above), we have: $$\dim_{k(x)}{\rm Ext}^1_{\rm cris}\big({\rm gr}_iD_{{\mathrm{rig}}}(r),D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq \ell}\big)=
\sum_{i_1=\ell+1}^{n}\dim_{k(x)}{\rm Ext}^1_{\rm cris}\big({\rm gr}_iD_{{\mathrm{rig}}}(r),{\rm gr}_{i_1}D_{{\mathrm{rig}}}(r)\big).$$ The result follows from (\[dimcris\]) below.
\[condsplit\] We have: $$\dim_{k(x)}\big( V_1\cap V_2\cap \cdots \cap V_{n-1}\big) = \dim_{k(x)}{\rm Ext}^1_{\mathcal{G}_K}(r,r)-\big([K:{\mathbb{Q}}_p]\frac{n(n-1)}{2}-\lg(w_x)\big).$$
To lighten notation in this proof, we write $D_{{\mathrm{rig}}}$ instead of $D_{{\mathrm{rig}}}(r)$ and drop the subscript $(\varphi,\Gamma_K)$. We first prove that, for $1\leq i\leq n$, we have an isomorphism of $k(x)$-vector spaces: $$\begin{gathered}
\label{isoi}
V_1\cap\cdots \cap V_{i-1}/V_1\cap\cdots \cap V_{i} \buildrel\sim\over\longrightarrow\\
{\rm Ext}^1\big({\rm gr}_iD_{{\mathrm{rig}}},D_{{\mathrm{rig}}}/D_{{\mathrm{rig}}}^{\leq i}\big)\big/{\rm Ext}^1\big({\rm gr}_iD_{{\mathrm{rig}}}/(t^{\Sigma_{i}({\bf k},w_x)}),D_{{\mathrm{rig}}}/D_{{\mathrm{rig}}}^{\leq i}\big)\end{gathered}$$ where $V_1\cap\cdots \cap V_{i-1}:={\rm Ext}^1(D_{{\mathrm{rig}}},D_{{\mathrm{rig}}})$ if $i=1$. We first define the map. We have the following commutative diagram: $$\begin{matrix}
&&\scriptstyle 0 &&\scriptstyle 0 &&\scriptstyle 0 &&\\
&&\downarrow &&\downarrow &&\downarrow &&\\
\!\!\!\scriptstyle{0}&\!\!\!\scriptstyle{\rightarrow} &\!\!\!\scriptstyle{{\rm Ext}^1({\rm gr}_iD_{{\mathrm{rig}}}/(t^{\Sigma_{i}({\bf k},w_x)}),D_{{\mathrm{rig}}}/D_{{\mathrm{rig}}}^{\leq i})}&\!\!\!\scriptstyle\rightarrow &\!\!\!\scriptstyle{\rm Ext}^1(D_{{\mathrm{rig}}}^{\leq i}/(t^{\Sigma_{i}({\bf k},w_x)}),D_{{\mathrm{rig}}}/D_{{\mathrm{rig}}}^{\leq i})&\!\!\!\scriptstyle\rightarrow &\!\!\!\scriptstyle{\rm Ext}^1(D_{{\mathrm{rig}}}^{\leq i-1}/(t^{\Sigma_{i}({\bf k},w_x)}),D_{{\mathrm{rig}}}/D_{{\mathrm{rig}}}^{\leq i})&\!\!\!\scriptstyle\rightarrow &\!\!\!\scriptstyle 0\\
&&\downarrow &&\downarrow &&\downarrow &&\\
\!\!\!\scriptstyle 0&\!\!\!\scriptstyle\rightarrow &\!\!\!\scriptstyle{\rm Ext}^1({\rm gr}_iD_{{\mathrm{rig}}},D_{{\mathrm{rig}}}/D_{{\mathrm{rig}}}^{\leq i})&\!\!\!\scriptstyle\rightarrow &\!\!\!\scriptstyle{\rm Ext}^1(D_{{\mathrm{rig}}}^{\leq i},D_{{\mathrm{rig}}}/D_{{\mathrm{rig}}}^{\leq i})&\!\!\!\scriptstyle\rightarrow &\!\!\!\scriptstyle{\rm Ext}^1(D_{{\mathrm{rig}}}^{\leq i-1},D_{{\mathrm{rig}}}/D_{{\mathrm{rig}}}^{\leq i})&\!\!\!\scriptstyle\rightarrow &\!\!\!\scriptstyle 0\\
&&\downarrow &&\downarrow &&\downarrow &&\\
\!\!\!\scriptstyle 0&\!\!\!\scriptstyle\rightarrow &\!\!\!\scriptstyle{\rm Ext}^1(t^{\Sigma_{i}({\bf k},w_x)}{\rm gr}_iD_{{\mathrm{rig}}},D_{{\mathrm{rig}}}/D_{{\mathrm{rig}}}^{\leq i})&\!\!\!\scriptstyle\rightarrow &\!\!\!\scriptstyle{\rm Ext}^1(t^{\Sigma_{i}({\bf k},w_x)}D_{{\mathrm{rig}}}^{\leq i},D_{{\mathrm{rig}}}/D_{{\mathrm{rig}}}^{\leq i})&\!\!\!\scriptstyle\rightarrow &\!\!\!\scriptstyle{\rm Ext}^1(t^{\Sigma_{i}({\bf k},w_x)}D_{{\mathrm{rig}}}^{\leq i-1},D_{{\mathrm{rig}}}/D_{{\mathrm{rig}}}^{\leq i})&\!\!\!\scriptstyle\rightarrow &\!\!\!\scriptstyle 0
\end{matrix}$$ where the injections on top and left and the surjections on the two bottom lines all follow from Definition \[veryreg\], and where the surjection on the top right corner follows from Corollary \[zero3\] below. Denote by $E_i$ the inverse image of ${\rm Ext}^1(D_{{\mathrm{rig}}}^{\leq i-1}/(t^{\Sigma_{i}({\bf k},w_x)}),D_{{\mathrm{rig}}}/D_{{\mathrm{rig}}}^{\leq i})$ in ${\rm Ext}^1(D_{{\mathrm{rig}}}^{\leq i},D_{{\mathrm{rig}}}/D_{{\mathrm{rig}}}^{\leq i})$, then we have an isomorphism: $$\begin{gathered}
\label{coim}
{\rm Ext}^1\big({\rm gr}_iD_{{\mathrm{rig}}},D_{{\mathrm{rig}}}/D_{{\mathrm{rig}}}^{\leq i}\big)\big/{\rm Ext}^1\big({\rm gr}_iD_{{\mathrm{rig}}}/(t^{\Sigma_{i}({\bf k},w_x)}),D_{{\mathrm{rig}}}/D_{{\mathrm{rig}}}^{\leq i}\big)\buildrel\sim\over\rightarrow \\
E_i\big/{\rm Ext}^1\big(D_{{\mathrm{rig}}}^{\leq i}/(t^{\Sigma_{i}({\bf k},w_x)}),D_{{\mathrm{rig}}}/D_{{\mathrm{rig}}}^{\leq i}\big).\end{gathered}$$ We consider the composition: $$\begin{gathered}
\label{mapi}
V_1\cap\cdots \cap V_{i-1}\hookrightarrow {\rm Ext}^1\big(D_{{\mathrm{rig}}},D_{{\mathrm{rig}}}\big)\twoheadrightarrow {\rm Ext}^1\big(D_{{\mathrm{rig}}}^{\leq i},D_{{\mathrm{rig}}}/D_{{\mathrm{rig}}}^{\leq i}\big)\twoheadrightarrow \\
{\rm Ext}^1\big(D_{{\mathrm{rig}}}^{\leq i},D_{{\mathrm{rig}}}/D_{{\mathrm{rig}}}^{\leq i}\big)\big/{\rm Ext}^1\big(D_{{\mathrm{rig}}}^{\leq i}/(t^{\Sigma_{i}({\bf k},w_x)}),D_{{\mathrm{rig}}}/D_{{\mathrm{rig}}}^{\leq i}\big)\end{gathered}$$ and note that the image of $V_1\cap\cdots \cap V_{i-1}$ falls in $E_i\big/{\rm Ext}^1(D_{{\mathrm{rig}}}^{\leq i}/(t^{\Sigma_{i}({\bf k},w_x)}),D_{{\mathrm{rig}}}/D_{{\mathrm{rig}}}^{\leq i})$ by Corollary \[zero2\] below. If $v\in V_1\cap\cdots \cap V_{i-1}$ is also in $V_i$, then its image in ${\rm Ext}^1(D_{{\mathrm{rig}}}^{\leq i},D_{{\mathrm{rig}}}/D_{{\mathrm{rig}}}^{\leq i})$ maps to $0$ in ${\rm Ext}^1(t^{\Sigma_{i}({\bf k},w_x)}D_{{\mathrm{rig}}}^{\leq i},D_{{\mathrm{rig}}}/D_{{\mathrm{rig}}}^{\leq i})$, hence belongs to ${\rm Ext}^1(D_{{\mathrm{rig}}}^{\leq i}/(t^{\Sigma_{i}({\bf k},w_x)}),D_{{\mathrm{rig}}}/D_{{\mathrm{rig}}}^{\leq i})$. By (\[coim\]), we thus have a canonical induced map: $$\begin{gathered}
\label{imap}
V_1\cap\cdots \cap V_{i-1}/V_1\cap\cdots \cap V_{i} \rightarrow \\
{\rm Ext}^1\big({\rm gr}_iD_{{\mathrm{rig}}},D_{{\mathrm{rig}}}/D_{{\mathrm{rig}}}^{\leq i}\big)\big/{\rm Ext}^1\big({\rm gr}_iD_{{\mathrm{rig}}}/(t^{\Sigma_{i}({\bf k},w_x)}),D_{{\mathrm{rig}}}/D_{{\mathrm{rig}}}^{\leq i}\big).\end{gathered}$$ Let us prove that (\[imap\]) is surjective. One easily checks that ${\rm Ext}^1(D_{{\mathrm{rig}}}/D_{{\mathrm{rig}}}^{\leq i-1},D_{{\mathrm{rig}}})\subseteq V_1\cap\cdots \cap V_{i-1}$ and that the natural map ${\rm Ext}^1(D_{{\mathrm{rig}}}/D_{{\mathrm{rig}}}^{\leq i-1},D_{{\mathrm{rig}}})\rightarrow {\rm Ext}^1\big({\rm gr}_iD_{{\mathrm{rig}}},D_{{\mathrm{rig}}}/D_{{\mathrm{rig}}}^{\leq i}\big)$ is surjective (again by Definition \[veryreg\]). This implies that [*a fortiori*]{} (\[imap\]) must also be surjective. Let us prove that (\[imap\]) is injective. If $v\in V_1\cap\cdots \cap V_{i-1}$ maps to zero, then the image of $v$ in ${\rm Ext}^1(D_{{\mathrm{rig}}}^{\leq i},D_{{\mathrm{rig}}}/D_{{\mathrm{rig}}}^{\leq i})$ belongs to ${\rm Ext}^1(D_{{\mathrm{rig}}}^{\leq i}/(t^{\Sigma_{i}({\bf k},w_x)}),D_{{\mathrm{rig}}}/D_{{\mathrm{rig}}}^{\leq i})$ by (\[coim\]), i.e. maps to zero in ${\rm Ext}^1(t^{\Sigma_{i}({\bf k},w_x)}D_{{\mathrm{rig}}}^{\leq i},D_{{\mathrm{rig}}}/D_{{\mathrm{rig}}}^{\leq i})$, i.e. $v\in V_i$ by (\[viker\]), hence $v\in V_1\cap\cdots \cap V_{i}$.
We now prove the statement of the proposition. From (\[isoi\]) and Lemma \[dim1\], we obtain for $1\leq i\leq n$: $$\begin{gathered}
\dim_{k(x)}\big(V_1\cap\cdots \cap V_{i-1}/V_1\cap\cdots \cap V_{i}\big)=\\
[K:{\mathbb{Q}}_p](n-i)-\sum_{\tau:\, K\hookrightarrow L}\big\vert\{j\in \{i+1,\dots,n\}, w_{x,\tau}^{-1}(j)<w_{x,\tau}^{-1}(i)\}\big\vert=\\
\sum_{\tau:\, K\hookrightarrow L}\big\vert\{j\in \{i+1,\dots,n\}, w_{x,\tau}^{-1}(i)<w_{x,\tau}^{-1}(j)\}\big\vert.\end{gathered}$$ Summing up $\dim_{k(x)}(V_1\cap\cdots \cap V_{i-1}/V_1\cap\cdots \cap V_{i})$ for $i=1$ to $n-1$ thus yields: $$\begin{gathered}
\dim_{k(x)}{\rm Ext}^1\big(D_{{\mathrm{rig}}},D_{{\mathrm{rig}}}\big)-\dim_{k(x)}(V_1\cap\cdots \cap V_{n-1})= \\
\sum_{\tau:\, K\hookrightarrow L}\big\vert\{1\leq i_1<i_2\leq n,\ w_{x,\tau}^{-1}(i_1)<w_{x,\tau}^{-1}(i_2)\}\big\vert.\end{gathered}$$ But $\vert\{1\leq i_1<i_2\leq n,\ w_{x,\tau}^{-1}(i_1)<w_{x,\tau}^{-1}(i_2)\}\vert= \frac{n(n-1)}{2}-\lg(w_{x,\tau})$ (see e.g. [@HumBGG §0.3]), and thus we get: $$\begin{aligned}
\dim_{k(x)}(V_1\cap\cdots \cap V_{n-1})&= &\dim_{k(x)}{\rm Ext}^1\big(D_{{\mathrm{rig}}},D_{{\mathrm{rig}}}\big)-\sum_{\tau:\, K\hookrightarrow L}\big(\frac{n(n-1)}{2}-\lg(w_{x,\tau})\big)\\
&= &\dim_{k(x)}{\rm Ext}^1\big(D_{{\mathrm{rig}}},D_{{\mathrm{rig}}}\big)-\big([K:{\mathbb{Q}}_p]\frac{n(n-1)}{2}-\lg(w_x)\big)\end{aligned}$$ which finishes the proof.
\[inter\] We have: $$\dim_{k(x)}\!\big(V \cap (V_1\cap \cdots \cap V_{n-1})\big)\!= \dim_{k(x)}{\rm Ext}^1_{\mathcal{G}_K}(r,r)-d_x-\big([K:{\mathbb{Q}}_p]\frac{n(n-1)}{2}-\lg(w_x)\big).$$
Consider the following cartesian diagram which defines $W_i$: $$\xymatrix{{\rm Ext}^1_{(\varphi,\Gamma_K)}\big(D_{{\mathrm{rig}}}(r),D_{{\mathrm{rig}}}(r)\big)\ar[r]&{\rm Ext}^1_{(\varphi,\Gamma_K)}\big(D_{{\mathrm{rig}}}(r)^{\leq i},D_{{\mathrm{rig}}}(r)\big)\\
W_i \ar[r]\ar@{^{(}->}[u]&
{\rm Ext}^1_{(\varphi,\Gamma_K)}\big(D_{{\mathrm{rig}}}(r)^{\leq i},D_{{\mathrm{rig}}}(r)^{\leq i}\big)\ar@{^{(}->}[u]},$$ then $W_i\subseteq V_i$, hence $W_1\cap \cdots \cap W_{n-1}\subseteq V_1\cap \cdots \cap V_{n-1}$. In fact, $W_1\cap \cdots \cap W_{n-1}$ is the $k(x)$-subvector space of ${\rm Ext}^1_{(\varphi,\Gamma_K)}(D_{{\mathrm{rig}}}(r),D_{{\mathrm{rig}}}(r))$ of extensions which respect the triangulation $(D_{{\mathrm{rig}}}(r)^{\leq i})_{1\leq i\leq n}$ on $D_{{\mathrm{rig}}}(r)$. A dévissage argument (using Definition \[veryreg\]) that we leave to the reader then shows that the composition: $$W_1\cap \cdots \cap W_{n-1}\hookrightarrow {\rm Ext}^1_{(\varphi,\Gamma_K)}\big(D_{{\mathrm{rig}}}(r),D_{{\mathrm{rig}}}(r)\big)\longrightarrow k(x)^{[K:{\mathbb{Q}}_p]n}$$ (where the second map is the Sen map in the proof of Proposition \[condweight\]) remains surjective. [*A fortiori*]{}, $V_1\cap \cdots \cap V_{n-1}\hookrightarrow {\rm Ext}^1_{(\varphi,\Gamma_K)}(D_{{\mathrm{rig}}}(r),D_{{\mathrm{rig}}}(r))\rightarrow k(x)^{[K:{\mathbb{Q}}_p]n}$ is also surjective. By the same proof as that of Proposition \[condweight\] we get: $$\dim_{k(x)}\!\big(V \cap (V_1\cap \cdots \cap V_{n-1})\big)\!= \dim_{k(x)}(V_1\cap \cdots \cap V_{n-1})-d_x$$ and the result follows from Proposition \[condsplit\].
\[ouf\] The image of $T_{X,x}$ in ${\rm Ext}^1_{\mathcal{G}_K}(r,r)$ has dimension $\leq \dim_{k(x)}{\rm Ext}^1_{\mathcal{G}_K}(r,r)-d_x-\big([K:{\mathbb{Q}}_p]\frac{n(n-1)}{2}-\lg(w_x)\big)$.
It follows from Theorem \[bergweight\] that the image of any $\vec{v}\in T_{X,x}$ in ${\rm Ext}^1_{\mathcal{G}_K}(r,r)$ is in $V$. It follows from Corollary \[inclv\] that the image of any $\vec{v}\in T_{X,x}$ in ${\rm Ext}^1_{\mathcal{G}_K}(r,r)$ is also in $V_1\cap \cdots \cap V_{n-1}$. One concludes with Proposition \[inter\].
By Lemma \[boundtx\] this finishes the proof of Theorem \[upperbound\].
[The collection of $(\varphi,\Gamma_K)$-submodules $(t^{\Sigma_i({\bf k},w_x)}D_{{\mathrm{rig}}}(r)^{\leq i})_{1\leq i\leq n}$ of $D_{{\mathrm{rig}}}(r)$ plays an important role in the proof of Corollary \[ouf\]. One can wonder if they “globalize” in a neighbourhood of the point $x$ in $X$? Note that those for which $\Sigma_i({\bf k},w_x)=0$ do by [@Bergdall Th.A].]{}
Calculation of some Ext groups
------------------------------
We prove several technical but crucial results of Galois cohomology that were used above.
For a continuous character $\delta:K^\times\rightarrow L^\times$ and $\tau:\, K\hookrightarrow L$, we define its (Sen) weight ${\rm wt}_\tau(\delta)\in L$ in the direction $\tau$ by taking the [*opposite*]{} of the weight defined in [@Bergdall §2.3]. For instance ${\rm wt}_\tau(\tau(z)^{k_\tau})=k_\tau$ ($k_\tau\in {\mathbb{Z}}$).
\[null\] Let $\tau:\, K\hookrightarrow L$ and $k_\tau\in {\mathbb{Z}}_{>0}$.\
(i) For $j\in \{0,1\}$ we have ${\rm Ext}^j_{(\varphi,\Gamma_K)}\big(\mathcal{R}_{L,K},\mathcal{R}_{L,K}(\delta)/(t_\tau^{k_\tau})\big)\ne 0$ if and only if ${\rm wt}_\tau(\delta)\in \{-(k_\tau-1),\dots,0\}$ and we have ${\rm Ext}^2_{(\varphi,\Gamma_K)}\big(\mathcal{R}_{L,K},\mathcal{R}_{L,K}(\delta)/(t_\tau^{k_\tau})\big)=0$ for all $\delta$.\
(ii) For $j\in \{1,2\}$ we have ${\rm Ext}^j_{(\varphi,\Gamma_K)}\big(\mathcal{R}_{L,K}(\delta)/(t_\tau^{k_\tau}),\mathcal{R}_{L,K}\big)\ne 0$ if and only if ${\rm wt}_\tau(\delta)\in \{-k_\tau,\dots,-1\}$ and we have ${\rm Ext}^0_{(\varphi,\Gamma_K)}\big(\mathcal{R}_{L,K}(\delta)/(t_\tau^{k_\tau}),\mathcal{R}_{L,K}\big)=0$ for all $\delta$.\
(iii) When either of these spaces is nonzero, it has dimension $1$ over $L$.
The first part of (i) is in [@Bergdall Prop.2.7] (and initially in [@Co Prop.2.18] for $K={\mathbb{Q}}_p$) and the second part in [@Liuduality Th.3.7(2)]. The second part of (ii) is obvious, let us prove the first. We have an exact sequence: $$\label{exactR}
0\longrightarrow \mathcal{R}_{L,K}(\delta^{-1})\longrightarrow \mathcal{R}_{L,K}(\tau(z)^{-k_\tau}\delta^{-1})\longrightarrow \mathcal{R}_{L,K}(\tau(z)^{-k_\tau}\delta^{-1})/(t_\tau^{k_\tau})\longrightarrow 0.$$ The cup product with (\[exactR\]) yields canonical morphisms of $L$-vector spaces: $$\begin{aligned}
\label{mapL}
{\rm Ext}^0_{(\varphi,\Gamma_K)}\big(\mathcal{R}_{L,K}/(t_\tau^{k_\tau}),\mathcal{R}_{L,K}(\tau(z)^{-k_\tau}\delta^{-1})/(t_\tau^{k_\tau})\big)&\rightarrow & {\rm Ext}^1_{(\varphi,\Gamma_K)}\big(\mathcal{R}_{L,K}/(t_\tau^{k_\tau}),\mathcal{R}_{L,K}(\delta^{-1})\big)\\
\nonumber {\rm Ext}^1_{(\varphi,\Gamma_K)}\big(\mathcal{R}_{L,K}/(t_\tau^{k_\tau}),\mathcal{R}_{L,K}(\tau(z)^{-k_\tau}\delta^{-1})/(t_\tau^{k_\tau})\big)&\rightarrow & {\rm Ext}^2_{(\varphi,\Gamma_K)}\big(\mathcal{R}_{L,K}/(t_\tau^{k_\tau}),\mathcal{R}_{L,K}(\delta^{-1})\big).\end{aligned}$$ There is an obvious isomorphism of $L$-vector spaces: $$\begin{aligned}
{\rm Ext}^0_{(\varphi,\Gamma_K)}\big(\mathcal{R}_{L,K},\mathcal{R}_{L,K}(\tau(z)^{-k_\tau}\delta^{-1})/(t_\tau^{k_\tau})\big)&\!\!\cong \!\!& {\rm Ext}^0_{(\varphi,\Gamma_K)}\big(\mathcal{R}_{L,K}/(t_\tau^{k_\tau}),\mathcal{R}_{L,K}(\tau(z)^{-k_\tau}\delta^{-1})/(t_\tau^{k_\tau})\big)\end{aligned}$$ and an analysis of the cokernel of the multiplication by $t_\tau^{k_\tau}$ map on a short exact sequence $0\rightarrow \mathcal{R}_{L,K}(\tau(z)^{-k_\tau}\delta^{-1})/(t_\tau^{k_\tau})\rightarrow {\mathcal E}\rightarrow \mathcal{R}_{L,K}\rightarrow 0$ of $(\varphi,\Gamma_K)$-module over $\mathcal{R}_{L,K}$ yields a canonical morphism of $L$-vector spaces: $$\begin{aligned}
{\rm Ext}^1_{(\varphi,\Gamma_K)}\big(\mathcal{R}_{L,K},\mathcal{R}_{L,K}(\tau(z)^{-k_\tau}\delta^{-1})/(t_\tau^{k_\tau})\big)&\!\!\!\rightarrow \!\!\!& {\rm Ext}^1_{(\varphi,\Gamma_K)}\big(\mathcal{R}_{L,K}/(t_\tau^{k_\tau}),\mathcal{R}_{L,K}(\tau(z)^{-k_\tau}\delta^{-1})/(t_\tau^{k_\tau})\big).\end{aligned}$$ Thus we have canonical morphisms of $L$-vector spaces: $$\begin{aligned}
\label{mapL2}
{\rm Ext}^0_{(\varphi,\Gamma_K)}\big(\mathcal{R}_{L,K},\mathcal{R}_{L,K}(\tau(z)^{-k_\tau}\delta^{-1})/(t_\tau^{k_\tau})\big)&\!\!\!\rightarrow \!\!\!& {\rm Ext}^1_{(\varphi,\Gamma_K)}\big(\mathcal{R}_{L,K}/(t_\tau^{k_\tau}),\mathcal{R}_{L,K}(\delta^{-1})\big)\\
\nonumber {\rm Ext}^1_{(\varphi,\Gamma_K)}\big(\mathcal{R}_{L,K},\mathcal{R}_{L,K}(\tau(z)^{-k_\tau}\delta^{-1})/(t_\tau^{k_\tau})\big)&\!\!\!\rightarrow \!\!\!& {\rm Ext}^2_{(\varphi,\Gamma_K)}\big(\mathcal{R}_{L,K}/(t_\tau^{k_\tau}),\mathcal{R}_{L,K}(\delta^{-1})\big).\end{aligned}$$ It is then a simple exercise of linear algebra to check that the morphisms in (\[mapL\]) fit into a natural morphism of complexes of $L$-vector spaces from the long exact sequence of ${\rm Ext}^j_{(\varphi,\Gamma_K)}(\mathcal{R}_{L,K},\cdot)$ applied to the short exact sequence (\[exactR\]) to the long exact sequence of ${\rm Ext}^j_{(\varphi,\Gamma_K)}(\cdot,\mathcal{R}_{L,K}(\delta^{-1}))$ applied to the short exact sequence $0\rightarrow t_\tau^{k_\tau}\mathcal{R}_{L,K}\rightarrow \mathcal{R}_{L,K}\rightarrow \mathcal{R}_{L,K}/(t_\tau^{k_\tau})\rightarrow 0$ (note that there is a shift in this map of complexes). Since all the morphisms are obviously isomorphisms except possibly the morphisms (\[mapL2\]), we deduce that the latter are also isomorphisms. Twisting by $\mathcal{R}_{L,K}(\delta)$ on the right hand side of (\[mapL2\]) and using (i) applied to the left hand side, the first part of (ii) easily follows. Finally (iii) follows from [@Bergdall Prop.2.7] and from the previous isomorphisms (\[mapL2\]).
Recall that for $i,\ell\in \{1,\dots,n\}$ we have an exact sequence: $$\begin{gathered}
\label{ses}
0\rightarrow {\rm Ext}^1_{(\varphi,\Gamma_K)}\big(D_{{\mathrm{rig}}}(r)^{\leq i}/t^{\Sigma_{\ell}({\bf k},w_x)}D_{{\mathrm{rig}}}(r)^{\leq i},D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq \ell}\big)\rightarrow \\
{\rm Ext}^1_{(\varphi,\Gamma_K)}\big(D_{{\mathrm{rig}}}(r)^{\leq i},D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq \ell}\big)\rightarrow \\
{\rm Ext}^1_{(\varphi,\Gamma_K)}\big(t^{\Sigma_{\ell}({\bf k},w_x)}D_{{\mathrm{rig}}}(r)^{\leq i},D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq \ell}\big)\end{gathered}$$ where the injection on the left follows as usual from Definition \[veryreg\].
\[cris\] For $1\leq i\leq \ell\leq n$, we have an isomorphism of subspaces of ${\rm Ext}^1_{(\varphi,\Gamma_K)}(D_{{\mathrm{rig}}}(r)^{\leq i},D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq \ell})$: $$\begin{gathered}
{\rm Ext}^1_{(\varphi,\Gamma_K)}\big(D_{{\mathrm{rig}}}(r)^{\leq i}/t^{\Sigma_{\ell}({\bf k},w_x)}D_{{\mathrm{rig}}}(r)^{\leq i},D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq \ell}\big)\cong \\
{\rm Ext}^1_{\rm cris}\big(D_{{\mathrm{rig}}}(r)^{\leq i},D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq \ell}\big).\end{gathered}$$
To lighten notation, we write $D_{{\mathrm{rig}}}$ instead of $D_{{\mathrm{rig}}}(r)$ and drop the subscript $(\varphi,\Gamma_K)$. By the exact sequence (\[ses\]) and a dévissage on $D_{{\mathrm{rig}}}^{\leq i}$ and $D_{{\mathrm{rig}}}/D_{{\mathrm{rig}}}^{\leq \ell}$ (recall from Definition \[veryreg\] and the discussion preceding Lemma \[dim1\] that ${\rm Ext}^1_{\rm cris}$ respects short exact sequences here), it is enough to prove (i) that the composition: $${\rm Ext}^1_{\rm cris}\big(D_{{\mathrm{rig}}}^{\leq i},D_{{\mathrm{rig}}}/D_{{\mathrm{rig}}}^{\leq \ell}\big)\subseteq {\rm Ext}^1\big(D_{{\mathrm{rig}}}^{\leq i},D_{{\mathrm{rig}}}/D_{{\mathrm{rig}}}^{\leq \ell}\big)\longrightarrow {\rm Ext}^1\big(t^{\Sigma_{\ell}({\bf k},w_x)}D_{{\mathrm{rig}}}^{\leq i},D_{{\mathrm{rig}}}/D_{{\mathrm{rig}}}^{\leq \ell}\big)$$ is zero and (ii) that: $${\rm Ext}^1\big({\rm gr}_{\ell'}D_{{\mathrm{rig}}}/(t^{\Sigma_{\ell}({\bf k},w_x)}),{\rm gr}_{\ell''}D_{{\mathrm{rig}}}\big)\cong {\rm Ext}^1_{\rm cris}\big({\rm gr}_{\ell'}D_{{\mathrm{rig}}},{\rm gr}_{\ell''}D_{{\mathrm{rig}}}\big)$$ (inside ${\rm Ext}^1({\rm gr}_{\ell'}D_{{\mathrm{rig}}},{\rm gr}_{\ell''}D_{{\mathrm{rig}}})$) for all $\ell',\ell''$ such that $\ell'\leq \ell$ and $\ell''\geq \ell+1$.
We prove (i). The map clearly factors through: $${\rm Ext}^1_{\rm cris}\big(t^{\Sigma_{\ell}({\bf k},w_x)}D_{{\mathrm{rig}}}^{\leq i},D_{{\mathrm{rig}}}/D_{{\mathrm{rig}}}^{\leq \ell}\big),$$ let us prove that the latter vector space is zero. By dévissage again, it is enough to prove that: $${\rm Ext}^1_{\rm cris}\big(t^{\Sigma_{\ell}({\bf k},w_x)}{\rm gr}_{\ell'}D_{{\mathrm{rig}}},{\rm gr}_{\ell''}D_{{\mathrm{rig}}}\big)=0$$ for $\ell',\ell''$ such that $\ell'\leq \ell$ and $\ell''\geq \ell+1$. It is enough to prove that, for all $\tau:\, K\hookrightarrow L$, we have ${\rm wt}_\tau(t^{\Sigma_{\ell}({\bf k},w_x)}{\rm gr}_{\ell'}D_{{\mathrm{rig}}})\geq {\rm wt}_\tau({\rm gr}_{\ell''}D_{{\mathrm{rig}}})$ (using Definition \[veryreg\] when these two weights are equal). This is equivalent to: $$\begin{aligned}
\label{bound}
\sum_{j=1}^\ell(k_{\tau,j} - k_{\tau,w_{x,\tau}^{-1}(j)})+k_{\tau,w_{x,\tau}^{-1}(\ell')}\geq k_{\tau,w_{x,\tau}^{-1}(\ell'')}\end{aligned}$$ which indeed holds for $\ell'$, $\ell''$ as above because $k_{\tau,1}>k_{\tau,2}>\cdots>k_{\tau,n}$.
We prove (ii). From (i) we have in particular an inclusion: $$\begin{aligned}
\label{incl}
{\rm Ext}^1_{\rm cris}\big({\rm gr}_{\ell'}D_{{\mathrm{rig}}},{\rm gr}_{\ell''}D_{{\mathrm{rig}}}\big)\subseteq {\rm Ext}^1\big({\rm gr}_{\ell'}D_{{\mathrm{rig}}}/(t^{\Sigma_{\ell}({\bf k},w_x)}),{\rm gr}_{\ell''}D_{{\mathrm{rig}}}\big).\end{aligned}$$ It is an easy (and well-known) exercise that we leave to the reader to check that: $$\begin{aligned}
\label{dimcris}
\dim_{k(x)}{\rm Ext}^1_{\rm cris}\big({\rm gr}_{\ell'}D_{{\mathrm{rig}}},{\rm gr}_{\ell''}D_{{\mathrm{rig}}}\big)=\big\vert\{\tau:\, K\hookrightarrow L, w_{x,\tau}^{-1}(\ell'')<w_{x,\tau}^{-1}(\ell')\}\big\vert.\end{aligned}$$ On the other hand, from (ii) and (iii) of Lemma \[null\], using (\[bound\]) and $\mathcal{R}_{L,K}(\delta)/(t_\tau^{k_\tau}t_\sigma^{k_\sigma})\cong \mathcal{R}_{L,K}(\delta)/(t_\tau^{k_\tau})\times \mathcal{R}_{L,K}(\delta)/(t_\sigma^{k_\sigma})$ if $\tau\ne\sigma$, we deduce: $$\begin{gathered}
\label{dimrig}
\dim_{k(x)}{\rm Ext}^1\big({\rm gr}_{\ell'}D_{{\mathrm{rig}}}/(t^{\Sigma_{\ell}({\bf k},w_x)}),{\rm gr}_{\ell''}D_{{\mathrm{rig}}}\big)=\big\vert\{\tau:\, K\hookrightarrow L, w_{x,\tau}^{-1}(\ell'')<w_{x,\tau}^{-1}(\ell')\}\big\vert.\end{gathered}$$ (\[incl\]), (\[dimcris\]) and (\[dimrig\]) imply ${\rm Ext}^1_{\rm cris}({\rm gr}_{\ell'}D_{{\mathrm{rig}}},{\rm gr}_{\ell''}D_{{\mathrm{rig}}})\cong {\rm Ext}^1({\rm gr}_{\ell'}D_{{\mathrm{rig}}}/(t^{\Sigma_{\ell}({\bf k},w_x)}),{\rm gr}_{\ell''}D_{{\mathrm{rig}}})$ which finishes the proof.
\[zero2\] Let $i\in \{1,\dots,n\}$, ${\mathcal E}\in {\rm Ext}^1_{(\varphi,\Gamma_K)}(D_{{\mathrm{rig}}}(r),D_{{\mathrm{rig}}}(r))$ and assume that the image of $\mathcal E$ (by pullback and pushforward) in: $${\rm Ext}^1_{(\varphi,\Gamma_K)}\big(t^{\Sigma_{i-1}({\bf k},w_x)}D_{{\mathrm{rig}}}(r)^{\leq i-1},D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq i-1}\big)$$ is zero. Then the image of $\mathcal E$ in: $${\rm Ext}^1_{(\varphi,\Gamma_K)}\big(t^{\Sigma_{i}({\bf k},w_x)}D_{{\mathrm{rig}}}(r)^{\leq i-1},D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq i}\big)$$ is also zero.
By Proposition \[cris\] applied with $(i,\ell)=(i-1,i-1)$, the image of $\mathcal E$ in ${\rm Ext}^1_{(\varphi,\Gamma_K)}(D_{{\mathrm{rig}}}(r)^{\leq i-1},D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq i-1})$ sits in: $${\rm Ext}^1_{\rm cris}\big(D_{{\mathrm{rig}}}(r)^{\leq i-1},D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq i-1}\big).$$ Hence its image in ${\rm Ext}^1_{(\varphi,\Gamma_K)}(D_{{\mathrm{rig}}}(r)^{\leq i-1},D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq i})$ sits in: $${\rm Ext}^1_{\rm cris}\big(D_{{\mathrm{rig}}}(r)^{\leq i-1},D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq i}\big).$$ It follows from Proposition \[cris\] again applied with $(i,\ell)=(i-1,i)$ that it maps to zero in ${\rm Ext}^1_{(\varphi,\Gamma_K)}(t^{\Sigma_{i}({\bf k},w_x)}D_{{\mathrm{rig}}}(r)^{\leq i-1},D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq i})$.
\[zero3\] For $2\leq i\leq n$ we have a surjection: $$\begin{gathered}
{\rm Ext}^1_{(\varphi,\Gamma_K)}\big(D_{{\mathrm{rig}}}(r)^{\leq i}/t^{\Sigma_{i}({\bf k},w_x)}D_{{\mathrm{rig}}}(r)^{\leq i},D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq i}\big)\twoheadrightarrow \\
{\rm Ext}^1_{(\varphi,\Gamma_K)}\big(D_{{\mathrm{rig}}}(r)^{\leq i-1}/t^{\Sigma_{i}({\bf k},w_x)}D_{{\mathrm{rig}}}(r)^{\leq i-1},D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq i}\big)\end{gathered}$$ where the map is the pullback along: $$D_{{\mathrm{rig}}}(r)^{\leq i-1}/t^{\Sigma_{i}({\bf k},w_x)}D_{{\mathrm{rig}}}(r)^{\leq i-1}\hookrightarrow D_{{\mathrm{rig}}}(r)^{\leq i}/t^{\Sigma_{i}({\bf k},w_x)}D_{{\mathrm{rig}}}(r)^{\leq i}.$$
This follows from Proposition \[cris\] (applied with $(i,\ell)=(i,i)$ and $(i,\ell)=(i-1,i)$) and the fact that the map: $${\rm Ext}^1_{\rm cris}\big(D_{{\mathrm{rig}}}(r)^{\leq i},D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq i}\big)\longrightarrow
{\rm Ext}^1_{\rm cris}\big(D_{{\mathrm{rig}}}(r)^{\leq i-1},D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq i}\big)$$ is surjective.
Modularity and local geometry of the trianguline variety {#modularity}
========================================================
We prove that the main conjecture of [@BHS] (see [@BHS Conj.3.22]), and thus the classical modularity conjectures by [@BHS Prop.3.26], imply Conjecture \[mainconj\] when ${\overline{r}}$ “globalizes” and $x$ is very regular.
A closed embedding {#closed}
------------------
Assuming the main conjecture of [@BHS] and using Theorem \[intercompanion\] below we construct a certain closed embedding in the trianguline variety (Proposition \[companion\]).
We fix a continuous representation ${\overline{r}}:\mathcal{G}_K\rightarrow {\mathrm{GL}}_n(k_L)$ as in §\[begin\] and keep the local notation of §\[localpart1\] and §\[phiGammacohomology\]. We also assume that there exist number fields $F/F^+$, a unitary group $G/F^+$, a tame level $U^p$, a set of finite places $S$ and an irreducible representation ${\overline{\rho}}$ as in §\[classic\] such that all the assumptions in §\[classic\] and §\[firstclassical\] are satisfied, and such that for each place $v\in S_p$ there is a place $\tilde v$ of $F$ dividing $v$ satisfying $F_{\tilde v}\cong K$ and ${\overline{\rho}}_{\tilde v}\cong {\overline{r}}$. Note that this implies in particular $(2n,p)=1$ (as $p>2$ and as $(n,p)=1$ by the proof of [@GHTT Th.9]). Assuming $(2n,p)=1$, it follows from [@CEGGPS Lem.2.2] and [@CEGGPS §2.3] that such $(F/F^+,G,U^p,S,{\overline{\rho}})$ always exist if $n=2$ or if ${\overline{r}}$ is (absolutely) semi-simple (increasing $L$ if necessary).
We recall the statement of [@BHS Conj.3.22] (see §\[weyl\] for $\widetilde X_{\rm tri}^\square({\overline{r}})$).
\[BHS\] The rigid subvariety $X_{\rm tri}^{{\mathfrak{X}}^p \rm-aut}({\overline{\rho}}_p)$ of $X_{\rm tri}^\square({\overline{\rho}}_p)$ doesn’t depend on ${\mathfrak{X}}^p$ and is isomorphic to $\widetilde X_{\rm tri}^\square({\overline{\rho}}_p):= \prod_{v\in S_p}\widetilde X_{\rm tri}^\square({\overline{\rho}}_{\tilde v})$.
\[conjvariant\] [(i) By (\[union\]), Conjecture \[BHS\] is thus equivalent to $X_p({\overline{\rho}})\buildrel{\sim}\over \rightarrow {\mathfrak{X}}_{{\overline{\rho}}^p}\times \widetilde X_{\rm tri}^\square({\overline{\rho}}_p)\times {\mathbb{U}}^g$.\
(ii) The authors do not know if $\widetilde X_{\rm tri}^\square({\overline{\rho}}_p)$ is really strictly smaller than $X_{\rm tri}^\square({\overline{\rho}}_p)$.\
(iii) Finally, recall that Conjecture \[BHS\] is [*implied*]{} by the classical modularity lifting conjectures for ${\overline{\rho}}$ (in all weights with trivial inertial type), see [@BHS Prop.3.26].]{}
Let ${\bf k}:=({\bf k}_i)_{1\leq i\leq n}$ where ${\bf k}_i:=(k_{\tau,i})_{\tau:\, K\hookrightarrow L}\in\mathbb{Z}^{{\mathrm{Hom}}(K,L)}$ is such that $k_{\tau,i}> k_{\tau,i+1}$ for all $i$ and $\tau$. For $w=(w_{\tau})_{\tau:\, K\hookrightarrow L}\in W=\prod_{\tau: K\hookrightarrow L}{\mathcal{S}}_n$, denote by $\mathcal{W}^n_{w,{\bf k},L}\subset \mathcal{W}^n_L$ the Zariski-closed (reduced) subset of characters $(\eta_1,\dots,\eta_n)$ defined by the equations: $$\label{equations}
{\rm wt}_\tau(\eta_{w_\tau(i)}\eta_i^{-1})=k_{\tau,i}-k_{\tau,w_\tau^{-1}(i)},\ \ 1\leq i\leq n,\ \ \tau:\, K\hookrightarrow L.$$ For instance one always has: $$\label{point}
(z^{{\bf k}_{w^{-1}(1)}}\chi_1,\dots,z^{{\bf k}_{w^{-1}(n)}}\chi_n)\in \mathcal{W}^n_{w,{\bf k},L}$$ where $\chi_i\in \mathcal{W}_L$ are finite order characters. Note that $\mathcal{W}^n_{1,{\bf k},L}=\mathcal{W}^n_L$. We define an automorphism $\jmath_{w,{\bf k}}:\mathcal{T}^n_L\buildrel\sim\over\rightarrow \mathcal{T}^n_L$, $\eta=(\eta_1,\dots,\eta_n)\mapsto \jmath_{w,{\bf k}}(\eta)=\jmath_{w,{\bf k}}(\eta_1,\dots,\eta_n)$ by: $$\jmath_{w,{\bf k}}(\eta_1,\dots,\eta_n):= (z^{{\bf k}_1-{\bf k}_{w^{-1}(1)}}\eta_1,\dots,z^{{\bf k}_n-{\bf k}_{w^{-1}(n)}}\eta_n)$$ which we extend to an automorphism $\jmath_{w,{\bf k}}:\mathfrak{X}_{{\overline{r}}}^\square\times \mathcal{T}^n_L\buildrel\sim\over\rightarrow\mathfrak{X}_{{\overline{r}}}^\square\times \mathcal{T}^n_L$, $(r,\eta)\mapsto (r,\jmath_{w,{\bf k}}(\eta))$. We will be particularly interested in applying $\jmath_{w,{\bf k}}$ to points whose image in $\mathcal{W}^n_L$ lies in $\mathcal{W}^n_{w,{\bf k},L}$.
\[example\] [Consider the case $[K:{\mathbb{Q}}_p]=2$ (so ${\mathrm{Hom}}(K,L)=\{\tau,\tau'\}$), $n=3$ and $w=(w_\tau,w_{\tau'})$ with $w_{\tau}=s_1s_2s_1$, $w_{\tau'}=s_2s_1$ ($s_1,s_2$ being the simple reflections in ${\mathcal S}_3$). Then $\mathcal{W}^3_{w,{\bf k},L}$ is the set of characters of the form: $$\eta=(\eta_1,\eta_2,\eta_3) = \big(\tau(z)^{k_{\tau,3}}\tau'(z)^{k_{\tau',2}}\chi_1, \ \tau(z)^{k_{\tau,2}}\tau'(z)^{k_{\tau',3}}\chi_2, \ \tau(z)^{k_{\tau,1}}\tau'(z)^{k_{\tau',1}}\chi_3\big)$$ where ${\rm wt}_\tau(\chi_1)={\rm wt}_\tau(\chi_3)$ and ${\rm wt}_{\tau'}(\chi_1)={\rm wt}_{\tau'}(\chi_2)={\rm wt}_{\tau'}(\chi_3)$. Note that there is no condition on ${\rm wt}_\tau(\chi_2)$ (so one could as well rewrite the middle character as just $\tau'(z)^{k_{\tau',3}}\chi_2 $). One has (when the $\eta_i$, or equivalently the $\chi_i$, come from characters in $\mathcal{T}_L$): $$\jmath_{w,{\bf k}}(\eta)= \big(\tau(z)^{k_{\tau,1}}\tau'(z)^{k_{\tau',1}}\chi_1,\ \tau(z)^{k_{\tau,2}}\tau'(z)^{k_{\tau',2}}\chi_2,\ \tau(z)^{k_{\tau,3}}\tau'(z)^{k_{\tau',3}}\chi_3\big).$$]{}
Let $\widetilde U_{\rm tri}^\square({\overline{r}}):=U_{\rm tri}^\square({\overline{r}})\cap \widetilde X_{\rm tri}^\square({\overline{r}})$ (a union of connected components of $U_{\rm tri}^\square({\overline{r}})$), then $\widetilde U_{\rm tri}^\square({\overline{r}})\times_{\mathcal{W}^n_L}\mathcal{W}^n_{w,{\bf k},L}$ is reduced (since smooth over $\mathcal{W}^n_{w,{\bf k},L}$) and Zariski-open (but not necessarily Zariski-dense) in $(\widetilde X_{\rm tri}^\square({\overline{r}})\times_{\mathcal{W}^n_L}\mathcal{W}^n_{w,{\bf k},L})^{\mathrm{red}}$ where $(-)^{{\mathrm{red}}}$ means the associated reduced closed analytic subvariety. We denote by $\overline{\widetilde U_{\rm tri}^\square({\overline{r}})\times_{\mathcal{W}^n_L}\mathcal{W}^n_{w,{\bf k},L}}$ its Zariski-closure, so that we have a chain of Zariski-closed embeddings: $$\begin{gathered}
\overline{\widetilde U_{\rm tri}^\square({\overline{r}})\times_{\mathcal{W}^n_L}\mathcal{W}^n_{w,{\bf k},L}}\subseteq (\widetilde X_{\rm tri}^\square({\overline{r}})\times_{\mathcal{W}^n_L}\mathcal{W}^n_{w,{\bf k},L})^{\mathrm{red}}\subseteq \widetilde X_{\rm tri}^\square({\overline{r}})\times_{\mathcal{W}^n_L}\mathcal{W}^n_{w,{\bf k},L}\subseteq \widetilde X_{\rm tri}^\square({\overline{r}})\\
\subseteq X_{\rm tri}^\square({\overline{r}})\subseteq \mathfrak{X}_{{\overline{r}}}^\square\times \mathcal{T}^n_L.\end{gathered}$$
\[companion\] Assume Conjecture \[BHS\], then for $w\in W$ the automorphism $\jmath_{w,{\bf k}}:\mathfrak{X}_{{\overline{r}}}^\square\times \mathcal{T}^n_L\buildrel\sim\over\rightarrow\mathfrak{X}_{{\overline{r}}}^\square\times \mathcal{T}^n_L$ induces a closed embedding of reduced rigid analytic spaces over $L$: $$\jmath_{w,{\bf k}}: \overline{\widetilde U_{\rm tri}^\square({\overline{r}})\times_{\mathcal{W}^n_L}\mathcal{W}^n_{w,{\bf k},L}}\hookrightarrow \widetilde X_{\rm tri}^\square({\overline{r}})\subseteq X_{\rm tri}^\square({\overline{r}}).$$
Since $\widetilde U_{\rm tri}^\square({\overline{r}})\times_{\mathcal{W}^n_L}\mathcal{W}^n_{w,{\bf k},L}$ is Zariski-dense in $\overline{\widetilde U_{\rm tri}^\square({\overline{r}})\times_{\mathcal{W}^n_L}\mathcal{W}^n_{w,{\bf k},L}}$, it is enough to prove $\jmath_{w,{\bf k}}(\widetilde U_{\rm tri}^\square({\overline{r}})\times_{\mathcal{W}^n_L}\mathcal{W}^n_{w,{\bf k},L})\subseteq \widetilde X_{\rm tri}^\square({\overline{r}})$, i.e. that any point $x'=(r',\delta')$ in $\widetilde U_{\rm tri}^\square({\overline{r}})$ with $\omega(x')\in \mathcal{W}^n_{w,{\bf k},L}$ is such that $\jmath_{w,{\bf k}}(x')$ is still in $\widetilde X_{\rm tri}^\square({\overline{r}})$.
Recall that by assumption: $$\label{rappelconj}
X_p({\overline{\rho}}) \buildrel{\substack{(\ref{union})\\ \sim}}\over \longrightarrow {\mathfrak{X}}_{{\overline{\rho}}^p}\times \widetilde X_{\rm tri}^\square({\overline{\rho}}_p)\times {\mathbb{U}}^g\subseteq {\mathfrak{X}}_{{\overline{\rho}}^p}\times ({\mathfrak{X}}_{{\overline{\rho}}_p}\times \widehat T_{p,L})\times {\mathbb{U}}^g.$$ Let $y'\in X_p({\overline{\rho}})$ be any point such that its image in $\widetilde X_{\rm tri}^\square({\overline{\rho}}_p)$ by (\[rappelconj\]) is $(x')_{v\in S_p}$. Write again $w$ for the element $(w)_{v\in S_p}\in \prod_{v\in S_p}(\prod_{F_{\tilde v}\hookrightarrow L}{\mathcal{S}}_n)$ (that is, for each $v$ we have the same element $w=(w_{\tau})_{\tau:\, K\hookrightarrow L}\in \prod_{\tau: K\hookrightarrow L}{\mathcal{S}}_n$), ${\bf k}$ for $({\bf k})_{v\in S_p}\in \prod_{v\in S_p}{\mathbb{Z}}^{{\mathrm{Hom}}(K,L)}$ (ibid.), $\jmath_{w,{\bf k}}$ for the automorphism $(\jmath_{w,{\bf k}})_{v\in S_p}$ of $\widehat T_{p,L}\cong \prod_{v\in S_p}\widehat T_{v,L}\cong \prod_{v\in S_p}\mathcal{T}^n_L$ and (again) $\jmath_{w,{\bf k}}$ for the automorphism ${\mathrm{id}}\times ({\mathrm{id}}\times \jmath_{w,{\bf k}})\times {\mathrm{id}}$ of ${\mathfrak{X}}_{{\overline{\rho}}^p}\times ({\mathfrak{X}}_{{\overline{\rho}}_p}\times \widehat T_{p,L})\times {\mathbb{U}}^g$. Then it is enough to prove that $\jmath_{w,{\bf k}}(y')\in X_p({\overline{\rho}})$ (via (\[rappelconj\])). Writing $y'=(\mathfrak{m}',\epsilon')\in X_p({\overline{\rho}})\subseteq {\mathfrak{X}}_\infty\times \widehat T_{p,L}$ where $\mathfrak{m}'\subset R_\infty[1/p]$ is the maximal ideal corresponding to the projection of $y'$ in ${\mathfrak{X}}_\infty$ and $\epsilon'=(\imath_v(\delta'))_{v\in S_p}$, we have ${\mathrm{Hom}}_{T_p}(\epsilon',J_{B_p}(\Pi_\infty^{R_\infty-\rm an}[\mathfrak{m}']\otimes_{k(\mathfrak{m}')}k(y')))\ne 0$ (see (\[pointpatching\])) and we have to prove (note that $\jmath_{w,{\bf k}}\circ\imath_v^{-1}=\imath_v^{-1}\circ\jmath_{w,{\bf k}}$ on $\widehat T_{v,L}$ and that $k(y')=k(\jmath_{w,{\bf k}}(y'))$): $$\label{nonzero}
{\mathrm{Hom}}_{T_p}\big(\jmath_{w,{\bf k}}(\epsilon'),J_{B_p}(\Pi_\infty^{R_\infty-\rm an}[\mathfrak{m}']\otimes_{k(\mathfrak{m}')}k(y'))\big)\ne 0.$$
From Theorem \[intercompanion\] below, it is enough to prove $\epsilon' \uparrow \jmath_{w,{\bf k}}(\epsilon')$ in the sense of Definition \[stronglink\] below. Since $\epsilon'\jmath_{w,{\bf k}}(\epsilon')^{-1}$ is clearly an algebraic character of $T_p$ by definition of $\jmath_{w,{\bf k}}$, it is enough to prove $\imath_v(\delta') \uparrow_{{\mathfrak t}_v} \jmath_{w,{\bf k}}(\imath_v(\delta'))$ (see §\[jw\] for the notation) for one, or equivalently all here, $v\in S_p$. From (\[equations\]), we see that we can write: $$\delta'=(z^{{\bf k}_{w^{-1}(1)}}\chi_1,\dots,z^{{\bf k}_{w^{-1}(n)}}\chi_n)\ \ {\rm and}\ \ \jmath_{w,{\bf k}}(\imath_v(\delta'))=(z^{{\bf k}_1}\chi_1,\dots,z^{{\bf k}_n}\chi_n)$$ where ${\rm wt}_\tau(\chi_i)={\rm wt}_\tau(\chi_{w_\tau(i)})$ for $1\leq i\leq n$ and $\tau:\, K=F_{\tilde v}\hookrightarrow L$ (compare Example \[example\]). As we only care about the ${\mathfrak t}_{v,L}$-action, setting $s_{\tau,i}:={\rm wt}_\tau(\chi_i)\in L$ and using usual additive notation, we can write $\imath_v(\delta')\vert_{{\mathfrak t}_{v,L}}=(\imath_v(\delta')_\tau)_{\tau:\, F_{\tilde v}\hookrightarrow L}$ and $\jmath_{w,{\bf k}}(\imath_v(\delta'))\vert_{{\mathfrak t}_{v,L}}=(\jmath_{w,{\bf k}}(\imath_v(\delta'))_\tau)_{\tau:\, F_{\tilde v}\hookrightarrow L}$ with: $$\begin{aligned}
\imath_v(\delta')_\tau&=&(k_{\tau,w_{\tau}^{-1}(1)}+s_{\tau,1},k_{\tau,w_{\tau}^{-1}(2)}+s_{\tau,2}+1,\dots,k_{\tau,w_{\tau}^{-1}(n)}+s_{\tau,n}+n-1)\\
\jmath_{w,{\bf k}}(\imath_v(\delta'))_\tau&=&(k_{\tau,1}+s_{\tau,1},k_{\tau,2}+s_{\tau,2}+1,\dots,k_{\tau,n}+s_{\tau,n}+n-1)\end{aligned}$$ (see the beginning of §\[jw\]). Since $s_{\tau,i}=s_{\tau,w_\tau^{-1}(i)}$ for all $i,\tau$, we can rewrite: $$\imath_v(\delta')_\tau=(k_{\tau,w_{\tau}^{-1}(1)}+s_{\tau,w_{\tau}^{-1}(1)},k_{\tau,w_{\tau}^{-1}(2)}+s_{\tau,w_{\tau}^{-1}(2)}+1,\dots,k_{\tau,w_{\tau}^{-1}(n)}+s_{\tau,w_{\tau}^{-1}(n)}+n-1)$$ hence we have $\imath_v(\delta')_\tau=w_\tau\cdot \jmath_{w,{\bf k}}(\imath_v(\delta'))_\tau$ for the “dot action” $\cdot$ with respect to the upper triangular matrices in ${\mathrm{GL}}_{n,F_{\tilde v}}\times_{F_{\tilde v},\tau}L$ (see [@HumBGG §1.8]). Let us write the permutation $w_\tau$ on $\{1,\dots,n\}$ as a product of commuting cycles $c_1\circ\cdots \circ c_m$ with pairwise disjoint support ${\rm supp}(c_i)\subseteq \{1,\dots,n\}$. Let us denote by ${\mathcal S}_{n,i}\subseteq {\mathcal S}_{n}$ the subgroup of permutations which fixes the elements in $\{1,\dots,n\}$ [*not*]{} in ${\rm supp}(c_i)$ and set ${\mathcal S}_{n,w_\tau}:= \prod_{i=1}^m{\mathcal S}_{n,i}\subseteq {\mathcal S}_n$. Then, arguing in each ${\rm supp}(c_i)$, it is not difficult to see that one can write $w_\tau$ as a product: $$w_\tau = s_{\alpha_d}s_{\alpha_{d-1}}\cdots s_{\alpha_1}$$ where the $\alpha_i$ are (not necessarily simple) roots of the upper triangular matrices in ${\mathrm{GL}}_{n,F_{\tilde v}}\times_{F_{\tilde v},\tau}L$, the associated reflections $s_{\alpha_i}$ are in ${\mathcal S}_{n,w_\tau}$ and where $s_{\alpha_i+1}s_{\alpha_{i}}\cdots s_{\alpha_1}> s_{\alpha_i}\cdots s_{\alpha_1}$ for the Bruhat order in ${\mathcal S}_n$ ($1\leq i\leq n-1$). By an argument analogous [*mutatis mutandis*]{} to the one in [@HumBGG §5.2], it then follows from the above assumptions (in particular $s_{\tau,i}=s_{\tau,w_\tau^{-1}(i)}$ for all $i$) that we have for $1\leq i\leq n-1$ with obvious notation: $$(s_{\alpha_i+1}\cdots s_{\alpha_1})\cdot \jmath_{w,{\bf k}}(\imath_v(\delta'))_\tau \leq (s_{\alpha_i}\cdots s_{\alpha_1})\cdot \jmath_{w,{\bf k}}(\imath_v(\delta'))_\tau.$$ By definition this implies that $w_\tau\cdot\jmath_{w,{\bf k}}(\imath_v(\delta'))_\tau$ is [*strongly linked*]{} to $\jmath_{w,{\bf k}}(\imath_v(\delta'))_\tau$ ([@HumBGG §5.1]). As this holds for all $\tau$, we have $\imath_v(\delta') \uparrow_{{\mathfrak t}_v} \jmath_{w,{\bf k}}(\imath_v(\delta'))$.
[It would be very interesting to find a purely local proof of the local statement of Proposition \[companion\] without assuming Conjecture \[BHS\].]{}
Companion points on the patched eigenvariety {#jw}
--------------------------------------------
We prove that the existence of certain points on the patched eigenvariety $X_p({\overline{\rho}})$ implies the existence of others (Theorem \[intercompanion\]). This result is crucially used in the proof of Proposition \[companion\] above.
We use the notation of §\[globalpart\]. We denote by ${\mathfrak g}$ (resp. ${\mathfrak b}$, resp. ${\mathfrak t}$) the ${\mathbb{Q}}_p$-Lie algebra of $G_p$ (resp. $B_p$, resp. $T_p$). We also denote by ${\mathfrak n}$ (resp. $\overline{\mathfrak n}$) the ${\mathbb{Q}}_p$-Lie algebra of the inverse image $N_p$ in $B_p$ (resp. $\overline N_p$ in $\overline B_p$) of the subgroup of upper (resp. lower) unipotent matrices of $\prod_{v\in S_p}{\mathrm{GL}}_n(F_{\tilde v})$. We add an index $L$ for the $L$-Lie algebras obtained by scalar extension $\cdot \otimes_{{\mathbb{Q}}_p}L$ (e.g. ${\mathfrak g}_L$, etc.) and we denote by $U(\cdot)$ the corresponding enveloping algebras.
For $v\in S_p$ we denote by ${\mathfrak t}_v$ the ${\mathbb{Q}}_p$-Lie algebra of the torus $T_v$, so that ${\mathfrak t}=\prod_{v\in S_p}{\mathfrak t}_v$. Recall that ${\mathfrak t}_v$ is an $F_{\tilde v}$-vector space, and thus ${\mathfrak t}_{v,L}={\mathfrak t}_v\otimes_{{\mathbb{Q}}_p}L\cong \prod_{\tau:\, F_{\tilde v}\hookrightarrow L}{\mathfrak t}_v\otimes_{F_{\tilde v},\tau}L$. We can see any $\eta=(\eta_v)_{v\in S_p}=(\eta_{v,1},\dots,\eta_{v,n})_{v\in S_p}\in \widehat T_{p,L}$ as an $L$-valued additive character of ${\mathfrak t}$, and thus of ${\mathfrak t}_{L}$ by $L$-linearity, via the usual derivative action $({\mathfrak z}_{v,1},\dots,{\mathfrak z}_{v,n})_{v\in S_p}\mapsto \sum_{v\in S_p}\sum_{i=1}^n\frac{d}{dt}\eta_{v,i}(\exp(t{\mathfrak z}_{v,i}))\vert_{t=0}$. Recall that the character ${\mathfrak z}_{v,i}\in F_{\tilde v}\mapsto \frac{d}{dt}\eta_{v,i}(\exp(t{\mathfrak z}_{v,i}))\vert_{t=0}$ is nothing else than $\sum_{\tau:\, F_{\tilde v}\hookrightarrow L}\tau({\mathfrak z}_{v,i}){\rm wt}_\tau(\eta_{v,i})\in L$.
In what follows we use notation and definitions from [@STdist] concerning $L$-Banach representations of $p$-adic Lie groups and their locally ${\mathbb{Q}}_p$-analytic vectors. If $\Pi$ is an admissible continuous representation of $G_p$ on a $L$-Banach space we denote by $\Pi^{\rm an}\subseteq \Pi$ its invariant subspace of locally ${\mathbb{Q}}_p$-analytic vectors.
\[injective\] Let $\Pi$ be an admissible continuous representation of $G_p$ on a $L$-Banach space and assume that the continuous dual $\Pi'$ is a finite projective $\mathcal{O}_L{{\mathchoice{\mbox{\rm [\hspace{-0.15em}[}}
{\mbox{\rm [\hspace{-0.15em}[}}
{\mbox{\scriptsize\rm [\hspace{-0.15em}[}}
{\mbox{\tiny\rm [\hspace{-0.15em}[}}}}K_p{{\mathchoice{\mbox{\rm ]\hspace{-0.15em}]}}
{\mbox{\rm ]\hspace{-0.15em}]}}
{\mbox{\scriptsize\rm ]\hspace{-0.15em}]}}
{\mbox{\tiny\rm ]\hspace{-0.15em}]}}}}[1/p]$-module. Let $\lambda$, $\mu$ be $L$-valued characters of ${\mathfrak t}_L$ that we see as $L$-valued characters of ${\mathfrak b}_L$ by sending ${\mathfrak n}_L$ to $0$. If $U(\mathfrak{g}_{L})\otimes_{U(\mathfrak{b}_L)}\mu\hookrightarrow U(\mathfrak{g}_{L})\otimes_{U(\mathfrak{b}_L)}\lambda$ is an injection of $U(\mathfrak{g}_{L})$-modules, then the map: $${\mathrm{Hom}}_{U(\mathfrak{g}_{L})}\big(U(\mathfrak{g}_{L})\otimes_{U(\mathfrak{b}_L)}\lambda,\Pi^{\rm an}\big)\longrightarrow{\mathrm{Hom}}_{U(\mathfrak{g}_{L})}\big(U(\mathfrak{g}_{L})\otimes_{U(\mathfrak{b}_L)}\mu,\Pi^{\rm an}\big)$$ induced by functoriality is surjective.
We have as in [@STdist Prop.6.5] a $K_p$-equivariant isomorphism: $$\label{limr}
\Pi^{\rm an}\cong {\lim_{\stackrel{r\rightarrow 1}{r<1}}}\ \Pi_r$$ where each $\Pi_r\subseteq \Pi^{\rm an}$ is a Banach space over $L$ endowed with an admissible locally ${\mathbb{Q}}_p$-analytic action of $K_p$. In particular each $\Pi_r$ is stable under $U(\mathfrak{g}_L)$ in $\Pi^{\rm an}$. If $f:U(\mathfrak{g}_{L})\otimes_{U(\mathfrak{b}_L)}\lambda\rightarrow\Pi^{\rm an}$ is a $U(\mathfrak{g}_L)$-equivariant morphism, the source, being of finite type over $U(\mathfrak{g}_{L})$, factors through some $\Pi_r$ by (\[limr\]). Moreover the action of $U(\mathfrak{g}_{L})$ on $\Pi_r$ extends to an action of the $L$-Banach algebra $U_r(\mathfrak{g}_{L})$ which is the topological closure of $U(\mathfrak{g}_{L})$ in the completed distribution algebra $D_r(K_p,L)$ (see [@STdist §5]). Consequently $f$ extends to a $U_r(\mathfrak{g}_{L})$-equivariant morphism: $$f_r:\, U_r(\mathfrak{g}_{L})\otimes_{U(\mathfrak{g}_{L})}(U(\mathfrak{g}_{L})\otimes_{U(\mathfrak{b}_L)}\lambda)\cong
U_r(\mathfrak{g}_{L})\otimes_{U_r(\mathfrak{b}_L)} \lambda\longrightarrow\Pi_r$$ where $U_r(\mathfrak{b}_L)$ is the closure of $U(\mathfrak{b}_L)$ in $D_r(K_p,L)$ and the first isomorphism follows from $\lambda\buildrel\sim\over\rightarrow U_r(\mathfrak{b}_L)\otimes_{U(\mathfrak{b}_L)}\lambda$ (as $\lambda$ is both finite dimensional with dense image). We deduce from [@OrlikStrauch Prop.3.4.8] (applied with $w=1$) that the injection $U(\mathfrak{g}_{L})\otimes_{U(\mathfrak{b}_L)}\lambda\hookrightarrow U(\mathfrak{g}_{L})\otimes_{U(\mathfrak{b}_L)}\mu$ extends to an injection of $U_r(\mathfrak{g}_{L})$-modules $U_r(\mathfrak{g}_{L})\otimes_{U_r(\mathfrak{b}_L)}\lambda\hookrightarrow U_r(\mathfrak{g}_{L})\otimes_{U_r(\mathfrak{b}_L)}\mu$. Moreover, as $U_r(\mathfrak{g}_{L})\otimes_{U_r(\mathfrak{b}_L)}\lambda$ and $U_r(\mathfrak{g}_{L})\otimes_{U_r(\mathfrak{b}_L)}\mu$ are $U_r(\mathfrak{g}_{L})$-modules of finite type, they have a unique topology of Banach module over $U_r(\mathfrak{g}_{L})$ and every $U_r(\mathfrak{g}_{L})$-linear map of one of them into $\Pi_r$ is automatically continuous (see [@STdist Prop.2.1]). We deduce from all this isomorphisms: $$\begin{aligned}
{\mathrm{Hom}}_{U(\mathfrak{g}_{L})}\big(U(\mathfrak{g}_{L})\otimes_{U(\mathfrak{b}_L)}\lambda,\Pi^{\rm an}\big)&\buildrel\sim\over\longrightarrow &\varinjlim_r{\mathrm{Hom}}_{U(\mathfrak{g}_{L})}\big(U(\mathfrak{g}_{L})\otimes_{U(\mathfrak{b}_L)}\lambda,\Pi_r\big)\\
&\buildrel\sim\over\longrightarrow &\varinjlim_r{\mathrm{Hom}}_{U_r(\mathfrak{g}_{L})-{\rm cont}}\big(U_r(\mathfrak{g}_{L})\otimes_{U_r(\mathfrak{b}_L)}\lambda,\Pi_r\big)\end{aligned}$$ where ${\mathrm{Hom}}_{U_r(\mathfrak{g}_{L})-{\rm cont}}$ means continuous homomorphisms of $U_r(\mathfrak{g}_{L})$-Banach modules, and likewise with $\mu$ instead of $\lambda$. By exactitude of $\varinjlim_r$, we see that it is enough to prove that $\Pi_r$ is an injective object (with respect to injections which have closed image) in the category of $U_r(\mathfrak{g}_{L})$-Banach modules with continuous maps.
By assumption the dual $\Pi'$ is a projective module of finite type over $\mathcal{O}_L{{\mathchoice{\mbox{\rm [\hspace{-0.15em}[}}
{\mbox{\rm [\hspace{-0.15em}[}}
{\mbox{\scriptsize\rm [\hspace{-0.15em}[}}
{\mbox{\tiny\rm [\hspace{-0.15em}[}}}}K_p{{\mathchoice{\mbox{\rm ]\hspace{-0.15em}]}}
{\mbox{\rm ]\hspace{-0.15em}]}}
{\mbox{\scriptsize\rm ]\hspace{-0.15em}]}}
{\mbox{\tiny\rm ]\hspace{-0.15em}]}}}}[1/p]$, hence a direct summand of $\mathcal{O}_L{{\mathchoice{\mbox{\rm [\hspace{-0.15em}[}}
{\mbox{\rm [\hspace{-0.15em}[}}
{\mbox{\scriptsize\rm [\hspace{-0.15em}[}}
{\mbox{\tiny\rm [\hspace{-0.15em}[}}}}K_p{{\mathchoice{\mbox{\rm ]\hspace{-0.15em}]}}
{\mbox{\rm ]\hspace{-0.15em}]}}
{\mbox{\scriptsize\rm ]\hspace{-0.15em}]}}
{\mbox{\tiny\rm ]\hspace{-0.15em}]}}}}[1/p]^{\oplus s}$ for some $s>0$. From the proof of [@STdist Prop.6.5] together with [@STdist Th.7.1(iii)], we also know that $\Pi_r$ is the continuous dual of the $D_r(K_p,L)$-Banach module: $$\Pi_r':=D_r(K_p,L)\otimes_{\mathcal{O}_L{{\mathchoice{\mbox{\rm [\hspace{-0.15em}[}}
{\mbox{\rm [\hspace{-0.15em}[}}
{\mbox{\scriptsize\rm [\hspace{-0.15em}[}}
{\mbox{\tiny\rm [\hspace{-0.15em}[}}}}K_p{{\mathchoice{\mbox{\rm ]\hspace{-0.15em}]}}
{\mbox{\rm ]\hspace{-0.15em}]}}
{\mbox{\scriptsize\rm ]\hspace{-0.15em}]}}
{\mbox{\tiny\rm ]\hspace{-0.15em}]}}}}[1/p]}\Pi'.$$ We get that the $D_r(K_p,L)$-module $\Pi_r'$ is a direct summand of $D_r(K_p,L)^{\oplus s}$. Now it easily follows from the results in [@Ko §1.4] that $D_r(K_p,L)$ is itself a free $U_r(\mathfrak{g}_{L})$-module of finite rank. Dualizing, we finally obtain that there is a finite dimensional $L$-vector space $W$ such that the left $U_r(\mathfrak{g}_{L})$-Banach module $\Pi_r$ is a direct factor of the left $U_r(\mathfrak{g}_{L})$-Banach module ${\rm Hom}_{\rm cont}(U_r(\mathfrak{g}_{L})\otimes_L W,L)$ (which is seen as a left $U_r(\mathfrak{g}_{L})$-module via the automorphism on $U_r(\mathfrak{g}_{L})$ extending the multiplication by $-1$ on $\mathfrak{g}_{L}$). Since direct summands and finite sums of injective modules are still injective, it is enough to prove the injectivity of ${\rm Hom}_{\rm cont}(U_r(\mathfrak{g}_{L}),L)$ in the category of $U_r(\mathfrak{g}_{L})$-Banach modules with continuous maps.
If $V$ is any $U_r(\mathfrak{g}_L)$-Banach module, it is not difficult to see that there is a canonical isomorphism of Banach spaces over $L$: $$\label{dualdual}
{{\mathrm{Hom}}}_{U_r(\mathfrak{g}_{L})-{\rm cont}}\big(V,{{\mathrm{Hom}}}_{\rm cont}(U_r(\mathfrak{g}_L),L)\big)\buildrel\sim\over\longrightarrow {{\mathrm{Hom}}}_{\rm cont}(V,L)$$ so that the required injectivity property is a consequence of the Hahn-Banach Theorem (see for example [@Schneidernfa Prop.9.2]).
We go on with two technical lemmas which require more notation. Fix a compact open uniform normal pro-$p$ subgroup $H_p$ of $K_p$ such that $H_p=(\overline N_p\cap H_p)(T_p\cap H_p)(N_p\cap H_p)$. For example $H_p$ can be chosen of the form $\prod_{v\in S_p}H_{v}$ where $H_v$ is the inverse image in $K_v$ of matrices of ${\mathrm{GL}}_n(\mathcal{O}_{F_{\tilde{v}}})$ congruent to $1$ mod $p^m$ for $m$ big enough. Let $N_0:=N_p\cap H_p$, $T_{p,0}:=T_p\cap H_p$, $\overline{N}_0:=\overline{N}_p\cap H_p$ (which are still uniform pro-p-groups) and $T_p^+:=\{ t\in T_p\ {\rm such\ that}\ tN_0t^{-1}\subseteq N_0\}$ (which is a multiplicative monoid in $T_p$). We also fix $z\in T_p^+$ such that $zN_0z^{-1}\subseteq N_0^p$ and we assume moreover $z^{-1}H_pz\subseteq K_p$ so that the elements of $z^{-1}H_pz$ normalize $H_p$ (as $H_p$ is normal in $K_p$). Note that such a $z$ always exists, for instance take $z$ such that $zN_0z^{-1}\subseteq N_0^p$, choose $r$ such that $H_p^{p^r}\subseteq zK_pz^{-1}$ and replace $H_p$ by $H_p^{p^r}$: with this new choice we still have $zN_0z^{-1}\subseteq N_0^p$.
For any uniform pro-$p$-group $H$ we denote by $\mathcal{C}(H,L)$ the Banach space of continuous $L$-valued functions on $H$ and, if $h\geq 1$, by $\mathcal{C}^{(h)}(H,L)$ the Banach space of $h$-analytic $L$-valued functions on $H$ defined in [@CD §0.3]. We have $\mathcal{C}(H_p,L)\cong \mathcal{C}(\overline{N}_0,L)\widehat{\otimes}_L\mathcal{C}(T_{p,0},L)\widehat{\otimes}_L\mathcal{C}(N_0,L)$ and likewise with $\mathcal{C}^{(h)}(\cdot,L)$.
\[restriction\] Let $f\in\mathcal{C}(H_p,L)$ such that for each left coset $(zH_pz^{-1}\cap H_p)n\subset H_p$, there exists $f_n\in\mathcal{C}^{(h)}(H_p,L)$ such that $f(gn)=f_n(z^{-1}gz)$ for $g\in zH_pz^{-1}\cap H_p$. Then we have: $$f\in\mathcal{C}^{(h-1)}(\overline{N}_0,L)\widehat{\otimes}_L\mathcal{C}^{(h)}(T_{p,0},L)\widehat{\otimes}_L\mathcal{C}(N_0,L).$$
Representatives of the quotient $(zH_pz^{-1}\cap H_p)\backslash H_p$ can be chosen in $N_0$, whence the above notation $n$ (do not confuse with the $n$ of ${\mathrm{GL}}_n$!). Restricting $f$ to the left coset $(zH_pz^{-1}\cap H_p)n$ for some $n\in N_0$ and translating on the right by $n$ we can assume that the support of $f$ is contained in $zH_pz^{-1}\cap H_p$. Then if $g\in zH_pz^{-1}\cap H_p$, we have by assumption $f(g)=F(z^{-1}gz)$ for some $F\in\mathcal{C}^{(h)}(H_p,L)$. Consequently $f|_{zH_pz^{-1}\cap H_p}$ can be extended to an $h$-analytic function on $zH_pz^{-1}$ and $f$ can be extended (by $0$) on $zH_pz^{-1}N_0=z\overline N_0z^{-1} T_{p,0} N_0$ as an element of: $$\mathcal{C}^{(h)}(z\overline{N}_0z^{-1},L)\widehat{\otimes}_L\mathcal{C}^{(h)}(T_{p,0},L)\widehat{\otimes}_L\mathcal{C}(N_0,L).$$ We deduce that $f$ is in the image of the restriction map (note that $zN_0z^{-1}\subseteq N_0$ implies $\overline{N}_0\subseteq z\overline{N_0}z^{-1}$): $$\mathcal{C}^{(h)}(z\overline{N}_0z^{-1},L)\widehat{\otimes}_L\mathcal{C}^{(h)}(T_{p,0},L)\widehat{\otimes}_L\mathcal{C}(N_0,L)\longrightarrow\mathcal{C}^{(h)}(\overline{N}_0,L)\widehat{\otimes}_L\mathcal{C}^{(h)}(T_{p,0},L)\widehat{\otimes}_L\mathcal{C}(N_0,L).$$ Now the stronger condition $zN_0z^{-1}\subseteq N_0^p$ implies $\overline{N}_0\subseteq z\overline{N_0}^pz^{-1}=(z\overline{N}_0z^{-1})^p$. But by [@CD Rem.IV.12] the restriction to $(z\overline{N}_0z^{-1})^p$ (and [*a fortiori*]{} to $\overline N_0$) of an $h$-analytic function on $z\overline{N}_0z^{-1}$ is $(h-1)$-analytic and we can conclude.
If $\Pi$ is an admissible continuous representation of $G_p$ on a $L$-Banach space and if $h\geq1$, we denote by $\Pi_{H_p}^{(h)}$ the $H_p$-invariant Banach subspace of $\Pi^{\rm an}$ defined in [@CD §0.3]. If $V$ is any (left) $U(\mathfrak{t}_L)$-module over $L$ and $\lambda:\mathfrak{t}_L\rightarrow L$ is a character, we let $V_\lambda$ be the $L$-subvector space of $V$ on which $\mathfrak{t}_L$ acts via the multiplication by $\lambda$. Recall that if $V$ is any $L[G_p]$-module, the monoid $T_p^+$ acts on $V^{N_0}$ via $v\longmapsto t\cdot v:=\delta_{B_p}(t)\sum_{n_0\in N_0/tN_0t^{-1}}n_0tv$ ($v\in V^{N_0}$, $t\in T_p^+$, see §\[classic\] for $\delta_{B_p}$). This $T_p^+$-action respects the subspace $(\Pi_\lambda^{\rm an})^{N_0}$ of $(\Pi^{\rm an})^{N_0}$ (use that $tN_0t^{-1}=N_0$ for $t\in T_p^0$).
We don’t claim any originality on the following lemma which is a variant of classical results (see e.g. [@EmertonJacquetI]), however we couldn’t find its exact statement in the literature.
\[compact\] Let $\Pi$ be an admissible continuous representation of $G_p$ on a $L$-Banach space, $\lambda$ an $L$-valued character of $\mathfrak{t}_L$ and $h\geq1$. Then the action of $z$ on $(\Pi^{\rm an})^{N_0}$ preserves the subspace $(\Pi^{(h)}_{H_p})^{N_0}_\lambda=(\Pi^{(h)}_{H_p})^{N_0}\cap \Pi^{\rm an}_\lambda$ and is a [compact]{} operator on this subspace.
Let $l_1,\dots,l_s$ be a system of generators of the continous dual $\Pi'$ as a module over the algebra $\mathcal{O}_L{{\mathchoice{\mbox{\rm [\hspace{-0.15em}[}}
{\mbox{\rm [\hspace{-0.15em}[}}
{\mbox{\scriptsize\rm [\hspace{-0.15em}[}}
{\mbox{\tiny\rm [\hspace{-0.15em}[}}}}H_p{{\mathchoice{\mbox{\rm ]\hspace{-0.15em}]}}
{\mbox{\rm ]\hspace{-0.15em}]}}
{\mbox{\scriptsize\rm ]\hspace{-0.15em}]}}
{\mbox{\tiny\rm ]\hspace{-0.15em}]}}}}[1/p]$. Define a closed embedding of $\Pi$ into $\mathcal{C}(H_p,L)^{\oplus s}$ via the map $v\mapsto (g\mapsto l_i(gv))_{1\leq i\leq s}$. This embedding is $H_p$-equivariant for the left action of $H_p$ on $\mathcal{C}(H_p,L)$ by right translation on functions. By [@CD Prop.IV.5], we have $\Pi_{H_p}^{(h)}=\Pi\cap \mathcal{C}^{(h)}(H_p,L)^{\oplus s}$. If $v\in\Pi_{H_p}^{(h)}$, $n\in N_0$ and $g\in H_p\cap z H_pz^{-1}$, we have $l_i(gnzv)=l_i(z(z^{-1}gz)(z^{-1}nz)v)$. Let $v\in\Pi_{H_p}^{(h)}$ and $n\in N_0$. As $z^{-1}N_0z$ normalizes $H_p$ (by the choice of $z$) we have $w:=(z^{-1}nz)v\in\Pi_{H_p}^{(h)}$ (see [@CD Prop.IV.16]). As $l_i(z\cdot)$ is a continuous linear form on $\Pi$, using [@CD Thm.IV.6(i)] the function $f_n:H_p\rightarrow L$, $g\mapsto f_n(g):=l_i(zgw)$ is in $\mathcal{C}^{(h)}(H_p,L)$ and $l_i(gnzv)=f_n(z^{-1}gz)$ for $g\in zH_pz^{-1}\cap H_p$. We deduce from Lemma \[restriction\] applied to the functions $f:H_p\rightarrow L$, $g\mapsto l_i(gzv)$ for $1\leq i\leq s$ that: $$\label{incl(h)}
z\Pi_{H_p}^{(h)}\subset \big(\mathcal{C}^{(h-1)}(\overline{N}_0,L)\widehat{\otimes}_L\mathcal{C}^{(h)}(T_{p,0},L)\widehat{\otimes}_L\mathcal{C}(N_0,L)\big)^{\oplus s}.$$ Let $v\in(\Pi_{H_p}^{(h)})^{N_0}$, the space on the right hand side of (\[incl(h)\]) being stable under $N_0$ (acting by right translation on functions), it still contains $z\cdot v=\sum_{n_0\in N_0/tN_0t^{-1}}n_0zv$. Since $z\cdot v$ is fixed under $N_0$, we deduce: $$z\cdot v\ \in\ \big(\mathcal{C}^{(h-1)}(\overline{N}_0,L)\widehat{\otimes}_L\mathcal{C}^{(h)}(T_{p,0},L)\big)^{\oplus s}\subseteq \ \mathcal{C}^{(h)}(H_p,L)^{\oplus s}.$$ In particular $z\cdot(\Pi_{H_p}^{(h)})^{N_0}_\lambda \subseteq (\Pi^{\rm an}_\lambda)^{N_0}\cap \mathcal{C}^{(h)}(H_p,L)^{\oplus s}=(\Pi_{H_p}^{(h)})^{N_0}_\lambda$ which shows the first statement. We also deduce: $$z\cdot(\Pi_{H_p}^{(h)})^{N_0}_\lambda\subseteq\big(\mathcal{C}^{(h-1)}(\overline{N}_0,L)\widehat{\otimes}_L\mathcal{C}^{(h)}(T_{p,0},L)\big)_{\!\lambda}^{\oplus s}\cong \mathcal{C}^{(h-1)}(\overline{N}_0,L)\widehat{\otimes}_L\big(\mathcal{C}^{(h)}(T_{p,0},L)_\lambda^{\oplus s}\big).$$ But by [@CD Prop. IV.13.(i)], we have $\mathcal{C}^{(h)}(T_{p,0},L)'\simeq D_r(T_{p,0},L)$ for $r=p^{-1/p^h}$ where $D_r(T_{p,0},L)$ is as in [@STdist §4]. Let $U_r(\mathfrak{t}_L)$ be the closure of $U(\mathfrak{t}_L)$ in $D_r(T_{p,0},L)$, then (as in the proof of Lemma \[injective\]) $D_r(T_{p,0},L)$ is a finite free $U_r(\mathfrak{t}_L)$-module ([@Ko §1.4]). Using $\lambda\buildrel\sim\over\rightarrow U_r(\mathfrak{t}_L)\otimes_{U(\mathfrak{t}_L)}\lambda$, it follows that $(\mathcal{C}^{(h)}(T_{p,0},L)_\lambda)'$, and hence $\mathcal{C}^{(h)}(T_{p,0},L)_\lambda^{\oplus s}$, are finite dimensional $L$-vector spaces. We denote the latter by $W_\lambda$.
We thus have $z\cdot(\Pi_{H_p}^{(h)})^{N_0}_\lambda\subseteq \mathcal{C}^{(h-1)}(\overline{N}_0,L)\otimes_L W_\lambda$: the endomorphism induced by $z$ on $(\Pi_{H_p}^{(h)})^{N_0}_\lambda$ factors through the subspace $(\Pi_{H_p}^{(h)})^{N_0}_\lambda\cap\big(\mathcal{C}^{(h-1)}(\overline{N}_0,L)\otimes_L W_\lambda\big)$. As the inclusion of $\mathcal{C}^{(h-1)}(\overline{N}_0,L)$ into $\mathcal{C}^{(h)}(\overline{N}_0,L)$ is compact and $W_\lambda$ is finite dimensional over $L$, the inclusion of $(\Pi_{H_p}^{(h)})^{N_0}_\lambda\cap\big(\mathcal{C}^{(h-1)}(\overline{N}_0,L)\otimes_L W\big)$ into $(\Pi_{H_p}^{(h)})^{N_0}_\lambda$ is compact, which proves the result.
If $\delta_v,\epsilon_v\in \widehat T_{v,L}$, we write $\epsilon_v \uparrow_{{\mathfrak t}_v} \delta_v$ if, seeing $\delta_v,\epsilon_v$ as $U({\mathfrak t}_{v,L})$-modules, we have $\epsilon_v \uparrow \delta_v$ in the sense of [@HumBGG §5.1] with respect to the roots of the upper triangular matrices in $({\mathrm{Res}}_{F_{\tilde v}/{\mathbb{Q}}_p}{\mathrm{GL}}_{n,F_{\tilde v}})_L$. Likewise if $\delta,\epsilon\in \widehat T_{p,L}$, we write $\epsilon \uparrow_{{\mathfrak t}} \delta$ if, seeing $\delta,\epsilon$ as $U({\mathfrak t}_{L})$-modules, we have $\epsilon \uparrow \delta$ in the sense of [@HumBGG §5.1] with respect to the roots of the upper triangular matrices in $\prod_{v\in S_p}({\mathrm{Res}}_{F_{\tilde v}/{\mathbb{Q}}_p}{\mathrm{GL}}_{n,F_{\tilde v}})_L$. Thus writing $\delta=(\delta_v)_{v\in S_p},\epsilon=(\epsilon_v)_{v\in S_p}$, we have $\epsilon \uparrow_{{\mathfrak t}} \delta$ if and only if $\epsilon_v \uparrow_{{\mathfrak t}_v} \delta_v$ for all $v\in S_p$.
\[stronglink\] Let $\delta,\epsilon\in \widehat T_{p,L}$, we write $\epsilon \uparrow \delta$ if $\epsilon \uparrow_{{\mathfrak t}} \delta$ and if $\epsilon\delta^{-1}$ is an algebraic character of $T_p$, i.e. $\epsilon\delta^{-1}=\delta_{\lambda}$ for some ${\lambda}=({\lambda}_v)_{v\in S_p}\in\prod_{v\in S_p}(\mathbb{Z}^n)^{{\mathrm{Hom}}(F_{\tilde v},L)}$.
We can now prove the main theorem of this section.
\[intercompanion\] Let ${\mathfrak m}\subseteq R_\infty[1/p]$ be a maximal ideal, $\delta,\epsilon\in \widehat T_{p,L}$ such that $\epsilon \uparrow \delta$ and $L'$ a finite extension of $L$ containing the residue fields $k(\delta)=k(\epsilon)$ and $k({\mathfrak m})$. Then we have: $${\mathrm{Hom}}_{T_p}\big(\epsilon,J_{B_p}(\Pi_\infty^{R_\infty-\rm an}[\mathfrak{m}]\otimes_{k({\mathfrak m})}L')\big)\ne 0\Longrightarrow {\mathrm{Hom}}_{T_p}\big(\delta,J_{B_p}(\Pi_\infty^{R_\infty-\rm an}[\mathfrak{m}]\otimes_{k({\mathfrak m})}L')\big)\ne 0.$$
We assume first $k(\delta)=k(\epsilon)=L$ and $L'=k({\mathfrak m})$, so that we can forget about $L'$. Let $\Pi$ be a locally ${\mathbb{Q}}_p$-analytic representation of $B_p$ over $L$. The subspace $\Pi^{\mathfrak{n}_L}$ of vectors killed by $\mathfrak{n}_L$ is a smooth representation of the group $N_0$ and we denote by $\pi_{N_0}:\Pi^{\mathfrak{n}_L}\twoheadrightarrow \Pi^{N_0}\subseteq \Pi^{\mathfrak{n}_L}$ the unique $N_0$-equivariant projection on its subspace $\Pi^{N_0}$. It is preserved by the action of $T_p$ inside $\Pi$, hence also by the action of $\mathfrak{t}_L$ and one easily checks that: $$\label{commutt}
\pi_{N_0}\circ \mathfrak{x}=\mathfrak{x}\circ \pi_{N_0}\ \ \ \ (\mathfrak{x}\in \mathfrak{t}_L)$$ (use $tN_0t^{-1}=N_0$ for $t\in T_p^0$). The subspace $\Pi_{\lambda}^{\mathfrak{n}_L}:=\Pi_\lambda\cap \Pi^{\mathfrak{n}_L}\subseteq \Pi^{\mathfrak{n}_L}$ is still preserved by $T_p$ and by (\[commutt\]) the projection $\pi_{N_0}$ sends $\Pi_{\lambda}^{\mathfrak{n}_L}$ onto $\Pi_{\lambda}^{N_0}:=\Pi^{N_0}\cap \Pi_{\lambda}^{\mathfrak{n}_L}\subseteq \Pi_{\lambda}^{\mathfrak{n}_L}$. We have $t\cdot v=\pi_{N_0}(tv)$ for $t\in T_p^+$, $v\in \Pi^{N_0}_\lambda$ and in the rest of the proof we view $\Pi^{N_0}_\lambda$ as an $L[T_p^+]$-module via this monoid action.
The locally ${\mathbb{Q}}_p$-analytic character $\delta:T_p\rightarrow L^\times$ determines a surjection of $L$-algebras $L[T_p]\twoheadrightarrow L$ and we denote its kernel by $\mathfrak{m}_\delta$ (a maximal ideal of the $L$-algebra $L[T_p]$). We still write $\mathfrak{m}_\delta$ for its intersection with $L[T_p^+]$, which is then a maximal ideal of $L[T_p^+]$. Let $\lambda:\mathfrak{t}_L\rightarrow L$ be the derivative of $\delta$, arguing as in [@EmertonJacquetI Prop.3.2.12] we get for $s\geq 1$: $$\label{jm}
J_{B_p}(\Pi)[\mathfrak{m}_\delta^s]\cong \Pi^{N_0}[\mathfrak{m}_{\delta}^s]\cong \Pi_\lambda^{N_0}[\mathfrak{m}_{\delta}^s],$$ (in particular ${\mathrm{Hom}}_{T_p}(\delta,J_{B_p}(\Pi))\cong \Pi^{N_0}[\mathfrak{m}_{\delta}]\cong \Pi_\lambda^{N_0}[\mathfrak{m}_{\delta}]$). Likewise we have $J_{B_p}(\Pi)[\mathfrak{m}_{\epsilon}^s]\cong \Pi^{N_0}[\mathfrak{m}_{\epsilon}^s]\cong \Pi_\mu^{N_0}[\mathfrak{m}_{\epsilon}^s]$ if $\mu:\mathfrak{t}_L\rightarrow L$ is the derivative of $\epsilon$.
Let $\mathfrak{I}\subset S_\infty[1/p]$ be an ideal such that $\dim_L(S_\infty[1/p]/\mathfrak{I})<\infty$ and define $\Pi_{\mathfrak{I}}:=\Pi_\infty[\mathfrak{I}]$. As the continuous dual $\Pi_\infty'$ is a finite projective $S_\infty{{\mathchoice{\mbox{\rm [\hspace{-0.15em}[}}
{\mbox{\rm [\hspace{-0.15em}[}}
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{\mbox{\tiny\rm [\hspace{-0.15em}[}}}}K_p{{\mathchoice{\mbox{\rm ]\hspace{-0.15em}]}}
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{\mbox{\scriptsize\rm ]\hspace{-0.15em}]}}
{\mbox{\tiny\rm ]\hspace{-0.15em}]}}}}[1/p]$-module (property (ii) in §\[firstclassical\]), the continuous dual $\Pi_\infty[\mathfrak{I}]'$ of the $G_p$-representation $\Pi_\infty[\mathfrak{I}]$, which is isomorphic to $\Pi_\infty'/\mathfrak{I}$ by the discussion at the end of [@BHS §3.1], is a finite projective $S_\infty{{\mathchoice{\mbox{\rm [\hspace{-0.15em}[}}
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{\mbox{\tiny\rm [\hspace{-0.15em}[}}}}K_p{{\mathchoice{\mbox{\rm ]\hspace{-0.15em}]}}
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{\mbox{\tiny\rm ]\hspace{-0.15em}]}}}}[1/p]/\mathfrak{I}S_\infty{{\mathchoice{\mbox{\rm [\hspace{-0.15em}[}}
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{\mbox{\tiny\rm ]\hspace{-0.15em}]}}}}[1/p]\cong \mathcal{O}_L{{\mathchoice{\mbox{\rm [\hspace{-0.15em}[}}
{\mbox{\rm [\hspace{-0.15em}[}}
{\mbox{\scriptsize\rm [\hspace{-0.15em}[}}
{\mbox{\tiny\rm [\hspace{-0.15em}[}}}}K_p{{\mathchoice{\mbox{\rm ]\hspace{-0.15em}]}}
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{\mbox{\scriptsize\rm ]\hspace{-0.15em}]}}
{\mbox{\tiny\rm ]\hspace{-0.15em}]}}}}[1/p]$-module (in particular it is an admissible continuous representation of $G_p$ over $L$). Moreover it is immediate to check that $\Pi_\infty^{R_\infty-\rm an}[\mathfrak{I}]$ is isomorphic to the subspace $\Pi_{\mathfrak{I}}^{\rm an}$ of locally ${\mathbb{Q}}_p$-analytic vectors of $\Pi_{\mathfrak{I}}$.
Taking the image of a vector in (the underlying $L$-vector space of) $\lambda$ or $\mu$ gives natural isomorphisms: $${\mathrm{Hom}}_{U(\mathfrak{g}_L)}\big(U(\mathfrak{g}_L)\otimes_{U(\mathfrak{b}_L)}\lambda,\Pi_{\mathfrak{I}}^{\rm an}\big)\buildrel\sim\over\longrightarrow (\Pi_{\mathfrak{I}}^{\rm an})_\lambda^{\mathfrak{n}_L}\ {\rm and}\ {\mathrm{Hom}}_{U(\mathfrak{g}_L)}\big(U(\mathfrak{g}_L)\otimes_{U(\mathfrak{b}_L)}\mu,\Pi_{\mathfrak{I}}^{\rm an}\big)\buildrel\sim\over\longrightarrow (\Pi_{\mathfrak{I}}^{\rm an})_\mu^{\mathfrak{n}_L}.$$ As $\mu$ is strongly linked to $\lambda$ by assumption, [@HumBGG Th.5.1] implies the existence of a unique (up to $L$-homothety) $U(\mathfrak{g}_L)$-equivariant injection: $$\label{mulambdau}
\iota_{\mu, \lambda}:\,U(\mathfrak{g}_L)\otimes_{U(\mathfrak{b}_L)}\mu\hookrightarrow U(\mathfrak{g}_L)\otimes_{U(\mathfrak{b}_L)}\lambda$$ which induces an $L$-linear map: $$\label{imulambda}
\iota^*_{\mu, \lambda}:\,(\Pi_{\mathfrak{I}}^{\rm an})_\lambda^{\mathfrak{n}_L}\longrightarrow (\Pi_{\mathfrak{I}}^{\rm an})_\mu^{\mathfrak{n}_L}.$$
We claim that (\[imulambda\]) maps the subspace $(\Pi_{\mathfrak{I}}^{\rm an})_\lambda^{N_0}$ to the subspace $(\Pi_{\mathfrak{I}}^{\rm an})_\mu^{N_0}$. It is enough to prove: $$\label{mulambda}
\iota^*_{\mu, \lambda}\circ\pi_{N_0}=\pi_{N_0}\circ \iota^*_{\mu, \lambda}.$$ Let $\mathfrak{v}$ be the image by (\[mulambdau\]) of a nonzero vector $v$ in the underlying $L$-vector space of $\mu$. Writing $U(\mathfrak{g}_L)\otimes_{U(\mathfrak{b}_L)}\lambda\cong U(\mathfrak{n}_L^-)$ we see that $\mathfrak{v}\in U(\mathfrak{n}_L^-)_{\mu-\lambda}$ (with obvious notation), that $\mathfrak{v}$ is killed by $\mathfrak{n}_L$ in $U(\mathfrak{g}_L)\otimes_{U(\mathfrak{b}_L)}\lambda$ and that the morphism $\iota^*_{\mu, \lambda}$ is given by the action (on the left) by $\mathfrak{v}$. To get (\[mulambda\]) it is enough to prove $\pi_{N_0}\circ\mathfrak{v}=\mathfrak{v}\circ\pi_{N_0}$ on $(\Pi_\mathfrak{I}^{\rm an})^{\mathfrak{n}_L}_{\lambda}$, which itself follows from $n\circ \mathfrak{v}=\mathfrak{v}\circ n$ for $n\in N_0$ ($n$ acting via the underlying $G_p$-action on $\Pi_{\mathfrak{I}}^{\rm an}$), or equivalently $\mathrm{Ad}(n)(\mathfrak{v})=\mathfrak{v}$ on $(\Pi_\mathfrak{I}^{\rm an})^{\mathfrak{n}_L}_{\lambda}$. Writing $n=\exp(\mathfrak{m})$ with $\mathfrak{m}\in\mathfrak{n}_L$ (do not confuse here with the maximal ideal $\mathfrak{m}$!), recall we have $\mathrm{Ad}(n)(\mathfrak{v})=\exp(\mathrm{ad}(\mathfrak{m}))(\mathfrak{v})$ (using standard notation). Since $\mathfrak{v}$ is killed by left multiplication by $\mathfrak{n}_L$ in $U(\mathfrak{g}_L)\otimes_{U(\mathfrak{b}_L)}\lambda$, we have: $$\exp(\mathrm{ad}(\mathfrak{m}))(\mathfrak{v})\in\mathfrak{v}+U(\mathfrak{g}_L)(\mathfrak{n}_L+\ker(\lambda))$$ where $\ker(\lambda):=\ker(U(\mathfrak{g}_L)\rightarrow U(\mathfrak{g}_L)\otimes_{U(\mathfrak{b}_L)}\lambda, \ \mathfrak{x}\mapsto \mathfrak{x}\otimes v)$. The action of $\mathrm{Ad}(n)(\mathfrak{v})$ on $(\Pi_\mathfrak{I}^{\rm an})^{\mathfrak{n}_L}_{\lambda}$ is thus the same as that of $\mathfrak{v}$.
We still write: $$\label{mulambda0}
\iota^*_{\mu, \lambda}:\,(\Pi_{\mathfrak{I}}^{\rm an})_\lambda^{N_0}\longrightarrow (\Pi_{\mathfrak{I}}^{\rm an})_\mu^{N_0}$$ for the map induced by (\[imulambda\]). Using $\mathfrak{v}\in U(\mathfrak{n}_L^-)_{\mu-\lambda}$ together with (\[mulambda\]), it is easy to check that $\iota^*_{\mu, \lambda}\circ t=(\delta\epsilon^{-1})(t)(t\circ \iota^*_{\mu, \lambda})$ for $t\in T_p^+$ (for the previous $L[T_p^+]$-module structure). Moreover, it follows from Lemma \[injective\] that (\[imulambda\]) is surjective, hence the top horizontal map and the two vertical maps are surjective in the commutative diagram: $$\begin{xy}\xymatrix{(\Pi_{\mathfrak{I}}^{\rm an})_\lambda^{\mathfrak{n}_L}\ar[d]_{\pi_{N_0}}\ar[r]^{(\ref{imulambda})} & (\Pi_{\mathfrak{I}}^{\rm an})_\mu^{\mathfrak{n}_L}\ar[d]^{\pi_{N_0}}\\(\Pi_{\mathfrak{I}}^{\rm an})_\lambda^{N_0}\ar[r]^{(\ref{mulambda0})} & (\Pi_{\mathfrak{I}}^{\rm an})_\mu^{N_0}}
\end{xy}$$ which implies that (\[mulambda0\]) is also surjective. Note also that both (\[imulambda\]) and (\[mulambda0\]) trivially commute with the action of $R_\infty$ (which factors through $R_\infty/\mathfrak{I}R_\infty$).
From [@CD §0.3] we have $\Pi_{\mathfrak{I}}^{\rm an}\cong \lim_{h\rightarrow +\infty}\Pi_{\mathfrak{I},H_p}^{(h)}$ and thus: $$(\Pi_{\mathfrak{I}}^{\rm an})_\lambda^{N_0}\cong \lim_{h\rightarrow +\infty}(\Pi_{\mathfrak{I},H_p}^{(h)})_\lambda^{N_0}\ \ {\rm and}\ \ (\Pi_{\mathfrak{I}}^{\rm an})_\mu^{N_0}\cong \lim_{h\rightarrow +\infty}(\Pi_{\mathfrak{I},H_p}^{(h)})_\mu^{N_0}.$$ By Lemma \[compact\] there is $z\in T_p^+$ which acts compactly on $(\Pi_{\mathfrak{I},H_p}^{(h)})_\lambda^{N_0}$ and $(\Pi_{\mathfrak{I},H_p}^{(h)})_\mu^{N_0}$. We deduce from this fact together with [@SerreCC Prop.9] and [@SerreCC Prop.12] that the map $\iota^*_{\mu,\lambda}$ in (\[mulambda0\]) remains surjective at the level of [*generalized eigenspaces*]{} for the action of $T_p^+$ (twisting this action by the character $\delta\epsilon^{-1}$ on the right hand side). Consequently $\iota^*_{\mu,\lambda}$ induces a surjective map: $$\bigcup_{s\geq 1}(\Pi_{\mathfrak{I}}^{\rm an})_{\lambda}^{N_0}[\mathfrak{m}_{\delta}^s]\twoheadrightarrow \bigcup_{s\geq 1}(\Pi_{\mathfrak{I}}^{\rm an})^{N_0}_\mu[\mathfrak{m}_{\epsilon}^s].$$ As both the source and target of this map are unions of finite dimensional $L$-vector spaces (as follows from the admissibility of $\Pi_{\mathfrak{I}}^{\rm an}$, [@EmertonJacquetI Th.4.3.2] and (\[jm\])) which are stable under $R_\infty$ and as $\iota^*_{\mu,\lambda}$ is $R_\infty$-equivariant, the following map induced by $\iota^*_{\mu,\lambda}$ remains surjective: $$\label{cupcup}
\bigcup_{s,t\geq 1}(\Pi_\mathfrak{I}^{\rm an})_{\lambda}^{N_0}[\mathfrak{m}_{\delta}^s, \mathfrak{m}^t]\twoheadrightarrow \bigcup_{s,t\geq 1}(\Pi_\mathfrak{I}^{\rm an})^{N_0}_\mu[\mathfrak{m}_{\epsilon}^s,\mathfrak{m}^t].$$ Since $\mathfrak{m}^t$ is an ideal of cofinite dimension in $R_\infty[1/p]$, the inverse image $\mathfrak{I}$ of $\mathfrak{m}^t$ in $S_\infty[1/p]$ is [*a fortiori*]{} of cofinite dimension in $S_\infty[1/p]$ and we can apply (\[cupcup\]) with such an $\mathfrak I$. But we have for this $\mathfrak{I}$: $$(\Pi_{\mathfrak{I}}^{\rm an})^{N_0}_\lambda[\mathfrak{m}_\delta^s,\mathfrak{m}^t]=(\Pi_\infty^{R_\infty-\rm an})^{N_0}_\lambda[\mathfrak{m}_\delta^s,\mathfrak{m}^t,\mathfrak{I}]=(\Pi_\infty^{R_\infty-\rm an})^{N_0}_\lambda[\mathfrak{m}_{\delta}^s,\mathfrak{m}^t]$$ and likewise with $\mathfrak{m}_{\epsilon}$, so that (\[cupcup\]) is a surjection: $$\bigcup_{s,t\geq 1}(\Pi_\infty^{R_\infty-\rm an})_{\lambda}^{N_0}[\mathfrak{m}_{\delta}^s, \mathfrak{m}^t]\twoheadrightarrow \bigcup_{s,t\geq 1}(\Pi_\infty^{R_\infty-\rm an})_{\mu}^{N_0}[\mathfrak{m}_{\epsilon}^s, \mathfrak{m}^t].$$ Looking at the eigenspaces on both sides, we obtain ${\mathrm{Hom}}_{T_p}(\delta,J_{B_p}(\Pi_\infty^{R_\infty-\rm an}[\mathfrak{m}]))\ne 0$ if ${\mathrm{Hom}}_{T_p}(\epsilon,J_{B_p}(\Pi_\infty^{R_\infty-\rm an}[\mathfrak{m}]))\ne 0$.\
Finally, when $k(\delta)=k(\epsilon)$ is larger than $L$, we replace $\Pi_\infty$ by $\Pi_\infty':=\Pi_\infty\otimes_LL'$, $S_\infty[1/p]$ by $S_\infty[1/p]\otimes_LL'$, ${\mathfrak m}$ by ${\mathfrak m}':=\ker(R_\infty[1/p]\otimes_LL'\twoheadrightarrow k({\mathfrak m})\otimes_LL'\twoheadrightarrow L')$ (the last surjection coming from the inclusion $k({\mathfrak m})\subseteq L'$) and the reader can check that all the arguments of the previous proof go through [*mutatis mutandis*]{}.
Tangent spaces on the trianguline variety {#geometry}
-----------------------------------------
We prove that Conjecture \[BHS\] implies Conjecture \[mainconj\] (when ${\overline{r}}$ “globalizes” and $x$ is very regular) and give one (conjectural) application.
We keep the notation and assumptions of §\[closed\]. We fix $x=(r,\delta)\in \widetilde X_{\rm tri}^\square({\overline{r}})\subseteq X_{\rm tri}^\square({\overline{r}})$ which is crystalline strictly dominant very regular. Recall from Lemma \[paramofcrystpt\] that $\delta=(\delta_1\dots,\delta_n)$ where $\delta_i=z^{{\bf k}_i}{\rm unr}(\varphi_i)$ with ${\bf k}_i=(k_{\tau,i})_{\tau:\, K\hookrightarrow L}\in\mathbb{Z}^{{\mathrm{Hom}}(K,L)}$ and $\varphi_i\in k(x)^\times$. The following result immediately follows from Proposition \[acconZaut\] and Theorem \[upperbound\] applied to $X=\widetilde X_{\rm tri}^\square(\overline r)$.
\[easybound\] Assume Conjecture \[BHS\], then we have: $$\dim_{k(x)}T_{\widetilde X_{\rm tri}^\square(\overline r),x}\leq \lg(w_x)-d_x+\dim X_{\rm tri}^\square({\overline{r}})=\lg(w_x)-d_x+n^2+[K:{\mathbb{Q}}_p]\frac{n(n+1)}{2}.$$
The rest of this section is devoted to the proof of the converse inequality (still assuming Conjecture \[BHS\]).
As in the proof of Proposition \[inter\], we consider for $1\leq i\leq n$ the cartesian diagram which defines $W_i$ (with the notation of §\[wedgesection\]): $$\xymatrix{{\rm Ext}^1_{(\varphi,\Gamma_K)}\big(D_{{\mathrm{rig}}}(r),D_{{\mathrm{rig}}}(r)\big)\ar[r]&{\rm Ext}^1_{(\varphi,\Gamma_K)}\big(D_{{\mathrm{rig}}}(r)^{\leq i},D_{{\mathrm{rig}}}(r)\big)\\
W_i \ar[r]\ar@{^{(}->}[u]&
{\rm Ext}^1_{(\varphi,\Gamma_K)}\big(D_{{\mathrm{rig}}}(r)^{\leq i},D_{{\mathrm{rig}}}(r)^{\leq i}\big).\ar@{^{(}->}[u]}$$ We define $W_{{\rm cris},i}\subseteq {\rm Ext}^1_{\rm cris}(D_{{\mathrm{rig}}}(r),D_{{\mathrm{rig}}}(r))$ as $W_i$ but replacing everywhere ${\rm Ext}^1_{(\varphi,\Gamma_K)}$ by its subspace ${\rm Ext}^1_{\rm cris}$. Note that $W_{{\rm cris},i}\subseteq W_i$ for $1\leq i\leq n$.
\[gradcris\] For $1\leq i\leq n$, we have isomorphisms of $k(x)$-vector spaces: $$\begin{aligned}
W_1\cap\cdots \cap W_{i-1}/W_1\cap\cdots \cap W_{i} &\buildrel\sim\over\longrightarrow &
{\rm Ext}_{(\varphi,\Gamma_K)}^1\big({\rm gr}_iD_{{\mathrm{rig}}}(r),D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq i}\big)\\
\nonumber W_{{\rm cris},1}\cap\cdots \cap W_{{\rm cris},i-1}/W_{{\rm cris},1}\cap\cdots \cap W_{{\rm cris},i} &\buildrel\sim\over\longrightarrow &
{\rm Ext}_{\rm cris}^1\big({\rm gr}_iD_{{\mathrm{rig}}}(r),D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq i}\big)\end{aligned}$$ where $W_1\cap\cdots \cap W_{i-1}:={\rm Ext}_{(\varphi,\Gamma_K)}^1(D_{{\mathrm{rig}}}(r),D_{{\mathrm{rig}}}(r))$ (resp. $W_{{\rm cris},1}\cap\cdots \cap W_{{\rm cris},i-1}:={\rm Ext}_{\rm cris}^1(D_{{\mathrm{rig}}}(r),D_{{\mathrm{rig}}}(r))$) if $i=1$.
We write $D_{{\mathrm{rig}}}$ instead of $D_{{\mathrm{rig}}}(r)$ and drop the subscript $(\varphi,\Gamma_K)$ in this proof. We start with the first isomorphism, the proof of which is analogous to (though simpler than) the proof of (\[isoi\]) in §\[endofproof\]. We have the exact sequence (using Definition \[veryreg\]): $$\label{exactwcris}
\!\!0\rightarrow {\rm Ext}^1\big({\rm gr}_iD_{{\mathrm{rig}}},D_{{\mathrm{rig}}}/D_{{\mathrm{rig}}}^{\leq i}\big)\rightarrow {\rm Ext}^1\big(D_{{\mathrm{rig}}}^{\leq i},D_{{\mathrm{rig}}}/D_{{\mathrm{rig}}}^{\leq i}\big)\rightarrow {\rm Ext}^1\big(D_{{\mathrm{rig}}}^{\leq i-1},D_{{\mathrm{rig}}}/D_{{\mathrm{rig}}}^{\leq i}\big)\rightarrow 0.$$ The composition: $$W_1\cap\cdots \cap W_{i-1}\hookrightarrow {\rm Ext}^1\big(D_{{\mathrm{rig}}},D_{{\mathrm{rig}}}\big)\twoheadrightarrow {\rm Ext}^1\big(D_{{\mathrm{rig}}}^{\leq i},D_{{\mathrm{rig}}}/D_{{\mathrm{rig}}}^{\leq i}\big)$$ lands in ${\rm Ext}^1({\rm gr}_iD_{{\mathrm{rig}}},D_{{\mathrm{rig}}}/D_{{\mathrm{rig}}}^{\leq i})$ by (\[exactwcris\]). If $v\in W_1\cap\cdots \cap W_{i-1}$ is also in $W_i$, then its image in ${\rm Ext}^1(D_{{\mathrm{rig}}}^{\leq i},D_{{\mathrm{rig}}}/D_{{\mathrm{rig}}}^{\leq i})$ is $0$. We thus deduce a canonical induced map: $$\label{wimap}
W_1\cap\cdots \cap W_{i-1}/W_1\cap\cdots \cap W_{i} \rightarrow {\rm Ext}^1\big({\rm gr}_iD_{{\mathrm{rig}}},D_{{\mathrm{rig}}}/D_{{\mathrm{rig}}}^{\leq i}\big).$$ Let us prove that (\[wimap\]) is surjective. One easily checks that ${\rm Ext}^1(D_{{\mathrm{rig}}}/D_{{\mathrm{rig}}}^{\leq i-1},D_{{\mathrm{rig}}})\subseteq W_1\cap\cdots \cap W_{i-1}$ and that the natural map ${\rm Ext}^1(D_{{\mathrm{rig}}}/D_{{\mathrm{rig}}}^{\leq i-1},D_{{\mathrm{rig}}})\rightarrow {\rm Ext}^1({\rm gr}_iD_{{\mathrm{rig}}},D_{{\mathrm{rig}}}/D_{{\mathrm{rig}}}^{\leq i})$ is surjective (again by Definition \[veryreg\]). This implies that [*a fortiori*]{} (\[wimap\]) must also be surjective. Let us prove that (\[wimap\]) is injective. If $w\in W_1\cap\cdots \cap W_{i-1}$ maps to zero, then the image of $w$ in ${\rm Ext}^1(D_{{\mathrm{rig}}}^{\leq i},D_{{\mathrm{rig}}}/D_{{\mathrm{rig}}}^{\leq i})$ is also zero, i.e. $w\in W_i$ hence $w\in W_1\cap\cdots \cap W_{i}$.
The proof for the second isomorphism is exactly the same replacing everywhere $W_j$ by $W_{{\rm cris},j}$ and ${\rm Ext}^1_{(\varphi,\Gamma_K)}$ by ${\rm Ext}^1_{\rm cris}$.
\[condcris\] We have: $$\begin{aligned}
\dim_{k(x)}\big(W_{1}\cap\cdots \cap W_{n-1}\big) &=& \dim_{k(x)}{\rm Ext}_{(\varphi,\Gamma_K)}^1\big(D_{{\mathrm{rig}}}(r),D_{{\mathrm{rig}}}(r)\big) - [K:{\mathbb{Q}}_p]\frac{n(n-1)}{2}\\
\dim_{k(x)}\big(W_{{\rm cris},1}\cap\cdots \cap W_{{\rm cris},n-1}\big) &=& \dim_{k(x)}{\rm Ext}^1_{\rm cris}\big(D_{{\mathrm{rig}}}(r),D_{{\mathrm{rig}}}(r)\big)-\lg(w_x).\end{aligned}$$
This follows from Proposition \[gradcris\] together with (\[dimcrisi\]) and Lemma \[dim1\] (both for $\ell=i$) by the same argument as at the end of the proof of Proposition \[condsplit\].
\[intercris\] [Note that $W_1\cap\cdots \cap W_{n-1} \cap {\rm Ext}^1_{\rm cris}(D_{{\mathrm{rig}}}(r),D_{{\mathrm{rig}}}(r))=W_{{\rm cris},1}\cap\cdots \cap W_{{\rm cris},n-1}$.]{}
Now consider $x':=(r,\delta')=(r,\delta'_1,\dots,\delta'_n)$ with $\delta'_i:=z^{{\bf k}_{w_x^{-1}(i)}}{\rm unr}(\varphi_i)$, then $x'\in \widetilde U_{\rm tri}^\square({\overline{r}})$ by (\[triang\]). We also have $\omega(x')\in \mathcal{W}^n_{w_x,{\bf k},L}$ by (\[point\]), thus $x'\in \widetilde U_{\rm tri}^\square({\overline{r}})\times_{\mathcal{W}^n_L}\mathcal{W}^n_{w_x,{\bf k},L}\subseteq \widetilde U_{\rm tri}^\square({\overline{r}})$ and $\jmath_{w,{\bf k}}(x')=x$. Recall from §\[wedgesection\] and the smoothness of $U_{\rm tri}^\square({\overline{r}})$ over $\mathcal{W}^n_{L}$ that the weight map $\omega$ induces a $k(x)$-linear surjection on tangent spaces (note that $k(x')=k(x)$): $$\label{sen}
d\omega : T_{\widetilde X_{\rm tri}^\square({\overline{r}}),x'}\cong T_{X_{\rm tri}^\square({\overline{r}}),x'}\twoheadrightarrow T_{{\mathcal W}^n_L,\omega(x')}\cong k(x)^{[K:{\mathbb{Q}}_p]n}, \ \ \vec{v} \longmapsto (d_{\tau,i,\vec{v}})_{1\leq i\leq n,\tau:\, K\hookrightarrow L}.$$
\[tgtx’\] We have an isomorphism of $k(x)$-subvector spaces of $T_{\widetilde X_{\rm tri}^\square({\overline{r}}),x'}$: $$\begin{gathered}
T_{(\widetilde X_{\rm tri}^\square({\overline{r}})\times_{\mathcal{W}^n_L}\mathcal{W}^n_{w_x,{\bf k},L})^{\mathrm{red}},x'}\buildrel\sim\over\longrightarrow \\
\left\{\vec{v}\in T_{\widetilde X_{\rm tri}^\square({\overline{r}}),x'}\ {\rm such\ that}\ d_{\tau,i,\vec{v}}=d_{\tau,w_{x,\tau}^{-1}(i),\vec{v}},\ 1\leq i\leq n,\ \tau:\, K\hookrightarrow L\right\}.\end{gathered}$$ In particular $\dim_{k(x)}T_{(\widetilde X_{\rm tri}^\square({\overline{r}})\times_{\mathcal{W}^n_L}\mathcal{W}^n_{w_x,{\bf k},L})^{\mathrm{red}},x'}=\dim X_{\rm tri}^\square({\overline{r}})-d_x$.
We write ${\mathrm{Hom}}$ instead of ${\mathrm{Hom}}_{k(x)-{\rm alg}}$ in this proof. Let $\widetilde U_{{\rm tri},w_x,{\bf k}}^\square({\overline{r}}):=\widetilde U_{\rm tri}^\square({\overline{r}})\times_{\mathcal{W}^n_L}\mathcal{W}^n_{w_x,{\bf k},L}$, we have: $$\label{tensorlocal}
{\mathcal O}_{\widetilde U_{{\rm tri},w_x,{\bf k}}^\square({\overline{r}}),x'}\cong {\mathcal O}_{\widetilde U_{\rm tri}^\square({\overline{r}}),x'}\otimes_{{\mathcal O}_{\mathcal{W}^n_L,\omega(x')}}{\mathcal O}_{\mathcal{W}^n_{w_x,{\bf k},L},\omega(x')}$$ and note that $T_{(\widetilde X_{\rm tri}^\square({\overline{r}})\times_{\mathcal{W}^n_L}\mathcal{W}^n_{w_x,{\bf k},L})^{\mathrm{red}},x'}=T_{\widetilde U_{{\rm tri},w,{\bf k}}^\square({\overline{r}}),x'}$. Recall that, if $A,B,C,D$ are commutative $k(x)$-algebras with $B,C$ being $A$-algebras, we have: $$\label{abcd}
{\mathrm{Hom}}(B\otimes_AC,D)\buildrel\sim\over\longrightarrow {\mathrm{Hom}}(B,D)\times_{{\mathrm{Hom}}(A,D)}{\mathrm{Hom}}(C,D).$$ From (\[tensorlocal\]) and (\[abcd\]) we deduce: $$\begin{gathered}
T_{\widetilde U_{{\rm tri},w,{\bf k}}^\square({\overline{r}}),x'}={\mathrm{Hom}}\big({\mathcal O}_{\widetilde U_{{\rm tri},w_x,{\bf k}}^\square({\overline{r}}),x'},k(x)[\varepsilon]/(\varepsilon^2)\big)\cong T_{\widetilde X_{\rm tri}^\square({\overline{r}}),x'}\times_{T_{{\mathcal W}^n_L,\omega(x')}}T_{{\mathcal W}^n_{w_x,{\bf k},L},\omega(x')}.\end{gathered}$$ But from (\[equations\]) we have: $$T_{{\mathcal W}^n_{w_x,{\bf k},L},\omega(x')}=\left\{(d_{\tau,i})_{1\leq i\leq n,\tau:\, K\hookrightarrow L}\in T_{{\mathcal W}^n_{L},\omega(x')}\ {\rm such\ that}\ d_{\tau,i}=d_{\tau,w_{x,\tau}^{-1}(i)},\ \forall\ i,\ \forall\ \tau\right\}$$ whence the first statement. The last statement comes from $\dim_{k(x)}T_{\widetilde X_{\rm tri}^\square({\overline{r}}),x'}=\dim \widetilde X_{\rm tri}^\square({\overline{r}})=\dim X_{\rm tri}^\square({\overline{r}})$ (since $x'$ is smooth on $X_{\rm tri}^\square({\overline{r}})$ as $x'\in U_{\rm tri}^\square({\overline{r}})$), the surjectivity of $T_{\widetilde X_{\rm tri}^\square({\overline{r}}),x'}\rightarrow T_{{\mathcal W}^n_{L},\omega(x')}$ (since the morphism $\widetilde U_{\rm tri}^\square({\overline{r}})\rightarrow {\mathcal W}^n_{L}$ is smooth by [@BHS Th.2.6(iii)]) and the same argument as in the proof of Proposition \[condweight\].
Recall from the discussion just before Conjecture \[mainconj\] that we have a closed embedding $\iota_{\bf k}:\widetilde{\mathfrak{X}}_{\overline r}^{\square,{\bf k}\rm -cr}\hookrightarrow \widetilde X_{\rm tri}^\square(\overline r)$ with $x\in \iota_{\bf k}(\widetilde{\mathfrak{X}}_{\overline r}^{\square,{\bf k}\rm -cr})$. We deduce an injection of $k(x)$-vector spaces: $$T_{\iota_{\bf k}(\widetilde {\mathfrak{X}}_{\overline r}^{\square,{\bf k}\rm -cr}),x}\hookrightarrow T_{\widetilde X_{\rm tri}^\square(\overline r),x}.$$ Likewise we deduce from Proposition \[companion\] (assuming Conjecture \[BHS\]) another injection of $k(x)$-vector spaces: $$T_{\jmath_{w_x,{\bf k}}\big(\overline{\widetilde U_{\rm tri}^\square({\overline{r}})\times_{\mathcal{W}^n_L}\mathcal{W}^n_{w,{\bf k},L}}\big),x}\hookrightarrow T_{\widetilde X_{\rm tri}^\square(\overline r),x}.$$ Taking the sum in $T_{\widetilde X_{\rm tri}^\square({\overline{r}}),x}$ of these two subspaces of $T_{\widetilde X_{\rm tri}^\square({\overline{r}}),x}$, we have an injection of $k(x)$-vector spaces: $$\label{injtang}
T_{\jmath_{w_x,{\bf k}}\big(\overline{\widetilde U_{\rm tri}^\square({\overline{r}})\times_{\mathcal{W}^n_L}\mathcal{W}^n_{w,{\bf k},L}}\big),x}+T_{\iota_{\bf k}(\widetilde{\mathfrak{X}}_{\overline r}^{\square,{\bf k}\rm -cr}),x}\hookrightarrow T_{\widetilde X_{\rm tri}^\square({\overline{r}}),x}.$$
\[dimsum\] Assume Conjecture \[BHS\], then we have: $$\dim_{k(x)}\big(T_{\jmath_{w_x,{\bf k}}\big(\overline{\widetilde U_{\rm tri}^\square({\overline{r}})\times_{\mathcal{W}^n_L}\mathcal{W}^n_{w,{\bf k},L}}\big),x}+T_{\iota_{\bf k}(\widetilde{\mathfrak{X}}_{\overline r}^{\square,{\bf k}\rm -cr}),x}\big)=\lg(w_x)-d_{x}+\dim X_{\rm tri}^\square({\overline{r}}).$$
The composition $\widetilde X_{\rm tri}^\square({\overline{r}})\hookrightarrow \mathfrak{X}_{{\overline{r}}}^\square\times \mathcal{T}^n_L\twoheadrightarrow \mathfrak{X}_{{\overline{r}}}^\square$ induces a $k(x)$-linear morphism $T_{\widetilde X_{\rm tri}^\square({\overline{r}}),x'}\!\rightarrow T_{\mathfrak{X}_{{\overline{r}}}^\square,r}$. Since $x'\in \widetilde U_{\rm tri}^\square({\overline{r}})\subseteq U_{\rm tri}^\square({\overline{r}})$, it follows from [@KPX Th.Cor.6.3.10] (arguing e.g. as in the proof of [@BHS Lem.2.11]) that the triangulation $(D_{{\mathrm{rig}}}^{\leq i})_{1\leq i\leq n}$ “globalizes” in a small neighbourhood of $x'$ in $U_{\rm tri}^\square({\overline{r}})$, or equivalenty in $\widetilde U_{\rm tri}^\square({\overline{r}})$. In particular, for any $\vec{v}\in T_{\widetilde U_{\rm tri}^\square({\overline{r}}),x'}\cong T_{\widetilde X_{\rm tri}^\square({\overline{r}}),x'}$ we have a triangulation of $D_{{\mathrm{rig}}}(r_{\vec{v}})$ by free $(\varphi,\Gamma_K)$-submodules over $\mathcal{R}_{k(x)[\varepsilon]/(\varepsilon^2),K}$ such that the associated parameter is $(\delta_{1,\vec{v}},\dots,\delta_{n,\vec{v}})$ (see §\[wedgesection\] for the notation). This has two consequences: (1) the proof of Lemma \[inj\] goes through and the above map $T_{\widetilde X_{\rm tri}^\square({\overline{r}}),x'}\rightarrow T_{\mathfrak{X}_{{\overline{r}}}^\square,r}$ is an injection of $k(x)$-vector spaces and (2) the image of the composition $T_{\widetilde X_{\rm tri}^\square({\overline{r}}),x'}\hookrightarrow T_{\mathfrak{X}_{{\overline{r}}}^\square,r}\twoheadrightarrow {\rm Ext}_{(\varphi,\Gamma_K)}^1(D_{{\mathrm{rig}}}(r),D_{{\mathrm{rig}}}(r))$ (see Lemma \[ext1\]) lies in $W_{1}\cap\cdots \cap W_{n-1}\!\subseteq \!{\rm Ext}_{(\varphi,\Gamma_K)}^1(D_{{\mathrm{rig}}}(r),D_{{\mathrm{rig}}}(r))$. From Lemma \[ext1\] we thus obtain an exact sequence: $$0\rightarrow K(r)\cap T_{\widetilde X_{\rm tri}^\square({\overline{r}}),x'}\rightarrow T_{\widetilde X_{\rm tri}^\square({\overline{r}}),x'}\rightarrow W_{1}\cap\cdots \cap W_{n-1}.$$ But $\dim_{k(x)}T_{\widetilde X_{\rm tri}^\square({\overline{r}}),x'}=\dim X_{\rm tri}^\square({\overline{r}})$ since $\widetilde X_{\rm tri}^\square({\overline{r}})$ is smooth at $x'$, and from Lemma \[ext1\], Lemma \[dim\] and Corollary \[condcris\], we have: $$\dim_{k(x)}K(r) + \dim_{k(x)}\big(W_{1}\cap\cdots \cap W_{n-1}\big) = n^2 + [K:{\mathbb{Q}}_p]\frac{n(n+1)}{2}= \dim X_{\rm tri}^\square({\overline{r}})$$ which forces a short exact sequence $0\rightarrow K(r)\rightarrow T_{\widetilde X_{\rm tri}^\square({\overline{r}}),x'}\rightarrow W_{1}\cap\cdots \cap W_{n-1}\rightarrow 0$. It then follows from Proposition \[tgtx’\] that we have a short exact sequence of $k(x)$-vector spaces: $$\label{se1}
0\rightarrow K(r)\rightarrow T_{(\widetilde X_{\rm tri}^\square({\overline{r}})\times_{\mathcal{W}^n_L}\mathcal{W}^n_{w_x,{\bf k},L})^{\mathrm{red}},x'}\rightarrow W_{1}\cap\cdots \cap W_{n-1}\cap V\rightarrow 0$$ where $V\subseteq {\rm Ext}^1_{(\varphi,\Gamma_K)}(D_{{\mathrm{rig}}}(r),D_{{\mathrm{rig}}}(r))$ is as in §\[endofproof\] (the intersection on the right hand side being in ${\rm Ext}^1_{(\varphi,\Gamma_K)}(D_{{\mathrm{rig}}}(r),D_{{\mathrm{rig}}}(r))$).
Arguing as in [@KisinModularity §2.3.5] we also have a short exact sequence (see [@Kisindef (3.3.5)]): $$\label{se2}
0\rightarrow K(r)\rightarrow T_{{\mathfrak{X}}_{\overline r}^{\square,{\bf k}\rm -cr},r}\rightarrow {\rm Ext}^1_{\rm cris}(D_{{\mathrm{rig}}}(r),D_{{\mathrm{rig}}}(r))\rightarrow 0.$$ Using $T_{\iota_{\bf k}(\widetilde{\mathfrak{X}}_{\overline r}^{\square,{\bf k}\rm -cr}),x}\cong T_{{\mathfrak{X}}_{\overline r}^{\square,{\bf k}\rm -cr},r}$ (which easily follows from the fact that the Frobenius eigenvalues $(\varphi_1,\dots,\varphi_n)$ are pairwise distinct) and: $$T_{(\widetilde X_{\rm tri}^\square({\overline{r}})\times_{\mathcal{W}^n_L}\mathcal{W}^n_{w,{\bf k},L})^{\mathrm{red}},x'}\cong T_{\widetilde U_{\rm tri}^\square({\overline{r}})\times_{\mathcal{W}^n_L}\mathcal{W}^n_{w,{\bf k},L},x'}\cong T_{\overline{\widetilde U_{\rm tri}^\square({\overline{r}})\times_{\mathcal{W}^n_L}\mathcal{W}^n_{w,{\bf k},L}},x'}\buildrel\sim\over\rightarrow T_{\jmath_{w_x,{\bf k}}(\overline{\widetilde U_{\rm tri}^\square({\overline{r}})\times_{\mathcal{W}^n_L}\mathcal{W}^n_{w,{\bf k},L}}),x},$$ we deduce from (\[se1\]) and (\[se2\]) a short exact sequence of $k(x)$-vector spaces: $$\begin{gathered}
0\rightarrow K(r)\rightarrow T_{\jmath_{w_x,{\bf k}}\big(\overline{\widetilde U_{\rm tri}^\square({\overline{r}})\times_{\mathcal{W}^n_L}\mathcal{W}^n_{w,{\bf k},L}}\big),x}\cap T_{\iota_{\bf k}(\widetilde{\mathfrak{X}}_{\overline r}^{\square,{\bf k}\rm -cr}),x}\rightarrow \\
W_{1}\cap\cdots \cap W_{n-1}\cap V\cap {\rm Ext}^1_{\rm cris}(D_{{\mathrm{rig}}}(r),D_{{\mathrm{rig}}}(r))\rightarrow 0,\end{gathered}$$ the intersection in the middle being in $T_{\mathfrak{X}_{{\overline{r}}}^\square,r}$. But we have: $$\begin{gathered}
W_{1}\cap\cdots \cap W_{n-1}\cap V\cap {\rm Ext}^1_{\rm cris}(D_{{\mathrm{rig}}}(r),D_{{\mathrm{rig}}}(r))\buildrel\sim\over\rightarrow W_{{\rm cris},1}\cap\cdots \cap W_{{\rm cris},n-1}\cap V\\
\buildrel\sim\over\rightarrow W_{{\rm cris},1}\cap\cdots \cap W_{{\rm cris},n-1}\end{gathered}$$ where the first isomorphism is Remark \[intercris\] and the second follows from ${\rm Ext}^1_{\rm cris}(D_{{\mathrm{rig}}}(r),D_{{\mathrm{rig}}}(r))\subseteq V$ (since the Hodge-Tate weights don’t vary at all in ${\rm Ext}^1_{\rm cris}(D_{{\mathrm{rig}}}(r),D_{{\mathrm{rig}}}(r))$). From Corollary \[condcris\] we thus get: $$\begin{gathered}
\label{dimker}
\dim_{k(x)}\Big(T_{\jmath_{w_x,{\bf k}}\big(\overline{\widetilde U_{\rm tri}^\square({\overline{r}})\times_{\mathcal{W}^n_L}\mathcal{W}^n_{w,{\bf k},L}}\big),x}\cap T_{\iota_{\bf k}(\widetilde{\mathfrak{X}}_{\overline r}^{\square,{\bf k}\rm -cr}),x}\big)=\\
\dim_{k(x)}K(r)+\dim_{k(x)}{\rm Ext}^1_{\rm cris}\big(D_{{\mathrm{rig}}}(r),D_{{\mathrm{rig}}}(r)\big)-\lg(w_x).\end{gathered}$$ We now compute using Proposition \[tgtx’\], (\[se2\]) and (\[dimker\]): $$\begin{gathered}
\dim_{k(x)}\big(T_{\jmath_{w_x,{\bf k}}\big(\overline{\widetilde U_{\rm tri}^\square({\overline{r}})\times_{\mathcal{W}^n_L}\mathcal{W}^n_{w,{\bf k},L}}\big),x}+ T_{\iota_{\bf k}(\widetilde{\mathfrak{X}}_{\overline r}^{\square,{\bf k}\rm -cr}),x}\big)=\big(\dim X_{\rm tri}^\square({\overline{r}})-d_x\big)+\\
\big(\dim_{k(x)}K(r)+\dim_{k(x)}{\rm Ext}^1_{\rm cris}\big(D_{{\mathrm{rig}}}(r),D_{{\mathrm{rig}}}(r)\big)\big)-\\
\big(\dim_{k(x)}K(r)+\dim_{k(x)}{\rm Ext}^1_{\rm cris}\big(D_{{\mathrm{rig}}}(r),D_{{\mathrm{rig}}}(r)\big)-\lg(w_x)\big)=\\
\dim X_{\rm tri}^\square({\overline{r}})-d_x+\lg(w_x).\end{gathered}$$
\[bonnedim\] Conjecture \[BHS\] implies Conjecture \[mainconj\] for ${\overline{r}}={\overline{\rho}}_{\tilde v}$ ($v\in S_p$), i.e.: $$\dim_{k(x)}T_{\widetilde X_{\rm tri}^\square({\overline{r}}),x}=\lg(w_x)-d_{x}+\dim X_{\rm tri}^\square({\overline{r}}).$$ In particular $x$ is smooth on $\widetilde X_{\rm tri}^\square({\overline{r}})$ if and only if $w_x$ is a product of distinct simple reflections.
It follows from (\[injtang\]) and Proposition \[dimsum\] that we have $\lg(w_x)-d_{x}+\dim X_{\rm tri}^\square({\overline{r}})\leq \dim_{k(x)}T_{\widetilde X_{\rm tri}^\square({\overline{r}}),x}$. The equality follows from Corollary \[easybound\] which gives the converse inequality. Note that we also deduce $T_{\jmath_{w_x,{\bf k}}\big(\overline{\widetilde U_{\rm tri}^\square({\overline{r}})\times_{\mathcal{W}^n_L}\mathcal{W}^n_{w,{\bf k},L}}\big),x}+T_{\iota_{\bf k}(\widetilde {\mathfrak{X}}_{\overline r}^{\square,{\bf k}\rm -cr}),x}\buildrel\sim\over\longrightarrow T_{\widetilde X_{\rm tri}^\square({\overline{r}}),x}$. Finally, as we have already seen, the last statement follows from Lemma \[coxeter\].
We end up with an application of Corollary \[bonnedim\] (thus assuming Conjecture \[BHS\]) to the classical eigenvariety $Y(U^p,{\overline{\rho}})$ of §\[classic\]. We keep the notation and assumptions of §\[classic\] and §\[firstclassical\] and we consider a point $x\in Y(U^p,{\overline{\rho}})$ which is crystalline strictly dominant very regular. In a recent on-going work ([@Bergdraft]), Bergdall, inspired by the upper bound in Theorem \[upperbound\], proved an analogous upper bound for $\dim_{k(x)}T_{Y(U^p,{\overline{\rho}}),x}$, and obtained in particular that $Y(U^p,{\overline{\rho}})$ is smooth at $x$ when the Weyl group element $w_x$ in (\[wx\]) is a product of distinct simple reflections and when some Selmer group (which is always conjectured to be zero) vanishes. As a consequence of Corollary \[bonnedim\] we prove that this should [*not*]{} remain so when $w_x$ is [*not*]{} a product of distinct simple reflections.
\[singhecke\] Assume Conjecture \[BHS\] and assume that $w_x$ is [not]{} a product of distinct simple reflections. Then the eigenvariety $Y(U^p,{\overline{\rho}})$ is singular at $x$.
For $v\in S_p$ denote by $x_v$ the image of $x$ in $X_{\rm tri}^\square({\overline{\rho}}_{\tilde v})$ via (\[eigenvartotrianguline\]). Since $Y(U^p,{\overline{\rho}})\hookrightarrow X_p({\overline{\rho}})$, we have $x_v\in \widetilde X_{\rm tri}^\square({\overline{\rho}}_{\tilde v})$. It follows from Corollary \[bonnedim\] that it is enough to prove the following: if $Y(U^p,{\overline{\rho}})$ is smooth at $x$ then $\widetilde X_{\rm tri}^\square({\overline{\rho}}_{\tilde v})$ is smooth at $x_v$ for all $v\in S_p$, or equivalently $\widetilde X_{\rm tri}^\square({\overline{\rho}}_p)\cong \prod_{v\in S_p}\widetilde X_{\rm tri}^\square({\overline{\rho}}_{\tilde v})$ is smooth at $(x_v)_{v\in S_p}$. Recall from §\[firstclassical\] that we have: $$\label{basechange}
Y(U^p,{\overline{\rho}})\cong X_p({\overline{\rho}})\times_{({\mathrm{Spf}}\, S_\infty)^{{\mathrm{rig}}}}{\mathrm{Sp}}\, L$$ where the map $S_\infty\twoheadrightarrow L$ is $S_\infty\twoheadrightarrow (S_\infty/{\mathfrak a})[1/p]$ and where $X_p({\overline{\rho}})\rightarrow {\mathfrak X}_\infty\rightarrow ({\mathrm{Spf}}\, S_\infty)^{{\mathrm{rig}}}$ is induced by the morphism $S_\infty\rightarrow R_\infty$. Let $\omega_{\infty}(x)$ be the image of $x$ in $({\mathrm{Spf}}\, S_\infty)^{{\mathrm{rig}}}$, by an argument similar to the one in the proof of Proposition \[tgtx’\] we deduce from (\[basechange\]): $$T_{Y(U^p,{\overline{\rho}}),x}\cong \left\{\vec{v}\in T_{X_p({\overline{\rho}}),x}\ {\rm mapping\ to}\ 0\ {\rm in}\ T_{({\mathrm{Spf}}\, S_\infty)^{{\mathrm{rig}}},\omega_\infty(x)}\otimes_{k(\omega_\infty(x))}k(x)\right\}.$$ This obviously implies: $$\label{tangentxp}
\dim_{k(x)}T_{Y(U^p,{\overline{\rho}}),x}\geq \dim_{k(x)}T_{X_p({\overline{\rho}}),x} - \dim_{k(\omega_\infty(x))}T_{({\mathrm{Spf}}\, S_\infty)^{{\mathrm{rig}}},\omega_\infty(x)}.$$ But $\dim_{k(\omega_\infty(x))}T_{({\mathrm{Spf}}\, S_\infty)^{{\mathrm{rig}}},\omega_\infty(x)}=g+[F^+:{\mathbb{Q}}]\tfrac{n(n-1)}{2}+|S|n^2$ (see beginning of §\[firstclassical\]) and $\dim_{k(x)}T_{Y(U^p,{\overline{\rho}}),x}=\dim Y(U^p,{\overline{\rho}})=n[F^+:{\mathbb{Q}}]$ since $x$ is assumed to be smooth on $Y(U^p,{\overline{\rho}})$, hence we deduce from (\[tangentxp\]): $$\dim_{k(x)}T_{X_p({\overline{\rho}}),x}\leq g+[F^+:{\mathbb{Q}}]\tfrac{n(n+1)}{2}+|S|n^2=\dim X_p({\overline{\rho}})$$ where the last equality follows from [@BHS Cor.3.11]. We thus have $\dim_{k(x)}T_{X_p({\overline{\rho}}),x}=\dim X_p({\overline{\rho}})$ which implies that $x$ is smooth on $X_p({\overline{\rho}})$, and thus by (i) of Remark \[conjvariant\] that $(x_v)_{v\in S_p}$ is smooth on $\widetilde X_{\rm tri}^\square({\overline{\rho}}_p)$.
[Singular crystalline strictly dominant points on eigenvarieties are already known to exist by [@Bel §6]. However, the singular points of [*loc. cit.*]{} are different from the points $x$ of Corollary \[singhecke\] since they have reducible associated global Galois representations.]{}
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abstract: 'We construct explicit minimal models for the (hyper)operads governing modular, cyclic and ordinary operads, and wheeled properads, respectively. Algebras for these models are homotopy versions of the corresponding structures.'
address:
- 'Macquarie University, NSW 2109, Australia'
- 'Mathematical Institute of the Academy, [Ž]{}itn[á]{} 25, 115 67 Prague 1, The Czech Republic'
- 'Mathematical Institute of the Academy, [Ž]{}itn[á]{} 25, 115 67 Prague 1, The Czech Republic'
author:
- Michael Batanin
- Martin Markl
- Jovana Obradović
title: 'Minimal models for graphs-related operadic algebras'
---
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Introduction {#introduction .unnumbered}
============
The fundamental feature of Batanin-Markl’s theory of operadic categories [@duodel] is that the objects under study are viewed as algebras over (generalized) operads in a specific operadic category, cf. also the introduction to [@Sydney]. Thus, for instance, ordinary operads arise as algebras over the terminal operad ${\sf 1}_\RTr$ in the operadic category $\RTr$ of rooted trees, modular operads are algebras over the terminal operad $\termggGrc$ in the operadic category $\ggGrc$ of genus-graded connected graphs, &c.
.5em [**Our aim**]{} is to construct explicit minimal models for the (hyper)operads governing modular, cyclic and ordinary operads, and wheeled properads. We believe that the methods developed here can be easily modified to obtain minimal models for operads governing other common operad- or PROP-like structures. According to general philosophy [@haha], algebras for these models describe strongly homotopy versions of the corresponding objects whose salient feature is the transfer property over weak homotopy equivalences. This might be compared to the following classical situation.
Associative algebras are algebras over the non-$\Sigma$ operad $\Ass$. Algebras over the minimal model of $\Ass$ are Stasheff’s strongly homotopy associative algebras, also called $A_\infty$-algebras, This situation fits well into the framework of the current article, since $\Ass$ is the terminal non-$\Sigma$ operad or, which is the same, the terminal operad in the operadic category of finite ordered sets and their order-preserving epimorphisms.
The case of strongly homotopy cyclic operads was treated by the third author in [@JO], while modular operads were addressed by B. Ward in [@ward]. Both articles use the language of colored operads while the operadic category lingo used here is, as we believe, more concise and efficient, after the necessary preparatory material developed in [@Sydney] has been available.
In a follow-up to this article we prove that the minimal models described in the present paper are the bar constructions over Koszul duals of the (hyper)operads that they resolve, in the sense of , which by definition means that those (hyper)operads are Koszul. This was already established in [@perm] for the operad ${\sf 1}_{\tt Per}$ governing permutads.
.5em [**The models.**]{} Here we point to the places where the advertised constructions can be found.
$\bullet$ The minimal model $\minggGrc$ of the operad $\termggGrc$ governing modular operads is constructed in Subsection \[Podari se mi koupit to auto?\]. Algebras for this minimal model are strongly homotopy modular operads.
$\bullet$ The minimal model $\minTr$ of the operad $\termTr$ governing cyclic operads is constructed in Subsection \[Dnes prvni vylet na kole\]. Algebras for this minimal model are strongly homotopy cyclic operads.
$\bullet$ The minimal model $\minWhe$ of the operad $\termWhe$ governing wheeled properads is constructed in Subsection \[Poslu Jarce obrazky kyticek.\]. Algebras for this minimal model are strongly homotopy wheeled properads.
$\bullet$ There are two operadic categories such that the algebras for their terminal operads are ordinary operads – the category $\RTr$ of rooted trees and its full subcategory $\SRTr$ of strongly rooted trees. The minimal models $\minRTr$ resp.$\minSRTr$ of the corresponding terminal operads $\termRTr$ resp. $\termSRTr$ are constructed in Subsections \[Prvni tyden v Berkeley konci.\] resp. \[Ze by byla monografie uz konecne dokoncena?\]. Both $\minRTr$ and $\minSRTr$ have the same algebras, namely strongly homotopy ordinary operads. The reason why we consider two categories governing the same structures is explained below.
.5em [**Methods used.**]{} We begin with the particular case of the operadic category $\Grc$ of connected graphs. Algebras for the terminal operad $\termGrc$ in that category are modular operads without the genus grading. We explicitly define, in Section \[hadrova\_panenka\], a minimal $\Grc$-operad $\minGrc = (\Free(D),\pa)$ and a map $\minGrc\stackrel\rho\longrightarrow \termGrc$ of differential graded $\Grc$-operads. Theorem \[Woy-Woy\] states that $\rho$ is a level-wise homological isomorphism, meaning that $\minGrc$ is a minimal model of $\termGrc$. Proof of Theorem \[Woy-Woy\] is a combination of the following facts.
On one hand, using the apparatus developed in [@Sydney], we describe, in Subsection \[Dnes\_je\_Michalova\_oslava.\], the piece $\Free(D)(\Gamma)$, $\Gamma \in \Grc$, of the free operad $\Free(D)$ as a colimit over the poset ${\tt gTr}(\Gamma)$ of graph-trees associated to $\Gamma$, which are abstract trees whose vertices are decorated by graphs from $\Grc$ and which fulfill suitable compatibility conditions involving $\Gamma$.
On the other hand, to each $\Gamma \in \Grc$ we associate, in Subsection \[Srni\], a hypergraph $\bfH_\Gamma$ and to that hypergraph a poset ${\mathcal
A}(\bfH_\Gamma)$ of its constructs, which are certain abstract trees with vertices decorated by subsets of the set of internal edges of $\Gamma$. We prove, in Proposition \[jovanica\], that the poset ${\tt gTr}(\Gamma)$ is order-isomorphic to the poset ${\mathcal A}(\bfH_\Gamma)$.
Lemma \[abspol\], based on the results of [@CIO], asserts that ${\mathcal
A}(\bfH_\Gamma)$ is in turn order-isomorphic to the face lattice of a convex polytope ${\mathcal G}(\bfH_\Gamma)$. Finally, using an ‘ingenious’ Lemma \[Jeste\_ani\_nevim\_kde\_budu\_v\_Melbourne\_bydlet.\], we show that the faces of ${\mathcal G}(\bfH_\Gamma)$ can be oriented so that the cellular chain complex of ${\mathcal
G}(\bfH_\Gamma)$ is isomorphic, as a differential graded vector space, to $(\Free(D)(\Gamma),\partial)$. Since ${\mathcal G}(\bfH_\Gamma)$ is acyclic in positive dimension, the same must be true for $(\Free(D)(\Gamma),\partial)$. It remains to show that $\rho$ induces an isomorphism of degree $0$ homology, but this is simple. The conclusion is that $\minGrc$ is indeed a minimal model of $\termGrc$.
In constructing the minimal models of the terminal operads $\termggGrc$, $\termTr$ and $\termWhe$ in the operadic categories $\ggGrc$ of genus-graded connected graphs, $\Tr$ of trees and $\Whe$ of ordered (‘wheeled’) connected graphs, respectively, we use the fact observed in [@Sydney Section 5] that these categories are discrete operadic opfibrations over $\Grc$. Their minimal models are then, thanks to Corollary \[grantova\_zprava\], the restrictions of the minimal model for $\termGrc$ along the corresponding opfibration map.
The situation of the terminal operad $\termRTr$ in the operadic category $\RTr$ of rooted trees is different, since this category is not an opfibration over $\Grc$. It is, however, a discrete operadic fibration with finite fibers, so Corollary \[grantova\_zprava\] of Section \[Mourek a Terezka\] applies as well.
We finally introduce a full subcategory $\SRTr \subset \RTr$ consisting of strongly rooted trees. The algebras for the terminal $\SRTr$-operad $\termSRTr$ are the same as $\termRTr$-algebras, i.e.ordinary operads. We consider this subcategory since it is the most economic description of ordinary operads. Although it is neither a fibration, nor an opfibration over $\Grc$, we show in Subsection \[Prvni tyden v Berkeley konci.\] that the minimal model for $\termSRTr$ can be obtained by a straightforward modification of the construction of the minimal model for $\termGrc$ given in Section \[hadrova\_panenka\].
.5em [**Plan of the paper.**]{} In Section \[mexicke pivo\] we recall necessary facts about hypergraph polytopes, and free operads in operadic categories. Section \[hadrova\_panenka\] is devoted to the construction of the minimal model for the terminal $\Grc$-operad, and presentation of the necessary preparatory material. Section \[Mourek a Terezka\] addresses minimal models for terminal operads in the operadic categories of genus-graded graphs, trees, wheeled graphs and strongly rooted trees.
.5em [**Conventions.**]{} Unless stated otherwise, all algebraic objects will be considered over a field $\bbk$ of characteristic zero. By $|X|$ we denote either the cardinality if $X$ is a finite set, or the geometric realization if $X$ is a graph. If not specified otherwise, (hyper)operads featured here will live in the monoidal category of differential graded $\bfk$-vector spaces. The [ *terminal operad*]{} in a given operadic category is the one whose all components equal $\bbk$ and whose structure operations are the identities. These operads are linearizations of the corresponding terminal set-operads, which hopefully justifies our relaxed terminology.
Recollections {#mexicke pivo}
=============
This section contains a preparatory material regarding hypergraph polytopes and operadic categories. The basic references are [@CIO; @JO] for the former and [@Sydney; @duodel] for the latter.
Hypergraph polytopes
--------------------
They are abstract polytopes whose geometric realization can be obtained by truncating the vertices, edges and other faces of simplices, in any finite dimension. In particular, the family of $n$-dimensional hypergraph polytopes consists of an interval of simple polytopes starting with the $n$-simplex and ending with the $n$-dimensional permutohedron.
### Hypergraph terminology {#hypergraph-terminology .unnumbered}
A hypergraph is a pair ${\bf H}=(H,{\bf H})$ of a finite set $H$ of [*vertices*]{} and a subset ${\bf H}\subseteq {\mathcal P}(H)\backslash\emptyset$ of [ *hyperedges*]{}, such that $\bigcup {\bf H}=H$ and such that, for all $x\in H$, $\{x\}\in {\bf H}$ (note that this property justifies the convention to use the bold letter ${\bf H}$ for both the hypergraph itself and its set of hyperedges). A hypergraph ${\bf H}$ is [ *connected*]{} if there are no non-trivial partitions $H=H_1\cup H_2$, such that $${\bf H}=\{X\in {\bf H}\,|\, X\subseteq H_1\}\cup\{Y\in {\bf H}\,|\,
Y\subseteq H_2\}.$$ A hypergraph ${\bf H}$ is [*saturated*]{} when, for every $X,Y\in{\bf H}$ such that $X\cap Y\neq\emptyset$, we have that $X\cup Y\in {\bf H}$. Every hypergraph can be saturated by adding the missing (unions of) hyperedges. Let us introduce the notation $${\bf{H}}_X:=\{Z\in {\bf{H}}\,|\, Z\subseteq X\},$$ for a hypergraph ${\bf H}$ and $X\subseteq H$. The [*saturation*]{} of ${\bf{H}}$ is then formally defined as the hypergraph $${\it Sat}({\bf{H}}):=\{ X\,|\, {\emptyset\subsetneq X\subseteq H\;\mbox{and}\;{\bf{H}}_X\;\mbox{is connected}}\}.$$
For a hypergraph ${\bf H}$ and $X\subseteq H$, we also set $${\bf H}\backslash X:={\bf{H}}_{H\backslash X}.$$ Observe that for each finite hypergraph there exists a partition $H=H_1\cup\ldots\cup H_m$, such that each hypergraph ${\bf{H}}_{H_i}$ is connected and ${\bf{H}}=\bigcup({\bf{H}}_{H_i})$. The ${\bf{H}}_{H_i}$’s are called the [*connected components*]{} of ${\bf{H}}$. We shall write ${\bf{H}}_i$ for ${\bf{H}}_{H_i}$. We shall use the notation
${\bf{H}}\backslash X \leadsto {\bf H}_1,\ldots,{\bf H}_n$
to indicate that ${\bf H}_1,\ldots,{\bf H}_n$ are the connected components of ${\bf H}\backslash X$.
### Abstract polytope of a hypergraph {#abstract-polytope-of-a-hypergraph .unnumbered}
We next recall from [@CIO] the definition of the abstract polytope $${\mathcal A}({\bf H})=(A({\bf H})\cup\{\emptyset\},\leq_{\bf H})$$ associated to a connected hypergraph ${\bf H}$.
The elements of the set $A({\bf H})$, to which we refer as the [*constructs*]{} of ${\bf H}$, are the non-planar, vertex-decorated rooted trees defined recursively as follows.
- If ${\bf H}$ is the empty hypergraph, then $A({\bf H})=\{\emptyset\}$, i.e. ${\mathcal A}({\bf H})$ is the singleton poset containing $\emptyset$.
Otherwise, let $\emptyset\neq X\subseteq H$ be a subset of the set of vertices of ${\bf H}$.
- If $X=H$, then the abstract rooted tree with a single vertex labeled by $X$ and without any inputs, is a construct of ${\bf H}$; we denote it by $H$.
- If $X\subsetneq H$, if ${\bf H}\backslash X \leadsto {\bf H}_1,\ldots,{\bf H}_n$, and if $C_1,\ldots,C_n$ are constructs of ${\bf H}_1,\ldots,{\bf H}_n$, respectively, then the tree whose root vertex is decorated by $X$ and that has $n$ inputs, on which the respective $C_i\,$’s are grafted, is a construct of ${\bf H}$; we denote it by $X\{C_1,\ldots,C_n\}$.
In what follows, we shall refer to the vertices of constructs by the sets decorating them, since they are a fortiori all distinct. The notation $C:\bf{H}$ will mean that $C$ is a construct of $\bf{H}$.
The partial order $\leq_{\bf H}$ on non-empty constructs is generated by the edge-contraction: $$Y\{X\{C_{11},\ldots,C_{1m}\},C_2,\ldots,C_n\}\leq_{\bf H}(Y\cup
X)\{C_{11},\ldots,C_{1m},C_2,\ldots,C_n\}$$ and the relation $$\hbox{if $C'_1 \leq_{\bf H_1} C''_1$ \ then \ $X\{\rada {C_1'}{C_n}\}
\leq_{\bf H}X\{\rada {C_1''}{C_n}\}$}.$$ In addition, for each construct $C$ of ${\bf H}$, we have that $\emptyset\leq_{\bf H} C$.
The faces of ${\mathcal A}({\bf H})$ are ranked by integers ranging from $-1$ to $|H|-1$. The face $\emptyset$ is the unique face of rank $-1$, whereas the rank of a construct $C$ is $|H|-|\mbox{vert}(C)|$. In particular, constructs whose vertices are all decorated with singletons are faces of rank $0$, whereas the construct $H$ is the unique face of rank $|H|-1$.
\[abspol\] The poset ${\mathcal A}({\bf H})$ is order-isomorphic to the face lattice of a convex polytope ${\mathcal G}({\bf H})$ obtained as a truncation of the -dimensional simplex. In particular, ${\mathcal A}({\bf H})$ is an abstract polytope of rank .
The polytope ${\mathcal G}({\bf H}) \subset {\mathbb R}^{n}$, where $n= |H|$, is constructed as follows. Assume that $H=\{x_1,\dots,x_n\}$, and define, for each $I\subseteq \{1,\dots,n\}$, the subsets $\pi^+_I$ and $\pi_I$ of ${\mathbb R}^{n}$ as
$\pi^+_I:=\{(x_1,\dots,x_n)\,|\, \sum_{i\in I}x_i\geq 3^{|I|}\}$ and $\pi_I:=\{(x_1,\dots,x_n)\,|\, \sum_{i\in I}x_i =3^{|I|}\}$,
respectively. Then $${\mathcal G}({\bf H}):=\bigcap\{\pi^+_Y\,|\, Y\in {\it Sat}({\bf
H})\backslash\{H\}\}\cap \pi_H.$$ The order-isomorphism between the poset of constructs of ${\bf H}$ and the poset of geometric faces of ${\mathcal G}(H)$ is defined in [@CIO Section 3.3].
The fact that ${\mathcal A}({\bf H})$ is an abstract polytope also follows from the results of [@DP-HP] which preceded [@CIO].
Free operads in the operadic category of graphs {#Dnes_je_Michalova_oslava.}
-----------------------------------------------
The basic operadic category in this section will be the category $\Grc$ of connected graphs introduced in [@Sydney Section 3] and Example 5.7 loc. cit. for which we also refer for terminology and notation. Results for other categories of graphs will be straightforward modifications of this situation. Recall that the objects $\GAmma$ of $\Grc$ are connected [*directed*]{} graphs. The adjective [*directed*]{} means that the (finite) set of vertices of $\GAmma$ is (linearly) ordered, as well as are the (finite) sets of half-edges adjacent to each vertex of $\GAmma$, and that also the (finite) set of legs of $\GAmma$ is ordered. To simplify the terminology, by a [*graph*]{} we always mean in this section an object of $\Grc$. As the first step in describing the component $\Free(E)(\GAmma)$, $\GAmma \in \Grc$, of the free operad $\Free(E)$ generated by a $1$-connected collection $E$ we identify, in Theorem \[Uz\_nechci\_letat\_zakladni\_vycvik.\] below, the set $\pi_0(\lTw(\GAmma))$ of connected components of the groupoid $\lTw(\GAmma)$ of labelled towers [@Sydney Section 10] with a certain class of trees defined below. Recall that we work with a skeletal version of the category of finite ordered sets, therefore arbitrary two order-isomorphic finite sets are the same.
Before we continue, we introduce a particular class of maps between graphs, called [*canonical contractions*]{} (or [*cc’s*]{} for short) of a subgraph. The informal definition is the following.
Let $\Gamma \subset \Gamma'$ be a subgraph and $\Gamma''$ be obtained from $\Gamma'$ by contracting all internal edges of $\Gamma$ into a vertex. The canonical contraction $\pi: \Gamma' \to \Gamma''$ is then the ‘obvious projection.’ We however need to specify labellings and orders of the vertices and flags of $\Gamma$ and $\Gamma''$, so a more formal definition is needed.
Assume that $\Gamma'= (V',F')\in \Grc$ is a graph with the set of vertices $V'$, the set of flags $F'$ and the structure map $g': F' \to V'$, see [@Sydney Definition 3.1]. Choose a nonempty subset $V \subset V'$ and a nonempty set $E$ of edges of $\Gamma'$ formed by the half-edges in $g'^{-1}(V)
\subset F'$ such that the subgraph of $\Gamma'$ spanned by $E$ is connected. Let us denote by $V'/V$ the ordered set $$V'/V := (V' \setminus V) \cup \{\min (V)\};$$ the notation being justified by the canonical set-isomorphism of $V/V'$ as above with the set-theoretic quotient $V'$ by the subset $V$. Let finally $V'' := V'/V$ and $$\label{Dnes_s_Jarkou_a_Oliverem.}
\phi : V' \to V'' = V'/V$$ be the ‘projection’ that is the identity on $V' \setminus V$ while it sends all elements of $V$ to $\min (V)$.
We construct $\Gamma''$ as the graph whose set of vertices is $V''$ and whose set of flags is $F'':=
F' \setminus E$. The defining map $g'':F''\to V''$ is the restriction of the composite $\phi \circ g'$, as in $$\label{Treti den v Berkeley}
\xymatrix@C=3.5em{F' \ar[d]_{g'} & \ F'':=
F' \setminus E \ar@{_{(}->}[l]_(.7){\psi}%)
\ar[d]^{g''}
\\
V' \ar@{->>}[r]^{\phi} &V''.
}$$ The involution $\sigma'' : F'' \to F''$ is the restriction of the involution $\sigma' : F' \to F'$ of $\Gamma'$. The map $g''$ defined by (\[Treti den v Berkeley\]) is however not order-preserving as required by the definition of a graph. We therefore reorder $F''$ by imposing the lexicographic order requiring that, for $a,b \in
F''$, $$a< b \hbox { if and only if }
\begin{cases}
\hbox{$g''(a) < g''(b)$ in $V''$, or}
\\
\hbox{$g''(a) = g''(b)$ and $a < b$ in $F''$}.
\end{cases}$$ This formula obviously does not change the local orders of flags in $F''$ around a given vertex.
We finally define the cc $\pi:\Gamma'\to \Gamma''$ as the couple $(\psi,\phi)$ with $\psi:F''\hookrightarrow F'$ the inclusion. The unique nontrivial fiber of $\pi$ is the graph $\Gamma$ given by the restriction $F\stackrel{g}{\longrightarrow} V$ of $g'$ to $F := g'^{-1}(V)$ whose involution is trivial everywhere except for the flags forming the edges in $E$, in which case it coincides with the involution of $\Gamma'$. A simple example of a canonical contraction can be found in Figure \[Vcera jsem si koupil kolo.\] below.
We may sometimes loosely denote $\Gamma'' := \Gamma'/\Gamma$. Canonical contractions in the above sense are modifications of pure contractions of [@Sydney Definition 3.4] in that that here we do not require the map of vertices to be order-preserving, which is compensated by introducing the lexicographic order on the flags of $\Gamma''$. Canonical contractions are elementary morphisms in the sense of [@Sydney Section 2].
Let us return to the main topics of this section. A [*graph-labelled tree*]{}, or [*graph-tree*]{} for short, is a rooted tree $T$ whose input leaves as well as internal edges are labelled by a finite ordered set $V$ subject to the condition that an internal edge $e$ of $T$ is labelled by the minimum of the labels of the input leaves of the subtree of $T$ ‘below’ $e$, i.e. of the maximal subtree of $T$ whose root vertex is $e$. Moreover, vertices of a graph tree $T$ are labelled by graphs in $\Grc$. This labelling shall satisfy two conditions.
[*Compatibility 1.*]{} The ordered set of vertices of $\Gamma_u$ labelling a vertex $u$ of $T$ equals the ordered set of the labels of the input edges of $u$.
[*Compatibility 2.*]{} Let $e$ be an internal edge of $T$ pointing from (the vertex labelled by) $\Gamma_u$ to (the vertex labelled by) $\Gamma_v$. Then the ordered set of the half-edges of $\Gamma_v$ adjacent to its vertex corresponding to $e$ is the same as the ordered set of the legs of $\Gamma_u$.
Since we are going to study free operads generated by $1$-connected collections only, we assume that the graphs labelling the vertices of a graph-tree have at least one internal edge.
A portrait of a graph-tree is given in Figure \[V nedeli jsem tahal na Kelimkovi.\]. The set $V$ equals in in this case to $\{a,b,c,d,e,f\} $ with some (linear) order. The graph $\Gamma_4$ has three vertices labelled by the elements of the subset $$\big\{\min\{a,b,c\}, \min\{d\}=d,\min\{e,f\}\big\} \subset \{a,b,c,d,e,f\}$$ with the induced linear order. The graph $\Gamma_5$ has only one vertex labelled by $d$.
Let $e$ be an internal edge of a graph-tree $T$ pointing from $\Gamma_u$ to $\Gamma_v$. Then the tree $T/e$ obtained by contracting the edge $e$ has an induced structure of a graph-tree given as follows. The leaves and internal edges of $T/e$ bear the same labels as they did in $T$. Also the vertices of $T/e$ except of the one, say $x$, created by the collapse of $e$, are labelled by the same graphs as in $T$. Finally, the vertex $x$ is labelled by the graph $\Gamma_x$ given by the vertex insertion of $\Gamma_u$ into the vertex of $\Gamma_v$ labelled by $e$. Since, by Compatibility 2, the ordered set of legs of $\Gamma_u$ is the same as the ordered set of the half-edges adjacent to the vertex of $\Gamma_v$ labelled by $e$, the vertex insertion is uniquely and well-defined. One clearly has $$\Vert(\Gamma_x) = (\Vert(\Gamma_v) \setminus \{\hbox{the vertex
labelled by } e\}) \cup \Vert(\Gamma_u),$$ where the union in the right hand side is disjoint thanks to Compatibility 1. The set $\Vert(\Gamma_x)$ bears an order induced from the inclusion $\Vert(\Gamma_x) \subset V$.
Repeating the collapsings described above we finally obtain a graph-tree with one vertex (i.e. a rooted corolla) whose only vertex is labelled by some graph $\GAmma \in \Grc$ with the ordered set of vertices $V$. We denote the graph $\GAmma$ thus obtained, which clearly does not depend on the order in which we contracted the edges of $T$, by $\gr(T)$.
\[Uz\_nechci\_letat\_zakladni\_vycvik.\] The set of connected components of the groupoid $\lTw(\GAmma)$ is canonically isomorphic to the set $\gTr(\GAmma)$ of graph-trees $T$ with $\gr(T) = \GAmma$.
(0,-4.0582557)(13.51,4.0582557) (4.4,1.9762975)(4.4,4.1512976) (7.3,-4.0487027)(8.0,-2.7612026) (8.7,-4.0487027)(8.0,-2.7612026) (5.9,-4.0487027)(4.4,1.9762975) (8.1,-2.8487024)(4.4,1.9762975) (1.7,-4.0487027)(2.7,0.7512975) (2.7,0.7512975)(4.4,1.9512974) (3.1,-4.0487027)(3.7,-1.6487025) (4.5,-4.0487027)(3.7,-1.6487025) (3.7,-1.6487025)(2.7,0.7512975) (3.7,-1.8487025)[$\Gamma_1$]{} (9.1,-2.8487024)(1.3,-2.8487024) (1.3,-1.6487025)(9.1,-1.6487025) (1.3,-0.44870254)(9.1,-0.44870254) (1.3,0.7512975)(9.1,0.7512975) (1.3,1.9512974)(9.1,1.9512974) (8.1,-2.8487024) (3.6,-1.7487025) (4.5,1.9512974) (2.7,0.7512975) (5.0,-0.34870255) (4.3,1.95)[$\Gamma_4$]{} (2.5,0.7512975)[$\Gamma_2$]{} (4.8,-0.355)[$\Gamma_5$]{} (3.4,-1.7487025)[$\Gamma_1$]{} (7.9,-2.8487024)[$\Gamma_3$]{} (8.5,1.3512975)(10.3,1.3512975)(10.3,2.9512975) (9.1,0.15129745)(11.1,0.15129745)(11.1,1.7512975) (9.1,-1.0487026)(11.9,-1.0487026)(11.9,0.35129747) (9.1,-2.2487025)(12.7,-2.2487025)(12.7,-0.84870255) (9.1,-3.4487026)(13.5,-3.4487026)(13.5,-1.8487025) (0.5,-2.8487024)[[level 1]{}]{} (0.5,-1.5487026)[[level 2]{}]{} (0.5,-0.44870254)[[level 3]{}]{} (0.5,0.7512975)[[level 4]{}]{} (0.5,1.9512974)[[level 5]{}]{} (13.1,-3.2487025)[$\Delta_0$]{} (12.3,-2.0487025)[$\Delta_1$]{} (11.5,-0.84870255)[$\Delta_2$]{} (10.7,0.35129747)[$\Delta_3$]{} (9.9,1.5512974)[$\Delta_4$]{} (9.5,0.35129747)(9.5,1.1512975) (9.5,-0.84870255)(9.5,-0.048702545) (9.5,-2.0487025)(9.5,-1.2487025) (9.5,-3.2487025)(9.5,-2.4487026) (9.9,-2.8487024)[$\tau_1$]{} (9.9,-1.6487025)[$\tau_2$]{} (9.9,-0.44870254)[$\tau_3$]{} (9.9,0.7512975)[$\tau_4$]{}
Recall from [@Sydney Section 10] that the objects of $\lTw(\GAmma)$ are labelled towers $$\label{t1}
\bfT = (\bfT,\ell): \
\GAmma \stackrel \ell\longrightarrow
\Delta_0 \stackrel {\tau_1} \longrightarrow \Delta_1 \stackrel {\tau_2} \longrightarrow
\Delta_2\stackrel {\tau_3} \longrightarrow
\cdots \stackrel {\tau_{k-1}} \longrightarrow \Delta_{k-1},$$ where $\rada \Delta{\Delta_{k-1}}$ are graphs in $\Grc$, $\ell$ an isomorphism, and $\Rada \tau1{k-1}$ elementary maps, i.e. maps with precisely one nontrivial fiber. We will construct a map $$A : \gTr(\GAmma) \longrightarrow \pi_0\big( \lTw(\GAmma)\big)$$ of sets as follows. Assume that $T \in \gTr(\GAmma)$ is a graph-tree with $k$ vertices. We distribute the vertices of $T$ to levels such that each level contains precisely one vertex, see Figure \[Zase\_pojedu\_proti\_vetru!\] for an example. Let $T_{i-1}$, $1 \leq i \leq k$, be the graph-tree obtained from $T$ by truncating everything above level $i$, level $i$ included, see Figure \[Zase\_pojedu\_proti\_vetru!\] again. Denote $\Delta_{i-1} :=
\gr(T_{i-1})$, $1 \leq i \leq k$. Notice that $\Delta_0 = \GAmma$ by definition. One then has the labelled tower $$\label{Jaruska_ma_jizdy.}
a(T) := \GAmma \stackrel \id\longrightarrow
\Delta_0 \stackrel {\tau_1} \longrightarrow \Delta_1 \stackrel {\tau_2} \longrightarrow
\Delta_2\stackrel {\tau_3} \longrightarrow
\cdots \stackrel {\tau_{k-1}} \longrightarrow \Delta_{k-1}\in \lTw(\GAmma),$$ in which the map $\tau_i : \Delta_{i-1} \to \Delta_i$, $1 \leq i \leq
k-1$, is defined as follows. Let $u$ be the only vertex on the $i$th level and $e$ its out-going edge. Then $\tau_i$ is the map that contracts the subgraph $\Gamma_u$ of $\Delta_{i-1}$ into the vertex of $e$ labelled by $e$. In other words, $\tau_i$ is the canonical contraction $\Delta_{i-1} \to \Delta_i = \Delta_{i-1}/\Gamma_u$.
In the situation of Figure \[Zase\_pojedu\_proti\_vetru!\], the graph $\Delta_0$ has vertices $\{a,b,c,d,e,f\}$ and $$\Vert(\Delta_1) = \big\{a,b,c,d,\min\{e,f\}\big\}.$$ The map $\tau_1$ contacts the subgraph $\Gamma_3$ of $\Delta_0$ into the vertex $\min\{e,f\}$ of $\Delta_1$. Likewise, $$\Vert(\Delta_2) = \big\{a,\min\{b,c\},d,\min\{e,f\}\big\}$$ and $\tau_2$ contacts the subgraph $\Gamma_1$ into the vertex $\min\{b,c\}$ of $\Delta_2$.
The actual value of $a(T)$ might depend on the choice of levels of $T$, but any two such values are related by an isomorphism of the 2nd type in the sense of [@Sydney Section 10]. The connected component of $a(T)$ therefore does not depend on the choices of levels, so one may define $A(T) := \pi_0( a(T)) \in \pi_0\big( \lTw(\GAmma)\big)$. Let us proceed to the inverse $$B : \pi_0\big( \lTw(\GAmma)\big) \longrightarrow \gTr(\GAmma)$$ of $A$. Suppose that we have a labelled tower $\bfT$ as in (\[t1\]). Our strategy will be to modify it within its isomorphism class, using the isomorphisms of the first type recalled in Figure \[f1\] below, into the form where $\ell$ is the identity and the remaining maps are canonical contractions.
Recall that, by [@Sydney Proposition 10.8], each such a tower can be functorially replaced within its isomorphisms class with a tower whose labelling $\ell$ is a quasibijection. We may thus assume this particular form from the beginning. Isomorphisms of the first type for such labelled towers are diagrams as in Figure \[f1\].
$$\xymatrix{
&\GAmma \ar[ld]_{\ell'}^\sim \ar[rd]^{\ell''}_\sim &
\\
\Delta_0' \ar[d]_{\tau'_1} \ar[rr]^{\sigma_1}_\sim & & \Delta_0'' \ar[d]^{\tau''_1}
\\
\Delta_1' \ar[d]_{\tau'_2} \ar[rr]^{\sigma_2}_\cong & & \Delta_1'' \ar[d]^{\tau''_2}
\\
\vdots \ar[d]_{\tau'_{k-1}} && \vdots \ar[d]^{\tau''_{k-1}}
\\
\Delta_{k-1}' \ar[rr]^{\sigma_{k}}_\cong && \Delta_{k-1}''
}$$
An important fact that holds in the category $\Grc$ is that quasibijections are local isomorphisms, i.e. automorphisms relabeling the vertices without changing the local orders of the adjacent flags. Consequently a composition of a quasibijection with an elementary map is elementary again. One may therefore absorb $\ell$ into $\tau_1$ in (\[t1\]) by replacing it with $$\label{Projedu_se_jeste_na_kole?}
\GAmma \stackrel \id\longrightarrow
\Delta_0 \stackrel {\widehat\tau_1} \longrightarrow \Delta_1 \stackrel {\tau_2} \longrightarrow
\Delta_2\stackrel {\tau_3} \longrightarrow
\cdots \stackrel {\tau_{k-1}} \longrightarrow \Delta_{k-1},$$ where $\widehat\tau_1 := \tau_1 \circ \ell$, which is isomorphic to (\[t1\]) via a diagram as in Figure \[f1\] with $\sigma_1 = \ell^{-1}$ and all other $\sigma$’s the identities. So we may assume in (\[t1\]) that $\ell = \id_\GAmma$.
Now we proceed by modifying the elementary map $\tau_1 : \Delta_1\to
\Delta_2$ in (\[Projedu\_se\_jeste\_na\_kole?\]) into a map that acts on vertices as a canonical contraction. Assume that $\Delta_i = (V_i,F_i)$, $i=1,2$, and that $\tau_1$ is given by the pair $(\phi_1,\psi_1)$ of maps in the diagram $$\xymatrix@C=3.5em{F_0 \ar[d]_{g_0} & F_1\ar@{_{(}->}[l]_{\psi_1}
\ar[d]^{g_1}
\\
V_0 \ar@{->>}[r]^{\phi_1} &\ V_1.
}$$ Let the only nontrivial fiber of $\tau_1$ be the one over some $x_1\in
V_1$, and $V := \phi_1^{-1}(\{x_1\}) \subset V_0$. Define the map $\vartheta :V_1 \to V_0$ by the commutativity of the diagram $$\xymatrix@C=3.5em{V_0 \ar@{->>}[d]_{\phi_1} \ar[rd] &
\\
V_1 \ar[r]^(.4){\vartheta}
& V_0/V
}$$ in which the diagonal arrow is the ‘contraction’ (\[Dnes\_s\_Jarkou\_a\_Oliverem.\]). Finally, let $\widehat \Delta_1 := (V_0/V,F_1)$ be the graph with the structure map the composition $\widehat g : F_1 \stackrel{g_1}\longrightarrow V_1
\stackrel\vartheta\longrightarrow V_0/V$ and $F_1$ ordered lexicographically.
It is clear that the couple $\theta := (\vartheta,\id_{F_1})$ defines a map $\Delta_1 \to \widehat
\Delta_1$. Since $\vartheta$ is a local isomorphism, its post- or precomposition with an elementary map is elementary again. We may therefore replace $\Delta_1$ by $\widehat
\Delta_1$, $\tau_1$ by $\theta\circ \tau_1$ and $\tau_2$ by $\tau_2
\circ \vartheta^{-1}$. This modification is isomorphic, via the diagram in Figure \[f1\] with all $\sigma$’s the identities except $\sigma_1
:= \vartheta^{-1}$, with the original tower. We apply the same process to all remaining $\tau$’s. The result will be a tower in which all $\tau$’s act on vertices as canonical contractions.
It remains to modify $\tau$’s so that also their local actions will be that of cc’s, starting with $\tau_1$ again. Let $\Gamma_1$ be its only nontrivial fiber over some $x_1 \in V_0$. By the definition of the fiber, the set of legs of $\Gamma_1$ is isomorphic to the set of flags at $x_1$, and we change the local order of flags at $x_1$ according to that isomorphism. After this change whose result is isomorphic to the original $\Delta_1$ via a local reordering, $\tau_1$ becomes a cc and this modification clearly has not changed the isomorphism class of the tower. We similarly modify the remaining $\tau$’s.
We thus modified the tower $\bfT$ in (\[t1\]) within its isomorphism class so that $\ell = \id_\GAmma$ and all $\tau$’s are canonical contractions. We will say that the resulting tower has the [*canonical form*]{}.
Denote by $V_i$ the set of vertices of $\Delta_i$, $0 \leq i \leq k\!-\!1$. It follows from the definition of canonical contractions that $V_0 \supset V_1 \supset \cdots \supset V_{k-1}$. Moreover, each $V_i$ contains a distinguished element $x_i$ over which the unique nontrivial fiber of $\tau_i$ lives. We extend the notation by putting $V_k := \{*\}$, the one-point set, and $x_k := *$. The vertex parts of $\tau$’s give rise to the sequence $$\label{Je_cas_chystat_se_do_Australie.}
\xymatrix@1{
V_0 \ \ar@{->>}[r]^{\phi_1} &
\ V_1 \ \ar@{->>}[r]^{\phi_2}
&\ V_2\ \ar@{->>}[r]^{\phi_3} &\ \cdots \ \ar@{->>}[r]^{\phi_{k-1}}
&\ V_{k-1}\ \ar@{->>}[r]^{\phi_k}
&\ V_k = \{*\}
}$$ of epimorphisms with the property that $\min(\phi_1^{-1} \circ \cdots \circ
\phi_i^{-1}(x_i)) = x_i$, $1 \leq i \leq k\-1$. Such a sequence of epimorphisms of finite ordered sets determines in the standard manner a rooted tree with levels, with its leaves labelled by $V_0$, with the root $*$ and the remaining vertices $\Rada x1{k-1}$. Forgetting the levels, decorating the root by $\Delta_0$ and $x_i$ by the fiber $\Gamma_i$ of $\tau_i$, $1\leq i \leq
k\-1$, leads to a graph-tree $B(\bfT) \in \gTr(\GAmma)$.
The reason why $B(\bfT)$ is well-defined, that is, $B(\bfT') = B(\bfT'')$ if $\bfT'$ and $\bfT''$ are isomorphic labelled towers, is that for isomorphisms of the second type, see [@Sydney Section 10] for terminology, the difference disappears after forgetting the levels of the tree corresponding to (\[Je\_cas\_chystat\_se\_do\_Australie.\]), while it is not difficult to see that the canonical forms of labelled towers related by an isomorphisms of the first type are the same.
It is clear that $\hbox{$(B \!\circ\! A)$}(T) = T$ for $T\in
\gTr(\GAmma)$. Given a labelled tower $\bfT \in \lTw(\GAmma)$, the concrete form of the tower $a(
B(\bfT)) \in \lTw(\GAmma)$ representing $\hbox{$(A\! \circ\!
B)$}(\bfT) \in \pi_0(\lTw(\GAmma))$ depends on the choice of levels for the tree $B(\bfT)$. But any two such towers are related by a type two isomorphism. Since modifying a tower into its canonical form does not change its isomorphism class, we established that $B$ is also a left inverse of $A$.
The set $\gTr(\GAmma)$ and therefore also the set $\pi_0(\lTw(\GAmma))$ of connected components of the category $\lTw(\GAmma)$ has a natural poset structure induced by the relation $T \prec T/e$ for a graph-tree $T\in \gTr(\GAmma)$ and its edge $e$. Its categorical origin is the following.
Let us denote, only for the purpose of this explanation, by $\ttC$ the category whose objects are the same as the objects of $\lTw(\GAmma)$, i.e. the labelled towers $\bfT$ as in (\[t1\]). We postulate that there is a unique morphism $\bfT \to
\bfS$, $\bfT \not=
\bfS$, in $\ttC$ if and only if $\bfS$ is obtained from $\bfT$ by composing two or more adjacent morphisms $\tau_i$’s that have mutually joint fibers, in the sense of [@Sydney Definition 2.12]. The only other morphisms in $\ttC$ are the identities.
We denote by $\lTw(\GAmma) \int \ttC$ the category with the objects of $\lTw(\GAmma)$ whose morphisms are formal compositions of a morphism of $\lTw(\GAmma)$ with a morphism of $\ttC$. The poset considered in the standard manner as a category is then canonically isomorphic to the pushout in [Cat]{} of the diagram $$\xymatrix{\lTw(\GAmma) \ar[d]\ar@{^(->}[r] & \lTw(\GAmma) \int \ttC
\\
\pi_0(\lTw(\GAmma))
}$$ in which $\pi_0(\lTw(\GAmma))$ is taken as a discrete category.
.
We are finally going to give an explicit formula for the free $\Grc$-operad $\Free(E)$ generated by a $1$-connected collection $E$ evaluated at a graph $\GAmma$. Recall that $E$ is a representation, in the category of graded vector spaces, of the groupoid $\QV(e)$ whose objects are graphs in $\Grc$ and morphisms are virtual isomorphisms which are, in this specific case, isomorphisms of graphs which need not respect the orders of the legs. The $1$-connectivity means that $E(\GAmma) \not= 0$ implies that $\GAmma \in \Grc$ has at least one internal edge.
\[Dnes jdu s Jaruskou na koncert.\] Let us consider the classical free non-$\Sigma$ operad $\Free(E) =
\{\Free(E)(n)\}_{n \geq 1}$ generated by a collection $E$ of graded vector spaces. A common mistake is to assume that the elements of $\Free(E)$ are (represented by) trees with vertices decorated by elements of $E$. This is true only when $E$ is concentrated in even degrees. Otherwise we need one more piece of information, namely a choice of levels of the underlying tree.
(0,-1.2070711)(12.014142,1.2070711) (10.6,1.2070711)(10.6,0.20707108)(9.2,-1.1929289) (10.6,0.20707108)(12.0,-1.1929289) (6.2,1.2070711)(6.2,0.20707108)(4.8,-1.1929289) (6.2,0.20707108)(7.6,-1.1929289) (1.8,1.2070711)(1.8,0.20707108)(0.4,-1.1929289) (1.8,0.20707108)(3.2,-1.1929289) (5.8,-0.19292893)(5.8,-1.1929289) (7.0,-0.59292895)(6.6,-1.1929289) (2.2,-0.19292893)(2.2,-1.1929289) (1.0,-0.59292895)(1.4,-1.1929289) (10.0,-0.39292893)(10.2,-1.1929289) (11.2,-0.39292893)(11.0,-1.1929289) (0.0,-0.19292893)(3.6,-0.19292893) (0.0,-0.59292895)(3.6,-0.59292895) (0.0,0.20707108)(3.6,0.20707108) (4.4,0.20707108)(8.0,0.20707108) (4.4,-0.19292893)(8.0,-0.19292893) (4.4,-0.59292895)(8.0,-0.59292895) (2.0,0.40707108)[$s$]{} (6.4,0.40707108)[$s$]{} (10.8,0.40707108)[$s$]{} (0.8,-0.39292893)[$s$]{} (2.4,0.0070710755)[$s$]{} (5.6,0.0070710755)[$s$]{} (7.2,-0.39292893)[$s$]{} (9.8,-0.19292893)[$s$]{} (11.4,-0.19292893)[$s$]{} (1.8,0.20707108) (1.0,-0.59292895) (2.2,-0.19292893) (5.8,-0.19292893) (6.2,0.20707108) (7.0,-0.59292895) (10.6,0.20707108) (10.0,-0.39292893) (11.2,-0.39292893)
Assume for instance that $s \in E(2)$ is a degree $1$ generator. The leftmost tree in Figure \[Dnes\_prijede\_Andulka.\] represents $(s \circ_2 s)
\circ_1 s \in \Free(E)(4)$ while the middle one $(s \circ_1 s)
\circ_3 s$ in the same piece of $\Free(E)$. By the parallel associativity of the $\circ_i$-operations $$(s \circ_2 s)
\circ_1 s = - (s \circ_1 s)
\circ_3 s,$$ thus the two decorated trees represent [*different*]{} elements. If we do not specify the levels in the rightmost tree in Figure \[Dnes\_prijede\_Andulka.\], we do not know to which one we refer to. The same caution is necessary also in case of free $\Grc$-operads.
Let us return to our description of the free operad $\Free(E)$. For a graph-tree $T$ we denote by $\Lev(T)$ the chaotic groupoid whose objects are all possible arrangements of levels of $T$. For a given $\lambda \in \Lev(T)$, let $\Gamma_i$, $1 \leq i \leq k\-1$, be the fiber of $\tau_i$ in the tower (\[Jaruska\_ma\_jizdy.\]) associated to $T$ with levels $\lambda$. We extend the notation by $\Gamma_k := \Delta_k$. For a 1-connected collection we define $$\label{Krtek_na_mne_kouka.}
E(T,\lambda) := E(\Gamma_1) \ot \cdots \ot E(\Gamma_k).$$ For different $\lambda$’s this expression differs only by the order of the factors, so we may, using the commutativity constraint for graded vector spaces, promote formula (\[Krtek\_na\_mne\_kouka.\]) into a functor $$\label{Zviratka_mi_pomahaji.}
E: \Lev(T) \longrightarrow \Vect$$ into the category of graded vector spaces.
\[podlehl\_jsem\] Given a $1$-connected collection $E$, one has the following description of the arity $\GAmma$ piece of the free operad $\Free(E)$: $$\label{Zbijecky duni.}
\Free(E)(\GAmma) \cong
\begin{cases}
\displaystyle \bigoplus_{T \in \gTr(\GAmma)} \ \colim{\lambda \in
\Lev(T)} E(T,\lambda)&\hbox {if $\GAmma$ has at least one internal edge,
and}
\\
\bfk&\hbox {if $\GAmma$ has no internal edges.}
\end{cases}$$
The statement is proved by applying the formulas of [@Sydney Section 10] to the particular case of $\Grc$. Notice that $\GAmma$ has no internal edges if and only if $\gTr(\GAmma) = \emptyset$.
Let us describe the operad structure of $\Free(E)$ given in (\[Zbijecky duni.\]). Recall first that the local terminal objects in the category $\Grc$ are directed graphs with no internal edges, i.e. directed corollas. The operad $\Free(E)$ is strictly extended unital in the sense of [@Sydney Section 7], with the transformation $\eta$ in [@Sydney eqn. (53)] given by the defining identity $$\Free(E)(\GAmma) = \bfk \hbox { if $\GAmma$ is local terminal.}$$ We describe next the action of the groupoid $\QV(e)$ generated by local isomorphisms, local reorderings and morphisms changing the global orders of legs of graphs. Let us start with the latter.
Let $T \in \gTr(\GAmma)$ be a graph tree and $\vartheta : \GAmma \to \UPsilon$ be an isomorphism changing the global orders of the legs. In other words, the graph $\UPsilon$ differs from $\GAmma$ only by the order of its legs. Since the legs of $\GAmma$ are the same as the legs of the graph $\Gamma_1$ decorating the root of $T$, one also has the induced isomorphism $\vartheta_1 : \Gamma_1 \to \Delta_1 \in \QV(e)$, where $\Delta_1$ is obtained from $\Gamma_1$ by reordering its legs according to $\vartheta$.
We define $S\in \gTr(\UPsilon)$ to be the graph-tree whose underlying tree is the same as the underlying tree of $T$, its edges have the same decorations as the corresponding edges in $T$, and also the vertices have the same decorations as in $T$ except for the root vertex of $S$ which is decorated by $\Delta_1$. If $T$ has levels $\lambda \in \Lev(T)$, we equip $S$ with the same levels. One then has the action $$\xymatrix@1@C=5em{
E(T,\lambda) = E(\Gamma_1) \ot \cdots \ot E(\Gamma_k)\
\ar[r]^{E(\vartheta_1) \ot \id^{\ot{k-1}}} & \ E(\Delta_1) \ot \cdots \ot E(\Gamma_k) = E(S,\lambda)
}$$ induced by the $\QV(e)$-action $E(\vartheta_1): E(\Gamma_1) \to E(\Delta_1)$ on the generating collection. The above actions assemble into an action $\Free(E)(\GAmma)
\to \Free(E)(\UPsilon)$ on the colimits (\[Zbijecky duni.\]).
The actions of local isomorphisms and local reorderings are defined similarly, so we can be brief. Given $T \in \gTr(\GAmma)$, a local reordering of $\GAmma$ induces in the obvious way local reorderings of the graphs decorating the vertices of $T$, and therefore also on the products (\[Krtek\_na\_mne\_kouka.\]).
Local isomorphisms act by reorderings of the set $V$ of vertices of $\GAmma$. Note that, by the definition of a graph-tree, the set $V$ and its order determine the labels of the edges of $T$, so a reordering of $V$ may change the labels of the edges of $T$. Thus, according to Compatibility 1 for graph-trees, it induces local isomorphisms of the graphs decorating the vertices of $T$ which in turn act on the products (\[Krtek\_na\_mne\_kouka.\]).
Let us finally attend to the operad composition. That is, for an elementary morphism $F \fib \GAmma \stackrel\phi\to \UPsilon$ in $\Grc$, we must describe a map $$\label{Vcera_mne_natacel_Oliver_a_pak_jsme_se_spolu_opili.}
\circ_\phi:
\Free(E)(\UPsilon) \ot \Free(E)(F) \longrightarrow \Free(E)(\GAmma) .$$ Given such a $\phi$, one can find as in the previous pages a canonical contraction $\widehat F \fib \GAmma
\stackrel{\widehat \phi}\to
\widehat \UPsilon$ and an isomorphism $\sigma : \UPsilon \to
\widehat \UPsilon$ in the commutative diagram $$\xymatrix@C1em{&\GAmma \ar[ld]_\phi \ar[rd]^{\widehat \phi} &
\\
\UPsilon\ar[rr]^\sigma_\cong &&\ \widehat \UPsilon.
}$$ Using the equivariance [@Sydney eqn. (57)] with $\omega = \id$, $\phi'=\phi$ and $\phi'' = \widehat \phi$, we see that $\circ_\phi$ is uniquely determined by $\circ_{\widehat \phi}$. So we may assume that $\phi$ in (\[Vcera\_mne\_natacel\_Oliver\_a\_pak\_jsme\_se\_spolu\_opili.\]) is a canonical contraction.
Let $S \in \gTr(\UPsilon), R \in \gTr(F), \lambda' \in \Lev(S)$ and $\lambda''
\in \Lev(R)$. Let also $x \in \Vert(\UPsilon)$ be the vertex over which the unique nontrivial fiber of $\phi$ lives. We define $T \in \gTr(\GAmma)$ as the graph-tree whose underlying tree is obtained by grafting the root of the underlying tree of $R$ to the leg of the underlying tree of $S$ labelled by $x$. The decorations of $T$ is inherited from the decorations of its graph-subtrees $S$ and $R$. It is simple to check that, since $\phi$ is a cc, $T$ is indeed a graph-tree.
We finally define $\lambda = \lambda' \circ_\phi \lambda''\in \Lev(T)$ by postulating that all vertices of $R$ are below the vertices of $S$ and that the restriction of $\lambda$ to the subtrees $S$ resp. $R$ is $\lambda'$ resp. $\lambda''$. The map (\[Vcera\_mne\_natacel\_Oliver\_a\_pak\_jsme\_se\_spolu\_opili.\]) is then the colimit of the obvious canonical isomorphisms $$E(T,\lambda) \cong E(S,\lambda') \ot E(R,\lambda'').$$
\[Je\_streda\_a\_melu\_z\_posledniho.\] When the generating collection is evenly graded, the elements of the product (\[Krtek\_na\_mne\_kouka.\]) represents the [*same*]{} elements of $\Free(E)(\GAmma)$ regardless the choice of $\lambda$, thus (\[Zbijecky duni.\]) can be replaced by a more friendly formula $$\Free(E)(\GAmma) \cong \bigoplus_{T \in \gTr(\GAmma)}
\bigotimes_{v \in \Vert(T)} E(\Gamma_v).$$ As illustrated in Warning \[Dnes jdu s Jaruskou na koncert.\], this simplification is not possible for general collections. Yet, since the input edges of each graph-tree are ordered, there exists a preferred choice of the levels specified by the following lexicographic rule.
Assume that $a < b$ are (the labels of) two input edges of a vertex $v
\in \Vert(T)$. Then all levels of the subtree of $T$ with the root $a$ are [*below*]{} the levels of the subtree with the root $b$. Denoting by $\lambda_\lex$ the above arrangement, then $$\label{Prvni_na_Vivat_tour.}
\Free(E)(\GAmma) \cong \bigoplus_{T \in \gTr(\GAmma)} \
E(T,\lambda_\lex).$$ One must however keep in mind that the combination $\lambda_\lex'
\circ_\phi \lambda_\lex''$ of two lexicographic arrangements may not be lexicographic. Thus, if we want to use (\[Prvni\_na\_Vivat\_tour.\]) the operadic composition based on the isomorphism $$E(T,\lambda_\lex'
\circ_\phi \lambda_\lex'') \cong E(S,\lambda_\lex') \ot E(R,\lambda_\lex'')$$ must be followed by bringing the result back into the preferred form.
Minimal model for $\termGrc$. {#hadrova_panenka}
=============================
The aim of this section is to construct an explicit minimal model of the terminal $\Grc$-operad $\termGrc$ governing non-genus graded modular operads.
Free operads and derivations.
-----------------------------
Free $\Grc$-operads are graded, $$\Free(E)(\GAmma) = \bigoplus_{n \geq 0} \Free^n(E)(\GAmma),\ \Gamma
\in \Grc,$$ where $\Free(E)^0(\GAmma) = \bbk$ and the higher pieces are given by the modification of (\[Zbijecky duni.\]): $$\label{Athalia}
\Free^n(E)(\GAmma) \cong
\bigoplus_{T \in \gTr^n(\GAmma)} \ \colim{\lambda \in
\Lev(T)} E(T,\lambda),$$ in which $\gTr^n(\GAmma)$ is, for $n \geq 1$, the subset of $\gTr(\GAmma)$ consisting of graph-trees $T$ with exactly $n$ vertices. Clearly $\Free^1(E)(\GAmma) \cong E(\GAmma)$. To describe $\Free^2(E)(\GAmma)$, we realize that there is precisely one way to introduce levels into a graph-tree $T \in \gTr^2(\GAmma)$, so (\[Athalia\]) takes the form $$\label{Athalia1}
\Free^2(E)(\GAmma) \cong
\bigoplus_{T \in \gTr^2(\GAmma)} \ E(\Gamma_v) \ot E(\Gamma_u),$$ where $\Gamma_v$ (resp. $\Gamma_u$) is the graph decorating the vertex $v$ at the top level of $T$ (resp. the vertex $u$ at the bottom level of $T$). We also have the obvious
A degree $s$ linear map $\varpi
: \Free(E) \to \Free(E)$ of collections is a degree $s$ [*derivation*]{} if $$\varpi\,\circ_\phi = \circ_\phi (\varpi \ot \id) + \circ_\phi(\id \ot
\varpi),$$ for every elementary morphism $F \fib \GAmma \stackrel\phi\to \UPsilon$ and $\circ_\phi$ as in (\[Vcera\_mne\_natacel\_Oliver\_a\_pak\_jsme\_se\_spolu\_opili.\]).
As expected, every derivation $\varpi$ is determined by its restriction $
\varpi|_E : E = \Free^1(E) \to \Free(E),
$ and every such a map extends to a derivation.
\[sempre\_dolens\] Given a linear map $\omega: E \to \Free(E)$, its extension $\varpi :
\Free(E) \to \Free(E)$ into a derivation is obtained by subsequent applications of $\omega$ to the factors $E(\Gamma_i)$, $1 \leq i \leq k$, of $E(T,\lambda)$ in (\[Krtek\_na\_mne\_kouka.\]), replacing each of these factors by its $\omega$-image.
Minimal models
--------------
They came to life, for dg associative commutative resp. dg Lie algebras, as the Sullivan resp. Quillen minimal models of rational homotopy types, see [@T] and citations therein. Minimal models for (classical) operads were introduced and studied in [@zebrulka], while minimal models for (hyper)operads governing permutads were treated in [@perm]. Below we give a definition for $\Grc$-operads, definitions for other types of (hyper)operads featuring in this paper are obvious modifications and we will thus not spell them out explicitly.
The [*minimal model*]{} of a dg $\Grc$-operad $\oP$ is dg $\Grc$-operad $\Min$ together with a dg $\Grc$-operad morphism $\rho : \Min \to \oP$, such that
- the component $\rho(\Gamma) : \Min(\Gamma) \to
\oP(\Gamma)$ of $\rho$ is a homology isomorphism of dg vector spaces for each $\Gamma \in
\Grc$, and
- the underlying non-dg $\Grc$-operad of $\Min$ is free, and the differential $\pa$ of $\Min$ has no constant and linear terms (the [*minimality condition*]{}).
One can prove, adapting the proof of Theorem II.3.127 in [@MSS], that minimal models are unique up to isomorphism. Our construction of the minimal model for $\termGrc$ begins by describing its generating $1$-connected collection. For a vector space $A$ of dimension $k$, we denote by $\det(A) :=
\hbox {\large$\land$}^k(A)$ the top-dimensional piece of its Grassmann algebra. If $S$ is a non-empty finite set, we let $\det(S)$ to be the determinant of the vector space spanned by $S$. Given two finite sets $S_1 = \{e^1_1,\ldots,e^1_a\}$, $S_2 = \{e^2_1,\ldots,e^2_b\}$, we define $$\omega_{S_1,S_2}:
\det(S_1 \sqcup S_2) \to \det(S_1) \ot \det(S_2).$$ by $$\omega_{S_1,S_2}(e^1_1 \land \cdots \land e^1_a \land
e^2_1 \land \cdots \land e^2_b) := (e^1_1 \land \cdots \land e^1_a) \ot
(e^2_1 \land \cdots \land e^2_b).$$ Let, for $\Gamma \in \Grc$, $\Edg(\Gamma)$ denote the set of its internal edges, and $\det(\Gamma) := \det(\Edg(\Gamma))$. With this notation, the generating collection of the minimal model for $\termGrc$ is defined as the one-dimensional vector space $$\label{Flicek}
D(\Gamma) := \det(\Gamma), \ \Gamma \in \Grc,$$ placed in degree $|\Gamma| := {\rm card}(\Edg(\Gamma)) -1$ if $\Gamma$ has at least one internal edge, while $D(\Gamma) := 0$ if $\Gamma$ is a corolla. Notice that for $\Gamma$ with exactly one internal edge, $\det(\Gamma)$ is canonically isomorphic to $\bbk$.
The degree $-1$ differential $\pa$ will be determined by its restriction (denoted by the same symbol) $$\pa : D \to \Free^2(D) \subset \Free(D)$$ as follows. Given $T \in \gTr^2(\Gamma)$, let $\Gamma_v,\Gamma_u \in \Grc$ have the same meaning as in (\[Athalia1\]), and $E_v := \Edg(\Gamma_v)$, $E_u := \Edg(\Gamma_u)$. For $\mu \in D(\Gamma) = \det(\Gamma)$ we put
$$\label{Flicek_na_mne_kouka.}
\pa_T(\mu) := \bigoplus_{T \in \gTr^2(\Gamma)} \pa_T(\mu),$$
where $$\label{Posledni patek v Sydney}
\partial_T(\mu) := (-1)^{|\Gamma|} \omega_{E_v,E_u} (\mu) \in
D(\Gamma_v) \ot D(\Gamma_u) \subset \Free^2(D)(\Gamma).$$
The derivation $\pa$ defined above is a differential, i.e. $\pa^2=0$.
It is simple to see that $\partial^2$ is a derivation as well, so it suffices only to verify that $\partial^2$ vanishes on the generating collection. We leave this as an exercise to the reader.
Let $\rho : \Free(D) \to \termGrc$ be the unique map of $\Grc$-operads whose restriction $\rho|_{D(\Gamma)}$ is, for $\Gamma \in \Grc$, given by $$\label{Obehnu_to_dnes?}
\rho|_{D(\Gamma)} :=
\begin{cases}
\id_\bbk : D(\Gamma) = \bbk \to \bbk =
\termGrc(\Gamma), &\hbox {if $|\Edg(\Gamma)| = 1$,
while}
\\
0,& \hbox {if $|\Edg(\Gamma)| \geq 2$.}
\end{cases}$$ Having all this, we formulate:
\[Woy-Woy\] The object $\minGrc := (\Free(D),\pa) \stackrel\rho\to (\termGrc,\pa
=0)$ is a minimal model of the terminal $\Grc$-operad.
The rest of this section is devoted to the proof of Theorem \[Woy-Woy\] and of the necessary auxiliary material.
Constructs represent graph-trees. {#Srni}
---------------------------------
The material of this subsection is based on modification and generalization of [@JO]. We start by associating to each object $\Gamma$ of ${\tt Grc}$ a hypergraph ${\bf H}_{\Gamma}$ defined as follows: the vertices of ${\bf H}_{\Gamma}$ are the internal edges of $\Gamma$ and two vertices are connected by an edge in ${\bf H}_{\Gamma}$ whenever, as edges of $\Gamma$, they share a common vertex. Observe that the leaves of $\Gamma$ play no role in the definition of ${\bf H}_{\Gamma}$.
\[ex1\] Here is an example of the association of a hypergraph to a graph:
(E)\[circle,draw=black,thick,fill=white,minimum size=2mm,inner sep=0.2mm\] at (0.5,0) [$1$]{}; (F) \[circle,draw=black,thick,fill=white,minimum size=2mm,inner sep=0.2mm\] at (-0.5,1.5) [$2$]{}; (A) \[circle,draw=black,thick,fill=white,minimum size=2mm,inner sep=0.2mm\] at (1.5,1.5) [$3$]{}; (x) \[draw=none,minimum size=4mm,inner sep=0.1mm\] at (0.5,2.075) [ $x$]{}; (y) \[draw=none,minimum size=4mm,inner sep=0.1mm\] at (0.5,0.85) [$y$]{}; (z) \[draw=none,minimum size=4mm,inner sep=0.1mm\] at (2.8,1.5) [$z$]{}; (u) \[draw=none,minimum size=4mm,inner sep=0.1mm\] at (-0.15,0.65) [ $u$]{}; (v) \[draw=none,minimum size=4mm,inner sep=0.1mm\] at (1.15,0.65) [$v$]{}; (F) to\[out=40,in=140\](A) node ; (A) to\[out=40,in=-40,looseness=21\](A) node ; (F) to\[out=-40,in=-140\](A) node ; (E)–(F) node ; (E)–(A) node ; (F)–(-1.3,1.75) node ; (F)–(-1.3,1.25) node ;
\(x) \[circle,draw=black,fill=black,inner sep=0mm,minimum size=1.3mm,label=[\[xshift=-0.22cm,yshift=-0.28cm\][$x$]{}]{}\] at (0,1) ; (y) \[circle,draw=black,fill=black,inner sep=0mm,minimum size=1.3mm,label=[\[xshift=0.2cm,yshift=-0.14cm\][$y$]{}]{}\] at (1,1) ; (z) \[circle,draw=black,fill=black,inner sep=0mm,minimum size=1.3mm,label=[\[xshift=0.22cm,yshift=-0.28cm\][$z$]{}]{}\] at (2,1) ; (u) \[circle,draw=black,fill=black,inner sep=0mm,minimum size=1.3mm,,label=[\[xshift=-0.22cm,yshift=-0.28cm\][ $u$]{}]{}\] at (0.5,0.1) ; (v) \[circle,draw=black,fill=black,inner sep=0mm,minimum size=1.3mm,,label=[\[xshift=0.22cm,yshift=-0.28cm\][$v$]{}]{}\] at (1.5,0.1) ; (z)–(v)–(u)–(x)–(y)–(z); (u)–(y)–(v)–(x); (x) to\[out=40,in=140\] (z);
In the lemma that follows, the notion of a subgraph of a graph $\Gamma$ is taken with respect to a connected set of internal edges (and not vertices). For example, the subgraph of the graph $\Gamma$ from Example \[ex1\] determined by the internal edge $x$ is
\(F) \[circle,draw=black,thick,fill=white,minimum size=2mm,inner sep=0.2mm\] at (-0.5,1.5) [$2$]{}; (A) \[circle,draw=black,thick,fill=white,minimum size=2mm,inner sep=0.2mm\] at (1.5,1.5) [$3$]{}; (x) \[draw=none,minimum size=4mm,inner sep=0.1mm\] at (0.5,2.075) [ $x$]{}; (F) to\[out=40,in=140\](A) node ; (2.3,1.25)–(A)–(2.3,1.75); (0.7,1.3)–(A)–(0.9,0.9); (0.3,1.3)–(F)–(0.1,0.9); (F)–(-1.3,1.75) node ; (F)–(-1.3,1.25) node ;
Therefore, in such a subgraph, an internal edge of the original graph may be cut into two half-edges.
\[connectededges\] The connected subgraphs of a graph $\Gamma$ that have at least one internal edge are in one-to-one correspondence with the connected subsets of ${H}_{\Gamma}$, i.e. with the non-empty subsets $X$ of vertices of ${\bf H}_{\Gamma}$ such that the hypergraph $({\bf H}_{\Gamma})_{X}$ is connected.
Thanks to Lemma \[connectededges\], for a graph $\Gamma$ and $\emptyset\neq X\subseteq \Edg(\Gamma)$, we can index the connected components of ${\bf H}_{\Gamma}\backslash X$ by the corresponding subgraphs of $\Gamma$, by writing $${\bf H}_{\Gamma}\backslash X\leadsto {\bf H}_{{\Gamma}_1},\dots,{\bf H}_{{\Gamma}_n}.$$ Observe that the subgraphs ${\Gamma}_1,\dots,{\Gamma}_n$ of ${\Gamma}$ do not in general make a decomposition of ${\Gamma}$, in the sense that the removal of the edges from the set $X$ may result in a number of subgraphs of ${\Gamma}$ reduced to a corolla without internal edges.
\[jovanica\] There exists a natural isomorphism $\alpha_\Gamma : {\mathcal A}({\bf
H}_{\Gamma}) \stackrel\cong\longrightarrow {\tt gTr}(\Gamma)$ between the abstract polytope ${\mathcal A}({\bf H}_{\Gamma})$ of constructs of the hypergraph $\H_\Gamma$ and the poset ${\tt gTr}(\Gamma)$ of graph-trees such that $\gr(T)
=\Gamma$.
We define the announced one-to-one correspondence $\alpha_{\Gamma}$ between constructs and graph-trees $T \in \gTr(\Gamma)$ by induction on the number of vertices of $C$. If $C$ is the maximal construct $\Edg(\Gamma): \bfH_\Gamma$, then $\alpha_\Gamma(T)$ is the planar rooted corolla
(0,0.0582557)(13,4.0582557) (4.5,1.9762975)(4.5,4.1512976) (4.5,1.9512974)(2,0) (4.5,1.9512974)(3,0) (4.5,1.9512974)(7,0) (4.5,1.9512974) (4.3,1.95)[$\,\Gamma$]{} (4.75,0)[$\dots$]{}
with the vertex decorated by $\Gamma$ and legs labelled by the ordered set $\Vert(\Gamma)$.
Suppose that $C=X\{C_1,\dots,C_p\}$, $X \subset \Edg(\Gamma)$, ${\bf H}_{\Gamma}\backslash X\leadsto {\bf H}_{1},\dots, {\bf H}_{p}$ and for $1 \leq i \leq p$. By Lemma \[connectededges\], there are connected subgraphs $\Gamma_i$ of $\Gamma$ such that $\bfH_i = {\bf H}_{\Gamma_i}$. There, moreover, exists a graph $\Gamma_X \in \Grc$ such that $\Rada \Gamma1p$ are the fibers of the iterated canonical contraction $\Gamma \to {\Gamma}_X$. This understood, we are in the situation when ${\bf H}_{\Gamma}\backslash X\leadsto {\bf H}_{{\Gamma}_1},\dots, {\bf
H}_{{\Gamma}_p}$ and , $1 \leq i
\leq p$.
The root vertex of the graph-tree $\alpha_{\Gamma}(C)$ will be decorated by $\Gamma_X$. We already have, by induction, the graph-trees $\alpha_{\Gamma_i}(C_i)$, and each of these trees is connected with the root of $\alpha_{\Gamma}(C)$ by the edge bearing the label of the vertex of $\Gamma_X$ to which $\Gamma_i$ has been contracted. We believe that Figure \[Zviratka-se-fotila-ve-Woy-Woy.\] makes this construction clear. The inductive step is finished by joining to the root of the graph-tree $\alpha_{\Gamma}(C)$ the legs indexed by the remaining vertices of $\Gamma_X$.
(0,-3.1006334)(7.8153715,3.1006334) (3.7053716,1.0186751)(3.6,3.193675) (3.0053718,-1.806325)(3.9053717,1.0186751) (6.4053717,-0.6063249)(3.9053717,1.0186751) (1.2053717,-0.6063249)(3.9053717,0.99367505) (6.6053715,-0.70632493) (3.0053718,-2.106325) (1.0053717,-0.70632493) (3.6053717,0.8936751) (1,-0.70632493)[$\alpha_{\Gamma_1}(C_1)$]{} (2.3,-2.106325)[$\alpha_{\Gamma_2}(C_2)$]{} (5.9,-0.70632493)[$\alpha_{\Gamma_p}(C_p)$]{} (3.3718,0.9367505)[$\Gamma_X$]{} (4.4053717,-0.8063249)[$\cdots$]{}
The inverse of $\alpha_{\Gamma}$ is defined by extracting the construct from a graph-tree $T$ in the following way. First, remove all the leaves of $T$ and then, for each vertex of $T$, replace the graph that decorates that vertex by the maximal construct of its associated hypergraph. In more detail, assume that $T \in \gTr(\Gamma)$, $\Gamma \in \Grc$. The underlying rooted tree of the construct $\alpha^{-1}_{\Gamma}(T)$ is obtained from the underlying tree of $T$ by amputating its legs. The vertex of $\alpha^{-1}_{\Gamma}(T)$ corresponding to a vertex $v \in \Vert(T)$ decorated by $\Gamma_v\in \Grc$ is decorated by the set $\Edg(\Gamma_v) \subset \Edg(\Gamma)$ of edges of $\Gamma_v$.
There is the following inductive, alternative construction of $\alpha^{-1}_\Gamma(T)$ that leads manifestly to a construct of $\bfH_\Gamma$. Assume that $e_1,\ldots,e_s \in V$ are the labels of the incoming internal edges of a vertex $v \in \Vert(T)$, and that $v_1,\ldots,v_s \in \Vert(T)$ are the initial vertices of these edges. Further, let $T_i$ be the maximal rooted graph-subtree of $T$ with the root $v_i$ and $\Gamma_i := \gr(T_i)$, $1 \leq i \leq s$. Then the corresponding subtree of $\alpha^{-1}_{\Gamma}(T)$ is the construct $$\Edg(\Gamma_v)\{\alpha^{-1}_{\Gamma_1}(T_1),\ldots,\alpha^{-1}_{\Gamma_s}(T_s)\}.$$ Notice that that the construct $\alpha^{-1}_{\Gamma}(T)$ inherits the planar structure of $T$. It is easy to verify that the correspondence $$\label{semper}
\gTr(\Gamma) \ni T \longleftrightarrow
\alpha_\Gamma(T) \in \{C\ |\ C: \H_\Gamma\}$$ preserves the poset structures.
For the graph $\Gamma$ from Example \[ex1\], the graph-tree $\alpha_\Gamma(C)$ associated to the construct $C=\{x,y\}\{\{u,v,z\}\}$ of the hypergraph ${\bf H}_{\Gamma}$ is shown in Figure \[patek\].
For an object $\Gamma$ of ${\Grc}$ and a construct $C:{\bf H}_{\Gamma}$, let ${\Lev}(C)$ denote the chaotic groupoid whose objects are all possible arrangements of levels of $C$, whereby a level of a construct is defined analogously as the one of a graph tree. It is clear that the correspondence (\[semper\]) defines a canonical isomorphism between ${\Lev}(C)$ and $\Lev(\alpha_\Gamma(C))$, thus each $1$-connected collection $E$ promotes into a functor $E: \Lev(C) \longrightarrow \Vect$ in the diagram $$\xymatrix{&\Vect
\\
\Lev(C)\ar[r]^\cong \ar[ur]^E&\ar[u]_E\Lev(\alpha_\Gamma(C))
}$$ where the vertical up-going arrow is (\[Zviratka\_mi\_pomahaji.\]). The following reformulation of Theorem \[podlehl\_jsem\] is a direct consequence of Proposition \[jovanica\].
For a $1$-connected collection $E$, the arity $\Gamma$ piece of the free operad $\Free(E)$ is given by $$\label{Free constructs.}
\Free(E)(\Gamma) \cong
\begin{cases}
\displaystyle \bigoplus_{C:{\H}_\Gamma} \ \colim{\varsigma \in
{\Lev}(C)} E(C,\varsigma)&\hbox {if $\Gamma$ has at least one internal edge,
and}
\\
\bfk&\hbox {if $\Gamma$ has no internal edges.}
\end{cases}$$
A chain complex. {#Za_chvili_volam_Mikesovi.}
----------------
In this subsection we recall a chain complex associated to a convex polyhedron featuring in Lemma \[Jeste\_ani\_nevim\_kde\_budu\_v\_Melbourne\_bydlet.\] below. Let therefore $K$ be such an $n$-dimensional polyhedron realized as the convex hull of finitely many points in $\bbR^n$. Each $k$-dimensional face $e$ of $K$, $0 \leq k \leq n$, is then embedded canonically into a $k$-dimensional affine subspace $\A_e$ of $\bbR^n$, namely into the span of its vertices. By an [*orientation*]{} of $e$ we understand an orientation of $\A_e$. For $k > 0$, that orientation is given by choice of a frame in $\A_e$. If $k=0$, $\A_e$ is a point, and the orientation is a sign assigned to that point. We say that $K$ is [*oriented*]{}, if an orientation of each face has been specified.
Assume that $a$ is a codimension one subface of $e$ and that the dimension of $a$ is $\geq 1$. Clearly $\A_a$ divides $\A_e$ into two half-spaces. Denote by $\A^a_e
\subset \A_e$ the one having non-empty intersection with $K$. Let the orientation of $a$ be given by linearly independent vectors $(\Rada v1{k-1})$ in $\A_a$. We say that an orientation of $a$ is [*compatible*]{} with the orientation of $e$ if the frame $(\Rada v1{k-1}, n)$ in $\A^a_e$, where $n$ is a vector normal to $\A_a \subset \A^a_e$, defines the orientation of $e$, cf. Figure \[S\_Jarkou\_u\_Pakousu.\] (left) where $k=2$. A modification of this notion to $0$-dimensional $a$’s is obvious.
$$\psscalebox{.8 .8} % Change this value to rescale the drawing.
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\rput[bl](-.2,0.30034882){$v_1$} \rput[bl](-.2,2.30034882){$\A_a$}
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\rput[bl](8.8,-0.4996512){$a$}
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\rput[bl](8.6,0.90034884){$h$}
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We assign to $K$ a chain complex $(C_*(K),\pa)$ of free abelian groups whose $k$th piece $C_k(K)$ is generated by $k$-dimensional faces of $K$. The value of the differential on a $k$-dimensional generator $\lambda$ is defined by $$\pa(\lambda) = \sum \eta^\delta_\lambda \cdot \delta,$$ where $\delta$ runs over all codimension one faces of $\lambda$ and $$\eta^\delta_\lambda :=
\begin{cases}
+1,&\hbox {if $\delta$ is oriented compatibly with $\lambda$, and}
\\
-1,&\hbox {otherwise.}
\end{cases}$$ It follows from standards methods of algebraic topology that $(C_*(K),\pa)$ is acyclic in positive dimensions while its $0$th homology equals ${\mathbb Z}$.
An ingenious lemma
------------------
Let $L = (L,\prec)$ be the face poset of an $n$-dimensional polyhedron $K$, ordered by the inclusion. Assume that $K$ is such that the following ‘diamond’ condition is satisfied.
Let $0< k < n$ and let $a$ be a $(k-1)$-dimensional face of $K$ which is a common boundary of two $k$-dimensional faces $e',e''$. Then there exists a $(k+1)$-dimensional face $h$ with $e'$ and $e''$ in its boundary.
A concise way to formulate the diamond condition is to say that the existence of $e'$ and $e''$ with $a \prec e',e''$ implies the existence of some $h$ with $e',e'' \prec h$, diagrammatically $$\xymatrix@C=1em@R=.7em{&h&
\\
e' \ar@{^{(}->}[ur] && \ e'', \ar@{_{(}->}[ul]
\\
&a \ar@{_{(}->}[ur] \ar@{^{(}->}[ul] &
}\tag{$\Diamond$}$$ hence the name. It follows from the properties of abstract polytopes that $e'$ and $e''$ are the only faces in the interval $[a,h]$, but the diamond condition need not be satisfied in a general polytope.
Assume that $(C_*(L),\pa)$ is a chain complex such that each $C_k(L)$ is the free abelian group generated by $k$-dimensional elements of $L$, $0
\leq k \leq n$. Suppose moreover that, for each $\lambda \in L$, $\pa(\lambda)$ is of the form $$\pa(\lambda) = \sum \eta^\delta_\lambda \cdot \delta,$$ where $\eta^\delta_\lambda \in \{-1,+1\}$ and $\delta$ runs over all codimension one faces of $\lambda$. Then one has:
\[Jeste\_ani\_nevim\_kde\_budu\_v\_Melbourne\_bydlet.\] The faces of $K$ could be oriented so that $(C_*(L),\pa)$ is the chain complex $(C_*(K),\pa)$ recalled in Subsection \[Za\_chvili\_volam\_Mikesovi.\].
The lemma will be proved by downward induction on the dimension of the faces of $K$. We start by choosing an orientation of the unique $n$-dimensional face of $K$ arbitrarily.
Assume that we have oriented all faces of $K$ of dimensions $\geq k$ for some $n > k \geq 0$. Let $a$ be a dimensional face of $K$, and choose some $k$ dimensional face $e$ such that $a \prec e$. This is always possible, since otherwise the face $a$ would be maximal, which contradicts the properties of a polytope. If $a$ occurs in $\pa (e)$ with the $+1$ sign, we equip it with the compatible orientation, if it occurs with the $-1$ sign, we equip it with the orientation opposite to the compatible one. We need to show that this recipe does not depend on the choice of $e$.
Assume therefore that $e'$ and $e''$ are two faces of $K$ with the properties described above. Let $h$ be a cell required by the diamond property. Then $$\begin{aligned}
\pa(h)& = \eta' \cdot e' + \eta'' \cdot e'' + \hbox{other terms,}
\ \eta', \eta'' \in \{-1,+1\},
\\
\pa(e')&= \varepsilon' \cdot a + \hbox{other terms,}
\ \varepsilon' \in \{-1,+1\},\ \hbox {and}
\\
\pa(e'')&= \varepsilon'' \cdot a + \hbox{other terms,} \
\varepsilon'' \in \{-1,+1\}.\end{aligned}$$ The condition $\pa^2(h) = 0$ together with the fact that $e'$ and $e''$ are the only faces in the interval $[a,h]$ imply $$\label{Dnes_zmena_casu.}
\eta'\varepsilon' + \eta''
\varepsilon'' =0.$$ The configuration of the relevant cells is indicated in Figure \[S\_Jarkou\_u\_Pakousu.\] (right) which shows a section of $h$ with a hyperplane orthogonal to $\A_a$.
Assume e.g. that $\eta' = \eta'' = 1$. Then both $e'$ and $e''$ have the orientation compatible with the orientation of $h$. By (\[Dnes\_zmena\_casu.\]) one has $\varepsilon'
= - \varepsilon''$; assume for instance that $\varepsilon' = 1,
\varepsilon'' = -1$. Then $a$ gets from $e'$ the compatible orientation, and from $e''$ the orientation opposite to the compatible one. It easily follows from the local geometry of the section in Figure \[S\_Jarkou\_u\_Pakousu.\] that these two orientations of $a$ are the same. The remaining cases can be analyzed similarly.
Splits and collapses
--------------------
The proof of Lemma \[Panenka\_na\_mne\_kouka.\] below relies on the actions of [*splitting*]{} the vertices and [*collapsing*]{} the edges of constructs of a hypergraph ${\bf H}$. We formalize the corresponding constructions below. Let $C:{\bf H}$.
#### *Splitting the vertices of $C$.*
Let $V\in \mbox{vert}(C)$ be such that $|V|\geq 2$. Let ${\bf H}\!-\!V$ be the hypergraph defined by $${\bf H}\!-\!V:=\{X\backslash V\,|\, X\in {\it Sat}({\bf
H})\}\backslash\{\emptyset\}.$$ Observe that, in general, ${\bf H}\!-\!V\neq {\bf H}\backslash V$. For example, for the hypergraph ${\bf H}_{\Gamma}$ from Example \[ex1\], we have that ${\bf H}_{\Gamma}\!-\!\{x,y\}$ is the complete graph on the vertex set $\{z,u,v\}$, whereas ${\bf H}_{\Gamma}\backslash\{x,y\}$ can be obtained from ${\bf H}_{\Gamma}\!-\!\{x,y\}$ by removing the edge $\{u,z\}$ and, hence, is a linear graph. Let $\{X,Y\}$ be a partition of $V$ such that the tree $X\{Y\}$ is a construct of ${\bf H}\!-\!V$. We define the construct $C[X\{Y\}/X\cup Y]:{\bf H}$, obtained from C by splitting the vertex $V$ into the edge $X\{Y\}$, by induction on the number of vertices of $C$, as follows. If $C=H$, we set $C[X\{Y\}/X\cup Y]:=X\{Y\}$.
Suppose that, for $Z\subset H$, $C=Z\{C_1,\dots,C_p\}$, ${\bf H}\backslash Z\leadsto {\bf H}_{1},\dots, {\bf H}_{p}$ and $C_i:{\bf H}_{i}$. If there exists an index $i$, $1\leq i\leq p$, such that $V\in \mbox{vert}(C_i)$, we define $$C[X\{Y\}/X\cup Y]:=Z\{C_1,\dots,C_{i-1},C_i[X\{Y\}/X\cup Y]
,C_{i+1},\dots,C_p\}.$$ Assume that $V=Z$ and let $\{i_1,\dots,i_q\}\cup \{j_1,\dots,j_r\}$ be the partition of the set $\{1,\dots,p\}$ such that the hypergraphs ${\bf H}_{i_s}$, for $1\leq s\leq q$, contain a vertex adjacent to some vertex of $Y$, while the hypergraphs ${\bf H}_{i_t}$, for $1\leq t\leq q$, have no vertices adjacent to a vertex of $Y$. We define $$C[X\{Y\}/X\cup Y]:=X\{Y\{C_{i_1},\dots
C_{i_q}\},C_{j_1},\dots,C_{j_r}\}.$$ If, exceptionally, $\{i_1,\dots,i_q\}=\emptyset$ resp. $\{j_1,\dots,j_r\}=\emptyset$, we set $$C[X\{Y\}/X\cup Y]:=X\{Y,C_1,\dots,C_p\}
\hbox { resp.} \ C[X\{Y\}/X\cup Y]:=X\{Y\{C_1,\dots,C_p\}\}.$$
The proof that the non-planar rooted tree $C[X\{Y\}/X\cup Y]$ is indeed a construct of ${\bf H}$ goes easily by induction on the number of vertices of $C$, the only interesting case being $C=Z\{C_1,\dots,C_p\}$. In that case, the argument is based on the fact that the set of vertices $Y\cup \bigcup_{i\in \{i_1,\dots,i_q\}}\mbox{vert}({\bf H}_i)$ determines a connected component ${\bf H}'$ of ${\bf H}$ and, furthermore, that $Y\{C_{i_1},\dots C_{i_q}\}:{\bf H}'$.
#### *Collapsing the edges of $C$.*
One can similarly define the construct , obtained from C by collapsing the edge $X\{Y\}$ into the vertex $X\cup Y$.
\[Panenka\_na\_mne\_kouka.\] The polyhedron ${\mathcal G}({\bf H})$ that realizes the abstract polytope ${\mathcal A}(\H)$ (see Lemma \[abspol\]) of a hypergraph $\H$ satisfies the diamond property.
We prove the lemma by constructing, for each construct $C:{\bf H}$ of rank $k-1$ for which there exist constructs $C'$ and $C''$ of rank $k$ such that $$\label{joca}
C\leq_{{\bf H}} C'\quad \mbox{and} \quad C\leq_{{\bf H}} C'',$$ a construct $D:{\bf H}$ of rank $k+1$ such that $C'\leq_{{\bf H}} D$ and $C''\leq_{{\bf H}} D$.
By definition of the partial order $\leq_{{\bf H}}$ of ${\mathcal
A}({\bf H})$, the relations , together with the fact that the rank of $C$ differs by $1$ from the rank of $C'$ and $C''$, mean that there exists a vertex $X\cup Y$ of $C'$ and a vertex $U\cup
V$ of $C''$, such that $$C=C'[X\{Y\}/X\cup Y]=C''[U\{V\}/U\cup V].$$ As vertices of $C$, the sets $X$, $Y$, $U$ and $V$ satisfy one of the following relations: they can either be mutually disjoint, or it can be the case that $X=U$ and $Y\cap V=\emptyset$, or it can be the case that $Y=U$ and $X\cap V=\emptyset$, plus the ‘mirror’ reflection of the last case, namely $X=V$ and $U\cap V = \emptyset$.
It is easily seen that other possible relations are forbidden. For example, the relation $Y=V$ would imply that $C$ is not a rooted tree. Depending on the mutual relation of the vertices $X$, $Y$, $U$ and $V$ of $C$, the above equality implies that the action of collapsing a particular edge of $C'$ and a particular edge of $C''$ leads to the same construct. Indeed, if $X$, $Y$, $U$ and $V$ are mutually disjoint, then $$C'[U\cup V/U\{V\}]=C''[X\cup Y/X\{Y\}],$$ if $X=U$ and $Y\cap V=\emptyset$, then $$C'[(X\cup Y)\cup V/(X\cup Y)\{V\}]=C''[(X\cup V)\cup Y/(X\cup
V)\{Y\}],$$ and if $Y=U$ and $X\cap V=\emptyset$, then $$C'[(X\cup Y)\cup V/(X\cup Y)\{V\}]=C''[X\cup (Y\cup V)/X\{Y\cup
V\}].$$
We define $D$ to be precisely the construct obtained from $C'$ (or, equivalently, from $C''$) by such a collapse. The three diamonds corresponding to the three possible constructions of $D$ can be pictured respectively as follows:
\(r) \[rectangle,draw=none\] at (0.8,-0.5) ; (k1)\[rectangle,draw=none\] at (2,-2.55) ; (jo) at (2,1.15) ; (mq1) \[rectangle,draw=none\] at (3.2,-0.5) ; (k1)–(r); (k1)–(mq1); (mq1)–(jo); (r)–(jo);
where we only display the edges involved in the construction. By definition, the construct $D$ satisfies the required properties.
Proof of Theorem \[Woy-Woy\]. {#Posledni tyden.}
-----------------------------
We establish first that $\minGrc$ is acyclic in positive dimensions and that $H_0(\minGrc) \cong \bbk$. By Proposition \[jovanica\], each construct $C : \H_\Gamma$ is, for $\Gamma \in \Grc$ with at least one internal edge, of the form $\alpha_\Gamma(T)$ for some graph-tree $T \in
\gTr(\Gamma)$. It is therefore supported by a rooted planar tree, so we may introduce the lexicographic arrangement $\varsigma_{\,\lex}$ of levels of its underlying tree. Consequently we get from (\[Free constructs.\]) an analog $$%\label{Dnes_uklid_pokoje_uz_v_8!}
\Free(E)(\GAmma) \cong \bigoplus_{C : \H_\GAmma} \
E(C,\varsigma_{\,\lex})$$ of formula (\[Prvni\_na\_Vivat\_tour.\]).
The case which interests us is when $E$ is the collection $D$ in (\[Flicek\]) generating $\minGrc$. A vertex $v$ of $C$ is decorated by a subset $X_v \subset \Edg(\Gamma)$, thus it contributes to $D(C,\varsigma_{\,\lex})$ by the multiplicative factor $\det(X_v)$. Let us fix an order of $\Edg(\Gamma)$. Then each $X_v$ bears an induced order, hence $\det(X_v)$ has a preferred basis element $$x_1\land \cdots\land x_r \in \det(X_v),\ x_1< \cdots< x_r,\ X_v = \{\Rada x1r\},$$ so it is canonically isomorphic to $\bfk$ placed, according to our conventions, in degree $|X_v| - 1$. Combining the above facts, we arrive at the canonical isomorphism $$\label{Flicek_a_Misa_s_motylkem}
\Free(D)(\GAmma) \cong \bigoplus_{C : \H_\GAmma} \Span(\{e_C\}),$$ where $\Span(\{e_C\})$ is the vector space spanned by a generator $e_C$ placed in degree that equals the rank of $C$, which in this case equals $|\Edg(\Gamma)|-|\Vert(C)|$.
The differential $\pa$ of the minimal model transfers, via isomorphism (\[Flicek\_a\_Misa\_s\_motylkem\]), into a differential denoted by the same symbol of the graded vector space at the right hand side of (\[Flicek\_a\_Misa\_s\_motylkem\]). It is straightforward to verify that the transferred differential has the form required by Lemma \[Jeste\_ani\_nevim\_kde\_budu\_v\_Melbourne\_bydlet.\], i.e. $$\label{Uz_jedou_fukary.}
\pa(e_C) = \sum \eta^F_C \cdot e_F,$$ where $\eta_F \in \{-1,+1\}$ and $F$ runs over all $F : \H_\Gamma$ such that $\grad(F) = \grad(C)-1$.
It is possible to establish the explicit values of the coefficients $\eta_C^F$ in (\[Uz\_jedou\_fukary.\]), but the ingenuity of Lemma \[Jeste\_ani\_nevim\_kde\_budu\_v\_Melbourne\_bydlet.\] makes it unnecessary.
Now we invoke that the poset ${\mathcal A}({\H_\Gamma})$ of constructs of $\H_\Gamma$ is, by Lemma \[abspol\], the poset of faces of a convex polytope $K$ which moreover fulfills the diamond property by Lemma \[Panenka\_na\_mne\_kouka.\]. By Lemma \[Jeste\_ani\_nevim\_kde\_budu\_v\_Melbourne\_bydlet.\], the cells of $K$ can be oriented so that $$\left( \bigoplus_{C : \H_\GAmma}
\Span(\{e_C\}),\pa \right)$$ is the cell complex $C_*(K)$. It is thus acyclic in positive dimension, and so is $(\Free(D)(\GAmma),\pa) = \minGrc(\Gamma)$, for each $\Gamma\in \Grc$. By the same reasoning, $$\label{Lisegacev1}
H_0(\minGrc)(\Gamma) \cong \bbk\ \hbox { for each $\Gamma \in \Grc$.}$$
The next step is to prove that the operad morphism $\rho : \Free(D)
\to \termGrc$ commutes with the differentials, which clearly amounts to proving that $\rho(\pa x) = 0$ for each degree $1$ element $\mu\in
\Free(D)(\Gamma)_1$. By the derivation property of $\pa$, it is in fact enough to address only the case when $\mu$ is a generator of degree $1$, i.e. an element of $D(\Gamma) = \det(\Edg(\Gamma))$ with $\Gamma$ having exactly two internal edges.
Let thus $\Gamma$ be such a graph and $a,b$ its two internal vertices. There are precisely two graph-trees $T',T''\in
\gTr^2(\Gamma)$, both with two vertices and one internal edge. The root vertex of $T'$ is decorated by some graph $\Gamma_v'$ with the only internal edge $a$, and the other vertex of $T'$ by $\Gamma_u'$ with the only internal edge $b$. The graph-tree $T''$ has similar decorations $\Gamma_v''$ and $\Gamma_u''$, but this time $\Edg(\Gamma_v'') =
\{b\}$ and $\Edg(\Gamma_u'') = \{a\}$. For a generator $\mu := a \land b \in D(\Gamma) = \det(\{a,b\})$ formula (\[Flicek\_na\_mne\_kouka.\]) gives $$\pa( a \land b) = a \ot b - b \ot a \in
(D(\Gamma'_v) \ot D(\Gamma'_u)) \oplus( D(\Gamma''_v) \ot D(\Gamma''_u))
\subset \Free^2(D)(\Gamma).$$ By the definition (\[Obehnu\_to\_dnes?\]) of the morphism $\rho$, $$\rho(\pa( a \land b)) = \rho( a \ot b - b \ot a) = 1\cdot 1 - 1\cdot 1 =0$$ as required.
The last issue that has to be established is that $\rho$ induces an isomorphism $$H_0(\rho) : H_0(\minGrc) \stackrel\cong\longrightarrow \termGrc.$$ To this end, in view of (\[Lisegacev1\]), it is enough to prove that $$H_0(\rho)(\Gamma) : H_0(\minGrc)(\Gamma)
\longrightarrow \termGrc(\Gamma) = \bbk$$ is nonzero for each $\Gamma\in \Grc$. Equation (\[Athalia\]) readily gives $$\label{Jeptha}
\Free(D)(\GAmma)_0 \cong
\bigoplus_{T \in \gTr_0(\GAmma)} \ \colim{\lambda \in
\Lev(T)} D(T,\lambda),$$ in which $\gTr_0(\GAmma)$ is the subset of $\gTr(\GAmma)$ consisting of graph-trees for which each decorating graph $\Gamma_v$, $v \in \Vert(\Gamma)$, has exactly one internal edge. For such a graph, $D(\Gamma_v) = \det(\Edg(\Gamma_v))$ is canonically isomorphic to $\bfk$ placed in degree $0$. The groupoid $\Lev(T)$ therefore acts trivially on $D(T,\lambda)$ which is canonically isomorphic to $\bfk$, so (\[Jeptha\]) leads to $$\label{Lisegacev}
\Free(D)(\GAmma)_0 \cong \Span(\gTr_0(\GAmma)),$$ in which each $T \in \gTr_0(\GAmma)$ corresponds to a vertex of the polytope $K$ associated to ${\mathcal A}(\H_\Gamma)$ and therefore represents a cycle that linearly generates $H_0(\minGrc)$. We will show that .
Under isomorphism (\[Lisegacev\]), each $T$ is an operadic composition of graph trees in $\gTr_0^1(\GAmma)$, i.e. graph trees whose underlying tree has one vertex which is decorated by a graph with one internal edge. By (\[Obehnu\_to\_dnes?\]), $\rho(S) = 1
\in \bbk$ for $S\in \gTr_0^1(\GAmma)$. Since all operadic compositions in $\termGrc$ are the identities $\id : \bbk \ot
\bbk \to \bbk$, $\rho(T) = 1$ for the composite $T$ as well. This finishes the proof of Theorem \[Woy-Woy\].
Other cases {#Mourek a Terezka}
===========
As diagram in (46) of [@Sydney] teaches us, many operadic categories of interest are obtained from the basic category $\Grc$ of directed connected graphs by iterated discrete operadic fibrations or opfibrations. This is in particular true for the category $\ggGrc$ of genus-graded graphs, the category $\Tr$ of trees, and the category $\Whe$ of wheeled graphs; they all are discrete operadic opfibrations over $\Grc$. Moreover, the inclusion $\RTr \hookrightarrow \Grc$ of the operadic category of rooted trees is a discrete operadic fibration with finite fibers. Corollary \[grantova\_zprava\] of Subsection \[Dnes\_hori\_bus.\] below states that the restrictions along discrete operadic opfibrations or fibrations with finite fibers preserve minimal models of the terminal operads. Therefore the minimal models of the terminal operads in the above mentioned categories are suitable restrictions of the minimal model $\minGrc$ of the terminal $\Grc$-operad constructed in Section \[hadrova\_panenka\]. We close this section by describing the minimal model of the terminal operad in the category $\SRTr$ of strongly rooted trees.
Operadic (op)fibrations and minimal models {#Dnes_hori_bus.}
------------------------------------------
The following material uses the terminology of [@Sydney]. All operadic categories in this subsection will be factorizable, graded, and such that all quasibijections are invertible, the blow up and unique fiber axioms are fulfilled, and a morphism is an isomorphisms if it is of grade $0$. These assumptions are fulfilled by all operadic categories discussed in the present paper.
Assume that $\ttO$ is such an operadic category. As argued in [@Sydney Section 10], one has the natural forgetful functor $\zap_\ttO : \OperV\ttO \to\CollectV\ttO$ from the category of $1$-connected strictly extended unital Markl’s $\ttO$-operads with values in a symmetric monoidal category $\ttV$ to the category of $1$-connected $\ttO$-collections in $\ttV$. Its left adjoint $\Free_\ttO : \CollectV\ttO \to
\OperV\ttO$ is the free operad functor.
Each strict operadic functor $p : \ttO \to \ttP$ induces the restriction $p^*: \OperV\ttP \to \OperV\ttO$ acting on objects by the formula $$\label{Zitra mam treti prednasku.}
p^*(\oP)(t) := \oP(p(t)), \ \oP \in \OperV\ttP , \ t \in \ttO.$$ The restriction $p^*$ may or may not have a right adjoint $p_*: \OperV\ttO \to \OperV\ttP$ and even if if it exists its form may not be simple unless $p$ has some special properties.
Recall the following general categorical definition. Assume we are given a commutative diagram of right adjoints $$\label{lrbc}
\xymatrix@C=3em@R=1.5em{\ttA
\ar[dd]^{u^*}
& &
\ar[dd]_{v^*}\ttB \ar[ll]_{p^*}
\\
& &
\\
\ttC%\ar@/_1em/[rr]_{p_*}
& & \ttD
\ar[ll]_{q^*}
}$$ in which $p^*$ and $q^*$ are also left adjoints. These functors can be organized into the following diagram of adjunctions $$\label{vseadj}
\xymatrix@C=4em{\ttA\ar@/_1.5em/[rr]_{p_*} \ar@/^1.5em/[rr]^{p_!}\ar@/^1em/[dd]^{u^*} & &
\ar@/_1em/[dd]_{v^*}\ttB \ar[ll]_{p^* \hspace{0.5mm} \perp}^{\perp} \\
\dashv & &\vdash
\\
\ar@/^1em/[uu]^{u_!}
\ttC\ar@/_1.5em/[rr]_{q_*} \ar@/^1.5em/[rr]^{q_!}& & \ttD
\ar[ll]_{q^* \hspace{0.5mm} \perp}^{\perp}
\ar@/_1em/[uu]_{v_!}
}$$ The square (\[lrbc\]) is called [*right Beck-Chevalley square*]{} if the following composite $$u_! q^* \to u_! q^* v^* v_! =
u_! u^* p^* v_! \to p^* v_!$$ is an isomorphism. Symmetrically, (\[lrbc\]) is [*a left Beck-Chevalley square*]{} if the composite $$q_!u^* \to q_! u^*p^* p_! =
q_! q^* v^* p_! \to v^*p_!$$ is an isomorphism.
\[left and right BC\] The following two conditions are equivalent:
- the mate $q_* u^* \leftarrow q_* u^* p^*p_*
= q_* q^* v^* p_* \leftarrow
v^* p_*$ is an isomorphism and
- the square (\[lrbc\]) is a right Beck-Chevalley square.
If $p_!$ is also a right adjoint to $p^*$ (that is, $p_!\cong p_*$) and $q_!$ is a right adjoint to $q^*$ then (\[lrbc\]) is a right Beck-Chevalley square if and only if it is a left Beck-Chevalley square.
Condition (i) just says that the right adjoints commute up to isomorphism. It follows that the left adjoint commute up to isomorphism as well, which is the right Beck-Chevalley condition (ii). The converse is clearly true as well.
If $p_!$ is also a right adjoint to $p^*$ and $q_!$ is a right adjoint to $q^*$ then obviously the left Beck-Chevalley condition is again about commutation of right adjoints, hence their left adjoints commute and the right Beck-Chevalley condition holds. The inverse implication is similar.
In the following proposition, is the restriction functor defined by (\[Zitra mam treti prednasku.\]) and $p_0^*: \CollectV\ttP \to \CollectV\ttO$ is the obvious similar restriction between the categories of collections.
\[Koronavirus se siri.\] The square $$\label{rightBC}
\xymatrix{\OperV\ttO
\ar[dd]^{\zap_\ttO}
& &
\ar[dd]_{\zap_\ttP}\OperV\ttP \ar[ll]_{p^*}
\\
& &
\\
\CollectV\ttO%\ar@/_1em/[rr]_{p_*}
& & \CollectV\ttP
\ar[ll]_{p_0^*}
}$$ is a right Beck-Chevalley square provided any of the two following conditions hold:
1. $p$ is a discrete operadic opfibration and $\ttV$ a cocomplete symmetric monoidal category;
2. $p$ is a discrete operadic fibration with finite fibers and $\ttV$ an additive cocomplete symmetric monoidal category.
The right adjoint $(p_0)_*: \CollectV\ttO \to
\CollectV\ttP$ to the restriction $p_0^*: \CollectV\ttP \to \CollectV\ttO$ is given on objects by
$$\label{Jolom}
(p_0)_*(E)(T) := \prod_{p(t) = T} E(t), \ E \in \CollectV{\ttO}, \
T \in \ttP.$$
Assume that $p : \ttO \to \ttP$ is a discrete operadic opfibration. By dualizing [@duodel Theorem 2.4] one verifies that the right adjoint is defined on objects by $$\label{dnes_moje_posledni_prednaska_v_MSRI}
p_*(\oO)(T) := \prod_{p(t) = T} \oO(t), \ \oO \in \Oper{\ttO} \
T \in \tt P.$$
Comparing (\[Jolom\]) with (\[dnes\_moje\_posledni\_prednaska\_v\_MSRI\]) we see that $(p_0)_*\, \zap_\ttO = \zap_\ttP p_*$, which is condition (i) of Lemma \[left and right BC\]. Thus (\[rightBC\]) is right Beck-Chevalley by the same lemma. This finishes the proof of the case of a discrete opfibration.
Let us assume that $p : \ttO \to \ttP$ is a discrete operadic fibration with finite fibers. We want to verify the assumptions of the second part of Lemma \[left and right BC\], i.e. to check that $(p_0)_!$ is a right adjoint to $p_0^*$ and that $p_!$ is a right adjoint to $p^*$.
It is clear that $(p_0)_!$ is for an arbitrary $p: \ttO \to \ttP$ given on objects by the formula $$(p_0)_!(E)(T) := \bigoplus_{p(t) = T} E(t), \ E \in \CollectV{\ttO},\
T \in \ttP.$$ Since $V$ is additive and $p$ has finite fibers, this functor coincides with the right adjoint $(p_0)_*$ described in (\[Jolom\]). On the other hand, [@duodel Theorem 2.4] gives the following formula for the underlying collection of $p_!(\oO)$: $$p_!(\oO)(T) := \bigoplus_{p(t) = T} \oO(t), \ \oO \in \OperV{\ttO}, \
T \in \ttP.$$ It is not hard to see, using the additivity of $V$ and the finiteness of the fibers of $p$, that this formula describes also a right adjoint to $p^*$, which completes the proof for operadic fibrations.
In the rest of this section, the coefficient category $\ttV$ will be that of differential graded vector spaces. It clearly satisfies all assumptions required in Proposition \[Koronavirus se siri.\].
Assume that (\[rightBC\]) is a right Beck-Chevalley square and $\rho : \minP \to \termP$ is the minimal model of the terminal $\ttP$-operad $\termP$. Then $$\xymatrix@1{
\minO := p^*(\minP)\ \ar[r]^(.55){p^*(\rho)}& \ p^*(\termP) = \termO
}$$ is the minimal model of the terminal $\ttO$-operad $\termO$.
It is clear that $p^*(\termP) = \termO$. Let $\minP =
(\Free_\ttP(E_\ttP),\pa_\ttP)$. Diagram (\[rightBC\]) is, by definition, a right Beck-Chevalley square if $p^*\, \Free_\ttP \cong \Free_\ttO\, p_0^*$. In particular, $$p^*(\Free_\ttP(E_\ttP)) \cong \Free_\ttO(p_0^*(E_\ttP)),$$ thus $p^*(\minP)$ is the free operad generated by the collection $E_\ttO := p_0^*(E_\ttP)$. It is easy to verify that $p^*$ brings derivations to derivations and differentials to differentials. We therefore conclude that $$p^*(\minP) \cong (\Free_\ttO(E_\ttO),\pa_\ttO),$$ where the minimality of $\pa_\ttO$ can also be established easily.
It remains to prove that $p^*(\rho)$ induces a component-wise isomorphism of homology. This however follows immediately from the definition of the restriction functor requiring that $$p^*(\rho)(t) = \rho(p(t)) : \minP(p(t)) \to \termP(p(t)) = \bbk, \ t \in \ttO,$$ where $\rho(p(t))$ is a homology isomorphism since $\rho : \minP \to
\termP$ is the minimal model of $\termP$ by assumption.
\[grantova\_zprava\] Let $p: \ttO \to \ttP$ be either a discrete operadic opfibration, or a discrete operadic fibration with finite fibers, and $\rho : \minP \to \termP$ the minimal model of the terminal $\ttP$-operad. Then $$\xymatrix@1{
\minO := p^*(\minP)\ \ar[r]^(.55){p^*(\rho)}& \ p^*(\termP) = \termO
}$$ is the minimal model of the terminal $\ttO$-operad.
The assumptions and conclusion of Corollary \[grantova\_zprava\] were verified in the context of operadic categories related to permutads in [@perm].
Minimal model for $\termggGrc$. {#Podari se mi koupit to auto?}
-------------------------------
The operadic category $\ggGrc$ consists of graphs $\Gamma \in \Grc$ equipped with a [*genus grading*]{}, which is a non-negative integer $g(v) \in \bbN$ specified for each $v \in \Vert(\Gamma)$. The genus of the entire graph $\Gamma$ is defined by $$g(\Gamma):=
\sum_{v \in \Vert(\Gamma)} g(v) + \dim(H^1(|\Gamma|; {\mathbb Z})),$$ where $|\Gamma|$ is the obvious geometric realization of $\Gamma$. As shown in [@Sydney Section 12], algebras for $\termggGrc$ are modular operads.
Assume that $\Gamma \in \ggGrc$ and that $T \in \Tr(\Gamma)$ is a graph-tree. Then there exists a unique genus grading of each of the graphs $\Gamma_v$ decorating the vertices of $T$ subject, along with the compatibilities required in Subsection \[Dnes\_je\_Michalova\_oslava.\], also to
[*Genus compatibility.*]{} Let $e$ be an internal edge of $T$ pointing from the vertex labelled by $\Gamma_u$ to the vertex labelled by $\Gamma_v$. By Compatibility 1, $e$ is also (the label of) a vertex of $\Gamma_v$. With this convention in mind we require that $$g(e) = g(\Gamma_u).$$ In words, the vertex of $\Gamma_v$ to which $\Gamma_u$ is contracted bears the genus $ g(\Gamma_u)$.
The statement can be verified directly, which we leave as an exercise to the reader. It can also be established by inductive applications of
\[Musim psat grantovou zpravu\] Let $\phi : \Gamma \to \Gamma''$ be an elementary morphism in $\Grc$ with fiber $\Gamma'$, in shorthand $$\label{V sobotu letim do Melbourne}
\Gamma' \fib \Gamma \stackrel\phi\longrightarrow \Gamma''.$$ Assume moreover that $\Gamma$ bears a genus grading. Then there are unique genus gradings of $\Gamma'$ and $\Gamma''$ such that (\[V sobotu letim do Melbourne\]) becomes a diagram, in $\ggGrc$, of an elementary map and its fiber.
A consequence of the fact that the obvious projection $p: \ggGrc \to \Grc$ is a discrete operadic opfibration, though it can also be verified directly.
For $\Gamma \in \ggGrc$ having at least one internal edge and for a $1$-connected $\ggGrc$-collection $E$, the right hand side of $$\label{Za tyden do SaFra}
\Freegg(E)(\GAmma) :=
\bigoplus_{T \in \gTr(\GAmma)} \ \colim{\lambda \in
\Lev(T)} E(T,\lambda),$$ makes sense because, as explained above, each of the graphs $\Gamma_i$, $1 \leq i \leq k$, in (\[Krtek\_na\_mne\_kouka.\]) where $E(T,\lambda)$ was defined, bears a unique genus grading induced by the genus grading of $\Gamma$.
Let $p: \ggGrc \to \Grc$ be as before the canonical projection that forgets the genus grading, and $p^*: \Oper\Grc \to \Oper\ggGrc$ resp. $p^*: \Collect\Grc \to
\Collect\ggGrc$ the induced restrictions. The values of the $\ggGrc$-collection $\Dgg \in
\Collect\ggGrc$ given by $$\Dgg(\Gamma) := \det(\Gamma), \ \Gamma \in \ggGrc,$$ do not depend on the genus grading, thus $\Dgg = p^*(D)$, where $D \in \Collect\Grc$ is as in (\[Flicek\]). For the same reasons $$\Freegg(\Dgg) = p^* \Free (D),$$ so, since $p: \ggGrc \to \Grc$ is a discrete operadic opfibration, $\Freegg(\Dgg)$ defined by (\[Za tyden do SaFra\]) with $E = \Dgg$ represents the free $\ggGrc$-operad on $\Dgg$ by Proposition \[Koronavirus se siri.\]. The differential $\pa$ on $\Freegg(\Dgg)$ is given by an obvious analog of (\[Posledni patek v Sydney\]).
As expected, we define $\rho : \Freegg(\Dgg) \to \termggGrc$ as the unique map of $\ggGrc$-operads whose restriction $\rho|_{\Dgg(\Gamma)}$ is, for $\Gamma \in \ggGrc$, given by a modification of (\[Obehnu\_to\_dnes?\]), namely by $$\rho|_{D(\Gamma)} :=
\begin{cases}
\id_\bbk : D(\Gamma) = \bbk \to \bbk =
\termggGrc(\Gamma), &\hbox {if $|\Edg(\Gamma)| = 1$,
while}
\\
0,& \hbox {if $|\Edg(\Gamma)| \geq 2$.}
\end{cases}$$
\[neco na jazyku\] The object $\minggGrc = (\Freegg(\Dgg),\pa) \stackrel\rho\longrightarrow
(\termggGrc,\pa =0)$ is a minimal model of the terminal $\ggGrc$-operad $\termggGrc$.
A consequence of Corollary \[grantova\_zprava\], though the acyclicity of $\minggGrc$ in positive dimensions follows directly from the acyclicity of $\minGrc$ proven in Subsection \[Posledni tyden.\], thanks to the isomorphism $$\minggGrc(\Gamma) \cong \minGrc(\widehat \Gamma),\ \Gamma \in \ggGrc,$$ of dg vector spaces, where $\widehat \Gamma \in
\Grc$ is $\Gamma$ stripped of the genus grading.
Minimal model for $\termTr$ {#Dnes prvni vylet na kole}
---------------------------
Let $\Tr \subset \Grc$ be the full subcategory of contractible, i.e. simply connected graphs. Algebras over the terminal $\Tr$-operad $\termTr$ are cyclic operads. Although it was not stated in [@Sydney], the inclusion $p: \Tr
\hookrightarrow \Grc$ is a discrete operadic opfibration as well, we thus are still in the comfortable situation of Subsection \[Dnes\_hori\_bus.\]. Also an analog of Lemma \[Musim psat grantovou zpravu\] is obvious: if $\Gamma \in \Grc$ is contractible, then $\Gamma'$, as a connected subgraph of $\Gamma$, is contractible too, and so is the quotient $\Gamma''$. The minimal model for $\termTr$ can therefore be constructed by mimicking the methods of Subsection \[Podari se mi koupit to auto?\], so we will be telegraphic.
For a graph $\Gamma \in \Tr$ having at least one internal edge and a $1$-connected $\Tr$-collection $E$, the expression in the right hand side of $$\label{Jaruska jede za M1.}
\FreeTr(E)(\GAmma) :=
\bigoplus_{T \in \gTr(\GAmma)} \ \colim{\lambda \in
\Lev(T)} E(T,\lambda)$$ makes sense, since each of the graphs $\Rada \Gamma1k$ in the definition (\[Krtek\_na\_mne\_kouka.\]) of $E(T,\lambda)$ is connected. Let $\DTr \in \Collect\Tr$ be the collection with $$\DTr(\Gamma) := \det(\Gamma), \ \Gamma \in \Tr.$$ For $\DTr$ in place of $E$, formula (\[Jaruska jede za M1.\]) describes the pieces of the free operad $\FreeTr(\DTr)$. The differential $\pa$ on $\FreeTr(\DTr)$ is given by an obvious modification of formula (\[Posledni patek v Sydney\]). Also the definition of $\rho : \FreeTr(\DTr) \to \termTr$ is the expected one. We have
\[Druhou panenku jsem nechal v Praze.\] The object $\minTr = (\FreeTr(\DTr),\pa) \stackrel\rho\longrightarrow (\termTr,\pa
=0)$ is a minimal model of the terminal $\Tr$-operad $\termTr$.
Verbatim modification of the proof of Theorem \[neco na jazyku\].
Minimal model for $\termWhe$. {#Poslu Jarce obrazky kyticek.}
-----------------------------
We say, following [@Sydney Example 2.19], that a directed connected graph $\Gamma \in \Gr$ is [*oriented*]{} if
- each internal edge if $\Gamma$ is oriented, meaning that one of the half-edges forming this edge is marked as the input one, and the other as the output, and
- also the legs of $\Gamma$ are marked as either input or output ones.
Oriented directed graphs form an operadic category $\Whe$. Algebras for the terminal $\Whe$-operad $\termWhe$ are wheeled properads introduced in [@mms]. As noted in Example 2.19 loc. cit., the functor $p : \Whe\to\Grc$ that forgets the orientation is a discrete operadic opfibration, thus the constructions of the previous two subsections, including the description of the minimal model for $\termWhe$, translate verbatim. We leave the details to the reader.
Minimal model for $\termRTr$ {#Prvni tyden v Berkeley konci.}
----------------------------
We will call the leg of $\Gamma \in \Tr$, minimal in the global order, the [*root*]{} of $\Gamma$. Let us orient edges of $\Gamma \in \Tr$ so that they point to the root. We say that $\Gamma$ is [*rooted*]{} if the outgoing half-edge of each vertex is the smallest in the local order at that vertex. In [@Sydney] we considered the full subcategory $\RTr$ of $\Tr$ consisting of rooted trees and identified algebras over the terminal $\RTr$ operad $\termRTr$ with ordinary, classical operads. The inclusion $p: \RTr \hookrightarrow \Tr$ is, however, a discrete operadic [*fibration*]{}, not an opfibration, . Nevertheless, the fibers of $p$ are finite, being either empty or an one-point set, thus Corollary \[grantova\_zprava\] applies, so we can construct an explicit minimal model for $\termRTr$ by obvious modifications of the methods used in the previous subsections.
\[Myska a Tucinek\] Figure \[Vcera jsem si koupil kolo.\] illustrates the failure of Lemma \[Musim psat grantovou zpravu\] for $\Tr$ in place of $\Grc$ and $\RTr$ in place of $\ggGrc$. The graph $\Gamma$ in that figure has vertices (indexed by) $\{1,2,3\}$ and half-edges $\{1,2,3,4,5,6\}$, the graph $\Gamma''$ has vertices $\{1,2\}$ and half-edges $\{1,2,3,4\}$. The map $\phi : \Gamma \to \Gamma''$ sends the vertices $1$ and $3$ of $\Gamma$ to the vertex $1$ (the fat one) of $\Gamma''$, and the vertex $2$ of $\Gamma$ to the vertex of $\Gamma''$ with the same label. The labels in the circles indicate the global orders. While $\Gamma$ is rooted, $\Gamma''$ is not, although $\phi$ is even a canonical contraction.
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Minimal model for $\termSRTr$ {#Ze by byla monografie uz konecne dokoncena?}
-----------------------------
It turns out that the operadic category $\RTr$ contains much smaller subcategory which still captures the classical operads in the same way $\RTr$ does. It is defined as follows. We say that a rooted tree $\Gamma \in \RTr$ is [*strongly rooted*]{}, if the order of its set $V$ of vertices is compatible with the rooted structure. By this we mean that, if $v\in V$ lies on the path connecting $u \in V$ with the root, then $v < u$ in $V$. We denote by $\SRTr \subset \RTr$ the full subcategory of strongly rooted trees. It is easy to show that all fibers of a map $\phi :
\Gamma' \to \Gamma''$ between strongly rooted trees are strongly rooted, and also that all rooted corollas are clearly strongly rooted. Consequently, $\SRTr$ is an operadic category.
We claim that algebras over the terminal $\SRTr$-operad $\termSRTr$ are the same as $\termRTr$-algebras, i.e. that they are ordinary operads. This might sound surprising, since $\SRTr$ has less objects than $\RTr$, therefore $\termSRTr$-algebras have less operations than $\termRTr$-algebras. Each operation of a $\termRTr$-algebra can however be obtained from an operation of a $\termSRTr$-algebra via certain permutation of inputs, since each rooted tree is isomorphic with a strongly rooted tree, by a local isomorphism.
Consider the rooted trees in Figure \[Zitra se podivam na Golden Bridge.\]. The left one belongs to $\SRTr$ and represents the operation
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$${\mathcal O}_{\Gamma'} : P(3) \ot P(2) \longrightarrow P(4)$$ given by ${\mathcal O}_{\Gamma'}(x \ot y) = x \circ_2 y$, where $\circ_2$ is the standard $\circ$-operation in a unital operad $P$, while $${\mathcal O}_{\Gamma''} : P(2) \ot P(3) \longrightarrow P(4)$$ is given by ${\mathcal O}_{\Gamma''}(a\ot b) = b \circ_2 a$. Thus ${\mathcal O}_{\Gamma''} = {\mathcal O}_{\Gamma'} \circ \sigma$ with $\sigma \in \Sigma_2$ the transposition.
Neither the inclusion $\SRTr \hookrightarrow \RTr$, nor the composite $\SRTr \hookrightarrow \RTr \hookrightarrow \Tr$ is a fibration or opfibration, but the category $\SRTr$ is, unlike $\RTr$, closed under canonical contractions. It can indeed be easily verified that, if $\Gamma' \in \SRTr$ and if $\pi :
\Gamma' \to \Gamma''$ is the canonical contraction, then $\Gamma''$ and also the fiber of $\pi$ belongs to $\SRTr$. The methods developed in Subsection \[Dnes\_je\_Michalova\_oslava.\] can therefore be used with $\SRTr$ in place of $\Grc$. Namely, each tower (\[t1\]) in $\SRTr$ can be brought into the canonical form where $\ell = \id_\GAmma$ and all $\tau$’s are canonical contractions, and as such be represented by a graph tree in $\gTr(\GAmma)$. The right hand side of formula (\[Zbijecky duni.\]) then, for $\Gamma \in \SRTr$ and $E \in
\Collect\SRTr$, expresses the component of the free $\SRTr$-operad $\FreeSRTr(E)$.
Our description of a minimal model for $\termSRTr$ is the expected one. We define the collection $\DSRTr \in \Collect\SRTr$ by $$\DSRTr(\Gamma) := \det(\Gamma), \ \Gamma \in \SRTr,$$ and the differential $\pa$ on the free operad $\FreeSRTr(\DSRTr)$ whose components are $$\label{Jaruska}
\FreeSRTr(\DSRTr)(\GAmma) :=
\bigoplus_{T \in \gTr(\GAmma)} \ \colim{\lambda \in
\Lev(T)} \DSRTr(T,\lambda)$$ by the verbatim version of formula (\[Posledni patek v Sydney\]). The morphism $\rho : \FreeSRTr(\DSRTr) \to \termSRTr$ is given by an obvious analog of (\[Obehnu\_to\_dnes?\]). One has
The object $\minSRTr := (\FreeSRTr(\DSRTr),\pa) \stackrel\rho\to (\termSRTr,\pa
=0)$ is a minimal model of the terminal $\SRTr$-operad.
The only possibly nontrivial issue is the acyclicity $\minSRTr$ in positive dimensions. Comparing the formula $$\Free(D)(\GAmma) :=
\bigoplus_{T \in \gTr(\GAmma)} \ \colim{\lambda \in
\Lev(T)} D(T,\lambda)$$ defining the component of the minimal model $\minGrc$ for $\termGrc$ with (\[Jaruska\]) we notice the [*equality*]{} $$(\FreeSRTr(\DSRTr)(\Gamma),\pa) = (\Free(D)(\Gamma),\pa)$$ for $\Gamma \in \SRTr$. In other words $$\minSRTr(\Gamma) = \minGrc(\Gamma), \ \hbox { for }
\Gamma \in \SRTr \subset \Grc.$$ The acyclicity of $\minSRTr$ thus follows from the acyclicity of $\minGrc$ established in the proof of Theorem \[Woy-Woy\].
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[^1]: The first author acknowledges the financial support of Praemium Academiæ of M. Markl during his scientific visit of Prague.
[^2]: The second author was supported by grant GA ČR 18-07776S, Praemium Academiæ and RVO: 67985840.
[^3]: The third author was supported by Praemium Academiæ of M. Markl and RVO: 67985840.
|
---
abstract: |
This work is motivated by the mostly unsolved task of parsing biological images with multiple overlapping articulated model organisms (such as worms or larvae). We present a general approach that separates the two main challenges associated with such data, individual object shape estimation and object groups disentangling. At the core of the approach is a deep feed-forward *singling-out network (SON)* that is trained to map each local patch to a vectorial descriptor that is sensitive to the characteristics (e.g. shape) of a central object, while being invariant to the variability of all other surrounding elements. Given a SON, a local image patch can be matched to a gallery of isolated elements using their SON-descriptors, thus producing a hypothesis about the shape of the central element in that patch. The image-level optimization based on integer programming can then pick a subset of the hypotheses to explain (parse) the whole image and disentangle groups of organisms.
While sharing many similarities with existing “analysis-by-synthesis” approaches, our method avoids the need for stochastic search in the high-dimensional configuration space and numerous rendering operations at test-time. We show that our approach can parse microscopy images of three popular model organisms (the C.Elegans roundworms, the Drosophila larvae, and the E. Coli bacteria) even under significant crowding and overlaps between organisms. We speculate that the overall approach is applicable to a wider class of image parsing problems concerned with crowded articulated objects, for which rendering training images is possible.
author:
- |
Victor Yurchenko\
Skoltech\
Russia\
[victor.yurchenko@skoltech.ru]{}
- |
Victor Lempitsky\
Skoltech\
Russia\
[lempitsky@skoltech.ru]{}
title: |
Parsing Images of Overlapping Organisms with\
Deep Singling-Out Networks
---
|
---
abstract: 'A system is offered for imitation resistant transmitting of encrypted information in wireless communication networks on the basis of redundant residue polynomial codes. The particular feature of this solution is complexing of methods for cryptographic protection of information and multi-character codes that correct errors, and the resulting structures (crypt-code structures) ensure stable functioning of the information protection system in the conditions simulating the activity of the adversary. Such approach also makes it possible to create multi-dimensional ‘‘crypt-code structures’’ to conduct multi-level monitoring and veracious restoration of distorted encrypted information. The use of authentication codes as a means of one of the levels to detect erroneous blocks in the ciphertext in combination with the redundant residue polynomial codes of deductions makes it possible to decrease the introduced redundancy and find distorted blocks of the ciphertext to restore them.'
author:
- Dmitry Samoylenko
- Mikhail Eremeev
- Oleg Finko
- Sergey Dichenko
title: 'Protection of Information from Imitation on the Basis of Crypt-Code Structures'
---
Introduction
============
The drawback of many modern ciphers used in wireless communication networks is the unresolved problem of complex balanced support of traditional requirements: cryptographic security, imitation resistance and noise stability. It is paradoxical that the existing ciphers have to be resistant to random interference, including the effect of errors multiplication [@Ferg1; @Men2; @Bur3]. However, such regimes of encrypting as cipher feedback mode are not only the exception, but, on the contrary, initiate the process of error multiplication. The existing means to withstand imitated actions of the intruder, which are based on forming authentication codes and the hash-code – only perform the indicator function to determine conformity between the transmitted and the received information [@Ferg1; @Men2; @Chr4], and does not allow restoring the distorted data.
In some works [@MC5; @Nid6; @SAm7; @SAm8] an attempt was made to create the so-called ‘‘noise stability ciphers’’. However, these works only propose partial solutions to the problem (solving only particular types of errors ‘‘insertion’’, ‘‘falling out’’ or ‘‘erasing’’ symbols of the ciphertext etc.), or insufficient knowledge of these ciphers, which does not allow their practical use.
Imitation Resistant Transmitting of Encrypted Information on the Basis of Crypt-Code Structures
===============================================================================================
The current strict functional distinction only expects the ciphers to solve the tasks to ensure the required cryptographic security and imitation resistance, while methods of interference resistant coding is expected to ensure noise stability. Such distinction between the essentially inter-related methods to process information to solve inter-related tasks will decrease the usability of the system to function in the conditions of destructive actions of the adversary, the purpose of which is to try to impose on the receiver any (different from the transmitted) message (imposition at random). At the same time, if these methods are combined, we can obtain both new information ‘‘structures’’ – crypt-code structures, and a new capability of the system for protected processing of information – *imitation resistance* [@Petl8], which we consider to be the ability of the system for *restoration* of veracious encrypted data in the conditions of simulated actions of the intruder, as well as unintentional interference.
The synthesis of crypt-code structures is based on the procedure of complexing of block cypher systems and multi-character correcting codes [@Fin8; @Fin9; @SAM8]. In one of the variants to implement crypt-code structures as a multi-character correcting code, *redundant residue polynomial codes* (RRPC) can be used, whose mathematical means is based on fundamental provisions of the Chinese remainder theorem for polynomials (CRT) [@Boss9; @Mandel10; @YuJ-H11].
Chinese Remainder Theorem for Polynomials and Redundant Residue Polynomial Codes
--------------------------------------------------------------------------------
Let $F[z]$ be ring of polynomials over some finite field $\bbbf_q,$ $q=p^s.$ For some integer $k>1,$ let $m_1(z),\, m_2(z),\, \ldots,\, m_k(z)\in~F[z]$ be relatively prime polynomials sorted by the increasing degrees, i.e. $\deg m_1(z) \leq \deg m_2(z) \leq \ldots \leq \deg m_k(z)$, where $\deg m_i(z)$ is the degree of the polynomial. Let us assume that $P(z) =\prod_{i=1}^{k}m_i(z).$ Then the presentation of $\varphi$ will establish mutually univocal conformity between polynomials $a(z),$ that do not have a higher degree than $P(z)$ $\bigl(\deg a(z) < \deg P(z)\bigr)$, and the sets of residues according to the above-described system of bases of polynomials (modules): $$\begin{gathered}
\varphi : {{}^{F[z]}\!/_{(P(z))}} \rightarrow {{}^{F[z]}\!/_{(m_1(z))}} \times \ldots \times {{}^{F[z]}\!/_{(m_k(z))}}:\\
: a(z) \mapsto \varphi\bigl(a(z)\bigr) := \bigl(\varphi_1\bigl(a(z)\bigr), \varphi_2\bigl(a(z)\bigr), \ldots, \varphi_k\bigl(a(z)\bigr)\bigr),\end{gathered}$$ where $\varphi_i\bigl(a(z)\bigr) := a(z)\mod m_i(z)$ $(i = 1,\, 2,\, \ldots,\, k)$.
In accordance with the CRT, there is a reverse transformation $\varphi^{-1}$, that makes it possible to transfer the set of residues by the system of bases of polynomials to the positional representation: $$\begin{gathered}
\label{1}
\varphi^{-1} : {{}^{F[z]}\!/_{(m_1(z))}} \times \ldots \times {{}^{F[z]}\!/_{(m_k(z))}} \rightarrow {{}^{F[z]}\!/_{(P(z))}}:\\
: \bigl(c_1(z), \ldots, c_k(z)\bigr) \mapsto a(z) = \sum_{i=1}^{k}c_{i}(z)B_{i}(z)~~{\rm modd}~\bigl(p,~P(z)\bigr),\end{gathered}$$ where $B_i(z) = k_i(z)P_i(z)$ are polynomial orthogonal bases, $k_i(z) = P^{-1}_i(z)\mod m_i(z),$ $P_i(z) =m_1(z)m_2(z) \ldots m_{i-1}(z)m_{i+1}(z) \ldots m_k(z)$ $(i = 1,\, 2,\, \ldots,\, k).$
Let us also introduce, in addition to the existing number $k,$ the number $r$ of redundant bases of polynomials while observing the condition of sortednes: $$\begin{aligned}
\label{2}
\deg m_1(z)\leq \ldots \leq \deg m_k(z) \leq \deg m_{k+1}(z) \leq \ldots \leq \deg m_{k+r}(z),\end{aligned}$$ and $$\begin{aligned}
\label{2.1}
\gcd\bigl(m_i(z),\, m_j(z)\bigr) = 1,\end{aligned}$$ for $i \neq j;$ $ i, j = 1,\, 2,\, \ldots,\, k+r$, then we obtain the expanded RRPC — an array of the kind: $$\begin{aligned}
\label{3}
C := \bigl(c_1(z),\, \ldots,\, c_k(z),\, c_{k+1}(z),\, \ldots,\, c_n(z)\bigr) : c_i(z) \equiv a(z)\mod m_i(z),\end{aligned}$$ where $n = k + r$, $c_i(z) \equiv a(z)\mod m_i(z)$ $(i = 1,\, 2,\, \ldots,\, n)$, $a(z) \in {{}^{F[z]}\!/_{(P(z))}}$.
Elements of the code $c_i(z)$ will be called symbols, each of which is the essence of polynomials from the quotient ring of polynomials over the module $m_i(z) \in {{}^{F[z]}\!/_{(m_i(z))}}.$ At the same time, if $a(z) \not\in {{}^{F[z]}\!/_{(P(z))}},$ then it is considered that this combination contains an error. Therefore, the location of the polynomial $a(z)$ makes it possible to establish if the code combination $a(z) = \bigl(c_1(z),\, \ldots,\, c_k(z),\, c_{k+1}(z),\, \ldots,\, c_n(z)\bigr)$ is allowed or it contains erroneous symbols.
Crypt-Code Structures on Based RRPC
-----------------------------------
Now, the sender-generated message $M$ shall be encrypted and split into blocks of the fixed length $M=\{M_1\|M_2\|\ldots\|M_k\},$ where ‘‘$\|$’’ is the operation of concatenation. Introducing a formal variable $z$ number $i$ block of the open text $M_i,$ we will represent in the polynomial form: $$\begin{aligned}
M_i(z) = \sum_{j=0}^{s-1}m_j^{(i)}z^j = m_{s-1}^{(i)}z^{s-1} + \ldots+m_{1}^{(i)}z+ m_0^{(i)},\end{aligned}$$ where $m_j^{(i)} \in \{0,\, 1\}\ \ \ (i=1,\, 2,\, \ldots,\, k;\ \ \ j = s-1,\, s-2,\, \ldots,\, 0).$
In order to obtain the sequence of blocks of the ciphertext $\Omega_1(z),\, \Omega_2(z),\, \ldots\\\ldots,\, \Omega_k(z)$ we need to execute $k$ number of encrypting operations, and to obtain blocks of the open text $M_1(z),\, M_2(z),\, \ldots,\, M_k(z),$ we need to execute $k$ number of decrypting operations. The procedures of encrypting and decrypting correspond to the following presentations: $$\begin{cases}
\Omega_1(z)\rightarrow{ E_{\kappa_{{\rm e},\,1}}}~:~M_1(z),\\
\Omega_2(z)\rightarrow{ E_{\kappa_{{\rm e},\,2}}}~:~M_2(z),\\
\ldots\ \ldots\ \ldots\ \ldots\ \ldots\ \ldots\ \\
\Omega_k(z)\rightarrow{ E_{\kappa_{{\rm e},\,k}}}~:~M_k(z);\\
\end{cases}~~~
\begin{cases}
M_1(z)\rightarrow{ D_{\kappa_{{\rm d},\,1}}}~:~M_1(z),\\
M_2(z)\rightarrow{ D_{\kappa_{{\rm d},\,2}}}~:~M_2(z),\\
\ldots\ \ldots\ \ldots\ \ldots\ \ldots\ \ldots\ \\
M_k(z)\rightarrow{ D_{\kappa_{{\rm d},\,k}}}~:~M_k(z),\\
\end{cases}$$ where $\kappa_{{\rm e},\,i}, \kappa_{{\rm d},\,i}$ are keys (general case) for encrypting and decrypting $(i=1,\, 2,\, \ldots,\, k)$; if $\kappa_{{\rm e},\,i}= \kappa_{{\rm d},\,i}$ — the cryptosystem is symmetric, if $\kappa_{{\rm e},\,i} \neq\kappa_{{\rm d},i}$ — it is asymmetric.
We will express the adopted blocks of the ciphertext and blocks of the open text correspondingly as $\Omega_i^{*}(z)$ and $M_i^{*}(z)$ $(i=1,\, 2,\, \ldots,\, k)$, as they can contain distortions. The formed blocks of the ciphertext $\Omega_i(z)$ will be represented as the minimum residues (deductions) on the pairwise relatively prime polynomials (bases) $m_i(z).$ Here, $\deg \Omega_i(z)<\deg m_i(z).$ The set of blocks of the ciphertext $\Omega_1(z),\, \Omega_2(z),\, \ldots,\, \Omega_k(z)$ will be represented as a single super-block of elements of the RRPC by the system of bases-polynomials $m_1(z), m_2(z), \ldots, m_k(z).$ In accordance with CRT for the set array of polynomials $m_1(z),\, m_2(z),\, \ldots,\, m_k(z)$, that meet the condition that $\gcd\bigl(m_i(z),\, m_j(z)\bigr) = 1$, and polynomials $\Omega_1(z),\, \Omega_2(z),\, \ldots,\, \Omega_k(z)$, such that $\deg \Omega_i(z)<\deg m_i(z)$, the system of congruences $$\label{4}
\begin{cases}
\Omega(z)\equiv \Omega_1(z)\mod m_1(z),\\
\Omega(z)\equiv\Omega_2(z)\mod m_2(z),\\
\ldots\ \ldots\ \ldots\ \ldots\ \ldots\ \ldots\ \ldots\ \\
\Omega(z)\equiv\Omega_k(z)\mod m_k(z)
\end{cases}$$ has the only one solution $\Omega(z)$.
Then, we execute the operation of expansion (Base Expansion) of the RRPC by introducing $r$ of redundant bases-polynomials $m_{k+1}(z),\, m_{k+2}(z),\, \ldots\\\ldots, \, m_{k+r}(z)$ that meet the condition (\[2\]), (\[2.1\]) and obtaining in accordance with Eq. (\[3\]) redundant blocks of data (residues), which we will express as $\omega_{k+1}(z),\, \omega_{k+2}(z),\, \ldots,\, \omega_{n}(z)$ $(n=k+r)$. The combination of ‘‘informational’’ blocks of the ciphertext and redundant blocks of data form crypt-code structures identified as a code word of the expanded RRPC: $\bigl\{\Omega_1(z),\, \ldots,\, \Omega_k(z),\, \omega_{k+1}(z),\,\ldots,\,\omega_{n}(z)\bigr\}_{\text{RRPC}}$.
Here, we define a single error of the code word of RRPC as a random distortion of one of the blocks of the ciphertext; correspondingly the $b$-fold error is defined as a random distortion of $b$ blocks. At the same time, it is known that RRPC detects $b$ errors, if $r\geq b$, and will correct $b$ or less errors, if $2b\leq r$ [@Fin8; @Boss9; @Mandel10].
The adversary, who affects communication channels, intercepts the information or simulates false information. At the same time, in order to impose false, as applied to the system under consideration, the adversary has to intercept a set of information blocks of the ciphertext to detect the redundant blocks of data.
In order to eliminate the potential possibility that the adversary may impose false information, we need to ensure the ‘‘mathematical’’ gap of the procedure (uninterrupted function) of forming redundant elements of code words of the RRPC. Moreover, code words of RRPC have to be distributed randomly, i.e. uniform distribution of code words in the set array of the code has to be ensured. In order to achieve that, the formed sequence of redundant blocks of data $\omega_{j}(z)$ $(j=k+1,\, k+2,\, \ldots,\, n)$ undergoes the procedure of encrypting: $$\begin{cases}
\vartheta_{k+1}(z)\rightarrow{ E_{\kappa_{{\rm e},k+1}}}~:~\omega_{k+1}(z),\\
\vartheta_{k+2}(z)\rightarrow{ E_{\kappa_{{\rm e},k+2}}}~:~\omega_{k+2}(z),\\
\ldots\ \ldots\ \ldots\ \ldots\ \ldots\ \ldots\ \\
\vartheta_{n}(z)\rightarrow{ E_{\kappa_{{\rm e},n}}}~:~\omega_{n}(z),\\
\end{cases}$$ where $\kappa_{{\rm e},\,j}\ \ \ (j=k+1,\, k+2,\, \ldots,\, n)$ are the keys for encrypting.
The process of encrypting of redundant symbols of the code word of the RRPC executes transposition of elements of the vector $\bigl \{\omega_{k+1}(z),\, \omega_{k+2}(z),\, \ldots\\ \ldots,\, \omega_{n}(z)\bigl \}\in \mathcal{A}$ onto the formed elements of the vector of redundant encrypted symbols $\left\{\vartheta_{k+1}(z),\, \vartheta_{k+2}(z),\, \ldots,\, \vartheta_{n}(z)\right\}\in \mathcal{B}$, where $\mathcal{A}$ is the array of blocks of the ciphertext, $\mathcal{B}$ is a finite array.
The operation of transposition excludes the mutually univocal transformation and prevents the adversary from interfering on the basis of the intercepted informational super-block of the RRPC (the ‘‘informational’’ constituent) $\Omega_{i}(z)$ $(i=1,\, 2,\, \ldots,\, k)$ by forming a verification sequence $\omega_{j}(z)$ $(j=k+1,\, k+2,\, \ldots,\, n)$ for overdriving the protection mechanisms and inserting false information. At the same time, it is obvious that, for the adversary, the set of keys $\kappa_{{\rm e},\,j}$ and functions of encrypting $E_{i}(\bullet)$ of the vector of redundant blocks of data forms a certain array $\mathcal{X}$ of the transformation rules, out of whose many variants, the sender and the addressee will only use a certain one [@Chr4; @Sim12; @Zub2017].
We should also note the exclusive character of the operation of encrypting the sequence of redundant blocks of data, due to this, its implementation requires a special class of ciphers that do not alter the lengths of blocks of the ciphertext (endomorphic ones) and not creating distortions (like omissions, replacements or insertions) of symbols, for example, ciphers of permutation.
Imitation Resistant Transmitting of Encrypted Information on the Basis of Multidimensional Crypt-Code Structures
================================================================================================================
A particular feature of the above-described system is the necessity to introduce redundant encrypted information in accordance with the RRPC characteristics and specified requirements to the repetition factor of the detected or corrected distortions in the sent data. The theory of coding tells us of solutions to obtain quite long interference-resistant codes with good correct ability on the basis of composing shorter codes that allow simpler implementation and are called composite codes [@BL13]. Such solutions can be the basis for the procedure to create multidimensional crypt-code structures.
Similarly to the previous solution, the open text $M$ undergoes the procedure of encrypting. The formed sequence of blocks of the ciphertext $\Omega_1(z),\, \Omega_2(z),\, \ldots,\, \Omega_k(z)$ is split into $k_2$ number of sub-blocks, contain $k_1$ number of blocks of the ciphertext $\Omega_i(z)$ in each one and it is expressed in the form of a matrix $\mathbf{W}$ sized $k_1\times k_2$: $$\begin{aligned}
\label{44}
\mathbf{W}=\left[
\begin{array}{ccccc}
\Omega_{1,\, 1}(z) & \Omega_{1,\, 2}(z) & \ldots & \Omega_{1,\, k_2}(z) \\
\Omega_{2,\, 1}(z) & \Omega_{2,\, 2}(z) & \ldots & \Omega_{2,\, k_2}(z) \\
\vdots & \vdots & \ddots & \vdots \\[-0.2em]
\Omega_{k_1,\, 1}(z) & \Omega_{k_1,\, 2}(z) & \ldots & \Omega_{k_1,\, k_2}(z) \\
\end{array}
\right],\end{aligned}$$ where the columns of the matrix $\mathbf{W}$ are sub-blocks made of $k_1$ number of blocks of the ciphertext $\Omega_i(z).$
For each line of the matrix $\mathbf{W,}$ redundant blocks of data are formed, for example, using non-binary codes of Reed-Solomon (code RS \[particular case\]) over $\bbbf_q,$ that allow the *2-nd* level of monitoring.
The mathematical means of the RS codes is explained in detail in [@FJ14], where one of the ways to form it is based on the deriving polynomial $g(z)$. In $\bbbf_q$ the minimal polynomial for any element $\alpha^i$ is equal to $M^{(i)}=z-\alpha^i$, then, the polynomial $g(z)$ of the RS code corresponds to the equation: $$\label{44}
g(z)=\bigl(z-\alpha^{t}\bigr)\bigl(z-\alpha^{t}\bigr)\ldots\bigl(z-\alpha^{t+2b-1}\bigr),$$ where $2b=n-k$; usually $t=0$ or $t=1$.
At the same time, the RS code is cyclic and the procedure of forming the systematic RS code is described by the equation: $$\begin{aligned}
\label{45}
C(z)=U(z)z^{n-k}+R(z),\end{aligned}$$ where $U(z) = u_{k-1}z^{k-1} + \ldots + u_{1}z+u_0$ informational polynomial, and $\{u_{k-1},\, \ldots,\, u_{1},\, u_0\}$ informational code blocks; $R(z)=h_{r-1}z^{r-1} + \ldots +h_{1}z + h_0$ the residue from dividing the polynomial $U(z)z^{n-k}$ by $g(z)$, a $\{h_{r-1},\,\ldots,\, h_{1},\, h_0\}$ the coefficients of the residue. Then the polynomial $C(z)=c_{n-1}z^{n-1} + \ldots +c_{1}z + c_0$ and, therefore $\{c_{n-1},\, \ldots,\, c_{1},\, c_0\}= \{u_{k-1},\, \ldots,\, u_{1},\, u_0,\, h_{r-1},\, \ldots,\, h_{1},\, h_0\}$ a code word.
Basing on the primitive irreducible polynomial, setting the characteristic of the field $\bbbf_q$ in accordance with the Eq. (\[44\]) a deriving polynomial $g(z)$ of the RS code is formed.
Blocks of the ciphertext $\Omega_{i,\,1}(z),$ $\Omega_{i,\,2}(z), \ldots,$ $\Omega_{i,\,k_2}(z)$ are elements $\mathbf{W}$ expressed as elements of the sorted array, at the same time a formal variable $x$ is introduced and a set of ‘‘informational’’ polynomials is formed: $$\begin{aligned}
\mho_i(x) = \sum_{j=1}^{k_2}\bigl(\Omega_{i,\,j}(z)\bigr)x^{j-1} = \bigl(\Omega_{i,\,k_2}(z)\bigr)x^{k_2-1}+ \ldots + \bigl(\Omega_{i,\,2}(z)\bigr)x + \Omega_{i,\,1}(z),\end{aligned}$$ where $i=1,\, 2, \ldots,\, k_1$.
For $\mho_i(x)$ $(i=1,\, 2,\, \ldots,\, k_1)$ in accordance with the Eq. (\[45\]) a sequence of residues is formed $$\begin{aligned}
R_i(x) = \sum_{j=1}^{r_2}\bigl(\omega_{i,\,j}(z)\bigr)x^{j-1} =\bigl(\omega_{i,\,r_2}(z)\bigr)x^{r_2-1} + \ldots +\bigl(\omega_{i,\, 2}(z)\bigr)x + \omega_{i,\,1}(z),\end{aligned}$$ where $\omega_{i,\,j}(z)$ are coefficients of the polynomial $R_i(x)$ $(i=1,\, 2,\, \ldots,\, k_1)$ assumed as redundant blocks of data of the *2-nd* level of monitoring; $n_2$ is the length of the RS code, $k_2$ is the number of ‘‘informational’’ symbols (blocks) of the RS code, $r_2$ is the number of redundant symbols (blocks) of the RS code; $n_2=k_2+r_2$.
Matrix $\mathbf{W}$ with generated redundant blocks of data of the *2-nd* level of monitoring will take the form: $$\begin{gathered}
\mathbf{\Psi} = \mspace{-3mu}\begin{bmatrix}\mathbf{W}_{k_1\times k_2} | \mathbf{\Upsilon}_{k_1\times r_2}\end{bmatrix}=\mspace{-3mu}
\begin{matrix}
\begin{bmatrix}
~{ \makebox[0pt][l]{$\smash{\overbrace{\phantom{ \begin{matrix}{ \Omega_{1,\,1}(z)} \mspace{-3mu}& { \ldots}& \mspace{-2mu} {\Omega_{1,\,k_2}(z)}\end{matrix}}}^{\text{$~~~~k_2~~~~$}}}$}{ \Omega_{1,\,1}(z)} \mspace{-3mu}& { \ldots}& \mspace{-2mu} {\Omega_{1,\,k_2}(z)}} \mspace{-2mu}& { \makebox[0pt][l]{$\smash{\overbrace{\phantom{ \begin{matrix}{ \omega_{1,\,k_2+1}(z)} \mspace{-2mu}& { \ldots} & \mspace{-3mu}{\omega_{1,\,n_2}(z)}\end{matrix}}}^{\text{$r_2$}}}$}{ \omega_{1,\,k_2+1}(z)} \mspace{-2mu}& { \ldots} & \mspace{-3mu}{\omega_{1,\,n_2}(z)}} \\
{ \Omega_{2,\,1}(z)} \mspace{-3mu}&{ \ldots}&\mspace{-2mu} {\Omega_{2,\,k_2}(z)} \mspace{-2mu} & { \omega_{2,\,k_2+1}(z)} \mspace{-2mu}& {\ldots} &\mspace{-3mu} { \omega_{2,\,n_2}(z)} \\
{ \cdots } \mspace{-3mu}&{ \cdots}& \mspace{-2mu} { \cdots } & { \cdots} \mspace{-2mu}& { \cdots} & \mspace{-3mu}{ \cdots } \\[0.1em]
{ \Omega_{k_1,\,1}(z)} \mspace{-3mu}&{ \ldots}& \mspace{-2mu}{\Omega_{k_1,\,k_2}(z)} \mspace{-2mu} & { \omega_{k_1,\,k_2+1}(z)}\mspace{-2mu} & {\ldots} &\mspace{-3mu} { \omega_{k_1,\,n_2}(z)} \\
\end{bmatrix}
\begin{aligned}
&\mspace{-13mu}\left.\begin{matrix}
\\[0.1em]
\\[0.10em]
\\[0.10em]
\\[0.10em]
\end{matrix} \right\} {\scriptstyle k_1}\\
\end{aligned}.
\end{matrix}
\end{gathered}$$
The lines of the matrix $\mathbf{\Upsilon}$ are redundant blocks of data of the *2-nd* level of monitoring that undergo the procedure of encrypting: $$\begin{cases}
\vartheta_{1,\,\gamma}(z)\rightarrow E_{\kappa_{{\rm e}_{1,\,\gamma}}}~:~ \omega_{1,\,\gamma}(z),\\[-0.2em]
\vartheta_{2,\,\gamma}(z)\rightarrow E_{\kappa_{{\rm e}_{2,\,\gamma}}}~:~ \omega_{2,\,\gamma}(z),\\[-0.2em]
\ldots\ \ldots\ \ldots\ \ldots\ \ldots\ \ldots\ \ldots\ \\[-0.2em]
\vartheta_{k_1,\,\gamma}(z)\rightarrow E_{\kappa_{{\rm e}_{k_1,\,\gamma}}}~:~ \omega_{k_1,\,\gamma}(z),\\[-0.2em]
\end{cases}$$ where $\kappa_{{\rm e}_{i,\,\gamma}}~(i=1,\, 2,\, \ldots,\, k_1;\ \ \ \gamma=k_2+1,\, k_2+2,\, \ldots,\, n_2)$ are the keys for encrypting.
The generated sequence of blocks of the redundant ciphertext of the *2-nd* level of monitoring $\vartheta_{i,k_2+1}(z),$ $\vartheta_{i,k_2+2}(z),$ $\ldots,$ $\vartheta_{i,n_2}(z)$ $(i=1, 2, \ldots, k_1)$ form a matrix $\mathbf{V}$ sized $k_1\times r_2$ redundant blocks of the ciphertext of the *2-nd* level of monitoring: $$\begin{aligned}
\mathbf{V}=\left[
\begin{array}{ccccc}
\vartheta_{1,\, k_2+1}(z) & \vartheta_{1,\, k_2+2}(z) & \ldots & \vartheta_{1,\, n_2}(z) \\[-0.1em]
\vartheta_{2,\, k_2+1}(z) & \vartheta_{2,\, k_2+2}(z) & \ldots & \vartheta_{2,\, n_2}(z) \\[-0.1em]
\ldots& \ldots & \ldots &\ldots \\[-0.1em]
\vartheta_{k_1,\, k_2+1}(z) & \vartheta_{k_1,\, k_2+2}(z)& \ldots & \vartheta_{k_1,\, n_2}(z) \\[-0.1em]
\end{array}
\right].\end{aligned}$$
Now, each column of the matrix $\mathbf{W}$ and $\mathbf{V}$ as a sequence of blocks of the ciphertext $\Omega_{1,\,j}(z),\, \Omega_{2,\,j}(z),\, \ldots,\, \Omega_{k_1,\,j}(z)\ \ (j=1,\, 2,\, \ldots,\, k_2)$ and $\vartheta_{1,\,\gamma}(z),\, \vartheta_{2,\,\gamma}(z),\, \ldots,\,\vartheta_{k_1,\,\gamma}(z)\ \ (\gamma=k_2+1,\, k_2+2,\, \ldots,\, n_2)$ are expressed in the form of minimal residues on the bases-polynomials $m_i(z)$, such that $\gcd\bigl(m_i(z),\, m_j(z)\bigr)= 1$ $(i\neq j;\ \ \ i,\,j = 1,\, 2,\, \ldots,\, k_1).$ At the same time $\deg \Omega_{i,\,j}(z) < \deg m_i(z)$, and $\deg \vartheta_{i,\,\gamma}(z) < \deg m_{i}(z)$. Then, as we have noted above, the arrays of blocks of the ciphertext $\Omega_{1,\,j}(z), \Omega_{2,\,j}(z),\,\ldots,\, \Omega_{k_1,\,j}(z)$ $(j=1,\, 2,\, \ldots,\, k_2)$ and $\vartheta_{1,\,\gamma}(z),\, \vartheta_{2,\,\gamma}(z),\,\ldots,\, \vartheta_{k_1,\,\gamma}(z)$ $(\gamma=k_2+1,\, k_2+2,\, \ldots,\, n_2)$ are expressed as united informational super-blocks of RRPC on the system of bases $m_1(z), m_2(z),\ldots, m_{k_1}(z)$. In accordance with CRT for the specified array of polynomials $m_1(z),\, m_2(z),\,\ldots, m_{k_1}(z)$ that meet the condition $\gcd\bigl(m_i(z),\, m_j(z)\bigr)= 1$, polynomials $\Omega_{1,\,j}(z), \Omega_{2,\,j}(z),\,\ldots,\, \Omega_{k_1,\,j}(z)$ $(j=1, 2, \ldots, k_2)$ and $\vartheta_{1,\gamma}(z), \vartheta_{2,\,\gamma}(z),\ldots, \vartheta_{k_1,\,\gamma}(z)$ $(\gamma=k_2+1, k_2+2, \ldots, n_2)$ such that $\deg \Omega_{i,\,j}(z)<\deg m_i(z)$, $\deg \vartheta_{i,\,\gamma}(z)<\deg m_i(z)$, the system of congruences (\[4\]) will take the form: $$\label{7}
\begin{cases}
\begin{cases}
\Omega_1(z)\equiv \Omega_{1,\,1}(z)\mod m_1(z), \\[-0.2em]
\Omega_1(z)\equiv \Omega_{2,\,1}(z)\mod m_2(z), \\[-0.2em]
\ldots\ \ldots\ \ldots\ \ldots\ \ldots\ \ldots\ \ldots\ \ldots \\[-0.2em]
\Omega_{1}(z)\equiv \Omega_{k_1,\,1}(z)\mod m_{k_1}(z);\\[-0.2em]
\end{cases}\\
~~~~\ldots\ \ldots\ \ldots\ \ldots\ \ldots\ \ldots\ \ldots\ \ldots \\[-0.2em]
\begin{cases}
\Omega_{k_2}(z)\equiv \Omega_{1,\,k_2}(z)\mod m_1(z), \\[-0.2em]
\Omega_{k_2}(z)\equiv \Omega_{2,\, k_2}(z)\mod m_2(z), \\[-0.2em]
\ldots\ \ldots\ \ldots\ \ldots\ \ldots\ \ldots\ \ldots\ \ldots \\[-0.2em]
\Omega_{k_2}(z)\equiv \Omega_{k_1,\,k_2}(z)\mod m_{k_1}(z);\\[-0.2em]
\end{cases}
\end{cases}$$ $$\label{8}
\begin{cases}
\begin{cases}
\vartheta_{k_2+1}(z)\equiv \vartheta_{1,\,k_2+1}(z)\mod m_1(z), \\[-0.2em]
\vartheta_{k_2+1}(z)\equiv \vartheta_{2,k_2+1}(z)\mod m_2(z), \\[-0.2em]
\ldots\ \ldots\ \ldots\ \ldots\ \ldots\ \ldots\ \ldots\ \ldots\ \ldots \\[-0.2em]
\vartheta_{k_2+1}(z)\equiv \vartheta_{k_1,\,k_2+1}(z)\mod m_{k_1}(z);\\[-0.2em]
\end{cases}\\
~~~~\ldots\ \ldots\ \ldots\ \ldots\ \ldots\ \ldots\ \ldots\ \ldots \\[-0.2em]
\begin{cases}
\vartheta_{n_2}(z)\equiv \vartheta_{1,\,n_2}(z)\mod m_1(z), \\[-0.2em]
\vartheta_{n_2}(z)\equiv \vartheta_{2,\, n_2}(z)\mod m_2(z), \\[-0.2em]
\ldots\ \ldots\ \ldots\ \ldots\ \ldots\ \ldots\ \ldots\ \ldots \\[-0.2em]
\vartheta_{n_2}(z)\equiv \vartheta_{k_1,\,n_2}(z)\mod m_{k_1}(z),\\[-0.2em]
\end{cases}
\end{cases}$$ where $\Omega_j(z)$, $\vartheta_\gamma(z)$ are the only solutions for $j = 1, 2, \ldots, k_2;
\gamma = k_2+1, \ldots, n_2$.
Now, according to the additionally formed $r_1$ redundant bases of polynomials $m_{k_1+1}(z),\, m_{k_1+2}(z),\, \ldots,\, m_{n_1}(z)$ $(n_1=k_1+r_1)$, meeting the condition (\[2\]), (\[2.1\]) and in accordance with the Eq. (\[3\]) redundant blocks of data are formed, that belong to the *1-st* level of monitoring, expressed as $\omega_{k_1+1,\,j}(z)$, $\omega_{k_1+2,\,j}(z),\, \ldots,\,\omega_{n_1,\,j}(z)$ $(j = 1,\, 2,\, \ldots,\, k_2)$, as well as reference blocks of data $\omega_{k_1+1,\,\gamma}(z)$, $\omega_{k_1+2,\,\gamma}(z),\, \ldots,\, \omega_{n_1,\,\gamma}(z)$ $(\gamma = k_2+1,\, k_2+2\, \ldots,\, n_2)$.
The formed redundant blocks of data o the *1-st* level of monitoring $\omega_{k_1+1,\,j}(z)$, $\omega_{k_1+2,\,j}(z),\, \ldots,\, \omega_{n_1,\,j}(z)$ $(j = 1,\, 2,\, \ldots,\, k_2)$ are encrypted: $$\begin{cases}
\vartheta_{k_1+1,\,\gamma}(z)\rightarrow E_{\kappa_{{\rm e}_{k_1+1,\,\gamma}}}~:~ \omega_{k_1+1,\,\gamma}(z),\\
\vartheta_{k_1+2,\,\gamma}(z)\rightarrow E_{\kappa_{{\rm e}_{k_1+2,\,\gamma}}}~:~ \omega_{k_1+2,\,\gamma}(z),\\
\ldots\ \ldots\ \ldots\ \ldots\ \ldots\ \ldots\ \ldots\ \ldots\ \\
\vartheta_{n_1,\,\gamma}(z)\rightarrow E_{\kappa_{{\rm e}_{n_1,\,\gamma}}}~:~~ \omega_{n_1,\,\gamma}(z),
\end{cases}$$ where $\kappa_{{\rm e}_{\iota,\,\gamma}}~(\iota=k_1+1,\, k_1+2,\, \ldots,\, n_1; \gamma=k_2+1,\, k_2+2,\, \ldots,\, n_2)$ are the keys for encrypting.
Now, the arrays of informational blocks of the ciphertext $\Omega_1(z),\,\Omega_2(z),\, \ldots\\\ldots, \,\Omega_k(z)$, blocks of the redundant encrypted text of the *1-st* and *2-nd* levels of monitoring $\vartheta_{k_1+1,\,j}(z)$, $\vartheta_{k_1+2,\,j}(z),\, \ldots,\, \vartheta_{n_1,\,j}(z)$ $(j=1,\, 2,\, \ldots,\, k_2)$ and $\vartheta_{i,\,k_2+1}(z)$, $\vartheta_{i,\,k_2+2}(z),\, \ldots,\,\vartheta_{i,\,n_2}(z)$ $(i=1,\, 2,\, \ldots,\, k_1)$, as well as reference blocks of data $\omega_{k_1+1,\,\gamma}(z),\, \omega_{k_1+2,\,\gamma}(z),\, \ldots,\, \omega_{n_1,\,\gamma}(z)$ $(\gamma = k_2+1, k_2+2\ldots, n_2)$ form multidimensional crypt-code structures, whose matrix representation correspond to the expression:\
$$\mathbf{\Phi}=
\begin{matrix}
\begin{bmatrix}
~{ \makebox[0pt][l]{$\smash{\overbrace{\phantom{ \begin{matrix}~~~{ \Omega_{1,\,1}(z)} &{ \ldots} & {\Omega_{1,\,k_2}(z)~~~}\end{matrix}}}^{\text{$~~~~~~~~~k_2~~~~~~~~~$}}}$}~~~{ \Omega_{1,\,1}(z)} &{ \ldots} & {\Omega_{1,\,k_2}(z)~~~}} & { \makebox[0pt][l]{$\smash{\overbrace{\phantom{ \begin{matrix}{ \vartheta_{1,\,k_2+1}(z)} & { \ldots} & {\vartheta_{1,\,n_2}(z)~~~~}\end{matrix}}}^{\text{$~~~~r_2~~~~$}}}$}{ \vartheta_{1,\,k_2+1}(z)} & { \ldots} & {\vartheta_{1,\,n_2}(z)~~~~}} \\
{ \ldots} &{ \ldots}& { \ldots} & { \ldots} & { \ldots} & { \ldots} \\
{ \Omega_{k_1,\,1}(z)} &{ \ldots} & {\Omega_{k_1,\,k_2}(z)} &{\vartheta_{k_1,\,k_2+1}(z)} & { \ldots} & {\vartheta_{k_1,\,n_2}(z) } \\
\\
{\vartheta_{k_1+1,\,1}(z)} &{ \ldots}& {\vartheta_{k_1+1,\,k_2}(z)} & {\omega_{k_1+1,\,k_2+1}(z)} & { \ldots} & { \omega_{k_1+1,\,n_2}(z)} \\
{ \ldots} &{\ldots}& {\ldots} & { \ldots} & { \ldots} & { \ldots} \\
{\vartheta_{n_1,\,1}(z)} &{\ldots}& {\vartheta_{n_1,\,k_2}(z)} & {\omega_{n_1,\,k_2+1}(z)} & { \ldots} & {\omega_{n_1,\,n_2}(z)} \\
\end{bmatrix}
\begin{aligned}
\mspace{-1mu} &\left.\begin{matrix}
\\[0.2em]
\\[0.2em]
\\[0.2em]
\end{matrix} \right\} {\scriptstyle k_1}\\
\mspace{-1mu} &\left.\begin{matrix}
\\[0.2em]
\\[0.2em]
\\[0.2em]
\end{matrix} \right\} {\scriptstyle r_1}\\
\end{aligned}
\end{matrix}.$$
The formed multidimensional crypt-code structures correspond to the following parameters (a particular case for 2 levels of monitoring): $$\begin{aligned}
\label{172}
\left\{
\begin{aligned}
&n=n_1n_2, \\[-0.2em]
&k=k_1k_2, \\[-0.2em]
&r=r_1n_2 + r_2n_1 -r_1r_2,\\[-0.2em]
&d_{\min}=d_{\min_1}d_{\min_2},\\[-0.2em]
\end{aligned}
\right.\end{aligned}$$ where $n,$ $k,$ $r,$ $d_{\min}$ are generalized monitoring parameters; $n_i,$ $k_i,$ $r_i,$ $d_{\min_i}$ are parameters of the level of monitoring number $i$ $(i=1,\, 2)$ [@BL13].
On the receiving side, multidimensional crypt-code structures undergo the procedure of reverse transformation. In order to achieve that, the received sequence of blocks of the ciphertext $\Omega_{i}(z)$ $(i = 1,\, 2,\, \ldots,\, k)$ is split into $k_2$ number of sub-blocks containing $k_1$ blocks of the ciphertext and expressed in the form of the matrix $\mathbf{W}^*$ with the parameters identical to the parameters of the sending side: $$\label{448}
\mathbf{W^*}=\left[
\begin{array}{ccccc}
\Omega^*_{1,\, 1}(z) & \Omega^*_{1,\, 2}(z) & \ldots & \Omega^*_{1,\, k_2}(z) \\
\Omega^*_{2,\, 1}(z) & \Omega^*_{2,\, 2}(z) & \ldots & \Omega^*_{2,\, k_2}(z) \\
\vdots & \vdots & \ddots & \vdots \\[-0.1em]
\Omega^*_{k_1,\, 1}(z) & \Omega^*_{k_1,\, 2}(z) & \ldots & \Omega^*_{k_1,\, k_2}(z) \\
\end{array}
\right],$$ where the columns of the matrix $\mathbf{W}^*$ are sub-blocks of $k_1$ blocks of the ciphertext $\Omega_i^*(z)$. The arrays of blocks of the redundant ciphertext of the *1-st* and *2-nd* levels of monitoring $\vartheta^*_{k_1+1,\,j}(z),\, \vartheta^*_{k_1+2,\,j}(z),\, \ldots,\, \vartheta^*_{n_1,\,j}(z)$ $(j=1,\, 2,\, \ldots,\, k_2)$, $\vartheta^*_{i,\,k_2+1}(z),\,\vartheta^*_{i,\,k_2+2}(z),\, \ldots,\, \vartheta^*_{i,\,n_2}(z)$ $(i=1,\, 2,\, \ldots,\, k_1)$ that were obtained in the parallel process undergo procedure of decrypting: $$\begin{cases}
\omega^{*}_{k_1+1,\,j}(z)\rightarrow D_{\kappa_{{\rm d}_{k_1+1,\,j}}}~:~ \vartheta_{k_1+1,\,j}^*(z),\\
\omega^{*}_{k_1+2,\,j}(z)\rightarrow D_{\kappa_{{\rm d}_{k_1+2,\,j}}}~:~ \vartheta_{k_1+2,\,j}^*(z),\\
\ldots\ \ldots\ \ldots\ \ldots\ \ldots\ \ldots\ \ldots\ \ldots\ \\[-0.2em]
\omega^{*}_{n_1,\,j}(z)\rightarrow D_{\kappa_{{\rm d}_{n_1,\,j}}}~:~ \vartheta_{n_1,\,j}^*(z);\\
\end{cases}
\begin{cases}
\omega^{*}_{1,\,\gamma}(z)\rightarrow D_{\kappa_{{\rm d}_{1,\,\gamma}}}~:~ \vartheta_{1,\,\gamma}^*(z),\\
\omega^{*}_{2,\,\gamma}(z)\rightarrow D_{\kappa_{{\rm d}_{2,\,\gamma}}}~:~ \vartheta_{2,\,\gamma}^*(z),\\
\ldots\ \ldots\ \ldots\ \ldots\ \ldots\ \ldots\ \ldots\ \\[-0.2em]
\omega^{*}_{k_1,\,\gamma}(z)\rightarrow D_{\kappa_{{\rm d}_{k_1,\,\gamma}}}~:~ \vartheta_{k_1,\,\gamma}^*(z),\\
\end{cases}$$ where $\kappa_{{\rm d}_{\iota,\,j}}$ and $\kappa_{{\rm d}_{i,\,\gamma}}$ $(\iota=k_1+1,\, k_1+2,\, \ldots,\, n_1;\ \ j=1,\, 2,\, \ldots,\, k_2)$,\
$(i=1,\, 2,\, \ldots,\, k_1;\ \ \gamma=k_2+1,\, k_2+2,\, \ldots,\, n_2)$ are the keys for decrypting.
Now, every column $\Omega^*_{1,\,j}(z),\, \Omega^*_{2,\,j}(z),\, \ldots\, \Omega^*_{k_1,\,j}(z)$ of the matrix $\mathbf{W}^*$ that is interpreted as an informational super-block of the RRPC is put into the conformity to the sequence of redundant blocks of data of the *1-st* level of monitoring $\omega^*_{k_1+1,\,j}(z),\, \omega^*_{k_1+2,\,j}(z)\,\ldots,\, \omega^*_{n_1,\,j}(z)$ $(j=1,\, 2,\, \ldots,\, k_2)$ on the bases-polynomials $m_i(z)\ \ (i=1,\,2,\, \ldots,\, n_1)$ resulting in forming the code vector of the expanded RRPC $\bigl\{\Omega^*_{1,\,j}(z),\, \ldots,\, \Omega^*_{k_1,\,j}(z),\, \omega^*_{k_1+1,\,j}(z),\, \ldots,\, \omega^*_{n_1,\,j}(z)\bigr\}_{\text{RRPC}}$.
Besides that, the columns of the *2-nd* level of monitoring $\vartheta^*_{1,\gamma}(z), \ldots, \vartheta^*_{k_1,\gamma}(z)$ are put into the conformity to the reference blocks of data $\omega^*_{k_1+1,\gamma}(z), \ldots, \omega^*_{n_1,\gamma}(z)$ $(\gamma=k_2+1, \ldots, n_2)$ on the bases-polynomials $m_i(z) (i=1,2, \ldots, n_1)$ and a code vector of the expanded RRPC $\bigl\{\vartheta^*_{1,\gamma}(z), \ldots, \vartheta^*_{k_1,\gamma}(z), \omega^*_{k_1+1,\gamma}(z), \ldots\\
\ldots, \omega^*_{n_1,\gamma}(z)\bigr\}_{\text{RRPC}}$ is formed. Then, the procedure is started to detect the RRPC elements distorted (simulated) by the adversary, basing on the detection capability conditioned by the equation $d_{\min_1}-1$. At the same time, if $\Omega_j^*(z), \vartheta_\gamma^*(z)\in{{}^{F[z]}\!/_{(P(z))}},$ then we assume that there are no distorted blocks of the ciphertext, where $\Omega_j^*(z), \vartheta_\gamma^*(z)$ solution of the comparison system (\[7\]), (\[8\]) in accordance with the Eq. (\[3\]), for $j=1, 2, \ldots, k_2; \gamma=k_2+1, \ldots, n_2$. Considering the condition $\left\lfloor(d_{\min_1}-1)2^{-1}\right\rfloor$, the procedure of restoring the distorted elements of RRPC can be executed with the help of calculating the minimal residues or with any other known method of RRPC decoding. The corrected (restored) elements number $j$ of the sequence of the ciphertext blocks $\Omega^{**}_{1,j}(z), \Omega^{**}_{2,j}(z), \ldots, \Omega^{**}_{k_1,j}(z)$ ‘‘replace’’ the distorted number $i$ (of the ciphertext blocks) of the lines $\Omega^{*}_{i,1}(z), \Omega^{*}_{i,2}(z), \ldots, \Omega^{*}_{i,k_2}(z)$ $(i=1,2,\ldots,k_1)$ of the matrix $\mathbf{W}^*.$ The symbols ‘‘\*\*’’ indicate the stochastic character of restoration.
Now, each line $\Omega^{*}_{i,1}(z), \Omega^{*}_{i,2}(z), \ldots, \Omega^{*}_{i,k_2}(z)$ is put into conformity of the blocks of the redundant ciphertext of the *2-nd* level of monitoring $\omega^{*}_{i,k_2+1}(z),$ $\omega^{*}_{i,k_2+2}(z), \ldots,$ $\omega^{*}_{i,n_2}(z)$ $(i=1,2,\ldots,k_1)$ and code vectors are formed for the RS code $\bigl\{\Omega^*_{i,1}(z), \ldots,$ $\Omega^*_{i,k_2}(z), \omega^*_{i,k_2+1}(z),$ $\ldots,$ $ \omega^*_{i,n_2}(z)\bigr\}_{\text{RS}}$.
According to the code vectors, polynomials are formed $$\begin{aligned}
\mathcal{C}_i^{*}(x)=\mho_i^{*}(x)+R_i^{*}(x) = \sum_{j=1}^{k_2}\bigl(\Omega_{i,j}^{*}(z)\bigr)x^{j-1}+\sum_{\gamma=k_2+1}^{n_2}\bigl(\omega_{i,\gamma}^{*}(z)\bigr)x^{\gamma-1}\end{aligned}$$ and their values are calculated for the degrees of the primitive element of the field $\alpha^\ell:$ $$\begin{aligned}
\mathcal{S}_{i,\ell}=\mathcal{C}_i^{*}(\alpha^\ell)= \sum_{j=1}^{k_2}\Bigl(\Omega_{i,j}^{*}(z)\Bigr)\Bigl(\alpha^{(j-1)}\Bigr)^\ell+\sum_{\gamma=k_2+1}^{n_2}\Bigl(\omega_{i,\gamma}^{*}(z)\Bigr)\Bigl(\alpha^{(\gamma-1)}\Bigr)^\ell,\end{aligned}$$ where $i=1,\, 2,\, \ldots,\, k_1;~~ \ell=0,\, 1,\, \ldots,\, r_2-1$, $r_2=n_2-k_2$.
At the same time, if the values of checksums $\mathcal{S}_{i,\ell}$ with $\alpha^\ell$ for each vector of the line are equal to zero, then we assume that there are no distortions. Otherwise, the values $\mathcal{S}_{i,\,0}, \mathcal{S}_{i,\,1}, \ldots, \mathcal{S}_{i,\,r_2-1}$ for $i=1,\, 2,\, \ldots,\, k_1$ are used for further restoration of the blocks of the ciphertext $\Omega_{i,\,1}^{*}(z),$ $\Omega_{i,\,2}^{*}(z), \ldots,$ $\Omega_{i,\,k_2}^{*}(z)$ with the help of well-known algorithms for decoding RS codes (of Berlekamp-Massey, Euclid, Forney and etc.).
The corrected (restored) sequences of redundant blocks of the ciphertext of the *2-nd* level of monitoring $\vartheta_{1,\gamma}^{**}(z), \ldots, \vartheta_{k_1,\gamma}^{**}(z)$ are subject of the second transformation (decryption) of redundant blocks of the ciphertext of the *2-nd* level of monitoring into redundant blocks of data of the *2-nd* level of monitoring $\omega_{1,\gamma}^{**}(z), \ldots, \omega_{k_1,\gamma}^{**}(z)$. The redundant blocks of data of the *2-nd* level of monitoring $\omega_{1,\gamma}^{**}(z), \ldots, \omega_{k_1,\gamma}^{**}(z)$ $(\gamma=k_2+1, k_2+2, \ldots, n_2)$ that have been formed again are used for forming code combinations of the RS code and their decoding.
Imitation Resistant Transmitting of Encrypted Information on the Basis of Crypt-Code Structures and Authentication Codes
========================================================================================================================
Currently, to detect simulation by the adversary in the communication channel, an additional encryption regime is used to simulate imitated insertion (forming an authentication code \[Message Authentication Code\]) [@Ferg1; @Men2; @Chr4]. A drawback of this method to prevent imitation by the adversary is the lack of possibility to restore veracious information in the systems for transmitting information. Complexing the method to protect from imitating of data on the basis of message authentication codes (MAC) and the above-described solution based on expanding the RRPC with encrypting the redundant information, it shall make it possible to overcome the drawback of the known solution. Let us assume that MAC are formed as usual from the sequence consisting of $k_2$ number of sub-blocks containing $k_1$ blocks each of the ciphertext $\Omega_i(z)$ in each one. Then the procedure of generation of MAC $H_i(z)$ $(i=1, \ldots, k_1)$ can be expressed: $$\left\{
\begin{array}{ll}
H_1(z) \rightarrow I_{h_{1}}: {\rm {\bf \Omega}}_1, \\[-0.2em]
H_2(z) \rightarrow I_{h_{2}}: {\rm {\bf \Omega}}_2, \\[-0.2em]
\ldots\ \ldots\ \ldots\ \ldots\ \ldots\ \\[-0.2em]
H_{k_1}(z) \rightarrow I_{h_{k}}: {\rm {\bf \Omega}}_{k_1},\\[-0.2em]
\end{array}
\right.$$ where $I_{h_{i}}$ is the operator of generation of an MAC on the key $h_{i}$ $(i=1, \ldots, k_1),$ ${\rm {\bf \Omega}}_i=\bigl\{\Omega_{i, 1}(z), \ldots, \Omega_{i, k_2}(z)\bigr\}$ is a vector equation of the super-block of the ciphertext, $k_2$ is the length of the super-block. Purposeful interfering of the adversary into the process of transmitting super-blocks of the ciphertext with the MAC calculated from them can cause their distorting. Correspondingly, on the receiving side, the super-blocks ${\rm {\bf \Omega}}_i^{*}=\bigl\{\Omega_{i, 1}^{*}(z), \ldots, \Omega_{i, k_2}^{*}(z)\bigr\}$ of the ciphertext are the source for calculating MAC: $$\left\{
\begin{array}{ll}
\widetilde{H}_1(z) \rightarrow I_{h_{1}}: {\rm {\bf \Omega}}_1^*, \\
\widetilde{H}_2(z) \rightarrow I_{h_{2}}: {\rm {\bf \Omega}}_2^*, \\
\ldots\ \ldots\ \ldots\ \ldots\ \ldots\ \\
\widetilde{H}_{k_1}(z) \rightarrow I_{h_{k}}: {\rm {\bf \Omega}}_{k_1}^*,\\
\end{array}
\right.$$ where ${\rm {\bf \Omega}}_i^*=\bigl\{\Omega_{i, 1}^*(z), \ldots, \Omega_{i, k_2}^*(z)\bigr\}$ is the received super-block of the ciphertext; $\widetilde{H}_i(z)$ are MAC from the received blocks of the ciphertext, for $i=1, 2, \ldots, k_1.$
Similarly to the previous solution for restoring the messages simulated by the adversary from the transmitted sequence of blocks of the ciphertext with MAC $\Bigl\{ \bigl\{{\rm {\bf \Omega}}_{1}, H_{1}(z)\bigr\}; \ldots; \bigl\{{\rm {\bf \Omega}}_{k_1}, H_{k_1}(z)\bigr\}; \bigl\{ \vec{\vartheta}_{k_1+1}, H_{k_1+1}(z)\bigr\}; \ldots; \bigl\{ \vec{\vartheta}_{n_1}, H_{n_1}(z)\bigr\}\Bigr\}_{\text{RRPC}}$, an extended RRPC is formed.
The sub-system of imitation-resistant reception of encrypted information on the basis of the RRPC and using MAC implements the following algorithm.
*Input:* the received sequence of vectors of encrypted message blocks with MAC: $\Bigl\{ \bigl\{{\rm {\bf \Omega}}_{1}^{*}, H_{1}^{*}(z)\bigr\}; \ldots; \bigl\{{\rm {\bf \Omega}}_{k_1}^{*}, H_{k_1}^{*}(z)\bigl\}; \bigl\{ \vec{\vartheta}_{k_1+1}^{*}, H_{k_1+1}^{*}(z)\bigr\}; \ldots; \bigl\{ \vec{\vartheta}_{n_1}^{*}, H_{n_1}^{*}(z)\bigr\}\Bigr\}_{\text{RRPC}}$.
*Output:* a corrected (restored) array of super-blocks of the ciphertext ${\bf \Omega}^{**}_1, {\bf \Omega}^{**}_2, \ldots, {\bf \Omega}^{**}_{k_1}$.
*Step 1.* Detection of the possible simulation by the adversary in the received sequence of blocks of the ciphertext with localization of the number $i$ row vector with the detected false blocks of the ciphertext, is executed by comparing the MAC received from the communication channel $H_{1}^{*}(z), \ldots, H_{k_1}^{*}(z),$ $H_{k_1+1}^{*}(z), \ldots,
H_{n_1}^{*}(z)$ and MAC $\widetilde{H}_{1}^{*}(z), \ldots, \widetilde{H}_{k_1}^{*}(z),$ $\widetilde{H}_{k_1+1}^{*}(z), \ldots,
\widetilde{H}_{n_1}^{*}(z)$ calculated in the sub-system of data reception. Next, a comparison procedure is performed for all row vectors $(i=1,\, \ldots,\, k_1$, $k_1+1,\, \ldots,\, n_1)$: $$\left\{
\begin{array}{ll}
1,~~ \textit{if} ~~\ H_i^*(z)=\widetilde{H}_i(z);\\
0,~~ \textit{if} ~~\ H_i^*(z)\neq \widetilde{H}_i(z).
\end{array}
\right.$$
*Step 2.* Restoring veracious data by solving the congruences systems: $$\label{11}
\left\{
\begin{array}{l}
\left\{
\begin{array}{l}
\Omega^{**}_1(z)\equiv \Omega^{*}_{{J_1},\,1}(z)\mod m_{{J_1}}(z),\\[-0.1em]
\ldots\ \ldots\ \ldots\ \ldots\ \ldots\ \ldots\ \ldots\ \ldots\ \ldots\ \\[-0.1em]
\Omega^{**}_{1}(z)\equiv \Omega^{*}_{{J_{k_1}},\,1}(z)\mod m_{{J_{k_1}}}(z),\\[-0.1em]
\Omega^{**}_1(z)\equiv \omega^{*}_{{J_{k_1+1}},\,1}(z)\mod m_{{J_{k_1+1}}}(z),\\[-0.1em]
\ldots\ \ldots\ \ldots\ \ldots\ \ldots\ \ldots\ \ldots\ \ldots\ \ldots\ \\[-0.1em]
\Omega^{**}_1(z)\equiv \omega^{*}_{{J_{n_1}},\,1}(z)\mod m_{{J_{n_1}}}(z);\\[-0.1em]
\end{array}
\right. \\[-0.1em]
\left.
\begin{array}{l}
\ldots\ \ldots\ \ldots\ \ldots\ \ldots\ \ldots\ \ldots\ \ldots\ \ldots\ \\
\end{array} \right. \\[-0.2em]
\left\{
\begin{array}{l}
\Omega^{**}_{k_2}(z)\equiv \Omega^{*}_{{J_1},k_2}(z)\mod m_{{J_1}}(z),\\
\ldots\ \ldots\ \ldots\ \ldots\ \ldots\ \ldots\ \ldots\ \ldots\ \ldots\ \\[-0.1em]
\Omega^{**}_{k_2}(z)\equiv \Omega^{*}_{{J_{k_1}},k_2}(z)\mod m_{J_{k_1}}(z),\\
\Omega^{**}_{k_2}(z)\equiv \omega^{*}_{{J_{k_1+1}},k_2}(z)\mod m_{{J_{k_1+1}}}(z),\\
\ldots\ \ldots\ \ldots\ \ldots\ \ldots\ \ldots\ \ldots\ \ldots\ \ldots\ \\[-0.1em]
\Omega^{**}_{k_2}(z)\equiv \omega^{*}_{{J_{n_1}},k_2}(z)\mod m_{{J_{n_1}}}(z),\\
\end{array} \right.
\end{array}
\right.$$ where $J_1, J_2, \ldots, J_{n_1}$ are row vector numbers, if the comparison result for these MAC showed absence of distortions in sequence of blocks of the ciphertext ${\rm {\bf \Omega}}^{*}_{j}(z)=\bigl\{\Omega_{j, 1}^*(z), \Omega_{j, 2}^*(z), \ldots, \Omega _{j, k_2}^*(z)\bigr\}$. In accordance with the CRT solutions of systems (\[11\]) is the following: $$\begin{gathered}
\Omega_{j}^{**}=\Omega_{J_{1}, j}^{*}(z)B_{J_1}(z)+\ldots+\Omega_{J_{k_1}, j}^{*}(z)B_{J_{k_1}}(z)+\ldots\\
\ldots+\omega_{J_{k_1+1}, k}^{*}(z)B_{J_{k_1+1}}(z)+\ldots+\omega_{J_{n_1}, k}^{*}(z)B_{J_{n_1}}(z)~~{\rm modd}~\bigl(p, P_{k_v}(z)\bigr),
\end{gathered}$$ where\
$\displaystyle B_{J_i}(z)=k_{J_i}(z)P_i(z)$ are polynomial orthogonal bases; $P_{k_v}(z)=\prod_{\begin{subarray}{l}
i=1, \ldots,k; \\ i\neq v
\end{subarray}}m_i(z)$; $v$ is the number of the detected ‘‘distorted’’ row vector; $P_{J_i}(z)=P_{k_v}(z)m_i^{-1}(z)$; $ k_{J_i}(z)=P_{J_i}^{-1}(z)\mod m_{J_{i}}(z)$ $(j=1, \ldots, k_2; i=1, \ldots, n_1)$.
The values of polynomial orthogonal bases are calculated beforehand and are stored in the memory of the RRPC decoder. Restoring veracious blocks can be done by calculating the minimal deductions or by any other known method.
In a comparative evaluation of the effectiveness of the methods under consideration for providing imitation resistant transmission of encrypted information, we will assume that the adversary distorts the ciphertext blocks in the generated crypt-code structures with probability $p_{adv}=2\cdot10^{-2}.$ Probability $p_{adv}$ distortion of each ciphertext block is constant and does not depend on the results of receiving the preceding elements of crypt-code structures. The probability $P(b)$ of reception crypt-code structures with $b$ and more errors are presented in the table \[tab1\], in accordance with which a higher recovery power is provided multidimensional crypt-code structures (RRP codes and RS codes). At what at the given values $k_1, k_2,$ the closer the matrix being formed $\mathbf{\Phi}_{n_1 \times n_2}$ to the square shape, the less the level of redundancy introduced.
----------------------------- -------------------------- ------- ------- -------------- ----------------- -----------
Method of construction Structures $n$ $k$ $d_{\min}$ $\frac{k}{n}$ $P(b)$
Crypt-code structures (6, 3, 4) 6 3 4 0.5 0.114158
(RRPC) (8, 4, 5) 8 4 5 0.5 0.010336
Multidimensional crypt-code (6, 3, 4); (11, 5, 7) 66 15 28 0.227 0.000133
structures: (RRPC); (RS) (8, 4, 5); (8, 4, 5) 64 16 25 0.25 0.000106
Multidimensional crypt-code (4, 3, 2); (6, 3, 4) 24 9 8 0.375 0.008862
structures: (MAC); (RRPC) (4, 3, 2); (8, 4, 5) 32 12 10 0.375 0.000802
----------------------------- -------------------------- ------- ------- -------------- ----------------- -----------
: Effectiveness crypt-code structures []{data-label="tab1"}
Conclusion
==========
The methods of information protection examined in this article (against simulation by the adversary) are based on the composition of block ciphering system and multi-character codes that correct errors by forming crypt-code structures with some redundancy. This redundancy is usually small and it makes it possible to express all the possible states of the protected information. Forming multidimensional crypt-code structures with several levels of monitoring makes it possible to not only detect simulating actions of the intruder but also, if necessary, to restore the distorted encrypted data with the set probability and their preliminary localization.
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|
---
author:
- 'T. Grenman'
- 'G. F. Gahm'
title: 'The tiny globulettes in the Carina Nebula [^1] '
---
[Small molecular cloudlets are abundant in many regions surrounding newborn stellar clusters. In optical images these so-called globulettes appear as dark silhouettes against the bright nebular background.]{} [We aim to make an inventory of the population of globulettes in the Carina Nebula complex, and to derive sizes and masses for comparisons with similar objects found in other regions. ]{} [The globulettes were identified from H$\alpha$ images collected at the Hubble Space Telescope.]{} [We have located close to 300 globulettes in the Carina complex, more than in any other region surveyed so far. The objects appear as well-confined dense clumps and, as a rule, lack thinner envelopes and tails. Objects with bright rims are in the minority, but more abundant than in other regions surveyed. Some globulettes are slightly elongated with their major axes oriented in the direction of young clusters in the complex. Many objects are quite isolated and reside at projected distances $>$ 1.5 pc from other molecular structures in the neighbourhood. No globulette coincides in position with recognized pre-main-sequence objects in the area. The objects are systematically much smaller, less massive, and much denser than those surveyed in other regions. Practically all globulettes are of planetary mass, and most have masses less than one Jupiter mass. The average number densities exceed 10$^{5}$ cm$^{-3}$ in several objects. We have found a statistical relation between density and radius (mass) in the sense that the smallest objects are also the densest.]{} [The population of small globulettes in Carina appears to represent a more advanced evolutionary state than those investigated in other regions. The objects are subject to erosion in the intense radiation field, which would lead to a removal of any thinner envelope and an unveiling of the core, which becomes more compact with time. We discuss the possibility that the core may become gravitationally unstable, in which case free-floating planetary mass objects can form. ]{}
Introduction {#sec:intro}
============
Optical images of galactic regions show a mix of bright and dark nebulosity. Foreground cold dust clouds obscure the bright background of warm and ionized gas. The warm plasma in these nebulæ accelerates outwards through the interaction with radiation and winds from hot and massive stars. The molecular gas is swept up and forms expanding shells, which are sculpted into complex filamentary formations, like elephant trunks, pillars that point at O stars in the nebula. Blocks of cold gas can detach from the shells and trunks and may fragment into smaller clouds that appear as dark patches on optical images of these nebulæ as noted long ago by Bok & Reilly ([@bok47]) and Thackeray ([@tha50]). The clumps may be round or shaped like tear drops, some with bright rims facing the central cluster (Herbig [@her74]).
A number of such regions have been subject to more detailed studies, and it has been found that many regions contain distinct, but very small clumps, extending over less than one to a few arcseconds. Several studies have focused on the so-called proplyds, which are photoevaporating discs surrounding very young stars (e.g. O’Dell et al. [@ode93]; O’Dell & Wen [@ode94]; McCaughrean & O’Dell [@mcc96]; Bally et al. [@bal00]; Smith et al. [@smi03]). In these studies small cloudlets without any obvious central stellar objects were also found, as also recognized by Hester et al. ([@hes96]) and Reipurth et al. ([@rei97; @rei03]) from Hubble Space Telescope images of nebular regions.
More systematic studies of such star-less cloudlets followed, and from the surveys of more than 20 regions by De Marco et al. ([@mar06]), Grenman ([@gre06]), and Gahm et al. ([@gah07]; hereafter Paper 1) it can be concluded that most of the objects have radii $<$10 kAU with size distributions that peak at $\sim$ 2.5 kAU. In Paper 1 masses were derived from extinction measures indicating that most objects have masses $<$ 13 M$_{J}$ (Jupiter masses), which currently is taken to be the domain of planetary-mass objects. This class of tiny clouds in regions were called [*globulettes*]{} in Paper 1 to distinguish them from proplyds and the much larger globules spread throughout interstellar space. We define globulettes as cloudlets with round or slightly elongated shapes with or without bright rims and/or tails.
Some globulettes are connected by thin filaments to larger molecular blocks and it is then natural to assume that isolated globulettes once detached from shells and trunks. They may also survive in this harsh environment for long times, as concluded in Paper 1. Follow-up 3D numerical simulations in Kuutmann ([@kuu07]) predict lifetimes of $\sim$ $10^{4}$ years, increasing with mass. Owing to the outer pressure exerted on the globulettes from surrounding warm gas, and the penetrating shock generated by photoionization, it was found that many globulettes may even collapse to form brown dwarfs or planetary-mass objects before evaporation has proceeded very far. The objects are protected against rapid photoevaporation by a screen of expanding ionized gas (e.g. Dyson [@dys68]; Kahn [@kah69]; Tenorio-Tagle [@ten77]). Consequently, the objects are expected to develop bright rims on the side facing the cluster because of the interaction with stellar light. In addition, the models predict that dusty tails emerge from the cloud cores. It is therefore puzzling that most globulettes lack any trace of bright rims in H$\alpha$, and that most are round, or only slightly elongated, without any trace of tails.
In a recent study by Gahm et al. ([@gah13]; hereafter Paper 2), based on NIR imaging and radio molecular line observations of globulettes in the Rosette Nebula, it was found that the objects contain dense cores, which strengthens the suggestion that many objects might collapse to form planetary-mass objects or brown dwarfs that are accelerated outwards from the nebular complex. The whole system of globulettes and trunks expands outwards from the central cluster with velocities of about 22 km s$^{-1}$. In the case where more compact objects are formed inside some globulettes, they will escape and become free-floating objects in the galaxy. In both the optical and radio/NIR surveys (Papers 1 and 2) it was concluded that the density is relatively high even close to the surface layers, which could explain why the objects lack extensive bright rims in H$\alpha$. Some of the optically completely dark objects were discovered to have thin rims manifested in P$\beta$ and H$_{2}$ emission. In a follow-up study of the NIR images, Mäkelä et al. ([@mak14]) found that some smaller globulettes are also crowned by thin bright rims that are not seen in H$\alpha$.
The present study is an inventory of globulettes in the Carina Nebula (NGC 3372) based on images taken from the [*Hubble Space Telescope*]{} (HST) through a narrow-band H$\alpha$ filter. Basic parameters, like size and mass, are derived and we compare the results to surveys of similar objects in other nebulae.
The Carina complex, with its extended network of bright and dark nebulosity, spans over several degrees in the sky and is one of the most prominent sites of star formation in the galaxy. More than 60 O-type stars and several young clusters (Tr 14, 15, and 16; Collinder 228 and 232; and Bochum 10 and 11) are located in the region, and more than a thousand pre-main sequence stars have been identified from optical, infrared, and X-ray surveys (e.g. Tapia et al. [@tap03]; Ascenso et al. [@asc07]; Sanchawala et al. [@san07a], [@san07b]; Smith et al. [@smi10a], [@smi10b]; Povich et al. [@pov11]; Gaczkowski et al. [@gac13]). The global properties of the nebular material was discussed in Smith et al. ([@smi00]), Smith & Brooks ([@smi07]), and references to studies based on observations of selected areas can be found in the comprehensive review by Smith ([@smi08]). Additional surveys from the submm range (Preibisch et al. [@pre11]; Pekruhl et al. [@pek13]) and the far IR (Preibisch et al. [@pre12]; Roccatagliata et al. [@roc13]) have been made more recently. The Carina Nebula, in all its glory, is presented in multicolour mosaics found at the Hubble Space Heritage webpage.
A number of small obscuring structures in the Carina Nebula were noted by Smith et al. ([@smi03]) from HST images, and were regarded as possible proplyds. However, the objects studied were found to be larger than the standard cases in the Orion Nebula (Bally et al. [@bal00]). More objects of this nature were recognized by Smith et al. ([@smi04]) who stated that their nature remains ambiguous: “analogues of Orion’s proplyds, starless cometary clouds, or something in between?” Ascenso et al. ([@asc07]), however, concluded from near-infrared imaging that these candidates do not harbour any stars. Most of these objects are globulettes by our definition and are thereby included in our list of nearly 300 globulettes. Thus the Carina complex is the richest known with regard to total number of globulettes. A number of Herbig-Haro jets emanating from embedded young stars in the region were found by Smith et al. ([@smi10a]). Most of these are related to trunks or larger fragments. However, HH 1006 is related to an isolated cloud with an embedded jet-driving source (Sahai et al. [@sah12]; Reiter & Smith [@reit13]). Tentative jet-signatures were also found for a few much smaller isolated clouds like HH 1011 and HHc-1.
The distance to the Carina complex has been estimated in several investigations with rather different results. A distance of 2.3 kpc has been adopted as a kind of standard (Smith [@smi08]). Recently, Hur et al. ([@hur12]) concluded that the main stellar clusters Tr 14 and 16 are located at a distance of 2.9 kpc. We have adopted this value in the present investigation, but will discuss the implications if the complex is closer.
The paper is organized as follows. We present the fields we have searched, the objects identified, and their measured properties in Section \[sec:obs\]. The results are analysed in Section \[sec:results\] and discussed further in Section \[sec:disc\]. We end with a summary in Section \[sec:conclude\].
$$\begin{array}{*{4}{p{0.1\textwidth}}}
\hline
\noalign{\smallskip}
\ Field/Target & R.A. (J2000.0) \hfill{} & Dec. (J2000.0)& Images \\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
1 / Pos 30 & 10:41:27 & -59:47:42 & J9dk09010 \\
2 / Pos 30 & 10:41:38 & -59:46:17 & J9dk09020 \\
3 / Pos 30 & 10:41:40 & -59:44:41 & J9dka9010 \\
4 / Pos 19 & 10:42:23 &-59:20:59 & J9dk12010 \\
5 / Pos 19 & 10:42:48 &-59:19:44 & J9dk32010 \\
6 / Tr 14 & 10:43:07 & -59:29:34 & J900c1010 \\
7 / Tr 14 & 10:43:23 & -59:32:06 & J900b1010 \\
8 / Tr 14& 10:43:24 & -59:27:55 & J900c2010 \\
9 / Tr 14 & 10:43:39 & -59:34:37 & J900a1010 \\
10 / Tr14 & 10:43:41 & -59:30:17 & J900b2010 \\
11 / Tr 14 & 10:43:47 & -59:35:53 & J90001020 \\
12 / Tr 14 & 10:43:55 & -59:37:09 & J90001010 \\
13 / HH 666& 10:43:58 & -59:54:39 & J900a9010 \\
14 / Tr 14 & 10:43:59 & -59:32:38 & J900a2010 \\
15 / Tr 14 & 10:44:00 & -59:28:05 & J900b3010 \\
16 / HH 666 & 10:44:01 & -59:58:42 & J900b9010 \\
17 / Tr 16 & 10:44:05 & -59:40:16 & J900c5010 \\
18 / Tr 14 & 10:44:07 & -59:33:53 & J90002020 \\
19 / Tr 14 & 10:44:15 & -59:35:09 & J90002010 \\
20 / Tr 14 & 10:44:17 & -59:30:27 & J900a3010 \\
21 / Tr 14 & 10:44:19 & -59:25:54 & J900b4010 \\
22 / Tr 16 & 10:44:20 & -59:42:48 & J900b5010 \\
23 / Tr 16 & 10:44:22 & -59:38:34 & J900c6010 \\
24 / Tr 14 & 10:44:35 & -59:33:09 & J90003010 \\
25 / Tr 16 & 10:44:36 & -59:45:19 & J900a5010 \\
26 / Pos 27 & 10:44:40 & -59:59:46 & J9dk07010 \\
27 / Pos 27& 10:44:43 & -59:56:34 & J9dk27010 \\
28 / Tr 16 & 10:44:44 & -59:46:35 & J90005020 \\
29 / Tr 16 & 10:44:49 & -59:37:35 & J900b7020 \\
30 / Tr 16 & 10:44:52 & -59:47:51 & J90005010 \\
31 / Tr 14 & 10:44:54 & -59:31:08 & J90004010 \\
32 / Tr 15 & 10:44:58 & -59:26:50 & J9dka0010 \\
33 / Tr 16& 10:44:58 & -59:38:47 & J900b7010 \\
34 / Tr 16 & 10:45:12 & -59:45:51 & J90006010 \\
35 / Tr 16 & 10:45:17 & -59:36:38 & J900b8010 \\
36 / Tr 15 & 10:45:23 & -59:26:59 & J9dk10010 \\
37 / Tr 16 & 10:45:44 & -59:40:34 & J90008020 \\
38 / Pos 23& 10:45:53 & -60:08:16 & j90010010 \\
39 / Pos 23 & 10:45:56 & -60:06:42 & J90010020 \\
40 / Pos 22& 10:46:32 & -60:05:14 & J9dk23010 \\
41 / Pos 21& 10:46:47 & -60:09:29 & J9dk22010 \\
42 / Pos 20& 10:46:58 & -60:06:26 & J9dk01010 \\
43 / Pos 20& 10:47:01 & -60:03:14 & J9dk21010 \\
\noalign{\smallskip}
\hline
\end{array}$$
Objects and measurements {#sec:obs}
========================
HST fields {#sec:fields}
----------
![Examples of globulettes found in the Carina Nebula numbered according to Tables A1-A5 in Appendix A. The most typical cases are the dark globulettes shown in the upper four rows followed by objects with bright halos. The last two rows show examples of elongated objects with tails, some with bright rims. We note the scales are different from panel to panel (the dimensions in arcsec are given for each object in the tables). []{data-label="mosaik"}](mosaik.jpg){width="9cm"}
The optical images of the Carina complex were downloaded from the HST archive, cycle 13 and 14 programs GO-10241 and 10475 (principal investigator N. Smith) based on observations with the ACS/WFI camera, which contains two CCDs of 2048 $\times$ 4096 pixels glued together with a small gap in between. The pixel size corresponds to $\approx$ 0.05 arcsec pixel$^{-1}$, and the field of view is $202 \times 202$ arcsec. All images selected were exposed for 1000 s through the narrow-band filter F658N, covering the nebular emission lines of H$\alpha$ and \[\].
Most of the HST fields contains globulettes, and only these are listed in Table \[HST\] with a running field number and references to target and image designations according to the HST archive. Figure \[map\] shows how the areas covered by HST are distributed over the region (see also Smith et al. [@smi10a]). Two regions on opposite sides of a large V-shaped dark cloud are rather well covered by HST. The total area covered is $\sim$ 700 arcmin$^2$, which is larger than covered by all HST-based surveys of regions together, but much smaller than the area covered of the Rosette Nebula in Paper 1. This ground-based survey was limited to objects with radii $\geq$ 0.8 arcsec, however.
The globulettes are easily recognized as dark patches against the bright background. Most are roundish without any bright rims or halos, similar to previous surveys. A number of the elongated objects with tear-drop forms are crowned with bright rims. The Carina complex is also rich in dark irregular blocks and fragments of all sizes, some of which are very elongated and shaped like worms or long, narrow cylinders, and some show very irregular shapes. These objects, which as a rule are much larger than typical globulettes, were not included in our list of globulettes, but in Sect. \[sec:pec\] we highlight some smaller cloudlets with peculiar shapes.
The Carina Nebula contains a large number of very small globulettes, down to the limit of resolution of HST, but we do not consider objects with dimensions $\le$ 3 pixels across. Some regions contain quite isolated globulettes that are located far away from any larger molecular block, while in others there are clusters of globulettes. Examples of different types of globulettes are shown in Fig. \[mosaik\], where the first four rows show round and dark globulettes, which are most abundant. Round objects with bright halos are found in the fifth row followed by objects that are more elongated or have developed pronounced tails, with or without bright rims.
Measurements {#sec:data}
------------
[lccccccccccl]{} CN & Field & x & y & R.A. & Dec. & $\alpha$ & $\beta$ & P.A. & $\bar r $ & Mass & Remarks\
&&&& (J2000.0) & (J2000.0) & (arcsec) & (arcsec) & (degr.) & (kAU) & (M$_{J}$) &\
\
1 & F1 & 360 & 4020 & 10:41:13.3 & -59:49:00 & 0.46 & 0.50 & & 1.39 & 2.4 &\
2 & F1 & 660 & 1084 & 10:41:18.9 & -59:46:38 & 0.30 & 0.36 & & 0.96 & 1.1 &\
3 & F1 & 1792 & 3251 & 10:41:23.6 & -59:48:35 & 0.38 & 0.40 & & 1.13 & 1.3 &\
4 & F1 & 1654 & 378 & 10:41:26.2 & -59:46:13 & 0.18 & 0.19 & & 0.54 & 0.4 &\
5 & F1 & 1819 & 395 & 10:41:27.2 & -59:46:15 & 0.19 & 0.27 & & 0.67 & 0.6 &\
6 & F1 & 2029 & 221 & 10:41:28.8 & -59:46:09 & 0.26 & 0.33 & 23 & 0.86 & 1.2 & T\
7 & F2 & 670 & 1907 & 10:41:28.9 & -59:45:53 & 0.23 & 0.25 & -43& 0.69 &*0.6 & BR,T\
8 & F1 & 2391 & 1935 & 10:41:29.1 & -59:47:36 & 0.20 & 0.22 & & 0.61 & 0.5 &\
9 & F1 & 2156 & 597 & 10:41:29.2 & -59:46:29 & 0.29 & 0.33 & & 0.90 & 1.0 &\
10 & F1 & 2752 & 1904 & 10:41:31.5 & -59:47:38 & 0.54 & 1.46 & 8 & 2.90 & *8.8 & BR,EL,T\
11 & F1 & 2604 & 373 & 10:41:32.4 & -59:46:22 & 0.31 & 0.51 & -38 & 1.19 & 2.0 & EL\
\
continued in Appendix A\
\
**
\[glob\]
Central positions were measured in terms of x and y coordinates and R.A. and Dec according to available HST readouts. The globulettes, designated CN (as Carina Nebula plus number), are listed in order of increasing R.A. in Table 2 showing only the first entries. The complete table is found in Tables A.1-A.5 in Appendix A. Finding charts for all fields containing globulettes are found in Figs. B.1-B.6 in Appendix B. In these charts we have also marked some objects that we do not consider to be regular globulettes, like some clumps with peculiar shapes (see Sect. \[sec:pec\]). Some larger fragments are marked as *Frag* and these features will be commented on in Sect. \[sec:disc\]. Most globulettes have circular or slightly elliptic shapes. The semi-major and semi-minor axes are given in arcseconds in Cols. 7 and 8. These quantites are defined from an outer contour where the intensity level has dropped to 95 % of the interpolated background nebular intensity. Outside this contour, the level of noise starts to affect the definition of the boundary, but as a rule very little matter resides in the outskirts. Column 9 gives the position angle of elongated objects, for which the ratio of semi-major and semi-minor axes is $>$ 1.5.
We derive the physical dimensions of the objects assuming a distance of 2.9 kpc (see Sect. \[sec:intro\]) and define a characteristic radius, $\bar r$, as the mean of the semi-major and semi-minor axes expressed in kAU (Column 10). For the determination of mass we strictly follow the procedure as described in detail in Paper 1 and Grenman ([@gre06]). In short, we measure the residual intensity for each pixel within a globulette relative to the interpolated bright background. This value relates to extinction due to dust at $\lambda$ 6563 [Å]{} ($A_{\alpha}$). Two extreme cases are considered: there is no foreground emission at all, or practically all the residual intensity in the darkest areas of each object is caused by foreground emission. We assume a standard interstellar reddening law (Savage & Mathis [@sav79]) to compute the visual extinction, $A_{V}$ = 1.20 $A_{\alpha}$, and the column densities of molecular hydrogen, $N(H_{2})$ = 9.4 10$^{20} A_V$, according to the relations in Bohlin et al. ([@boh78]) for each pixel assuming a standard mass ratio of gas to dust of 100, and that all hydrogen is in molecular form. The total column density is derived assuming a cosmic chemical composition. Finally, we sum over all pixels inside the contour defined above to obtain the total mass, and we select the mean of the two extreme cases defined above as a measure of the mass of each object. Column 11 gives the so derived mean mass of each globulette. The maximum and minimum masses rarely differ from the mean by more than a factor of two.
In the last column remarks about individual objects are found. Elongated globulettes are marked as $EL$. Some objects have developed tails or tear-drop forms and are marked $T$. Objects with pronounced bright rims are marked $BR$, and those with bright halos as $BH$. The derived masses for the $BH$ objects are lower limits, and their masses are set in italics in Column 11. Symbol $C$ indicates that the object is connected by a dark, thin filament to a larger structure, like a nearby trunk, or to another globulette (with number marked). Objects noted in Smith et al. ([@smi03]) are marked $S$ in Column 12 followed by the symbol they used, and two HH candidates recognized in Smith et al. ([@smi10a]) are also noted.
The derived masses are subject to other uncertainties as well. For instance, uniform density has been assumed, which is consistent (to a first approximation) with column densities derived as a function of radial position (see Paper 1). However, the objects may have developed dense cores that escape detection. Another concern is the use of a normal extinction law since larger-than-normal ratios of $R$ have been found in certain areas (e.g. Thé et al. [@the80]; Smith [@smi02]; Tapia et al. [@tap03]; Hur et al. [@hur12]). Since the globulettes may condense from larger clouds, they may contain larger dust grains than assumed for a normal extinction law. Finally, nebular H$\alpha$ photons entering a globulette may scatter into the line of sight to the observer (e.g. Mattila et al. [@mat07]). This effect would lead to an underestimation of mass. The effect is expected to be small, but cannot be evaluated further until more precise information exists on locations within the nebula and local radiation fields.
Results {#sec:results}
=======
We have found a total of 288 globulettes in the HST-images of the Carina complex. Most of the objects are dark without any bright rims or halos, just like those found in surveys of other regions. The globulettes are spread over the entire region, but are more abundant along the western part of the V-shaped dark cloud and in areas surrounding Tr 14 and 16. Examples of quite isolated globulettes can be found in Fields 10 and 25 in Figs. \[fields2\] and \[fields4\]. Clusters of globulettes are found in, for example, Fields 12 and 41 in Figs. \[fields2\] and \[fields6\]. The total number of globulettes found exceeds the number found in any other region. This large complex is comparatively well covered by HST observations and the number per unit area is comparable to the areas studied by De Marco et al. ([@mar06]).
Distributions of radii and masses {#sec:distribute}
---------------------------------
{width="16cm"}
The left panel in Fig. \[distribution\] shows the distribution of average radii of the Carina globulettes expressed in arcsec, and in kAU in the middle panel. The bulk of the Carina globulettes have radii $<$ 1000 AU, and the distribution increases steeply towards the detection limit. Hence, the Carina globulettes are, on the whole, significantly smaller than the accumulated distribution for the seven regions investigated by De Marco et al. ([@mar06]), which peaks at 2.5 kAU, and with detection limits similar to ours. We note that if we instead assume a distance of 2.3 kpc to the Carina complex, as advocated by Smith ([@smi08]), then the Carina globulettes would be even smaller by $\sim$ 20%.
The masses derived for the tiny globulettes in the Carina complex are consequently also, on the whole, considerably smaller than for other regions. Most of the globulettes have masses well within the domain of planetary masses. The right panel in Fig. \[distribution\] shows that the number of such objects increases rapidly below 3 M$_{J}$ towards the detection limit. Only 4 % of the Carina globulettes are more massive than 10 M$_{J}$, the most massive being CN 78 and 80 with $\sim$ 130 M$_{J}$. This is in sharp contrast to the corresponding distribution in the Rosette Nebula that hosts a large number of more massive clumps, some with masses of several hundred M$_{J}$ (Paper 1). However, even though this complex is at half the distance to the Carina complex, tiny objects with masses $<$ 2 M$_{J}$ escape detection in this ground-based survey. The largest objects with masses $>$ 20 M$_{J}$ are located close to and along the V-shaped dust feature (Fields 9, 11, and 12) and to the south in Field 3, Position 30 (see Fig. \[map\]). They may represent relatively recent detachments from the nearby shell structures.
The Carina globulettes not only differ in size from those in other regions, but also in density. Their average density amounts to $\rho$ = 2.8 10$^{-19}$ g cm$^{-3}$ compared to $\rho$ = 6.2 10$^{-20}$ g cm$^{-3}$ for those in the Rosette Nebula. In terms of number densities of molecular hydrogen they exceed 10$^{5}$ cm$^{-3}$ in several Carina globulettes.
Orientations {#sec:PAs}
------------
Elongated globulettes, with or without tails, line up in the same direction in certain areas, but are more randomly oriented in others. The lower left-panel in Fig. \[PAfields\] shows the location of three selected areas, I to III, projected on a strip composed from images obtained from Spitzer/IRAC 4.5 $\mu$m images (key no 23695360). The two upper panels show areas I and II on the 4.5 $\mu$m background, while for area III (bottom-right panel) the optical and Spitzer images are superimposed to better illustrate the locations of stars and bright and dark nebulosity. Included are also objects with pronounced bright rims and tails and a few round objects surrounded by bright halos that are distinctly brighter at one side. Obviously, some objects classified as round may in fact be elongated if they are oriented closer to the line of sight, and objects surrounded by bright halos flag the presence of bright rims on the remote side.
All of the objects depicted in areas I and II are oriented in about the same direction and point at the cluster Tr 14, located in area III, and it is clear that the objects have been sculpted by the interaction with photons coming from the bright stars in this cluster. It should be noted that there are also a large number of round, dark objects in these areas. In area III, elongated objects are more randomly oriented, particularly in the central part of the image. An example is CN 93 with 60 M$_{J}$ in Field 18, an isolated globulette seen in projection against the cluster Tr 14. However, both the direction of the tail and the bright rim indicates that the globulette is influenced by some object east of the cluster core. Globulettes just above and along the western extension of the V-shaped dust lane (in the right part of this panel), as well as a group to the left in the panel point in the general direction of the bright nebulosity surrounding $\eta$ Carina and Tr 16. Several O stars are spread over this nebulosity, and it is likely that their combined radiation caused the shaping and orientation of these elongated globulettes. In addition, we found the same predominant direction of objects in regions south of area III (not shown here).
Peculiar objects {#sec:pec}
----------------
The Carina Nebula hosts a large number of clouds of very irregular shape like dark worm-like filaments and larger fragments, such as the “Defiant Fingers” described by Smith et al. ([@smi04]). Some of these fragments are accompanied by smaller cloudlets with irregular shapes that most likely have eroded from the larger ones. We did not include these objects in the list of globulettes. In Figs. B1-B6 we have marked some larger fragments, since they might be important birth-places of globulettes (see Sect. \[sec:disc\]).
The border-line between what we define as a globulette or not is a bit arbitrary, but this is less important in our statistical analysis. Figure \[strange\] shows some examples of cloudlets with peculiar shapes, and their locations can be found in Figs. B1-B6, except for $j$ which is in an HST field void of globulettes. This object, like object $c$, was noted and labelled in Smith et al. ([@smi03]). Objects B and D could be similar to standard globulettes, but since B appears to be surrounded by strong H$\alpha$ emission, and D is in the background behind strong foreground emission, these objects could not be measured for extinction. The other objects have peculiar dusty tails, and object A even shows three such outgrowths. These tails can be the result of erosion of elongated objects with peculiar density structure, or they may be examples of dust-enshrouded jets from embedded sources. Objects CN 219 and 241 were recognized as HH-objetcs in Smith et al. ([@smi03]), but their nature remains unclear. The objects CN 219 has a bright rim with a detached bright spot just to the north and the thin, twisted dust-tail in CN 241 is remarkable as is the surrounding, faint, and very elongated bright halo. Some of the peculiar objects in Fig. \[strange\] deserve a closer inspection.
![Average density versus radius for all dark globulettes in the Carina complex.[]{data-label="RhoR"}](RhoRadius.pdf){width="10cm"}
Discussion {#sec:disc}
==========
None of the 288 globulettes coincides in position with any of the YSO candidates listed in Povich et al. ([@pov11]) and Gaczkowski et al. ([@gac13]). There are stars seen inside the boundaries of two globulettes in optical images, namely CN 138 (Fig. \[mosaik\]) and the object designated $j$ in Fig. 5 in Smith et al. ([@smi03]). These stars are likely to be foreground stars, since they show no sign of IR excess judging from the Two Micron All Sky Survey (Skrutskie et al. [@skr06]) or existing Spitzer images. There are no obvious proplyd candidates in our sample, except possibly for CN 219 with a jet-like extension (see Figure \[strange\]) and listed as a Herbig-Haro object (HH 1011) in Smith et al. ([@smi10a]).
The fraction of objects with bright rims and halos is 39%, which is large compared to findings from other regions (De Marco et al. [@mar06]; Paper 1). In the central parts of the Carina Nebula objects with bright rims even dominate over those without indicating that the interaction with the radiation field is more intense closer to the centre.
An important finding is the statistical relation between average density and radius shown in Fig. \[RhoR\], where we have selected only distinctly dark objects without bright rims and halos. The smallest objects with radii $<$ 1 kAU are on average four times denser than those with radii $>$2 kAU. The corresponding distribution including all objects in Tables A1-A5 shows the same general trend but with a larger scatter, and where the densities of BRs and BHs are systematically lower than the distribution in Fig. \[RhoR\]. This is expected since the masses for these objects are lower limits as pointed out in Sect. \[sec:data\]. It is likely that the distribution in Fig. \[RhoR\] reflects how globulettes evolve with time.
Origin and fate {#sec:origin}
---------------
Most dark formations seen in the optical images are located in front of the central regions of bright nebulosity. For the mass estimates we have considered two extreme cases: all residual emission in the darkest parts is due to foreground emission, or there is (practically) no foreground emission. The Carina region is very complex, and it is difficult to judge how deeply embedded a given globulette is in the warm nebulosity. For the same reason it is hard to determine the geometrical distance between a given globulette and other dust formations or clusters in the area. Some globulettes are quite isolated with projected distances to the closest dust complexes of more than 1.5 pc, for example the group of globulettes in Field 14 containing Tr 14. A possible scenario is that this group is the remnant of a larger cloud that gradually eroded in the intense radiation field from Tr 14.
It was inferred in Paper 1 that globulettes in the Rosette Nebula originate from condensations in elephant trunks and shell features. In the Carina complex there are a number of isolated, larger fragments that must have detached from shell structures long ago. Such fragments are marked in Fields 7, 9, 12, 24, and 29. Figure \[fragments\] shows examples of such fragments and all contain condensations with masses similar to those found in globulettes. Fragment 4 is surrounded by a cluster of smaller irregular fragments that appear to be leftovers from a presumably larger block that once eroded. We note that Fragment 2, which hosts several condensations, looks like a detached elephant trunk composed by a network of thin twisted filaments similar to the threaded elephant trunks discussed in Carlqvist et al. ([@car03]).
As discussed in Paper 2, the lack of distinct bright rims in H$\alpha$ may be traced to a combination of several circumstances. One is that the density distributions are rather flat and that the density is high even close to the surface where the gas is in molecular form as flagged by fluorescent H$_{2}$ emission. In addition, thin P$\beta$ emitting rims were discovered to be present in several objects, rims that in some cases appear to extend over the remote side of the objects. Such thin bright rims, not detected in H$\alpha$, were also found in several much smaller globulettes in the Rosette Nebula (Mäkelä et al. [@mak14]). Moreover, the Carina objects could be located at considerable distances from the bright UV-radiating stars, in which case the flux of exciting Lyman continuum photons is moderate producing only weak photodissociation in the outer layer at the dense surface.
The large number of quite isolated globulettes indicates that the objects have survived for a long time in the nebula. Most of these objects are tiny and dense, and unlike the larger objects they lack thin envelopes. It appears that the population of tiny globulettes in the Carina complex are in a more evolved state than those encountered in other regions, either because they have eroded faster in the intense radiation field, or because they are, on the whole, older. A likely scenario is that globulettes detach from larger molecular blocks, like shell structures, pillars, and fragments. Thin envelopes would gradually be lost with time, and the remnant cores may become denser with time. This scenario is further supported by the findings presented in Fig. \[RhoR\] showing that the average density is inversely proportional to radius.
In Paper 1 we applied simple virial arguments to conclude that most globulettes in the Rosette Nebula could be gravitationally unstable, especially after considering the influence of an outer pressure from the surrounding warm plasma. When applying the same analysis to the Carina globulettes we found that the globulettes are close to virial equilibrium but none is bound, even when assuming an outer pressure (thermal plus turbulence) of the same magnitude as in the Rosette complex. On the other hand, the radiation pressure exerted by light from the numerous O stars in Carina should be much higher. This pressure acts on one side of the globulettes. The derived masses are subject to uncertainties (see Sect. \[sec:obs\]), and we note that very little extra mass is needed to confine the objects, as would be the case if they contain denser yet unresolved cores, or more speculatively, even Jupiter-sized planets. This would clarify why the globulettes appear to have survived for such a long time, as can be seen by their distribution over the nebula, where many objects are quite isolated and reside far away from larger molecular structures.
The total number of unbound planets in the Milky Way could amount to several hundred billion (Sumi et al. [@sumi11]). Globulettes in regions may be an additional source of such free-floating planetary-mass objects besides an origin in circumstellar protoplanetary disks from where they are ejected (e.g. Veras et al. [@veras09]).
Conclusions {#sec:conclude}
===========
We have made an inventory of globulettes in the Carina Nebula complex based on existing HST narrow-band H$\alpha$ images. A total of 288 globulettes were listed and measured for size, mass, and density. Most objects are either round or slightly elongated, and many of the latter are oriented in the direction of massive young clusters in the area. We discuss why only a minority have developed bright H$\alpha$ emitting rims and/or tails, and we note that there is no evidence so far of any embedded young stars.
The Carina globulettes are, on the whole, much smaller and less massive than those recognized from HST surveys of a number of other regions. Practically all are of planetary mass, and most have masses less than one Jupiter mass. The corresponding mean densities are much higher than in other regions, exceeding number densities of 10$^{5}$ cm$^{-3}$ in several objects. We found a statistical relation between average density and size in the sense that the smallest globulettes are also the densest. Globulettes may detach from larger blocks of molecular gas, like isolated fragments, elephant trunks, and shell structures, after which their thinner envelopes evaporate and leave denser cores, which may become even more compressed with time.
From virial arguments we conclude that the objects are not bound unless they contain a bit more mass than inferred from the derived mean mass. Most of the tiny objects are quite isolated and located at projected distances of $>$ 1.5 pc from the closest larger molecular structures, which indicates that the objects can survive for long times in the nebula. We speculate that the objects might contain denser cores or even planetary-mass objects that already have formed in their interior.
We suggest that the Carina globulettes are a more evolved state than the larger and less dense objects that are abundant in other regions. Globulettes in regions may be one source of the large number of free-floating planetary-mass objects that has been estimated to exist in the Galaxy.
We thank the referee Bo Reipurth for valuable comments and suggestions. This work was supported by the Magnus Bergvall Foundation, the Längmanska Kulturfonden, and the Swedish National Space Board.
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Properties of the globulettes in the Carina complex {#AppendixA}
===================================================
[lccccccccccl]{} CN & Field & x & y & R.A. & Dec. & $\alpha$ & $\beta$ & P.A. & $\bar r $ & Mass & Remarks\
&&&& (J2000.0) &(J2000.0) & (arcsec) & (arcsec) & (degr.) & (kAU) & (M$_{J}$) &\
\
1 & F1 & 360 & 4020 & 10:41:13.3 & -59:49:00 & 0.46 & 0.50 & & 1.39 & 2.4 &\
2 & F1 & 660 & 1084 & 10:41:18.9 & -59:46:38 & 0.30 & 0.36 & & 0.96 & 1.1 &\
3 & F1 & 1792 & 3251 & 10:41:23.6 & -59:48:35 & 0.38 & 0.40 & & 1.13 & 1.3 &\
4 & F1 & 1654 & 378 & 10:41:26.2 & -59:46:13 & 0.18 & 0.19 & & 0.54 & 0.4 &\
5 & F1 & 1819 & 395 & 10:41:27.2 & -59:46:15 & 0.19 & 0.27 & & 0.67 & 0.6 &\
6 & F1 & 2029 & 221 & 10:41:28.8 & -59:46:09 & 0.26 & 0.33 & 23 & 0.86 & 1.2 & T\
7 & F2 & 670 & 1907 & 10:41:28.9 & -59:45:53 & 0.23 & 0.25 & -43 & 0.69 &*0.6 & BR,T\
8 & F1 & 2391 & 1935 & 10:41:29.1 & -59:47:36 & 0.20 & 0.22 & & 0.61 & 0.5 &\
9 & F1 & 2156 & 597 & 10:41:29.2 & -59:46:29 & 0.29 & 0.33 & & 0.90 & 1.0 &\
10 & F1 & 2752 & 1904 & 10:41:31.5 & -59:47:38 & 0.54 & 1.46 & 8 & 2.90 & *8.8 & BR,EL,T\
11 & F1 & 2604 & 373 & 10:41:32.4 & -59:46:22 & 0.31 & 0.51 & -38 & 1.19 & 2.0 & EL\
12 & F2 & 1207 & 1745 & 10:41:32.6 & -59:45:50 & 0.31 & 0.35 & -15 & 0.96 & 1.5 & T\
13 & F1 & 2808 & 667 & 10:41:33.4 & -59:46:38 & 0.36 & 0.70 & 13 & 1.54 & 2.7 & EL\
14 & F2 & 1694 & 1606 & 10:41:35.9 & -59:45:48 & 0.39 & 0.46 & & 1.23 & 1.8 &\
15 & F3 & 2659 & 851 & 10:41:44.9 & -59:43:44 & 1.47 & 1.67 & & 4.55 & 40 &\
16 & F3 & 2806 & 1494 & 10:41:45.2 & -59:44:17 & 0.52 & 0.71 & & 1.78 & 4.5 &\
17 & F3 & 2706 & 964 & 10:41:45.2 & -59:43:50 & 0.20 & 0.21 & & 0.59 & 0.5 &\
18 & F3 & 2972 & 933 & 10:41:46.9 & -59:43:51 & 0.69 & 1.82 & 12& 3.64 & 22 & EL,T\
19 & F3 & 3527 & 1546 & 10:41:49.8 & -59:44:26 & 0.31 & 0.44 & & 1.09 & 1.3 &\
20 & F3 & 3656 & 1501 & 10:41:50.7 & -59:44:25 & 0.36 & 0.67 & 6 & 1.49 & *2.5 & BR,EL\
21 & F2 & 3781 & 272 & 10:41:51.2 & -59:45:02 & 0.64 & 1.00 & 10& 2.38 & *5.6 & BR,EL\
22 & F4 & 2772 & 3493 & 10:42:20.7 & -59:19:45 & 0.35 & 0.40 & & 1.09 & *1.6 & BR\
23 & F5 & 2719 & 1339 & 10:42:43.7 & -59:20:16 & 0.18 & 0.20 & & 0.55 & 0.4 &\
24 & F5 & 1880 & 938 & 10:42:48.7 & -59:20:42 & 0.35 & 0.72 & -5 & 1.55 & *2.8 & BR,EL\
25 & F5 & 1242 & 524 & 10:42:52.4 & -59:21:08 & 0.25 & 0.47 & -3 & 1.04 & *1.5 & BR,EL\
26 & F8 & 1029 & 1467 & 10:43:15.9 & -59:28:07 & 0.13 & 0.14 & & 0.39 & 0.2 &\
27 & F6 & 3938 & 1861 & 10:43:16.1 & -59:28:31 & 0.52 & 1.22 & 33 & 2.52 & 8.5 & EL,T,C\
28 & F8 & 973 & 1754 & 10:43:16.8 & -59:28:19 & 0.21 & 0.27 & & 0.70 & 0.8 &\
29 & F6 & 2697 & 3970 & 10:43:17.6 & -59:30:33 & 0.26 & 0.28 & & 0.78 & 0.9 &\
30 & F7 & 1960 & 1480 & 10:43:19.9 & -59:31:45 & 0.24 & 0.38 & 20 & 0.90 & 1.5 & EL\
31 & F7 & 3105 & 479 & 10:43:22.1 & -59:30:31 & 0.36 & 0.61 & 23 & 1.41 & 4.3 & EL\
32 & F7 & 2177 & 3163 & 10:43:27.5 & -59:32:47 & 0.35 & 0.41 & & 1.10 & 2.4 &\
33 & F13 & 3571 & 682 & 10:43:31.4 & -59:55:20 & 0.49 & 0.75 & -13 & 1.80 & *4.0 & BH,EL\
34 & F7 & 2583 & 3776 & 10:43:32.0 & -59:33:00 & 0.23 & 0.26 & 36 & 0.71 & 0.7 & T\
35 & F7 & 2883 & 4143 & 10:43:35.1 & -59:33:06 & 0.19 & 0.25 & & 0.64 & 0.5 &\
36 & F7 & 3181 & 3863 & 10:43:35.6 & -59:32:46 & 0.23 & 0.24 & & 0.68 & 0.7 &\
37 & F7 & 3360 & 4002 & 10:43:37.1 & -59:32:46 & 0.11 & 0.13 & & 0.35 & 0.2 &\
38 & F9 & 3329 & 499 & 10:43:39.1 & -59:32:57 & 0.95 & 1.33 & & 3.31 & 34 & S$e$\
39 & F9 & 3552 & 393 & 10:43:39.8 & -59:32:46 & 0.39 & 0.51 & & 1.31 & 4.0 & S$e$\
40 & F10 & 1537 & 3364 & 10:43:43.8 & -59:31:24 & 0.19 & 0.21 & & 0.58 & 0.6 &\
41 & F11 & 1933 & 2248 & 10:43:46.4 & -59:36:05 & 0.18 & 0.25 & & 0.62 & 0.5 &\
42 & F11 & 1935 & 2353 & 10:43:46.8 & -59:36:09 & 0.35 & 0.46 & & 1.17 &2.6 &\
43 & F11 & 2066 & 2364 & 10:43:47.6 & -59:36:06 & 0.26 & 0.34 & -10 & 0.87 & 1.3 & T\
44 & F9 & 2972 & 3165 & 10:43:47.6 & -59:34:56 & 0.33 & 0.40 & & 1.06 & 2.1 &\
45 & F9 & 3255 & 2797 & 10:43:47.7 & -59:34:33 & 0.13 & 0.13 & & 0.38 & 0.2 &\
46 & F9 & 3325 & 2858 & 10:43:48.3 & -59:34:33 & 0.11 & 0.13 & & 0.35 & 0.2 &\
47 & F9 & 2914 & 3633 & 10:43:49.1 & -59:35:16 & 0.13 & 0.14 & & 0.39 & 0.2 &\
48 & F9 & 2893 & 3668 & 10:43:49.1 & -59:35:18 & 0.11 & 0.12 & & 0.33 & 0.1 &\
49 & F10 & 2378 & 3919 & 10:43:50.4 & -59:31:18 & 0.36 & 0.44 & & 1.16 & 3.2 &\
50 & F9 & 3229 & 3794 & 10:43:51.4 & -59:35:14 & 0.30 & 0.41 & & 1.03 & 2.0 &\
51 & F12 & 506 & 3746 & 10:43:52.5 & -59:39:03 & 0.56 & 0.78 & & 1.94 & 5.0 &\
52 & F16 & 3116 & 1570 & 10:43:52.9 & -59:58:49 & 0.27 & 0.34 & & 0.88 & 1.0 &\
53 & F11 & 2489 & 3191 & 10:43:53.0 & -59:36:27 & 0.21 & 0.30 & & 0.74 & 0.8 &\
54 & F11 & 2530 & 3187 & 10:43:53.2 & -59:36:25 & 0.33 & 0.48 & & 1.17 & 2.7 & C to 56\
55 & F11 & 2573 & 3152 & 10:43:53.3 & -59:36:23 & 0.12 & 0.16 & 39 & 0.41 & 0.3 & T\
56 & F11 & 2554 & 3196 & 10:43:53.3 & -59:36:25 & 0.26 & 0.35 & & 0.88 & 1.4 & C to 54\
57 & F11 & 2589 & 3155 & 10:43:53.4 & -59:36:22 & 0.12 & 0.13 & & 0.36 & 0.2 &\
58 & F11 & 2873 & 2777 & 10:43:53.5 & -59:35:59 & 0.25 & 0.29 & & 0.78 & 1.0 &\
59 & F11 & 2611 & 3162 & 10:43:53.6 & -59:36:22 & 0.13 & 0.15 & & 0.41 & 0.2 &\
********
\[globCSD\]
[lccccccccccl]{} CN & Field & x & y & R.A. & Dec. & $\alpha$ & $\beta$ & P.A. & $\bar r $ & Mass & Remarks\
&&&& (J2000.0) &(J2000.0) & (arcsec) & (arcsec) & (degr.) & (kAU) & (M$_{J}$) &\
\
60 & F11 & 2952 & 2770 & 10:43:53.9 & -59:35:56 & 0.15 & 0.17 & & 0.46 & 0.3 &\
61 & F11 & 2569 & 3315 & 10:43:53.9 & -59:36:29 & 0.27 & 0.70 & -6 & 1.38 & 2.7 & EL\
62 & F11 & 2874 & 2959 & 10:43:54.2 & -59:36:06 & 0.16 & 0.24 & & 0.58 & 0.4 &\
63 & F11 & 2722 & 3170 & 10:43:54.2 & -59:36:19 & 0.13 & 0.14 & & 0.39 & 0.2 &\
64 & F12 & 830 & 3731 & 10:43:54.2 & -59:38:53 & 0.44 & 0.47 & & 1.32 & 2.1 &\
65 & F11 & 2595 & 3360 & 10:43:54.2 & -59:36:31 & 0.15 & 0.16 & & 0.45 & 0.2 &\
66 & F16 & 2341 & 102 & 10:43:54.5 & -60:00:11 & 0.33 & 0.41 & & 1.07 & 2.2 &\
67 & F12 & 891 & 3782 & 10:43:54.7 & -59:38:54 & 0.17 & 0.18 & & 0.51 & 0.3 &\
68 & F12 & 827 & 3883 & 10:43:54.8 & -59:38:59 & 0.75 & 1.12 & -10 & 2.71 & 14 & EL\
69 & F12 & 1124 & 3501 & 10:43:54.9 & -59:38:35 & 0.36 & 0.42 & & 1.13 & 1.5 &\
70 & F16 & 2304 & 190 & 10:43:54.9 & -60:00:08 & 0.42 & 0.57 & & 1.44 & 4.2 &\
71 & F11 & 2766 & 3381 & 10:43:55.2 & -59:36:26 & 0.13 & 0.16 & & 0.42 & 0.2 &\
72 & F12 & 1251 & 3451 & 10:43:55.4 & -59:38:30 & 0.39 & 0.50 & & 1.29 & 2.5 &\
73 & F11 & 2833 & 3366 & 10:43:55.5 & -59:36:24 & 0.12 & 0.16 & & 0.41 & 0.2 &\
74 & F11 & 2790 & 3454 & 10:43:55.6 & -59:36:29 & 0.20 & 0.23 & & 0.62 & 0.5 &\
75 & F11 & 2796 & 3536 & 10:43:55.9 & -59:36:32 & 0.18 & 0.33 & -9 & 0.74 & 0.7 & EL\
76 & F11 & 2805 & 3527 & 10:43:56.0 & -59:36:31 & 0.18 & 0.26 & & 0.64 & 0.5 &\
77 & F12 & 1049 & 3910 & 10:43:56.1 & -59:38:54 & 0.73 & 1.34 & -16 & 3.00 & 17 & EL\
78 & F11 & 2606 & 3914 & 10:43:56.4 & -59:36:53 & 2.06 & 2.80 & & 7.05 & 128 & C\
79 & F17 & 1970 & 196 & 10:43:56.4 & -59:39:03 & 1.16 & 1.22 & & 3.45 & 29 &\
80 & F11 & 2844 & 3613 & 10:43:56.6 & -59:36:33 & 2.30 & 2.93 & & 7.58 & 132 &\
81 & F17 & 2057 & 149 & 10:43:56.7 & -59:38:58 & 0.63 & 0.65 & & 1.86 & 4.7 &\
82 & F14 & 2761 & 894 & 10:43:56.7 & -59:31:32 & 0.22 & 0.28 & & 0.73 & 1.1 &\
83 & F12 & 1222 & 3852 & 10:43:56.8 & -59:38:47 & 0.34 & 0.44 & & 1.13 & 1.3 &\
84 & F14 & 2839 & 813 & 10:43:56.8 & -59:31:26 & 0.13 & 0.15 & & 0.41 & 0.2 &\
85 & F12 & 1406 & 3654 & 10:43:56.9 & -59:38:33 & 0.70 & 0.82 & & 2.20 & 10 &\
86 & F17 & 2182 & 172 & 10:43:57.5 & -59:38:56 & 0.44 & 0.50 & & 1.36 & 3.0 &\
87 & F12 & 1314 & 3927 & 10:43:57.5 & -59:38:47 & 0.71 & 0.84 & & 2.25 & 12 &\
88 & F14 & 3211 & 691 & 10:43:58.1 & -59:31:10 & 0.13 & 0.21 & & 0.49 & 0.3 &\
89 & F11 & 3786 & 2786 & 10:43:58.4 & -59:35:32 & 0.13 & 0.15 & & 0.41 & 0.2 &\
90 & F17 & 2303 & 274 & 10:43:58.5 & -59:38:56 & 0.58 & 0.81 & & 2.02 & 5.8 &\
91 & F14 & 3214 & 863 & 10:43:58.9 & -59:31:16 & 0.12 & 0.12 & & 0.35 & 0.2 &\
92 & F14 & 2726 & 1454 & 10:43:58.9 & -59:31:54 & 0.22 & 0.27 & & 0.71 & 0.8 &\
93 & F18 & 2146 & 258 & 10:43:59.6 & -59:32:38 & 1.31 & 1.69 & 5 & 4.35 & 63 & T,S$i$\
94 & F18 & 384 & 3025 & 10:44:01.1 & -59:35:22 & 0.14 & 0.16 & & 0.44 & 0.2 &\
95 & F14 & 2903 & 1866 & 10:44:01.6 & -59:32:04 & 0.09 & 0.11 & & 0.29 & 0.1 &\
96 & F18 & 575 & 2990 & 10:44:01.9 & -59:35:15 & 0.18 & 0.26 & -40 & 0.64 & 0.8 & T\
97 & F14 & 2890 & 2055 & 10:44:02.4 & -59:32:12 & 0.12 & 0.13 & & 0.36 & 0.2 &\
98 & F14 & 2865 & 2085 & 10:44:02.4 & -59:32:14 & 0.37 & 0.46 & & 1.20 & *3.2 & BR,S$f$\
99 & F18 & 559 & 3151 & 10:44:02.5 & -59:35:22 & 0.20 & 0.22 & & 0.61 & 0.5 &\
100 & F14 & 2994 & 2059 & 10:44:02.8 & -59:32:08 & 0.36 & 0.54 & & 1.31 & *4.0 & S$f$\
101 & F18 & 660 & 3153 & 10:44:03.0 & -59:35:19 & 0.17 & 0.30 & -13 & 0.68 & 0.6 & EL,T\
102 & F12 & 3489 & 2374 & 10:44:03.1 & -59:36:40 & 0.12 & 0.23 & 3 & 0.51 & 0.3 & EL\
103 & F18 & 722 & 3146 & 10:44:03.3 & -59:35:17 & 0.15 & 0.23 & 11 & 0.55 & 0.5 & EL,T\
104 & F18 & 708 & 3183 & 10:44:03.4 & -59:35:19 & 0.17 & 0.21 & & 0.55 & 0.5 &\
105 & F12 & 3420 & 2634 & 10:44:03.8 & -59:36:53 & 0.10 & 0.10 & & 0.29 & 0.1 &\
106 & F12 & 3392 & 2729 & 10:44:03.9 & -59:36:57 & 0.10 & 0.23 & 27& 0.48 & *0.2 & BH,EL\
107 & F12 & 3409 & 2742 & 10:44:04.1 & -59:36:57 & 0.10 & 0.16 & 7 & 0.38 & *0.2 & BH,EL\
108 & F12 & 3447 & 2703 & 10:44:04.2 & -59:36:55 & 0.10 & 0.17 & 21& 0.38 & *0.2 & BH,EL\
109 & F12 & 3450 & 2789 & 10:44:04.5 & -59:36:58 & 0.12 & 0.19 & 22& 0.45 & *0.2 & BH,EL\
110 & F17 & 2657 & 1400 & 10:44:04.7 & -59:39:32 & 0.33 & 0.45 & & 1.13 & 2.3 &\
111 & F17 & 2603 &1490 & 10:44:04.8 & -59:39:37 & 0.49 & 0.62 & & 1.61 & 5.2 &\
112 & F17 & 2587 & 1545 & 10:44:04.9 & -59:39:40 & 0.43 & 0.50 & & 1.35 & 3.2 &\
113 & F14 & 2686 & 2944 & 10:44:05.2 & -59:32:52 & 0.26 & 0.69 & 43 & 1.38 & 3.2 & EL\
114 & F17 & 2737 & 1427 & 10:44:05.3 & -59:39:30 & 0.11 & 0.10 & & 0.30 & 0.1 &\
115 & F14 & 3623 & 2009 & 10:44:05.8 & -59:31:46 & 0.14 & 0.24 & 44 & 0.55 & 0.5 & EL\
116 & F15 & 2760 & 2864 & 10:44:06.4 & -59:28:13 & 0.17 & 0.35 & -31 & 0.75 & *0.8 & BH,EL\
117 & F14 & 2847 & 3192 & 10:44:07.0 & -59:32:56 & 0.26 & 0.38 & & 0.93 & 1.8 &\
118 & F14 & 3190 & 2832 & 10:44:07.2 & -59:32:31 & 0.08 & 0.09 & & 0.25 & 0.1 &\
119 & F17 & 2961 & 1710 & 10:44:07.6 & -59:39:35 & 1.22 & 1.68 & & 4.20 & 33 &\
*******
\[globCSD\]
[lccccccccccl]{} CN & Field & x & y & R.A. & Dec. & $\alpha$ & $\beta$ & P.A. & $\bar r $ & Mass & Remarks\
&&&& (J2000.0) &(J2000.0) & (arcsec) & (arcsec) & (degr.) & (kAU) & (M$_{J}$) &\
\
120 & F14 & 3189 & 2936 & 10:44:07.6 & -59:32:35 & 0.11 & 0.12 & & 0.33 & 0.2 &\
121 & F14 & 3203 & 3000 & 10:44:07.9 & -59:32:37 & 0.13 & 0.14 & & 0.39 & 0.2 &\
122 & F14 & 2861 & 3513 & 10:44:08.5 & -59:33:08 & 0.19 & 0.23 & & 0.61 & 0.6 &\
123 & F14 & 3194 & 3190 & 10:44:08.7 & -59:32:45 & 0.23 & 0.29 & & 0.75 & 1.1 &\
124 & F14 & 3272 & 3217 & 10:44:09.3 & -59:32:43 & 0.18 & 0.23 & & 0.59 & 0.5 &\
125 & F14 & 2906 & 3805 & 10:44:09.9 & -59:33:17 & 0.12 & 0.14 & & 0.38 & 0.2 &\
126 & F14 & 3011 & 3781 & 10:44:10.4 & -59:33:13 & 0.20 & 0.39 & 16 & 0.86 & 0.9 & EL\
127 & F14 & 2991 & 3827 & 10:44:10.5 & -59:33:16 & 0.28 & 0.49 & -5 & 1.12 & *2.3 & BH,EL\
128 & F14 & 2909 & 3956 & 10:44:10.6 & -59:33:23 & 0.11 & 0.12 & & 0.33 & 0.2 &\
129 & F14 & 2977 & 3907 & 10:44:10.7 & -59:33:19 & 0.15 & 0.39 & 43 & 0.78 & 0.8 & EL\
130 & F20 & 1107 & 2190 & 10:44:12.6 & -59:31:03 & 0.19 & 0.29 & 20 & 0.70 & *0.7 & BH,EL\
131 & F21 & 1458 & 1517 & 10:44:12.8 & -59:25:53 & 0.17 & 0.21 & & 0.55 & *0.4 & BR\
132 & F21 & 1441 & 1599 & 10:44:13.0 & -59:25:57 & 0.16 & 0.20 & & 0.52 & *0.4 &\
133 & F21 & 1347 & 1758 & 10:44:13.2 & -59:26:06 & 0.38 & 0.89 & -11 & 1.84 & *3.8 & BH,EL\
134 & F19 & 1781 & 2650 & 10:44:15.0 & -59:35:41 & 0.21 & 0.23 & & 0.64 & 0.6 &\
135 & F20 & 1747 & 2070 & 10:44:15.3 & -59:30:38 & 0.13 & 0.23 & 3 & 0.52 & *0.3 & BH,EL\
136 & F22 & 1814 & 1588 & 10:44:16.8 & -59:42:36 & 0.10 & 0.14 & & 0.35 & *0.2 & BH\
137 & F22 & 2295 & 1451 & 10:44:18.8 & -59:42:16 & 0.41 & 0.52 & -26 & 1.35 & *2.5 & BH,T\
138 & F20 & 3991 & 352 & 10:44:19.1 & -59:28:20 & 0.32 & 0.67 & -24 & 1.44 & *2.7 & BH,EL,T\
139 & F22 & 2338 & 1531 & 10:44:19.4 & -59:42:18 & 0.11 & 0.15 & & 0.38 & *0.2 & BH\
140 & F28 & 287 & 834 & 10:44:19.9 & -59:33:12 & 0.09 & 0.10 & & 0.28 & 0.1 &\
141 & F28 & 245 & 1067 & 10:44:20.6 & -59:33:23 & 0.13 & 0.16 & & 0.42 & 0.2 &\
142 & F28 & 439 & 1398 & 10:44:22.9 & -59:33:30 & 0.41 & 0.90 & -17 & 1.90 & *10 & BH,EL\
143 & F26 & 3864 & 1397 & 10:44:28.1 & -60:00:08 & 0.18 & 0.26 & & 0.64 & 0.5 &\
144 & F26 & 3874 & 1486 & 10:44:28.1 & -60:00:03 & 0.26 & 0.34 & & 0.87 & 1.2 &\
145 & F26 & 3875 & 1543 & 10:44:28.2 & -60:00:00 & 0.51 & 0.71 & & 1.77 & 4.0 &\
146 & F26 & 3811 & 1412 & 10:44:28.5 & -60:00:07 & 0.27 & 0.32 & & 0.86 & 1.1 &\
147 & F26 & 3810 & 1434 & 10:44:28.5 & -60:00:06 & 0.24 & 0.27 & & 0.74 & 0.8 &\
148 & F26 & 3759 & 1382 & 10:44:28.8 & -60:00:09 & 0.23 & 0.28 & & 0.74 & 0.8 &\
149 & F26 & 3623 & 506 & 10:44:28.9 & -60:00:53 & 0.25 & 0.32 & & 0.83 & 0.9 &\
150 & F23 & 2896 & 2906 & 10:44:29.1 & -59:38:40 & 0.11 & 0.12 & & 0.33 & *0.2 & BH\
151 & F23 & 3070 & 2960 & 10:44:30.2 & -59:38:37 & 0.13 & 0.16 & & 0.42 & *0.3 & BH\
152 & F28 & 552 & 3222 & 10:44:30.6 & -59:34:41 & 0.17 & 0.25 & & 0.61 & 0.5 &\
153 & F23 & 2756 & 3560 & 10:44:31.1 & -59:39:10 & 0.18 & 0.33 & & 0.74 & *0.6 & BH\
154 & F28 & 503 & 3487 & 10:44:31.3 & -59:34:53 & 0.12 & 0.13 & & 0.36 & 0.2 &\
155 & F26 & 3610 & 3069 & 10:44:31.4 & -59:58:47 & 0.23 & 0.23 & & 0.67 & 0.5 &C\
156 & F28 & 657 & 3347 & 10:44:31.6 & -59:34:43 & 0.12 & 0.16 & & 0.41 & *0.2 & BH\
157 & F23 & 2990 & 3452 & 10:44:31.9 & -59:39:01 & 0.11 & 0.11 & & 0.32 & *0.1 & BH\
158 & F23 & 3159 & 3340 & 10:44:32.2 & -59:38:49 & 0.12 & 0.22 & 24 & 0.49 & *0.3 & BH,EL\
159 & F28 & 650 & 3554 & 10:44:32.4 & -59:34:51 & 0.18 & 0.30 & -11 & 0.69 & 0.6 & EL\
160 & F27 & 3512 & 750 & 10:44:32.5 & -59:57:30 & 0.42 & 1.19 & 6 & 2.33 & 9.1 &EL,T\
161 & F28 & 1561 & 2363 & 10:44:32.6 & -59:33:36 & 0.10 & 0.12 & & 0.32 & *0.1 & BH\
162 & F28 & 671 & 3584 & 10:44:32.6 & -59:34:52 & 0.11 & 0.21 & 44 & 0.46 & 0.3 & EL\
163 & F29 & 156 & 490 & 10:44:32.6 & -59:37:36 & 0.15 & 0.20 & & 0.51 & *0.5 & BH\
164 & F26 & 3436 & 3148 & 10:44:32.6 & -59:58:44 & 0.24 & 0.23 & & 0.68 & 0.7 & C\
165 & F28 & 731 & 3521 & 10:44:32.7 & -59:34:47 & 0.25 & 0.50 & -5 & 1.09 & *1.6 & BH,EL\
166 & F28 & 1180 & 2970 & 10:44:32.9 & -59:34:12 & 0.11 & 0.15 & & 0.38 & *0.2 & BR\
167 & F28 & 960 & 3282 & 10:44:32.9 & -59:34:31 & 0.26 & 0.47 & -37 & 1.06 & *1.3 & BH,EL\
168 & F27 & 3434 & 856 & 10:44:33.1 & -59:57:25 & 0.14 & 0.16 & & 0.44 & 0.2 &\
169 & F26 & 2931 & 196 & 10:44:33.1 & -60:01:14 & 0.34 & 0.43 & & 1.12 & 1.8 &\
170 & F26 & 3400 & 863 & 10:44:33.3 & -59:57:25 & 0.11 & 0.12 & & 0.33 & 0.1 &\
171 & F27 & 989 & 3389 & 10:44:33.5 & -59:34:34 & 0.11 & 0.22 & -13 & 0.48 & *0.2 & BH,EL\
172 & F26 & 2859 & 463 & 10:44:33.9 & -60:01:01 & 0.50 & 0.55 & & 1.52 & 3.6 &\
173 & F28 & 1165 & 3251 & 10:44:33.9 & -59:34:24 & 0.18 & 0.27 & & 0.65 & *0.6 & BH\
174 & F28 & 1147 & 3298 & 10:44:34.0 & -59:34:26 & 0.14 & 0.32 &-4 & 0.67 & *0.4 & BH,EL\
175 & F26 & 2899 & 971 & 10:44:34.1 & -60:00:36 & 0.46 & 0.73 & 2 & 1.73 & 4.8 & EL,T\
176 & F28 & 1231 & 3217 & 10:44:34.1 & -59:34:20 & 0.12 & 0.24 & 21 & 0.52 &*0.3 & BH,EL\
177 & F28 & 1250 & 3203 & 10:44:34.2 & -59:34:19 & 0.31 & 0.78 & -29 & 1.58 & 3.3 & EL\
178 & F26 & 2733 & 168 & 10:44:34.4 & -60:01:17 & 0.15 & 0.16 & & 0.45 & 0.3 &\
179 & F29 & 455 & 586 & 10:44:34.5 & -59:37:30 & 0.13 & 0.15 & & 0.41 & 0.4 &\
**************************
\[globCSD\]
[lccccccccccl]{} CN & Field & x & y & R.A. & Dec. & $\alpha$ & $\beta$ & P.A. & $\bar r $ & Mass & Remarks\
&&&& (J2000.0) &(J2000.0) & (arcsec) & (arcsec) & (degr.) & (kAU) & (M$_{J}$) &\
\
180 & F28 & 1585 & 2904 & 10:44:34.8 & -59:33:57 & 0.11 & 0.13 & & 0.35 & 0.2 & BH\
181 & F28 & 1330 & 3301 & 10:44:35.0 & -59:34:21 & 0.10 & 0.24 & -9& 0.49 & *0.2 & BH, EL\
182 & F26 & 2599 & 499 & 10:44:35.6 & -60:01:01 & 0.18 & 0.19 & & 0.54 & 0.4 &\
183 & F28 & 1340 & 3499 & 10:44:35.8 & -59:34:29 & 0.12 & 0.15 & & 0.39 & *0.3 & BR\
184 & F28 & 1707 & 3005 & 10:44:35.9 & -59:33:58 & 0.19 & 0.26 & & 0.65 & *0.6 & BH\
185 & F29 & 306 & 1298 & 10:44:36.8 & -59:38:03 & 0.15 & 0.34 & 9 & 0.71 & *0.8 & BH,EL\
186 & F29 & 385 & 1214 & 10:44:36.8 & -59:37:57 & 0.12 & 0.19 & -42 & 0.45 & *0.3 & BH,EL\
187 & F29 & 325 & 1329 & 10:44:37.0 & -59:38:03 & 0.13 & 0.22 & -13 & 0.51 & *0.5 & BH,EL\
188 & F29 & 967 & 666 & 10:44:37.5 & -59:37:17 & 0.10 & 0.12 & & 0.32 & 0.1 &\
189 & F26 & 2389 & 1025 & 10:44:37.5 & -60:00:37 & 0.60 & 0.67 & & 1.84 & 3.5 &\
190 & F29 & 389 & 1433 & 10:44:37.8 & -59:38:05 & 0.18 & 0.34 & 21 & 0.75 & *1.0 & BH,EL,T\
191 & F29 & 416 & 1430 & 10:44:37.9 & -59:38:04 & 0.09 & 0.10 & & 0.28 & *0.1 & BH\
192 & F29 & 601 & 1217 & 10:44:37.9 & -59:37:50 & 0.13 & 0.17 & 9 & 0.44 & *0.3 & BR.EL\
193 & F29 & 466 & 1381 & 10:44:37.9 & -59:38:01 & 0.14 & 0.23 & 7 & 0.54 & *0.3 & BH,EL\
194 & F29 & 1203 & 581 & 10:44:38.3 & -59:37:06 & 0.12 & 0.14 & & 0.38 & 0.2 &\
195 & F28 & 2194 & 3098 & 10:44:38.8 & -59:33:47 & 0.09 & 0.10 & & 0.29 & *0.1 & BH\
196 & F29 & 763 & 1319 & 10:44:39.2 & -59:37:49 & 0.26 & 0.45 & 6 & 1.03 & *2.0 & BH,EL\
197 & F29 & 877 & 1190 & 10:44:39.2 & -59:37:40 & 0.10 & 0.12 & & 0.32 & *0.1 & BH\
198 & F29 & 1014 & 1365 & 10:44:40.6 & -59:37:43 & 0.15 & 0.20 & & 0.51 & 0.3 &\
199 & F25/28 & 1945 & 3616 & 10:44:41.3 & -59:46:26 & 1.22 & 1.46 & & 3.89 & 33 &\
200 & F25 & 3596 & 1558 & 10:44:42.1 & -59:44:14 & 0.35 & 0.59 & 2 & 1.36 & 2.3 & EL\
201 & F33 & 380 & 1177 & 10:44:45.2 & -59:39:07 & 0.15 & 0.25 & -26 & 0.58 & *0.4 & BH,EL\
202 & F32 & 1780 & 3807 & 10:44:46.8 & -59:26:25 & 0.18 & 0.30 & -1 & 0.70 & *0.7 & BR,EL\
203 & F32 & 1884 & 3764 & 10:44:47.0 & -59:26:30 & 0.25 & 0.33 & & 0.84 & *1.1 & BR\
204 & F33 & 1657 & 317 & 10:44:48.1 & -59:37:53 & 0.27 & 0.34 & & 0.88 & *1.1 & BR\
205 & F28 & 3952 & 3839 & 10:44:51.0 & -59:33:25 & 0.11 & 0.10 & & 0.30 & *0.1 & BH\
206 & F31 & 1558 & 3034 & 10:44:55.0 & -59:32:03 & 0.23 & 0.37 & 31 & 0.87 & *1.0 & BH,EL,C to 207\
207 & F31 & 1578 & 3041 & 10:44:55.2 & -59:32:03 & 0.18 & 0.41 & -7 & 0.81 & *0.8 &BH,EL,C to 206\
208 & F28 & 2749 & 4107 & 10:44:55.5 & -59:47:38 & 0.22 & 0.29 & & 0.74 & 0.6 &\
209 & F31 & 1729 & 2933 & 10:44:55.5 & -59:31:54 & 0.12 & 0.17 & & 0.42 & *0.2 & BH\
210 & F31 & 1744 & 2954 & 10:44:55.7 & -59:31:54 & 0.18 & 0.24 & & 0.61 & *0.5 & BH\
211 & F31 & 1776 & 2936 & 10:44:55.8 & -59:31:53 & 0.15 & 0.28 & -8& 0.62 & *0.5 & BH,EL\
212 & F31 & 1822 & 2927 & 10:44:56.0 & -59:31:51 & 0.11 & 0.13 & & 0.35 & *0.2 & BH\
213 & F31 & 1839 & 2938 & 10:44:56.1 & -59:31:51 & 0.24 & 0.50 & -10 & 1.07 & *1.3 & BH,EL\
214 & F33 & 2034 & 1833 & 10:44:56.4 & -59:38:40 & 0.23 & 0.29 & & 0.75 & 1.0 &\
215 & F30 & 2894 & 2278 & 10:44:57.1 & -59:47:36 & 0.37 & 0.42 & & 1.15 & 1.6 &\
216 & F34 & 634 & 493 & 10:44:58.1 & -59:45:30 & 0.24 & 0.31 & & 0.80 & *0.8 & BH\
217 & F33 & 2984 & 1488 & 10:44:59.7 & -59:37:56 & 0.21 & 0.26 & & 0.68 & *0.6 & BR\
218 & F33 & 3039 & 2209 & 10:45:03.0 & -59:38:22 & 0.17 & 0.20 & & 0.54 & 0.4 &\
219 & F32 & 2175 & 991 & 10:45:05.0 & -59:26:57 & 0.34 & 1.23 & 44 & 2.28 & *4.0 & BH,EL,HH 1011\
220 & F34 & 1926 & 694 & 10:45:05.8 & -59:45:00 & 0.27 & 0.33 & & 0.87 & 1.0 &\
221 & F32 & 1901 & 849 & 10:45:06.0 & -59:26:44 & 0.17 & 0.39 & 36& 0.81 & *0.6 & BH,EL,T\
222 & F32 & 1800 & 790 & 10:45:06.5 & -59:26:39 & 0.22 & 0.63 & 35 & 1.22 & *1.4 & BH,EL\
223& F35 & 1157 & 892 & 10:45:07.0 & -59:36:22 & 0.19 & 0.32 & -4 & 0.74 & *0.8 & BH,EL\
224 & F33 & 3463 & 2708 & 10:45:07.3 & -59:38:28 & 0.09 & 0.12 & 9 & 0.30 & *0.1 & BR,EL\
225 & F33 & 3876 & 2237 & 10:45:07.4 & -59:37:56 & 0.13 & 0.17 & & 0.43 & 0.2 &\
226 & F33 & 2807 & 3660 & 10:45:08.0 & -59:39:25 & 0.11 & 0.21 &23 & 0.46 & *0.2 & BR,EL\
227 & F34 & 1339 & 2233 & 10:45:08.7 & -59:46:19 & 0.83 & 0.93 & & 2.55 & 11 &\
228 & F36 & 3883 & 3930 & 10:45:10.2 & -59:28:18 & 0.12 & 0.31 & 41& 0.62 & *0.4 & BH,EL\
229 & F36 & 2492 & 3336 & 10:45:14.8 & -59:27:11 & 0.26 & 0.45 & 41 & 1.03 & *1.5 & BR,EL\
230 & F36 & 3665 & 3179 & 10:45:15.2 & -59:28:10 & 0.23 & 0.94 & 31 & 1.70 & *1.0 & BH,EL,T\
231 & F36 & 3991 & 3108 & 10:45:15.5 & -59:28:26 & 0.17 & 0.30 & -8 & 0.68 & *0.4 & BH,EL,T\
232 & F34 & 2953 & 1788 & 10:45:15.5 & -59:45:14 & 0.45 & 0.65 & & 1.60 & 2.5 &\
233 & F36 & 3003 & 3106 & 10:45:16.0 & -59:27:37 & 0.10 & 0.19 & 35 & 0.42 & *0.2 & BR,EL\
234 & F34 & 3018 & 1878 & 10:45:16.2 & -59:45:15 & 0.53 & 0.98 & -14 & 2.19 & 5.6 & EL\
235 & F36 & 2999 & 3076 & 10:45:16.2 & -59:27:37 & 0.11 & 0.23 & 35 & 0.49 & *0.3 & BR,EL\
236 & F36 & 2080 & 3045 & 10:45:16.9 & -59:26:52 & 0.12 & 0.21 & -4 & 0.49 & *0.3 & BR,EL\
237 & F36 & 2850 & 2967 & 10:45:17.0 & -59:27:03 & 0.09 & 0.18 & 36 & 0.39 & *0.2 & BR,EL\
238 & F36 & 2053 & 2993 & 10:45:17.3 & -59:26:50 & 0.11 & 0.18 & 37 & 0.42 & *0.2 & BR,EL\
239 & F36 & 2086 & 2949 & 10:45:17.6 & -59:26:52 & 0.19 & 0.34 & 38 & 0.77 & *1.0 & BR,EL\
*******************************************
\[globCSD\]
[lccccccccccl]{} CN & Field & x & y & R.A. & Dec. & $\alpha$ & $\beta$ & P.A. & $\bar r $ & Mass & Remarks\
&&&& (J2000.0) &(J2000.0) & (arcsec) & (arcsec) & (degr.) & (kAU) & (M$_{J}$) &\
\
240 & F36 & 3557 & 2638 & 10:45:18.8 & -59:28:07 & 0.43 & 0.49 & & 1.33 &*2.3 & BR\
241 & F34 & 4031 & 1315 & 10:45:19.4 & -59:44:23 & 0.86 & 1.29 & -15 & 3.12 & 14 & EL,T,HH 900\
242 & F36 & 3013 & 2526 & 10:45:19.8 & -59:27:40 & 0.11 & 0.20 & -5 & 0.45 & 0.5 & EL\
243 & F36 & 2815 & 2471 & 10:45:20.3 & -59:27:31 & 0.22 & 0.33 & -8 & 0.80 & 1.3 & EL\
244 & F35 & 3833 & 803 & 10:45:20.3 & -59:34:54 & 0.19 & 0.56 & -6 & 1.09 & *1.9 & BH,EL\
245 & F36 & 2956 & 2448 & 10:45:20.4 & -59:27:38 & 0.12 & 0.20 & -4 & 0.46 & *0.3 & EL\
246 & F35 & 3925 & 862 & 10:45:21.0 & -59:34:53 & 0.11 & 0.22 &-21 & 0.48 & *0.4 & BH,EL\
247 & F36 & 2947 & 2037 & 10:45:23.1 & -59:27:39 & 0.17 & 0.35 & 41 & 0.75 & *0.5 & BR,EL\
248 & F36 & 2438 & 2035 & 10:45:23.4 & -59:27:14 & 0.15 & 0.26 & 34 & 0.59 & *0.5 & BR,EL\
249 & F36 & 2642 & 2011 & 10:45:23.4 & -59:27:24 & 0.14 & 0.25 &-6 & 0.56 & *0.4 & BR,EL\
250 & F36 & 2426 & 1969 & 10:45:23.8 & -59:27:13 & 0.10 & 0.20 & 41 & 0.44 & *0.2 & BR,EL,\
251 & F36 & 3344 & 1606 & 10:45:25.7 & -59:28:01 & 0.11 & 0.19 & 44 & 0.43 & *0.2 & BH,EL,T\
252 & F36 & 3446 & 1332 & 10:45:27.4 & -59:28:07 & 0.12 & 0.41 & 36 & 0.77 & *0.4 & BR,EL\
253 & F36 & 3855 & 1259 & 10:45:27.6 & -59:28:28 & 0.20 & 0.57 & -7 & 1.12 & *0.9 & BR,EL\
254 & F36 & 3465 & 722 & 10:45:31.4 & -59:28:10 & 0.10 & 0.25 & 28 & 0.51 & *0.2 & BR,EL\
255 & F37 &1063 & 1293 & 10:45:43.3 & -59:41:48 & 0.17 & 0.32 & -8 & 0.71 & *0.5 & BH,EL\
256 & F37 & 1218 & 1373 & 10:45:49.3 & -59:41:20 & 0.19 & 0.22 & & 0.59 & 0.5 &\
257 & F37 & 2123 & 1452 & 10:45:49.6 & -59:41:23 & 0.24 & 0.32 & & 0.81 & 0.9 & T\
258 & F38 & 1950 & 772 & 10:45:52.4 & -60:09:22 & 0.94 & 1.20 & & 3.10 & 29 &\
259 & F39 & 2083 & 1920 & 10:45:56.4 & -60:06:50 & 0.25 & 0.37 & & 0.90 & 1.0 &\
260 & F39 & 1727 & 1822 & 10:45:58.5 & -60:07:00 & 0.39 & 0.54 & & 1.35 & 2.2 & C\
261 & F40 & 3160 & 2494 & 10:46:25.4 & -60:04:46 & 0.23 & 0.34 & 24 & 0.83 & *0.8 & BH,T\
262 & F40 & 1860 & 859 & 10:46:32.5 & -60:06:16 & 0.17 & 0.52 & 11 & 1.00 & *0.8 & BH,EL\
263 & F40 & 1829 & 2265 & 10:46:34.0 & -60:05:07 & 0.22 & 0.74 & 13 & 1.39 & *1.4 & BH,EL\
264 & F40 & 1684 & 2025 & 10:46:34.8 & -60:05:20 & 0.18 & 0.39 & 13 & 0.83 & *0.9 & BH,EL\
265 & F40 & 1595 & 3374 & 10:46:36.6 & -60:04:13 & 0.29 & 0.35 & & 0.93 & *0.9 & BH,C\
266 & F40 & 1099 & 2480 & 10:46:39.1 & -60:05:01 & 0.18 & 0.36 & 6 & 0.78 & *0.6 & BH,EL\
267 & F40 & 1183 & 3446 & 10:46:39.4 & -60:04:13 & 0.18 & 0.28 & & 0.67 & *0.4 & BH\
268 & F40 & 548 & 2567 & 10:46:42.8 & -60:05:01 & 0.12 & 0.21 & 10 & 0.48 & *0.3 & BH,EL\
269 & F40 & 572 & 3295 & 10:46:43.3 & -60:04:24 & 0.22 & 0.35 & 27 & 0.83 & *0.7 & BH,EL\
270 & F40 & 536 & 3228 & 10:46:43.5 & -60:04:28 & 0.15 & 0.31 & 24 & 0.67 & *0.5 & BH,EL\
271 & F41 & 2754 & 3454 & 10:46:44.0 & -60:08:16 & 0.35 & 0.64 & -38 & 1.44 & *2.0 & BH,EL\
272 & F42 & 4047 & 547 & 10:46:44.2 & -60:07:29 & 0.23 & 0.41 & -43 & 0.93 & *0.6 & BH,EL\
273 & F41 & 2206 & 3203 & 10:46:47.4 & -60:08:33 & 0.25 & 0.29 & & 0.78 & 0.6 &\
274 & F41 & 1750 & 609 & 10:46:48.0 & -60:10:44 & 0.18 & 0.28 & 40 & 0.67 & 0.5 & EL\
275 & F41 & 2193 & 3922 & 10:46:48.2 & -60:07:57 & 0.40 & 0.68 & -25 & 1.7 & *3.3 & BH,EL\
276 & F41 & 2118 & 3648 & 10:46:48.4 & -60:08:11 & 0.25 & 0.32 & & 0.83 & *0.7 & BR\
277 & F41 & 1974 & 3499 & 10:46:49.2 & -60:08:20 & 0.32 & 0.34 & & 0.96 & 1.0 &\
278 & F41 & 1870 & 3499 & 10:46:49.9 & -60:08:20 & 0.60 & 0.91 & 32 & 2.19 & *5.3 & BH,EL\
279 & F41 & 1788 & 3611 & 10:46:50.6 & -60:08:15 & 0.37 & 0.39 & & 1.10 & *1.4 & BH\
280 & F41 & 1664 & 2864 & 10:46:50.7 & -60:08:53 & 0.25 & 0.29 & & 0.78 & 0.7 &\
281 & F43 & 3479 & 1455 & 10:46:51.4 & -60:03:35 & 0.34 & 0.47 & 7& 1.17 & *2.9 & BR, EL,T\
282 & F41 & 1472 & 3476 & 10:46:52.5 & -60:08:24 & 0.20 & 0.26 & & 0.67 & 0.5 &\
283 & F43 & 2949 & 1487 & 10:46:54.9 & -60:03:37 & 0.31 & 0.50 & 26& 1.17 & 2.2 & EL,C to 268\
284 & F43 & 2933 & 1486 & 10:46:55.0 & -60:03:38 & 0.26 & 0.42 & 15& 0.99 & 1.5 & EL,C to 267\
285 & F43 & 2818 & 1519 & 10:46:55.8 & -60:03:37 & 0.13 & 0.14 & & 0.39 & 0.2 &\
286 & F42 & 2371 & 3032 & 10:46:57.7 & -60:05:38 & 0.82 & 1.47 & 35& 3.32 & 25 & EL\
287 & F43 & 2498 & 3796 & 10:47:00.0 & -60:01:46 & 0.25 & 0.45 & -23& 1.02 & *1.3 & BH,EL\
288 & F43 & 2411 & 4092 & 10:47:00.9 & -60:01:32 & 0.32 & 0.37 & & 1.00 & 1.4 &\
*******************************
\[globCSD\]
Atlas of fields with globulettes. {#AppendixB}
=================================
. \[fields3\]
[^1]: Based on observations collected with the NASA/ESA Hubble Space Telescope, obtained at the Space Telescope Science Institute.
|
---
abstract: 'Various forms of representations may arise in the many layers embedded in deep neural networks (DNNs). Of these, where can we find the most compact representation? We propose to use a pruning framework to answer this question: How compact can each layer be compressed, without losing performance? Most of the existing DNN compression methods do not consider the relative compressibility of the individual layers. They uniformly apply a single target sparsity to all layers or adapt layer sparsity using heuristics and additional training. We propose a principled method that automatically determines the sparsity of individual layers derived from the importance of each layer. To do this, we consider a metric to measure the importance of each layer based on the layer-wise capacity. Given the trained model and the total target sparsity, we first evaluate the importance of each layer from the model. From the evaluated importance, we compute the layer-wise sparsity of each layer. The proposed method can be applied to any DNN architecture and can be combined with any pruning method that takes the total target sparsity as a parameter. To validate the proposed method, we carried out an image classification task with two types of DNN architectures on two benchmark datasets and used three pruning methods for compression. In case of VGG-16 model with weight pruning on the ImageNet dataset, we achieved up to 75% (17.5% on average) better top-5 accuracy than the baseline under the same total target sparsity. Furthermore, we analyzed where the maximum compression can occur in the network. This kind of analysis can help us identify the most compact representation within a deep neural network.'
author:
- |
Hyun-Joo Jung$^1$[^1] Jaedeok Kim$^1$Yoonsuck Choe$^{1,2}$\
$^1$Machine Learning Lab, Artificial Intelligence Center, Samsung Research, Samsung Electronics Co.\
56 Seongchon-gil, Secho-gu, Seoul, Korea, 06765\
$^{2}$Department of Computer Science and Engineering, Texas A&M University\
College Station, TX, 77843, USA
bibliography:
- 'main.bib'
title: |
How Compact?: Assessing Compactness of Representations\
through Layer-Wise Pruning
---
INTRODUCTION
============
In recent years, DNN models have been used for a variety of artificial intelligence (AI) tasks such as image classification [@simonyan2014very; @he2016deep; @huang2017densely], semantic segmentation [@he2017mask], and object detection [@redmon2017yolo9000]. The need for integrating such models into devices with limited on-board computing power have been growing consistently. However, to extend the usage of large and accurate DNN models to resource-constrained devices such as mobile phones, home appliances, or IoT devices, compressing DNN models while maintaining their performance is an imperative.
A recent study [@zhu2017prune] demonstrates that making large DNN models sparse by pruning can consistently outperform directly trained small-dense DNN models. In pruning DNN models, we would like to address the following problem: *“How can we determine the target sparsity of individual layers?"*. Most existing methods set the layer-wise sparsity to be uniformly fixed to a single target sparsity [@zhu2017prune] or adopt layer-wise sparsity manually [@han2015learning; @he2017channel].
Starting from the assumption that not all layers in a DNN model have equal importance, we propose a new method that automatically computes the sparsity of individual layers according to layer-wise importance.
The contributions of the proposed method are as follows:
- The proposed method can analytically compute the layer-wise sparsity of DNN models. The computed layer-wise sparsity value enables us to prune the DNN model more efficiently than pruning with uniform sparsity for all layers. In our experiments, we validate that the proposed layer-wise sparsity scheme can compress DNN models more than the uniform-layer sparsity scheme while retaining the same classification accuracy.
- The proposed method can be combined with any pruning method that takes total target sparsity as an input parameter. Such a condition is general in compression tasks because users usually want to control the trade-off between compression ratio and performance. In our experiments, we utilized three different pruning approaches (weight pruning by [@han2015learning [-@han2015learning]]{}, random channel pruning, and channel pruning by [@li2016pruning [-@li2016pruning]]{}) for compression. Other pruning approaches are also applicable.
- The proposed method can be applied to any DNN architecture because it does not require any constraint on the DNN architectures (e.g., [@liu2017learning [-@liu2017learning]]{}; [@ye2018rethinking [-@ye2018rethinking]]{} requires batch normalization layer to prune DNN model).
- To compute the layer-wise sparsity, we do not require additional training or evaluation steps which take a long time.
RELATED WORKS
=============
Pruning is a simple but efficient method for DNN model compression. [@zhu2017prune ([-@zhu2017prune])]{} showed that making large DNN models sparse by pruning can outperform small-dense DNN models trained from scratch. There are many pruning methods according to the granularity of pruning from weight pruning [@han2015learning; @han2015deep] to channel pruning [@he2017channel; @liu2017learning; @li2016pruning; @molchanov2016pruning; @ye2018rethinking]. These approaches mainly focused on how to select the redundant weights/filters in the model, rather than considering how many weights/filters need to be pruned in each layer.
However, considering the role of each layer is critical for efficient compression. [@arora2018stronger ([-@arora2018stronger])]{} measured the noise sensitivity of each layer and tried to use it to compute the most effective number of parameters for each layer.
Recently, [@he2018adc ([-@he2018adc])]{} proposed Automated Deep Compression (ADC). They aimed to automatically find the sparsity ratio of each layer which is similar to our goal. However, they used reinforcement learning to find the sparsity ratio and characterized the state space as a layout of layer, i.e., kernel dimension, input size, and FLOPs of a layer, etc.
In this paper, we focus on directly measuring the importance of each layer for the given task and the model. We used the value of the weight matrix itself rather than the layout of the layer. We will show considering layer-wise importance in DNN compression is effective.
PROBLEM FORMULATION
===================
Our goal is to compute the sparsity of each layer in the model given the total target sparsity of the model. In this section we derive the relation between the total target sparsity $s$ of a model and the sparsity of each layer while considering the importance of each layer.
Let $s_l \in [0,1)$, $l=1,\cdots L$, be the layer sparsity of the $l$-th layer where $L$ is the total number of layers in the model. Then, the layer sparsity $s_l$ should satisfy the following condition: $$\begin{aligned}
\label{eqn:sparsity_condi}
\sum_{l=1}^{L} {s_l}{N_l}=sN,\end{aligned}$$ where $N_l$ is the number of parameters in the $l$-th layer and $N$ is the total number of parameters in the model.
A DNN model is usually overparameterized and it contains many redundant parameters. Pruning aims at removing such parameters from every layer and leaving only the important ones. So, if a large proportion of parameters in a layer is important for the given task, the layer should not be pruned too much while otherwise the layer should be pruned aggressively. Hence, it is natural to consider the importance of a layer when we determine the layer sparsity.
We assume that the number of remaining parameters in the $l$-th layer after pruning is proportional to the importance of the $l$-th layer, $\omega_l$. Because the number of remaining parameters for the $l$-th layer is equal to $(1-s_l)N_l$, we have,
$$\begin{aligned}
\label{eqn:remained_params}
(1-s_l)N_l = {{\alpha}}\omega_l,\end{aligned}$$
where ${\alpha}> 0$ is a constant. Because we want all layers to equally share the number of pruned parameters with respect to the importance of each layer, ${\alpha}$ is set to be independent of the layer index.
By summing both sides over all the layers, we obtain $$\begin{aligned}
\label{eqn:sum_remained}
\sum_{l=1}^{L} (1-s_l)N_l & = N-sN, \\
\label{eqn:sum_importance}
\sum_{l=1}^{L} {{\alpha}}\omega_l & = {{\alpha}}\Omega,\end{aligned}$$ where $\Omega$ is the sum of importance of all layers. From (Eq. \[eqn:sum\_remained\]) and (Eq. \[eqn:sum\_importance\]), we can easily compute ${\alpha}$, i.e., $$\begin{aligned}
\label{eqn:compute_alpha}
{{\alpha}} = \frac{(1-s)N}{\Omega}.\end{aligned}$$
The above equation can be seen as the ratio of the total number of remained parameters and the sum of importance of all layers which is proportional to the total number of effective parameters.
When ${\alpha}=1$, the number of remaining parameters determined by $s$ is the same as the number of effective parameters. In that case, we can readily prune parameters in each layer. When ${\alpha}<1$, then the number of remaining parameters becomes smaller than the number of effective parameters. Therefore, we need to prune some effective parameters. In this case, ${\alpha}$ acts as a distributor that equally allocates the number of pruning effective parameters to all layers. Thanks to the ${\alpha}$, the compression is prevented from pruning a specific layer too much. In the case of ${\alpha}>1$, it acts similar to the case of ${\alpha}<1$ but allocates the number of pruning redundant parameters instead.
Because all factors consisting ${\alpha}$ are automatically determined by the given model, we can control the above pruning cases by controlling $s$ only, which we desired.
By combining (Eq. \[eqn:remained\_params\]) and (Eq. \[eqn:compute\_alpha\]), we obtain the layer-wise sparsity as $$\begin{aligned}
\label{eqn:layer_wise_sparsity}
\nonumber s_l & = 1-{{\alpha}\frac{{\omega}_l}{N_l}} \\
& = 1-(1-s)\frac{N}{N_l}\frac{\omega_l}{\Omega}.\end{aligned}$$
The remaining problem is: *How can we compute the layer-wise importance $\omega_l$*? We propose metric to measure the importance of each layer based on the *layer-wise capacity*. In the subsection below, we will explain the metric in detail.
Layer-wise Capacity
-------------------
We measure the importance of a layer using the layer capacity induced by noise sensitivity [@arora2018stronger]. Based on the definition [@arora2018stronger], the authors proved that a matrix having low noise sensitivity has large singular values (i.e., low rank). They identified the effective number of parameters of a DNN model by measuring the capacity of mapping operations (e.g., convolution or multiplication with a matrix) which is inverse proportional to the noise sensitivity. The layer capacity has an advantage of directly counting the effective number of parameters in a layer. However, they only consider a linear network having fully connected layers or convolutional layers.
Motivated by the work, we use the concept of capacity to compute layer-wise sparsity. The layer capacity $\mu_l$ of the $l$-th layer is defined as $$\begin{aligned}
\label{eqn:capacity}
\mu_l := \max_{x^l\in S_l}\frac{||W^{l}x^{l}||}{||W^{l}||_{F} ||x^{l}||}, \end{aligned}$$ where $W^{l}$ is a mapping (e.g., convolution filter or multiplication with a weight matrix) of the $l$-th layer and $x^{l}$ is the input of the $l$-th layer. $||\cdot||$ and $||\cdot||_{F}$ are $l^2$ norm and the Frobenius norm of the operator, respectively. $S_l$ is a set of inputs of the $l$-th layer. In other words, $\mu_l$ is the largest number that satisfies $\mu_l ||W^{l}||_{F} ||x^{l}|| = ||W^{l}x^{l}||$.
According to the work [@arora2018stronger] and (Eq. \[eqn:capacity\]), the mapping having large capacity has low rank and hence small number of effective parameters. Therefore, we let the effective number of parameters to be inverse proportional to the layer-wise capacity. More specifically, the effective number of parameters $e_l$ of the $l$-th layer can be written as, $$\begin{aligned}
\label{eqn:effective_params}
e_l \propto \frac{{\beta}}{\mu_l^2},\end{aligned}$$ where ${\beta}$ is a constant.
In fact, ${\beta}$ might be different across layers because other attributes such as the depth of layers (distance from the input layer) would affect the number of effective parameters. In this paper, however, we set ${\beta}$ to be constant for simplicity and focus on the layer-wise capacity.
We assume that the layer having a large number of effective parameters (in other words, having small capacity) should not be pruned too much. Therefore, we can set the importance of the $l$-th layer $\omega_l$ to be the number of effective parameters $e_l$, i.e, $\omega_l = e_l$. Then, (Eq. \[eqn:layer\_wise\_sparsity\]) becomes, $$\begin{aligned}
\label{eqn:sparsity_capacity}
\nonumber s_l & = 1-(1-s)\frac{N}{N_l}\frac{e_l}{E}\\
& = 1-(1-s)\frac{N}{N_l}\frac{M}{\mu_{l}^{2}}, \end{aligned}$$ where $E$ and $M$ are the sum of $e_l$ and $1/{\mu_l^2}$ for all layers, respectively. Because the value of $s_l$ is independent of the value of ${\beta}$, $s_l$ can be obtained without any knowledge of the value of ${\beta}$.
PRUNING WITH LAYER-WISE SPARSITY
================================
In this section, we explain our proposed DNN model compression process with layer-wise sparsity. Given the total target sparsity, we first compute the layer-wise importance of each layer from the two proposed metrics (layer-wise performance gain and layer-wise capacity). According to (Eq. \[eqn:sparsity\_capacity\]), we can compute layer-wise sparsity easily.
However, there are some issues in computing the actual layer-wise sparsity. First, the user might want to control the minimum number of remaining parameters. In the worst case, the required number of pruned parameters for a layer might be equal or larger than the total number of parameters in the layer. Second, the exact number of pruned parameters would be different from the computed sparsity from (Eq. \[eqn:layer\_wise\_sparsity\]). For example, in channel pruning, the number of pruned parameters in the $(l-1)$th layer also affects the $l$-th layer.
To handle those problems, we re-formulate (Eq. \[eqn:layer\_wise\_sparsity\]) as an optimization problem. The objective function is defined as, $$\begin{aligned}
\label{eqn:optimization}
\min_{\epsilon} & \quad ||\epsilon||^2, \\
\nonumber
\textrm{such that }
& \quad \xi_l \leq {\alpha}(1+\varepsilon_l) \omega_l \leq N_l \textrm{ for all $l = 1, \cdots, L$},\\
\nonumber
& \quad \sum_{l} {\alpha}(1+\varepsilon_l) \omega_l = (1-s)N,\end{aligned}$$ where $\xi_l$ is the minimum number of remaining parameters after pruning. $\epsilon = (\varepsilon_1, \cdots, \varepsilon_L)$ is the vector of layer-wise perturbations on the constant ${\alpha}$. That is, though we set the number of pruned parameters of each layer equally proportional to the importance of the layer, we perturb the degree of pruning of each layer by adding $\varepsilon_l$ to ${\alpha}$ in unavoidable cases.
\[prop:sol\] If $\sum_{l=1}^L \xi_l \leq (1-s)N$, then a solution of the optimization problem (\[eqn:optimization\]) exists and it is unique.
Since ${\left\Vert\epsilon\right\Vert}^2$ is strictly convex in $\epsilon$, the optimization problem (\[eqn:optimization\]) has a unique solution if the problem is feasible. So it is enough to show that the constraint set of the problem is not empty. Denote the constraint set of the inequalities by $$\begin{aligned}
{\mathcal{D}}:= {\{\epsilon \in {\mathbb{R}}^L\colon\xi_l \leq {\alpha}(1+{\varepsilon}_l) {\omega}_l \leq N_l \textrm{ for $l=1,\cdots,L$}\}}.
\end{aligned}$$ Consider a continuous function $f\colon {\mathcal{D}}\to {\mathbb{R}}$ such that $f(\epsilon) := \sum_{l=1}^L {\alpha}(1 + {\varepsilon}_l) w_l$. If we take $\epsilon_{max} = (N_1 / {\alpha}w_1 - 1,\cdots, N_L / {\alpha}w_L - 1)$, then $\epsilon_{max} \in {\mathcal{D}}$ and $$\begin{aligned}
f(\epsilon_{max}) = \sum_{l=1}^L N_l = N \geq (1-s)N.
\end{aligned}$$ Similarly, by taking $\epsilon_{min} = (\xi_1 / {\alpha}w_1 - 1,\cdots, \xi_L / {\alpha}w_L - 1)$, we have $\epsilon_{min} \in {\mathcal{D}}$ and $$\begin{aligned}
f(\epsilon_{min}) = \sum_{l=1}^L \xi_l \leq (1-s)N.
\end{aligned}$$ By the provided condition, we know $\sum_{l=1}^L \xi_l \leq (1-s)N \leq N$. Then the continuity of $f$ yields that there exists a point $\tilde{\epsilon}\in{\mathcal{D}}$ such that $f(\tilde{\epsilon}) = (1 - s) N$, which completes our proof.
Intuitively, Proposition \[prop:sol\] says that the total target sparsity $s$ of pruning should not violate the constraint of the minimum number of remaining parameters. Let us assume $\sum_{l=1}^L \xi_l \leq (1-s)N$ to guarantee the feasibility of the optimization problem. We then can apply vairous convex optimization algorithms to obtain the optimal solution $\epsilon^*$ [@boyd2004convex].
Remark that our proposed method does not require model training or evaluation to find the values of layer sparsities. We obtain analytically the layer sparsity without additional training or evaluation. Many existing works [@zhong2018where; @he2018adc] use a trial and error approach for searching for the best combination of hyperparameters such as the layer sparsity $s_l$. Evaluating each combination through additional training or evaluation is necessary which makes these approaches not feasible on large datasets.
Layer Filter size Output size Activation
--------- --------------- -------------- ------------ -- -- -- --
Conv1 $(3,3,3,32)$ $(32,32,32)$ ReLU
Conv2 $(3,3,32,32)$ $(32,32,32)$ ReLU
Maxpool $(2,2)$ $(16,16,32)$
Conv3 $(3,3,32,64)$ $(16,16,64)$ ReLU
Conv4 $(3,3,64,64)$ $(16,16,64)$ ReLU
Maxpool $(2,2)$ $(8,8,32)$
FC1 $(2048,512)$ $(1,512)$ ReLU
FC2 $(512,10)$ $(1,10)$ Softmax
: Simple DNN model architecture. Conv and FC mean convolutional and fully connected layer, respectively.[]{data-label="tbl:keras-model"}
Our method requires convex optimization in a small dimensional space that has negligible computing time in general. The total calculation time mainly depends on the time spent by calculating the layer capacity $\mu_l$ by (Eq. \[eqn:capacity\]). Although it is required to compute the norm of each input vectors induced by the whole dataset, it has approximately similar computing costs as inferencing. Our approach has an advantage in terms of computation time.
With our implementation computing (Eq. \[eqn:capacity\]) took less than 3 hours for VGG16 model with the ImageNet dataset in a single GPU machine. For VGG-16 model with the CIFAR-10 dataset, the total computation time of (Eq. \[eqn:capacity\]) was less than 1 minute on the same hardware setup. The most time consuming part of our implementation was calculating the Frobenius norm of each layer. There could be more speed up if we use an approximate value of norms or use a subset of the data set instead of the whole data set. However, these are out of the scope of this paper so we will not discuss it any further.
Note that although here we assume that all layers in the model are compressed, selecting and compressing the subset of $1, \cdots, L$ can easily be handled. Moreover, the process described below can also be applied to any pruning methods such as channel pruning or weight pruning.
Selection of Target Sparsity for Channel Pruning
------------------------------------------------
In the experiments, we applied the proposed layer-wise sparsity method to both weight pruning and channel pruning.
However, in case of channel pruning, a pruned layer affects the input dimension of the next layer. So the actual number of pruned parameters may differ from our expectation. As a consequence, our proposed method does not achieve the exact total target sparsity $s$ (over-pruned in usual) by channel pruning. In fact, the exact number of remaining parameters by channel pruning depends on the topology of a neural network and properties of each layer, e.g. the kernel size of a convolution layer. So an exact formulation of the number of remaining parameters is not mathematically tractable.
To overcome such a limitation we need to consider a method to achieve the total target sparsity. Because the value of $s$ in our proposed method can be considered as the compression strength in pruning, we can find the exact sparsity $\hat{s}$ from the $s$. Let $C(s)$ be the actual number of remaining parameters in a model after applying channel pruning with sparsity $s$, i.e., the $l$-th layer is pruned by the sparsity $s_l$ induced by (\[eqn:optimization\]). Then the total target sparsity is achievable if we use the value $\hat{s}$ instead of the total target sparsity $s$. Finally, the problem becomes finding proper $\hat{s}$ that satisfies, $$\begin{aligned}
C(\hat{s}) = (1-s)N.\end{aligned}$$
Because both solving (\[eqn:optimization\]) and counting the number of parameters require low computational costs, the value of $\hat{s}$ can be obtained in reasonable time. We therefore can control the achieved sparsity of the model by using $\hat{s}$ in case of channel pruning.
![Classification accuracy comparison against the baseline and the proposed method using a simple DNN model on the CIFAR-10 dataset. For compression, weight pruning [@han2015learning] is used. (p) and (p + ft) mean pruning only and fine-tuning after pruning, respectively.[]{data-label="fig:cifar10simple_wemag"}](fig/cifar10simple_wemag.pdf){width="0.85\columnwidth"}
![Classification accuracy comparison against the baseline and the proposed method using simple DNN model on the CIFAR-10 dataset. For compression, random channel pruning is used. We plot the median value for 10 trials and the vertical bar at each point represents the max and min value. (p) and (p + ft) mean pruning only and fine-tuning after pruning, respectively.[]{data-label="fig:cifar10simple_chrand"}](fig/cifar10simple_chrand.pdf){width="0.85\columnwidth"}
![Classification accuracy comparison against the baseline and the proposed method using simple DNN model on the CIFAR-10 dataset. For compression, channel pruning [@li2016pruning] is used. (p) and (p + ft) mean pruning only and fine-tuning after pruning, respectively.[]{data-label="fig:cifar10simple_chmag"}](fig/cifar10simple_chmag.pdf){width="0.85\columnwidth"}
EXPERIMENTAL RESULTS
====================
In this section, we investigate how much our proposed scheme affects the performance of a pruned model. To do this, we carried out an image classification task with the simple DNN and VGG-16 [@simonyan2014very] model on the CIFAR-10 [@krizhevsky2014cifar] and ImageNet [@Deng09imagenet:a] dataset.
In all experiments, we compared our layer-wise sparsity method with the uniform sparsity method which we use as the baseline. To investigate how much robust our proposed scheme is to different pruning methods, we applied three pruning methods (magnitude-based weight pruning by [@han2015learning [-@han2015learning]]{}, random channel pruning, and magnitude-based channel pruning by [@li2016pruning [-@li2016pruning]]{}) to DNN architectures.
We set $\xi_l$ as $3\times w_l\times h_l\times c_{l-1}$ where $c_{l-1}$ is the number of input channels for the $l$-th layer and $w_l$ and $h_l$ is the spatial filter size of the $l$-th layer. In other words, we wanted to remain at least 3 channels for performance.
We implemented the proposed method using Keras [@chollet2015keras]. For the VGG-16 model on the ImageNet dataset, we used pre-trained weights in Keras, and for the VGG-16 model on the CIFAR-10 dataset, we used pre-trained weights from [@cifar-vgg]. The simple DNN model was designed and trained by ourselves.
Simple DNN Model on CIFAR-10 Dataset
------------------------------------
Table \[tbl:keras-model\] shows the architecture of the simple DNN model used in this experiment. To compress the model, we applied pruning to Conv2-4 and FC1 layers. After pruning is done, we can additionally apply (optional) fine-tuning to improve performance. Therefore we checked the accuracy of both pruned and fine-tuned models to investigate the resilience of the proposed method. For fine-tuning, we ran 3 epochs with learning rate = 0.0001.
Because the computation of layer-wise sparsity is not affected by the choice of pruning method, we need to compute layer-wise sparsity for the model only once. In the subsections below, therefore, we shared the computed layer-wise sparsity with all three pruning methods.
### Weight Pruning
To prune weights, we used magnitude-based pruning [@han2015learning]. In other words, we pruned weights that have small absolute values because we can consider that they do not contribute much to the output.
Figure \[fig:cifar10simple\_wemag\] shows the performance after compression. As we can see in the figure, the proposed method outperforms under all total target sparsities in pruning only case, and achieves similar or better results after fine-tuning. At large total target sparsity, the effect of the proposed method becomes apparent. For example, at total target sparsity 0.9, the accuracy of the proposed method drops by 0.179 after pruning only, while the baseline drops by 0.656.
### Random Channel Pruning
To eliminate the effect of the details of a pruning algorithm, we also applied random pruning in channel. Given the total target sparsity, we computed the layer-wise sparsities and the number of required parameters to be pruned in order. Then the number of pruned channels is determined by applying floor operation to the number of pruned parameters. According to the computed number of pruned channels, we *randomly* selected which channels are pruned and repeated the selection 10 times.
We compared the classification accuracy of the proposed layer-wise sparsity scheme against the baseline. Figure \[fig:cifar10simple\_chrand\] shows the results. Similar with weight pruning, the proposed layer-wise sparsity scheme also outperforms the baseline under all total target sparsities. As we can see in figure \[fig:cifar10simple\_chrand\], the proposed method conducts compression reliably compared with the baseline method (smaller height of min-max bar than the baseline). Interestingly, the result of the proposed method after pruning outperforms the baseline method after fine-tuning.
{width="1.8\columnwidth"}
![Classification accuracy comparison against the baseline and the proposed methods using the VGG-16 model on the CIFAR-10 dataset. For compression, random channel pruning is used. We plot the median value for 10 trials and the vertical bar at each point represents the max and min value. (p) and (p + ft) mean pruning only and fine-tuning after pruning, respectively.[]{data-label="fig:cifar10vgg_chrand"}](fig/cifar10vgg_chrand.pdf){width="0.9\columnwidth"}
![Classification accuracy comparison with the baseline and the proposed method of VGG-16 model on the CIFAR-10 dataset. For compression, channel pruning [@li2016pruning] is used. (p) and (p + ft) mean pruning only and fine-tuning after pruning, respectively.[]{data-label="fig:cifar10vgg_chmag"}](fig/cifar10vgg_chmag.pdf){width="0.9\columnwidth"}
### Channel Pruning
We used channel pruning [@li2016pruning] to compress the model. The authors used the sum of absolute weights in a channel as the criteria for pruning. In other words, channels that have small magnitude weights are pruned. Figure \[fig:cifar10simple\_chmag\] shows the results.
Numerically, the proposed method achieved up to 58.9% better classification accuracy than the baseline using the same total target sparsity. Please refer to the accuracy of the Baseline (p) and Proposed (p) at total target sparsity 0.6 in figure \[fig:cifar10simple\_chmag\]. In terms of compression ratio, the proposed method can prune up to 5 times more parameters than the baseline while retaining the same accuracy. Please compare the accuracy of the Baseline (p) at total target sparsity 0.1 and Proposed (p) at total target sparsity 0.5 in figure \[fig:cifar10simple\_chmag\].
VGG-16 on CIFAR-10 Dataset
--------------------------
In this experiment, we applied pruning to all Conv and FC layers except the first and the last layers (Conv1 and FC2) in the VGG-16 model. For fine-tuning, we ran 3 epochs with learning rate = 0.0001 in all pruning cases.
Figure \[fig:cifar10vgg\_sparsity\] shows the computed layer-wise sparsities under different total target sparsities, $s$. The computed sparsities confirm our assumption that all layers have different importance for the given task.
### Random Channel Pruning
Similar to the simple DNN model, we repeated the random channel pruning 10 times and the result of pruning is shown in figure \[fig:cifar10vgg\_chrand\]. Surprisingly, the proposed method achieves almost 90% maximum accuracy when total target sparsity is 0.8. Although the VGG-16 model is considered already quite overparameterized, we can still say that the proposed method efficiently compresses DNN models. The proposed method conduct compression reliably compared with the baseline method (smaller height of min-max bar than the baseline) as we can see in figure \[fig:cifar10vgg\_chrand\]. Though the performance after pruning is the same with the random guessing or worse than the baseline when $s > 0.2$, the performance is almost recovered (except the case $s=0.9$) after fine-tuning. This demonstrates that considering layer-wise sparsity helps not only pruning but also performance improvement with fine-tuning.
Method Error (%) $\Delta_{err}$ \# params. ratio(param) \# FLOPs ratio(FLOP)
------------------------- ----------- ---------------- ------------ -------------- ---------------------- -------------
Original 6.41 15.0M ${3.13 \times 10^8}$
Baseline 7.42 +1.01 5.4M 64.0% ${1.13 \times 10^8}$ 64.0%
[@li2016pruning] 6.60 -0.15 5.4M 64.0% ${2.06 \times 10^8}$ 34.2%
**Proposed** **6.25** **-0.16** 5.4M 64.0% ${2.46 \times 10^8}$ 21.6%
[@ayinde2018building]-A **6.33** +0.13 3.23M 78.1% ${1.86 \times 10^8}$ 40.5%
[@ayinde2018building]-B 6.70 +0.50 3.23M 78.1% ${1.86 \times 10^8}$ 40.5%
**Proposed** 6.53 **+0.12** 3.23M 78.5% ${2.13 \times 10^8}$ 32.1%
### Channel Pruning
Figure \[fig:cifar10vgg\_chmag\] shows the results. Though the accuracy values after pruning only are worse than the baseline, the degree of performance improvement after fine-tuning is better than the baseline when $s > 0.2$ except $s=0.9$.
Similar to the results of random channel pruning, the proposed method maintains the performance within 3% of the original model until the total target sparsity becomes 0.7.
Table \[tbl:cifarvgg\_results\] shows the performance comparison with other channel pruning methods. For fair comparison, we pruned Conv layers only. For fine-tuning, we ran 100 epochs with a constant learning rate 0.001 and select the best accuracy. As we can see in the Table, the proposed method won first place when the compressed model has 5.4M parameters and is in second place when the compressed model has 3.23M parameters. However, in performance drop ($\Delta_{err}$), the proposed method outperforms other methods in both cases. Our proposed method prunes more filters from the later layers than those from the former layers. It performs better in reducing the number of parameters, while it does not in reducing FLOPs as we can see in Table \[tbl:cifarvgg\_results\]. However, reducing FLOPs can be easily achievable by reformulating (Eq. \[eqn:sparsity\_condi\]).
{width="1.8\columnwidth"}
VGG-16 on ImageNet Dataset
--------------------------
In this experiment, we applied pruning to all Conv and FC layers except the first and the last layers (Conv1 and FC3) in the VGG-16 model on the ImageNet dataset.
![Classification accuracy comparison against the baseline and the proposed method using a VGG-16 model on the ImageNet dataset. ’w’ means weight pruning and ’ch’ means channel pruning.[]{data-label="fig:ImageNet_vgg"}](fig/ImageNet_vgg.png){width="\columnwidth"}
Figure \[fig:imagenetvgg\_sparsity\] shows the computed layer-wise sparsities under different total target sparsities. As we can see in the figure, the difference of layer-wise sparsity between layers. Surprisingly, the figure says that we only need prune two fully connected layers (FC1 and FC2) until the total target sparsity $s$ becomes 0.8. Such results are reasonable because more than 85% of total parameters in VGG are concentrated in the FC1 and FC2 layers. Therefore we can consider that there would be many redundant parameters in those layers. However, we can also see that the proposed method computes layer-wise sparsities not considering the number of parameters only, from the figure. For example, the number of parameters in FC1 layer is far lager than the FC2 layer but the sparsity of FC2 layer is larger than the FC1 layer. Conv11, Conv12, and Conv13 also represents similar results (all three layers have the same number of parameters).
### Weight Pruning
Figure \[fig:ImageNet\_vgg\] shows the performance after compression. The proposed method outperforms the baseline under all target sparsities both top-1 and top-5 accuracy. Both methods maintain the performance until $s=0.4$. But when $s$ becomes larger than 0.4, the proposed method shows consistently better performance.
### Channel Pruning
Figure \[fig:ImageNet\_vgg\] shows the compression results using channel pruning [@li2016pruning]. Because the channel pruning removes bunch of parameters, distributing the number of reqired parameters to be pruned to all layers according to the layer-wise sparsity is harder than weight pruning. Therefore the performance is worse than the weight pruning but still outperforms the baseline method in both top-1 and top-5 accuracy cases.
From the above results, the proposed layer-wise sparsity scheme outperforms the baseline method except for few cases. We can validate our claim that not all layers have the same importance for the given task and the proposed layer-wise sparsity scheme is highly effective in various DNN models for compression by pruning.
CONCLUSION
==========
In this paper, we proposed a new method that automatically computes the layer-wise sparsity from the layer-wise capacity for DNN model compression, especially pruning. Our proposed method does not require additional training or evaluation steps to compute the layer-wise sparsity, which has an advantage in terms of computation time. Experimental results validated the efficiency of the proposed layer-wise sparsity calculation in DNN model compression. Furthermore, the estimated layer-wise sparsity varied greatly across layers, suggesting that the information can be used to find where the most compact representation resides in the deep neural network.
[^1]: Equally contributed.
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abstract: 'We prove that for a symmetric, strictly log-convex density on the real line, there are four possible types of perimeter-minimizing triple bubbles. This extends the work of Bongiovanni et al. [@Bo], which shows that there are two possible types of perimeter-minimizing double bubbles.'
address: |
Courant Institute of Mathematical Sciences\
New York University\
251 Mercer St.\
New York, NY 10012, USA
author:
- Nat Sothanaphan
title: '1D Triple Bubble Problem with Log-Convex Density'
---
[**Keywords:** triple bubble; density; isoperimetric]{}
Introduction
============
The log-convex density theorem proved by Chambers [@Ch] states that on ${\mathbb{R}}^N$ with smooth, radially symmetric, log-convex density, a sphere centered at the origin is perimeter minimizing for a given volume. We seek the corresponding least-perimeter way to enclose and separate many given volumes, the perimeter-minimizing $n$-bubble. In our previous work [@Bo], we showed that for double bubbles on ${\mathbb{R}}$ with symmetric, strictly log-convex density, there are two possible types of perimeter minimizers shown in Figure \[fig:doubletriple\].
![The two types of perimeter-minimizing double bubbles on ${\mathbb{R}}$ with symmetric, strictly log-convex density for given volumes $V_1 \leq V_2$.[]{data-label="fig:doubletriple"}](doubleinterval "fig:"){width="180pt"} ![The two types of perimeter-minimizing double bubbles on ${\mathbb{R}}$ with symmetric, strictly log-convex density for given volumes $V_1 \leq V_2$.[]{data-label="fig:doubletriple"}](tripleinterval "fig:"){width="180pt"}
This paper extends the techniques of our previous work [@Bo] to characterize triple bubbles on ${\mathbb{R}}$ with symmetric, strictly log-convex density. Our main result is stated in Theorem \[thm:main\]: there are four possible types of perimeter-minimizing triple bubbles illustrated in Figure \[fig:4types\]. We also prove that minimizers of each of the four types exist using numerical computation.
![The four types—213, 3123, 2313, and 32123—of perimeter-minimizing triple bubbles with prescribed volumes $V_1 \leq V_2 \leq V_3$. The label $n$ means that the corresponding component is part of the bubble with volume $V_n$.[]{data-label="fig:4types"}](4types){width="350pt"}
\[thm:main\] On ${\mathbb{R}}$ with symmetric, log-convex density that is uniquely minimized at the origin and prescribed volumes $V_1 \leq V_2 \leq V_3$, a perimeter-minimizing triple bubble is, up to reflection, of one of the four types: $213$, $3123$, $2313$, and $32123$, as shown in Figure \[fig:4types\]. Moreover, there exists a symmetric, strictly log-convex, $C^1$ density such that each of the four types is perimeter minimizing for some prescribed volumes.
![Numerical proof of the existence of four types of minimizers for the density $f_2$ of Section \[sec:code\]. Colors correspond to types as: = 213; = 3123; = 2313; = 32123. See more information in Figure \[fig:seconddensity\].[]{data-label="fig:previewplot"}]({{2-0.01}} "fig:"){width="225pt"} ![Numerical proof of the existence of four types of minimizers for the density $f_2$ of Section \[sec:code\]. Colors correspond to types as: = 213; = 3123; = 2313; = 32123. See more information in Figure \[fig:seconddensity\].[]{data-label="fig:previewplot"}]({2-6} "fig:"){width="225pt"}
The isoperimetric problem with density was studied by Bobkov and Houdré [@BH] and Bayle [@Ba]. An important case is ${\mathbb{R}}^N$ with smooth, radially symmetric, log-convex density, where the log-convex density theorem, conjectured by Rosales et al. [@RCBM] and proved by Chambers [@Ch], states that spheres centered at the origin are isoperimetric.
The double bubble theorem [@Mo Chapt. 14] states that in ${\mathbb{R}}^N$, the standard double bubble consisting of three spherical caps meeting at 120 degrees is the least-perimeter double bubble. This was proven in ${\mathbb{R}}^3$ by Hutchings et al. [@HMRR] and in ${\mathbb{R}}^N$ by Reichardt [@Re]. There are results on double bubbles in the sphere ${\mathbb}{S}^N$, hyperbolic space ${\mathbb}{H}^N$, flat tori ${\mathbb}{T}^2$ and ${\mathbb}{T}^3$, and Gauss space (Euclidean space with density $e^{-r^2}$); see [@Mo Chapt. 19]. Recently, Milman and Neeman [@MN] proved the Gaussian double bubble conjecture, which states that the solution is three halfspaces meeting at 120 degrees.
For more than two bubbles, Wichiramala [@Wi] proved the triple bubble conjecture in ${\mathbb{R}}^2$ that the standard triple bubble is perimeter minimizing. Milman and Neeman [@MN2] also recently proved the Gaussian multi-bubble conjecture.
Bongiovanni et al. [@Bo] solved the double bubble problem on ${\mathbb{R}}$ with smooth, symmetric, strictly log-convex density $f$ and characterized the transitions of minimizers when $(\log f)'$ is unbounded. The author [@So] extended the characterization to the case where $(\log f)'$ is bounded.
This paper is organized as follows. Section \[sec:nstruct\] gives results on structures of $n$-bubbles on ${\mathbb{R}}$ that improve on those of Bongiovanni et al. [@Bo]. In particular, we show that a perimeter-minimizing $n$-bubble must be a “standard nested bubble,” defined in Definition \[def:nested\].
In Section \[sec:types\], under the mild condition that the density is nonincreasing on $(-\infty,0]$ and nondecreasing on $[0,\infty)$, we narrow down the types of perimeter-minimizing triple bubbles to ten possible types. Some arguments in this section are based on personal communication with Antonio Cañete.
Section \[sec:types2\] proves our main result that there are four types of perimeter-minimizing triple bubbles under the hypotheses of Theorem \[thm:main\]. We show this result by arguments based on rearrangements and equilibrium conditions and draw from tools developed by Bongiovanni et al. [@Bo].
Section \[sec:code\] gives numerical proof of the existence of all four types of minimizers for a certain symmetric, strictly log-convex density. The code used for this purpose is more efficient than the one used in our previous work [@Bo]. The plots obtained will also be useful in forming conjectures in Section \[sec:conj\].
Finally, we state in Section \[sec:conj\] conjectures on transitions of minimizers as volumes vary, which may serve as a basis for further work.
Acknowledgments {#acknowledgments .unnumbered}
---------------
I would like to thank Antonio Cañete, Eliot Bongiovanni, and Frank Morgan for discussions on this problem. Antonio Cañete initiated the discussion, contributed questions and arguments, and gave comments on a draft of this paper; Eliot Bongiovanni came up with conjectures and made progress in various directions; and Frank Morgan gave comments on my arguments and the writing of this paper.
$n$-Bubble Structure {#sec:nstruct}
====================
In this section, we study perimeter-minimizing $n$-bubbles on ${\mathbb{R}}$ with a density that is nonincreasing on $(-\infty,0]$ and nondecreasing on $[0,\infty)$. Our results show that a perimeter-minimizing $n$-bubble must be a “standard nested bubble”: standard multi-bubbles contained within one another and defined in Definition \[def:nested\]. This improves on the results of Bongiovanni et al. [@Bo Prop. 3.8]. Figure \[fig:nested2d\] shows an example of a 2D nested bubble, and Figure \[fig:nested1d\] gives an example of a 1D nested bubble.
![2D nested bubble of type (3,2,4).[]{data-label="fig:nested2d"}](nested2d){width="200pt"}
![1D standard nested bubble of type $(2,3,1)$, defined in Definition \[def:nested\].[]{data-label="fig:nested1d"}](ex1d){width="330pt"}
We start with definitions. Consider a perimeter-minimizing $n$-bubble on ${\mathbb{R}}$ with prescribed volumes $V_1 \leq V_2 \leq \cdots \leq V_n$. The bubble with volume $V_a$ is called “bubble $a$.” If bubble $a$ has one component, let $R_a$ denote this component. If bubble $a$ has two components, let $R_a^\ell$ and $R_a^r$ denote the left and right components and $V_a^\ell$ and $V_a^r$ denote their volumes, respectively. The following two definitions are only for 1D $n$-bubbles.
On ${\mathbb{R}}$ with density, a *standard $n$-bubble* is an $n$-bubble with contiguous components in equilibrium where every bubble has one component.
\[def:nested\] Let $k_1,\dots,k_m$ be positive integers. An $n$-bubble $B$ is a *standard nested bubble of type $(k_1,\dots,k_m)$* if there are standard $k_i$-bubbles $B_i$ with disjoint boundaries such that
1. The boundary points of $B$ consist exactly of the boundary points of the $B_i$’s;
2. $B_{i+1}$ is contained in a single component of $B_i$;
3. In $B$, the components to the left and right of $B_i$ for $i>1$ are a single bubble, and no other bubbles of $B$ have more than one component.
See Figure \[fig:nested1d\]. Notice that for a standard nested $n$-bubble of type $(k_1,\dots,k_m)$, the $k_i$’s are uniquely determined and sum to $n$. The triple interval [@Bo Def. 4.1] on the right-hand side of Figure \[fig:doubletriple\] is a standard nested bubble of type $(1,1)$.
It is easily seen that the requirements for an $n$-bubble $B$ to be a standard nested bubble is equivalent to the following.
1. $B$ consists of contiguous intervals;
2. Every bubble of $B$ has at most two components;
3. If bubbles $a$ and $b$ both have two components, then the order of their components is either $R_a^\ell,R_b^\ell,R_b^r,R_a^r$ or $R_b^\ell,R_a^\ell,R_a^r,R_b^r$.
We now prove the following proposition on the structure of 1D perimeter-minimizing $n$-bubbles.
\[prop:1dstruct\] On ${\mathbb{R}}$ with continuous density that is nonincreasing on $(-\infty,0]$ and nondecreasing on $[0,\infty)$, a perimeter-minimizing $n$-bubble is a standard nested bubble.
We show that any perimeter-minimizing $n$-bubble satisfies conditions (i)–(iii) stated above. Point (i) follows from [@Bo Prop. 3.5], and point (ii) follows from [@Bo Lemma 3.7]. It remains to verify point (iii).
Without loss of generality, let $R_a^\ell$ be the leftmost of the four components. By [@Bo Lemma 3.7], any left component must be to the left of any right component, so the order $R_a^\ell,R_a^r,R_b^\ell,R_b^r$ is impossible. Finally, we show that an $n$-bubble with the order $R_a^\ell,R_b^\ell,R_a^r,R_b^r$ is not perimeter-minimizing by an argument similar to the proof of [@Bo Prop. 3.8]. Move all components and boundary points between $R_a^\ell$ and $R_b^\ell$ to the right and all components and boundary points between $R_a^r$ and $R_b^r$ to the left in such a way as to preserve volumes of bubbles. By [@Bo Lemma 3.7], perimeter does not increase during the move. But when one of $R_b^\ell$ or $R_a^r$ disappears, the perimeter will decrease. Therefore the only possible order is $R_a^\ell,R_b^\ell,R_b^r,R_a^r$.
Proposition \[prop:1dstruct\] indeed improves on our previous characterizations of perimeter-minimizing $n$-bubbles, as it immediately implies [@Bo Prop. 3.8].
We close this section with two conjectures. First, we think that the characterization in Proposition \[prop:1dstruct\] is optimal in the sense that a standard nested bubble of every type can be perimeter minimizing. This is stated in Conjecture \[conj:structoptimal\]. Our previous work [@Bo Thm. 4.15] proved this for double bubbles and Section \[sec:code\] proves it for triple bubbles.
\[conj:structoptimal\] For any positive integers $n$ and $k_1,\dots,k_m$ that sum to $n$, a standard nested bubble of type $(k_1,\dots,k_m)$ is a perimeter-minimizing $n$-bubble for some symmetric, log-convex density on ${\mathbb{R}}$ and some prescribed volumes.
Finally, Proposition \[prop:1dstruct\] might be true in higher dimensions for smooth, radially symmetric, strictly log-convex densities, for an appropriate notion of “standard nested bubble.” This is stated in Conjecture \[conj:structhighdim\]. The case of single bubbles is the log-convex density theorem [@Ch], and the case of double bubbles was conjectured in our previous work [@Bo Conj. 7.1].
\[conj:structhighdim\] On ${\mathbb{R}}^N$ with smooth, radially symmetric, strictly log-convex density, a perimeter-minimizing $n$-bubble is a standard nested bubble.
Triple Bubbles Under Mild Conditions on Density {#sec:types}
===============================================
We now focus on perimeter-minimizing triple bubbles on ${\mathbb{R}}$. Let the notation $$\mid I_1 \mid I_2 \mid \dots \mid I_m \mid,$$ where $I_i$’s are compact intervals, denote a multi-bubble with contiguous components $I_1,I_2,\dots,I_m$ from left to right in this order. By Proposition \[prop:1dstruct\] applied to triple bubbles, the following is immediate.
\[prop:init\] On ${\mathbb{R}}$ with continuous density that is nonincreasing on $(-\infty,0]$ and nondecreasing on $[0,\infty)$, a perimeter-minimizing triple bubble is, up to reflection, of one of the forms: $$\begin{aligned}
&(1) \mid R_a \mid R_b \mid R_c \mid; &&(2) \mid R_a^\ell \mid R_b \mid R_c \mid R_a^r \mid; \\
&(3)\mid R_a \mid R_b^\ell \mid R_c \mid R_b^r \mid; &&(4) \mid R_a^\ell \mid R_b^\ell \mid R_c \mid R_b^r \mid R_a^r \mid,\end{aligned}$$ where ${\left\{a,b,c\right\}}={\left\{1,2,3\right\}}$.
To further reduce the number of possibilities, we will use the following lemma. The idea for this lemma comes from arguments of Antonio Cañete.
\[lem:squeeze\] Let $f$ be a positive function that is nonincreasing on $(-\infty,0]$ and nondecreasing on $[0,\infty)$. Suppose that $x_0 \leq x_1 \leq \dots \leq x_n$ with $y_i \in [x_{i-1},x_i]$. Then $f(x_0)+\dots+f(x_n) > f(y_1)+\dots+f(y_n)$.
We proceed by induction. The case $n=0$ is obvious. For $n \geq 1$, if $y_1 \geq 0$, then $f(x_i) \geq f(y_i)$ and $f(x_0)>0$, so the inequality holds. If $y_1 < 0$, then $f(x_0)\geq f(y_1)$, and we can apply the induction hypothesis.
By using Lemma \[lem:squeeze\], we can strengthen Proposition \[prop:init\] as follows.
\[prop:typemonotone\] On ${\mathbb{R}}$ with continuous density that is nonincreasing on $(-\infty,0]$ and nondecreasing on $[0,\infty)$, a perimeter-minimizing triple bubble is, up to reflection, of one of the forms:
1. $\mid R_2 \mid R_1 \mid R_3 \mid$, $\mid R_1 \mid R_2 \mid R_3 \mid$, $\mid R_1 \mid R_3 \mid R_2 \mid$;
2. $\mid R_3^\ell \mid R_1 \mid R_2 \mid R_3^r \mid$, $\mid R_2^\ell \mid R_1 \mid R_3 \mid R_2^r \mid$;
3. $\mid R_2 \mid R_3^\ell \mid R_1 \mid R_3^r \mid$, $\mid R_1 \mid R_3^\ell \mid R_2 \mid R_3^r \mid$, $\mid R_3 \mid R_2^\ell \mid R_1 \mid R_2^r \mid$;
4. $\mid R_3^\ell \mid R_2^\ell \mid R_1 \mid R_2^r \mid R_3^r \mid$, $\mid R_2^\ell \mid R_3^\ell \mid R_1 \mid R_3^r \mid R_2^r \mid$.
We need to rule out some cases from Proposition \[prop:init\] as not perimeter minimizing.
(2): We eliminate $\mid R_1^\ell \mid R_2 \mid R_3 \mid R_1^r \mid$. Let the boundary points be $x_0<x_1<x_2<x_3<x_4$. Switch $R_1^\ell$ with $R_2$ and $R_3$ with $R_1^r$ to get $\mid R_2 \mid R_1 \mid R_3 \mid$ with boundary points $x_0<y_1<y_2<x_4$. Then $y_1 \in [x_1,x_2]$ and $y_2 \in [x_2,x_3]$. By Lemma \[lem:squeeze\], $f(x_1)+f(x_2)+f(x_3) > f(y_1)+f(y_2)$, so the original configuration is not perimeter minimizing.
(3): We rule out $\mid R_a \mid R_b^\ell \mid R_c \mid R_b^r \mid$ with $b < c$. Switch $R_c$ and $R_b^r$ to get $\mid R_a \mid R_b \mid R_c \mid$. Then two boundary points disappear and one new boundary point appears between them. By Lemma \[lem:squeeze\], the new configuration has less perimeter than the original one.
(4): We rule out $\mid R_a^\ell \mid R_b^\ell \mid R_c \mid R_b^r \mid R_a^r \mid$ with $b<c$, and $\mid R_1^\ell \mid R_3^\ell \mid R_2 \mid R_3^r \mid R_1^r \mid$. For the first one, switch $R_c$ and $R_b^r$ and use a similar argument as in case (3). For the second one, assume by symmetry that $V_1^\ell \leq V_3^r$. Then compare with $\mid R_2 \mid R_3 \mid R_1 \mid$ with the same leftmost and rightmost boundary points using Lemma \[lem:squeeze\]. To make the comparison, ignore the right endpoint of $R_3^\ell$.
We do not know whether more types can be eliminated from Proposition \[prop:typemonotone\] without more conditions on the density. For completeness, we include a double bubble counterpart of Proposition \[prop:typemonotone\] that generalizes [@Bo Prop. 4.6].
\[prop:doublemonotone\] On ${\mathbb{R}}$ with continuous density that is nonincreasing on $(-\infty,0]$ and nondecreasing on $[0,\infty)$, a perimeter-minimizing double bubble is, up to reflection, of the form $$\mid R_1 \mid R_2 \mid \quad \text{or} \quad \mid R_2^\ell \mid R_1 \mid R_2^r \mid.$$
By Proposition \[prop:1dstruct\], a perimeter minimizer must be a standard nested bubble. Thus we only need to rule out $\mid R_1^\ell \mid R_2 \mid R_1^r \mid$. Indeed, switching $R_2$ and $R_1^r$ yields a configuration with less perimeter by Lemma \[lem:squeeze\].
Notice that in Propositions \[prop:typemonotone\] and \[prop:doublemonotone\], the smallest bubble always has one component. We suspect that this may be true for $n$-bubbles for any $n$, as stated in the following conjecture.
On ${\mathbb{R}}$ with continuous density that is nonincreasing on $(-\infty,0]$ and nondecreasing on $[0,\infty)$, in a perimeter-minimizing $n$-bubble, the smallest volume, if it is unique, is contained in one component.
Triple Bubbles Under Log-Convex Density {#sec:types2}
=======================================
Our goal in this section is to reduce the ten types of Proposition \[prop:typemonotone\] to the four types of Figure \[fig:4types\]. To do so, we assume that the density is symmetric, log-convex, and uniquely minimized at the origin. Symmetry and log-convexity are not enough: the type $\mid R_1 \mid R_2 \mid R_3 \mid$ is perimeter minimizing for a constant density. It is easily seen that this condition is weaker than symmetry and strict log-convexity. It is in fact strictly weaker, as the density $e^{{\left\lvertx\right\rvert}}$ satisfies it but is not strictly log-convex. An equivalent condition is that the density is symmetric, log-convex, strictly decreasing on $(-\infty,0]$, and strictly increasing on $[0,\infty)$.
Recall that a convex function $g$ has left and right derivatives, which we denote by $g'_L$ and $g'_R$. Given a density $f$, define the volume coordinate by $$V= \int_0^x f,$$ where $x$ is the positional coordinate. By [@Bo Lemma 4.2], $f$ is log-convex if and only if it is *convex* in volume coordinates, in which case $(\log f)'_L(x) = f'_L(V)$ and $(\log f)'_R(x) = f'_R(V)$.
The following lemma is a piece of logic that will be used in Proposition \[prop:typelogconvex\].
\[lem:summorethanzero\] Let $f$ be a symmetric and log-convex density on ${\mathbb{R}}$. In volume coordinates, $f'_R(v_1)+f'_R(v_2)>0$ implies $v_1+v_2 \geq 0$, and $v_1+v_2 > 0$ implies $f'_L(v_1)+f'_L(v_2) \geq 0$.
Observe that $f'_L(-v)=-f'_R(v)$ by symmetry and $v>w$ implies $f'_L(v) \geq f'_R(w)$ by convexity in volume coordinates. Hence $$\begin{aligned}
v_1+v_2<0 &\implies -v_2>v_1 \implies f'_L(-v_2) \geq f'_R(v_1) \\
&\implies -f'_R(v_2) \geq f'_R(v_1) \implies f'_R(v_1)+f'_R(v_2) \leq 0,\end{aligned}$$ implying the first conclusion. The second conclusion follows by replacing $v_1$ and $v_2$ with $-v_1$ and $-v_2$.
We now show that for a symmetric, log-convex density that is uniquely minimized at the origin, only one type from each group in Proposition \[prop:typemonotone\] can be perimeter minimizing.
\[prop:typelogconvex\] On ${\mathbb{R}}$ with symmetric, log-convex density that is uniquely minimized at the origin, a perimeter-minimizing triple bubble is, up to reflection, of one of the forms: $$\begin{aligned}
&(1) \mid R_2 \mid R_1 \mid R_3 \mid; && (2) \mid R_3^\ell \mid R_1 \mid R_2 \mid R_3^r \mid; \\
&(3) \mid R_2 \mid R_3^\ell \mid R_1 \mid R_3^r \mid; && (4) \mid R_3^\ell \mid R_2^\ell \mid R_1 \mid R_2^r \mid R_3^r \mid.\end{aligned}$$
We need to eliminate more types from Proposition \[prop:typemonotone\] as not perimeter minimizing. Let $f$ be the density. Recall that $f$ is strictly decreasing on $(-\infty,0]$ and strictly increasing on $[0,\infty)$. Notice that, in volume coordinates, $f'_L(v)>0$ if $v>0$ and $f'_R(v)<0$ if $v<0$.
(4): We rule out $\mid R_2^\ell \mid R_3^\ell \mid R_1 \mid R_3^r \mid R_2^r \mid$ with $V_2<V_3$. By [@Bo Lemma 3.7], $0 \in R_1$. We now try switching bubbles 2 and 3. Consider the configuration $\mid R_3^{\ell\prime} \mid R_2^{\ell\prime} \mid R_1 \mid R_2^{r\prime} \mid R_3^{r\prime} \mid$, where the outer boundary points and $R_1$ do not move, and the volumes are split so that $V_2^\ell <
V_3^{\ell\prime} < V_2^\ell+V_3^\ell$ and $V_2^r < V_3^{r\prime} < V_2^r +V_3^r$. It is easy to see that this configuration has less perimeter than the original one.
(3): We show that if $\mid R_a \mid R_b^\ell \mid R_c \mid R_b^r \mid$ is perimeter minimizing, then $V_b \geq V_a \geq V_c$. By [@Bo Lemma 3.7], $0 \in R_c$. Switching $R_a$ and $R_b^\ell$ shows that $V_a \geq V_b^\ell$. Switching $R_b^\ell$ with $R_c$ allows us to conclude by Lemma \[lem:squeeze\] that $V_b^\ell \geq V_c$. Thus $V_a \geq V_b^\ell \geq V_c$.
It remains to show that $V_b \geq V_a$. In volume coordinates, let the boundary points be $w_0,w_1,w_2,w_3,w_4$. By the equilibrium condition [@Bo Rmk. 3.4], $$f'_R(w_0)+f'_R(w_1)+f'_R(w_4) \geq 0.$$ Since $0 \in R_c$, $w_1 < 0$, so $f'_R(w_1)<0$ and hence $f'_R(w_0)+f'_R(w_4)>0$. By Lemma \[lem:summorethanzero\], $w_0+w_4 \geq 0$. Since $w_2 \leq 0$, $w_0 \leq -V_a-V_b^\ell$ and $w_4 \leq V_c+V_b^r$, and so $V_c+V_b^r \geq V_a+V_b^\ell$. Finally, $V_c \leq V_b^\ell$ implies that $V_b^r\geq V_a$, so $V_b \geq V_a$ as required.
(2): We show that if $\mid R_a^\ell \mid R_b \mid R_c \mid R_a^r \mid$ is perimeter minimizing, then $V_a \geq V_b$ and $V_a \geq V_c$. By [@Bo Lemma 3.7], $0 \in R_b \cup R_c$. Assume by symmetry that $0 \in R_b$. Switching $R_c$ and $R_a^r$ gives $V_a^r \geq V_c$, and so $V_a \geq V_c$.
It remains to show that $V_a \geq V_b$. Let the endpoints in volume coordinates be $w_0,w_1,w_2,w_3,w_4$. By switching $R_a^\ell$ and $R_b$ and noting that the new boundary point must be no closer to the origin than the old one, either $V_a^\ell \geq V_b$ or $w_2-V_a^\ell \geq -w_1$. In the first case, $V_a \geq V_b$, as desired. In the second case, $w_1+w_2\geq V_a^\ell >0$. By Lemma \[lem:summorethanzero\], $f'_L(w_1)+f'_L(w_2) \geq 0$. Because $w_3>0$, $f'_L(w_3)>0$. Combining these gives $f'_L(w_1)+f'_L(w_2)+f'_L(w_3)>0$, contradicting the equilibrium condition $f'_L(w_1)+f'_L(w_2)+f'_L(w_3) \leq 0$.
(1): We show that if $\mid R_a \mid R_b \mid R_c \mid$ is perimeter minimizing, then $V_a \geq V_b$ and $V_c \geq V_b$. Let the endpoints in volume coordinates be $w_0,w_1,w_2,w_3$. Assume by symmetry that $w_1+w_2 \geq 0$. In particular, $w_2 \geq 0$. Now switching $R_b$ and $R_c$ gives either $V_c \geq V_b$ or $w_1+V_c \leq -w_2$. But the second case implies that $w_1+w_2\leq -V_c<0$, contradiction. Hence $V_c \geq V_b$.
It remains to show that $V_a \geq V_b$. Suppose for contradiction that $V_b>V_a$. We know that $0 \in R_a \cup R_b$. If $0 \in R_a$, then moving $R_c$ to the left of $R_a$ strictly decreases perimeter. So $0 \in R_b$. By switching $R_a$ and $R_b$, either $V_a \geq V_b$ or $w_2-V_a \geq -w_1$. The first case is a contradiction, so the second case holds. Because $w_1+w_2 \geq V_a > 0$, Lemma \[lem:summorethanzero\] implies that $f'_L(w_1)+f'_L(w_2) \geq 0$. Since $V_c \geq V_b > V_a$, $w_3-w_2>w_1-w_0$, and so $w_0+w_3>w_1+w_2 > 0$. By Lemma \[lem:summorethanzero\], $f'_L(w_0)+f'_L(w_3) \geq 0$. Now the equilibrium condition $$f'_L(w_0)+f'_L(w_1)+f'_L(w_2)+f'_L(w_3) \leq 0$$ implies that $f'_L(w_1)+f'_L(w_2) = f'_L(w_0)+f'_L(w_3) = 0$. Thus $f'_R(-w_1)=f'_L(w_2)$ and $f'_R(-w_0)=f'_L(w_3)$.
If the density is strictly log-convex, then $-w_1=w_2$ and $-w_0=w_3$, and so $V_a=V_c \geq V_b$, contradiction. But further arguments are needed in the general case. By convexity, $f$ in volume coordinates is linear on the intervals $[-w_1,w_2]$ and $[-w_0,w_3]$. Recall that $w_0+w_3>w_1+w_2 > 0$. Move all boundary points to the left while preserving volumes until $w_1+w_2=0$. Along the way, if we use primes to indicate moved boundary points, $-w_1\leq -w_1' \leq w_2' \leq w_2$ and $-w_0 \leq -w_0' \leq w_3' \leq w_3$. By linearity of $f$, perimeter does not change during the move. But now switching $R_a$ and $R_b$ after the move strictly decreases perimeter, a contradiction. Therefore $V_a \geq V_b$.
Observe that the conclusion of Proposition \[prop:typelogconvex\] does not hold for a symmetric, log-convex density that is not uniquely minimized at the origin. Indeed, such a density attains its minimum on some open interval containing the origin. So the type $\mid R_1 \mid R_2 \mid R_3 \mid$ with all boundary points near the origin is perimeter minimizing.
Numerical Plots of Minimizers {#sec:code}
=============================
In the previous section, we show that there can only be four types of perimeter-minimizing triple bubbles for a symmetric, log-convex density that is uniquely minimized at the origin. We prove in this section that no more types can be ruled out, completing the proof of Theorem \[thm:main\]. Specifically, we show via numerical computation that for a certain symmetric, strictly log-convex, $C^1$ density, each of the four types of Proposition \[prop:typelogconvex\] is perimeter minimizing for some prescribed volumes. We designate the four types as 213, 3123, 2313, and 32123, as in Figure \[fig:4types\].
Implementation
--------------
We outline how to compute equilibria of the four types. The following proposition shows how to compute a standard $n$-bubble for given volumes.
\[prop:computestandard\] On ${\mathbb{R}}$ with a symmetric, strictly log-convex, $C^1$ density $f$, a standard $n$-bubble enclosing volumes $V_1,\dots,V_n$ from left to right in this order exists and is unique. In volume coordinates, its leftmost boundary point ${\widetilde{V}}$ is the unique solution to $$\label{eq:equigeneral}
f'({\widetilde{V}})+f'({\widetilde{V}}+V_1)+\dots+f'({\widetilde{V}}+V_1+\dots+V_n)=0, \quad {\widetilde{V}}\in [-V_1-\dots-V_n,0].$$
Equation is the equilibrium condition [@Bo Cor. 3.3]. The conclusion follows from the fact that its left-hand side is strictly increasing in ${\widetilde{V}}$ due to strict convexity, is negative for ${\widetilde{V}}= -V_1-\dots-V_n$, and is positive for ${\widetilde{V}}= 0$.
From Proposition \[prop:computestandard\], if $f'$ is given symbolically in volume coordinates, we can effciently solve for a standard $n$-bubble using bisection search.
The next proposition considers existence and uniqueness of equilibria of the four types.
\[prop:fourtypesexist\] Consider ${\mathbb{R}}$ with a symmetric, strictly log-convex, $C^1$ density $f$ and prescribed volumes $V_1\leq V_2 \leq V_3$. An equilibrium of each of the types $213$, $3123$, and $32123$ exists and is unique. An equilibrium of type $2313$ exists and is unique if, in volume coordinates, $$f'{\left(V_3+\frac{V_1}{2}\right)} > f'{\left(V_2+\frac{V_1}{2}\right)} + f'{\left(\frac{V_1}{2}\right)},$$ and does not exist otherwise.
Notice that each equilibrium is a standard nested bubble. By Proposition \[prop:computestandard\], its constitutent standard bubbles exist and are unique. So an equilibrium exists and is unique if the standard bubbles are properly nested, and does not exist otherwise.
Equilibria of types $213$ and $32123$ are easily seen to be properly nested. To check that an equilibrium of type $3123$ is properly nested, apply [@Bo Lemma 4.8] to the inner standard bubble. Finally, to derive conditions for an equilibrium of type $2313$ to be properly nested, use the equilibrium condition for the outer standard bubble and the fact that its left-hand side is strictly increasing in ${\widetilde{V}}$.
Given a density and prescribed volumes, we can compute perimeters of equilibria of the four types by first checking for existence using Proposition \[prop:fourtypesexist\] and then solving for the constituent standard bubbles using Proposition \[prop:computestandard\]. We can then generate 3D plots of types of perimeter minimizers as prescribed volumes vary, as shown in Figures \[fig:firstdensity\] and \[fig:seconddensity\]. A 3D plot is represented as an animation of 2D plots, with $V_2$ and $V_3$ being spatial axes and $V_1$ varying in time.
We plot only the region where $V_1 \leq V_2 \leq V_3$, and the complement of this region is left as an empty space. To prevent rounding errors, we plot a point only if the perimeter-minimizing type has perimeter at least $10^{-4}$ less than those of the other types, and we leave the point an empty space otherwise. In the top left plot of Figure \[fig:seconddensity\], the white stripe between the and regions is a result of this.
Solving for equilibria using volume coordinates is more efficient than doing so using positional coordinates as in our previous paper [@Bo]. Indeed, a plot in this paper has more than $10^7$ points and the plot in our previous paper [@Bo Fig. 9] has $10^4$ points, but both take on the order of 30 minutes to generate. We also use a faster language, Julia, while our previous paper used Mathematica. This accounts for some of the speedup.
Results
-------
We generate plots of two densities, to illustrate two phenomena that can occur. The first density is $$\label{eq:density1}
f_1(V)={\left\lvertV\right\rvert}\sqrt{\log({\left\lvertV\right\rvert}+1)}+1.$$ We can check that $f_1$ is strictly log-convex and $C^1$. The idea behind $f_1$ is that we are trying to simulate the Borell density $f(x)=e^{x^2}$ in the asymptotics. By the Fundamental Bounding Lemma 6.3 in [@Bo], the Borell density grows asymptotically as ${\left\lvertV\right\rvert}\sqrt{\log{{\left\lvertV\right\rvert}}}$ in volume coordinates. Then we put in the “plus 1” to make the density well-defined and positive. For $f_1$, only 3 types of minimizers, except 2313, can be visually identified. (This is not a proof that the other type does not occur). Figure \[fig:firstdensity\] provides snapshots of the plot and their descriptions.
![Snapshots from the animation (<https://github.com/natso26/triple-bubbles/raw/master/triple.mp4>) of 3D triple bubble type as a function of the three prescribed volumes $V_1$, $V_2$, $V_3$ for the density $f_1$ . Note that the $V_2$ and $V_3$ axes are on different scales. Colors correspond to types as: = 213; = 3123; = 32123. The white areas near the axes are regions where $V_1 \leq V_2 \leq V_3$ is not satisfied and so nothing is drawn. As $V_1$ increases, the blue region pushes the yellow region to the right, while the purple region slowly rises.[]{data-label="fig:firstdensity"}]({{1-0.5}} "fig:"){width="225pt"} ![Snapshots from the animation (<https://github.com/natso26/triple-bubbles/raw/master/triple.mp4>) of 3D triple bubble type as a function of the three prescribed volumes $V_1$, $V_2$, $V_3$ for the density $f_1$ . Note that the $V_2$ and $V_3$ axes are on different scales. Colors correspond to types as: = 213; = 3123; = 32123. The white areas near the axes are regions where $V_1 \leq V_2 \leq V_3$ is not satisfied and so nothing is drawn. As $V_1$ increases, the blue region pushes the yellow region to the right, while the purple region slowly rises.[]{data-label="fig:firstdensity"}]({1-6} "fig:"){width="225pt"}
From the plot of the density $f_1$, one might wonder whether the type 2313 can ever be perimeter minimizing. It turns out that this can happen if the density at the origin decreases. Consider the second density $$\label{eq:density2}
f_2(V)={\left\lvertV\right\rvert}\sqrt{\log({\left\lvertV\right\rvert}+1)}+0.01,$$ which is a translate of $f_1$. For $f_2$, all four types in Proposition \[prop:typelogconvex\] can be perimeter minimizing. Nevertheless, the type 2313 can only be seen for small $V_1$, so the $V_1$ (time) axis is plotted on a logarithmic scale. Figure \[fig:seconddensity\] shows snapshots of the plot and their descriptions. Based on the animation, I also attempt to sketch the 3D plot in Figure \[fig:sketch\].
Table \[tab:fourtypes\] shows that all four types can be perimeter minimizing for the density $f_2$.
$(V_1,V_2,V_3)$ Type Type Type Type
--------------------- ------------------ ------------------ ------------------ ------------------
$(5, 100, 500)$ **1479.6294773** 1667.8737745 Not exist 1661.4875997
$(5, 40, 2000)$ 5608.7794571 **5467.6249803** Not exist 5469.4347271
$(0.01, 100, 1500)$ 4271.5195673 4351.3210336 **4271.5168203** 4335.5242035
$(2, 80, 2500)$ 7167.5032872 7080.5694767 Not exist **7071.1211666**
: Prescribed volumes $(V_1,V_2,V_3)$ for which each of the four types are perimeter minimizing for the density $f_2$ . The numbers in the tables are perimeters of equilibria of each type.[]{data-label="tab:fourtypes"}
\
In Figures \[fig:firstdensity\] and \[fig:seconddensity\], the boundaries between the and regions are straight lines. In the $(V_1,V_2)$ coordinates, they are tie curves between the double bubbles $12$ and $212$ as studied in [@Bo].
Code and animation for these two densities can be found at my GitHub repository <https://github.com/natso26/triple-bubbles>. The code is written in Julia. To run the code, follow the instructions in the repository to run it in <http://www.juliabox.com>.
![Snapshots from the animation (<https://github.com/natso26/triple-bubbles/raw/master/triple2.mp4>) of 3D triple bubble type as a function of the three prescribed volumes $V_1$, $V_2$, $V_3$ for the density $f_2$ . The $V_2$ and $V_3$ axes are on different scales. Colors correspond to types as: = 213; = 3123; = 2313; = 32123. The white areas near the axes are regions where $V_1 \leq V_2 \leq V_3$ is not satisfied and so nothing is drawn. The white stripes near the transition boundaries are regions where the difference between a perimeter-minimizing type and another type does not exceed $10^{-4}$ and so nothing is drawn to guard against rounding errors. In this animation, $V_1$ increases exponentially. As $V_1$ increases, the purple region first squeezes the green region out of existence; then the blue region emerges and pushes the yellow region to the right while the purple region slowly rises. The green region can only be seen when $V_1$ is small (less than 0.05).[]{data-label="fig:seconddensity"}]({{2-0.01}} "fig:"){width="225pt"} ![Snapshots from the animation (<https://github.com/natso26/triple-bubbles/raw/master/triple2.mp4>) of 3D triple bubble type as a function of the three prescribed volumes $V_1$, $V_2$, $V_3$ for the density $f_2$ . The $V_2$ and $V_3$ axes are on different scales. Colors correspond to types as: = 213; = 3123; = 2313; = 32123. The white areas near the axes are regions where $V_1 \leq V_2 \leq V_3$ is not satisfied and so nothing is drawn. The white stripes near the transition boundaries are regions where the difference between a perimeter-minimizing type and another type does not exceed $10^{-4}$ and so nothing is drawn to guard against rounding errors. In this animation, $V_1$ increases exponentially. As $V_1$ increases, the purple region first squeezes the green region out of existence; then the blue region emerges and pushes the yellow region to the right while the purple region slowly rises. The green region can only be seen when $V_1$ is small (less than 0.05).[]{data-label="fig:seconddensity"}]({{2-0.03}} "fig:"){width="225pt"} ![Snapshots from the animation (<https://github.com/natso26/triple-bubbles/raw/master/triple2.mp4>) of 3D triple bubble type as a function of the three prescribed volumes $V_1$, $V_2$, $V_3$ for the density $f_2$ . The $V_2$ and $V_3$ axes are on different scales. Colors correspond to types as: = 213; = 3123; = 2313; = 32123. The white areas near the axes are regions where $V_1 \leq V_2 \leq V_3$ is not satisfied and so nothing is drawn. The white stripes near the transition boundaries are regions where the difference between a perimeter-minimizing type and another type does not exceed $10^{-4}$ and so nothing is drawn to guard against rounding errors. In this animation, $V_1$ increases exponentially. As $V_1$ increases, the purple region first squeezes the green region out of existence; then the blue region emerges and pushes the yellow region to the right while the purple region slowly rises. The green region can only be seen when $V_1$ is small (less than 0.05).[]{data-label="fig:seconddensity"}]({2-2} "fig:"){width="225pt"} ![Snapshots from the animation (<https://github.com/natso26/triple-bubbles/raw/master/triple2.mp4>) of 3D triple bubble type as a function of the three prescribed volumes $V_1$, $V_2$, $V_3$ for the density $f_2$ . The $V_2$ and $V_3$ axes are on different scales. Colors correspond to types as: = 213; = 3123; = 2313; = 32123. The white areas near the axes are regions where $V_1 \leq V_2 \leq V_3$ is not satisfied and so nothing is drawn. The white stripes near the transition boundaries are regions where the difference between a perimeter-minimizing type and another type does not exceed $10^{-4}$ and so nothing is drawn to guard against rounding errors. In this animation, $V_1$ increases exponentially. As $V_1$ increases, the purple region first squeezes the green region out of existence; then the blue region emerges and pushes the yellow region to the right while the purple region slowly rises. The green region can only be seen when $V_1$ is small (less than 0.05).[]{data-label="fig:seconddensity"}]({2-6} "fig:"){width="225pt"}
![A rough sketch of 3D triple bubble type as a function of the three prescribed volumes $V_1$, $V_2$, $V_3$ for the density $f_2$ . The axes are not on the same scale, and the scale for the $V_1$ axis is nonlinear just as in the animation. I do not claim any accuracy of this sketch. The transition boundaries seem to be made up of three surfaces stitched together. The green (2313) region is possibly bounded; if so, this is the first instance of a bounded region for 1D $n$-bubbles. Section \[sec:conj\] provides more conjectures.[]{data-label="fig:sketch"}](surfsketch){width="400pt"}
Therefore, based on Proposition \[prop:typelogconvex\] and the work in this section, we conclude that Theorem \[thm:main\] holds.
Further Work {#sec:conj}
============
In this work, we completely determined the possible types of perimeter-minimizing triple bubbles for symmetric, strictly log-convex densities. Nevertheless, there remains many unanswered questions about transitions between different types. In particular, many conjectures can be formulated based on the sketch in Figure \[fig:sketch\]. To list some questions and conjectures:
1. **Possible types.** For some densities, there are only three types of minimizers (all except 2313), as conjectured to be the case for the density $f_1$ ; for other densities, all four types of minimizers occur, as is the case for the density $f_2$ . Can there be other sets of possible perimeter-minimizing types?
2. **Nature of tie surfaces.** The tie surfaces, that is, the set where at least two types of minimizers have equal perimeter, are really surfaces: multiple 2D manifolds stitched together. Moreover, for $C^k$ densities, they are $C^k$ manifolds (except where they meet). Furthermore, are there only three surfaces stitched together as in Figure \[fig:sketch\], or is this an illusion created by some surfaces meeting at nearly 180 degrees?
3. **Boundedness of regions.** The region for type 2313, if it exists, is bounded. The regions for other types are never bounded.
4. **When some volumes are equal.** If $V_2=V_3$, then the minimizer is of type 213. If $V_1=V_2$, then as $V_3$ increases, the minimizer transitions from type 213 to type 3123.
5. **When volumes are large.** For each $V_1$, for $V_2$ large (as a function of $V_1$), for $V_3$ large (as a function of $V_1$ and $V_2$), the minimizer is of type 32123.
6. **Transitions when volumes increase.** There is a constant $\lambda_0$, a function $\lambda_1(V_1)$, and functions $\lambda_{2313}(V_1)$ and $\lambda_{2313}'(V_1)$ for $V_1 \leq \lambda_0$ with the following properties:
- If $V_2 < \lambda_1(V_1)$, then as $V_3$ increases, the minimizer transitions from type 213 to type 3123.
- If $V_2 > \lambda_1(V_1)$, $V_1<\lambda_0$, and $\lambda_{2313}(V_1) < V_2 < \lambda_{2313}'(V_1)$, then as $V_3$ increases, the minimizer transitions from type 213 to type 2313 and then to type 3123.
- If $V_2 > \lambda_1(V_1)$ and either $V_1>\lambda_0$, $V_2 < \lambda_{2313}(V_1)$, or $V_2 > \lambda_{2313}'(V_1)$, then the minimizer transitions only from type 213 to type 3123.
See Figure \[fig:sketch\].
[10]{}
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|
---
abstract: '[*Random utility theory*]{} models an agent’s preferences on alternatives by drawing a real-valued score on each alternative (typically independently) from a parameterized distribution, and then ranking the alternatives according to scores. A special case that has received significant attention is the Plackett-Luce model, for which fast inference methods for maximum likelihood estimators are available. This paper develops conditions on general random utility models that enable fast inference within a Bayesian framework through MC-EM, providing concave log-likelihood functions and bounded sets of global maxima solutions. Results on both real-world and simulated data provide support for the scalability of the approach and capability for model selection among general random utility models including Plackett-Luce.'
author:
- |
Hossein Azari Soufiani\
SEAS, Harvard University\
azari@fas.harvard.edu David C. Parkes\
SEAS, Harvard University\
parkes@eecs.harvard.edu Lirong Xia\
SEAS, Harvard University\
lxia@seas.harvard.edu
title: Random Utility Theory for Social Choice
---
Introduction
============
Problems of learning with rank-based error metrics [@Liu11:Learning] and the adoption of learning for the purpose of rank aggregation in social choice [@Conitzer05:Common; @Conitzer09:Preference; @Xia10:Aggregating; @Xia11:Maximum; @Roos11:How; @Procaccia12:Maximum] are gaining in prominence in recent years. In part, this is due to the explosion of socio-economic platforms, where opinions of users need to be aggregated; e.g., judges in crowd-sourcing contests, ranking of movies or user-generated content.
In the problem of social choice, users submit ordinal preferences consisting of partial or total ranks on the alternatives and a single rank order must be selected to be representative of the reports. Since Condorcet [@Condorcet1785:Essai], one approach to this problem is to formulate social choice as the problem of estimating a true underlying world state (e.g., a true quality ranking of alternatives), where the individual reports are viewed as noisy data in regard to the true state. In this way, social choice can be framed as a problem of inference.
In particular, Condorcet assumed the existence of a true [*ranking*]{} over alternatives, with a voter’s preference between any pair of alternatives $a, b$ generated to agree with the true ranking with probability $p>1/2$ and disagree otherwise. Condorcet proposed to choose as the outcome of social choice the ranking that maximizes the likelihood of observing the voters’ preferences. Later, Kemeny’s rule was shown to provide the maximum likelihood estimator (MLE) for this model [@Young95:Optimal].
But Condorcet’s probabilistic model assumes identical and independent distributions on pairwise comparisons. This ignores the strength in agents’ preferences (the same probability $p$ is adopted for all pairwise comparisons), and allows for cyclic preferences. In addition, computing the winner through the Kemeny rule is $\Theta_2^P$-complete [@Hemaspaandra05:Complexity].
To overcome the first criticism, a more recent literature adopts the [*random utility model*]{} (RUM) from economics [@Thurstone27:Law]. Consider ${\mathcal C}=\{c_1,..,c_m\}$ alternatives. In RUM, there is a ground truth utility (or score) associated with each alternative. These are real-valued parameters, denoted by $\vec \theta=(\theta_1,\ldots,\theta_m)$. Given this, an agent independently samples a random utility ($X_j$) for each alternative $c_j$ with conditional distribution $\mu_j(\cdot|\theta_j)$.
Usually $\theta_j$ is the mean of $\mu_j(\cdot|\theta_j)$.[^1] Let $\pi$ denote a permutation of $\{1,\ldots,m\}$, which naturally corresponds to a linear order: $[c_{\pi(1)}\succ c_{\pi(2)}\succ\cdots\succ
c_{\pi(m)}]$. Slightly abusing notation, we also use $\pi$ to denote this linear order. Random utility $(X_1,\ldots,X_m)$ generates a distribution on preference orders, as $$\begin{aligned}
\label{Order1}
{\text{Pr}}(\pi\ |\ \vec{\theta})={\text{Pr}}(X_{\pi(1)}>X_{\pi(2)}>\ldots>X_{\pi(m)})\end{aligned}$$
The generative process is illustrated in Figure \[fig:rum\].
{width="100.00000%"}
0 The systematic study of such models (known as [*choice theory*]{}) has been an important topic in psychology and economics since Thurstone’s seminal work in 1927 [@Thurstone27:Law], which is well-known as the [*random utility theory*]{} in economics.
In social choice we have a set of [*alternatives*]{}, ${\mathcal C}=\{c_1,..,c_m\}$.
Adopting RUMs rules out cyclic preferences, because each agent’s outcome corresponds to an order on real numbers, and it also captures the strength of preference, and thus overcomes the second criticism, by assigning a different parameter ($\theta_j$) to each alternative.
A popular RUM is Plackett-Luce (P-L) [@Luce59:Individual; @Plackett75:Analysis], where the random utility terms are generated according to Gumbel distributions with fixed shape parameter [@Block60:Random; @Yellott77:Relationship]. For P-L, the likelihood function has a simple analytical solution, making MLE inference tractable. P-L has been extensively applied in econometrics [@McFadden74:Conditional; @Berry95:Automobile], and more recently in machine learning and information retrieval (see [@Liu11:Learning] for an overview). Efficient methods of EM inference [@Hunter04:MM; @Caron12:Efficient], and more recently expectation propagation [@Guiver09:Bayesian], have been developed for P-L and its variants.
In application to social choice, the P-L model has been used to analyze political elections [@Gormley09:Grade]. EM algorithm has also been used to learn the [*Mallows*]{} model, which is closely related to the Condorcet’s probabilistic model [@Lu11:Learning].
Although P-L overcomes the two difficulties of the Condorcet-Kemeny approach, it is still quite restricted, by assuming that the random utility terms are distributed as Gumbel, with each alternative is characterized by one parameter, which is the mean of its corresponding distribution. In fact, little is known about inference in RUMs beyond P-L. Specifically, we are not aware of either an analytical solution or an efficient algorithm for MLE inference for one of the most natural models proposed by Thurstone [@Thurstone27:Law], where each $X_j$ is normally distributed.
Our Contributions
-----------------
In this paper we focus on RUMs in which the random utilities are independently generated with respect to distributions in the [*exponential family*]{} (EF) [@Morris12:Natural]. This extends the P-L model, since the Gumbel distribution with fixed shape parameters belonging to the EF. Our main theoretical contributions are Theorem \[thm1\] and Theorem \[thm2\], which propose conditions such that the log-likelihood function is concave and the set of global maxima solutions is bounded for the [*location family*]{}, which are RUMs where the shape of each distribution $\mu_j$ is fixed and the only latent variables are the locations, i.e., the means of $\mu_j$’s. These results hold for existing special cases, such as the P-L model, and many other RUMs, for example the ones where each $\mu_j$ is chosen from Normal, Gumbel, Laplace and Cauchy.
We also propose a novel application of MC-EM. We treat the random utilities ($\vec{X}$) as latent variables, and adopt the Expectation Maximization (EM) method to estimate parameters $\vec{\theta}$. The E-step for this problem is not analytically tractable, and for this we adopt a Monte Carlo approximation. We establish through experiments that the Monte-Carlo error in the E-step is controllable and does not affect inference, as long as numerical parameterizations are chosen carefully. In addition, for the E-step we suggest a parallelization over the agents and alternatives and a Rao-Blackwellized method, which further increases the scalability of our method.
We generally assume that the data provides total orders on alternatives from voters, but comment on how to extend the method and theory to the case where the input preferences are [*partial*]{} orders.
We evaluate our approach on synthetic data as well as two real-world datasets, a public election dataset and one involving rank preferences on sushi. The experimental results suggest that the approach is scalable despite providing significantly improved modeling flexibility over existing approaches.
For the two real-world datasets we have studied, we compare RUMs with normal distributions and P-L in terms of four criteria: log-likelihood, predictive log-likelihood, Akaike information criterion (AIC), and Bayesian information criterion (BIC). We observe that when the amount of data is not too small, RUMs with normal distributions fit better than P-L. Specifically, for the log-likelihood, predictive log-likelihood, and AIC criteria, RUMs with normal distributions outperform P-L with 95% confidence in both datasets.
RUMs and Exponential Families {#rumef}
=============================
In social choice, each agent $i\in \{1,\ldots,n\}$ has a strict preference order on alternatives. This provides the data for an inferential approach to social choice. In particular. let $L({\mathcal C})$ denote the set of all linear orders on ${\mathcal C}$. Then, a [*preference-profile*]{}, $D$, is a set of $n$ preference orders, one from each agent, so that $D \in L({\mathcal C})^n$.
A [*voting rule*]{} $r$ is a mapping that assigns to each preference-profile a set of winning rankings, $r: L({\mathcal C})^n\mapsto (2^{L({\mathcal C})}\setminus\emptyset)$. In particular, in the case of ties the set of winning rankings may include more than a singleton ranking. In the maximum likelihood (MLE) approach to social choice, 0
it is assumed that there is an unobserved ground truth, reflecting a true ranking of alternatives, and that each agent’s preference order is drawn independently. The MLE approach finds the ground truth that maximizes the probability of the preference profile (as provided by the reports, or votes of participants). This estimated ground truth is then used to determine the winning alternative. for a preference-profile (viewed as the the preference profile is viewed as [*data*]{}, $D=\{\pi^1,\ldots,\pi^n\}$.
Given this, the probability (likelihood) of the data given ground truth $\vec\theta$ (and for a particular $\vec\mu$) is $\Pr(D\ |\ \vec\theta)=\prod^n_{i=1} \Pr(\pi^i\ |\ \vec\theta),$ where, $$\begin{aligned}
\label{int}
\!\!P(\pi|\vec{\theta})\!=\!\!\!\int^{\infty}_{x_{\pi(n)}=-\infty}\int^{\infty}_{x_{\pi(n-1)}=x_{\pi(n)}}\!\!\!..\int^{\infty}_{x_{\pi(1)}=x_{\pi(2)}} \!\!\!\!\!\!\!\!\mu_{\pi(n)}(x_{\pi(n)})..\mu_{\pi(1)}(x_{\pi(1)}) dx_{\pi(1)}dx_{\pi(2)}..dx_{\pi(n)}\end{aligned}$$ The MLE approach to social choice selects as the winning ranking that which corresponds to the $\vec\theta$ that maximizes $\Pr(D\ |\
\vec\theta)$. In the case of multiple parameters that maximize the likelihood then the MLE approach returns a set of rankings, one ranking corresponding to each parameterization.
In this paper, we focus on probabilistic models where each $\mu_j$ belongs to the [*exponential family (EF)*]{}. The density function for each $\mu$ in EF has the following format: $$\begin{aligned}
\label{expf}
{\text{Pr}}(X= x)&=\mu(x)=e^{\eta(\theta) T(x)-A(\theta)+B(x)},\end{aligned}$$ where $\eta(\cdot)$ and $A(\cdot)$ are functions of $\theta$, $B(\cdot)$ is a function of $x$, and $T(x)$ denotes the sufficient statistics for $x$, which could be multidimensional.
In the RUM, let $\mu_j$’s be Gumbel distributions. That is, for alternative $j\in \{1,\ldots,m\}$ we have $\mu_j(x_j|\theta_j)=e^{-(x_j-\theta_j)}e^{-e^{-(x_j-\theta_j)}}$. Then, we have: ${\text{Pr}}(\pi\ |\ \vec \lambda)={\text{Pr}}(x_{\pi(1)}> x_{\pi(2)}>..> x_{\pi(m)})\notag= \prod^m_{j=1}{\lambda_{\pi(j)}\over\sum^m_{j'=j} \lambda_{\pi(j')}}$ , where $\eta({\theta_j})=\lambda_j=e^{\theta_j}$, $T(x_j)=-e^{-x_j}$, $B(x_j)=-x_j$ and $A(\theta_j)=-\theta_j$.This gives us the Plackett-Luce model.
Global Optimality and Log-Concavity
===================================
In this section, we provide a condition on distributions that guarantees that the likelihood function is log-concave in parameters $\vec \theta$. We also provide a condition under which the set of MLE solutions is bounded when any one latent parameter is fixed. Together, this guarantees the convergence of our MC-EM approach to a global mode with an accurate enough E-step. We focus on the [*location family*]{}, which is a subset of RUMs where the shapes of all $\mu_j$’s are fixed, and the only parameters are the means of the distributions. For the location family, we can write $X_j=\theta_j+\zeta_j$, where $X_j\sim\mu_j(\cdot|\theta_j)$ and $\zeta_j=X_j-\theta_j$ is a random variable whose mean is $0$ and models an agent’s [*subjective noise*]{}. The random variables $\zeta_j$’s do not need to be identically distributed for all alternatives $j$; e.g., they can be normal with different fixed variances.
We focus on computing solutions ($\vec \theta$) to maximize the log-likelihood function, $$\begin{aligned}
l(\vec{\theta};D)=\sum^n_{i=1} \log \Pr(\pi^i\ |\ {\vec\theta})\end{aligned}$$
\[thm1\] For the location family, if for every $j\leq m$ the probability density function for $\zeta_j$ is log-concave, then $l(\vec{\theta};D)$ is concave.
The theorem is proved by applying the following lemma, which is Theorem 9 in [@Prekopa80:Logarithmic].
\[thm9\] Suppose $g_1(\vec{\theta},\vec{\zeta}),...,g_R(\vec{\theta},\vec{\zeta})$ are concave functions in $\mathbb{R}^{2m}$ where $\vec{\theta}$ is the vector of $m$ parameters and $\vec{\zeta}$ is a vector of $m$ real numbers that are generated according to a distribution whose pdf is logarithmic concave in $\mathbb{R}^m$. Then the following function is log-concave in $\mathbb{R}^m$. $$\begin{aligned}
\label{joint}
L_i(\vec{\theta},G)=\Pr(g_1(\vec{\theta},\vec{\zeta})\ge0,...,g_R(\vec{\theta},\vec{\zeta})\ge0), \ \ \vec{\theta}\in \mathbb{R}^m
\end{aligned}$$
To apply Lemma \[thm9\], we define a set $G^i$ of function $g^i$’s that is equivalent to an order $\pi^i$ in the sense of inequalities implied by RUM for $\pi^i$ and $G^i$ (the joint probability in (\[joint\]) for $G^i$ to be the same as the probity of $\pi^i$ in RUM with parameters $\vec{\theta}$). Suppose $g^i_{r}(\vec{\theta},\vec{\zeta})=\theta_{\pi^i(r)}+\zeta^i_{\pi^i(r)}-\theta_{\pi^i{(r+1)}}-\zeta^i_{{\pi^i({r+1})}}$ for $r=1,..,m-1$.
Then considering that the length of order $\pi^i$ is $R+1$, we have: $$\begin{aligned}
L_i(\vec{\theta},\pi^i)=L_i(\vec{\theta},G^i)=\Pr(g^i_1(\vec{\theta},\vec{\zeta})\ge0,...,g^i_R(\vec{\theta},\vec{\zeta})\ge0), \ \ \vec{\theta}\in \mathbb{R}^m\end{aligned}$$
This is because $g^i_{r}(\vec{\theta},\vec{\zeta})\ge0$ is equivalent to that in $\pi^i$ alternative $\pi^i(r)$ is preferred to alternative $\pi^i(r+1)$ in the RUM sense.
To see how this extends to the case where preferences are specified as partial orders, we consider in particular an interpretation where an agent’s report for the ranking of $m_i$ alternatives implies that all other alternatives are worse for the agent, in some undefined order. Given this, define $g^i_{r}(\vec{\theta},\vec{\zeta})=\theta_{\pi^i(r)}+\zeta^i_{\pi^i(r)}-\theta_{\pi^i{(r+1)}}-\zeta^i_{{\pi^i({r+1})}}$ for $r=1,..,m_i-1$ and $g^i_{r}(\vec{\theta},\vec{\zeta})=\theta_{\pi^i(m_i)}+\zeta^i_{\pi^i(m_i)}-\theta_{\pi^i{(r+1)}}-\zeta^i_{{\pi^i({r+1})}}$ for $r=m_i,..,m-1$. Considering that $g^i_r(\cdot)$s are linear (hence, concave) and using log concavity of the distributions of $\vec {\zeta^i}=(\zeta^i_1,\zeta^i_2,..,\zeta^i_m)$’s, we can apply Lemma \[thm9\] and prove log-concavity of the likelihood function.
It is not hard to verify that pdfs for normal and Gumbel are log-concave under reasonable conditions for their parameters, made explicit in the following corollary.
For the location family where each $\zeta_j$ is a normal distribution with mean zero and with fixed variance, or Gumbel distribution with mean zeros and fixed shape parameter, $l(\vec{\theta};D)$ is concave. Specifically, the log-likelihood function for P-L is concave.
The concavity of log-likelihood of P-L has been proved [@Ford57:Solution] using a different technique. Using Fact 3.5. in [@Proschan89:Log], the set of global maxima solutions to the likelihood function, denoted by $S_D$, is convex since the likelihood function is log-concave. However, we also need that $S_D$ is bounded, and would further like that it provides one unique order as the estimation for the ground truth.
For P-L, Ford, Jr. [@Ford57:Solution] proposed the following necessary and sufficient condition for the set of global maxima solutions to be bounded (more precisely, unique) when $\sum_{j=1}^m e^{\theta_j}=1$.
\[cond1\] Given the data $D$, in every partition of the alternatives ${\mathcal C}$ into two nonempty subsets ${\mathcal C}_1\cup {\mathcal C}_2$, there exists $c_1\in {\mathcal C}_1$ and $c_2\in {\mathcal C}_2$ such that there is at least one ranking in $D$ where $c_1\succ c_2$.
We next show that Condition \[cond1\] is also a necessary and sufficient condition for the set of global maxima solutions $S_D$ to be bounded in location families, when we set one of the values $\theta_j$ to be $0$ (w.l.o.g., let $\theta_1=0$). If we do not bound any parameter, then $S_D$ is unbounded, because for any $\vec\theta$, any $D$, and any number $s\in \mathbb R$, $l(\vec \theta;D)=l(\vec \theta+s;D)$.
\[thm2\] Suppose we fix $\theta_1=0$. Then, the set $S_D$ of global maxima solutions to $l(\theta;D)$ is bounded if and only if the data $D$ satisfies Condition \[cond1\].
If Condition \[cond1\] does not hold, then $S_D$ is unbounded because the parameters for all alternatives in $C_1$ can be increased simultaneously to improve the log-likelihood. For sufficiency, we first present the following lemma.
\[lem1\] If alternative $j$ is preferred to alternative $j'$ in at least in one ranking then the difference of their mean parameters $\theta_{j'}-\theta_j$ is bounded from above ($\exists Q \ where \ \theta_{j'}-\theta_j<Q$) for all the $\vec{\theta}$ that maximize the likelihood function.
Suppose that $j\succ j'$ in rank $i$, then for any $\vec{\theta}\in \mathbb{R}^m$: $$\begin{aligned}
&L_i(\vec{\theta},\pi^i)=L_i(\vec{\theta},G^i)=\Pr(g_1(\vec{\theta},\vec{\zeta})\ge0,...,g_R(\vec{\theta},\vec{\zeta})\ge0)\nonumber\\
\le & \Pr(g_{\pi^i(r)}(\vec{\theta},\vec{\zeta})\ge0, g_{\pi^i(r+1)}(\vec{\theta},\vec{\zeta})\ge0, \ldots, g_{\pi^i(r')}(\vec{\theta},\vec{\zeta})\ge0) \le
\Pr( \zeta_{j}-\zeta_{j'}\ge \theta_{j'}-\theta_{j}),\end{aligned}$$ where $j=\pi^i(r)$ and $j'=\pi^i(r')$.
Let $K=l(\vec 0; D)$. Since the log-likelihood is always smaller than $0$, it follows that for any $\vec \theta\in S_D$ and any $i\leq n$, $L_i(\vec \theta;\pi^i)\geq K$.
Hence, $\Pr(\zeta_{j}-\zeta_{j'}\ge \theta_{j'}-\theta_{j})\geq K$.
Therefore, there exists $K'$ such that $\theta_{j'}-\theta_{j}< K'$, where $K'$ depends on the fixed $\zeta_{j'}$ and $\zeta_{j}$.
Now consider a directed graph $G_D$, where the nodes are the alternatives, and there is an edge between $c_{j}$ to $c_{j'}$ if in at least one ranking $c_{j}\succ c_{j'}$. By Condition \[cond1\], for any pair $j\neq j'$, there is a path from $c_{j}$ to $c_{j'}$ (and conversely, a path from $c_{j'}$ to $c_j$). To see this, consider building a path between $j$ and $j'$ by starting from a partition with ${\mathcal C}_1= \{j\}$ and following an edge from $j$ to $j_1$ in the graph where $j_1$ is an alternatives in ${\mathcal C}_2$ for which there must be such an edge, by Condition \[cond1\]. Consider the partition with ${\mathcal C}_1=\{j,j_1\}$, and repeat until an edge can be followed to vertex $j'\in {\mathcal C}_2$. It follows from Lemma \[lem1\] that for any $\vec{\theta}\in S_D$ we have $|\theta_j-\theta_{j'}|<Qm$, using the telescopic sum of bounded values of the difference of mean parameters along the edges of the path, since the length of the path is no more than $m$ (and tracing the path from $j$ to $j'$ and $j'$ to $j$), meaning that $S_D$ is bounded.
Now that we have the log concavity and bounded property, we need to declare conditions under which the bounded convex space of estimated parameters corresponds to a unique order. The next theorem provides a necessary and sufficient condition for all global maxima to correspond to the same order on alternatives. Suppose that we order the alternatives based on estimated $\theta$’s (meaning that $c_j$ is ranked higher than $c_{j'}$ iff $\theta_j>\theta_{j'}$).
\[thm:sameorder\] The order over parameters is strict and is the same across all $\vec\theta\in S_D$ if, for all $\vec\theta\in S_D$ and all alternatives $j\neq j'$, $\theta_j\neq \theta_{j'}$.
Suppose for the sake of contradiction there exist two maxima, $\vec \theta, \vec\theta^*\in S_D$ and a pair of alternatives $j\neq j'$ such that $\theta_j>\theta_{j'}$ and $\theta^*_{j'}>\theta_{j}^*$. Then, there exists an $\alpha<1$ such that the $j$th and $j'$th components of $\alpha\vec \theta+(1-\alpha)\vec\theta^*$ are equal, which contradicts the assumption.
Hence, if there is never a tie in the scores in any $\vec \theta\in S_D$, then any vector in $S_D$ will reveal the unique order.
Monte Carlo EM for Parameter Estimation
=======================================
In this section, we propose an MC-EM algorithm for MLE inference for RUMs where every $\mu_j$ belongs to the EF.[^2]
The EM algorithm determines the MLE parameters $\vec{\theta}$ iteratively, and proceeds as follows. In each iteration $t+1$, given parameters $\vec{\theta}^t$ from the previous iteration, the algorithm is composed of an E-step and an M-step. For the E-step, for any given $\vec\theta=(\theta_1,\ldots,\theta_m)$, we compute the conditional expectation of the complete-data log-likelihood (latent variables $\vec x$ and data $D$), where the latent variables $\vec x$ are distributed according to data $D$ and parameters $\vec{\theta}^t$ from the last iteration.
For the M-step, we optimize $\vec \theta$ to maximize the expected log-likelihood computed in the E-step, and use it as the input $\vec\theta^{t+1}$ for the next iteration: $$\begin{aligned}
&\mbox{E-Step :} \notag \ \ \ Q(\vec{\theta},\vec{\theta}^t)=E_{\vec X}\left\{\log \prod_{i=1}^n \Pr(\vec{x}^i,\pi^i \ |\ \vec{\theta})\ |\ D, \vec{\theta}^{t}\right\}
\\
&\mbox{M-step :}\notag \ \ \ \vec{\theta}^{t+1}\in\arg\max_{\vec{\theta}}
Q(\vec{\theta},\vec{\theta}^t)\end{aligned}$$
Monte Carlo E-step by Gibbs sampler
-----------------------------------
The E-step can be simplified using as follows: $$\begin{aligned}
&E_{\vec X}\{\log \prod_{i=1}^n \Pr(\vec{x}^i,\pi^i\ |\ \vec{\theta})\ |\ {D, \vec{\theta}^{t}} \}=E_{\vec X}\{\log \prod_{i=1}^n \Pr(\vec{x}^i|\ \vec{\theta})\Pr(\pi^i|\vec{x}^i)\ |\ {D, \vec{\theta}^{t}} \}\end{aligned}$$ $$\begin{aligned}
&=\sum_{i=1}^n \sum_{j=1}^m E_{X^i_j}\{\log \mu_j(x^i_j|\theta_j)\ |\ \pi^i,\vec{\theta}^{t}\}
=\sum_{i=1}^n \sum_{j=1}^m (\eta(\theta_j)E_{X^i_j}\{T(x^i_j)\ |\ \pi^i,\vec{\theta}^{t}\}-A(\theta_j)+W,
$$ where $W=E_{X^i_j}\{B(x^i_j)\ |\ \pi^i,\vec{\theta}^{t} \}$ only depends on $\vec\theta_t$ and $D$ (not on $\vec \theta$), which means that it can be treated as a constant in the M-step. Hence, in the E-step we only need to compute $S^{i,t+1}_j=E_{X_j^i}\{T(x^i_j)\ |\ \pi^i,\vec{\theta}^t\}$ where $T(x^i_j)$ is the sufficient statistic for the parameter $\theta_j$ in the model. We are not aware of an analytical solution for $E_{X^i_j}\{T(x^i_j)\ |\ \pi^i,\vec{\theta}^t\}$. However, we can use a Monte Carlo approximation, which involves sampling $\vec{x}^i$ from the distribution $\Pr(\vec{x}^i\ |\ \pi^i,\vec{\theta^t})$ using a Gibbs sampler, and then approximates $S^{i,t+1}_j$ by ${1\over N}\sum^N_{k=1}T(x^{i,k}_j)$ where $N$ is the number of samples in the Gibbs sampler.
In each step of our Gibbs sampler for voter $i$, we randomly choose a position $j$ in $\pi^i$ and sample $x_{\pi^i(j)}^i$ according to a [*TruncatedEF*]{} distribution $\Pr(\cdot |\ x_{\pi^i(-j)}, \vec{\theta^t}, \pi^i)$, where $\ x_{\pi^i(-j)}=(\ x_{\pi^i(1)},\ldots, \ x_{\pi^i(j-1)}, \ x_{\pi^i(j+1)},\ldots,\ x_{\pi^i(m)})$. The TruncatedEF is obtained by truncating the tails of $\mu_{\pi^i(j)}(\cdot|\theta^t_{\pi^i(j)})$ at $x_{\pi^i(j-1)}$ and $x_{\pi^i(j+1)}$, respectively. For example, a truncated normal distribution is illustrated in Figure \[figreTN\].
![A truncated normal distribution. \[figreTN\]](newfigure2.jpg){width=".9\textwidth"}
**Rao-Blackwellized**: To further improve the Gibbs sampler, we use Rao-Blackwellized [@Brooks11:Handbook] estimation using $E\{T(x^{i,k}_j)\ |\ x^{i,k}_{-j},\pi^i,\vec{\theta}^t\}$ instead of the sample $x^{i,k}_j$, where $x^{i,k}_{-j}$ is all of $\vec{x}^{i,k}$ except for $x^{i,k}_j$. Finally, we estimate $E\{T(x^{i,k}_j)\ |\ x^{i,k}_{-j},\pi^i,\vec{\theta}^t\}$ in each step of the Gibbs sampler using $M$ samples as $S_j^{i,t+1}\simeq{1\over N}\sum^N_{k=1} E\{T(x^{i,k}_j)\ |\ x^{k}_{-j},\pi^i,\vec{\theta}^t\}\simeq{1\over NM}\sum^N_{k=1} \sum^M_{l=1} T(x^{i^l,k}_j),$ where $x^{i^l,k}_j\sim \Pr(x^{i^l,k}_j\ |\ x^{i,k}_{-j}, \pi^i,\vec{\theta})$. Rao-Blackwellization reduces the variance of the estimator because of conditioning and expectation in $E\{T(x^{i,k}_j)\ |\ x^{i,k}_{-j},\pi^i,\vec{\theta}^t\}$.
M-step
------
In the E-step we have (approximately) computed $S_j^{i,t+1}$. In the M-step we compute $\vec\theta^{t+1}$ to maximize $\sum_{i=1}^n \sum_{j=1}^m (\eta(\theta_j)E_{X^i_j}\{T(x^i_j)\ |\ \pi^i,\vec{\theta}^{t}\}-A(\theta_j)+E_{X^i_j}\{B(x^i_j)\ |\ \pi^i,\vec{\theta}^{t}\})$. Equivalently, we compute $\theta_j^{t+1}$ for each $j\leq m$ separately to maximize $\sum_{i=1}^n\{\eta(\theta_j)E_{X^i_j}\{T(x^i_j)\ |\ \pi^i,\vec{\theta}^{t}\}-A(\theta_j)\}=\eta(\theta_j)\sum_{i=1}^nS_{j}^{i,t+1}-nA(\theta_j)$.
For the case of the normal distribution with fixed variance, where $\eta(\theta_j)=2 \theta_j$ and $A(\theta_j)=(\theta_j)^2$, we have $\theta_{j}^{t+1}={1\over n}\sum^n_{i=1} S^{i,t+1}_j$. The algorithm is illustrated in Figure \[figalg\].
{width="400pt"}
Convergence
-----------
In the last section we showed that if the RUM satisfies the premise in Theorem \[thm1\] and Theorem \[thm2\] the data satisfies Condition \[cond1\], then the log-likelihood function is concave, and the set of global maxima solutions is bounded. This guarantee the convergence of MC-EM for an exact E-step.
In general, MC-EM methods do not have the uniform convergence property of EM methods. In order to control the error of approximation in the MC-E step we can increase the number of samples with the iterations [@Wei90:Monte]. However, in our application, we are not concerned with the exact estimation of $\vec{\theta}$, as we are only interested in their orders relative to each-other. Therefore, as long as the approximation error remains relatively small, such that the differences of $\theta_j$s are much larger than the error, we are safe to stop.
A known problem with Gibbs sampling is that it can introduce correlation among samples. To address this, we sub-sample the samples to reduce the correlation, and call the ratio of sub-sampling the [*thinning factor*]{} ($0< F\le1$). A suitable thinning ratio can be set using empirical results from the sampler.
With an approach similar to [@Booth99:Maximizing], we can derive a relationship between the variance of error in $\vec{\theta}^{t+1}$ and the Monte-Carlo error in the E-step approximation: $$\begin{aligned}
\mathit{Var}({\theta_j}^{t+1})={1\over n^2}\sum^n_{i=1} \mathit{Var}(S^{i,t+1}_j)={1\over MNn^2}\sum^n_{i=1} \mathit{Var}(x^{i}_j)\le {F V\over MNn},\end{aligned}$$ where $N$ is number of samples in Gibbs sampler, $M$ is the number of samples for Rao-Blackwellization, $n$ is number of agents, $F$ is the thinning factor and $V=\max_j (\mathit{Var}_{x \sim \mu_j}(x))$, and samples $x^{i}_j$ are assumed to be independent. Given, $T$, $V$ and $n$, we can make $\mathit{Var}({\theta_j}^{t+1})$ arbitrarily small by increasing $MN$.
Experimental Results
====================
We evaluate the proposed MC-EM algorithm on synthetic data as well as two real world data sets, namely an election data set and a dataset representing preference orders on sushi. For simulated data we use the Kendall correlation [@Grzegorzewski09:Kendall] between two rank orders (typically between the true order and the method’s result) as a measure of performance.
Experiments for Synthetic Data
------------------------------
We first generate data from Normal models for the random utility terms, with means $\theta_j=j$ and equal variance for all terms, for different choices of variance ($\mathit{Var}=2,4$). We evaluate the performance of the method as the number of agents $n$ varies. The results show that a limited number of iterations in the EM algorithm (at most 3), and samples $MN=4000$ (M=5, N=800) are sufficient for inferring the order in most cases. The performance in terms of Kendall correlation for recovering ground truth improves for larger number of agents, which corresponds to more data. See Figure \[figcomp\], which shows the asymptotic behavior of the maximum likelihood estimator in recovering the true parameters. Figure \[figcomp\] left and middle panels show that the more the size of dataset the better the performance of the method.
Moreover, for large variances in data generation, due to increasing noise in the data, the rate that performance gets better is slower than that for the case for smaller variances. Notice that the scales on the y-axis are different in the left and middle panels.
![Left and middle panel: Performance for different number of agents $n$ on synthetic data for $m=5,10$ and $\mathit{Var}=2,4$, with specifications $MN=4000$, $EM iterations=3$. Right panel: Performance given access to sub-samples of the data in the public election dataset, x-axis: size of sub-samples, y-axis: Kendall Correlation with the order obtained from the full data-set. Dashed lines are the 95% confidence intervals. \[figcomp\]](figure3.jpg){width="100.00000%"}
Experiments for Model Robustness
--------------------------------
We apply our method to a public election dataset collected by Nicolaus Tideman [@Tideman06:Collective], where the voters provided partial orders on candidates. A partial order includes comparisons among a subset of alternative, and the non-mentioned alternatives in the partial order are considered to be ranked lower than the lowest ranked alternative among mentioned alternatives.
The total number of votes are $n=280$ and the number of alternatives $m=15$. For the purpose of our experiments, we adopt the order on alternatives obtained by applying our method on the entire dataset as an assumed ground truth, since no ground truth is given as part of the data. After finding the ground truth by using all 280 votes (and adopting a normal model), we compare the performance of our approach as we vary the amount of data available. We evaluate the performance for sub-samples consisting of $10,20,\ldots,280$ of samples randomly chosen from the full dataset. For each sub-sample size, the experiment is repeated $200$ times and we report the average performance and the variance. See the right panel in Figure \[figcomp\]. This experiment shows the robustness of the method, in the sense that the result of inference on a subset of the dataset shows consistent behavior with the case that the result on the full dataset. For example, the ranking obtained by using half of the data can still achieve a fair estimate to the results with full data, with an average Kendall correlation of greater than 0.4.
Experiments for Model Fitness
-----------------------------
In addition to a public election dataset, we have tested our algorithm on a sushi dataset, where 5000 users give rankings over 10 different kinds of sushi [@Kamishima03:Nantonac]. For each experiment we randomly choose $n\in\{10, 20, 30, 40,
50\}$ rankings, apply our MC-EM for RUMs with normal distributions where variances are also parameters.
In the former experiments, both the synthetic data generation and the model for election data, the variances were fixed to $1$ and hence we had the theoretical guarantees for the convergence to global optimal solutions by Theorem \[thm1\] and Theorem \[thm2\]. When we let the variances to be part of parametrization we lose the theoretical guarantees. However, the EM algorithm can still be applied, and since the variances are now parameters (rather than being fixed to $1$), the model fits better in terms of log-likelihood. For this reason, we adopt RUMs with normal distributions in which the variance is a parameter that is fit by EM along with the mean. We call this model a [*normal model*]{}. We compute the difference between the normal model and P-L in terms of four criteria: log-likelihood (LL), predictive log-likelihood (predictive LL), AIC, and BIC. For (predictive) log-likelihood, a positive value means that normal model fits better than P-L, whereas for AIC and BIC, a negative number means that normal model fits better than P-L. Predictive likelihood is different from likelihood in the sense that we compute the likelihood of the estimated parameters for a part of the data that is not used for parameter estimation.[^3] In particular, we compute predictive likelihood for a randomly chosen subset of $100$ votes. The results and standard deviations for $n=10,50$ are summarized in Table \[tab:difference\].
--------- ------------------------------------------------------------------------------------------------------------- ---------- ----- ----- ---- ---------- ----- ----- --
Dataset LL Pred. LL AIC BIC LL Pred. LL AIC BIC
Sushi **8.8(4.2)&-56.1(89.5) &-7.6(8.4) &5.4(8.4)&**22.6(6.3)&**40.1(5.1)&**-35.2(12.6)&-6.1(12.6)\
Election& 9.4(10.6)&91.3(103.8)&-8.8(21.2) &4.2(21.2) &**44.8(15.8)&**87.4(30.5)&**-79.6(31.6)&-50.5(31.6)\
**************
--------- ------------------------------------------------------------------------------------------------------------- ---------- ----- ----- ---- ---------- ----- ----- --
: Model selection for the sushi dataset and election dataset. Cases where the normal model fits better than P-L statistically with 95% confidence are in bold.\[tab:difference\]
When $n$ is small $(n=10)$, the variance is high and we are unable to obtain statistically significant results in comparing fitness. When $n$ is not too small ($n=50$), RUMs with normal distributions fit better than P-L. Specifically, for log-likelihood, predictive log-likelihood, and AIC, RUMs with normal distributions outperform P-L with 95% confidence in both datasets.
Implementation and Run Time
---------------------------
The running time for our MC-EM algorithm scales linearly with number of agents on real world data (Election Data) with slope 13.3 second per agent on an Intel $i5$ $2.70$GHz PC. This is for 100 iterations of EM algorithm with Gibbs sampling number increasing with iterations as $2000+300*iteration \ steps$.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work is supported in part by NSF Grant No. CCF- 0915016. Lirong Xia is supported by NSF under Grant \#1136996 to the Computing Research Association for the CIFellows Project. We thank Craig Boutilier, Jonathan Huang, Tyler Lu, Nicolaus Tideman, Paolo Viappiani, and anonymous NIPS-12 reviewers for helpful comments and suggestions, or help on the datasets.
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[^1]: $\mu_j(\cdot|\theta_j)$ might be parameterized by other parameters, for example variance.
[^2]: Our algorithm can be naturally extended to compute a maximum [*a posteriori*]{} probability (MAP) estimate, when we have a prior over the parameters $\vec{\theta}$. Still, it seems hard to motivate the imposition of a prior on parameters in many social choice domains.
[^3]: The use of predictive likelihood allows us to evaluate the performance of the estimated parameters on the rest of the data, and is similar in this sense to cross validation for supervised learning.
|
---
abstract: 'In a recent paper, “Reexamining $f\left(R,T\right)$ gravity”, by S. B. Fisher and E. D. Carlson, Phys. Rev. D 100, 064059 (2019), the authors claim that for the particular $f(R,T)$ modified gravity model, with $f(R,T)=f_1(R)+f_2(T)$, the term $f_2(T)$ must be included in the matter Lagrangian and therefore it does not have any physical significance. We carefully reexamine the line of reasoning presented in the paper, and we show that there are several major conceptual problems related to the author’s physical interpretations, as well as in the physical and mathematical approaches used to derive the energy-momentum tensor of the theory. These problems raise some serious concerns about the validity of most of the results presented in the paper.'
author:
- Tiberiu Harko
- 'Pedro H. R. S. Moraes'
title: 'Comment on “Reexamining $f\left(R,T\right)$ gravity”, Phys. Rev. D 100, 064059 (2019)'
---
Introduction
============
In an interesting and thought provoking paper, “Reexamining $f\left(R,T\right)$ gravity” by S. B. Fisher and E. D. Carlson [@1], the authors propose a mathematical, as well as a physical reformulation of a specific version of the $f(R,T)$ gravity theory [@2], in which the gravitational Lagrangian $f(R,T)$ can be decomposed as $f(R,T)=f_1(R)+f_2(T)$. Here $R$ is the Ricci scalar, $T$ denotes the trace of the energy-momentum tensor and $f_1(R)$ and $f_2(T)$ are functions specifically dependent on $R$ and $T$, respectively. In the present Comment, we carefully reexamine the results of the paper [@1] and we show that there are several conceptual problems in their physical analysis and interpretation of the $T$-dependence of the $f(R,T)$ gravity.
We will first briefly summarize in the following what the authors of [@1] are essentially doing.
Let us begin with the gravitational action $$\begin{aligned}
S&=&\int{\left[\frac{1}{2\kappa ^2}f(R,T)+L_m\right]\sqrt{-g}d^4x} \notag
\\
&=&\int{\left[\frac{1}{2\kappa ^2}f_1(R)+f_2(T)+L_m\right]\sqrt{-g}d^4x},\end{aligned}$$ where $\kappa^ 2=8\pi G$, $f_1(R)$ and $f_2(T)$ are arbitrary (analytical) functions of $R$ and $T$, respectively, $L_m$ is the matter action, and $g$ is the metric determinant. Since both $f_2(T)$ and $L_m$ are functions of the same argument, that is, of the thermodynamical parameters of the system, one can obviously combine them into a single term $L_m^{(\text{eff})}$, $$\label{1}
L_m^{(\text{eff})}=f_2(T)+L_m,$$ so that the gravitational action of the $f(R,T)$ theory takes an *effective* form, given by $$\label{1a}
S=\int{\left[\frac{1}{2\kappa ^2}f_1(R)+L_m^{(\text{eff})}\right]\sqrt{-g}d^4x}.$$
Now the main *physical claim* of the authors of [@1] is that the effective matter Lagrangian (\[1\]) is the true matter Lagrangian of the system and that “$f_2(T)$ is not physically meaningful”. But, before going into the discussion of the claimed physical usefulness of $f_2(T)$, let us consider firstly the simple example of the scalar field, as discussed in [@1].
The case of the scalar field
============================
For a scalar field $\phi$ with potential $V(\phi)$, the Lagrangian $L_{\phi}$ and its trace $T_{\phi}$ are (in the notations of [@1]) $$\label{3}
L_{\phi}=\frac{1}{2}\left[\nabla _{\mu}\phi \nabla ^{\mu} \phi-V(\phi)\right], T_{\phi}=-\nabla _{\nu}\phi \nabla ^{\nu}\phi+2V(\phi).$$
Then, in the framework of the $f(R,T)=f_{1}(R)+f_{2}(T_{\phi})$ theory, a scalar field Lagrangian of the form $$L_{\phi }^{({\text{eff}})}(\phi )=L_{\phi }+\alpha \ln \left\vert T_{\phi
}\right\vert , \label{4}$$where $\alpha $ is a constant, is perfectly justified.
Let us analyze now the possibility of reducing this scalar field Lagrangian to its standard form (\[3\]). For this we consider a general transformation of the scalar field given by $\phi =F(\Phi )$, where $F$ is an arbitrary function of a new scalar field $\Phi$. Since we have \_= \_, the scalar field Lagrangian becomes $$\begin{aligned}
\label{5}
L_{\Phi }(\Phi )&=&\frac{1}{2}\left[\frac{dF\left( \Phi \right) }{d\Phi }\right] ^{2}\left[ \nabla _{\mu }\Phi \nabla ^{\mu }\Phi -U\left( \Phi
\right) \right] \notag \\
&+&\alpha \ln \left\vert -\nabla _{\nu }\Phi \nabla ^{\nu }\Phi +2U(\Phi
)\right\vert +2\alpha \ln \frac{dF\left( \Phi \right) }{d\Phi }, \notag \\\end{aligned}$$ where $U\left( \Phi \right) =V(\Phi )/\left[ dF(\Phi )/d\Phi \right] ^{2}$. Hence Eq. (\[4\]) is almost form invariant with respect to an arbitrary transformation of the scalar field and therefore the claim “For more complicated functions $f_{2}(T)$, the resulting terms will of course not be simply a rescaling of the field, but will change the free field into an interacting field.” in [@1] does not look to be true ([*in a natural way*]{}) for arbitrary functions $f_2\left(T_{\phi}\right)$.
The energy momentum tensor in $f(R,T)=f_1(R)+f_2(T)$ gravity
============================================================
Next we need to clarify the concept of matter, “physical pressure” and “physical energy density”, respectively. By matter we usually understand a system of interacting particles, whose structure and dynamics are determined by the known (or lesser known) laws of nature. Based on these laws, the physical parameters of fluids, like the four-velocity, for example, and the set of thermodynamic scalars, such as energy density, pressure, temperature and specific enthalpy, can be defined uniquely in an instantaneous Lorentz frame carried by the fluid, and determined accordingly experimentally.
However, in order to have a correct understanding of the terms we are using in our Comment it is necessary to explain the definitions of physical and effective thermodynamic quantities. In a broad sense, we mean by physical quantities those defined in standard textbooks of physics, like, for example, [@Lan]. In a more restricted sense [*we define the physical (thermodynamical) quantities as the quantities that are obtained from the microscopic distribution functions of the particles (Fermi-Dirac, Bose-Einstein, Boltzmann etc.)*]{}.
The presence of the gravitational field modifies the distribution function at the microscopic level, and at the level of the total energy. The problem of the gravitational energy and of its localization is a complex one (not yet considered in $f(R,T)$ gravity), but once we succeed in including it in the distribution functions of the particles, we can obtain from them the thermodynamic potentials, energy density, pressure etc.. The quantities obtained in this way indeed correspond to the “true” physical and thermodynamical variables in a gravitational field. However, they will depend essentially on the metric, and any role played by $f_2(T)$, if any, is uncertain. For a discussion of the problem of the energy-momentum pseudotensor of the gravitational field in standard general relativity see [@Lan1].
On the other hand we can construct thermodynamic like quantities by simply combining (additively) the physical pressure with other similar quantities of different origins. We call these kind of quantities [*effective quantities*]{}, and they are not “true” physical quantities in the sense previously defined, since (generally) [*they cannot be derived from microscopic distribution functions of particles*]{}.
Hence the scalar physical thermodynamic quantities [*cannot be rescaled arbitrarily by adding to them some functions of the same thermodynamic parameters*]{}. One cannot claim that the pressure of the degenerate fermionic relativistic gas, given by $p\propto \rho ^{4/3}$ is not physical, and cannot be determined experimentally, and that the true pressure of the degenerate Fermi gas is (let us say), $p\propto \rho ^{4/3}+f_2(T=\rho-3p)$. [*One should clearly point out that the standard thermodynamic quantities (density, temperature, pressure etc.) are real physical quantities that can be obtained from the microscopic distribution function of particles, and as such, they are at the theoretical foundations for the description of gravitational processes involving the presence of matter*]{}. On the other hand the thermodynamic quantities considered in [@1] can be considered only [*effective quantities*]{}, and they are definitely not the “true” physical pressures or energy densities of any (real) physical system, since they cannot be obtained from any (known) classical or quantum distribution function of particles.
Constructing the energy-momentum tensor
---------------------------------------
In order to extract some useful information from the action (\[1a\]), and to construct an energy-momentum tensor for the theory, the authors of [1]{} adopt the formalism developed in [@3], by using for the matter Lagrangian density the standard expression (16) in their paper. Then the authors arrive at the “on shell” current densities and particles number, defined in Eqs. (21a) and (21b), which include the function $f_2(T)$ and which are conserved “on shell”. However, since in this approach [*the physical current $J^{\mu}$ and the physical particle number $n$ are not conserved*]{}, the authors reach the controversial conclusion that in this model there may be some “true” energy density, given by their Eqs. (27) and (29), and which also may represent the experimentally measurable energy density.
The problem of the construction of the energy-momentum tensor in modified theories of gravity with geometry-matter coupling was discussed in [@4] and [@5] (see also [@6]).
However, in the present Comment we use a different, and more physical approach [@7; @8]. First we require that the variations of the entropy density $s$ and of the ordinary matter number flux vector density, n\^=nu\^, where $n$ is the particle number, defined as n=, satisfy the constraints s=0, and n\^=0, respectively, which maintain unchanged the entropy and particle production rates. Therefore, the entropy and particle number currents satisfy the conservation equations $\delta \left(
n^{\mu }\partial _{\mu }s\right) =0$ and $\nabla _{\mu }\left( nu^{\mu
}\right) =0$, respectively.
Let the equation of state for matter be given as $\rho =\rho \left(
n,s\right) $. Then, since $\delta s=0$, from the thermodynamic relation $\left( \partial \rho /\partial n\right) _{s}=w=\left( \rho +p\right) /n$, we obtain $\delta \rho =w\delta n$. By taking the variation of the particle number $n$ we find [@8] $$\begin{aligned}
\delta n&=&\frac{n}{2}\left( -g\right) u^{\mu }u^{\nu }\left( \frac{\delta
g_{\mu \nu }}{g}-\frac{g_{\mu \nu }}{g^{2}}\delta g\right) = \notag \\
&&-\frac{n}{2}\left( u^{\mu }u^{\nu }+g^{\mu \nu }\right) \delta g_{\mu \nu
}.\end{aligned}$$
For the sake of concreteness, and for simplicity, we assume that the ordinary matter Lagrangian is $L_{m}=-\rho $ [@8], and we introduce the effective matter action as $$S_{m}^{(eff)}=-\int \left[ \rho -f_{2}\left( T\left( n,s\right) \right) \right] \sqrt{-g}d^{4}x.$$
By taking the variation of $S_{m}^{(eff)}$, we obtain $$\begin{aligned}
\delta S_{m}^{(eff)} &=&-\int \Bigg\{\left[ \delta \rho -\frac{df_{2}(T)}{dT}\frac{dT\left( n,s\right) }{dn}\delta n\right] \sqrt{-g}+ \notag \\
&&\left[ \rho -f_{2}\left( T\left( n,s\right) \right) \right] \delta \sqrt{-g}\Bigg\}d^{4}x= \notag \\
&&-\int \left[ w-\frac{df_{2}(T)}{dT}\frac{dT\left( n,s\right) }{dn}\right] \delta n\sqrt{-g}- \notag \\
&&\frac{1}{2}\left[ \rho -f_{2}\left( T\left( n,s\right) \right) \right]
\sqrt{-g}g^{\mu \nu }\delta g_{\mu \nu },\end{aligned}$$immediately giving $$\begin{aligned}
&&\hspace{-0.5cm}\delta S_{m}^{(eff)}=\frac{1}{2}\int \Bigg\{\left[ \rho +p-n\frac{df_{2}(T)}{dT}\frac{dT(n,s)}{dn}\right] u^{\mu }u^{\nu }+ \notag \\
&&\hspace{-0.5cm}\left[ p+f_{2}\left( T\left( n,s\right) \right) -n\frac{df_{2}(T)}{dT}\frac{dT(n,s)}{dn}\right] g^{\mu \nu }\Bigg\}\delta g_{\mu \nu }\sqrt{-g}d^{4}x. \notag \\\end{aligned}$$
Hence the [*effective matter energy-momentum tensor*]{} that can be constructed from the action (\[1a\]) is given by \^[(eff)]{}T\^&=&u\^u\^+\
&&g\^,\
and it obviously reduces to the perfect fluid form of standard general relativity when $f_{2}(T)=0$. $^{(eff)}T^{\mu \nu }$ can also be written in terms of [*an effective energy density $\rho _{eff}$ and pressure $p_{eff}$*]{}, defined as $$\rho _{eff}(n,s)=\rho (n,s)-f_{2}\left( T\left( n,s\right) \right) ,$$ $$p_{eff}(n,s)=p(n,s)+f_{2}\left( T\left( n,s\right) \right) -n\frac{df_{2}(T)}{dT}\frac{dT(n,s)}{dn},$$in the form \^[(eff)]{}T\^=( \_[eff]{}+p\_[eff]{}) u\^u\^+p\_[eff]{}g\^.
This energy-momentum tensor is different from the one obtained in [@1] on both conceptual and mathematical levels. For the fluid description of matter, and in standard general relativity, [*the same expression for the energy-momentum tensor can be obtained if one assumes for the matter Lagrangian the expression $L_m=p$, and by decomposing the velocity in terms of scalar potentials*]{} [@8]. If the degeneracy of the matter Lagrangian can be removed in $f(R,T)$ gravity, due to the presence of the matter-geometry coupling, and if yes, how this can be done, is a problem that goes beyond the topics of the present Comment, and thus we will not consider it.
We must also point out that in the present approach [*the ordinary matter satisfies all the standard conservation laws*]{}, without the necessity of introducing any “on shell” quantities, and conservation laws, or modifying the physical interpretation of the ordinary thermodynamical quantities.
Conclusions
===========
We can now summarize our main findings as follows. The claim in [@1] that the function $f_2(T)$ can be just simply included in the matter action in a [*physical*]{} way is questionable. The matter energy density and pressure are two fundamental thermodynamic quantities that are obtained from microscopic particle distribution functions, and they cannot be arbitrarily modified without completely changing the content of the physical laws, and of the corresponding theories. However, such a construction is perfectly valid from the point of view of the construction of [*effective*]{} physical quantities. The mathematical/physical approach employed by the authors of [@1] to derive the energy-momentum tensor of the $f(R,T)=f_1(R)+f_2(T)$ gravity theory, [*even correct mathematically*]{}, raises some questions about its physical interpretation, since matter satisfies the conservation equations of the current and entropy only “on shell” (that is, they are not conserved in the true physical sense). The “true” physical quantities describing matter in any physical theory are $\rho $ and $p$, and they are not equivalent in any sense (be it mathematical or physical) with [*the effective quantities that also include $f_2(T)$*]{}. Hence the search for the functional form of $f_2(T)$ is a valid one, and finding observational restrictions/constraints on the function $f_2(T)$, as done, for example, in [@9] and [@10], is an important field of research that could lead to some new insights and a better understanding of the mathematical structure and astrophysical and cosmological implication of the $f(R,T)$ gravity.
Acknowledgments {#acknowledgments .unnumbered}
===============
T. H. would like to thank the Yat-Sen School of the Sun Yat-Sen University in Guangzhou, P. R. China, for the kind hospitality offered during the preparation of this work. P. H. R. S. M. would like to thank São Paulo Research Foundation (FAPESP), grant 2015/08476-0, for financial support. He also thanks the financial support of FAPESP under the thematic project 2013/26258-4.
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---
abstract: 'We tackle the novel problem of navigational voice queries posed against an entertainment system, where viewers interact with a voice-enabled remote controller to specify the program to watch. This is a difficult problem for several reasons: such queries are short, even shorter than comparable voice queries in other domains, which offers fewer opportunities for deciphering user intent. Furthermore, ambiguity is exacerbated by underlying speech recognition errors. We address these challenges by integrating word- and character-level representations of the queries and by modeling voice search sessions to capture the contextual dependencies in query sequences. Both are accomplished with a probabilistic framework in which recurrent and feedforward neural network modules are organized in a hierarchical manner. From a raw dataset of 32M voice queries from 2.5M viewers on the Comcast Xfinity X1 entertainment system, we extracted data to train and test our models. We demonstrate the benefits of our hybrid representation and context-aware model, which significantly outperforms models without context as well as the current deployed product.'
author:
- 'Jinfeng Rao$^{1,2}$, Ferhan Ture$^{1}$, Hua He$^{2}$, Oliver Jojic$^{1}$, and Jimmy Lin$^{3}$'
title: |
Talking to Your TV: Context-Aware Voice Search\
with Hierarchical Recurrent Neural Networks
---
Introduction
============
Voice-based interactions with computing devices are becoming increasingly prevalent, driven by several convergent trends. The ubiquity of smartphones and other mobile devices with restrictive input methods makes voice an attractive modality for interaction: Apple’s Siri, Microsoft’s Cortana, and the Google Assistant are prominent examples. Google observed that there are more searches taking place from mobile devices than from traditional desktops,[^1] and that 20% of mobile searches are voice queries.[^2] The success of these products has been enabled by advances in automatic speech recognition (ASR), thanks mostly to deep learning.
Increasing comfort with voice-based interactions, especially with AI-agents, feeds into the emerging market on “smart homes”. Products such as Amazon Echo and Google Home allow users to control a variety of devices via voice (e.g., “turn on the TV”, “play music by Adele”), and to issue voice queries (e.g., “what’s the weather tomorrow?”). The market success of these products demonstrates that people do indeed want to control smart devices in their environment via voice.
In this paper, we tackle the problem of navigational voice queries posed against an entertainment system, where viewers interact with a voice-enabled remote controller to specify the program (TV shows, movies, sports games) they wish to watch. If a viewer wishes to watch the popular series “Game of Thrones”, saying the name of the program should switch the television to the proper channel. This is simpler and more intuitive than scrolling through channel guides or awkwardly trying to type in the name of the show on the remote controller. Even if the viewer knows that Game of Thrones is on HBO, finding the right channel may still be challenging, since entertainment packages may have hundreds of channels.
Our problem is challenging for a few reasons. Viewers have access to potentially tens of thousands of programs, especially if we include on-demand titles. Program names can be highly ambiguous. For instance, the query “Chicago Fire” could refer to either the television series or a soccer team. Even with recent advances, ASR errors can exacerbate the ambiguity by transcribing queries like “Caillou” (a Canadian children’s education television series) as “you”. Based on our analysis of 32M voice queries in this domain, we find that they are shorter (average of 2.04 words) than published statistics about voice queries on smartphones and computers [@guy2016searching; @schalkwyk2010your]. Short queries make the prediction problem more difficult because there is less signal to extract.
[**Contributions.**]{} We tackle the above challenges using two key ideas to infer user intent: hybrid query representations and modeling search sessions. Specifically, our contributions are as follows:
- To our knowledge, we are the first to systematically study voice queries in the entertainment context. We propose a technique to automatically collect ground truth labels for voice query sessions from real-world usage data by examining viewing behaviors following the sessions.
- Our probabilistic model has two key features: First, we integrate word- and character-level representations of the queries. Second, we model voice search sessions to understand the contextual dependencies in query sequences. Both are accomplished with a probabilistic framework in which recurrent and feedforward neural network modules are organized in a hierarchical manner.
- Evaluations on a large real-world dataset demonstrate the effectiveness of our hybrid query representation and context-aware models, significantly outperforming strong baselines as well as the current deployed system. Detailed analyses clarify [*how*]{} our models are better able to understand user intent.
Background and Related Work
===========================
The context of our work is voice search on the Comcast Xfinity X1 entertainment platform, by one of the largest cable companies in the United States with approximately 22 million subscribers in 40 states. X1 is essentially a software package distributed as part of the X1 cable box, which has been deployed to 17 million customers since around 2015. X1 can be controlled via the “voice remote”, which is a remote controller that has an integrated microphone to receive voice queries from viewers. The current deployed system is based on a combination of hand-crafted rules and machine-learned models to arrive at a final response. The system has a diverse set of capabilities, which increases query ambiguity and magnifies the overall challenge of understanding user intent. These capabilities range from channel change to entity search (e.g., sports team, person, movie, etc.). In addition, voice queries may involve general questions, from home security control to troubleshooting the wifi network, or may be ultimately directed to external apps such as Pandora. In this paper, we focus on navigational voice queries where viewers specify the program they wish to watch.
In our particular application, we receive as input the one-best result of the ASR system, which is a text string. We do not have access to the acoustic signal, as the ASR system is a black box. Although it would be ideal if we could build joint models over both the acoustic signals, transcription lattice, and user intent, in many operational settings this is not practical or even possible. In the case of X1, for a variety of reasons, the ASR is outsourced to a third party—a scenario not uncommon in many organizations who do not wish to invest in ASR from the ground up. Thus, as we described in the introduction, transcription error compounds the ambiguity in the queries and introduces additional complexity that our models need to handle.
We are, of course, not the first to tackle voice search [@wang2008introduction; @acero2008live; @feng2009effects; @chelba2013empirical; @shan2010search], although to our knowledge we are the first to focus on voice queries directed at an entertainment system. How is this particular domain different? The setting is obviously different—in our case, viewers are clearly sitting in front a television with an entertainment intent. To compare and contrast viewers’ actual utterances, we can turn to previously-published work that studied the characteristics of voice search logs, especially in comparison to text search data [@crestani2006written; @yi2011mobile; @guy2016searching; @schalkwyk2010your]. Schalkwyk et al. [@schalkwyk2010your] reported statistics of queries collected from Google Voice search logs, which found that short queries, in particular 1-word and 2-word queries, were more common in the voice search setting, while long queries were much rarer. In contrast, in a more recent study, Guy [@guy2016searching] reported that voice queries tend to be longer than text queries, based on a half-million query dataset from the Yahoo mobile search application. In addition, Guy studied the characteristics of voice queries in a more comprehensive way, including query term frequencies, popularity, syntax, post-click behaviors, etc. The average length across 32M voice queries is 2.04 in our dataset, much shorter than the reported average of 4.2 for Yahoo voice search[^3] [@guy2016searching].
We note another important difference between our entertainment context and voice search applications on smartphones: on a mobile device, it is common to back off to a web search if the query intent is not identified with high confidence. For Yahoo, Guy reported that less than half of voice queries (43.3%) are handled by a pre-defined card. While we are not aware of any scientific study about web browsing behavior on a TV, our intuition is that a list of search results is less useful to TV viewers than it might be for smartphone users, since subsequent interactions are much more awkward: it is difficult for users to scroll and they have limited input methods for follow-up interactions. Furthermore, televisions are not optimized for browsing webpages at a distance.
Personalization can help disambiguate queries [@zweig2011personalizing; @bennett2012modeling], since user preference is an important signal in deciphering user intent. However, since the TV is usually shared amongst the household, the feasibility of reliable personalization is not as clear as on a smartphone or computer (i.e., not obvious low-hanging fruit). This makes it even more important to exploit other signals.
There is also research on voice query reformulations that is relevant to our work on modeling sessions [@jiang2013users; @hassan2015characterizing; @shokouhi2014mobile; @shokouhi2016did]. For example, Jiang et al. [@jiang2013users] analyzed different types of voice recognition errors and users’ corresponding reformulation strategies. Hassan et al. [@hassan2015characterizing] built classifiers to differentiate between reformulated and non-reformulated query pairs. The study by Shokouhi et al. [@shokouhi2014mobile] suggested that users don’t prefer to switch between voice and text when reformulating queries. A more recent paper [@shokouhi2016did] proposed an automatic way to label voice queries by examining post-click and reformulation behaviors, which produced a large amount of “free” training data to reduce ASR errors. These papers provide a source of inspiration for our models.
Our approach to tackling the challenges associated with ambiguous voice queries is to take advantage of context. Our fundamental assumption is that when the viewer is not satisfied with the results of a query, she will issue another query in rapid succession and continue until the desired program is found or until she gives up. Note that these sequences often represent refinement of user intent: part of the process is the viewer deciding what to watch. By modeling voice search sessions (i.e., sequences of successive voice queries), we can better understand the viewer’s underlying intent. For example, compare two sessions: \[“tv shows”, “ncis”, “cargo fire”, “chicago fire”\] and \[“espn”, “chicago sports”, “chicago fire”\]. Although both end in the same query, it is fairly clear that in the first case, the viewer is interested in the TV drama series “Chicago Fire” (since previous queries all mention other drama series), whereas in the second session, it is clear that the viewer is interested in the sports team with the same name. This idea, of course, is not novel, and there is a large body of literature focused on exploiting web search sessions (e.g., [@jones2008beyond; @cao2009context; @bennett2012modeling; @guan2013utilizing; @liu2010personalizing; @luo2015session; @zhang2015information], just to mention a few). A comprehensive survey is beyond the scope of this paper, but previous work is concerned with text-based web search, which differs both in modality and in domain.
Model Architecture
==================
Problem Formulation
-------------------
Given a voice query session $[q_1, \ldots, q_n]$, our task is to predict the program $p$ that the user intends to watch. We perform this prediction cumulatively at each time step $t \in [1, n]$ on each successive new voice query $q_t$, exploiting all previous queries in the session, $[q_1, \ldots , q_{t-1})$. For example, consider a three-query session $s_i=[q_{i_1}, q_{i_2}, q_{i_3}]$, there will be three separate predictions: first with $[q_{i_1}]$, second with $[q_{i_1}, q_{i_2}]$, and third with $[q_{i_1}, q_{i_2}, q_{i_3}]$. We sessionize the voice query logs heuristically based on a time gap (in this case, 45 seconds—more details later), similar to how web query logs are sessionized based on inactivity. As described above, each query is a text string, the output of a third-party “black box” ASR system that we do not have internal access to.
We aim to learn a mapping function $\Theta$ from a query sequence to a program prediction, modeled using a probabilistic framework:
$$\begin{aligned}
\textrm{Data: } D &= \{(s_i, p_i)\ |\ s_i = [q_{i_1}, ..., q_{i_{|s_i|}}],\ p_i \in \Phi\}_1^{|D|}\\
\textrm{Model: } \hat{\theta} &= \arg \max_\theta \prod_{i=1}^{|D|} \prod_{t=1}^{|s_i|} P(p_i | q_{i_1}, ... , q_{i_t}; \theta)
\label{eq:problem}\end{aligned}$$
where $D$ denotes a set of labeled sessions ($s_i$ denotes the $i$-th session with $|s_i|$ queries), $p_i$ is the intended program for session $i$, $\Phi$ is the global set of programs, and $\theta$ is the set of parameters in the mapping function $\Theta$. Our goal is to maximize the product of prediction probabilities.
We decompose the program prediction task into learning three mapping functions: a query embedding function $\mathds{F}(x;\theta_{\mathds{F}})$, a contextual function $\mathds{G}(x; \theta_{\mathds{G}})$, and a classification function $\mathds{H}(x; \theta_{\mathds{H}})$. The query embedding function $\mathds{F}(\cdot)$ takes the text of the query as input and produces a semantic representation of the query; the contextual function $\mathds{G}(\cdot)$ considers representations of all the preceding queries as context and maps them to a high-dimensional embedding vector to capture both semantic and contextual features; finally, the classification function $\mathds{H}(\cdot)$ predicts possible programs from the learned contextual vector. We adopt the following decomposition:
$$\begin{split}
P(p_i | q_{i_1}, ... , q_{i_t}) \sim\ &P(p_i | c_{i_t}) \cdot P(c_{i_t} | v_{i_1}, ... , v_{i_t}) \\
&\cdot P(v_{i_1}, ... , v_{i_t} | q_{i_1}, ... , q_{i_t})
\end{split}
\label{eq:decompose}$$
where $c_{i_t}$ denotes the *contextual* embedding of the first $t$ queries in the $i\textrm{-th}$ session and $v_{i_t}$ denotes the embedding of the $t\textrm{-th}$ query of the $i\textrm{-th}$ session. The relationship between these embeddings can be formulated using the three mapping functions above: $\mathds{F}$ maps a query $q_{i_j}$ to its embedding $v_{i_j}$ in vector space; $\mathds{G}$ maps a sequence of query embeddings $[v_{i_1}, ... ,
v_{i_t}]$ to a contextual embedding $c_{i_t}$; and $\mathds{H}$ maps the contextual embedding to a program $p_i$:
$$\begin{split}
v_{i_t} & \sim \mathds{F}(q_{i_t}; \theta_{\mathds{F}}) \\
c_{i_t} & \sim \mathds{G}(v_{i_1}, ... , v_{i_t}; \theta_{\mathds{G}}) \\
p_i & \sim \mathds{H}(c_{i_t}; \theta_{\mathds{H}}) \\
& 1 \le t \le |s_i|
\end{split}$$
By assuming that each query is embedded independently, we can reduce the last term in Equation (\[eq:decompose\]) as follows:
$$\begin{split}
P(p_i | q_{i_1}, ... , q_{i_t})= P(p_i | c_{i_t}) \cdot P(c_{i_t} | v_{i_1}, ... , v_{i_t})
\cdot \prod_{j=1}^t P(v_{i_j} | q_{i_j})
\end{split}$$
We model the query embedding function $\mathds{F}(\cdot)$ and the contextual function $\mathds{G}(\cdot)$ by organizing two Long Short-Term Memory (LSTM) [@hochreiter1997long] models in a hierarchical manner. The decision function $\mathds{H}(\cdot)$ is represented as a feedforward neural network layer. Before we introduce the details of our model architecture, we provide an overview of the LSTM model.
Long Short-Term Memory Networks
-------------------------------
Long Short-Term Memory (LSTM) [@hochreiter1997long] networks are well-known for being able to capture long-range contextual dependencies over input sequences. This is accomplished by using a sequence of memory cells to store and memorize historical information, where each memory cell contains three gates (input gate, forget gate, and output gate) to control the information flow. The gating mechanism enables the LSTM to handle the gradient vanishing/explosion problem for long sequences of inputs.
Given an input sequence $\textbf{x} = (x_1, ... , x_T)$, an LSTM model outputs a sequence of hidden vectors $\textbf{h} = (h_1, ... ,
h_T)$. A memory cell at position $t$ digests the input element $x_t$ and previous state information $h_{t-1}$ to produce updated state $h_t$ as follows:
$$\begin{split}
i_t & = \sigma(W_{xi} x_t + W_{hi} h_{t-1} + b_i) \\
f_t & = \sigma(W_{xf} x_t + W_{hf} h_{t-1} + b_f) \\
o_t & = \sigma(W_{xo} x_t + W_{ho} h_{t-1} + b_o) \\
c_t & = f_t \cdot c_{t-1} + i_t \cdot \sigma(W_{xc} x_t + W_{hc} h_{t-1} + b_c) \\
h_t & = o_t \cdot \tanh(c_t)
\end{split}$$
where the $W$ terms are weight matrices, the $b$ terms represent bias vectors, $\sigma$ is the sigmoid activation function, and $i$, $f$, $o$, and $c$ are respectively the input gate, forget gate, output gate, and cell vectors, with each having the same size as the output vector $h$. In this paper, we refer to the size of the output vector $h$ as the LSTM size.
In many application scenarios, the input sequence $\textbf{x}$ can vary in length for different instances (i.e., queries can have different numbers of words and characters in our task). There are two standard ways to handle this variable length issue. One way is to perform an initial scan over a single batch or the entire dataset to obtain the maximum sequence length, then create an array of memory cells with the maximum length. Whenever a sequence element $x_t$ arrives, the memory cell at index $t$ will digest the input element and produce the hidden state $h_t$. The other way is to dynamically allocate space for storing new memory cells only when the arriving instance $\textbf{x}$ has a greater length than all previous instances. The created LSTM memory cells all share the same parameters. We use the second strategy (what we call *dynamic allocation policy*) in our implementations to avoid needing an initial scan.
Query Representation {#sec:query}
--------------------
Since query strings serve as the sole input in our model, an expressive query representation is essential to accurate predictions. We represent each query as a sequence of elements (words or characters); each element is passed through a lookup layer and projected into a $d$-dimensional vector, thereby representing the query as an $m\times d$ matrix ($m$ is the number of elements in the query). We consider three variations of this representation:
1. **Character-level** representation, which encodes a query as a sequence of characters and the lookup layer converts each character to a one-hot vector. In this case, $m$ would be the number of characters in the query and $d$ would be the size of the character dictionary of the entire dataset.
2. **Word-level** representation, which encodes the query as a sequence of words, and the word vectors are read from a pre-trained word embedding, e.g., word2vec [@Mikolov:2013aa]. In this case, $d$ would be the dimensionality of the word embedding.
3. **Combined** representation, which combines both the character-level and word-level representations by feeding the representations to two separate query embedding functions $\mathds{F}_c$ and $\mathds{F}_w$, respectively, then concatenating the two learned vectors $v_c$ and $v_w$ as the combined query embedding vector.
Our intuition for these different representations is as follows: Based on our analysis of voice query logs, we observe many unsatisfactory responses due to speech recognition errors. For example, voice queries intended for the program “Caillou” (a Canadian children’s education television series) are often recognized as “Cacio” or “you”. Capturing such variations with a word-level representation would likely suffer from data sparsity issues. On the other hand, initializing a query through word embedding vectors would encode words in a semantic vector space, which would help in matching queries to programs based on semantic relatedness (e.g., the query “Portland Trail Blazers” is semantically similar to the intended program “NBA basketball” without any words in common). Word embeddings are also useful for recognizing semantically-similar contextual clues such as “Search”, “Find” or “Watch”. With a character-level representation, such similarities would need to be learned from scratch. Whether the benefits of either representation balance the drawbacks is an empirical question we study through experiments, but we hypothesize that a combined representation would capture the best of both worlds.
Basic Model {#sec:basic}
-----------
![Architecture of the Basic Model[]{data-label="independent"}](single.pdf){width="0.5\linewidth"}
In the basic context-independent model, queries in a session are assumed to be independent and thus we do not attempt to model context. That is, each query is treated as a complete sample for model inference and prediction. The mapping function $\Theta$ from query to program can be simplified as follows:
$$\begin{split}
\Theta & \sim \arg \max_\theta \prod_{i=1}^{|D|} \prod_{t=1}^{|s_i|} P(p_i | q_{i_t}) \\
& = \arg \max_\theta \prod_{i=1}^{|D|} \prod_{t=1}^{|s_i|} P(p_i | v_{i_t}) P(v_{i_t} | q_{i_t})
\end{split}
\label{equation:independent}$$
$$v_{i_t} \sim \mathds{F}(q_{i_t}; \theta_{\mathds{F}}), \quad p_i \sim \mathds{H}(v_{i_t}; \theta_{\mathds{H}}), \quad 1 \le t \le |s_i|$$
Here, the program $p_i$ is only dependent on the current query $q_{i_t}$. The contextual function $\mathds{G}(\cdot)$ is modeled as an identity function since there is no context from our assumption.
The architecture of the basic model is shown in Figure \[independent\]. In the bottom, we use an LSTM as our query embedding function $\mathds{F}(\cdot)$. The text query is projected into an $m\times d$ dimensional matrix through the lookup layer, then fed to the LSTM, which has $m$ memory cells and each cell processes an element vector. The hidden state at the last time step $h_m$ is used as the query embedding vector $v$. At the top, there is a fully-connected layer followed by a soft-max layer for learning the classification function $\mathds{H}(\cdot)$. The fully-connected layer consists of two linear layers with one element-wise activation layer in between. Given the query embedding vector $v$ as input, the fully-connected layer computes the following:
$$\begin{split}
&l = \sigma(W_{h_1} \cdot v + b_{h_1}) \\
&l_2 = W_{h_2} \cdot l + b_{h_2}
\end{split}$$
where the $W$ terms are the weight matrices and the $b$ terms are bias vectors. We use the $\textrm{tanh}$ function as the non-linear activation function $\sigma$, which is commonly adopted in many neural network architectures. The soft-max layer normalizes the vector $l_2$ to a $L1$ norm vector $o$, with each output score $o[p_j]$ denoting the probability of producing program $p_j$ as output:
$$o[p_j] = \frac{\exp(l_2[p_j] - \textrm{shift})}{\sum_{p_k=1}^{|\Phi|} \exp(l_2[p_k] - \textrm{shift})}$$
where $\textrm{shift} = \max_{p_k=1}^{|\Phi|} l_2[p_k]$, $|\Phi|$ is the total number of programs in the dataset.
$e_{i_t} = \textrm{encode}(q_{i_t})$ \[line:encode\] $h_{1,...,m} = \textrm{LSTM:forward}(e_{i_t})$ \[line:lstm\] $l_2 = \textrm{FC:forward}(h_m)$ \[line:fc\] $o = \textrm{softmax:forward}(l_2)$ $loss = \textrm{criterion:forward}(o, p_i)$ $\textrm{grad$\_$criterion} = \textrm{criterion:backward}(o, p_i)$ $\textrm{grad$\_$soft} = \textrm{softmax:backward}(l_2, \textrm{grad$\_$criterion})$ $\textrm{grad$\_$linear} = \textrm{FC:backward}(h_m, \textrm{grad$\_$soft})$ $\textrm{grad$\_$lstm} = \textrm{zeros}(m, \textrm{lstm$\_$size})$ \[line:gradlstm1\] $\textrm{grad$\_$lstm}[m] = \textrm{grad$\_$linear}$ $\textrm{LSTM:backward}(e_{it}, \textrm{grad$\_$lstm})$ \[line:gradlstm2\] $\textrm{update$\_$parameters}()$ \[line:updatesimple\]
We adopt the negative log likelihood loss function to train the model, which is derived from Equation (\[equation:independent\]):
$$\begin{split}
L & = -\sum_{i=1}^{|D|} \sum_{t=1}^{|s_i|} \log P(p_i | q_{i_t}) + \lambda \cdot \|<\theta_\mathds{F}, \theta_\mathds{G}, \theta_\mathds{H}>\|^2 \\
&= -\sum_{i=1}^{|D|} \sum_{t=1}^{|s_i|} \log o_{i_t}[p_i] + \lambda \cdot \|<\theta_\mathds{F}, \theta_\mathds{G}, \theta_\mathds{H}>\|^2
\end{split}$$
where $o_{i_t}$ is the score vector computed from query $q_{i_t}$ and $p_i$ is the true program for session $i$; $\lambda$ is the regularization weight and $<\theta_\mathds{F}, \theta_\mathds{G},
\theta_\mathds{H}>$ is the set of model parameters. The optimization goal is to minimize the loss criterion $L$.
The training process is shown in Algorithm \[alg:basic\]. The overall structure is to iterate over each query in all sessions to perform the forward prediction and backward propagation operations. The *forward* phase follows our model architecture in Figure \[independent\]. A query is first encoded as a matrix in Line \[line:encode\] by specifying the encoding method (i.e., character, word, or combined). Line \[line:lstm\] computes the output states $h$ by feeding an input matrix to the LSTM model. FC in Line \[line:fc\] denotes the fully-connected layer. In the *backward* phase, each module requires the original inputs and the gradients propagated from its upper layer to compute the gradients with respect to the inputs and its own parameters. It is worth noting that in Lines \[line:gradlstm1\]-\[line:gradlstm2\] the gradients $\textrm{grad$\_$lstm}$ are initialized to zero for the first $m-1$ cells. This is because in the forward phase, we only use the last LSTM state $h_m$ as the query embedding vector for the upper layers and throw away the other states $h_{1, ...,
m-1}$. Line \[line:updatesimple\] performs gradient descent to update model parameters. All the forward and backward functions used here are written as black box operations, and we refer interested readers elsewhere [@Goodfellow-et-al-2016] for more details.
Full Context Model {#sec:context}
------------------
![Architecture of the Full Context Model[]{data-label="fig:context"}](context.pdf){width="0.9\linewidth"}
We propose two approaches to modeling context: the full context model (presented here) and the constrained context model (presented next). The architecture of the full context model is shown in Figure \[fig:context\], which uses the basic model as a building block. We use another LSTM (the dotted rectangle in the middle of Figure \[fig:context\]) to learn the contextual function $\mathds{G}(v_1, ..., v_t; \theta_{\mathds{G}})$. Previous query embedding vectors $[v_1, ..., v_{t-1}]$ are encoded as a context vector $c_{t-1}$, which is combined with the current query embedding vector $v_t$ and fed to the LSTM memory cell at time $t$. This allows the LSTM to find an optimal combination of signals from the previous context and the current query. For sessions with a single underlying intent (i.e., the user is consistently looking for a specific program), the model can learn the intrinsic relatedness between successive queries and continuously reinforce confidence in the true intent. In reality, context can sometimes be irrelevant (e.g., user zapping through channels), which might introduce noise. When the context diverges too much from the current query embeddings, the model should be able to ignore the noisy signals to reduce their negative impact.
We adopt a many-to-many hierarchical architecture. A query embedding $\mathds{F}(\cdot)$ and classification layer $\mathds{H}(\cdot)$ is applied over each query for program prediction at each time step $t$. We hope to find the true user intent as early as possible to reduce the interactions between the user and our voice product. The parameters of the query embedding layer $\mathds{F}(\cdot)$, as well as the classification layer $\mathds{H}(\cdot)$, are shared by all queries regardless of their position in the session. For instance, two identical queries with different positions in a session will have the same query embedding vector. Except for the contextual layer $\mathds{G}(\cdot)$, all other modules (e.g., query embedding, fully-connected, soft-max layers, loss function) remain same as in the basic (context-independent) model.
$v = \textrm{zeros}(|s_i|, \textrm{lstm$\_$size})$ \[line:fwd1\] $e_{i_t} = \textrm{encode}(q_{i_t})$ $h_{1,...,m} = \textrm{LSTM[t]:forward}(e_{i_t})$ \[line:lstmarray\] $v_t = h_m$ \[line:fwdlstm\] \[line:fwd2\] $c_{1, ..., |s_i|} = \textrm{C$\_$LSTM:forward}(v_{1, ..., |s_i|})$ \[line:fwdcontext\] $\textrm{session$\_$loss} = 0 $ \[line:fwdbwd1\] $\textrm{grad$\_$linear} = \textrm{zeros}(|s_i|, \textrm{lstm$\_$size})$ $l_2 = \textrm{FC:forward}(c_t)$ $o = \textrm{softmax:forward}(l_2)$ $\textrm{loss} = \textrm{criterion:forward}(o, p_i)$ $\textrm{session$\_$loss} = \textrm{session$\_$loss} + loss $ \[line:sessionloss\] $\textrm{grad$\_$criterion} = \textrm{criterion:backward}(o, p_i)$ $\textrm{grad$\_$soft} = \textrm{softmax:backward}(l_2, \textrm{grad$\_$criterion})$ $\textrm{grad$\_$linear}[t] = \textrm{FC:backward}(c_t, \textrm{grad$\_$soft})$ \[line:fwdbwd2\] $\textrm{grad$\_$context} = \textrm{C$\_$LSTM:backward}(v_{1, ..., |s_i|}, \textrm{grad$\_$linear})$ \[line:bwdcontext1\] \[line:bwdembed\] $\textrm{grad$\_$lstm} = \textrm{zeros}(m, \textrm{lstm$\_$size})$ $\textrm{grad$\_$lstm}[m] = \textrm{grad$\_$context}[t]$ $\textrm{LSTM[t]:backward}(e_{i_t}, \textrm{grad$\_$lstm})$ \[line:bwdcontext2\] $\textrm{update$\_$parameters}()$ \[line:update\]
The training process for this model (Algorithm \[alg:context\]) starts with forward predictions for multiple queries in the session (Lines \[line:fwd1\]-\[line:fwd2\]). Similar to Algorithm \[alg:basic\], only the last LSTM state $h_m$ is selected as the query embedding vector (Line \[line:fwdlstm\]). Since sessions can have a variable number of queries, we use the dynamic allocation policy to create a list of LSTMs with each LSTM ingesting a query (i.e., LSTM\[t\] in Line \[line:lstmarray\]). Line \[line:fwdcontext\] utilizes another LSTM model to compute the context from sequential query embeddings. Lines \[line:fwdbwd1\]-\[line:fwdbwd2\] perform forward predictions and backward propagations for multiple queries in the classification layer. The queries are processed in a sequential manner such that for each query all forward operations are immediately followed by all backward operations before moving to the next query. Lines \[line:bwdcontext1\]-\[line:bwdcontext2\] propagate the gradients through the contextual and embedding LSTMs. Line \[line:update\] updates model parameters for each session by optimizing the session loss in Line \[line:sessionloss\].
It is important to note that the prediction task is applied at the query level: our model tries to predict the program after [*each*]{} query in the session. The alternative is to optimize for program prediction given [*all*]{} queries in the session—this is a much easier task, since the entire session has been observed. It also defeats the purpose of our setup since we wish to satisfy viewer intents as soon as possible.
Constrained Context Model {#sec:c-context}
-------------------------
In addition to the full context model described above, we explore a variant that we call the *constrained* context model. The model architecture is the same as the full context model (Figure \[fig:context\]). The difference, however, lies in how we learn the model. For the constrained context model, we adopt a pre-training strategy as follows: we first train the basic model (Algorithm \[alg:basic\]) and then use the learned LSTM parameters to initialize the constrained context model’s query embedding layer. The embedding layer is then fixed and purely used for generating query embeddings. That is, lines \[line:bwdembed\]-\[line:bwdcontext2\] are removed from Algorithm \[alg:context\].
Our intuition behind this model is to restrict the search space during model inference, aiming to reduce the complexity of optimization compared to the full context model
. Whether this reduction in optimization complexity is beneficial to the prediction task is an empirical question we study in the following section.
Experimental Setup
==================
[**Data Preparation.**]{} We collected raw data from voice queries submitted to Xfinity X1 voice-enabled remote controllers during the week of Feb. 22 to 28, 2016. The dataset contains 32.3M queries from 2.5M unique viewers. Based on preliminary analyses, we selected 45 seconds as the threshold for dividing successive queries into sessions, yielding 20.0M sessions in total.
In order to build a training set for supervised learning, we need to acquire the true user intent for each session. We automatically extracted noisy labels by examining what the viewer watched after the voice session. This idea exactly parallels inferring user intent from clickthrough data in the web domain. No doubt that for web search, the process of gathering labels is highly refined given the amount of effort invested by commercial web search engine companies. By comparison, our heuristics may seem relatively crude, but given the paucity of work in this domain, they represent a good initial attempt to tackle the problem.
If the viewer began watching a program $p$ at most $K$ seconds after the last query in the voice session $v$ and kept watching it for at least $L$ seconds, we label the session with $p$. The selection of $K$ and $L$ represents a balance between the quantity and quality of collected labels. Reducing $K$ and/or increasing $L$ increases the confidence in the correctness of collected labels but also reduces the number of labels we obtain, and vice versa. After some initial exploration, we set $K$ to a relatively small value (30 seconds) and $L$ to a relatively large value (150 seconds)—which yields a good balance between data quantity and quality (based on manual spot-checking). Using these parameters, we extracted 13.0M session-program pairs. Note that in reality viewers navigate with a combination of voice queries and keypad entry, so it is [*not*]{} the case that our gathered sessions reflect only successful voice interactions with our X1 platform.
Without any restriction on these sessions, some voice queries might reflect arbitrary intent (e.g., “closed caption on”, “the square root of eighty one”, “change to channel 36”, or even complete gibberish). In order to limit ourselves to voice sessions with a single clear intent, we used two heuristic strategies to discard sessions with multiple or unclear intents. First, we define a way to reliably predict whether a query is program-related (i.e., the query is primarily associated with a TV series, movie, video, or sports program). We obtain this from the deployed X1 system, which categorizes each query into one of many action types. We say a query is program-related if it is categorized as one of the following: {SERIES, MOVIE, MUSICVIDEO, SPORTS}. Based on this knowledge, we restricted our data to sessions in which over 2/3 of queries are program-related and the final query in the session is also program-related. Since channel changes are a large portion of the data, this reduces the number of labeled pairs to 2.1M.
Second, we computed the normalized Levenshtein distance [@yujian2007normalized] between each query pair in the session, and kept only those sessions where [*any*]{} pair of queries has a distance less than 0.5. Our goal here is to ensure that there is at least [*some*]{} cohesion in the sessions. This heuristic has a relatively minor effect: the resulting filtered dataset contains 1.96M sessions in total.
From this data, we created five splits: a training set used in all experiments, and two groups of development and test sets. The first development and test sets contain only single-query sessions, called SingleDev and SingleTest. These are used to study whether the context-based models hurt accuracy in sessions without context. The second group contains only multiple-query sessions (i.e., at least two queries in each session), called MultiDev and MultiTest. In order to build the global set of programs $\Phi$, we only kept programs if there are at least 50 associated sessions in the training set, yielding 471 programs. In other words, our task is 471-way classification. Statistics for each of the splits are summarized in Table \[tab:stat\].
Dataset \# sessions \# queries avg. session len. avg. query len.
------------ ------------- ------------ ------------------- -----------------
Train 126016 181058 1.44 2.34
SingleDev 24792 24792 1.00 2.40
SingleTest 24572 24572 1.00 2.36
MultiDev 28427 82828 2.91 2.30
MultiTest 28173 82272 2.92 2.30
: Dataset Statistics.[]{data-label="tab:stat"}
[0.45]{}
**ID** **Model** **Query** **P@1** **P@5** **MRR**
-------- ---------------------------------------- ----------- -------------------------------------------- ------------------------------------------ --------------------------------------------
1 EditDist - $0.8148^{7}$ $0.8520^{7}$ 0.8354
2 Deployed X1 - $0.8743^{1,7}$ - -
3 $\textnormal{SVM}^{\textnormal{rank}}$ - $0.9131^{1,2,7}$ $0.9309^{1,7}$ $0.9267^{1,7}$
4 Basic char $0.9438^{\textnormal{1-3,7}}$ $0.9526^{\textnormal{1,7}}$ $0.9617^{\textnormal{1,3,7}}$
5 Basic word $0.9434^{\textnormal{1-3,7}}$ $0.9526^{\textnormal{1,7}}$ $0.9615^{\textnormal{1,3,7}}$
6 Basic comb $\boldsymbol{0.9466^{\textnormal{1-3,7}}}$ $\boldsymbol{0.9551^{\textnormal{1,7}}}$ $0.9637^{\textnormal{1,3,7}}$
7 Context-f char 0.7526 0.7936 0.8371
8 Context-f word $0.9262^{\textnormal{1,2,7}}$ $0.9416^{\textnormal{1,7}}$ $0.9590^{\textnormal{1,3,7}}$
9 Context-f comb $0.9315^{\textnormal{1,2,7}}$ $0.9474^{\textnormal{1,7}}$ $\boldsymbol{0.9669^{\textnormal{1,3,7}}}$
10 Context-c char $0.9378^{\textnormal{1-3,7}}$ $0.9499^{\textnormal{1,7}}$ $0.9626^{\textnormal{1,3,7}}$
11 Context-c word $0.9428^{\textnormal{1-3,7}}$ $0.9502^{\textnormal{1,7}}$ $0.9608^{\textnormal{1,3,7}}$
12 Context-c comb $0.9435^{\textnormal{1-3,7}}$ $0.9532^{\textnormal{1,7}}$ $0.9627^{\textnormal{1,3,7}}$
[0.45]{}
**ID** **Model** **Query** **P@1** **P@5** **MRR** **QR**
-------- ---------------------------------------- ----------- ------------------------------------------ -------------------------------------------- -------------------------------------------- -------------------------------------------
1 EditDist - 0.4708 0.5297 0.5033 0.834
2 Deployed X1 - 0.4544 - - -
3 $\textnormal{SVM}^{\textnormal{rank}}$ - $0.5280^{1,2,7}$ $0.5985^{1,7}$ $0.5606^{1,7}$ $0.949^{1,7}$
4 Basic char $0.6052^{\textnormal{1-3,7}}$ $0.6471^{\textnormal{1,3,7}}$ $0.6901^{\textnormal{1,3,7}}$ $1.108^{\textnormal{1,3,7}}$
5 Basic word $0.6085^{\textnormal{1-3,7}}$ $0.6437^{\textnormal{1,3,7}}$ $0.6773^{\textnormal{1,3,7}}$ $1.086^{\textnormal{1,3,7}}$
6 Basic comb $0.6135^{\textnormal{1-3,7}}$ $0.6510^{\textnormal{1,3,7}}$ $0.6868^{\textnormal{1,3,7}}$ $1.113^{\textnormal{1,3,7-9}}$
7 Context-f char 0.4818 0.5316 $0.5803^{1}$ 0.856
8 Context-f word $0.5989^{\textnormal{1-3,7}}$ $0.6384^{\textnormal{1,3,7}}$ $0.6868^{\textnormal{1,3,7}}$ $1.075^{\textnormal{1,3,7}}$
9 Context-f comb $0.5982^{\textnormal{1-3,7}}$ $0.6428^{\textnormal{1,3,7}}$ $0.6883^{\textnormal{1,3,7}}$ $1.039^{\textnormal{1,3,7}}$
10 Context-c char $0.6394^{\textnormal{1-9}}$ $0.6842^{\textnormal{1,3-9}}$ $0.7306^{\textnormal{1,3-9}}$ $1.117^{\textnormal{1,3,5,7-9}}$
11 Context-c word $0.6387^{\textnormal{1-9}}$ $0.6826^{\textnormal{1,3-9}}$ $0.7290^{\textnormal{1,3-9}}$ $1.112^{\textnormal{1,3,7-9}}$
12 Context-c comb $\boldsymbol{0.6427^{\textnormal{1-9}}}$ $\boldsymbol{0.6872^{\textnormal{1,3-9}}}$ $\boldsymbol{0.7343^{\textnormal{1,3-9}}}$ $\boldsymbol{1.128^{\textnormal{1,3-9}}}$
**Model Training**. In total, we have three options for query representation, *char*, *word*, *combined* (Section \[sec:query\]), and three options for the model, *basic*, *full context*, or *constrained context* (Sections \[sec:basic\]-\[sec:c-context\]). Therefore, we have a total of nine experimental settings, by crossing the three representations with the three models.
The entire dataset contains 80 distinct characters in total, which means the size of the one-hot vector used in the *char* setting is 80. For the word representation, we used 300-dimensional GloVe [@pennington2014glove] word embeddings to encode each word, which is trained on 840 billion tokens and freely available. The word vocabulary of our dataset is 20.4K, with 1759 words not found in the GloVe word embeddings. Unknown words were randomly initialized with values uniformly sampled from \[-0.05, 0.05\].
During training, we used the stochastic gradient descent algorithm together with $\textrm{RMS-PROP}$ [@tieleman2012lecture] to iteratively update the model parameters. The learning rate was initially set to $10^{-3}$, and then decreased by a factor of three when the development set loss stopped decreasing for three epochs. The maximum number of training epochs was 50. For the constrained context model, the number of pre-train epochs was selected as 15. The output size of the LSTMs was set to 200 and the size of linear layer was set to 150. The regularization weight $\lambda$ was chosen as $10^{-4}$. At test time, we selected the model that obtained the highest P@1 accuracy on the development set for evaluation. Our models were implemented using the Torch framework. We ran all experiments on a server with two 8-core processors (Intel Xeon E5-2640 v3 2.6GHz) and 1TB RAM, with each experiment running on 6 CPU threads.
**Baselines.** We considered three baselines for comparison. The first baseline is the edit distance algorithm, in which we compared each candidate program’s title to the issued query and returned the program with the smallest edit distance to the query as the predicted label. Second, we obtained responses from our production X1 system, which combines statistical machine learning models with hand-crafted rules to produce the best response.
Third, we built a learning-to-rank baseline using the $\textnormal{SVM}^{\textnormal{rank}}$ library. We first used the edit distance baseline and tf-idf algorithm to find the top 10 closest programs, and merged them as ranking candidates. We then designed two types of features: (1) the edit distance and tf-idf score between the query and the candidate programs, and (2) we first computed cosine similarities of all pairs of word vectors between query and candidate programs (word vectors were initialized from GloVe embeddings), and then we took the maximum/mean/minimum values of the word pair similarities as features.
**Evaluation Metrics.** We used four metrics to evaluate our models: precision at one (P@1), precision at five (P@5), Mean Reciprocal Rank (MRR), and Query Reduction (QR). The first three are standard retrieval metrics that are averaged over all queries, but the last requires some explanation. Query reduction is a measure of how many queries a viewer has “saved”. For a session with $n$ queries, the number of reductions is $n-i$ if the model returns the correct prediction at the $i$-th query, which means that the viewer does not need to issue the next $n-i$ queries, hence a *reduction* of $n-i$. We average this metric over all sessions. Note that QR is not applicable to single-query sessions.
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Results
=======
Results for the single-query and multiple-query sessions, on the SingleTest and MultiTest splits, respectively, are shown in Table \[tab:results\]. Each row represents an experimental condition (numbered for convenience); the second column specifies the model condition: “Context-f” denotes the *full* context model and “Context-c” represents the *constrained* context model. The third column indicates the query representation, and the remaining columns list the various evaluation metrics. Superscripts indicate the row indexes for which the metric difference is statistically significant ($p < 0.01$) based on Fisher’s two-sided, paired randomization test [@smucker2007comparison]. A dash symbol “-” connecting two integer indices “$a$-$b$” is shorthand for $a, a+1,
\ldots, b-1, b$.
Let’s first consider the baselines: the current deployed X1 system achieves fairly high accuracy (P@1 of 0.8743) on single-query sessions. Since our internal APIs only return the top prediction, we cannot compute P@5 or MRR. The good accuracy of X1 suggests that viewers are already fairly satisfied with the current deployed system, since for single-query sessions they reach their intended programs in a single shot. In a sense, this is not surprising because, by definition, these are the “easy queries”. The edit distance baseline also performs fairly well for these easy cases. The $\textnormal{SVM}^{\textnormal{rank}}$ predictor achieves better accuracy than the deployed system because it takes advantage of word embeddings to consider semantic relatedness at both the character and the word levels in a supervised setting. Thus, our learning-to-rank approach forms a reasonably strong baseline.
Turning our attention to baselines on the multiple-query sessions in Table \[tab:results\](b), we see that accuracy drops significantly for all baselines. These sessions represent information needs that weren’t satisfied in a single shot, which of course makes them more challenging. We note that the accuracy of the deployed X1 system falls below even that of edit distance, which is not surprising as there would not have been multiple queries if X1 provided the correct response on the first try.
There are two important questions our experiments are designed to answer: First, which is the most effective query representation (character, word, or combined)? And second, which is the most effective context model (no context, full, or constrained)?
For the first question, we observe that in the basic and constrained context model, the word-level query representation is quite close to the character-level query representation. However, in nearly all conditions, across nearly all metrics, the combined condition further improves (albeit only slightly) upon both representations, which shows that character-level and word-level representations provide signals that supplement each other.
In terms of the context models, it seems clear that the constrained context model significantly outperforms all other models, including the basic and full context models. Considering that the constrained context model copies its query embedding layer from the basic model, we conclude that contextual information does help with the prediction task. On the other hand, note that the full and constrained models share exactly the same architecture. The only difference lies in whether or not we back-propagate to the query embedding layer during training—the constrained model was designed to restrict the model search space during model inference. The effectiveness gap on the multiple-query session dataset demonstrates that the query embedding layer obtained by the constrained context model through pre-training is of higher quality. This is likely due to insufficient data for the full context model to effectively learn parameters for both LSTM levels. However, a caveat: it is conceivable that with even more training data, the full context model will improve. But as it currently stands, the constrained context model displays a better ability to exploit contextual information for predicting viewers’ intent. Overall, for the multiple-query sessions, the constrained model with the combined representation yields a 41% relative improvement over the deployed X1 system for P@1 and a 22% relative improvement over $\textnormal{SVM}^{\textnormal{rank}}$.
Looking at the accuracy of the context models for single-query sessions, we want to make sure that the more sophisticated models do not “screw up” the easy queries. We confirm that this is indeed the case. It is no surprise that the basic model performs the best on single-query sessions: since there is no context to begin with, all the “contextual machinery” of the richer models can only serve as distractors. We find that the constrained model with the combined representation (best condition above) still performs well—slightly worse (but not significantly so) than the basic model with the combined representation. It is also interesting to note that the full context model with the character representation is terrible, which provides additional evidence that the search space is probably too large with the combination of much longer query representations and optimization at the session level.
---------------------------------------- ----------------------------- ------------------------------------------------------------------------ ---------------------------------------------------------
Cacio : You : You : Caillou Sienna cover : Color : Casey undercover
Caillou K.C. Undercover
**Model** **Query** **Example 1** **Example 2**
EditDist - $\star$ : House : House : $\star$ Bee Movie : Room : $\star$
Deployed X1 - NA : House : House : $\star$ NA : In Living Color : $\star$
$\textnormal{SVM}^{\textnormal{rank}}$ - $\star$ : Now You See Me : Now You See Me : $\star$ CSI Cyber : Room : $\star$
Basic char $\star$ (0.81) : $\star$ (0.80) : $\star$ (0.80) : $\star$ (1.0) $\star$ (0.76) : Carolina (0.07) : $\star$ (0.99)
Basic word Child Genius (0.03) : $\star$ (0.57) : $\star$ (0.57) : $\star$ (1.0) Recovery Road (0.48) : $\star$ (0.08) : $\star$ (0.75)
Basic comb Paw Patrol (0.17) : $\star$ (0.83) : $\star$ (0.83) : $\star$ (1.0) $\star$ (0.37) : Magic Mike XXL (0.31) : $\star$ (0.98)
Context-f char Lego Ninjago (0.30) : $\star$ (0.79) : $\star$ (0.90) : $\star$ (0.99) $\star$ (0.43) : $\star$ (0.67) : $\star$ (0.89)
Context-f word Paw Patrol (0.30) : $\star$ (0.62) : $\star$ (0.98) : $\star$ (1.0) $\star$ (0.29) : $\star$ (0.65) : $\star$ (1.0)
Context-f comb Lego Ninjago (0.03) : $\star$ (0.60) : $\star$ (0.98) : $\star$ (1.0) $\star$ (0.41) : $\star$ (0.54) : $\star$ (0.99)
Context-c char $\star$ (0.96) : $\star$ (0.99) : $\star$ (0.99) : $\star$ (1.0) $\star$ (0.81) : $\star$ (0.96) : $\star$ (0.99)
Context-c word Wallykazam (0.07) : $\star$ (0.59) : $\star$ (0.86) : $\star$ (1.0) $\star$ (0.89) : $\star$ (0.80) : $\star$ (1.0)
Context-c comb Paw Patrol (0.17) : $\star$ (0.93) : $\star$ (1.0) : $\star$ (1.0) $\star$ (0.65) : $\star$ (0.83) : $\star$ (0.97)
---------------------------------------- ----------------------------- ------------------------------------------------------------------------ ---------------------------------------------------------
Context Analysis
----------------
To better understand how our models take advantage of context, we focused on multiple-query sessions and examined how accuracy evolves during the course of a session. Results are shown in Figure \[fig:context-analysis\]. The leftmost plot shows the histogram of session lengths (i.e., number of queries in a session) in the MultiTest split (each bar annotated with the actual count). In Figures \[fig:context-analysis\](b)-(d), we show the average P@1 score from MultiTest at different positions in the session (on the $x$ axis), i.e., at the first query in the session, the second query, etc. For illustrative purposes we focus on “short” sessions with a length of three (8441 sessions), “medium” sessions with a length of six (850 sessions), and “long” sessions with a length of nine (157 sessions). In each plot, we compared our context models with the baselines. For clarity, in all cases the models used the combined query representation.
We observe several interesting patterns in Figures \[fig:context-analysis\](b)-(d). First, for the non-context models (EditDist, X1, $\textnormal{SVM}^{\textnormal{rank}}$, and Basic), the accuracy of all queries before the final query is essentially the same (with small fluctuations due to noise). Accuracy for the final query rises significantly because the viewer finally found what she was looking for (and thus is likely to be an “easy” query). However, for the context-aware models (Context-f and Context-c), we observe a consistent increase in the accuracy curves as the session progresses. This demonstrates that as the model accumulates more context, it can better identify the viewer’s true intent. The full context model performs consistently worse than the basic model at the first query, since there is no context. Similarly, the full context model performs slightly worse than the basic model for the final query in each session. This finding is consistent with the results in Table \[tab:results\], since for single-query sessions, the basic model beats the full context model slightly.
In Table \[tab:case-study\], we provide two real example sessions to illustrate how each model responds to the sequence of viewer queries. The session is shown in the first row, where each query is separated by a colon. The second row shows the viewer’s intent (i.e., ground truth label). The remaining rows show the output of each model; due to space limitations, we only show the top predictions along with their confidence scores for our models. Each prediction in the sequence is also separated by a colon. To save space, we use the symbol $\star$ to indicate that the prediction is correct.
In the first example (left), the viewer is consistently looking for the program “Caillou”, but the query fails three times in a row due to ASR errors. For the first query “Cacio”, the edit distance algorithm can find the intended program because there are many characters in common. However, X1 failed and labeled this query as NA (i.e., no answer). For our models, both the Basic/char and Context-c/char models can predict the correct program from the query “Cacio” with high confidence. However, models with word-level representations all fail for this query. This is not a surprise as the word “Cacio” is a rare mis-transcription of the word “Caillou” and thus rarely seen in the training set.[^4] For the next two successive queries “You”, all baselines failed. The basic models still succeed with two identical queries having the same confidence scores. However, for both the full and constrained context models, confidence on the second query “You” is higher (due to the previous context). This is an example of how contextual clues can help, and confirms our intuitions. Since the second example behaves quite similarly, we omit a description in prose for space considerations.
Manual Evaluation
-----------------
As a final summative evaluation to verify our findings, we tested our models on 100 manually-labeled queries. For these, we randomly selected queries from multi-query sessions for which the deployed X1 system produced “no answer” (NA), which is by construction the most challenging queries. Results of this evaluation are shown in Table \[tab:golden\]. For brevity, we only examined models with the combined query representation; “Prec.” indicates P@1 over all predictions. In this experiment, we allowed the model to not give an answer by setting a confidence threshold—this allows the model to trade off coverage and precision. “Cov.” indicates the percentage of queries that yield a response for that threshold, and “Prec.” indicates the precision. For example, with a threshold of 0.9, the constrained context model can answer 33% of the queries at 91% precision.
We observe that the relative effectiveness of the models is generally consistent with previous experiments, although for these queries we see that the full context model beats the basic model. The constrained context model is able to correctly respond to about half the queries that the deployed system completely failed on, which represents a substantial, real gain. Furthermore, by adjusting the confidence threshold, we can achieve very high precision at the cost of coverage. For our best model (constrained context), we can answer 43% of the queries at 84% precision.
---------------------------------------- ----------- ---------- ----------- ---------- ----------- ---------- -----------
**Model** **Prec.** **Cov.** **Prec.** **Cov.** **Prec.** **Cov.** **Prec.**
EditDist 18% - - - - - -
Deployed X1 0 - - - - - -
$\textnormal{SVM}^{\textnormal{rank}}$ 29% - - - - - -
Basic 38% 21% 86% 24% 88% 27% 85%
Context-f 45% 19% 100% 25% 93% 32% 88%
Context-c 50% 33% 91% 38% 87% 43% 84%
---------------------------------------- ----------- ---------- ----------- ---------- ----------- ---------- -----------
: Results on 100 manually-labeled queries. “Prec.” indicates P@1. We can tradeoff coverage “Cov.” with precision “Prec.” by giving the model the option of not providing an answer, for a particular confidence threshold.[]{data-label="tab:golden"}
Efficiency Analysis
-------------------
Having obtained significant improvements against strong baselines in terms of prediction accuracy, we wonder if our neural network models can achieve sufficiently low latencies for production deployment. To this end, we studied the training time and test time (i.e., prediction latency) of our models, shown in Table \[tab:efficiency\]. The first two columns show the experimental setting as before. Column “\#Params” shows the total number of parameters in the model, column “Training” denotes the training time for each epoch, and column “Test” shows the prediction latency per query. “Avg.” indicates the average value of training/test times, and “Conf.” indicates the 95% confidence interval of training/test times (both the averaged over 30 epochs). Overall, the training time of all models is less than around 100 minutes per epoch, and the per-query prediction latency is within 8 milliseconds. Most model configurations converge in the first 20 epochs. This suggests that our models can be re-trained with a quick turnaround given new data, and that predictions can be made with low latency. Both are crucial considerations in production environments.
Comparing the different query representations, we observe that *combined* has the most number of parameters and was also the slowest to train and test; *char* has the least number of parameters but consumed more time in both training and test compared to the *word* model. For characters, the size of the one-hot vectors is smaller than that of the word embedding vectors, resulting in fewer parameters in the character-level LSTM at the query embedding layer. However, character-level representations are much longer than word-level representations, which consumes more time when producing query embeddings.
----------- ----------- -------------- ---------- ------------------ ---------- --------------
**Model** **Query** **\#Params** **Avg.** **Conf.** **Avg.** **Conf.**
Basic char 326,871 62.1 \[59.7, 64.7\] 6.4 \[6.2, 6.6\]
Basic word 502,871 32.6 \[32.2, 33.0\] 3.0 \[3.0, 3.1\]
Basic comb 758,471 94.5 \[91.4, 97.2\] 6.9 \[6.6, 7.0\]
Context-f char 648,471 72.1 \[68.4, 74.5\] 6.6 \[6.4, 7.0\]
Context-f word 824,471 58.8 \[57.2, 60.8\] 4.0 \[4.0, 4.1\]
Context-f comb 1,210,071 102.4 \[100.8, 103.8\] 7.0 \[6.8, 7.2\]
Context-c char 648,471 32.1 \[30.9, 33.7\] 6.6 \[6.4, 6.8\]
Context-c word 824,471 30.1 \[29.5, 31.2\] 4.0 \[3.9, 4.1\]
Context-c comb 1,210,071 42.5 \[41.8, 43.1\] 6.9 \[6.8, 7.0\]
----------- ----------- -------------- ---------- ------------------ ---------- --------------
: Model efficiency comparisons. Column “Training” denotes the training time for each epoch, and column “Test” shows the prediction latency per query. “Avg.” indicates the average value of training/test times, and “Conf.” indicates the 95% confidence interval of training/test times.[]{data-label="tab:efficiency"}
With the same query representation, the full context models have more parameters and took longer to train and test than the corresponding basic models. The extra parameters and training/test latencies come from the contextual LSTM layer. The constrained context models have the same number of parameters and similar prediction latencies as the full context models since they share the same architecture. However, the training time of the constrained context model is less than half of the full context models, suggesting that most of the training effort is spent on the query embedding layer in the full context models.
[0.23]{} ![Training loss and testing accuracy for each epoch; $\blacktriangle$ denotes epochs where the learning rate was reduced.[]{data-label="fig:loss-curves"}](training-loss-eps-converted-to.pdf "fig:"){width="1.0\linewidth"}
[0.23]{} ![Training loss and testing accuracy for each epoch; $\blacktriangle$ denotes epochs where the learning rate was reduced.[]{data-label="fig:loss-curves"}](testing-p1-eps-converted-to.pdf "fig:"){width="1.0\linewidth"}
We plot training loss and testing accuracy curves in Figure \[fig:loss-curves\]: (a) shows the training loss curve as a function of epoch, (b) shows the P@1 curve in the MultiTest set at each epoch. The symbol $\blacktriangle$ denotes the epochs where the learning rate was reduced by three because development loss had not decreased for the three epochs. We see that most models converged within 20 epochs. In the basic and full context models, the *char* representation took longer to converge than *word* or *combined*, which shows that a character-level representation is more difficult to learn. It is also interesting that the gap in training loss between the basic and full context models is larger than the gap in test accuracy, which means that although the full context model is difficult to train, the benefit of context enables it to generalize well. The constrained context model walks a middle ground in terms of model complexity and the ability to capture context information, leading to both lower training loss and higher test accuracy.
Conclusion
==========
Our vision is that future entertainment systems should behave like intelligent agents and respond to voice queries. As a first step, we tackle a specific problem, voice navigational queries, to help users find the program they are looking for. We articulate the challenges associated with this problem, which we tackle with two ideas: by combining word- and character-level representations of queries, and by modeling session context, both using hierarchically-arranged neural network modules. Empirically results on a large real-world voice query log show that our techniques can effectively cope with ambiguity and compensate for underlying ASR errors. Indeed, we allow viewers to talk to their TVs, and for customers who learn of this feature for the first time, it is a delightful experience!
[00]{}
\#1 \#1[[DOI:]{}0[\#1]{} ]{} \#1 \#1 \#1 \#1 \#1[\#1]{} \#1[\#1]{} \#1[\#1]{}
[acero2008live]{} . . In .
[bennett2012modeling]{} . . In .
[cao2009context]{} . . In .
[chelba2013empirical]{} . . In .
[crestani2006written]{} . . , (), .
[feng2009effects]{} . . In .
[Goodfellow-et-al-2016]{} . . .
[guan2013utilizing]{} . . In .
[guy2016searching]{} . . In .
[hassan2015characterizing]{} . . In .
[hochreiter1997long]{} . . , (), .
[jiang2013users]{} . . In .
[jones2008beyond]{} . . In .
[liu2010personalizing]{} . . In .
[luo2015session]{} . . In .
[Mikolov:2013aa]{} . . ().
[pennington2014glove]{} . . In .
[schalkwyk2010your]{} . . In .
[shan2010search]{} . . In .
[shokouhi2014mobile]{} . . In .
[shokouhi2016did]{} . . In .
[smucker2007comparison]{} . . In .
[tieleman2012lecture]{} . . ().
[wang2008introduction]{} . . , (), .
[yi2011mobile]{} . . In .
[yujian2007normalized]{} . . , (), .
[zhang2015information]{} . . In .
[zweig2011personalizing]{} . . In .
[^1]: <http://selnd.com/1c1tKXg>
[^2]: Stated by Google CEO Sundar Pichai during Google I/O 2016.
[^3]: Similar conclusions follow for other length-based statistics: median was 2 (vs. 4), maximum was 69 (vs. 109), and standard deviation was 1.23 (vs. 2.96).
[^4]: “Cacio” was found in the GloVe word embeddings, thus its word vector was not randomly initialized.
|
---
abstract: 'We present the Usadel theory describing the superconducting proximity effect in heterostructures with a half-metallic layer. It is shown that the full spin polarization inside the half-metals gives rise to the giant triplet spin-valve effect in superconductor (S) – ferromagnet (F) – half-metal (HM) trilayers as well as to the $\varphi_0$-junction formation in the S/F/HM/F/S systems. In addition, we consider the exactly solvable model of the S/F/HM trilayers of atomic thickness and demonstrate that it reproduces the main features of the spin-valve effect found within the Usadel approach. Our results are shown to be in a qualitative agreement with the recent experimental data on the spin-valve effect in ${\rm MoGe/Cu/Ni/CrO_2}$ hybrids \[A. Singh [*et al.*]{}, Phys. Rev. X **5**, 021019 (2015)\].'
author:
- 'S. Mironov'
- 'A. Buzdin'
title: 'Triplet proximity effect in superconducting heterostructures with a half-metallic layer'
---
Introduction
============
Spin-polarized superconducting states attracts growing interest since they are expected to provide powerful mechanisms for controlling the current and magnetization in the devices of superconducting spintronics.[@Linder_Review] Although polarized states are not supported in conventional s-wave superconductors, they can emerge in artificial heterostructures consisting of a superconductor (S) and several ferromagnetic (F) layers with different orientations of magnetic moments. [@Bergeret_2001; @Kadigrobov; @Buzdin_RMP; @Bergeret_RMP] The non-collinear exchange field in the F-layers destroys the spin-singlet structure of Cooper pairs penetrating from the superconductor. This results in the appearance of spin-triplet superconducting correlations with all possible spin projections $S$, both polarized states with $S=\pm 1$ and non-polarized one with $S=0$.
The correlations with $S\pm 1$ have two distinctive features which give an insight into their experimental observation and practical utilization.[@Bergeret_2001; @Kadigrobov] First, such correlations are unsensitive to the exchange field parallel to the spin quantization axis. As a result, they become long-range: in diffusive systems their decay length inside the ferromagnet is comparable with the one in normal metal while the correlations with $S=0$ decay at much shorter distances from the superconductor. Second, these long-range triplet correlations (LRTC) appear only if the ferromagnet has a non-collinear distribution of magnetization. Thus, to control the amplitude of the LRTC in the system one can use ferromagnetic bilayer with tunable mutual orientation of magnetic moments while the effects coming from the correlations with $S=0$ can be damped by increasing the F-layer thickness.
Experimental observation of the LRTC is mostly based on probing the long-range Josephson current in the ${\rm S/F^\prime/F/F^\prime/S}$ junctions.[@Khaire; @Sosnin; @Robinson2; @Sprungmann; @Wang; @Khasawneh] If the thickness of the central F layer is believed to exceed the decay length of the short-range non-polarized correlations the observation of non-zero critical current can be attributed to the presence of the LRTC.[@Houzet_Buzdin] Other observations of the LRTC are based on the so-called triplet spin-valve effect in ${\rm S/F_1/F_2}$ and ${\rm F_1/S/F_2}$ systems revealing in non-monotonic dependence of the S-layer critical temperature $T_c$ on the angle $\theta$ between the magnetic moments in the ${\rm F_1}$ and ${\rm F_2}$ layers.[@Leksin; @Zdravkov; @Jara; @Flokstra; @Dybko] The LRTC open an additional channel for the “leakage" of the Cooper pairs from the superconductor. As a result, $T_c$ can have the minimum at $\theta\not= 0,\pi$.[@Fominov_SFF; @Mironov_FSF] However this effect in $T_c(\theta)$ is typically washed out by the monotonically increasing contribution from the correlations with $S=0$, which shifts the minimum of $T_c$ from $\theta=\pi/2$.
During the past few years the focus in the studies of LRTC is moving towards the heterostructures containing half-metallic (HM) layers (e.g. ${\rm CrO_2}$).[@Pickett; @Coey] A recent progress in fabrication of such structures has resulted in several breakthrough experiments manifesting the long-range Josephson current through the layer of ${\rm CrO_2}$[@Keizer; @Anwar] and triplet spin-valve effect in ${\rm MoGe/Cu/Ni/CrO_2}$ structures.[@Aarts] The importance of these experiments is connected with the fact that in half-metals the energy bands for electrons with spin up and down are separated at a distances comparable with the Fermi energy. As a result, only spin-polarized correlations with $S=+1$ can penetrate into the HM-layer while all other correlations should vanish at its boundary and cannot influence the Josephson current in ${\rm S/F/HM/F/S}$ junctions or the triplet spin-valve effect in S/F/HM systems. Thus, half-metals provide a unique possibility to probe the phenomena caused by LRTC independently from other effects.
However up to now there is no convenient and commonly accepted theoretical model describing the superconducting proximity effect with half-metals. Most common approaches for the treatment of the proximity effect in multilayered structures are based on the quasiclassical approximation.[@Buzdin_RMP; @Bergeret_RMP] This approximation becomes broken near the interfaces which implies using some sort of boundary conditions matching the quasiclassical Green functions in different layers. In contrast with the S/F interfaces where near the critical temperature one may use linear Kupriyanov-Lukichev boundary conditions for the anomalous Green function,[@Kupriyanov] the interfaces with half-metals require more sophisticated boundary conditions since the number of the Green function components at the opposite sides of the interface is different due to the large energy separation of the spin-up and spin-down bands. There are a lot of papers where the authors made attempts to overcome this problem using different versions of the scattering matrix approach. The resulting boundary conditions were extensively used for the description of half-metals within the Blonder-Tinkham-Klapwijk,[@Zheng; @Feng; @Linder_HM; @Enoksen] Bogoliubov-de-Gennes,[@Asano_HM; @Sawa; @Beri] Eilenberger,[@Tokuyasu; @Eschrig_2003; @Eschrig_2008; @Eschrig_2009; @Eschrig_2010] and Usadel[@Cottet_U; @Bergeret_HM_R; @Bergeret_HM; @Eschrig2015] formalisms. Despite the fact that these models provide a number of generic qualitative predictions, most of the results strongly depend on the microscopical mechanisms of the singlet-triplet conversion and on the concrete form of the scattering matrices,[@Eschrig_2008; @Takahashi; @Kupferschmidt] which are not available form the experimental data. An alternative phenomenological approach based on the circuit theory[@Nazarov] does not contain any information about the particular geometry of the system and, thus, can hardly be applied for the quantitative description of real heterostructures.
In the present paper we propose the phenomenological model of the superconducting proximity effect with half-metals based on the Usadel equation in the diffusive limit. Our model is based on the following three key assumptions. (i) The impurities are non-magnetic and do not cause spin-flip processes, which allows to introduce the anomalous Green function inside the half-metal. The Green function component with $S=+1$ satisfies the Usadel equation equivalent to the one in normal metal while all other components are zero. (ii) The are no barriers at the boundaries of the HM layers which results in the continuity of the component with $S=+1$. (iii) The components with $S=0$ and $S=-1$ cannot penetrate the HM layer and vanish at its outer boundaries. Our simple model is shown to explain all main features of the recently observed triplet spin-valve effect in ${\rm MoGe/Cu/Ni/CrO_2}$ structures [@Aarts] and predict several unusual phenomena manifesting the differences between the influence of weak and strong ferromagnets on the proximity effect.
To verify our main conclusions we also considered the situation when the layers of the S/F/HM spin-valve have atomic thickness and are separated by the tunnel barriers. The advantage of such microscopical model is the possibility to find the exact solution of the Gor’kov equations without any prior assumptions about the profiles of the Green functions. Using this model we obtained the analytical dependencies $T_c(\theta)$ which appear to reproduce all main features found within the Usadel approach.
The paper is organized as follows. In Sec. \[Sec\_Model\] we introduce our model. In Sec. \[Sec\_SFHM\] we apply it for the description of the triplet spin-valve effect in S/F/HM systems and compare our results with the experimental observations of Ref. . In Sec. \[Sec\_SFHMFS\] we study the anomalous Josephson effect in dirty S/F/HM/F/S structures with non-coplanar magnetic moments and demonstrate that such systems support the states with the spontaneous phase difference, previously predicted for the ballistic limit.[@Eschrig_2009] In Sec. \[Sec\_Atomic\] we consider the spin-valve effect for the S/F/HM systems of atomic thickness and compare the results with the conclusions of Sec. \[Sec\_SFHM\]. In Sec. \[Sec\_Conc\] we summarize our results and discuss their possible applications.
Model {#Sec_Model}
=====
Let us consider a multilayered structure consisting of superconductors, ferromagnets and half-metals with the interfaces perpendicular to the $x$-axis. Two examples of such structures are shown in Fig. \[Fig\_SFHM\] and Fig. \[Fig\_SFHMFS\]. We assume that the systems is in the diffusive limit and the temperature $T$ is close to the critical temperature $T_c$ of the superconducting transition. In this case outside the half-metallic layers the superconducting properties of the system can be described in terms of the linearized Usadel equation[@Eschrig] $$\label{Main_System}
\frac{D}{2}\partial_x^2\hat{f}-\omega_n \hat{f}-\frac{i}{2}\left(\mathbf{h}
\mathbf{\hat{\sigma}} \hat{f}+ \hat{f}\mathbf{h}\mathbf{\hat{\sigma}}\right)+
\hat{\Delta}=0,$$ where the quasiclassical Green function $$\label{Matrix_f}
\hat{f}=\left(
\begin{array}{cc}
f_{\uparrow\uparrow} & f_{\uparrow\downarrow} \\
f_{\downarrow\uparrow} & f_{\downarrow\downarrow} \\
\end{array}
\right)
=\left(f_s+\mathbf{f}_t\hat{\sigma}\right)i\hat{\sigma}_y$$ is the $2\times 2$ matrix in the spin space, $\hat{\Delta}=\Delta i\hat{\sigma}_y$ is the superconducting pairing potential inside the S layers, $\omega_n=\pi T(2n+1)>0$ are the Matsubara frequencies, ${\bf h}$ is the exchange field in the ferromagnets, and $D$ is the diffusion constant.
To describe the superconducting correlations inside the half-metallic layers we assume that (i) electron scattering on the impurities does not cause spin-flips and (ii) the barriers are not spin-active. In this case we may introduce the Usadel Green function with only one nonzero component $f_{\uparrow\uparrow}$ meaning that the spin polarization in the half-metal is directed along the $z$-axis.
![(Color online) The sketch of the ${\rm S/F/HM}$ spin valve. The exchange field in the F-layer makes the angle $\theta$ with the spin quantization axis in the half-metal.[]{data-label="Fig_SFHM"}](Fig1.eps){width="25.00000%"}
To match the solutions of the Eq. (\[Main\_System\]) in different layers one should put the boundary conditions at each interface. We assume all interfaces between the layers to be transparent for electrons. In this case at the interfaces which separate two non-halfmetallic material the boundary conditions come to the continuity of $\hat{f}$ and the combination $\sigma\partial_x\hat{f}$ ($\sigma$ is the Drude conductivity of the corresponding layer).[@Kupriyanov] Similarly, for the interfaces with the half-metal we demand only the continuity of $f_{\uparrow\uparrow}$ and $\sigma\partial_x f_{\uparrow\uparrow}$ while all other components $f_{\downarrow\downarrow}$, $f_{\uparrow\downarrow}$ and $f_{\downarrow\uparrow}$ should vanish at the HM-layer boundaries. At the outer boundaries of the heterostructure we demand $\partial_x\hat{f}=0$.
The physical meaning of the boundary conditions at the interface with half-metal is very clear: the interface with the HM layer plays the role of spin filter, which is absolutely transparent for the correlations with $S=+1$ and opaque for all other correlations.
Triplet spin-valve effect in S/F/HM systems {#Sec_SFHM}
===========================================
The geometry of the S/F/HM system under consideration is shown schematically in Fig. \[Fig\_SFHM\]. We choose the origin of the $x$-axis in a way that the superconductor is at $-d_s<x<0$, the ferromagnet is at $0<x<d_f$, and the HM layer occupies the region $d_f<x<d_f+d_h$. The exchange field ${\bf h}$ in the F-layer is assumed to be rotated on the angle $\theta$ in the $xz$-plane and have two components: $h_{z}=h \cos\theta$ and $h_{x}=h \sin\theta$. The spin quantization axis in the HM-layer coincides with the $z$ one.
To demonstrate the key difference between the S/F/HM system and classical ${\rm S/F_2/F_2}$ spin valves it is instructive to write the explicit form of Eq. (\[Main\_System\]) inside the F-layer for all components $f_s$ and ${\bf f}_t$: $$\label{Usadel_F}
\begin{array}{l}{{\displaystyle}(D_f/2)\partial_x^2 f_s=\omega_n f_s+ih\cos\theta f_{tz}+ih\sin\theta f_{tx},
}\\{}\\{{\displaystyle}(D_f/2)\partial_x^2 f_{tz}=\omega_n f_{tz}+ih\cos\theta f_s,}\\{}\\{{\displaystyle}(D_f/2) \partial_x^2 f_{tx}=\omega_n f_{tx}+ih\sin\theta f_s,}\\{}\\{{\displaystyle}(D_f/2)\partial_x^2 f_{ty}=\omega_n f_{ty},} \end{array}$$ where $D_f$ is the diffusion constants in the ferromagnet. Clearly the spin-singlet component $f_s$ of the anomalous function coming from the superconductor induces the triplet components $f_{tz}$ and $f_{tx}$ while the equation for $f_{ty}$ remains independent. In the ${\rm S/F_2/F_2}$ structures the boundary conditions do not mix different components of the function $\hat{f}$ and, thus, the absence of the source in equation for $f_{ty}$ immediately leads to $f_{ty}(x)= 0$ in the whole heterostructure.
However, in the S/F/HM systems the situation is completely different. The penetration of the non-zero component $f_{\uparrow\uparrow}=-f_{tx}+if_{ty}$ into the HM-layer together with the vanishing of the component $f_{\downarrow\downarrow}=f_{tx}+if_{ty}$ is possible only if inside the half-metal $$\label{Fxy}
f_{ty}=if_{tx}\not= 0.$$ Consequently, this results in the appearance of $f_{ty}$ also in the S and F layers.
For the further analysis it is convenient to exclude $f_{ty}$ from (\[Main\_System\]) by substituting the solution of the equation for $f_{ty}$ into the boundary conditions. As a result, all information about the component $f_{ty}$ becomes included into the effective boundary condition for $f_{tx}$, which reads (see Appendix \[App\_BoundCond\]) $$\label{fx_bound}
\left.\frac{\partial_x f_{tx}}{f_{tx}}\right|_{x=d_f}=-q_f\Gamma$$ with $$\label{Gamma_def}
\Gamma=2\mu_h+\frac{\mu_s+\mu_f}{1+\mu_s \mu_f}.$$ In this expression $$\label{mu_def}
\mu_j=\frac{\sigma_j}{\sigma_f}\sqrt{\frac{D_f}{D_j}}\tanh(q_jd_j),$$ where the index $j\in\left\{s,f,h\right\}$ corresponds to the S, F and HM layers respectively, $q_j=\sqrt{2\omega_n/D_j}$, $D_j$ and $\sigma_j$ are the diffusion constant and the normal conductivity in the $j$-th layer.
Note that in the case when the diffusion constants and conductivities of all layers are equal to each other the expression for $\Gamma$ takes the form $$\label{fx_bound_ident}
\Gamma=2\tanh(q_hd_h)+\tanh\left[q_f(d_s+d_f)\right].$$ Clearly, this expression reflects the fact that at the F/HM interface the component $f_{tx}$ of the Green function induces two components ($f_{tx}$ and $f_{ty}$) in the half-metal and also the component $f_{ty}$ in the S/F bilayer.
Now let us analyze the dependence of the S-layer critical temperature on the angle $\theta$. To simplify the calculations we assume that the thickness of the S-layer is much smaller than the superconducting coherence length $\xi_{s0}=\sqrt{D_s/4\pi T_{c0}}$, which enables neglecting the spatial variation of the pairing potential $\Delta$ across the superconducting film (here $T_{c0}$ is the critical temperature of the isolated superconductor). For convenience we choose $\Delta$ to be real. Then the dependence $T_c(\theta)$ is defined by the self-consistency equation $$\label{SelfConsistencyEquation}
\Delta\ln\frac{T_c\left(\theta\right)}{T_{c0}}+\sum\limits_{n=0}^{\infty}
\left[\frac{\Delta}{n+1/2}-2\pi T_c\left(\theta\right) f_s\left(\theta\right)\right]=0.$$
The equation (\[Main\_System\]) with the boundary conditions described above allows us to obtain the analytical expressions for the anomalous Green function in each layer (the details of the calculations are presented in Appendix \[App\_Calc\]). Inside the superconductor the component $f_s$ has the form $$\label{fs_res}
f_s=\Re\left(\frac{\Delta}{\omega_n+\tau_{\pi}^{-1}}\right)- \frac{\Delta}{\omega_n} \frac{\mu_sW^2\sin^2\theta}{Q\sin^2\theta +\Gamma+\left(\Gamma-2\mu_h\right)\cos^2\theta},$$ where $q=\sqrt{2\left(\omega_n+ih\right)/D_f}$, $p=\mu_sq_f$, $$\label{W_def_main}
W=\Im\left\{\frac{q}{q\cosh(qd_f)+p\sinh(qd_f)}\right\},$$ $$\label{Q_def_main}
Q=\Re \left\{\frac{q}{q_f}\frac{p+q\tanh(qd_f)}{q+p\tanh(qd_f)}\right\},$$ and the pair-breaking parameter $$\label{tau_pi_def}
\tau_\pi^{-1}=\frac{\sigma_f}{\sigma_s}\frac{D_s}{2d_s}q\coth(qd_f)$$ is the same as for the S/F/S junction with the F-layer thickness $2d_f$ in the $\pi$-state (see, e.g., Ref. ).
![(Color online) The dependencies of the critical temperature on the angle $\theta$ for different thicknesses $d_f$ of the ferromagnetic layer. The system parameters are: $d_s=0.5\xi_{s0}$, $h/4\pi T_{c0}=2$, $d_h=\sqrt{D_h/4\pi T_{c0}}$, $D_f/D_s=\sigma_f/\sigma_s=10^{-3}$, and $D_h/D_f=\sigma_h/\sigma_f=10^{-3}$.[]{data-label="Fig_Tcphi"}](Fig2.eps){width="38.00000%"}
The expression (\[fs\_res\]) allows to analyze the main features of the critical temperature behavior. First, in contrast with the case of ${\rm S/F_1/F_2}$ spin valve[@Fominov_SFF] the dependence $T_c(\theta)$ is always symmetric, i.e. $T_c(\pi-\theta)=T_c(\theta)$. Second, $f_s\left(\sin^2\theta\right)$ is a monotonically decreasing function and, thus, the minimum of the critical temperature corresponds to $\theta=\pi/2$. Note that these two features were clearly observed in recent experiments with spin valves containing the half-metallic ${\rm CrO_2}$ layer.[@Aarts]
The typical dependencies $T_c(\theta)$ are shown in Fig. \[Fig\_Tcphi\]. One sees that S/F/HM system with $d_f\sim \xi_f$ reveal giant triplet spin-valve effect originating due to the LRTC: for the chosen parameters the increase of $\theta$ results in the damping of $T_c$ from $0.4T_{c0}$ to zero. In Fig. \[Fig\_Tcdf\] we also plot the dependencies of $T_c$ on the F-layer thickness $d_f$ for $\theta=0$ and $\theta=\pi/2$. For $d_f\ll \xi_f$ the superconductivity is fully suppressed by the proximity with half-metal. When increasing $d_f$ above a certain threshold, which depends on $\theta$, the critical temperature becomes non-zero and grows rapidly. Interestingly, the difference in the thresholds for $\theta=0$ and $\theta=\pi/2$ makes it possible to simultaneously reach the absolute maximum of $T_c$ for $\theta=0$ and have $T_c(\pi/2)=0$.
![(Color online) The dependencies of the critical temperature on the thickness of the ferromagnetic layer for $\theta=0$ (blue curve) and $\theta=\pi/2$ (red curve). The system parameters are the same as in Fig. \[Fig\_Tcphi\].[]{data-label="Fig_Tcdf"}](Fig3.eps){width="38.00000%"}
Anomalous Josephson effect in S/F/HM/F/S systems {#Sec_SFHMFS}
================================================
![(Color online) The sketch of the ${\rm S/F_1/HM/F_2/S}$ Josephson junction. The exchange field vectors in the ferromagnets and half-metal are perpendicular to each other.[]{data-label="Fig_SFHMFS"}](Fig4.eps){width="35.00000%"}
In this section we analyze the behavior of the Josephson current through the ${\rm F_1/HM/F_2}$-trilayer (see Fig. \[Fig\_SFHMFS\]). In particular, we show that the peculiar mixing of different spin-triplet $\hat{f}$ components at F/HM interfaces results in the $\varphi_0$-junction formation provided the magnetic moments in the F-layers and the spin quantization axis in the HM-layer are the non-coplanar vectors. Previously this effect was noted within the circuit-theory for the junctions consisting of two ferromagnetic superconductors separated by the half-metal.[@Nazarov] The model of the ferromagnetic superconductors[@Nazarov] allows to simplify the calculations but can hardly be applied for the real systems in which the ferromagnetic order strongly suppresses superconductivity. Here we consider a more realistic situation when the regions with superconducting and ferromagnetic orders are separated in space.
To make the physical origin of the effects under consideration more transparent we restrict ourselves to the simplest case when the exchange field in the ${\rm F_1}$-layer is directed along the $x$-axis while the magnetic moment of the ${\rm F_2}$-layer has only $y$-component. For simplicity we consider equal magnitude $h$ of the exchange field in the ${\rm F_1}$ and ${\rm F_1}$-layers. Similar to the previous sections we assume that in the half-metal the spins are quantized in the $z$-direction. We choose the $x$-axis perpendicular to the interfaces so that the ${\rm F_1}$, ${\rm HM}$ and ${\rm F_2}$-layers occupies the regions $-(d_f+d_h/2)<x<-d_h/2$, $-d_h/2<x<d_h/2$ and $d_h/2<x<d_f+d_h/2$ respectively.
Before proceeding with the calculation of the Josephson current let us briefly point out the main difference of the described system from the ${\rm S_1/F_1/F'/F_2/S_2}$ junctions with the same magnetic configuration extensively studied before. In the latter system if the magnetic moments in the three ferromagnets are perpendicular to each other and the thickness of the central layer strongly exceeds the coherence length $\sqrt{D_2/h_2}$ the Josephson current is negligibly small. Indeed, the ${\rm F_1}$ layer produces the component $f_{tx}$ of the anomalous Green function which becomes long-range in the ${\rm F'}$ layer and reaches the ferromagnet ${\rm F_2}$. However, this component do not contribute to the Josephson current since the magnetic moment in the ${\rm F_2}$ directed along the $y$-axis cannot convert this $f_{tx}$ component into the singlet $f_s$ one.
The situation becomes completely different if instead of the ferromagnet ${\rm F'}$ one has half-metallic layer. In this case at the ${\rm F_1/HM}$ interface both $f_{tx}$ and $f_{ty}$ components are produced. As a result, in the ${\rm F_2}$-layer the $f_{ty}$ component can be effectively converted into the $f_s$ one and, thus, produce the non-vanishing Josephson current.
The current-phase relation $j(\varphi)$ of the Josephson junction is defined by the sum $$\label{Curr_def}
j\left(\varphi\right)=8\pi e D N T \sum\limits_{\omega_n>0}\Im\left(f_s^*\partial_x f_s-{\bf f}_t^*\partial_x {\bf f}_t\right),$$ where $e$ is the charge of electron, $N$ is the electronic density of states, $D$ is the diffusion coefficient. Since the current does not depend on the position across the junction the anomalous Green function can be taken in arbitrary point. Practically it is convenient to choose this point, e.g., at the ${\rm S/F_1}$ interface (at $x=-d_f-d_h/2$) where only the singlet component $f_s$ is non-zero.
To simplify the further calculations we assume that the normal conductivity of the superconducting electrodes strongly exceed the ones in the ferromagnets, so that at both S/F interfaces the rigid boundary conditions are fulfilled: $f_s=(\Delta/\omega_n) e^{\pm i\varphi/2}$ (the signs $+$ and $-$ in the phase factor correspond to the right and left S-layers respectively) while ${\bf f}_t=0$. Also we assume that the thickness of the HM-layer is much less than the coherence length $\sqrt{D_h/4\pi T_{c0}}$ which allows to neglect the spatial variations of the function $f_{\uparrow\uparrow}$ across this layer. Solving the Usadel equation with the boundary conditions discussed above we obtain the analytical expression for the derivative $\partial_x f_s$ at $x=-d_f-d_h/2$ (see Appendix \[App\_CPR\]) and the resulting current-phase relation: $$\label{CPR_res}
j\left(\varphi\right)=4\pi e D_f N T \sin\left(\varphi+\frac{\pi}{2}\right)\sum\limits_{\omega_n>0} \frac{\Delta^2}{\omega_n^2}J_n,$$ where $$\label{CPR_res}
J_n=\frac{1}{q_f\coth(q_fd_f)+ \Re\left[q\coth(qd_f)\right]}\Im^2\left[\frac{q}{\sinh(qd_f)}\right]$$ and the definition of $q$ and $q_f$ are the same as in Sec. \[Sec\_SFHM\].
From Eq. (\[CPR\_res\]) one can see that the non-coplanarity of the magnetic moments in the magnetic layers results in the appearance of the spontaneous Josephson phase difference (so-called $\varphi_0$-junction). In contrast with the ordinary S/F/S systems where the formation of $\varphi_0$-junction requires strong spin-orbit, coupling[@Buzdin_Phi; @Yokoyama; @Mironov_BdG; @Bergeret_Phi0] here the spontaneous phase arises due to spin polarization in the HM-layer. Note that in case of arbitrary mutual orientation of the magnetic moments in the F-layers one can expect the spontaneous Josephson phase $\varphi_0$ to be equal to the angle between the projections of these magnetic moments to the $xy$-plane (see Ref. ).
S/F/HM spin valve of the atomic thickness {#Sec_Atomic}
=========================================
![(Color online) S/F/HM spin valve of the atomic thickness. The angle between the exchange field in the ferromagnet and the spin quantization axis in the half-metal is denoted as $\theta$. The coupling between the layers is described by the transfer integral $t$.[]{data-label="Fig_SFHM_atomic"}](Fig5.eps){width="15.00000%"}
In this section we study the spin-valve effect in the S/F/HM trilayers of atomic thickness. Experimentally such kind of systems can be realized, e.g., on the basis of the ${\rm RuSr_2GdCu_2O_8}$ and ${\rm La_{0.7}Ca_{0.3}MnO_3}$ compounds. The first one is the ferromagnetic superconductor with the alternating S-layers of ${\rm CuO_2}$ and the magnetically ordered ${\rm RuO_2}$ layers. The second compound is a half-metal[@CoeyAdv] which is recently shown to have a strong influence on the properties of the superconducting systems.[@Villegas]
Here we use the microscopical Gor’kov formalism to calculate how the critical temperature of the S/F/HM trilayer depends on the angle $\theta$ between the exchange field ${\bf h}=h\cos\theta\hat{\bf z}+h\sin\theta\hat{\bf x}$ in the ferromagnet and the spin quantization axis $\hat{\bf z}$ in the HM-layer (see Fig. \[Fig\_SFHM\_atomic\]). We assume that the neighboring layers are coupled by the electron tunneling describing by the transfer integral $t$. Previously this approach was successfully applied for the description of the spin-valve effect in F/S/F structures (see \[\] and references therein). The main advantage of the atomic layers model is the possibility to obtain the exact solutions for the Green functions and analyze their properties. Below we obtain such solutions for the case of the S/F/HM structures and demonstrate that they reproduce all main features of the spin-valve effect described within the phenomenological Usadel model in Sec. \[Sec\_SFHM\].
Let us denote the two-component electronic operators in the S, F and HM layers as $\varphi$, $\psi$ and $\eta$ respectively. For simplicity we assume that the quasiparticle motion in the plane of the S and F-layers is described by the same energy spectrum $\xi({\bf p})$. At the same time, in the half-metal the energy is strongly spin dependant: we assume that for the spin-up quasiparticles $\xi_\uparrow=\xi({\bf p})$ while for the spin-down ones $\xi_\downarrow=+\infty$, which implies $\eta_\downarrow=0$.
The Hamiltonian of the system under consideration has the form[@TollisAtomic] $$\label{Ham}
\hat{H}=\hat{H}_0+\hat{H}_{S}+\hat{H}_t,$$ where $$\label{H0}
\hat{H}_0=\sum\limits_{{\bf p};\alpha=1,2}\left[\xi({\bf p})\varphi_\alpha^+\varphi_\alpha + \psi_\alpha^+\hat{C}(\theta)\psi_\alpha+ \eta_\alpha^+\hat{P}\eta_\alpha\right],$$ $$\label{HS}
\hat{H}_S=\sum\limits_{{\bf p}}\left(\Delta^*\varphi_{{\bf p},2}\varphi_{-{\bf p},1} + \Delta\varphi_{{\bf p},1}^+\varphi_{-{\bf p},2}^+\right),$$ $$\label{Ht}
\hat{H}_t=\sum\limits_{{\bf p};\alpha=1,2}t\left(\varphi_\alpha^+\psi_\alpha+ \psi_\alpha^+\varphi_\alpha +\psi_\alpha^+\eta_\alpha+ \eta_\alpha^+\psi_\alpha\right).$$ In Eq. (\[H0\]) we introduces two matrices describing the effect of Zeeman coupling in the F-layer and the spin polarization in the half-metal:[@TollisAtomic] $$\label{HS}
\hat{C}(\theta)=\left(
\begin{array}{cc}
\xi-h\cos\theta & -h\sin\theta \\
-h\sin\theta & \xi+h\cos\theta \\
\end{array}
\right),
~~~\hat{P}=\left(
\begin{array}{cc}
\xi & 0 \\
0 & +\infty \\
\end{array}
\right).$$ Note that in contrast with the model of Ref. in our system the superconducting and ferromagnetic regions are separated in space which makes it possible to consider the exchange field $h$ of arbitrary magnitude.
The critical temperature $T_c$ of the S-layer is defined by the linear expansion of the anomalous Green function $F_{\alpha,\beta}^+=\left<T_\tau\left(\varphi_\alpha^+,\varphi_\beta^+\right)\right>$ over the gap potential $\left|\Delta\right|$, where $T_\tau$ denotes the time-ordered product for the imaginary time $\tau$. Writing and solving the systems of Gor’kov equations for the S/F/HM system we find (the details of calculations are presented in Appendix \[App\_Gorkov\]): $$\label{Gorkov_F_res}
\begin{array}{c}
{{\displaystyle}\frac{\hat{F}^+}{\Delta^*}=\left\{(i\omega+\xi)\hat{1}-t^2\left[(i\omega+\hat{C})- t^2(i\omega+\hat{P})^{-1}\right]^{-1}\right\}^{-1}}\\{}\\{{\displaystyle}\times\hat{I}\left\{(i\omega-\xi)\hat{1}-t^2\left[(i\omega-\hat{C})- t^2(i\omega-\hat{P})^{-1}\right]^{-1}\right\}^{-1}}.
\end{array}$$ To simplify the further calculations we assume the tunneling constant $t$ to be small and perform the power expansion of Eq. (\[Gorkov\_F\_res\]) over $t^2$. To obtain the non-trivial dependence $T_c(\theta)$ we should keep the terms up to $t^6$. Also it is convenient to represent the self-consistency equation[@DaumensAtomic] in the form $$\label{Self_cons}
T_c(\theta)=T_c(0)-2T_{c0}^2\sum\limits_{\omega_n>0}\int\limits_{-\infty}^{+\infty}{\rm Re}\frac{F^+_{12}(\theta)-F^+_{12}(0)}{\Delta^*}d\xi,$$ where $T_c(0)=T_{c0}\left[1-O(t^2)\right]$ is the critical temperature at $\theta=0$ \[for $t\ll T_{c0}$ one has $\left|T_c(\theta)-T_{c0}\right|\ll T_{c0}$\] and the sum is taken over the discrete set of positive Matsubara frequencies $\omega_n=\pi T_{c0}(2n+1)$.
Substituting the expression for $F^+_{12}$ into Eq. (\[Self\_cons\]) we found: $$\label{DF_res}
T_c(\theta)=T_c(0)+\sum\limits_{\omega_n>0}\int\limits_{-\infty}^{+\infty}\frac{4T_{c0}^2t^6h^2\sin^2\theta d\xi}{(i\omega-\xi)(i\omega+\xi)^3\left[(i\omega+\xi)^2-h^2\right]^3},$$ Taking the integral over $\xi$ we get: $$\label{GTc_res}
T_c(\theta)=T_c(0)-\sum\limits_{\omega_n>0}\frac{\pi T_{c0}^2t^6 h^2\sin^2\theta}{\omega^3(4\omega^2+h^2)^3},$$
First, one can clearly see that the deviation of the critical temperature from $T_c(0)$ is proportional to $\sin^2\theta$ and, thus, $T_c(\pi-\theta)=T_c(\theta)$ in the full accordance with the conclusion of Sec. \[Sec\_SFHM\].
Second, Eq. (\[GTc\_res\]) shows that the magnitude of the spin-valve effect which can be characterized by the value $\delta T_c=T_c(0)-T_c(\theta)$ has non-monotonic dependence on $h$ with the maximum $\delta T_c^{max}\propto t^6/T_{c0}^5$ at $h\sim T_{c0}$. Indeed, for $h\ll T_{c0}$ the exchange field weakly affect the system properties, and $$\label{GTc_res_h_small}
\delta T_c=\frac{511\zeta(9) t^6 h^2}{2^{15}\pi^8T_{c0}^7}\propto \frac{t^6 }{T_{c0}^5}\left(\frac{h}{T_{c0}}\right)^2.$$ In the opposite limit when $h\gg T_{c0}$ the strong Zeeman splitting of the energy bands inside the F-layer effectively damps the tunneling constant between the layers, and as a result the spin-valve effect is also weak: $$\label{GTc_res_h_big}
\delta T_c=\frac{7\zeta(3)t^6}{8\pi^2T_{c0}h^4}\propto \frac{t^6}{T_{c0}^5}\left(\frac{T_{c0}}{h}\right)^4.$$ In terms of the previously discussed Usadel theory the latter relation simply reflects the fact that for $h\gg T_{c0}$ the coherence length $\xi_f=\sqrt{D_f/h}$ in the F-layer becomes much smaller than its thickness, and superconducting correlations do not reach half-metal.
Conclusion {#Sec_Conc}
==========
To sum up, we proposed the phenomenological Usadel theory of the superconducting proximity effect in multilayered systems with a half-metallic layer. It is shown that the boundary between ferromagnet and half-metal serves as a source of additional triplet component of the anomalous Green function which is perpendicular to both exchange field ${\bf h}$ in the F-layer and the spin quantization axis in half-metal $\hat{\bf z}$. For the S/F/HM trilayes we analyzed the dependence of the critical temperature $T_c$ on the angle $\theta$ between ${\bf h}$ and $\hat{\bf z}$ and found that the discovered triplet component strongly enhances the spin valve effect compared to the traditional S/F/F structures: increasing the angle $\theta$ one can damp $T_c$ from the value comparable to the critical temperature of the isolated superconductor down to zero. Note that the described giant damping of $T_c$ appears [*only*]{} due to the long-range triplet correlations (LRTC) since the short range ones do not penetrate the HM-layer and, thus, are not sensitive to $\theta$. In addition, we showed that the full spin polarization in the HM-layer requires the symmetry $T_c(\pi-\theta)=T_c(\theta)$, which was clearly observed in recent experiments with the ${\rm MoGe/Cu/Ni/CrO_2}$ spin valves.[@Aarts]
To verify our main conclusions about the peculiarities of the spin-valve effect in the S/F/HM structures we considered the case when the layers of such system have the atomic thickness. For this case we obtained the exact analytical solution of the Gor’kov equations and calculated the dependencies $T_c(\theta)$. We found that if the tunneling rate between the layers is small the deviation of $T_c(\theta)$ from the critical temperature at $\theta=0$ is proportional to $\sin^2\theta$ which reproduces the symmetry relation found within the Usadel formalism.
Also we demonstrated that the new “perpendicular” triplet component of the anomalous Green function dramatically modifies the current-phase relation of the S/F/HM/F/S Josephson junctions provided the exchange field vectors in the F-layers and the spin quantization axis $\hat{\bf z}$ in half-metal are non-coplanar. First, we found that such systems support the $\varphi_0$-junction formation. This result is non-trivial since in the usual S/F/S and S/N/S structures the appearance of the spontaneous Josephson phase difference requires strong spin-orbit coupling. [@Buzdin_Phi; @Yokoyama; @Mironov_BdG; @Bergeret_Phi0] In contrast, here such phase emerges only due to the spin selectivity of the half-metal. Second, the critical current of the S/F/HM/F/S structures does not vanish when the exchange field vectors and $\hat{\bf z}$ are perpendicular to each other. This result directly originates from the presence of the additional Green function component. It is exactly this component, which makes it possible not only to generate the long-range triplet correlations near the S-lead but also to convert them back into the singlet ones near the opposite lead. Note that previously these effects were discussed for the system of two ferromagnetic superconductors and the half-metal separated with the tunnel barriers.[@Nazarov] However the model of ferromagnetic superconductor is valid only for extremely weak exchange field values which make it inapplicable to the real heterostructures.
Finally, we would like to mention that the appearance of the additional triplet component of the anomalous Green function should strongly influence the local density of states (LDOS) and the electromagnetic response of all considered heterostructures. Indeed, spin-triplet correlations make positive contribution into the LDOS[@Buzdin_DOS; @Kontos; @Cottet_DOS; @Nazarov] and, as a result, LDOS in the superconducting state can even exceed the one above $T_c$. Obviously, the new triplet component generated by the half-metal should provide more favorable conditions for the investigation of this unusual phenomenon as well as for the observation of the related Fulde-Ferrell-Larkin-Ovchinnikov instabilities.[@Mironov_FFLO]
ACKNOWLEDGMENTS {#acknowledgments .unnumbered}
===============
The authors thank A. S. Mel’nikov for useful discussions. This work was supported by the French ANR “MASH," NanoSC COST Action MP1201, and the Russian Presidential foundation (Grant SP-6340.2013.5).
Effective boundary conditions at F/HM interface {#App_BoundCond}
===============================================
Let us derive the effective boundary condition for the $f_{tx}$ component of the anomalous Green function at the F/HM interface of the S/F/HM heterostructure. From Eq. (\[Main\_System\]) using the condition $\partial_xf_{ty}=0$ at $x=-d_S$ and $x=d_F+d_H$ one finds the solution for the $f_{ty}$ in the S and F layers as well as the solution for $f_{\uparrow\uparrow}$ in the HM-layer: $$\label{fy_GenSol}
\begin{array}{l}{{\displaystyle}{\rm S:}~~~~~~ f_{ty}=A\cosh\left[q_s\left(x+d_s\right)\right],}\\{{\displaystyle}{\rm F:}~~~~~~ f_{ty}=B\cosh\left[q_f\left(x-d_f\right)\right]+C\sinh\left[q_f\left(x-d_f\right)\right],}\\{{\displaystyle}{\rm HM:}~~~f_{\uparrow\uparrow}=H\cosh\left[q_h\left(x-d_f-d_h\right)\right],}
\end{array}$$ where $q_j=\sqrt{2\omega_n/D_j}$ (the index $j=s,f,h$ corresponds to the S, F and HM layer respectively), $D_j$ is the diffusion constant in the $j$-th layer, and $A$, $B$, $C$ and $H$ are the integration constants. Introducing the parameter $\nu_{sf}=\left(\sigma_s/\sigma_f\right)\sqrt{D_f/D_s}$ and $\nu_{hf}=\left(\sigma_h/\sigma_f\right)\sqrt{D_f/D_h}$ and using the boundary conditions at $x=0$ and $x=d$ we obtain the systems of equations: $$\begin{array}{c}{{\displaystyle}B\cosh(q_fd_f)-C\sinh(q_fd_f)=A\cosh\left(q_sd_s\right),}\\{{\displaystyle}-B\sinh(q_fd_f)+C\cosh(q_fd_f)=\nu_{sf} A\sinh\left(q_sd_s\right),}\\{{\displaystyle}f_{tx}(d_f)+iB=0,}\\{{\displaystyle}-f_{tx}(d_f)+iB=H\cosh\left(q_hd_h\right),}\\{{\displaystyle}-\partial_xf_{tx}(d_f)+iq_fC=-q_f\nu_{hf}H\sinh\left(q_hd_h\right).}
\end{array}$$ Excluding the constants $A$, $B$, $C$ and $H$ from this system we obtain the effective boundary condition (\[fx\_bound\]) for the component $f_{tx}$.
Calculation of the anomalous Green function in the S/F/HM spin valve {#App_Calc}
====================================================================
As follows from Eq. (\[Main\_System\]) if the gap potential $\Delta$ is chosen to be real then the components $f_s$ and $f_{ty}$ of the anomalous Green function are also real while the components $f_{tz}$ and $f_{tx}$ are imaginary. Then it is convenient to introduce the complex function $$\label{F_def}
F(x)=f_s+f_{tz}\cos\theta+f_{tx}\sin\theta$$ and the real function $$\label{R_def}
R(x)=i\left(f_{tz}\sin\theta-f_{tx}\cos\theta\right),$$ so that $$\label{f_from_FR}
\begin{array}{l}
{{\displaystyle}f_s=\Re (F),}\\{{\displaystyle}f_{tz}=i\Im(F)\cos\theta-iR\sin\theta ,}\\{{\displaystyle}f_{tx}=i\Im(F)\sin\theta+iR\cos\theta.}
\end{array}$$ The introduced functions satisfy the systems of equations $$\label{FR_system}
\begin{array}{l}{{\displaystyle}\frac{D}{2}\partial_x^2 F=\left(\omega_n +ih\right)F-\Delta,}\\{}\\{{\displaystyle}\frac{D}{2}\partial_x^2 R=\omega_n R.}
\end{array}$$ The boundary conditions for $F$ and $R$ straightly follow from the corresponding conditions for the $\hat{f}$ components.
Since we assumed that $d_s\ll\xi_{s0}$ the solution of Eq. (\[FR\_system\]) inside the superconductor satisfying the boundary conditions at $x=-d_s$ can be represented in the form $$\label{FR_S_solution}
\begin{array}{l}{{\displaystyle}F=\frac{\Delta}{\omega_n}+F_S\frac{q_s^2(x+d_s)^2}{2},}\\{}\\{{\displaystyle}R=R_S\frac{q_s^2(x+d_s)^2}{2}.}
\end{array}$$ The analogous solution in the F layer reads $$\label{FR_F_solution}
\begin{array}{l}{{\displaystyle}F=F_1\cosh\left[q(x-d_f)\right]+F_2 \sinh\left[q(x-d_f)\right],}\\{}\\{{\displaystyle}R=R_1\cosh\left[q_f(x-d_f)\right]+R_2 \sinh\left[q_f(x-d_f)\right],}
\end{array}$$ where we have introduced the complex wave-vector $q=\sqrt{2\left(\omega_n+ih\right)/D_f}$.
To calculate the unknown amplitudes in the functions $F$ and $R$ let us first consider the boundary conditions at $x=d_f$. Since $f_s=f_{tz}=0$ at this interface one finds: $$\label{BC_FHM_sz}
\Re(F_1)=0,~~~ \Im(F_1)\cos\theta=R_1\sin\theta.$$ These two equations can be rewritten as $$\label{BC_FHM_sz_c}
F_1=iR_1\tan\theta.$$ The condition for the component $f_{tx}$ reads \[see (\[fx\_bound\])\] $$\label{BC_FHM_x}
\Im(qF_2)\sin\theta+q_fR_2\cos\theta=-q_f\Gamma\left[\Im(F_1)\sin\theta+R_1\cos\theta\right].$$
The boundary conditions at $x=0$ give the rest four equations: $$\label{BC_SF}
\begin{array}{l}{{\displaystyle}\frac{\Delta}{\omega_n}+F_S=F_1\cosh(qd_f)-F_2 \sinh(qd_f),}\\{}\\{{\displaystyle}pF_S=-q F_1\sinh(qd_f)+qF_2 \cosh(qd_f),}\\{}\\{{\displaystyle}R_S=R_1\cosh(q_fd_f)-R_2 \sinh(q_fd_f),}\\{}\\{{\displaystyle}pR_S=-q_fR_1\sinh(q_fd_f)+q_fR_2 \cosh(q_fd_f),}
\end{array}$$ where $$\label{mu_approx}
p=\mu_sq_f\approx\frac{\sigma_s}{\sigma_f}\frac{2d_s}{D_s}\omega_n.$$
To solve the system of equations (\[BC\_FHM\_sz\_c\])-(\[BC\_SF\]) it is convenient to exclude $F_S$ and $R_S$ from (\[BC\_SF\]) and then express $R_2$ and $F_2$ in terms of $R_1$ using (\[BC\_FHM\_sz\_c\]). The result is \[see Eq. (\[Gamma\_def\])\] $$\label{R2F2}
\begin{array}{l}{{\displaystyle}R_2=\left(\Gamma-2\mu_h\right)R_1,}\\{}\\{{\displaystyle}F_2=\frac{iR_1\tan\theta\left[p\cosh(qd_f)+q\sinh(qd_f)\right]-p\Delta/\omega_n}{q\cosh(qd_f)+p\sinh(qd_f)}.}
\end{array}$$ Finally, substituting (\[R2F2\]) into (\[BC\_FHM\_x\]) we obtain: $$\label{R1}
R_1=\frac{\Delta}{\omega_n}\frac{\mu_sW\sin\theta\cos\theta}{Q\sin^2\theta +\Gamma+\left(\Gamma-2\mu_h\right)\cos^2\theta},$$ where $$\label{W_def}
W=\Im\left\{\frac{q}{q\cosh(qd_f)+p\sinh(qd_f)}\right\},$$ $$\label{Q_def}
Q=\Re \left\{\frac{q}{q_f}\frac{p+q\tanh(qd_f)}{q+p\tanh(qd_f)}\right\}.$$ The obtained explicit expression for $R_1$ enables straightforward calculation of all other amplitudes in the anomalous Green function. In particular, $$\label{Fs_calc}
F_S=\frac{q\left[iR_1\tan\theta-(\Delta/\omega_n)\cosh(qd_f)\right]}{q\cosh(qd_f)+p\sinh(qd_f)}.$$
Calculation of the current-phase relation for the S/F/HM/F/S junction {#App_CPR}
=====================================================================
Let us denote the coordinates of the left and right boundaries of the HM-layer as $x_L=-d_h/2$ and $x_R=d_h/2$. It is convenient to represent the solution of the Usadel equation in the ${\rm F_1}$-layer in the form $$\label{JJ_F1_Gen_Sol}
\begin{array}{l}{{\displaystyle}f_s+ f_{tx}=A_1^+\sinh\left[q\left(x-x_L\right)\right]+ B_1\cosh\left[q\left(x-x_L\right)\right],}\\{}\\{{\displaystyle}f_s- f_{tx}=A_1^-\sinh\left[q^*\left(x-x_L\right)\right]- B_1\cosh\left[q^*\left(x-x_L\right)\right],}\\{}\\{{\displaystyle}f_{ty}= iB_1\frac{\sinh\left[q_f\left(x-x_L+d_f\right)\right]}{\sinh(q_fd_f)},}\\{}\\{{\displaystyle}f_{tz}=0,}
\end{array}$$ where $q_f=\sqrt{2\omega_n/D_f}$ and $q=\sqrt{2(\omega_n+ ih)/D_f}$ (we assume that the diffusion constants and normal conductivities in the F-layers are equal to each other). In Eq. (\[JJ\_F1\_Gen\_Sol\]) we took into account that $f_s=f_{\downarrow\downarrow}=0$ at $x=x_L$ and $f_{ty}=0$ at $x=x_L-d_f$. Analogously, the solution in the ${\rm F_2}$-layer can be written as $$\label{JJ_F2_Gen_Sol}
\begin{array}{l}{{\displaystyle}f_s+ f_{ty}=A_2^+\sinh\left[q\left(x-x_R\right)\right]+B_2\cosh\left[q\left(x-x_R\right)\right],}\\{}\\{{\displaystyle}f_s- f_{ty}=A_2^-\sinh\left[q^*\left(x-x_R\right)\right]-B_2\cosh\left[q^*\left(x-x_R\right)\right],}\\{}\\{{\displaystyle}f_{tx}= iB_2\frac{\sinh\left[q_f\left(x-x_R-d_f\right)\right]}{\sinh(q_fd_f)},}\\{}\\{{\displaystyle}f_{tz}=0.}
\end{array}$$
Inside the HM-layer the solution for the only non-zero component $f_{\uparrow\uparrow}=-f_{tx}+if_{ty}$ has the form $$\label{JJ_HM_Gen_Sol}
f_{\uparrow\uparrow}=P_1\sinh(q_hx)+P_2\cosh(q_hx),$$ with $q_h=\sqrt{2\omega_n/D_h}$. Taking derivation of Eq. (\[JJ\_HM\_Gen\_Sol\]) and excluding the constants $P_1$ and $P_2$ we obtain two equations which connect the values of $f_{\uparrow\uparrow}$ and $\partial_xf_{\uparrow\uparrow}$ on the left and right sides of the half-metal (the corresponding values are indicated by the upper indexes $L$ and $R$): $$\label{HM_LR}
\begin{array}{l}{{\displaystyle}q_hf_{\uparrow\uparrow}^R=\partial_xf_{\uparrow\uparrow}^L\sinh(q_hd_h)+q_hf_{\uparrow\uparrow}^L\cosh(q_hd_h),}\\{}\\{{\displaystyle}\partial_xf_{\uparrow\uparrow}^R=\partial_xf_{\uparrow\uparrow}^L\cosh(q_hd_h)+q_hf_{\uparrow\uparrow}^L\sinh(q_hd_h).}
\end{array}$$ Further for simplicity we will assume that $d_h\ll\sqrt{D_h/4\pi T_{c0}}$. Then the system (\[HM\_LR\]) transforms into $f_{\uparrow\uparrow}^R=f_{\uparrow\uparrow}^L$ and $\partial_xf_{\uparrow\uparrow}^R=\partial_xf_{\uparrow\uparrow}^L$. Taking this into account and substituting Eqs. (\[JJ\_F1\_Gen\_Sol\])-(\[JJ\_F2\_Gen\_Sol\]) into the boundary conditions at the S/F and F/HM interfaces we obtain: $$\label{JJ_E1}
-A_1^+\sinh(qd_f)+ B_1\cosh(qd_f) = \frac{\Delta }{\omega_n}e^{-i\varphi/2},$$ $$-A_1^-\sinh(q^*d_f)- B_1\cosh(q^*d_f) = \frac{\Delta }{\omega_n}e^{-i\varphi/2},$$ $$A_2^+\sinh(qd_f)+B_2\cosh(qd_f)=\frac{\Delta }{\omega_n}e^{i\varphi/2},$$ $$A_2^-\sinh(q^*d_f)-B_2\cosh(q^*d_f)=\frac{\Delta }{\omega_n}e^{i\varphi/2},$$ $$-B_1=iB_2,$$ $$-B_1=iB_2$$ $$\label{JJ_E2}
\begin{array}{c}
{{\displaystyle}-qA_1^++q^*A_1^--2B_1q_f\coth(q_fd_f)=~~~~~}\\{}\\{{\displaystyle}~~~~~~~-2iB_2q_f\coth(q_fd_f)+iqA_2^+-iq^*A_2^-.}
\end{array}$$
To calculate the Josephson current through the junction we need to find only the combination $\Im\left(f_s^*\partial_x f_s\right)$ at $x=-d_f-d_h/2$. Solving the system of equations (\[JJ\_E1\])-(\[JJ\_E2\]) we find: $$\label{B1_res}
B_1=\frac{1}{2}\frac{ie^{-i\varphi/2}+e^{i\varphi/2}}{q_f\coth(q_fd_f)+ \Re\left[q\coth(qd_f)\right]}\Im\left[\frac{q}{\sinh(qd_f)}\right]$$ and, as a consequence, $$\label{Der_res}
\Im\left(f_s^*\partial_x f_s\right)=\frac{1}{2}\frac{\sin\left(\varphi+\pi/2\right)\Im^2\left[\frac{q}{\sinh(qd_f)}\right]}{q_f\coth(q_fd_f)+ \Re\left[q\coth(qd_f)\right]}$$ Substituting this expression into Eq. (\[Curr\_def\]) we obtain the desired current-phase relation.
Calculation of the Green function for the S/F/HM system of atomic thickness {#App_Gorkov}
===========================================================================
Let us introduce the following Green functions in the imaginary time representation: $$\label{Green_def}
\begin{array}{c}{{\displaystyle}G_{\alpha,\beta}=-\left<T_\tau\left(\varphi_\alpha,\varphi_\beta^+\right)\right>, F_{\alpha,\beta}^+=\left<T_\tau\left(\varphi_\alpha^+,\varphi_\beta^+\right)\right>,}\\{}\\
{{\displaystyle}E_{\alpha,\beta}^{\psi}=-\left<T_\tau\left(\psi_\alpha,\varphi_\beta^+\right)\right>, F_{\alpha,\beta}^{\psi+}=\left<T_\tau\left(\psi_\alpha^+,\varphi_\beta^+\right)\right>,}\\{}\\
{{\displaystyle}E_{\alpha,\beta}^{\eta}=-\left<T_\tau\left(\eta_\alpha,\varphi_\beta^+\right)\right>, F_{\alpha,\beta}^{\eta+}=\left<T_\tau\left(\eta_\alpha^+,\varphi_\beta^+\right)\right>,}
\end{array}$$ Then performing the Fourier transform we obtain the following system of the matrix Gor’kov equations: $$\label{Gorkov1}
\left(i\omega-\xi\right)\hat{G}-t\hat{E}^\psi+\Delta\hat{I}\hat{F}^+=\hat{1},$$ $$\label{Gorkov2}
\left(i\omega+\xi\right)\hat{F}^++t\hat{F}^{\psi+}-\Delta^*\hat{I}\hat{G}=0,$$ $$\label{Gorkov3}
\left(i\omega-\hat{C}\right)\hat{E}^\psi-t\hat{G}-t\hat{E}^\eta=0,$$ $$\label{Gorkov4}
\left(i\omega+\hat{C}\right)\hat{F}^{\psi+}+t\hat{F}^++t\hat{F}^{\eta+}=0,$$ $$\label{Gorkov5}
\left(i\omega-\hat{P}\right)\hat{E}^\eta-t\hat{E}^\psi=0,$$ $$\label{Gorkov6}
\left(i\omega+\hat{P}\right)\hat{F}^{\eta+}+t\hat{F}^{\psi+}=0,$$ where $\hat{I}=i\hat{\sigma}_y$, and $\hat{1}$ is the unit matrix in the spin space. Solving the system (\[Gorkov1\])-(\[Gorkov6\]) and considering only the linear term in the expansion of the function $\hat{F}^+$ over $\left|\Delta\right|$ we obtain the expression (\[Gorkov\_F\_res\]).
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LA-UR-18-24011\
TUM-HEP-1137/18\
1804.06358\
\
**[ Next-to-Next-to-Leading Order $N$-Jettiness Soft Function\
for $tW$ Production]{}**
<span style="font-variant:small-caps;">Hai Tao Li$^{a}$, Jian Wang$^b$</span>\
*$^a$Los Alamos National Laboratory, Theoretical Division, Los Alamos, NM 87545, USA*\
*$^b$Physik Department T31, Technische Universität München, James-Franck-Straße 1, D–85748 Garching, Germany*\
Introduction
============
Precise calculation of cross sections for the processes at the Large Hadron Collider (LHC) or future high-energy hadron colliders is crucial for testing the Standard Model (SM) and for searching for new physics. In the last a few years, there is a burst of fully differential next-to-next-to-leading order (NNLO) results for a large number of processes in the SM; see a recent review in ref. [@Heinrich:2017una]. One of the main difficulties in the higher-order QCD calculations is to develop a systematical method to deal with the infrared singularities caused by double real emissions. The $N$-jettiness subtraction [@Boughezal:2015dva; @Gaunt:2015pea] has proven to be successful in computing the NNLO differential cross sections of processes with jets, for example, $W/Z/H/\gamma+j$ [@Boughezal:2015dva; @Boughezal:2015aha; @Boughezal:2015ded; @Campbell:2016lzl]. This subtraction method is based on the soft-collinear effective theory (SCET) [@Bauer:2000ew; @Bauer:2000yr; @Bauer:2001ct; @Bauer:2001yt; @Beneke:2002ph], which is an effective theory of QCD in the infrared regions. The $N$-jettiness $\mathcal{T}_N$ is an observable, proposed in [@Stewart:2010tn], to describe the event shape of jet processes or processes with initial-state hadrons, a generalization of thrust at lepton colliders and beam thrust at hadron colliders [@Stewart:2009yx]. The application of this observable to the NNLO calculations has been explored extensively for massive quark decay [@Gao:2012ja], and differential cross sections of processes at both hadron colliders [@Boughezal:2015dva; @Gaunt:2015pea; @Boughezal:2015aha; @Boughezal:2015ded; @Berger:2016oht; @Campbell:2016lzl; @Heinrich:2017bvg] and electron-hadron colliders [@Berger:2016inr; @Abelof:2016pby]. It is also used as a jet resolution variable in combining higher-order resummation with NLO calculations and parton showers [@Alioli:2012fc]. However, for more complicated processes, e.g., involving one massive and two massless partons, the results are still missing.
The $N$-jettiness event shape variable is defined by [@Stewart:2010tn] $$\begin{aligned}
\label{eq:tau}
{\mathcal{T}_N}= \sum_k \min_i\left\{n_i \cdot q_k \right\}~,\end{aligned}$$ where $n_i$ $(i=a,b,1,...,N)$ are light-like reference vectors representing the moving directions of massless external particles, and $q_k$ denotes the momentum of soft or collinear partons. Note that eq.(\[eq:tau\]) seems different from the original definition in ref.[@Stewart:2010tn] because the variable ${\mathcal{T}_N}$ in our definition is of mass dimension one while that in ref.[@Stewart:2010tn] is dimensionless. But they are actually the same up to a constant factor $Q$ after replacing $n_i$ by $2q_i/Q$. In the infrared divergent regions, the observable $\mathcal{T}_N \to 0$, and the cross section is approximated by [@Stewart:2009yx; @Stewart:2010tn] $$\begin{aligned}
\label{eq:sigma}
\frac{d\sigma}{d{\mathcal{T}_N}} \propto \int H\otimes B_1 \otimes B_2 \otimes S \otimes \bigg(\prod_{n=1}^{N}J_n \bigg)~.\end{aligned}$$ Here the hard function $H$ encodes all the information about hard scattering. The beam functions $B_i, ~(i=1,2),$ describe the perturbative and non-perturbative contributions from initial state, and have been obtained up to NNLO [@Stewart:2010qs; @Berger:2010xi; @Gaunt:2014xga; @Gaunt:2014cfa]. The jet function $J_n$ describes the final-state jet with a fixed invariant mass and has been calculated at NNLO [@Becher:2006qw; @Becher:2010pd]. The soft function $S$ contains soft interactions between all colored particles. It has been studied up to NNLO for massless processes [@Jouttenus:2011wh; @Kelley:2011ng; @Monni:2011gb; @Boughezal:2015eha; @Kang:2015moa; @Campbell:2017hsw].
The differential cross section for any observable ${\mathcal{O}}$ is given by $$\begin{aligned}
\label{eq:sigmatot}
\frac{d\sigma}{d{\mathcal{O}}}= \frac{d\sigma}{d{\mathcal{O}}}\Big|_{{\mathcal{T}_N}<\Delta} + \frac{d\sigma}{d{\mathcal{O}}}\Big|_{{\mathcal{T}_N}>\Delta} ~,\end{aligned}$$ where a small cut-off parameter $\Delta$ on the right-hand side is imposed. For the NNLO calculations the first term on the right-hand side at the leading power can be obtained by expanding eq. (\[eq:sigma\]) to the second order of the strong coupling $\alpha_s$. The second term, due to the phase-space constraint, can be dealt with the standard NLO subtraction method for the process with an extra parton in the final state.
The extension of the $N$-jettiness subtraction to more complicated processes requires the calculation of the corresponding soft and hard functions. We have calculated the $N$-jettiness soft function for one massive colored particle production up to NNLO in ref. [@Li:2016tvb], where we assume that it is produced at rest. In this paper, we present the result for more general situations, i.e., the massive colored particle can carry any possible momentum. Our result can be used to construct the $N$-jettiness subtraction terms for $tW$ production at hadron colliders.
This paper is organized as follows. In section \[sec:factor\], we briefly introduce the definition of the soft function in terms of soft Wilson lines. In section \[sec:Renor\], we study the renormalization group (RG) equation of the soft function and thus derive the structure of the soft function. We provide the details of the techniques in our calculations in section \[sec:tech\]. Then, in section \[sec:results\], we present the numerical results of the NLO and NNLO soft functions and compare the divergent terms with the predictions from RG equation. We conclude in section \[sec:conc\].
\[sec:factor\]Definition of the soft function
=============================================
In this section we first discuss the kinematics and the factorization of the cross section for $tW$ production. Then we present the definition of soft function.
We consider the process $$\begin{aligned}
P_1 + P_2 \to t/\bar{t} + W^\pm + X~,\end{aligned}$$ where $P_1$ and $P_2$ denote incoming hadrons, $t/\bar{t}$ and $W^\pm$ represent the top/anti-top quark and the $W$-boson in the final state, respectively. And $X$ includes any unobserved final state. The partonic process at leading order (LO) for $tW^-$ production is $$\begin{aligned}
\label{eq:process}
b(p_1) + g(p_2) \to t(p_3)+W^-(p_4)~.\end{aligned}$$ It is convenient to introduce two light-like vectors $$\begin{aligned}
\label{eq:nnb}
n^\mu&=(1,0,0,1), \qquad \bar{n}^\mu=(1,0,0,-1)~.\end{aligned}$$ Any momentum can be decomposed as $p^{\mu}=(p^+,p^-,p_{\perp})$ with $p^+= p\cdot n, p^- =p \cdot \bar{n}$. The momenta given in eq. (\[eq:process\]) can be written in the partonic center-of-mass frame as $$\begin{aligned}
p_1^{\mu} = \frac{\sqrt{\hat{s}}}{2}n^{\mu}~,\quad p_2^{\mu} = \frac{\sqrt{\hat{s}}}{2} \bar{n}^{\mu}, \quad
p_3^\mu =m_t v^\mu~,\end{aligned}$$ where $v^2=1$. Specifically, we parameterize $v$ by two variables, i.e., $\beta_t$ and $\theta_t$, which measure the magnitude and the direction of the velocity, $$\begin{aligned}
v^+ = \frac{1-\beta_t \cos \theta_t}{\sqrt{1-\beta_t^2}},\quad
v^- = \frac{1+\beta_t \cos \theta_t}{\sqrt{1-\beta_t^2}},\quad
|v_{\perp}|= \frac{\beta_t \sin \theta_t}{\sqrt{1-\beta_t^2}},\end{aligned}$$ where $\beta_t=\sqrt{1-m_t^2/E_t^2}$ with $E_t$ the top quark energy. The 0-jettiness event shape variable in this process is defined as $$\begin{aligned}
\label{eq:tau-def}
\mathcal{\tau} \equiv {\mathcal{T}_{0}}= & \sum_{k} \min\{n\cdot q_k, \bar{n}\cdot q_k \}~.\end{aligned}$$ Since $n\cdot q_k\equiv 2p_1\cdot q_k/\sqrt{\hat{s}}$ and $\bar{n}\cdot q_k\equiv 2p_2\cdot q_k/\sqrt{\hat{s}}$, this definition of $\mathcal{\tau}$ is Lorentz invariant. The explicit choice in eq.(\[eq:nnb\]) just makes our calculation easier, but the final result is general and independent of this choice.
In the limit $\tau \ll \sqrt{\hat{s}}$, the final state contains no hard radiations, only soft and collinear radiations allowed. In this limit the cross section admits a factorised form, which can be derived in the framework of SCET. For $tW$ production, the collinear singularities, which are only associated with the initial partons, and the soft singularities are all properly regularised by $\tau$ defined in eq. (\[eq:tau-def\]). Compared with processes without massive colored particles, the only difference is the soft function and the hard function. Following [@Stewart:2009yx; @Stewart:2010tn], we write $$\begin{gathered}
\label{eq:fact}
\frac{d\sigma}{dY d\tau} = \int d\Phi_2\frac{d\hat{\sigma}_0}{d\Phi_2} \int dt_a dt_b d\tau_s H(\beta_t,\cos\theta_t, \mu) B_1(t_a, x_a, \mu) B_2(t_b, x_b, \mu)
\\
\times S(\tau_s, \beta_t,\cos\theta_t,\mu)\delta\left(\tau-\tau_s - \frac{t_a+t_b}{\sqrt{\hat{s}}}\right)\left(1+\mathcal{O}\left(\frac{\tau}{\sqrt{\hat{s}}}\right)\right)~,\end{gathered}$$ where $\int d\Phi_2$ is the two-body phase space integral, $d\hat{\sigma}_0$ is the LO partonic differential cross section, $Y$ is the rapidity of the partonic colliding system in the laboratory frame, the momentum fractions $x_a=\sqrt{\hat{s}/s} e^{Y}$ and $x_b=\sqrt{\hat{s}/s}e^{-Y}$ with $\sqrt{s}$ the collider energy, and $\mu$ is the renormalization scale. In momentum space the soft function is defined as the vacuum matrix element $$\begin{gathered}
\label{eq:soft}
S(\tau,\beta_t,\cos\theta_t, \mu)=\sum_{X_s}
\Big\langle 0 \Big|\mathbf{\bar{T}} Y_n^{\dagger}Y_{\bar{n}}Y_{v} \Big| X_s \Big\rangle
\delta\bigg(\tau-\sum_{k} \min\left(n\cdot \hat{P}_k, \bar{n}\cdot \hat{P}_k\right)\bigg)
\Big\langle X_s\Big| \mathbf{T} Y_nY_{\bar{n}}^{\dagger}Y_{v}^{\dagger} \Big| 0 \Big\rangle,\end{gathered}$$ where $\mathbf{T}(\mathbf{\bar{T}})$ is the (anti-)time-ordering operator. And $Y_n$, $Y_{\bar{n}}$ and $Y_{v}$ are the soft Wilson lines defined explicitly as [@Bauer:2001yt; @Chay:2004zn; @Korchemsky:1991zp] $$\begin{aligned}
Y_n(x) & = \mathbf{P} \exp\left( ig_s\int^0_{-\infty}ds\, n\cdot A^a_s(x+sn)\mathbf{T}^a\right) ,\\
Y^{\dagger}_{\bar{n}}(x) & = \mathbf{\bar{P}} \exp\left( -ig_s\int^0_{-\infty}ds\, \bar{n}\cdot A^a_s(x+s \bar{n})\mathbf{T}^a\right), \\
Y^{\dagger}_v(x) & = \mathbf{P} \exp\left( ig_s\int_0^{\infty}ds\, v\cdot A^a_s(x+sv)\mathbf{T}^a\right)\end{aligned}$$ where $\mathbf{P}$ and $\mathbf{\bar{P}}$ are the path-ordering and the anti-path-ordering operators. $\hat{P}_k$ in eq.(\[eq:soft\]) is the operator extracting the momentum of each soft emission. The purpose of this paper is to calculate the soft function defined above for $tW$ production up to NNLO accuracy.
Renormalization {#sec:Renor}
===============
In SCET the bare soft function in eq.(\[eq:soft\]) contains ultra-violet divergences in perturbative calculations, which are cancelled by the counterterm defined in the standard renormalization procedure. The renormalized soft function is finite and can be used in the calculation of the cross section in eq. (\[eq:fact\]). The renormalization introduces the scale $\mu$ dependence in the soft function, as well as in the hard and beam function. Because of the fact that the physical cross section does not dependent on the intermediate scale, the RG equation of the soft function can be derived from the RG equations of the hard and beam function, which will be used to extract the anomalous dimension of the soft function. Given the anomalous dimension the divergences in the bare soft function, as well as the scale dependence of the renormalized soft function, can be predicted. In this section we briefly discuss the renormalization of the soft function and the expression of the soft anomalous dimension. We work in $d=4-2\epsilon$ dimensional space-time.
Based on dimensional analysis, the bare soft function, in perturbation theory, can be written as $$\begin{aligned}
\label{eq:baresoft}
S(\tau,\beta_t,\cos\theta_t, \mu) = \delta(\tau) + \frac{1}{\tau} \sum_{n=1}^{\infty} \left(\frac{Z_{\alpha_s} \alpha_s}{4\pi}\right)^n \left(\frac{\tau}{\mu}\right)^{-2n\epsilon} s^{(n)}(\beta_t,\cos\theta_t)~,\end{aligned}$$ where we use renormalized strong coupling $\alpha_s$ and its renormalization factor $Z_{\alpha_s}=1-\beta_0 \alpha_s/(4\pi \epsilon)+\mathcal{O}(\alpha_s^2)$. The soft function after the Laplace transformation can be written as $$\begin{aligned}
\label{eq:lapsoft}
\tilde{S}(L,\beta_t,\cos\theta_t,\mu) &= \int_{0}^{\infty} d\tau \exp\left( -\frac{\tau}{e^{\gamma_E}\mu e^{L/2}} \right) S(\tau,\beta_t,\cos\theta_t,\mu)
\nonumber \\
&= 1+\sum_{n=1}^{\infty} \left(\frac{Z_{\alpha_s}\alpha_s}{4\pi} \right)^n e^{-n(L+2\gamma_E)\epsilon}\Gamma(-2n\epsilon) s^{(n)}(\beta_t,\cos\theta_t)~.\end{aligned}$$ Then the corresponding renormalized soft function $\tilde{s} $ is defined as $$\begin{aligned}
\label{eq:soft_ren}
\tilde{s}(L,\beta_t,\cos\theta_t,\mu) = Z_s^{-1}(L,\beta_t,\cos\theta_t,\mu) \tilde{S}(L,\beta_t,\cos\theta_t,\mu)~,\end{aligned}$$ where the renormalization factor $Z_s$ satisfies the differential equation $$\begin{aligned}
\frac{ d\ln Z_s(L,\beta_t,\cos\theta_t,\mu)} {d\ln\mu} = -\gamma_s(L,\beta_t,\cos\theta_t,\mu)\end{aligned}$$ with $\gamma_s$ the anomalous dimension of the soft function. We will suppress the arguments of the renormalization factor, anomalous dimension and the soft function in the following text for convenience.
Given the soft anomalous dimension $\gamma_s$, following refs. [@Becher:2009cu; @Becher:2009qa], the closed expression for $Z_s$ is derived and can be written as $$\begin{aligned}
\ln Z_s &= \frac{\alpha_s}{4\pi}\left(\frac{\gamma_s^{(0)\prime}}{4\epsilon^2}+ \frac{\gamma_s^{(0)}}{2\epsilon} \right)
+ \left(\frac{\alpha_s}{4\pi}\right)^2 \left(-\frac{3 \beta_0 \gamma_s^{(0)\prime} }{16\epsilon^3} + \frac{\gamma_s^{(1)\prime} -4 \beta_0 \gamma_s^{(0)}}{16\epsilon^2}+ \frac{\gamma_s^{(1)}}{4\epsilon} \right)
+\mathcal{O}(\alpha_s^3).\end{aligned}$$ The expansion series and derivative of the soft anomalous dimension are given by $$\begin{aligned}
\gamma_s = \sum_{i=0} \left( \frac{\alpha_s}{4\pi}\right)^{i+1} \gamma_s^{(i)}
\qquad \text{and} \qquad \gamma_s^{(i)\prime} = \frac{d \gamma_s^{(i)}}{d\ln\mu}~.\end{aligned}$$ From eq. (\[eq:soft\_ren\]), we obtain the renormalized NLO and NNLO soft functions in Laplace space $$\begin{aligned}
\label{eq:s_div}
\tilde{s}^{(1)} =& \tilde{S}^{(1)}-Z_s^{(1)} ~,
\nonumber \\
\tilde{s}^{(2)} =& \tilde{S}^{(2)}-Z_s^{(2)} - \tilde{S}^{(1)} Z_s^{(1)}+ Z_s^{(1)2} -\frac{\beta_0}{\epsilon} \tilde{S}^{(1)}~.\end{aligned}$$ Since the renormalized soft function is finite, the divergent terms in the bare soft function $\tilde{S}$ is related to the renormalization factor $Z_s$ and can be derived from the above equations.
As discussed before, the soft anomalous dimension $\gamma_s$ can be derived from the independence of the cross section on the renormalization scale $\mu$, $$\begin{aligned}
\label{eq:s_rg}
\frac{d\ln \tilde{s}} {d\ln \mu} =\gamma_s = -\frac{d\ln H}{d\ln \mu} - \frac{d\ln \tilde{B}_1}{d\ln \mu} - \frac{d\ln\tilde{B}_2}{d\ln \mu}~,\end{aligned}$$ where $ \tilde{B}_i$ is the beam function in Laplace space, of which the NLO and NNLO results can be found in refs. [@Stewart:2010qs; @Berger:2010xi; @Gaunt:2014xga; @Gaunt:2014cfa]. And the RG equation of the beam function is exactly the same as the evolution equation of the jet function to all orders [@Stewart:2010qs], $$\begin{aligned}
\label{eq:beam}
\frac{d\tilde{B}_i}{d\ln\mu} = \left(- \mathbf{T}_i\cdot \mathbf{T}_i {\gamma_{\rm cusp}}\Big( \ln \frac{s_{12}}{\mu^2} +L\Big)+ \gamma_B^i\right) \tilde{B}_i~,\end{aligned}$$ where $\mathbf{T}_i$ is the color generator associated with the $i$-th parton [@Catani:1996jh; @Catani:1996vz] and the anomalous dimension $\gamma_B^i$ can be found in refs. [@Gaunt:2014xga; @Gaunt:2014cfa].
The RG equation for the hard function can be obtained from refs. [@Ferroglia:2009ep; @Ferroglia:2009ii] where the two-loop divergences have been calculated for massive scattering amplitudes in non-abelian gauge theories. It is straightforward to organize the RG equation for the hard function as $$\begin{aligned}
\label{eq:hard}
\frac{d\ln H}{d\ln\mu} =& (\mathbf{T}_1\cdot \mathbf{T}_1+\mathbf{T}_2\cdot \mathbf{T}_2) \ln \frac{s_{12}}{\mu^2}
- \mathbf{T}_1\cdot \mathbf{T}_3 {\gamma_{\rm cusp}}\ln \frac{s_{13}^2}{s_{12} m_t^2}
- \mathbf{T}_2\cdot \mathbf{T}_3 {\gamma_{\rm cusp}}\ln \frac{s_{23}^2}{s_{12} m_t^2}
{\nonumber}\\
& +2\gamma^1 + 2\gamma^2 + 2\gamma^Q~\end{aligned}$$ with $s_{12}=2p_1\cdot p_2 + i0,s_{13}=-2p_1\cdot p_3 + i0,s_{23}=-2p_2\cdot p_3 + i0$. The anomalous dimensions $\gamma^{1,2}$ and $\gamma^Q$, associated with the initial- and final-state particles, can be found in refs. [@Ferroglia:2009ep; @Ferroglia:2009ii] and references therein.
Inserting eqs. (\[eq:beam\]-\[eq:hard\]) to eq.(\[eq:s\_rg\]), the anomalous dimension of the soft function is obtained $$\begin{aligned}
\label{eq:gammas}
\gamma_s =& (\mathbf{T}_1\cdot \mathbf{T}_1 + \mathbf{T}_2\cdot \mathbf{T}_2){\gamma_{\rm cusp}}L
+ \mathbf{T}_1\cdot \mathbf{T}_3 {\gamma_{\rm cusp}}\ln \frac{s_{13}^2}{s_{12} m_t^2}
+ \mathbf{T}_2\cdot \mathbf{T}_3 {\gamma_{\rm cusp}}\ln \frac{s_{23}^2}{s_{12} m_t^2}{\nonumber}\\&
- 2 \gamma^Q
- \gamma_B^1-\gamma_B^2 - 2\gamma^1-2\gamma^2~,\end{aligned}$$ of which each ingredient is available up to NNLO.
Techniques in calculation {#sec:tech}
=========================
In the calculation of the NLO and NNLO soft function, we have to deal with one and two soft radiations, respectively. The phase space integration is $$\begin{aligned}
\int \frac{d^d q}{(2\pi)^d}\delta^+(q^2)&=\frac{1}{(2\pi)^d}\frac{\Omega_{d-3}}{4}\int dq^+ dq^- (q^+ q^-)^{-\epsilon}\int_0^{\pi} d\phi_q \sin^{-2{\epsilon}}\phi_q \\
\int \frac{d^dq_1d^dq_2}{(2\pi)^{2d}} \delta^+(q_1^2)\delta^+(q_2^2) &=\frac{1}{(2\pi)^{2d}}\frac{\Omega_{d-3}^2}{16}
\int dq_1^+ dq^-_1 (q^+_1 q^-_1)^{-\epsilon}
\int dq^+_2 dq^-_2 (q^+_2 q^-_2)^{-\epsilon} {\nonumber}\\ & \quad
d\phi_{1} \sin^{-2{\epsilon}}\phi_{1} d\phi_{2} \sin^{-2{\epsilon}}\phi_{2} \label{eq:phase2a} \end{aligned}$$ with $$\begin{aligned}
\Omega_{d-3} &= \frac{2\pi^{{\frac{1}{2}}-{\epsilon}}}{\Gamma({\frac{1}{2}}-{\epsilon})} =
2-3 \zeta(2) \epsilon ^2-\frac{14 \zeta (3) \epsilon ^3}{3}-\frac{15}{8} \zeta(4) \epsilon ^4+O\left(\epsilon ^5\right) .\end{aligned}$$ The $\phi$ angle is measured in the frame with the top quark $\phi_t=0$. More explicitly, we choose $$\begin{aligned}
p_{3\perp}&=|p_{3\perp}|(0; 0, 1) , {\nonumber}\\
q_{1\perp}&=|q_{1\perp}|(0; \sin\phi_1, \cos\phi_1) , {\nonumber}\\
q_{2\perp}&=|q_{2\perp}|( \sin\phi_2 \sin\beta\hat{n}_{{\epsilon}}; \sin\phi_2 \cos\beta, \cos\phi_2 ).\end{aligned}$$
For the integrand involving $1/(q_1\cdot q_2)$, the phase space integration is parameterize as $$\begin{aligned}
\int \frac{d^dq_1d^dq_2}{(2\pi)^{2d}} \delta^+(q_1^2)\delta^+(q_2^2) &=\frac{1}{(2\pi)^{2d}}\frac{\Omega_{d-3}\Omega_{d-4}}{16}
\int dq_1^+ dq^-_1 (q^+_1 q^-_1)^{-\epsilon}
\int dq^+_2 dq^-_2 (q^+_2 q^-_2)^{-\epsilon} {\nonumber}\\ & \quad
d\phi_{1} \sin^{-2{\epsilon}}\phi_{1} d\phi_{12} \sin^{-2{\epsilon}}\phi_{12} d\beta_{12} \sin^{-1-2{\epsilon}}\beta_{12}
\label{eq:phase2b}\end{aligned}$$ with $$\begin{aligned}
\Omega_{d-4} &= \frac{2\pi^{-{\epsilon}}}{\Gamma(-{\epsilon})} = \frac{-2{\epsilon}\pi^{-{\epsilon}} }{\Gamma(1-{\epsilon})}.\end{aligned}$$
At NLO, the measurement function is defined as $$\begin{aligned}
F(n,\bar{n},q) & =
\delta(q^+-\tau)\Theta(q^--q^+)
+ \delta(q^--\tau)\Theta(q^+-q^-)~,\end{aligned}$$ where $q^+=q\cdot n$ and $q^-=q\cdot \bar{n}$. At NNLO, the measurement function is defined as $$\begin{aligned}
\label{eq:measure}
F(n,\bar{n},q_1,q_2) &= \delta(q_1^++q_2^+-\tau)\Theta(q^-_1-q_1^+)\Theta(q_2^--q_2^+)
\nonumber \\ &
+\delta(q_1^++q_2^--\tau)\Theta(q^-_1-q_1^+)\Theta(q_2^+-q_2^-)
\nonumber \\ &
+\delta(q_1^-+q_2^--\tau)\Theta(q^+_1-q_1^-)\Theta(q_2^+-q_2^-)
\nonumber \\ &
+\delta(q_1^-+q_2^+-\tau)\Theta(q^+_1-q_1^-)\Theta(q_2^--q_2^+).\end{aligned}$$ One can see that at NNLO the whole phase space is partitioned to four pieces. We label them as Region-I, Region-II, Region-III and Region-IV, respectively.
In the hemisphere with $q_i^+=\tau_i$, we parameterize $q_i^- =\tau_i/t_i$ with $t_i \in (0,1)$ and $$\begin{aligned}
dq_i^- = dt_i \frac{\tau_i}{t_i^2},\quad (q_i^+ q_i ^-)^{-{\epsilon}} = (\tau_i^2/t_i)^{-{\epsilon}} .\end{aligned}$$ And then all those singularities at NLO will appear as $\tau_i^{-1-2{\epsilon}}$ and $t_i^{-1+{\epsilon}}$. In the end, we define $$\begin{aligned}
\tau_1 = \tau v, \quad \tau_2 =\tau\bar{v},\end{aligned}$$ with $\bar{v}\equiv 1-v$. We have $$\begin{aligned}
\int d\tau_1 d\tau_2 \delta(\tau-\tau_1-\tau_2) =\tau \int_0^1 dv.\end{aligned}$$ If the integrands do not involve $1/(q_1\cdot q_2)$, we perform the phase space integration straightforward after the parameterization. For the integrands involving $1/(q_1\cdot q_2)$, we have $$\begin{aligned}
q_1\cdot q_2
&= {\frac{1}{2}}\frac{ \tau_1 \tau_2} {t_1 t_2 }\left[ t_2 + t_1 -2 \sqrt{t_1 t_2} \cos \phi_{12} \right] ,\end{aligned}$$ in the Region-I or Region-III, and $$\begin{aligned}
q_1\cdot q_2
&= {\frac{1}{2}}\frac{ \tau_1 \tau_2} {t_1 t_2 }\left[ 1 + t_1 t_2 -2 \sqrt{t_1 t_2} \cos \phi_{12} \right] ,\end{aligned}$$ in the Region-II or Region-IV. Here $\phi_{12}$ is the angle between $q_{1\perp}$ and $q_{2\perp}$.
In the Region-I and Region-III, the double-real corrections contain a new kind of singularities that appear when $t_1 =t_2$ and $\phi_{12}=0$. Following the method in ref. [@Campbell:2017hsw], we change the integration variables from ${\phi_2, \beta}$ to ${\phi_{12},\beta_{12}}$. All dependence on $\phi_2$ (such as $q_{2\perp}\cdot p_{3\perp}$) can be expressed in terms of $\phi_{12} $ and $\beta_{12}$, $$\begin{aligned}
\cos\phi_2 = \cos\phi_1 \cos \phi_{12} - \sin \phi_1 \sin \phi_{12} \cos \beta_{12}.\end{aligned}$$ The $\beta_{12}$ angle integration can be transformed by defining $\cos\beta_{12}=1-2x$, $$\begin{aligned}
\int_0^{\pi} d\beta_{12} (\sin\beta_{12})^{-1-2{\epsilon}}&=2^{-1-2{\epsilon}} \int_0^1 dx [x(1-x)]^{-1-{\epsilon}}.
\label{eq:betaint}\end{aligned}$$ Define $\cos\phi_{12} = 1-2z$, $$\begin{aligned}
q_1\cdot q_2 &= {\frac{1}{2}}\frac{ \tau_1 \tau_2} {t_1 t_2 }\left[ (\sqrt{t_2} -\sqrt{ t_1})^2 +4z\sqrt{t_1 t_2} \right] {\nonumber}\\
&= {\frac{1}{2}}\frac{ \tau_1 \tau_2} {t_1 t_2 }\frac{(t_2-t_1)^2}{\left[ (\sqrt{t_2} -\sqrt{ t_1})^2 +4r \sqrt{t_1 t_2} \right]}
.
\label{eq:q1q2}\end{aligned}$$ By writing in this form, we have picked out the singular part as $t_2 \to t_1$. The parameter $r$ is solved to be $$\begin{aligned}
r =\frac{(\sqrt{t_2}-\sqrt{t_1})^2 (1 -z ) }{(\sqrt{t_2} -\sqrt{t_1})^2 + 4 z \sqrt{t_1 t_2}},
$$ and the Jacobian is $$\begin{aligned}
\frac{dz}{dr} = -\frac{(t_2-t_1)^2}{\left[ (\sqrt{t_2} -\sqrt{ t_1})^2 +4r \sqrt{t_1 t_2} \right]^2}.\end{aligned}$$ The $\phi_{12}$ angular integration is given by $$\begin{aligned}
\int_0^{\pi} d\phi_{12} \sin^{-2{\epsilon}}\phi_{12} &=4^{-{\epsilon}} \int_0^1 dz [z(1-z)]^{-{\frac{1}{2}}-{\epsilon}}
\label{eq:z}\\
&=4^{-{\epsilon}} \int_0^1 dr [r(1-r)]^{-{\frac{1}{2}}-{\epsilon}}
\frac{|t_2 - t_1|^{1-2{\epsilon}}}{\left[ (\sqrt{t_2} -\sqrt{ t_1})^2 +4r \sqrt{t_1 t_2} \right]^{1-2{\epsilon}}}.\end{aligned}$$ Combined with eq.(\[eq:q1q2\]), we see that the singular part of $(q_1\cdot q_2)^{-1}\sim |t_2 - t_1|^{-1-2{\epsilon}}$ and $(q_1\cdot q_2)^{-2}\sim |t_2 - t_1|^{-3-2{\epsilon}}$. However, we find that the coefficient of $(q_1\cdot q_2)^{-2}$ is proportional to $(t_1-t_2)^2$. Now we divide the integration region of $t_1 ,t_2$ to two sectors, i.e., $$\begin{aligned}
1. \quad t_1> t_2&: \quad t_2=t_1 (1-w), \quad w\in (0,1) , {\nonumber}\\
2. \quad t_1 < t_2&: \quad t_1=t_2 (1-w), \quad w\in (0,1) .\end{aligned}$$ In each sector, $|t_2 - t_1|$ has a definite sign and thus is easy to deal with.
In the Region-II and Region-IV, one can carry out the same procedure as above except the relation between $r$ and $z$ changes to $$\begin{aligned}
r =\frac{(1-\sqrt{t_1 t_2})^2 (1 -z ) }{(1 -\sqrt{t_1 t_2})^2 + 4 z \sqrt{t_1 t_2}}.\end{aligned}$$ With this parametisation above, all the divergences can be extracted through the expansion, $$\begin{aligned}
x^{-1+n{\epsilon}} = \frac{1}{n{\epsilon}}\delta(x) + \left(\frac{1}{x}\right)_+ + n{\epsilon}\left(\frac{\ln x}{x}\right)_+
+ \cdots.\end{aligned}$$ Then all the phase space integration can be performed numerically.
Results of the soft function {#sec:results}
============================
NLO soft function
-----------------
![Numerical results for the NLO soft function and the comparison of $A_{-1}$ and $A_0$ with the RG predictions with fixed $\cos\theta_t$ (left) and $\beta_t$ (right). []{data-label="fig:nlo"}](nlo_ct.pdf "fig:") ![Numerical results for the NLO soft function and the comparison of $A_{-1}$ and $A_0$ with the RG predictions with fixed $\cos\theta_t$ (left) and $\beta_t$ (right). []{data-label="fig:nlo"}](nlo_bt.pdf "fig:")
The LO soft function is trivial and has been given explicitly in eq.(\[eq:baresoft\]). In this section, we present its NLO result. Expanding the soft Wilson lines in eq.(\[eq:soft\]) in a series of the strong coupling, we obtain the NLO soft function $$\begin{aligned}
S^{(1)}(\tau) =\frac{ 2 e^{\gamma_E \epsilon}\mu^{2\epsilon} }{\pi^{1-\epsilon}}
\int d^dq \delta(q^2) J_a^{\mu (0)\dagger}d_{\mu\nu}(q) J_a^{\nu (0)}(q)F(n,\bar{n},q)~,\end{aligned}$$ where $e^{\gamma_E \epsilon}$ is inserted because we use $\overline{\rm MS}$ renormalization scheme. The factor $J_a^{\mu (0)}(q)$ is the LO one-gluon soft current, or the eikonal current, $$\begin{aligned}
J_a^{\mu (0)}(q) = \sum_{i=1}^{3} \mathbf{T}_i^{a} \frac{p_i^{\mu}}{p_i \cdot q}\end{aligned}$$ with $a$ the color index.
After performing the phase space integration, we obtain the NLO bare soft function $$\begin{aligned}
\label{eq:snlo}
s^{(1)} =
\frac{A_{-1}}{\epsilon} + A_{0} + A_{1} \epsilon + A_{2} \epsilon^2 + A_{3} \epsilon^3 + \mathcal{O}(\epsilon^4) ,
$$ where $A_i$ is a function of $\beta_t$ and $\cos\theta_t$. Figure 1 shows the numerical results for the NLO soft function and the comparison of the divergent coefficients between the numerical calculations and RG predictions with fixed $\cos\theta_t$ or $\beta_t$. The deviations are not larger than 0.2% except for the case of $|A_i|\to 0$ . The points at $\beta_t = 0$ just reproduce our previous results in ref. [@Li:2016tvb], as expected. It can also be seen that when $\beta_t \to 1$, i.e., the top quark is highly boosted, the coefficients $A_i,i=0,1,2,3,$ become divergent. This is due to the logarithmic structures such as $\ln^n (1-\beta_t)$ in the limit of $\beta_t \to 1$. In principle, this kind of logarithms can be predicted from effective field theory for boosted top productions. Because the top quark mass is small compared with its energy in the limit, the scale hierarchy of the process is $\tau\ll m_t \ll \sqrt{\hat{s}}$, and thus a different factorization formula should be derived. We leave the detailed discussion to a future work. Notice that in eq. (\[eq:snlo\]) and fig. \[fig:nlo\] we also show $A_2$ and $A_3$ which do not contribute to the NLO result. However, they will contribute to the renormalized NNLO soft function.
NNLO soft function
------------------
The NNLO contribution consists of two parts, i.e., $$\begin{aligned}
s^{(2)} = s_{\rm VR}^{(2)}+s_{\rm DR}^{(2)}.\end{aligned}$$ The first part is the virtual-real correction, i.e., the one-loop virtual corrections to LO soft gluon current $J^{\mu(1)}_a(q)$; the second part is the double-real correction, i.e., the corrections with a double-gluon soft current $J^{\mu\nu(0)}_{ab}(q_1,q_2)$ or a massless quark-pair emission. For the virtual-real contribution we use the soft limit of one-loop QCD amplitudes which has been studied in refs. [@Bern:1998sc; @Bern:1999ry; @Catani:2000pi] and ref. [@Bierenbaum:2011gg] for massless and massive external particles. As for the double-real contribution we make use of the results in refs. [@Catani:1999ss; @Czakon:2011ve] where the infrared behaviour of tree-level QCD amplitudes at NNLO has been analyzed. The details of the virtual-real and double-real matrix element can be found in our previous paper [@Li:2016tvb].
------------ ---------- ----- --------- --------- --------- --------- --------- ---------
Num RG Num RG Num RG Num RG
$C_A^2$ -8.0000 -8 2.4972 2.4968 84.3749 84.3784 147.222 147.233
$C_F^2$ -8.0004 -8 19.6943 19.6916 26.6838 26.6908 64.0611 63.9972
$C_A C_F$ -16.0000 -16 22.1903 22.1885 107.408 107.386 194.270 194.217
$C_A n_f $ 0 0 -1.3332 -1.3333 -3.0273 -3.0283 3.2803 3.2779
$C_F n_f $ 0 0 -1.3335 -1.3333 1.0599 1.0597 -1.0928 -1.0949
Max.devi.
------------ ---------- ----- --------- --------- --------- --------- --------- ---------
: Comparison between the numerical calculations and the RG predictions of the divergent terms in different color factors with $\beta_t=0.3$ and $\cos\theta_t = 0.5 $. In the last line, we show the maximum deviation of the numerical calculations with respect to the RG predictions.[]{data-label="tab:nnloo"}
![Numerical results for NNLO bare soft function and the comparison to the RG predictions with $\beta_t =$ 0.5 (left) and 0.9 (right). The color factors are $C_A=3$ and $C_F=4/3$ and the number of flavors is $n_f=5$. []{data-label="fig:nnlo_betat"}](nnlo_betat_05.pdf "fig:") ![Numerical results for NNLO bare soft function and the comparison to the RG predictions with $\beta_t =$ 0.5 (left) and 0.9 (right). The color factors are $C_A=3$ and $C_F=4/3$ and the number of flavors is $n_f=5$. []{data-label="fig:nnlo_betat"}](nnlo_betat_09.pdf "fig:")
![Numerical results for NNLO bare soft function and the comparison to the RG predictions with $\cos\theta_t =$ -0.6 (left) and 0.6 (right). The color factors are $C_A=3$ and $C_F=4/3$ and the number of flavors is $n_f=5$. []{data-label="fig:nnlo_ct"}](nnlo_ct_m06.pdf "fig:") ![Numerical results for NNLO bare soft function and the comparison to the RG predictions with $\cos\theta_t =$ -0.6 (left) and 0.6 (right). The color factors are $C_A=3$ and $C_F=4/3$ and the number of flavors is $n_f=5$. []{data-label="fig:nnlo_ct"}](nnlo_ct_06.pdf "fig:")
With the techniques discussed in section \[sec:tech\], the double-real part is calculated numerically after sector decomposition. The bare soft function at NNLO, defined in eq. (\[eq:baresoft\]), can be written as $$\begin{aligned}
s^{(2)} = \frac{B_{-3}}{\epsilon^3}
+\frac{B_{-2}}{\epsilon^2}
+\frac{B_{-1}}{\epsilon^1}
+B_0 + B_1 \epsilon + \mathcal{O}(\epsilon^2)~.\end{aligned}$$ Using eq. (\[eq:s\_rg\]) and the anomalous dimensions in eq. (\[eq:gammas\]) the divergent terms in the bare NNLO soft function can be predicted, which is an important cross check of our calculations.
Table \[tab:nnloo\] shows the comparison of the divergent terms in different color structures with fixed $\beta_t=0.3$ and $\cos\theta_t=0.5$. We see that the maximum deviation is less than $0.2\%$. Figures \[fig:nnlo\_betat\] and \[fig:nnlo\_ct\] show the numerical calculations and the RG predictions with $\cos\theta_t$ in the range of $(-1,1)$ but fixed $\beta_t$ and with $\beta_t$ in the range of $(0,1)$ but fixed $\cos \theta_t$, respectively. We find that the numerical results are consistent with the RG predictions. For most of the cases the deviations are less than $0.2\%$, while the deviations can be about $1\%$ only when the absolute values of the coefficient $B_i$ are close to zero. We have checked that the points at $\beta_t = 0$ reproduce our previous results in ref. [@Li:2016tvb]. Similar to the NLO results, in the highly boosted region, the NNLO coefficients contains logarithmic structures such as $\ln^n(1-\beta_t)$. They are divergent when $\beta_t \to 1$. This fact explains the behaviour of the distributions of the points near the end point of $\beta_t$ in fig.\[fig:nnlo\_ct\].
Conclusions {#sec:conc}
===========
The $N$-jettiness subtraction method is one of the efficient methods to perform differential calculations of the NNLO cross sections. In this paper, we present the calculation of NNLO soft function for one massive colored particle production which is one of the indispensable ingredients in $N$-jettiness subtraction method. Our calculation makes use of the one-loop soft current and infrared limit of the QCD matrix elements from refs. [@Catani:2000pi; @Bierenbaum:2011gg; @Catani:1999ss; @Czakon:2011ve] to construct the integrand. The phase space integrations are performed with the sector decomposition method and the techniques are discussed in details. The divergent terms of NLO and NNLO soft functions in our calculations are in very good agreement with those from the RG predictions. Though our result is general for a single massive colored particle production, we focus on $tW$ production in the discussion because it is one of the most important processes in the SM. Once the two-loop hard function is obtained, we can perform the NNLO calculation for the differential cross section of $tW$ production at hadron colliders. Our method can also be applied to the calculation of the $N$-jettiness soft function for top quark pair production, which provides another way to study the NNLO differential cross section for this process. We leave this application in future study.
Acknowledgements {#acknowledgements .unnumbered}
================
HTL would like to acknowledge the TU Munich for its hospitality during the completion of this work. We thank Xiaohui Liu for useful discussion. The work of HTL was supported by Department of Energy Early Career Program. The work of JW was supported by the BMBF project No. 05H15WOCAA.
|
---
abstract: 'The extension of the Standard Model by right handed neutrinos with masses in the GeV range can simultaneously explain the observed neutrino masses via the seesaw mechanism and the baryon asymmetry of the universe via leptogenesis. It has previously been claimed that the requirement for successful baryogenesis implies that the rate of neutrinoless double $\beta$ decay in this scenario is always smaller than the standard prediction from light neutrino exchange alone. In contrast, we find that the rate for this process can also be enhanced due to a dominant contribution from heavy neutrino exchange. In a small part of the parameter space it even exceeds the current experimental limit, while the properties of the heavy neutrinos are consistent with all other experimental constraints and the observed baryon asymmetry is reproduced. This implies that neutrinoless double $\beta$ decay experiments have already started to rule out part of the leptogenesis parameter space that is not constrained by any other experiment, and the lepton number violation that is responsible for the origin of baryonic matter in the universe may be observed in the near future.'
author:
- |
Marco Drewes$^a$, Shintaro Eijima$^b$\
\
\
\
bibliography:
- 'all.bib'
title: '[**Neutrinoless double $\beta$ decay and low scale leptogenesis** ]{}'
---
Introduction {#sec:introduction}
============
With the exception of neutrinos, all fermions in the Standard Model (SM) of particle physics are known to exist with both left handed (LH) and right handed (RH) chirality. If RH neutrinos exist, they can explain the observed neutrino flavour oscillations via the seesaw mechanism [@Minkowski:1977sc; @GellMann:seesaw; @Mohapatra:1979ia; @Yanagida:1980xy; @Schechter:1980gr; @Schechter:1981cv]. In addition, RH neutrinos may also explain the baryon asymmetry of the universe (BAU) [@Canetti:2012zc] via leptogenesis during their CP violating decays [@Fukugita:1986hr] or CP violating oscillations [@Akhmedov:1998qx; @Asaka:2005pn] in the early universe, or compose the Dark Matter (DM) [@Adhikari:2016bei]. In Refs. [@Asaka:2005pn; @Asaka:2005an] it has been proposed that all of these puzzles can be solved simultaneously by RH neutrinos alone, which was found to be feasible in Refs. [@Canetti:2012vf; @Canetti:2012kh]. A pedagogical review of this scenario, which is known as the *Neutrino Minimal Standard Model* ($\nu$MSM), can be found in Ref. [@Boyarsky:2009ix]. Finally, light RH neutrinos could also act as Dark Radiation in the early universe and explain the observed neutrino oscillation anomalies [@Abazajian:2012ys]. A general review on the role of RH neutrinos in particle physics and cosmology can e.g. be found in Ref. [@Drewes:2013gca]. In the present work we focus on the possibility that RH neutrinos $N_I$ with Majorana masses $M_I$ in the GeV range can simultaneously explain the observed neutrino oscillations and the baryon asymmetry of the universe without violating any of the known experimental or cosmological constraints on their properties [@Atre:2009rg; @Antusch:2014woa; @Drewes:2015iva; @deGouvea:2015euy; @Fernandez-Martinez:2016lgt].
Experimentally the GeV range is very interesting because the RH neutrinos can be searched for in meson decays at b-factories [@Canetti:2014dka; @Milanes:2016rzr] or fixed target experiments [@Gorbunov:2007ak], including NA62 [@Asaka:2012bb], the SHiP experiment proposed at CERN [@Anelli:2015pba; @Alekhin:2015byh; @Graverini:2015dka] or a similar setup proposed at the DUNE beam at FNAL [@Akiri:2011dv; @Adams:2013qkq]. With sufficient statistics, it might even be possible to measure the CP violation in the $N_I$ decay [@Cvetic:2015naa]. Theoretically the *low scale seesaw* is motivated by models based on classical scale invariance [@Khoze:2013oga], in the framework of the “inverse seesaw” [@Mohapatra:1986bd; @Mohapatra:1986aw] and other models with an approximate conservation of lepton number (e.g. [@Chikashige:1980ui; @Gelmini:1980re; @Wyler:1982dd; @GonzalezGarcia:1988rw; @Branco:1988ex; @Abada:2007ux; @Shaposhnikov:2006nn; @Gavela:2009cd; @Sierra:2012yy; @Racker:2012vw; @Fong:2013gaa]) or by applying Ockham’s razor to the number of new particles required to explain the known beyond the SM phenomena [@Asaka:2005pn]. Placing the seesaw scale in the GeV range can avoid the hierarchy problem of the Higgs mass [@Shaposhnikov:2007nj], to which superheavy RH neutrinos would contribute [@Vissani:1997ys], while avoiding cosmological constraints that disfavour heavy neutrino masses below 100 MeV [@Hernandez:2014fha].
It has been pointed out by different authors [@Asaka:2011pb; @LopezPavon:2012zg; @Lopez-Pavon:2015cga; @Gorbunov:2014ypa; @Drewes:2015iva] that the rate for neutrinoless double $\beta$ decay in the presence of RH neutrinos with GeV masses can significantly differ from the standard prediction from light neutrinos alone. In this work we address the question whether an large rate of neutrinoless double $\beta$ decay can be realised while simultaneously generating the observed BAU. Previous studies have found that this requirement suppresses the rate of neutrinoless double $\beta$ decay [@Bezrukov:2005mx; @Asaka:2011pb; @Asaka:2013jfa]. A key point in the line of argument was the assumption that a degeneracy in the heavy neutrino masses is required for leptogenesis if they lie in the GeV range. However, the mass degeneracy is not a necessary requirement for low scale leptogenesis if there are more than two heavy neutrinos [@Drewes:2012ma].
In this letter we show the rate of neutrinoless double beta decay in the scenario with three RH neutrinos can exceed that only from light neutrino exchange while explaining the BAU via leptogenesis. Furthermore we show in a numerical parameter scan that even in the scenario with two RH neutrinos, which is the minimal number to explain the observed neutrino oscillations, there exists a corner in parameter space in which this is possible.
The seesaw model
================
The (type I) seesaw model is defined by adding $n$ RH neutrinos $\nu_R$ to the SM, which leads to the Lagrangian $$\begin{aligned}
\label{L}
\mathcal{L} &=&\mathcal{L}_{SM}+
i \overline{\nu_R}{\displaystyle{\not}{\partial}}\nu_R-
\overline{\ell_{L}}F\nu_{R}\tilde{\Phi} -
\tilde{\Phi}^{\dagger}\overline{\nu_{R}}F^{\dagger}\ell_L -{\rm \frac{1}{2}}(\overline{\nu_R^c}M_{M}\nu_{R}
+\overline{\nu_{R}}M_{M}^{\dagger}\nu^c_{R}).
\end{aligned}$$ $\mathcal{L}_{SM}$ is the SM Lagrangian, $\ell_{L}=(\nu_{L},e_{L})^{T}$ are the SM lepton doublets and $\Phi$ is the Higgs doublet with $\tilde{\Phi}=\epsilon\Phi^*$. Here $\epsilon$ is the antisymmetric $SU(2)$-invariant tensor. $M_{M}$ a Majorana mass term for $\nu_{R}$ and $F$ is a matrix of Yukawa couplings. We have defined $\nu_R^c\equiv C\overline{\nu_R}^T$, where the charge conjugation matrix is $C=i\gamma_2\gamma_0$. We work in the heavy neutrino mass basis in flavour space, i.e., $(M_M)_{IJ}=\delta_{IJ}M_I$. Adding $n$ RH neutrinos to the SM introduces $7n-3$ new physical parameters. The relation between these parameters and the parameters constrained by neutrino oscillation data [@Gonzalez-Garcia:2015qrr] can be expressed in terms of the Casas-Ibarra parametrisation [@Casas:2001sr] $$\begin{aligned}
\label{CasasIbarraDef}
F=\frac{i}{v}U_\nu\sqrt{m_\nu^{\rm diag}}\mathcal{R}\sqrt{M^{\rm diag}}\,\end{aligned}$$ with $(m_\nu^{\rm diag})_{ij}=\delta_{ij} m_i$, where $m_i$ are the light neutrino masses. The matrix $U_\nu$ can be factorised as $$\begin{aligned}
\label{PMNS}
U_\nu=V^{(23)}U_\delta V^{(13)}U_{-\delta}V^{(12)}{\rm diag}(e^{i \alpha_1/2},e^{i \alpha_2 /2},1)\,,\end{aligned}$$ with $U_{\pm \delta}={\rm diag}(e^{\mp i \delta/2},1,e^{\pm i \delta /2})$. The non vanishing entries of the matrix $V=V^{(23)}V^{(13)}V^{(12)}$ are given by: $$\begin{aligned}
V^{(ij)}_{ii}=V^{(ij)}_{jj}=\cos \uptheta_{ij} \ , \
V^{(ij)}_{ij}=-V^{(ij)}_{ji}=\sin \uptheta_{ij} \ ,\
V^{(ij)}_{kk}=1 \quad \text{for $k\neq i,j$}.\end{aligned}$$ The parameters $\uptheta_{ij}$ are the light neutrino mixing angles, $\delta$ is referred to as the Dirac phase and $\alpha_{1,2}$ as Majorana phases. The complex orthogonal matrix $\mathcal{R}$ fulfils the condition $\mathcal{R}\mathcal{R}^T=1$. In case of $n=3$ it can be expressed as $$\begin{aligned}
\mathcal{R}=\mathcal{R}^{(23)}\mathcal{R}^{(13)}\mathcal{R}^{(12)}\end{aligned}$$ where the non-vanishing entries are given by the three complex “Euler angles” $\omega_{ij}$, $$\begin{aligned}
\mathcal{R}^{(ij)}_{ii}=\mathcal{R}^{(ij)}_{jj}=\cos \omega_{ij} \ , \
\mathcal{R}^{(ij)}_{ij}=-\mathcal{R}^{(ij)}_{ji}=\sin \omega_{ij} \ , \
\mathcal{R}^{(ij)}_{kk}=1 \quad \text{for $k\neq i,j$}.\end{aligned}$$ For two flavours there is only one complex angle $\omega$, and one has to distinguish between normal ordering (NO) and inverted ordering (IO): $$\begin{aligned}
\mathcal{R}^{\rm NO}=
\begin{pmatrix}
0 && 0\\
\cos \omega && \sin \omega \\
-\xi \sin \omega && \xi \cos \omega
\end{pmatrix}\,,\quad \quad
\mathcal{R}^{\rm IO}=
\begin{pmatrix}
\cos \omega && \sin \omega \\
-\xi \sin \omega && \xi \cos \omega \\
0 && 0
\end{pmatrix}
\,,\end{aligned}$$ where $\xi=\pm 1$. When the Higgs field obtains an expectation value $v(T)$, the Yukawa couplings lead to mixing between $\nu_R$ and $\nu_L$. This mixing can be quantified by the matrix $$\theta=v F M_M^{-1}.$$ In general, the mass eigenstates can be expressed in terms of the Majorana spinors $$\label{LightMassEigenstates}
\upnu_i=\left[ V_\nu^{\dagger}\nu_L-U_\nu^{\dagger}\theta\nu_{R}^c + V_\nu^{T}\nu_L^c-U_\nu^{T}\theta\nu_{R} \right]_i
$$ which can be identified with the light neutrinos with masses $m_i$, and $$N_I=\left[V_N^\dagger\nu_R+\Theta^{T}\nu_{L}^{c} + V_N^T\nu_R^c+\Theta^{\dagger}\nu_{L}\right]_I.$$ The observed light mass eigenstates $\upnu_i$ are connected to the active flavour eigenstates by the matrix $V_\nu$, which is related to $U_\nu$ via $V_\nu= (\mathbbm{1}-\frac{1}{2}\theta\theta^{\dagger})U_\nu$. $V_N$ and $U_N$ are their equivalents in the sterile sector; $U_N$ diagonalises the heavy neutrino mass matrix $M_N=M_M + \frac{1}{2}(\theta^{\dagger} \theta M_M + M_M^T \theta^T \theta^{*})$ after electroweak symmetry breaking, and $V_N= (1-\frac{1}{2}\theta^T\theta^*)U_N$. The mixing between the heavy and light states can is finally given by $$\Theta_{\alpha I}=(\theta U_N^*)_{\alpha I}.$$ The overall magnitude of the mixing is governed by the imaginary part of the complex angels $\omega$ or $\omega_{ij}$. For instance, for $n=2$ one finds $$\begin{aligned}
{\rm tr}[\Theta^\dagger \Theta]&=&\frac{M_2-M_1}{2M_1 M_2} (m_2-m_3)\cos(2 {\rm Re}\omega)+\frac{M_1+M_2}{2M_1 M_2}(m_2+m_3)\cosh(2 {\rm Im}\omega) $$ with normal ordering and $$\begin{aligned}
{\rm tr}[\Theta^\dagger \Theta]&=&\frac{M_2-M_1}{2M_1 M_2} (m_1-m_2)\cos(2 {\rm Re}\omega)+\frac{M_1+M_2}{2M_1 M_2}(m_1+m_2)\cosh(2 {\rm Im}\omega) $$ with inverted ordering.
Neutrinoless double $\beta$ decay
=================================
#### General case
- In the context of neutrino physics, constraints on the lifetime of neutrinoless double $\beta$ decay are commonly expressed in terms of the quantity $$\label{mee}
m_{\beta\beta}=\left|
\sum_i (U_\nu)_{ei}^2m_i + \sum_I \Theta_{eI}^2M_I f_A(M_I)
\right|.$$ The first term is the contribution due to the exchange of light neutrinos, $$m_{\beta\beta}^\nu=\sum_i (U_\nu)_{ei}^2m_i.$$ The second term comes from heavy neutrino exchange. For $M_I$ larger than the typical momentum exchange $\sim100$ MeV in neutrinoless double $\beta$ decay, the $N_I$ are virtual. The suppression due to this virtuality is parametrised by the function $f_A$, which suffers from some uncertainty due to uncertainties in the nuclear matrix elements that determine the exchanged momentum. For our purpose, we approximate it by $$f_A(M)\simeq \frac{\Lambda^2}{\Lambda^2+M^2}\Big|_{\Lambda^2= (0.159{\rm GeV})^2},$$ which corresponds to the “Argonne” model discussed in Ref. [@Faessler:2014kka]. Here $\Lambda$ is the typical momentum exchange in the decay. At tree level,[^1] we can use the unitarity relation $$\begin{aligned}
\label{SeesawConsistency}
\sum_i m_i (U_\nu)_{\alpha i}^2 + \sum_I M_I \Theta_{\alpha I}^2 = 0 $$ to rewrite (\[mee\]) as $$\begin{aligned}
m_{\beta\beta}&=&\left|
m_{\beta\beta}^\nu
+f_A(\bar{M})\sum_IM_I\Theta_{e I}^2
+\sum_IM_I\Theta_{e I}^2[f_A(M_I)-f_A(\bar{M})]
\right|\nonumber\\
&=&
\left|
[1-f_A(\bar{M})]m_{\beta\beta}^\nu+\sum_IM_I\Theta_{e I}^2[f_A(M_I)-f_A(\bar{M})]
\right|,\label{meerewritten}\end{aligned}$$ where $\bar{M}$ is an arbitrarily chosen mass scale. It is usually assumed that the contribution from $N_I$-exchange is negligible due to the suppression by the function $f_A$. Recently several authors have pointed out that this suppression is not efficient enough for $M_I$ in the GeV range [@Bezrukov:2005mx; @LopezPavon:2012zg; @Asaka:2011pb; @Asaka:2013jfa; @Lopez-Pavon:2015cga; @Gorbunov:2014ypa; @Drewes:2015iva], and that the exchange of $N_I$ may dominate neutrinoless double $\beta$ decay. This can significantly modify the allowed regions in the $m_{\rm lightest}$-$m_{\beta\beta}$ plane, which are based on the approximation $m_{\beta\beta}=m_{\beta\beta}^\nu$. Here $m_{\rm lightest}$ is the mass of the lightest neutrinos. So far it has been argued that this can only suppress the rate of neutrinoless double $\beta$ decay in models where the $N_I$ generate the BAU via leptogenesis because it was assumed that successful leptogenesis requires a degeneracy in the heavy neutrino masses [@Bezrukov:2005mx; @Asaka:2013jfa; @Asaka:2011pb; @Gorbunov:2014ypa]. Indeed, if the difference $f_A(M_I)-f_A(\bar{M})$ is negligible, Eq. (\[meerewritten\]) reduces to $$\begin{aligned}
m_{\beta\beta}\simeq \left|
[1-f_A(\bar{M})]m_{\beta\beta}^\nu
\right|,\end{aligned}$$ which is always smaller than $m_{\beta\beta}^\nu$.[^2] However, it has recently been pointed out [@Drewes:2012ma] and confirmed [@Shuve:2014zua; @Hernandez:2015wna] that the need for a mass degeneracy is specific to the scenarios with $n=2$ and that for $n>2$, leptogenesis from neutrino oscillations does not require a mass degeneracy.
#### The case $n=2$
- Moreover, one may wonder whether the mass degeneracy of order $10^{-3}$ that is required in the model with $n=2$ is sufficient to suppress the term $\sum_IM_I\Theta_{e I}^2[f_A(M_I)-f_A(\bar{M})]$ in Eq. (\[meerewritten\]) for $M_I$ moderately larger than 100 MeV. In absence of a strong mass degeneracy, this term can either increase or reduce $m_{\beta\beta}$. In the case $n=2$, $m_{\beta\beta}$ can be expressed in terms of the model parameters as $$\begin{aligned}
m_{\beta\beta}&=&
\bigg|m_2 \cos^2\uptheta_{13} \sin^2\uptheta_{12} e^{i\alpha_2}
+m_3 \sin^2\uptheta_{13} e^{-2i\delta}\nonumber\\
&&-f_A(M_2) \left[
\sqrt{m_3} \cos\omega \sin\uptheta_{13} e^{-i\delta}
+\sqrt{m_2} \sin\omega \sin\uptheta_{12} \cos\uptheta_{13} e^{i\alpha_2/2}
\right]^2\nonumber\\
&&-f_A(M_1)\left[
-\sqrt{m_3} \sin\omega \sin\uptheta_{13} e^{-i\delta}
+\sqrt{m_2} \cos\omega \sin\uptheta_{12} \cos\uptheta_{13} e^{i\alpha_2/2}
\right]^2\bigg|\end{aligned}$$ for normal ordering and $$\begin{aligned}
m_{\beta\beta}&=&
\cos^2\uptheta_{13}
\bigg|
m_1 e^{i\alpha_1}
\cos^2\uptheta_{12}
+ m_2 e^{i\alpha_2}\sin^2\uptheta_{12}\nonumber\\
&&- f_A(M_2)\left[
e^{i\alpha_2/2}\sqrt{m_2} \cos\omega \sin\uptheta_{12}
+ e^{i\alpha_1/2}\sqrt{m_1} \sin\omega \cos\uptheta_{12}
\right]^2\nonumber\\
&&- f_A(M_1)\left[
-e^{i\alpha_2/2}\sqrt{m_2} \sin\omega \sin\uptheta_{12}
+ e^{i\alpha_1/2}\sqrt{m_1} \cos\omega \cos\uptheta_{12}
\right]^2
\bigg|\end{aligned}$$ for inverted ordering. For $n=2$, it is convenient to choose $$\bar{M}=\frac{M_2 + M_1}{2}$$ and define $$\Delta M = \frac{M_2 - M_1}{2}.$$ Since leptogenesis with $n=2$ requires a mass degeneracy, $\bar{M}$ in this case has a physical meaning as the common mass of the heavy neutrinos. This allows to express Eq. (\[meerewritten\]) as $$\begin{aligned}
m_{\beta\beta}\simeq \left|
[1-f_A(\bar{M})]m_{\beta\beta}^\nu
+ 2 f_A^2(\bar{M})\frac{\bar{M}^2}{\Lambda^2}\Delta M \left(
\Theta_{e1}^2 - \Theta_{e2}^2
\right)
\right|,\end{aligned}$$ where we have neglected higher order terms in $\Delta M/\bar{M}$. In the term that is proportional to $m_{\beta\beta}^\nu$, the contribution from $N_I$ exchange interferes destructively and reduces $m_{\beta\beta}$. The second term can have either sign and can reduce or enhance $m_{\beta\beta}$. The largest effect is expected if the mass splitting $\Delta M$ is relatively large and the mixings $\Theta_{e I}$ of $N_1$ and $N_2$ with the electron flavour are maximally different. Using the fact that the lightest neutrino is massless for $n=2$ ($m_{\rm lightest}=0$) and one of the light neutrino mass splittings is much larger than the other ($\Delta m_{\rm atm}^2\gg \Delta m_{\rm sol}^2$), we can approximate $$\begin{aligned}
{\rm for} \ {\rm NO}: m_{\beta\beta}\simeq & \bigg|
[1-f_A(\bar{M})]m_{\beta\beta}^\nu
+
2 f_A^2(\bar{M})\frac{\bar{M}^2}{\Lambda^2}
\frac{\Delta M}{\bar{M}}|\Delta m_{\rm atm}|
e^{-2i\delta} \sin^2\uptheta_{13} \cos(2\omega)
\bigg|, \\
{\rm for} \ {\rm IO}: m_{\beta\beta}\simeq & \bigg|
[1-f_A(\bar{M})] m_{\beta\beta}^\nu
+
2 f_A^2(\bar{M})\frac{\bar{M}^2}{\Lambda^2}
\frac{\Delta M}{\bar{M}}
|\Delta m_{\rm atm}|\cos^2\uptheta_{13} \\
&\times\Big[\left(e^{i\alpha_2}\sin^2\uptheta_{12}
-e^{i\alpha_1}\cos^2\uptheta_{12}
\right)\cos(2\omega)
+e^{i(\alpha_1 + \alpha_2)/2}\xi\sin(2\uptheta_{12})\sin(2\omega)\nonumber
\Big]
\bigg|.\end{aligned}$$ This shows that, for given $\bar{M}$ and $\Delta M$, one can in principle make the term proportional to $\Delta M$ arbitrarily large by choosing a sufficiently large $|{\rm Im}\omega|$. In the limit ${\rm Im}\omega\gg 1$ one finds $$\begin{aligned}
{\rm for} \ {\rm NO}: m_{\beta\beta}\simeq & \bigg|
[1-f_A(\bar{M})]m_{\beta\beta}^\nu\\
&+
f_A^2(\bar{M})\frac{\bar{M}^2}{\Lambda^2}
\frac{\Delta M}{\bar{M}}|\Delta m_{\rm atm}| \sin^2\uptheta_{13}
e^{2{\rm Im}\omega}e^{-2i({\rm Re}\omega + \delta)} \bigg|, \nonumber\\
{\rm for} \ {\rm IO}: m_{\beta\beta}\simeq & \bigg|
[1-f_A(\bar{M})] m_{\beta\beta}^\nu\\
&+
f_A^2(\bar{M})\frac{\bar{M}^2}{\Lambda^2}
\frac{\Delta M}{\bar{M}}
|\Delta m_{\rm atm}|\cos^2\uptheta_{13} e^{2{\rm Im}\omega}e^{-2i{\rm Re}\omega}
\left(\xi e^{i\alpha_2/2}\sin\uptheta_{12}
+ i e^{i\alpha_1/2}\cos\uptheta_{12}
\right)^2
\bigg|.\nonumber\end{aligned}$$ Consistency with neutrino oscillation data at tree level is guaranteed by the use of the Casas Ibarra parameterisation. However, for masses in the GeV range, there exist various constraints on $\Theta_{e I}$ from direct searches for $N_I$ particles, indirect tests involving rare processes and precision observables as well as cosmology that impose upper bounds on $|\Theta_{e I}|^2$. These are e.g. summarised in Refs. [@Atre:2009rg; @Antusch:2014woa; @Drewes:2015iva; @deGouvea:2015euy; @Fernandez-Martinez:2016lgt] and references therein. In the following we use the analysis in Ref. [@Drewes:2015iva] as a basis.
The comparably strong sensitivity of the term involving $\Delta M$ to the shape of the function $f_A$ implies that the observation of neutrinoless double $\beta$ decay in different nuclei can possibly help to obtain information on the fundamental parameters and $L$ violation even if $\Delta M$ is too small to be resolved experimentally in direct searches for heavy neutrinos.
Baryogenesis
============
In leptogenesis, a matter-antimatter asymmetry is generated in the lepton sector and then partly transferred into a baryon number by weak sphalerons [@Kuzmin:1985mm], which violate $B+L$ and conserve $B-L$. Here $B$ is the total baryon number and $L$ is the total SM lepton number. In the SM, $B$ is conserved at temperatures $T$ below the temperature $T_{\rm sph}\simeq 130$ GeV [@D'Onofrio:2014kta] of sphaleron freezeout. Hence, the BAU is determined by the lepton asymmetry $L$ at $T=T_{\rm sph}$. In the framework of the seesaw mechanism, RH neutrinos with GeV masses must have Yukawa couplings smaller than that of the electron to be consistent with the smallness of the observed neutrino masses and constraints from experimental searches [@Drewes:2015iva]. As a result, they may not reach thermal equilibrium in the early universe before $T=T_{\rm sph}$, and the BAU is generated via CP violating flavour oscillations amongst the $N_I$ during their production [@Akhmedov:1998qx].[^3] Since the $N_I$ are highly relativistic at $T>T_{\rm sph}$, the violation of $L$ during this process by the Majorana masses is suppressed as $\sim M_I^2/T^2$. However, sizable asymmetries $L_\alpha$ are generated in the individual flavours $\alpha=e,\mu,\tau$. These are partly converted into a total $L\neq 0$[^4] by a flavour asymmetric washout that hides part of the CP-asymmetry from the sphalerons by storing them in helicity-odd occupation numbers of the $N_I$, which leads to the generation of a $B\neq 0$ by sphalerons. This process crucially relies on the Majorana masses $M_I$ of the heavy neutrinos $N_I$. At the same time, these Majorana masses are responsible for $L$ violation that makes neutrinoless double $\beta$ decay possible in the seesaw model. This immediately raises the question whether the regime in which the $L$ violation due to the masses of heavy neutrinos explain the origin of baryonic matter in the universe may be accessible to neutrinoless double $\beta$ decay searches. We now study the question whether a value of $m_{\beta\beta}>m_{\beta\beta}^\nu$ can be made consistent with successful leptogenesis via neutrino oscillations in low scale seesaw models.
#### The case $n=3$
- Since a positive contribution to $m_{\beta\beta}$ from $N_I$ exchange can only come from the term $\sum_IM_I\Theta_{e I}^2[f_A(M_I)-f_A(\bar{M})]$ in Eq. (\[meerewritten\]), the chances for this are the best in scenarios with $n>2$ that do not require a mass degeneracy. However, the parameter space of these scenarios is rather large. Though many authors have studied this process [@Akhmedov:1998qx; @Asaka:2005pn; @Shaposhnikov:2008pf; @Anisimov:2010dk; @Anisimov:2010aq; @Canetti:2010aw; @Garny:2011hg; @Garbrecht:2011aw; @Canetti:2012vf; @Canetti:2012kh; @Drewes:2012ma; @Canetti:2014dka; @Khoze:2013oga; @Shuve:2014zua; @Garbrecht:2014bfa; @Abada:2015rta; @Hernandez:2015wna; @Kartavtsev:2015vto; @Hernandez:2016kel; @Drewes:2016gmt], no complete scan of the parameter space has been performed to date, and such a parameter scan goes beyond the scope of this Letter. For the sake of a proof of principles, we restrict ourselves to a specific region in the parameter space of the scenario with $n=3$ in which the BAU can be estimated analytically [@Canetti:2014dka]. The rates at which heavy neutrino interaction eigenstates approach thermal equilibrium at temperatures $T\gg M_I$ are governed by the eigenvalues of the matrix $\Gamma_N\simeq F^\dagger F\gamma_{\rm av}T$, c.f. Eq. (\[rates1\]), where $\gamma_{\rm av}$ is a numerical coefficient that we set to $\gamma_{\rm av}=0.012$ here, corresponding to the value from Ref. [@Garbrecht:2014bfa] based on Refs. [@Besak:2012qm; @Garbrecht:2013urw]. The rate at which they oscillate is determined by the mass splittings $M_I^2-M_J^2$. If the CP violating oscillations that generate flavoured asymmetries $Y_\alpha$ occur long before one of the $N_I$ comes into thermal equilibrium, then the generation of the $Y_\alpha$ and the washout (which leads to a $B\neq 0$) can be treated as two separate processes. The condition for this reads $$\begin{aligned}
\label{conditon1}
\frac{||F^\dagger F||\gamma_{\rm av}a_R^{2/3}}{(M_I^2-M_J^2)^{2/3}}\ll 1,\end{aligned}$$ where $a_R=m_P(45/(4\pi^3 g_*))^{1/2}=T^2/H$ can be interpreted as the comoving temperature in a radiation dominated universe with Hubble parameter $H$. Here $m_P$ is the Planck mass, $g_*$ the number of degrees of freedom in the primordial plasma and $||F^\dagger F||$ refers to the largest eigenvalue of the matrix. Then the flavoured asymmetries can be estimated as [@Drewes:2016gmt] $$\begin{aligned}
\label{flavoured:asymmetries}
Y_\alpha
&\approx&-\sum\limits_{\overset{I,J,\beta}{I\not=I}}
\frac{{\rm Im}[F_{\alpha I} F_{I\beta}^\dagger F_{\beta J} F_{J\alpha}^\dagger]}{{\rm sign}(M_I^2-M_J^2)}
\left(\frac{m_{\rm Pl}^2}{|M_I^2-M_J^2|}\right)^{\frac 23}
3.4
\times 10^{-4}
\gamma_{\rm av}^2\
.\end{aligned}$$ Once some heavy neutrino interaction eigenstates approach equilibrium, the washout of the asymmetries $Y_\alpha$ begins. For $T\gg M_I$, the rate for this process is roughly given by $\Gamma^{\alpha}_L\simeq (F F^\dagger)_{\alpha\alpha} \gamma_{\rm av}T/g_w$ with $g_w=2$.[^5] If two SM flavours come into equilibrium before sphaleron freezeout,[^6] $$\label{conditon2}
\Gamma^{\beta\neq\alpha}_L/H \gg 1 \ {\rm at} \ T=T_{\rm sph},$$ then the BAU can be estimated as $$\label{BAU}
Y_B\simeq -\frac{28}{79}Y_\alpha\frac{3}{7}e^{-\Gamma^\alpha_L/H},$$ where $28/79$ is the sphaleron conversion factor, the factor $3/7$ comes from the equilibration of all charges except $Y_\alpha$ during their washout and the exponential describes the washout of $Y_\alpha$ itself. By plugging numbers into the parametrisation (\[CasasIbarraDef\]), it is straightforward to see that $m_{\beta\beta}>m_{\beta\beta}^\nu$ can be realised while producing a BAU that exceeds the observed value and respecting the conditions (\[conditon1\]) and (\[conditon2\]). We illustrate the parameter dependence of $m_{\beta\beta}$ and $Y_B$ on the observable Dirac phase $\delta$ and ${\rm Im}\omega_{23}$ in figures \[w\] and \[e\] to show that a large $m_{\beta\beta}$ can indeed be realised while explaining the observed BAU. The quantities ${\rm Im}\omega_{ij}$ determine the magnitude of the active-sterile mixing $U_{\alpha I}^2$ and can thereby be constrained experimentally if heavy neutrinos are found in the laboratory.
![ The BAU and $m_{\beta\beta}$ as a function of ${\rm Im}\omega_{13}$. We fix $M_1=0.22$ GeV, $M_2=0.85$ GeV, $M_3=0.63$ GeV, $m_1=23$ meV, $m_2=24.6$ meV, $m_3=54.6$ meV, $\alpha_1=11.88$, $\alpha_2=11.64$, $\omega_{12}=12.23 + 3.38i$, $\omega_{23}=11.39 - 0.21i$, $\delta=5.76$ and ${\rm Re}\omega_{13}=5.18$. In the dotted region the condition (\[conditon2\]) is not fulfilled. Here and in Fig. \[e\] we used the radiatively corrected Casas-Ibarra parameterisation introduced in Ref. [@Lopez-Pavon:2015cga] instead of the tree level formula (\[CasasIbarraDef\]) to ensure consistency with neutrino oscillation data at one loop level. \[w\] ](BAUw.pdf "fig:"){width="80.00000%"}\
![ The BAU and $m_{\beta\beta}$ as a function of ${\rm Im}\omega_{13}$. We fix $M_1=0.22$ GeV, $M_2=0.85$ GeV, $M_3=0.63$ GeV, $m_1=23$ meV, $m_2=24.6$ meV, $m_3=54.6$ meV, $\alpha_1=11.88$, $\alpha_2=11.64$, $\omega_{12}=12.23 + 3.38i$, $\omega_{23}=11.39 - 0.21i$, $\delta=5.76$ and ${\rm Re}\omega_{13}=5.18$. In the dotted region the condition (\[conditon2\]) is not fulfilled. Here and in Fig. \[e\] we used the radiatively corrected Casas-Ibarra parameterisation introduced in Ref. [@Lopez-Pavon:2015cga] instead of the tree level formula (\[CasasIbarraDef\]) to ensure consistency with neutrino oscillation data at one loop level. \[w\] ](meew.pdf "fig:"){width="80.00000%"}
![ The BAU and $m_{\beta\beta}$ as a function of $\delta$. We fix $M_1=0.22$ GeV, $M_2=0.85$ GeV, $M_3=0.63$ GeV, $m_1=23$ meV, $m_2=24.6$ meV, $m_3=54.6$ meV, $\alpha_1=11.88$, $\alpha_2=11.64$, $\omega_{12}=12.23 + 3.38i$, $\omega_{23}=11.39 - 0.21i$ and $\omega_{13}=5.18 - 1.62i$. \[e\] ](BAUd.pdf "fig:"){width="80.00000%"}\
![ The BAU and $m_{\beta\beta}$ as a function of $\delta$. We fix $M_1=0.22$ GeV, $M_2=0.85$ GeV, $M_3=0.63$ GeV, $m_1=23$ meV, $m_2=24.6$ meV, $m_3=54.6$ meV, $\alpha_1=11.88$, $\alpha_2=11.64$, $\omega_{12}=12.23 + 3.38i$, $\omega_{23}=11.39 - 0.21i$ and $\omega_{13}=5.18 - 1.62i$. \[e\] ](meed.pdf "fig:"){width="80.00000%"}
This treatment is of course very simplified and should be understood as a proof of principle. A detailed study of the parameter space in the region where the conditions do not apply requires a numerical solution of the quantum kinetic equations for each point in parameter space.
![ The function $v(T)$ used in our calculation.\[higgsvev\] ](Higgsvev.pdf){width="80.00000%"}
#### The case $n=2$
- For a more quantitative treatment we return to the scenario with $n=2$, where the lower dimensionality of the parameter space makes a numerical scan less expensive. It is well-known that leptogenesis in this scenario requires a mass degeneracy of order $|\Delta M|/\bar{M}\ll 1$ [@Canetti:2012vf; @Canetti:2012kh; @Shuve:2014zua; @Hernandez:2015wna]. We perform a numerical scan in order to address the question whether successful baryogenesis and $m_{\beta\beta}>m_{\beta\beta}^\nu$ can be realised simultaneously for $n=2$. Phenomenologically this is interesting because this scenario effectively describes baryogenesis in the $\nu$MSM. In order to identify the parameter region where baryogenesis is possible, we solve momentum integrated kinetic equations for the two helicity components $\rho_{N}$ ans $\rho_{\bar{N}}$ of the heavy neutrino density matrix and $Y_\alpha$ [@Asaka:2005pn; @Shaposhnikov:2008pf], $$\begin{aligned}
\label{kinequ1}
i \frac{1}{\mathcal{H}X}\frac{d\rho_{N}}{d X}&=&[H_N, \rho_{N}]-\frac{i}{2}\{\Gamma_N, \rho_{N} - \rho^{eq}\} +\frac{i}{2} Y_\alpha{\tilde\Gamma^\alpha}_N~,\\
i \frac{1}{\mathcal{H}X}\frac{d\rho_{\bar{N}}}{d X}&=& [H_N^*, \rho_{\bar{N}}]-\frac{i}{2}\{\Gamma^*_N, \rho_{\bar{N}} - \rho^{eq}\} -\frac{i}{2} Y_\alpha{\tilde\Gamma^{\alpha *}}_N~,\label{kinequ2}\\
i \frac{1}{\mathcal{H}X}\frac{dY_\alpha}{d X}&=&-i\Gamma^\alpha_L Y_\alpha +
i {\rm tr}\left[{\tilde \Gamma^\alpha}_L(\rho_{N} -\rho^{eq})\right] -
i {\rm tr}\left[{\tilde \Gamma^{\alpha*}}_L(\rho_{\bar{N}} -\rho^{eq})\right]
~.
\label{kinequ3}\end{aligned}$$ Here $\rho^{eq}$ is the equilibrium density matrix and $X=\bar{M}/T$ is a dimensionless time variable. The function $$\mathcal{H}\equiv -\frac{\partial}{\partial X}\sqrt{\frac{45}{4\pi^3 g_*}}\frac{m_P}{2M^2} X$$ can be identified with the Hubble parameter if the number of degrees of freedom $g_*$ is constant during the evolution, which is justified in the present context. The coefficients appearing in Eqns. (\[kinequ1\])-(\[kinequ2\]) can be expressed as $$\begin{aligned}
H_N&=&\frac{1}{4T}\left[
-2\bar{M}\Delta M \sigma_3 + \tilde{F}^\dagger \tilde{F}\frac{T^2}{4} + \tilde{F}^\dagger \tilde{F} v^2(T)
\right]\\
\Gamma_N&=&\phantom{i} \sum_{\alpha}\big(\tilde{F}^*_{\alpha I}\tilde{F}_{\alpha J}R(T,M)_{\alpha\alpha}+\tilde{F}_{\alpha I}\tilde{F}^*_{\alpha J}R_M(T,M)_{\alpha\alpha}\big),\label{rates1}\\
(\tilde{\Gamma}_L^\alpha)_{IJ} \simeq (\tilde{\Gamma}_N^\alpha)_{IJ}&=& \phantom{i} \big( \tilde{F}^*_{\alpha I}\tilde{F}_{\alpha J}R(T,M)_{\alpha\alpha}-\tilde{F}_{\alpha I}\tilde{F}^*_{\alpha J}R_M(T,M)_{\alpha\alpha}\big),\label{rates2} \\
\Gamma_L^\alpha
&=&\phantom{i} \frac{1}{g_w}\big((FF^\dagger)_{\alpha\alpha}\left(R(T,M)_{\alpha\alpha}+R_M(T,M)_{\alpha\alpha}\right)\big)\label{rates3}, \end{aligned}$$ with $\tilde{F}=FU_N\simeq F$. The function $v(T)$ is shown in figure \[higgsvev\]. We have assumed that the average momentum of heavy neutrinos is $|{\textbf{p}}|\simeq 2T$. In the limit $T\gg M_I$ one can approximate $R_M\simeq0$, $R\simeq \gamma_{\rm av} T$. The equations (\[kinequ1\])-(\[kinequ3\]) are the heavy neutrino equivalent of the density matrix equations commonly used in neutrino physics [@Sigl:1992fn] and are derived in the appendix of Ref. [@Canetti:2012kh]. Our scan comprises $5\times 10^7$ parameter choices for each neutrino mass ordering. We use a logarithmic prior for the mass splitting in the interval $-16 \leq \log(\Delta M/{\rm GeV})/\log10 \leq 0$ and flat priors in all other quantities in the the parametrisation (\[CasasIbarraDef\]). We considere the mass range $0.1 {\rm GeV} < \bar{M} < 5$ GeV. We accept a point when the generated BAU lies within a $5\sigma$ range of the observed value $\eta_B=(8.06 - 9.11)\times 10^{-11}$ [@Ade:2015xua]. At the same time, we require consistency with all direct and indirect constraints on the low scale seesaw that are summarised in Ref. [@Drewes:2015iva] (except the constraint on $m_{\beta\beta}$ of course). These include indirect experimental constraints from neutrino oscillation data, electroweak precision data, lepton universality, searches for rare lepton decays and tests of CKM unitarity with bounds from big bang nucleosynthesis and past direct searches at colliders and fixed target experiments.
The result of this scan is shown in figure \[scanfig\].
![\[scanfig\] The blue points correspond to values of $\bar{M}$ and $m_{\beta\beta}$ that are consistent with successful leptogenesis and the constraints on the low scale seesaw summarised in Ref. [@Drewes:2015iva]. The red band shows the upper limit on $m_{\beta\beta}$ from the KamLAND-Zen experiment [@KamLAND-Zen:2016pfg], where the width of the band comes from the theoretical uncertainty in the nuclear matrix elements that affects the translation from a bound on the lifetime into a bound on $m_{\beta\beta}$. The upper plot is for normal mass ordering, the lower for inverted mass ordering. ](mbb_rangeNH.jpg "fig:"){width="80.00000%"}\
![\[scanfig\] The blue points correspond to values of $\bar{M}$ and $m_{\beta\beta}$ that are consistent with successful leptogenesis and the constraints on the low scale seesaw summarised in Ref. [@Drewes:2015iva]. The red band shows the upper limit on $m_{\beta\beta}$ from the KamLAND-Zen experiment [@KamLAND-Zen:2016pfg], where the width of the band comes from the theoretical uncertainty in the nuclear matrix elements that affects the translation from a bound on the lifetime into a bound on $m_{\beta\beta}$. The upper plot is for normal mass ordering, the lower for inverted mass ordering. ](mbb_rangeIH.jpg "fig:"){width="80.00000%"}
The densely populated area corresponds to the standard prediction $m_{\beta\beta}^\nu$. For $\bar{M}>2$ GeV we find almost no points outside this region because the suppression of the heavy neutrino contribution due to $f_A$ is efficient. For lower masses, we find deviations from the standard prediction in both directions. For inverted ordering the value of $m_{\beta\beta}$ can exceed the present day experimental limit from the KamLAND-Zen [@KamLAND-Zen:2016pfg] and GERDA [@Agostini:2013mzu] experiments. This implies that neutrinoless double $\beta$ decay experiments have already started to rule out part of the leptogenesis parameter space that is not constrained by any other experiment. The allowed parameter region with $m_{\beta\beta}>m_{\beta\beta}^\nu$ is characterised by relatively large mass splitting and large $|{\rm Im}\omega|$, e.g. $\Delta M/\bar{M}\sim 10^{-3}$ and $|{\rm Im}\omega|>2$, see Fig. \[ImOmegaDeltaMplot\]. [^7] To the best of our knowledge, this parameter region is not singled out by any known symmetry, which seems to imply that a large value of $m_{\beta\beta}$ for $n=2$ requires considerable tuning. For $\bar{M}$ below the kaon mass the viable parameter space rapidly shrinks because $|\Theta_{\alpha I}|^2$ is constrained from below by the requirement that the $N_I$ decay before BBN and constrained from above by direct searches in fixed target experiments.
![\[ImOmegaDeltaMplot\] A representative distribution of parameter values that lead to successful baryogenesis and $m_{\beta\beta}>m_{\beta\beta}^\nu$ while being in agreement with all other direct and indirect constraints discussed in Ref. [@Drewes:2015iva]. The colour indicates the magnitude of $\bar{M}$, which ranges from values below the kaon mass (lightest) to values above the D-meson mass (darkest). ](ImOmegaDeltaM.pdf){width="80.00000%"}
Conclusions
===========
We conclude that the rate of neutrinoless double $\beta$ decay in low scale leptogenesis scenarios within the minimal seesaw model with Majorana masses in the GeV range can be both, smaller and larger than the expectation from light neutrino exchange alone, while respecting all known constraints on the properties of heavy neutrinos from experiments and cosmology. For inverted ordering the value of $m_{\beta\beta}$ can exceed the present day experimental limit, which implies that neutrinoless double $\beta$ decay experiments have already started to rule out part of the leptogenesis parameter space that is not constrained by any other experiment. The observation of a value of $m_{\beta\beta}$ that deviates from the standard prediction would contain valuable information about the heavy neutrino mass splitting and the CP-violating phases in their couplings. Together with a measurement of the Dirac phase $\delta$ in neutrino oscillation experiments, this would allow to impose strong constraints on the violation of lepton number and CP in the low scale seesaw model. If any heavy neutral leptons are discovered in future experiments and their mixings $|\Theta_{\alpha I}|^2$ with the SM neutrinos have been measured, this information will be crucial to decide whether theses particles are indeed responsible for the generation of baryonic matter in the universe.\
\
Acknowledgements {#acknowledgements .unnumbered}
================
We would like to thank Mikhail Shaposhnikov for helpful discussions in the initial phase of this project and for sponsoring MaD’s visit to Lausanne that made this project possible. We would also like to than Fedor Bezrukov for his comments on the final version of this manuscript. This work was supported by the Deutsche Forschungsgemeinschaft (DFG) and the Swiss National Science Foundation (SNF).
[^1]: Loop corrections are e.g. discussed in Refs. [@Fernandez-Martinez:2015hxa; @Drewes:2015iva].
[^2]: The possibility to reduce $m_{\beta\beta}$ below $m_{\beta\beta}^\nu$ is interesting because it means that even a non-observation of neutrinoless double $\beta$ decay at the level $m_{\beta\beta}<10^{-2}$ eV may not rule out the inverted ordering.
[^3]: An alternative mechanism with $M_I$ in the GeV range has been proposed in Ref. [@Hambye:2016sby].
[^4]: $L$ here refers to the SM lepton number. One can define a generalised lepton number that includes the helicity odd occupation numbers of the heavy neutrino mass eigenstates and remains in good approximation conserved during baryogenesis.
[^5]: The factor $g_w$ accounts for the fact that $\gamma_{\rm av}$ has been determined in the context of $\Gamma_N$, which interacts with both components of the SU(2) doublet $\ell_L$, while the $Y_\alpha$ violating interactions of $\ell_L$ only involve the singlet $\nu_R$.
[^6]: If the initial asymmetries $Y_\beta$ in flavours other than $\alpha$ are much larger than $Y_\alpha$, the stronger condition $|Y_\alpha e^{-\Gamma^\alpha_L/H}|\gg
|\sum_{\beta\neq\alpha} Y_\beta e^{-\Gamma^\beta_L/H}|$ should be used at $T=T_{EW}$.
[^7]: These results agree with what was found in the analyses in Refs. [@Asaka:2016zib; @Hernandez:2016kel], which were performed in parallel to our analysis and appeared on arxiv.org in the same week. The main results of Ref. [@Hernandez:2016kel] had been presented by Pilar Hernandez at the MIAPP workshop *Why is there more Matter than Antimatter in the Universe?* the week before.
|
---
author:
- |
Anton Kapustin, Kevin Setter\
[*California Institute of Technology*]{}
title: 'Geometry of Topological Defects of Two-dimensional Sigma Models'
---
Introduction {#sec:intro}
============
It is well-known that two-dimensional sigma models with differing amounts of left and right-moving supersymmetry (e.g. the (0,1) and (0,2) supersymmetric sigma models) cannot be consistently defined on worldsheets with nontrivial boundary. This is essentially because a boundary condition must set to zero a linear combination of left- and right-moving fermions.
This does not however rule out the possibility of defining these models on a worldsheet equipped with *defects*, i.e. one dimensional submanifolds $D$ along which the fields of two distinct CFTs are glued together consistently. (A boundary condition is a special case in which one of the CFTs is trivial.) Our goal is to study supersymmetric defects in the (0,1) and (0,2) sigma models.
Defects of two dimensional CFTs have attracted much interest recently, especially in the context of rational conformal field theory and as a means of elegantly implementing dualities [@Kapustin:2009av], [@Davydov:2010rm], [@Sarkissian:2008dq], [@Fuchs:2007tx]. In this paper, we will analyze criteria for preserving superconformal symmetry, disregarding any enhanced chiral symmetry which the theories may possess; moreover, we will focus on the target space geometry of defects, and shall only briefly comment on their role in implementing dualities.
Consider a two-dimensional worldsheet disconnected into two domains $\Sigma$ and ${{\widehat \Sigma}}$ by a defect line $D$. On the domain $\Sigma$, one defines a sigma model of maps $\Phi : \Sigma \to X$ and on the domain ${{\widehat \Sigma}}$ one defines a sigma model of maps ${{\widehat \Phi}}: {{\widehat \Sigma}}\to {{\widehat X}}$, where $X$ and ${{\widehat X}}$ are two compact, Riemannian target spaces. The restriction of these maps to $D$ defines a product map $\Phi \times {{\widehat \Phi}}|_D : D \to X \times {{\widehat X}}$. As observed in [@Fuchs:2007fw], gluing conditions will require the product map to takes values in some submanifold $Y \subseteq X \times {{\widehat X}}$ of the product of the targets.
One may also include a term in the action coupling the bulk fields to a line bundle with connection living on the worldvolume of $Y$ (equipping it with a closed 2-form $F$), exactly by analogy with Chan-Paton bundles on branes for the case of sigma model boundary conditions. Borrowing the terminology of [@Fuchs:2007fw], we refer to the pair $(Y,F)$ as a *bibrane*.
To ensure that the defect theory preserves a specified set of symmetries of the bulk theories, we impose certain requirements on the geometry of the embedding $Y$, as well as on the choice of the 2-form $F$. Let us describe the relevant set of symmetries we wish to preserve by listing the associated Noëther charges. The (0,1) supersymmetry algebra in 1+1 dimensions reads $$\{Q_+,Q_+\} = H + P$$ where $Q_+$ is the single, real, right-moving supercharge, $H$ is the worldsheet energy, and $P$ is the worldsheet momentum. We wish to find $(Y,F)$ such that $H$, $P$, and $Q_+$ remain as conserved charges of the defect theory; since $H$ and $P$ will be separately conserved, we will be studying examples of what are known in the literature as *topological defects* [@Fuchs:2007tx].
Already in the case of the bosonic sigma model (i.e. with no fermionic fields present) it is an interesting question which bibranes $(Y,F)$ define topological defects, and it is to this that we will first direct our attention. It turns out to be natural to choose the neutral signature metric $G = g \oplus - {{\widehat g}}$ on the (pseudo-Riemannian) product manifold $M = X \times {{\widehat X}}$, where $g$ and ${{\widehat g}}$ are the positive-definite metrics on $X$ and ${{\widehat X}}$.
Those $(Y,F)$ that supply topological defects turn out to bear a structural resemblance to A-branes of symplectic manifolds (exchanging the symplectic form on the ambient target space for the neutral signature metric defined above); for instance, just as A-branes are coisotropic submanifolds with respect to the symplectic form on the target, $Y$ will be required to be “coisotropic" with respect to $G$, in a sense that we will explain in section \[sec:geometry\]. Moreover, we will describe two special classes of topological defects (the “graphs of isometries" and “half para-Kähler" defects), which are the analogs of Lagrangian and space-filling A-branes.
Alternatively, it turns out to be natural to employ the language of Hitchin’s generalized geometry [@Gualtieri:2007ng], in terms of which we obtain the following simple characterization: $(Y,F)$ define a topological defect of the bosonic sigma model if and only if the “$F$-rotated generalized tangent bundle" of $Y$ is stabilized by the “generalized metric" on $X \times {{\widehat X}}$. We will explain what these terms mean in greater depth in section \[sec:geometry\].
We refer to $(Y,F)$ satisfying the above stabilization condition as *topological bibranes*; they can be defined for any neutral signature manifold $M$ and are, perhaps, mathematically interesting in their own right. However, we should point out that the neutral signature manifolds relevant to the current physics discussion are of a very restricted type: namely, manifolds that can be expressed as a global product $M = X \times {{\widehat X}}$ such that the metric restricted to $TX$ (resp. $T {{\widehat X}}$) is positive (resp. negative) definite.
In section \[sec:gluing\] we discuss defect gluing conditions in general and say what it means for a gluing condition to preserve a symmetry of the bulk theories. In section \[sec:bosonic\] we write down gluing conditions on the fields corresponding to the choice of $(Y,F)$ and analyze the topological defect requirement. In section \[sec:geometry\], we explain the analogy with A-branes and reformulate the topological bibrane condition on $(Y,F)$ terms of generalized geometry. In section \[sec:susy01\] we analyze defects of the (0,1) supersymmetric sigma model, supplementing the bosonic gluing conditions with an additional fermionic gluing condition and studying the geometry of a certain middle dimensional subbundle of $TY$. In section \[sec:susy02\] we treat topological defects of the (0,2) sigma model, which we will describe as those $(Y,F)$ that are simultaneously A-branes and B-branes with respect to a certain symplectic form and complex structure. Finally, in section \[sec:tduality\] we briefly comment on a subset of our topological defects which implement dualities relating the sigma model theories on $\Sigma$ and ${{\widehat \Sigma}}$.
Proofs of selected propositions discussed in the text are offered in the appendix.
K.S. thanks Ketan Vyas for useful discussions. This work was supported in part by the DOE grant DE-FG02-92ER-40701.
Defect gluing conditions {#sec:gluing}
========================
Before discussing defects in specific theories, let us discuss defect gluing conditions in general and say what it means for a defect to preserve a symmetry of the bulk theories.
The variation of the action will, in general, consist of three types of terms: $$\delta S = (\delta S)_\Sigma + (\delta S)_{{\widehat \Sigma}}+ (\delta S)_D$$ where $ (\delta S)_\Sigma$ and $(\delta S)_{{\widehat \Sigma}}$ are integrals over the domains $\Sigma$ and ${{\widehat \Sigma}}$ respectively, which vanish for field configurations solving the bulk equations of motion, and the third term, $(\delta S)_D$, is an integral over $D$, which in general, will not vanish unless we impose a gluing condition on the fields along $D$. A gluing condition constrains the values of the fields along $D$ and hence also the set of *allowed variations*, by which we mean variations mapping one solution of the gluing condition to another solution. Therefore, the first requirement of a good gluing condition is that it sets $(\delta S)_D$ to zero identically for all allowed variations and that it does so “minimally" (i.e. without overconstraining the data along $D$).
Additionally, we may wish for the defect theory to preserve a certain symmetry of the bulk. Let $(\sigma^0, \sigma^1)$ be worldsheet coordinates in which $D$ is described locally as the set of points with $\sigma^1 = 0$. In these coordinates, we say that a gluing condition classically preserves a symmetry of the bulk if and only if the 1- component of the associated Noëther current glues continuously across $D$, thereby ensuring the existence of a conserved Noëther charge in the composite theory. (The quantum theory may develop an anomaly, but we will confine our discussion to the classical problem.) This being satisfied, the symmetry variation will automatically be among the allowed variations.
For instance, conservation of the worldsheet energy $H$ requires that the off-diagonal components of the stress-energy tensors glue continuously: $$T^{\, 1}_{\;\;\; 0} - {\widehat T}^{\,1}_{\;\;\; 0} = 0$$ at points of $D$. Defects satisfying this condition are said to be *conformal defects* [@Bachas:2001vj].
If, in addition, the defect gluing condition ensures that the diagonal components of the stress-energy tensor glue continuously: $$T^{\, 1}_{\;\;\; 1} - {\widehat T}^{\,1}_{\;\;\; 1} = 0$$ then worldsheet momentum $P$ is also a conserved charge. Defects satisfying this condition are called *topological defects* [@Fuchs:2007tx], due to the fact that the location of $D$ on the worldsheet can be deformed smoothly without affecting the values of correlators (so long as it does not cross through the location of a local operator insertion).
In subsequent sections, we will write down an action, vary the action, and then systematically analyze what gluing conditions set $(\delta S)_D$ minimally and ensure continuous gluing across $D$ of the 1-components of relevant Noëther currents.
Topological defects of the bosonic sigma model {#sec:bosonic}
==============================================
To begin, we analyze topological defects of the bosonic sigma model.
Let us fix notation. As above, let $(\sigma^0, \sigma^1)$ be worldsheet coordinates in which the defect line $D$ is given locally by $\sigma^1 = 0$, and the worldsheet metric is taken to be flat, with signature (-, +). Let $\Sigma$ be the domain given by $\sigma^1 \geq 0$, and ${{\widehat \Sigma}}$ be the domain given by $\sigma^1 \leq 0$. The fields of the bosonic sigma model with defect consist of maps $\Phi : \Sigma \to X$ and ${{\widehat \Phi}}:{{\widehat \Sigma}}\to {{\widehat X}}$; in terms of local coordinates $\phi^i$ on $X$ and ${{\hat \phi}}^i$ on ${{\widehat X}}$, we can describe these maps by functions $\phi^i (\sigma)$ for $\sigma^1 \geq 0$ and ${{\hat \phi}}^i (\sigma)$ for $\sigma^1 \leq 0$. (Note that the target spaces are assumed to have the same dimensionality $n$ since we are looking for topological – not merely conformal – defects.) Likewise, the product map $\Phi \times {{\widehat \Phi}}|_D : D \to X \times {{\widehat X}}$ can be described by the functions $\phi^I = (\phi^i, {{\hat \phi}}^i) |_{(\sigma^0, 0)}$. Indices $i,j$ range from $1,\dots n$ and $I,J$ range from $1,\dots 2n$.
The total action for the theory with defect is the sum of bulk terms and a term coupling the bulk fields to a connection on a rank one vector bundle living on the submanifold $Y \subseteq X \times {{\widehat X}}$: $$\begin{aligned}
S &= \int_{\Sigma} d^2 \sigma \Big(-\frac{1}{2} g_{ij} \partial_\mu \phi^i \partial^\mu \phi^j
-\frac{1}{2} b_{ij} \epsilon^{\mu \nu} \partial_\mu \phi^i \partial_\nu \phi^j \Big) \\
&+ \int_{{{\widehat \Sigma}}} d^2 \sigma \Big(-\frac{1}{2} {{\widehat g}}_{ij} \partial_\mu {{\hat \phi}}^i \partial^\mu {{\hat \phi}}^j - \frac{1}{2} {{\widehat b}}_{ij} \epsilon^{\mu \nu} \partial_\mu {{\hat \phi}}^i \partial_\nu {{\hat \phi}}^j \Big)
+ \int_{D} d \sigma^0 {{\mathcal A}}_I \partial_0 \phi^I\end{aligned}$$ where ${{\mathcal A}}= {{\mathcal A}}_I (\phi) d \phi^I$ is the connection 1-form, $g_{ij} (\phi)$ and $b_{ij}(\phi)$ are the metric and B-field on $X$, and ${{\widehat g}}_{ij} (\phi)$ and ${{\widehat b}}_{ij}(\phi)$ are the metric and B-field on ${{\widehat X}}$. For simplicity we assume that both $b$ and ${{\widehat b}}$ are closed 2-forms.
Varying this action and picking out the term localized on $D$, one finds $$\begin{aligned}
(\delta S)_D = \int_{D} d \sigma^0 \Big( &-(g_{ij} \partial_1 \phi^i + b_{ij} \partial_0 \phi^i)\, \delta \phi^j + ({{\widehat g}}_{ij} \partial_1 {{\hat \phi}}^i + {{\widehat b}}_{ij} \partial_0 {{\hat \phi}}^i)\, \delta {{\hat \phi}}^j \notag \\
&- (\partial_I {{\mathcal A}}_J - \partial_J {{\mathcal A}}_I) \, \partial_0 \phi^I \, \delta \phi^J \; \;\Big)
\label{eq:deltaS}\end{aligned}$$ Let $\sigma = (\sigma^0, 0)$ be a point on $D$ and let us write down gluing conditions on the fields at $\sigma$ sufficient to ensure vanishing of this expression for all allowed variations.
As mentioned in the introduction, a choice of gluing condition corresponds to a choice of a submanifold $Y \subseteq X \times {{\widehat X}}$. Let $y = \Phi \times {{\widehat \Phi}}|_D (\sigma)$ be the image of $\sigma$ under the product map. We impose first the Dirichlet condition $$y \in Y$$ If $k$ is the dimension of $Y$, with $0 \leq k \leq 2n$, then the above represents $2n-k$ independent gluing constraints on the $2n$ bosonic fields $\phi^I$. In order to ensure vanishing of , we supplement these Dirichlet conditions with a set of $k$ Neumann conditions on the derivatives $\partial_1 \phi^I$. It is easiest to describe these in terms of three target space tangent vectors $u,v,w \in T_y ( X \times {{\widehat X}})$, with $$\begin{aligned}
u &= u^I \partial_I = (\partial_0 \phi^I ) \partial_I \\
v &= v^I \partial_I = (\partial_1 \phi^I ) \partial_I \\
w &= w^I \partial_I = (\delta \phi^I) \partial_I \end{aligned}$$ where $\partial_I \equiv \frac{\partial}{\partial \phi^I}$. Constraining points of $D$ to be mapped to the submanifold $Y$ requires $u \in TY$, the subspace of vectors tangent to $Y$; moreover, allowed variations $\delta \phi^I$ are those such that $w \in TY$ as well.
Here and throughout, we shall only have occasion to discuss vectors that are evaluated at points of $Y$; hence, we regard all vectors as lying in the pullback bundle ${{e^*TM}}$, where $e:Y \to M = X \times {{\widehat X}}$ is the embedding map, and shall regard $TY$ as a subbundle $TY \subseteq {{e^*TM}}$. Since the target space metrics $g$ and ${{\widehat g}}$ enter $(\delta S)_D$ with opposite signs, we find it useful to define a *neutral signature* metric and B-field on $X \times {{\widehat X}}$ by $$G_{I J} = \begin{pmatrix} g_{ij} & 0 \\ 0 & -{{\widehat g}}_{ij} \end{pmatrix}, \quad
B_{I J} = \begin{pmatrix} b_{ij} & 0 \\ 0 & -{{\widehat b}}_{ij} \end{pmatrix}$$ Moreover, we write $F = -{{\mathcal F}}- e^* B$ for closed 2-form on $Y$ obtained by combining the curvature ${{\mathcal F}}= d{{\mathcal A}}$ of the line bundle and the pullback of the B-field. Using these definitions, we may state the Neumann conditions as follows: $$G(u,w) = F(v,w)$$ for all $w \in TY$. As promised this represents $k$ independent conditions, one for each linearly independent tangent vector $w$.
Together, the pair $(u,v)$ satisfying the above gluing conditions are said to be an *allowed pair* of tangent vectors.
Having written down good gluing conditions on the fields, let us now analyze what additional constraints the topological defect condition places on allowed pairs; this will constrain the choice of submanifold $Y$ as well as the choice of 2-form $F$ on its worldvolume. First of all, the conformal defect condition $ T^{\, 1}_{\;\;\; 0} - {\widehat T}^{\,1}_{\;\;\; 0} = 0$ is equivalent to $$G(u,v) = 0$$ for all allowed pairs $(u,v)$. This is an automatic consequence of the antisymmetry of $F$ since, if $(u,v)$ is an allowed pair, then $ G(u,v) = G(v,u) = F(u,u) = 0$, where we have applied the definition of allowed pair, setting $w = u$. Therefore, $(Y,F)$ automatically defines a conformal defect of the bosonic sigma model.
Further requiring the topological condition $T^{\, 1}_{\;\;\; 1} - {\widehat T}^{\,1}_{\;\;\; 1} = 0 $ is equivalent to requiring $$G(u,u) = - G(v,v)$$ for all allowed pairs $(u,v)$. This condition turns out to be equivalent to an apparently stronger condition, as follows.
\[prop:stronger\] If $(Y,F)$ are such that $G(u,u) = -G(v,v)$ for all allowed pairs $(u,v)$, then $v \in TY$ and $(v,u)$ is an allowed pair as well.
(See appendix for the proof of this proposition and other proofs which are not immediate.) The converse is true as well, since if $(Y,F)$ satisfy the conditions of this proposition, then by setting $w=u$, one has $G(u,u) = F(v,u) = -F(u,v) = -G(v,v)$. Hence, we arrive at the following:
\[prop:topbibrane\] Suppose $Y \subseteq X \times {{\widehat X}}$ is a submanifold and $F$ is a closed 2-form on its worldvolume. Then $(Y,F)$ supplies a topological defect of the bosonic sigma model if and only if the following condition is met:
- If $(u,v)$ is a pair of vectors such that $u \in TY, v \in {{e^*TM}}$ and $G(v,w)=F(u,w)$ for all $w \in TY$, then
- $v \in TY$ as well, and $G(u,w) = F(v,w)$ for all $w \in TY$.
A pair $(Y,F)$ satisfying the conditions of the preceding theorem are said to define a *topological bibrane*.
Geometry of topological bibranes {#sec:geometry}
================================
Having obtained a characterization of topological bibranes $(Y,F)$ above, we reformulate this condition slightly to put it in a more understandable form.
We have chosen to equip the manifold $M = X \times {{\widehat X}}$ with a neutral signature metric $G$; let us therefore record some basic facts from the theory of submanifolds of indefinite signature spaces. See [@Bejancu:2006p569] for a more complete discussion.
Let $Y$ be an embedded submanifold of a $2n$-dimensional pseudo-Riemannian manifold $M$ equipped with a nondegenerate, symmetric metric $G$. The orthogonal subbundle ${{(TY)^\bot}}$ is defined to the be the set of vectors that are $G$-orthogonal to all of $TY$: $${{(TY)^\bot}}= \{ u \in {{e^*TM}}: G(u,v) = 0 \qquad \text{for all $v \in TY$}\}$$
If $TY$ is $k$-dimensional, then ${{(TY)^\bot}}$ is $(2n-k)$-dimensional. The main difference between the theory of pseudo-Riemannian submanifolds as compared with the theory of Riemannian submanifolds is that the restriction of the metric $G$ to $Y$ can develop degenerate directions. Hence, the subbundle $\Delta = TY \cap {{(TY)^\bot}}$ will in general be nontrivial. Borrowing terminology from symplectic geometry, we define the following three special classes of submanifolds:
Depending on whether the bundle $TY$, regarded as a subbundle of ${{e^*TM}}$, contains (or is contained by) its orthogonal bundle ${{(TY)^\bot}}$, we say that
- $TY$ is *isotropic* if $TY \subseteq {{(TY)^\bot}}$
- $TY$ is *coisotropic* if $TY \supseteq {{(TY)^\bot}}$
- $TY$ is *Lagrangian* if $TY = {{(TY)^\bot}}$
Hence, the dimension $k$ lies in the range $0 \leq k \leq n$ for isotropic subbundles, $n \leq k \leq 2n$ for coisotropic subbundles, and all Lagrangian subbundles are $n$ dimensional. (Lagrangian subbundles exist only when the signature of $G$ is $(n,n)$.) In the proof of proposition \[prop:stronger\] it was shown that the submanifolds $Y$ corresponding to topological bibranes have the property that orthogonal vectors are also tangent to the submanifold; in our classification, they are coisotropic submanifolds. Therefore, in the following we focus on coisotropic submanifolds (which include Lagrangian submanifolds as a special case).
It is convenient to define a particular frame for ${{e^*TM}}$ adapted to the submanifold $Y$. This is not quite as straightforward as in the Riemannian case, since a canonical splitting ${{e^*TM}}= TY \oplus {{(TY)^\bot}}$ is no longer available. However, for coisotropic submanifolds there exists [@Bejancu:1995p230] a splitting of the tangent bundle of the form $${{e^*TM}}= \underbrace{{{(TY)^\bot}}\oplus SY}_{TY} \oplus NY$$ where $SY$ is a complementary *screen* distribution to ${{(TY)^\bot}}$ within $TY$ and $NY$ is a complementary *transverse* distribution to $TY$ in ${{e^*TM}}$. We have $\dim {{(TY)^\bot}}= \dim NY = 2n-k$ and $\dim SY = 2k - 2n$.
Making use of this adapted frame, let us now write down an equivalent characterization of topological bibranes:
\[prop:coisotropic\] The pair $(Y,F)$ define a topological bibrane if and only if $Y \subseteq X \times {{\widehat X}}$ is a coisotropic submanifold such that $\ker F = {{(TY)^\bot}}$ (i.e. the degenerate directions of $F$ and $G|_{TY}$ coincide) and, additionally, $$({\widetilde}G^{-1} {\widetilde}F)^2 = +1$$ on $SY$, where ${\widetilde}G \equiv G|_{SY}$ and ${\widetilde}F \equiv F|_{SY}$.
This is remarkably similar to the characterization of A-brane boundary conditions of the $(2,2)$ supersymmetric sigma model given in [@Kapustin:2001ij], where the role of the antisymmetric symplectic form $\Omega$ on the target is exchanged for a neutral signature, symmetric metric $G$. Just as A-branes are required to be coisotropic with respect to $\Omega$, topological bibranes are required to be coisotropic with respect to $G$. Moreover, for A-branes the quotient bundle $TY / (TY)^\Omega$ is equipped with an endomorphism given by $\Omega^{-1} F$ and squaring to -1; similarly, for topological bibranes the quotient bundle $TY / {{(TY)^\bot}}$ is equipped with an endomorphism $G^{-1} F$ squaring to +1.
We consider two important special classes of topological bibranes. When $F=0$, the condition that the degenerate directions of $F$ and $G|_{TY}$ agree implies that $TY = {{(TY)^\bot}}$, i.e. $Y$ is a Lagrangian submanifold of $X \times {{\widehat X}}$. This in turn implies that, locally, $Y$ is the graph of an isometry $f: X \to {{\widehat X}}$. These *graph-of-isometry* type bibranes are the analogs of Lagrangian A-branes.
The other special class of topological bibranes are the *space-filling* bibranes with $Y = X \times {{\widehat X}}$. In this case, ${{(TY)^\bot}}$ is trivial and the screen distribution $SY$ is all of $TY$. The condition in proposition \[prop:coisotropic\] then implies that $F$ is a symplectic form such that $(G^{-1} F)^2 = 1$. Space-filling bibranes are the topological bibrane analog of space-filling A-branes.
Indeed, if a manifold $M$ carries a symplectic form $F$, an almost product structure $R$, and a neutral signature metric $G$ with the compatibility requirement $$F(u,v) = G(R u, v)$$ for all vector fields $u,v$, and if in addition one demands that $$dF = 0$$ then $M$ is said to be an *almost para-Kähler manifold* [@Alekseevsky:2009p273]. Alternatively, the triplet of structures $(F,R,G)$ satisfying the above is known as an *almost bi-Lagrangian structure* on the manifold [@Etayo:2006p347], which terminology is inspired by the fact that the +1 and -1 eigenbundles of $R$ form two complementary Lagrangian subbundles with respect to the symplectic form $F$. In case one of these subbundles is integrable, we call the manifold *half para-Kähler* and in case both subbundles are integrable, we call the manifold *para-Kähler*. Integrability of one of the eigenbundles need not imply integrability of the other eigenbundle. Indeed, in section \[sec:susy01\] we will require integrability of just the positive eigenbundle in order to ensure supersymmetry. The study of para-Kähler manifolds is a rich and developing subbranch of indefinite signature geometry; they have appeared in an unrelated physical context in [@Cortes:2003zd].
Another useful way to reformulate the topological bibrane condition is in terms of the language of *generalized geometry* (see [@Gualtieri:2007ng] for a review). In the framework of generalized geometry one only works with the sum $TM \oplus T^* M$ as well as sections thereof (rather than $TM$ or $T^*M$ in isolation). In particular, one speaks of *generalized tangent vectors* of $M$, defined as pairs $(u,\xi)$ with $u \in TM$, $\xi \in T^* M$.
The objects of ordinary of geometry have generalized counterparts; for instance, one defines the *generalized tangent subbundle* of a submanifold $Y \subseteq M$ to be sum of its tangent bundle and conormal bundle (regarded as a subbundle of $T^*M$): $$\tau Y = TY \oplus N^*Y$$ Generalized geometry provides a prescription for incorporating a nonzero 2-form $F$:
The *F-rotated generalized tangent subbundle* of a submanifold $Y \subseteq M$ is the set of generalized tangent vectors $(u,\xi)$ such that $$\tau^F Y = \{ (u,\xi): u \in TY, \xi \in T^*M \quad \text{such that $F u = e^*\xi$} \}$$ Here $e:Y \to M$ is the embedding map and $F u$ represents the 1-form produced by contracting the 2-form $F$ on the vector $u$.
What we have been calling allowed pairs of tangent vectors $(u,v)$ are nothing but sections of $\tau^F Y$ (after lowering the vector $v$ to a 1-form $\xi = G v$).
The generalized counterparts of complex structures are the *generalized complex structures*: endomorphisms $\mathcal J : TM \oplus T^*M \to TM \oplus T^*M$ with $\mathcal J ^2 = -1$. Symplectic manifolds with symplectic form $\Omega$ carry a generalized complex structure given by $$\mathcal J_\Omega = \begin{pmatrix} & -\Omega^{-1} \\ \Omega & \end{pmatrix}$$ (referring coordinates on $T^*M$ to a dual basis) and the A-branes are elegantly characterized as those $(Y,F)$ such that the F-rotated generalized tangent subbundle is stabilized by the $\mathcal J_\Omega$:
(in reference [@Gualtieri:2007ng]) (Y,F) define an A-brane of a symplectic manifold if and only if $\mathcal J_\Omega$ acting on a section of $\tau^F Y$ gives back another section of $\tau^F Y$, i.e. $$\mathcal J_\Omega (\tau^F Y) = \tau^F Y$$
Motivated by our theme of comparing topological bibranes of the bosonic sigma model with A-branes of the $(2,2)$ sigma model by replacing $\Omega$ with the neutral signature $G$ and replacing almost complex structures with almost product structures, let us define a *generalized (almost) product structure* to be an endomorphism $\mathcal R : TM \oplus T^*M \to TM \oplus T^*M$ with $\mathcal R ^2 = 1$. Indeed, the generalized geometry object encoding the metric is a generalized almost product structure:
As defined in [@Gualtieri:2007ng], the *generalized metric*[^1] is the particular generalized almost product structure given by $$\mathcal{R}_G = \begin{pmatrix} & G^{-1} \\ G & \end{pmatrix}.$$
Finally, we can compactly describe topological bibranes as follows.
\[prop:generalizedmetric\] The pair $(Y,F)$ with $Y \subseteq M$ and $F$ a closed 2-form on $Y$, satisfy the conditions for a topological bibrane (as defined in the previous section) if and only if $\mathcal R_G$ acting on a section of $\tau^F Y$ gives back another section of $\tau^F Y$, i.e. $$\mathcal R_G (\tau^F Y) = \tau^F Y$$
Topological defects of the (0,1) supersymmetric sigma model {#sec:susy01}
===========================================================
We now proceed to our main topic of interest: supersymmetric, topological defects of the (0,1) supersymmetric sigma model. In addition to the bosonic fields $\phi^i$ and ${{\hat \phi}}^i$ we now have right-moving fermionic fields ${\psi_+}^i$ on $\Sigma$ and ${\widehat \psi_+}^i$ on ${{\widehat \Sigma}}$.
The bosonic gluing conditions described in section \[sec:bosonic\] must be supplemented with fermionic gluing conditions on the fields $\psi^I = (\psi^i, {{\widehat \psi}}^i)$, setting half of the total fermionic degrees of freedom to zero the defect line $D$; we accomplish this by constraining $\psi^I$ to take values in an $n$-dimensional subbundle – which we denote ${{\mathcal R_+}}$ – of the $2n$-dimensional bundle ${{e^*TM}}$.
Indeed, topological bibranes $(Y,F)$ are equipped with a natural middle dimensional subbundle, which we describe as follows
Let $(Y,F)$ be a topological bibrane, as characterized in propositions \[prop:topbibrane\], \[prop:coisotropic\], and \[prop:generalizedmetric\]. Define ${{\mathcal R_+}}$ be the following bundle on $Y$: $$\label{eq:rplus}
{{\mathcal R_+}}= \{ u \in TY: G(u,w) = F(u,w) \quad \text{for all $w \in TY$} \}$$
Let us see what this subbundle corresponds to in the two special classes of topological bibranes we discussed previously. In case $(Y,F)$ is space-filling, ${{\mathcal R_+}}$ is the +1 eigenbundle of the almost product structure $G^{-1} F$. On the other hand, for graph-of-isometry type bibranes, this subbundle is simply the tangent bundle $TY$ itself.
\[prop:rplus\] The subbundle ${{\mathcal R_+}}$, as defined above, is Lagrangian with respect to ${{e^*TM}}$. In particular, it is $n$-dimensional.
Given $(Y,F)$ satisfying the geometric topological bibrane conditions, we impose the bosonic gluing conditions written previously, and we augment these with condition that the fermions take values in ${{\mathcal R_+}}$; moreover, we require that ${{\mathcal R_+}}$ be an integrable distribution on $Y$. For completeness, we record here the full set of gluing conditions: $$\begin{aligned}
\phi &\in Y \\
G(v,w) &= F(u,w) \\
G(s,w) &= F(s,w) \end{aligned}$$ for all $w \in TY$, where $u^I \equiv \partial_0 \phi^I$, $v^I \equiv \partial_1 \phi^I$, and $s^I \equiv \psi^I$.
Let us now show that these gluing conditions define a topological, supersymmetry-preserving defect. We write an explicit action for the $(0,1)$ supersymmetric sigma model with defect (see [@Melnikov:2003zv]): $$\begin{aligned}
S &= \int_{\Sigma} d^2 \sigma \Big(-\frac{1}{2} g_{ij} \partial_\mu \phi^i \partial^\mu \phi^j
-\frac{1}{2} b_{ij} \epsilon^{\mu \nu} \partial_\mu \phi^i \partial_\nu \phi^j + \frac{i}{2} g_{ij} \psi_+^i D_- \psi_+^j \Big) \\
&+ \int_{{{\widehat \Sigma}}} d^2 \sigma \Big(-\frac{1}{2} {{\widehat g}}_{ij} \partial_\mu {{\hat \phi}}^i \partial^\mu {{\hat \phi}}^j -\frac{1}{2} {{\widehat b}}_{ij} \epsilon^{\mu \nu} \partial_\mu {{\hat \phi}}^i \partial_\nu {{\hat \phi}}^j + \frac{i}{2} {{\widehat g}}_{ij} {{\widehat \psi}}_+^i {{\widehat D}}_- {{\widehat \psi}}_+^j \Big) + \int_{D} d \sigma^0 {{\mathcal A}}_I \partial_0 \phi^I\end{aligned}$$ where $\partial_\pm = \partial_0 \pm \partial_1$ and the covariant derivatives are given by $$\begin{aligned}
D_{\pm} {\psi_+}^i &= \partial_{\pm} {\psi_+}^i + \partial_{\pm} \phi^j \Gamma^i_{jk} {\psi_+}^k \\
{{\widehat D}}_{\pm} {\widehat \psi_+}^i &= \partial_{\pm} {\widehat \psi_+}^i + \partial_{\pm} {{\hat \phi}}^j \hat \Gamma^i_{jk} {\widehat \psi_+}^k\end{aligned}$$ Varying the action and picking out the term localized on $D$, one obtains $$(\delta S)_D = \int_{D} d \sigma^0 \Big( -G_{IJ} \partial_1 \phi^I \delta \phi^J
+ F_{IJ} \partial_0 \phi^I \delta \phi^J - \frac{i}{2} G_{IJ} {\psi_+}^I \delta^{\rm cov} {\psi_+}^J \Big)$$ where $G$ and $F$ are as before and we have introduced the notation $$\delta^{\rm cov} {\psi_+}^I = \delta {\psi_+}^I + \delta \phi^{J} \Gamma^I_{JK} {\psi_+}^K$$ for the *covariant variations*, which, unlike $\delta \psi^I$, transform covariantly under target space coordinate changes. Here $\Gamma^I_{JK}$ are the coefficients of the Levi-Civita connection of the neutral signature metric $G$ on $X \times {{\widehat X}}$.
The first two terms in $(\delta S)_D$ vanish identically since the bosonic gluing conditions are identical to those in section \[sec:bosonic\]. Moreover, the third term in $(\delta S)_D$ can be shown to vanish by appealing to the following
\[prop:parallelness\] Let $(M,G)$ be a (pseudo-)Riemannian manifold with Levi-Civita connection $\nabla$. Let $Y \subseteq M$ be a submanifold equipped with closed 2-form $F$ on its worldvolume, and let ${{\mathcal R_+}}$ be the subbundle defined in . If ${{\mathcal R_+}}$ is integrable, then we have the following: $$(e^*\nabla)_u s \in \Gamma({{\mathcal R_+}})$$ for all $u \in \Gamma(TY)$ and $s \in \Gamma({{\mathcal R_+}})$. Here $e^* \nabla$ is the pullback of $\nabla$ under the embedding map $e: Y \to M$. In words, the subbundle ${{\mathcal R_+}}$ is $\nabla$-parallel with respect to $TY$.
\[prop:vanishing\] The term $G_{IJ} {\psi_+}^I \delta^{\rm cov} {\psi_+}^J$ vanishes for all allowed variations.
Note that in case $F=0$, the subbundle ${{\mathcal R_+}}$ equals $TY$, and the above proposition reduces to the surprising result that the covariant derivative of sections of $TY$ are again in $TY$, i.e. Lagrangian submanifolds are automatically *totally geodesic*, a result also obtained in [@Bejancu:1995p230], [@Alekseevsky:2009p273].
We now check that the above gluing conditions define a topological defect and that they preserve (0,1) supersymmetry of the theory. The bosonic portions of the stress-energy tensors automatically glue smoothly by the analysis of section \[sec:bosonic\]. It remains to check that the contributions involving fermions glue smoothly; i.e. $$(T^{\, \mu}_{\;\;\; \nu})^f - ({\widehat T}^{\,\mu}_{\;\;\; \nu})^f = 0$$ for all $\mu, \nu$. This is equivalent to the condition $$G(\nabla_u s, s) = 0$$ where the vector fields $u,s$ are as we have defined previously. This equation is satisfied, once again by appeal to our parallelness proposition.
Finally, the condition for preservation of supersymmetry is that the 1-component of the supercurrents glue smoothly. This is equivalent to $$G(s, t) = 0$$ where $s \equiv {\psi_+}$ and $t \equiv \partial_+ \phi$. This is satisfied by virtue of the fact that both $\partial_+ \phi$ and ${\psi_+}$ are constrained to take values in ${{\mathcal R_+}}$ by the gluing conditions, and the fact that ${{\mathcal R_+}}$ is a Lagrangian subbundle.
Topological defects of the $(0,2)$ supersymmetric sigma model {#sec:susy02}
=============================================================
Consider now the case when the target spaces $X$ and ${{\widehat X}}$ are Kähler manifolds with Kähler forms $\omega$ and ${{\widehat \omega}}$, complex structures $j$ and ${{\widehat j}}$. In this case the supersymmetric sigma models on either side of the defects possess $(0,2)$ supersymmetry; let us geometrically classify those topological bibranes $(Y,F)$ that preserve both supercharges.
We impose the gluing conditions written in the previous section and assume that the conditions for $(0,1)$ supersymmetry are met. Then $(0,2)$ supersymmetry will follow if $$G'^1 - \widehat G'^1 = 0$$ along $D$, where $G'$ and $\widehat G'$ are the second supercurrents. This is the condition $$\label{eq:omega}
\Omega(s,t) = 0$$ for all $s,t \in {{\mathcal R_+}}$, where $$\Omega = G J$$ is the Kähler form on $X \times {{\widehat X}}$ with respect to complex structure $J = j \oplus {{\widehat j}}$. We can say the following about when is satisfied:
The following conditions on topological bibrane $(Y,F)$ are equivalent:
1. ${{\mathcal R_+}}$ is an $\Omega$-Lagrangian subbundle.
2. $J({{\mathcal R_+}}) = {{\mathcal R_+}},$ i.e. the leaves of the foliation by subbundle ${{\mathcal R_+}}$ are complex submanifolds with respect to complex structure $J = j \oplus {{\widehat j}}$.
3. $Y$ is a complex submanifold of $X \times {{\widehat X}}$ and $F$ has type $(1,1)$ with respect to the induced complex structure on $Y$.
4. In generalized geometry terms, $\mathcal J_J$ acting on a section of $\tau^F Y$ gives back another section of $\tau^F Y$, where $$\mathcal J_J = \begin{pmatrix} J & \\ & -J^{{\mathsf T}}\end{pmatrix}$$ is the generalized complex structure associated with complex structure $J$.
5. $(Y,F)$ is a B-brane with respect to complex structure $J$.
The generalized metric $\mathcal{R}_G$ and the generalized complex structures $\mathcal J_\Omega$ and $\mathcal J_J$ are related by $$\mathcal{R}_G = -\mathcal J_\Omega \mathcal J_J$$ This shows that if the bundle $\tau^F Y$ is stabilized by any two of these, then it is automatically stabilized by the third. In particular, we have the following
\[prop:02defects\] The bibrane $(Y,F)$ is a topological, supersymmetry preserving defect of the $(0,2)$ sigma model when it is both an A-brane with respect to symplectic form $\Omega = \omega \oplus -{{\widehat \omega}}$ and a B-brane with respect to complex structure $J = j \oplus {{\widehat j}}$ on $X \times {{\widehat X}}$ (with suitable integrability conditions).
Topological defects and T-duality {#sec:tduality}
=================================
We briefly comment on the role our topological defects play in implementing dualities of the bosonic sigma models with targets $X$ and ${{\widehat X}}$. For concreteness let $X$ and ${{\widehat X}}$ be flat tori and let $Y = X \times {{\widehat X}}$ be a space-filling defect equipped with a line bundle with curvature ${{\mathcal F}}= d{{\mathcal A}}$; we define the 2-form $F = -{{\mathcal F}}- B$, as in section 2.
Topological defects separating theories $X$ and ${{\widehat X}}$ can be fused with those separating ${{\widehat X}}$ and $X'$ to yield topological defects separating $X$ and $X'$. The invisible defect separating $X$ and $X$ (corresponding to the diagonal $Y \subset X \times X$ and $F=0$) is a unital element with respect to this operation of fusion and an *invertible defect* is one that can be fused with another defect to yield the invisible defect. On general grounds, an invertible topological defect implements a duality[^2].
Hence, we expect that if the 2-form ${{\mathcal F}}$ is chosen such that $(Y,F)$ is an invertible topological defect, then the sigma model with target $X$ will be related to the sigma model with target ${{\widehat X}}$ by a duality transformation. For tori, the group of duality transformations is $O(n,n;{{\mathbb Z}})$ (the group of automorphisms of a lattice of signature $(n,n)$) [@Giveon:1994fu].
Let us fix the background data $(g,b)$ and take $${{\mathcal F}}= \begin{pmatrix} f & h \\ -h^{{\mathsf T}}& {{\widehat f}}\end{pmatrix}.$$ where $f, {{\widehat f}}$ are antisymmetric $n \times n$ matrices. Since ${{\mathcal F}}$ is the curvature of a line bundle on the torus $X \times {{\widehat X}}$, its entries are constrained to be integers. Generically, the determinant of the off-diagonal matrix $h$ can take any integer value; however, let us now restrict the form of ${{\mathcal F}}$ by assuming that $\det h = \pm 1$, or equivalently $h \in GL(n,{{\mathbb Z}})$.
The topological defect condition for space-filling defects states that $(G^{-1} F)^2 = 1$; writing this out in terms of the blocks, this is equivalent to the following relationship between the background data $(g,b)$ for $X$ and $({{\widehat g}}, {{\widehat b}})$ for ${{\widehat X}}$: $$\begin{aligned}
{{\widehat g}}&= h^{{\mathsf T}}\big( g - (b+f) g^{-1} (b+f) \big)^{-1} h \\
0 &= (b+f) g^{-1} h + h {{\widehat g}}^{-1}({{\widehat b}}- {{\widehat f}}) \end{aligned}$$ We recognize this as the duality transformation consisting of a shift of the B-field by the matrix $f$, followed by a T-duality (inverting the data $g+b$), followed by a basis change of the lattice generating the torus (parametrized by unimodular matrix $h$), followed by another B-field shift (this time by ${{\widehat f}}$) (see [@Giveon:1994fu], eqs. (2.4.25), (2.4.26), (2.4.39)). The corresponding duality element is $$\begin{pmatrix}
{{\widehat f}}h^{-1} & h^{{\mathsf T}}+ {{\widehat f}}h^{-1} f \\
h^{-1} & h^{-1} f
\end{pmatrix}
\in O(n,n;{{\mathbb Z}})$$ (A similar calculation with $f = {{\widehat f}}= 0$ and $h=1$ is performed in [@Sarkissian:2008dq].) It is evident from this expression that quantization of the entries is implied by quantization of the entries of ${{\mathcal F}}$.
Conclusions {#sec:conclusions}
===========
We have provided several equivalent characterizations of topological defects of the bosonic sigma model (“topological bibranes") in propositions \[prop:topbibrane\], \[prop:coisotropic\], and \[prop:generalizedmetric\]: on the one hand, topological bibranes are closely related to the A-branes of symplectic geometry (exchanging the symplectic form for a neutral signature metric); on the other hand, one can provide an elegant description of topological bibranes in terms of the generalized metric on the product of the targets. Dimension $n$ topological bibranes correspond to graphs of isometries $X \to {{\widehat X}}$, and dimension $2n$ topological bibranes are equipped with a (half) para-Kähler structure, i.e. a neutral signature metric $G$ together with an endomorphism $R=G^{-1} F$ squaring to the identity, such that $R^{{\mathsf T}}G R = -G$.
Fermionic fields are incorporated by requiring the fermions to take values in a certain middle dimensional subbundle ${{\mathcal R_+}}$ with which topological bibranes are equipped. We have shown the key result that this subbundle is automatically parallel with respect to all of $TY$ (assuming integrability of ${{\mathcal R_+}}$ and $dF = 0$).
We remark that the class of topological defects studied in this paper is not broad enough to close under the operation of fusion, due to the fact that we have not allowed our defects to include extra degrees of freedom living on the interface. For this reason, we leave a detailed study of fusion to future work.
Another question is to what extent the preservation of the various symmetries (which we have analyzed here on the classical level) carry over to the quantum level. Presumably, one can argue that the invertible defects are exactly supersymmetric and topological, but it would be interesting to find out which other of our defects have this property.
The present work was motivated in part by a desire to understand whether something analogous to D-branes can be defined for the heterotic string. Since the worldsheet of the heterotic string possesses only right-moving fermions, a worldsheet-with-boundary is ruled out, but perhaps not a worldsheet-with-defect. It would be quite interesting to study applications of the present analysis of defects of the bosonic, (0,1), and (0,2) sigma models to heterotic string theory.
Proofs of selected propositions {#app:proofs}
===============================
In this appendix we offer proofs of selected propositions discussed in the text.
If $(Y,F)$ are such that $G(u,u) = -G(v,v)$ for all allowed pairs $(u,v)$, then $v \in TY$ and $(v,u)$ is an allowed pair as well.
First, we show that $Y$ is necessarily a coisotropic submanifold (in the terminology of section \[sec:geometry\]). This is because $v_0 \in {{(TY)^\bot}}$ implies that $(0,v_0)$ is an allowed pair. Then $$G(v_0, v_0) = -G(0,0) = 0$$ This, in turn, implies that ${{(TY)^\bot}}$ is an isotropic subspace, since $$G(v_0, v_0 ') = \frac{1}{2} G(v_0 + v_0', v_0 + v_0') = 0$$ for all $v_0, v_0' \in {{(TY)^\bot}}$. That is to say, ${{(TY)^\bot}}\subseteq (TY^\bot)^\bot = TY$, which is what it means for $Y$ to be coisotropic.
Next, we show that if $(u,v)$ is an allowed pair, then $v$ is in the subspace $TY$. (The definition of allowed pair requires $u \in TY$ but not, a priori, $v \in TY$.) Let $v_0 \in {{(TY)^\bot}}$. For any value of a real parameter $t$, it is easy to see that $(u, v+ t v_0)$ is also an allowed pair. Hence, $$-G(u,u) = G(v+ t v_0, v + t v_0)$$ for all $t$. Differentiating both sides of this equation with respect to $t$, we have $$0 = \frac{d}{dt} G(v + t v_0, v + t v_0) = 2 G(v, v_0)$$ (since $G(v,v)$ is $t$ independent and $G(v_0, v_0) = 0$). The choice of $v_0 \in {{(TY)^\bot}}$ was arbitrary, so $$v \in ({{(TY)^\bot}})^\bot = TY$$ Finally, we show that $(u,v)$ an allowed pair implies $(v,u)$ is an allowed pair as well. By linearity, if $(u,v)$ and $(u', v')$ are allowed pairs, then $(u+u', v+v')$ is also an allowed pair. By assumption, $$G(u+u', u+u') = -G(v+v',v+v')$$ which, using $G(u,u) = -G(u',u')$ and $G(u', u') = -G(v', v')$, implies that $$\label{eq:Guv}
G(u, u') = -G(v, v') .$$ Let $(u,v)$ be an allowed pair and $w \in TY$ be arbitrary. It is not hard to see that there exists a $v' \in TY$ such that $(w,v')$ is an allowed pair. Setting $u' = w$ in above, one finds that $$G(u,w) = -G(v,v') = -G(v', v) = -F(w,v) = F(v,w)$$ We conclude that $(v,u)$ is also an allowed pair.
The pair $(Y,F)$ define a topological bibrane if and only if $Y \subseteq X \times {{\widehat X}}$ is a coisotropic submanifold such that $\ker F = {{(TY)^\bot}}$ (i.e. the degenerate directions of $F$ and $G|_{TY}$ coincide) and, additionally, $$({\widetilde}G^{-1} {\widetilde}F)^2 = +1$$ on $SY$, where ${\widetilde}G \equiv G|_{SY}$ and ${\widetilde}F \equiv F|_{SY}$.
To prove “only if,” suppose that $(Y,F)$ is a topological bibrane. Theorem \[prop:stronger\] tells us that if $(u,0)$ is an allowed pair then $(0,u)$ is an allowed pair as well, and vice versa; this implies that $\ker F = {{(TY)^\bot}}$. Hence, the degenerate directions of $F$ and $G|_{TY}$ coincide and ${\widetilde}F$ and ${\widetilde}G$ (the restrictions of $G$ and $F$ to the screen distribution $SY$) are both nondegenerate.
Moreover, if ${\widetilde}u, {\widetilde}v \in SY$, then by definition, $({\widetilde}u, {\widetilde}v)$ is an allowed pair if and only if ${\widetilde}v = ({\widetilde}G^{-1} {\widetilde}F) {\widetilde}u$. Since $(u, ({\widetilde}G^{-1} {\widetilde}F) {\widetilde}u)$ is an allowed pair, $(({\widetilde}G^{-1} {\widetilde}F) {\widetilde}u,u )$ is an allowed pair as well (again, by theorem \[prop:stronger\]). This means that ${\widetilde}u = ({\widetilde}G^{-1} {\widetilde}F)^2 {\widetilde}u$ and since ${\widetilde}u \in SY$ was arbitrary, we have $$({\widetilde}G^{-1} {\widetilde}F)^2 = 1$$ To prove “if," let $(u,v)$ be an allowed pair; we wish to show that $v \in TY$ and $(v,u)$ is an allowed pair as well.
Let $v_0 \in {{(TY)^\bot}}$ be arbitrary. We then have $$G(v, v_0) = F(v, v_0) = 0$$ where the first equality is by the definition of allowed pair and the second equality is by the fact that $\ker F = {{(TY)^\bot}}$. Hence $v \in ({{(TY)^\bot}})^\bot = TY$.
Now, we write $u = u_0 + {\widetilde}u$ and $v = v_0 + {\widetilde}v$ for the decomposition of $TY$ vectors with respect to the splitting $TY = {{(TY)^\bot}}\oplus SY$. Since $F(u,w) = G(v,w)$ for all $w \in TY$, we have $F({\widetilde}u, {\widetilde}w) = G({\widetilde}v,{\widetilde}w)$ for all ${\widetilde}w \in SY$, or ${\widetilde}v = ({\widetilde}G^{-1} {\widetilde}F) {\widetilde}u$. But $ ({\widetilde}G^{-1} {\widetilde}F)^2 = 1$, so ${\widetilde}u = ({\widetilde}G^{-1} {\widetilde}F) {\widetilde}v$. Hence, $F({\widetilde}v, {\widetilde}w) = G({\widetilde}u,{\widetilde}w)$ for all ${\widetilde}w \in SY$, and finally $F(v,w) = G(u,w)$ for all $w \in TY$.
The subbundle ${{\mathcal R_+}}$, as defined above, is Lagrangian with respect to ${{e^*TM}}$. In particular, it is $n$-dimensional.
It is convenient to first refine the description of the tangent bundles of topological bibranes given in proposition \[prop:coisotropic\]. Let ${\widetilde}R$ be the endomorphism ${\widetilde}G^{-1} {\widetilde}F: SY \to SY$, which by proposition \[prop:coisotropic\], squares to the identity. Hence the screen distribution admits a splitting $SY = {\widetilde}{{{\mathcal R_+}}} \oplus {\widetilde}{\mathcal R_-}$ into the +1 and -1 eigenbundles of ${\widetilde}R$. Indeed, since ${\widetilde}G^{-1} {\widetilde}F$ is the product of a symmetric and an antisymmetric matrix, it is traceless, and in particular, exactly half of its eigenvalues are +1 and half are -1. Hence, $$\dim {\widetilde}{{{\mathcal R_+}}} = \dim {\widetilde}{\mathcal R_-} = \frac{1}{2}\dim SY = \frac{1}{2}(2k - 2n) = k-n$$ where $\dim M = 2n$ and $\dim Y = k$ (recall that since $Y$ is coisotropic, $k \geq n$). From the definition of the bundle ${{\mathcal R_+}}$, it follows straightforwardly that $${{\mathcal R_+}}= {{(TY)^\bot}}\oplus {\widetilde}{{{\mathcal R_+}}}$$ Hence, its dimension is $$\dim {{\mathcal R_+}}= \dim {{(TY)^\bot}}+ \dim {\widetilde}{{{\mathcal R_+}}} = (2n-k) + (k-n) = n$$ Moreover, if $s,t \in {{\mathcal R_+}}$, then $$G(s,t) = F(s,t) = -F(t,s) = -G(t,s) = -G(s,t) = 0$$ so ${{\mathcal R_+}}$ is an isotropic subbundle of ${{e^*TM}}$. (In the first and third equalities above, we have used the definition of ${{\mathcal R_+}}$ and in the second and fourth equalities we have used the symmetries of $F$ and $G$.) Since it is middle dimensional and isotropic, it follows that ${{\mathcal R_+}}$ is Lagrangian with respect to ${{e^*TM}}$.
\[\[prop:parallelness\]\] Let $(M,G)$ be a (pseudo-)Riemannian manifold with Levi-Civita connection $\nabla$. Let $Y \subseteq M$ be a submanifold equipped with closed 2-form $F$ on its worldvolume, and let ${{\mathcal R_+}}$ be the subbundle defined in . If ${{\mathcal R_+}}$ is integrable, then we have the following: $$(e^*\nabla)_u s \in \Gamma({{\mathcal R_+}})$$ for all $u \in \Gamma(TY)$ and $s \in \Gamma({{\mathcal R_+}})$. Here $e^* \nabla$ is the pullback of $\nabla$ under the embedding map $e: Y \to M$. In words, the subbundle ${{\mathcal R_+}}$ is $\nabla$-parallel with respect to $TY$.
We will use the involutivity of ${{\mathcal R_+}}$ (which follows from integrability) to establish the formula $$\label{eq:dFust}
2 G( (e^*\nabla)_u s, t) = dF(u,s,t)$$ for all $u \in \Gamma(TY)$ and $s,t \in \Gamma({{\mathcal R_+}})$, keeping in mind that we regard sections of $TY$ and ${{\mathcal R_+}}$ as sections of the pullback bundle ${{e^*TM}}$, equipped with pullback connection $e^*\nabla$. Since $dF=0$ by assumption, this formula will imply that $$(e^*\nabla)_u s \in \Gamma({{\mathcal R_+}}^\bot) = \Gamma({{\mathcal R_+}})$$ In the space-filling case, we have noted that our manifold is (half) para-Kähler, and this formula reduces to an equation appearing in Theorem 1 in [@Cortes:2003zd]; the proof will follow in a similar vein.
To show , we first write down the Kosczul formula $$2 G( (e^*\nabla)_u s, t) = u(G(s,t)) + s(G(u,t)) - t(G(u,s))
+ G([u,s],t) - G([u,t],s) - G([s,t],u)$$ Since $s,t \in \Gamma({{\mathcal R_+}})$. we can replace $G$ by $-F$ in all of the terms on the right hand side above except for the last one. However, as ${{\mathcal R_+}}$ is involutive, $[s,t] \in {{\mathcal R_+}}$, so that we can replace $G$ by $+F$ in the last term: $$2 G( (e^*\nabla)_u s, t) = -u(F(s,t)) - s(F(u,t)) + t(F(u,s))
- F([u,s],t) + F([u,t],s) - F([s,t],u)$$ On the right hand side, we have the right form for the exterior derivative of $F$, except for the sign of the first term $-u(F(s,t))$. However, note that ${{\mathcal R_+}}$ is an isotropic subbundle with respect to both $G$ and $F$, so this term is zero and we can flip its sign. The equation is proved.
The term $G_{IJ} {\psi_+}^I \delta^{\rm cov} {\psi_+}^J$ vanishes for all allowed variations.
We write the fermionic boundary conditions as $${R(\phi)^I}_J {\psi_+}^J = {\psi_+}^I$$ where $R:{{e^*TM}}\to {{e^*TM}}$ is a $\phi$-dependent matrix squaring to the identity. Varying the above and writing in terms of the covariant variations, one has $$(\nabla_K {R^I}_J) \delta \phi^K {\psi_+}^J + {R^I}_J \delta^{\rm cov} {\psi_+}^J = \delta^{\rm cov} {\psi_+}^I$$ or, since $\delta \phi$ is tangent for allowed variations and ${\psi_+}$ takes values in ${{\mathcal R_+}}$, $$(\nabla_K {R^I}_J) {\pi^K}_L {P^J}_M \delta \phi^L {\psi_+}^M + {R^I}_J \delta^{\rm cov} {\psi_+}^J = \delta^{\rm cov} {\psi_+}^I$$ where $\pi$ is a projector onto $TY$ and $P = \frac{1}{2}(1+R)$ is the projector onto ${{\mathcal R_+}}$. However, by proposition \[prop:parallelness\], the first term vanishes. We then have $$G_{IJ} {\psi_+}^I \delta^{\rm cov} {\psi_+}^J = (G_{IJ} {P^I}_K {P^J}_L) {\psi_+}^K \delta^{\rm cov} {\psi_+}^L = 0$$ since ${{\mathcal R_+}}$ is an isotropic subbundle.
[99]{}
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[^1]: Various authors disagree on what object is to be given the name “generalized metric"; we follow the convention of [@Gualtieri:2007ng].
[^2]: See [@Davydov:2010rm] for a recent discussion of this point in the context of rational CFTs.
|
---
abstract: 'We show that an interesting of pairing occurs for spin-imbalanced Fermi gases under a specific experimental condition—the spin up and spin down Fermi levels lying within the $p_x$ and $s$ orbital bands of an optical lattice, respectively. The pairs condense at a finite momentum equal to the sum of the two Fermi momenta of spin up and spin down fermions and form a $p$-orbital pair condensate. This $2k_F$ momentum dependence has been seen before in the spin- and charge- density waves, but it differs from the usual $p$-wave superfluids such as $^3$He, where the orbital symmetry refers to the relative motion within each pair. Our conclusion is based on the density matrix renormalization group analysis for the one-dimensional (1D) system and mean-field theory for the quasi-1D system. The phase diagram of the quasi-1D system is calculated, showing that the $p$-orbital pair condensate occurs in a wide range of fillings. In the strongly attractive limit, the system realizes an unconventional BEC beyond Feynman’s no-node theorem. The possible experimental signatures of this phase in molecule projection experiment are discussed.'
author:
- Zixu Zhang
- 'Hsiang-Hsuan Hung'
- Chiu Man Ho
- Erhai Zhao
- 'W. Vincent Liu'
bibliography:
- 'splusp.bib'
title: 'Modulated pair condensate of $p$-orbital ultracold fermions'
---
introduction
============
Pairing with mismatched Fermi surfaces has long fascinated researchers in the fields of heavy fermion and organic superconductors, color superconductivity in quark matter [@RevModPhys.76.263], and, most recently, ultracold Fermi gases with spin imbalance [@giorgini:1215; @Sheehy20071790; @0034-4885-73-7-076501; @PhysRevLett.99.250403]. In a classic two-component model for superconductivity, the mismatch arises from the spin polarization of fermions in the same energy band. Its effect was predicted to produce intriguing, unconventional superfluids such as the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) [@PhysRev.135.A550; @fflo2], deformed Fermi surface [@PhysRevLett.88.252503; @PhysRevA.72.013613], and breached pair phases [@PhysRevLett.90.047002; @PhysRevLett.94.017001]. The limiting case of large spin imbalance was also studied to explore the formation of Fermi polarons [@PhysRevLett.102.230402]. In parallel, the behavior of particles in the higher orbital bands of optical lattices, due to large filling factors, thermal excitations or strong interactions, is widely studied for novel orbital orderings of both bosons [@liu:013607; @PhysRevA.72.053604; @kuklov:110405] and fermions [@zhao:160403; @wu:200406] with repulsive interactions. Recently, interband pairing of unpolarized fermions was shown theoretically to give rise to Cooper pair density waves [@PhysRevB.81.012504].
In this article, we report a fermion pairing phase resulting from the interplay of Fermi surface mismatch and $p$-orbital band physics. In such a phase, the pair condensate wave function is spatially modulated and has a $p$-wave symmetry. This phase arises in an attractive two-component Fermi gas on anisotropic optical lattices under a previously unexplored condition of spin imbalance. Namely the majority ($\uparrow$) spin and the minority ($\downarrow$) spin occupy up to Fermi levels lying in the $p_x$ and $s$ bands, respectively. We show that pairings take place near the respective Fermi surfaces of the spin $\uparrow$ fermions in $p_x$ band and $\downarrow$ fermions in $s$ band. This induces a modulated $p$-orbital pair condensate that differs from the usual $p$-wave superfluids such as $^3$He. The state requires only an on-site isotropic contact interaction and the pair is a spin singlet, while the $^3$He $p$-wave superconductivity has to involve anisotropic interaction and spin triplet. The modulation wave vector of the order parameter is ${Q}\approx k_{F\uparrow}+k_{F\downarrow}$, where $k_{F\uparrow}$, $k_{F\downarrow}$ are Fermi momenta for spin $\uparrow$ and $\downarrow$ species, respectively. This $2k_F$ momentum dependence is an unprecedented signature in superfluids other than the spin- and charge- density waves. In the strongly attractive limit, tightly bounded pairs condense at finite momentum $Q$, which realizes an unconventional Bose-Einstein condensate beyond Feynman’s no-node theorem [@liu:013607; @PhysRevA.72.053604; @congjun:modphyslett; @kuklov:110405; @radzihovsky:095302].
model
=====
The system under consideration is at zero temperature and consists of two-component fermions in a three-dimensional (3D) cubic optical lattice with lattice constant $a$, described by the Hamiltonian $$\begin{aligned}
H&=&\sum_{\sigma} \int
d^3\mathbf{x}\psi^{\dag}_{\sigma}(\mathbf{x}) [-\frac{\hbar^2}{2m}\nabla^2+V(\mathbf{x})-\mu_{\sigma}] \psi_{\sigma}(\mathbf{x})\nonumber\\
&&+g\int
d^3\mathbf{x}\psi_{\uparrow}^{\dag}(\mathbf{x}) \psi_{\downarrow}^{\dag}(\mathbf{x})\psi_{\downarrow} (\mathbf{x})\psi_{\uparrow}(\mathbf{x}).
\label{eq:generalham}\end{aligned}$$ Here $\psi_{\sigma}(\mathbf{x})$ is the fermionic field operator at $\mathbf{x}$ with spin $\sigma=\uparrow, \downarrow$, $V(\mathbf{x})$ is the lattice potential, $\mu_{\sigma}$ is the chemical potential for spin $\sigma$ fermions, and $g<0$ is the contact attraction which can be tuned by the Feshbach resonance. In particular, we consider the case where the lattice potential in the $x$ (parallel) direction is much weaker than the other two (transverse) directions, so the system behaves quasi-one-dimensionally.
We expand $\psi_{\sigma}(\mathbf{x})=
\sum_{n\mathbf{r}}\phi_n(\mathbf{x}-\mathbf{r})c_{n\mathbf{r}}$, where $\phi_n(\mathbf{x}-\mathbf{r})$ is the $n$th band Wannier function at lattice site $\mathbf{r}$ with $c_{n\mathbf{r}}$ the annihilation operator in Wannier basis. As a result, we obtain the usual attractive Hubbard model with nearest-neighbor hopping between $i$th site with orbital band $\alpha$ and $j$th site with orbital band $\beta$ $$t_{\alpha \beta}=-\int d^3 \mathbf{x} \phi_{\alpha}^*(\mathbf{x}-\mathbf{r}_i) \left[-\frac{\hbar ^2 \nabla ^2}{2m} + V(\mathbf{x})\right]\phi_{\beta}(\mathbf{x}-\mathbf{r}_j)
\label{eq:hopping}$$ and on-site attraction between orbitals $$U_{\alpha \beta \gamma \eta}=g\int d^3\mathbf{x}\phi_{\alpha}^*(\mathbf{x}-\mathbf{r}_i) \phi_{\beta}^*(\mathbf{x}-\mathbf{r}_i) \phi_{\gamma}(\mathbf{x}-\mathbf{r}_i) \phi_{\eta}(\mathbf{x}-\mathbf{r}_i).
\label{eq:onsiteint}$$
The lowest two energy bands are the $s$ and $p_x$ band (the $p_y$ and $p_z$ band are much higher in energy because of tighter confinement in the transverse directions). For brevity the $p_x$ band is simply called $p$ band in the following. By filling fermions with spin $\uparrow$ to the $p$ band and spin $\downarrow$ to the $s$ band, the Hamiltonian becomes $$\begin{aligned}
H_{sp}&=&-\sum_{\langle\mathbf{r,r'}\rangle} (t_s^{\parallel}
S_{\mathbf{r}}^{\dag}S_{\mathbf{r'}}-t_p^{\parallel} P_{\mathbf{r}}^{\dag}
P_{\mathbf{r'}}+h.c.)-\mu_s\sum_{\mathbf{r}} n_{\mathbf{r}}^s\nonumber\\
&&-\sum_{\langle\mathbf{r,r''}\rangle} (t_{s}^{\perp}
S_{\mathbf{r}}^{\dag}S_{\mathbf{r''}}+t_p^{\perp} P_{\mathbf{r}}^{\dag}
P_{\mathbf{r''}}+h.c.)-\mu_p\sum_{\mathbf{r}} n_{\mathbf{r}}^p\nonumber\\
&&+\omega_b\sum_{\mathbf{r}} n_{\mathbf{r}}^p+U_{sp}\sum_{\mathbf{r}} n_{\mathbf{r}}^s
n_{\mathbf{r}}^p.
\label{eq:ham}\end{aligned}$$ Here, $\langle\mathbf{r,r'}\rangle$ and $\langle\mathbf{r,r''}\rangle$ denote the nearest neighboring lattice sites in parallel and transverse directions. $t_s^{\parallel}$ and $t_p^{\parallel}$ are the hopping amplitudes along the parallel direction for the $s$- and $p$-band fermions respectively, while $t_s^{\perp}=t_p^{\perp}=t^{\perp}$ are the hopping amplitudes in transverse directions. $S_{\mathbf{r}}$ ($P_{\mathbf{r}}$) is the annihilation operator at lattice site $\mathbf{r}$ for $s$-band $\downarrow$ ($p$-band $\uparrow$) fermions. $n_{\mathbf{r}}^s=S_{\mathbf{r}}^{\dag}S_{\mathbf{r}},
n_{\mathbf{r}}^p=P_{\mathbf{r}}^{\dag}P_{\mathbf{r}}$ are the number operators, and $\mu_s,\mu_p$ are the corresponding chemical potentials. $U_{sp}$ is the attractive on-site interaction between $s$- and $p$-band fermions and can be tuned by changing the scattering length using Feshbach resonance. $\omega_b$ is related to the band gap. In the tight binding region we assume $\omega_b\gg |U_{sp}|$, and consequently the $s$-band fully filled spin $\uparrow$ fermions are dynamically inert and not included in the $H_{sp}$.
DMRG calculation for 1D case
============================
First we consider the pairing problem in the simplest case of 1D ($t^{\perp}=0$), which is schematically shown in Fig. \[fig:cartoon\](a). The two relevant Fermi momenta are $k_{F\downarrow}$ (for $s$-band $\downarrow$ fermions) and $k_{F\uparrow}$ (for $p$-band $\uparrow$ fermions). From a weak coupling point of view, to pair fermions of opposite spin near their respective Fermi surfaces, the Cooper pairs have to carry finite center-of-mass momentum (CMM) due to Fermi surface mismatch. Furthermore, in order for all Cooper pairs to have roughly the same CMM, the only choice is to pair fermions of opposite chirality. Note that the dispersion of $p$ band is inverted with respect to the $s$ band, so pairing occurs between fermions with momenta of the same sign but opposite group velocities. These elementary considerations show that the CMM of the pair should be approximately the sum of two Fermi momenta, $$Q\approx k_{F\uparrow}+k_{F\downarrow}\,.
\label{eq:cartooncouple}$$ This result differs from that of the usual one-dimensional spin imbalanced fermions within the same band, where the FFLO pair momentum is the difference, ${Q}\approx|k_{F\uparrow}-k_{F\downarrow}|$, as found in a two-leg-ladder system [@feiguin:076403].
![(Color online) (a) A schematic illustration showing the pairing between $s$- and $p$-band fermions. The $s$ band is also fully occupied with $\uparrow$ fermions (not shown). (b) The spatial variation of the pairing correlation $C(x)$ for $N_s=49$, $N_p=15$ according to DMRG. The blue scatters are the DMRG result and the solid line is the fitting using function $a\cos(qx+b)/x^{\eta}+c$. The inset in (b) shows the $s$- and $p$-wave Wannier functions in momentum space, which are elongated in the transverse direction (in real space they are compressed in transverse direction). The $s$-wave Wannier function has even parity while the $p$-wave Wannier function has odd parity.[]{data-label="fig:cartoon"}](cartoon.eps){width="193pt" height="113pt"}
Mean-field theory and weak coupling consideration can provide only a qualitative picture for 1D problems. To unambiguously identify the nature of the ground state, we use density matrix renormalization group (DMRG) to compute the pair correlation function. In the numerical calculations, we used parameters $t_s^{\parallel}=1$ as the unit of energy, $t_p^{\parallel}=8$, $\mu_s=1.7$, $\mu_p-\omega_b=-11$, in which the ratio between $t_s$ and $t_p$ is chosen according to typical tight-binding bandwidth ratio. $U_{sp}$ is tunable with Feshbach resonance and in the following calculation we will focus on $U_{sp}=-9$ [@footnote]. The truncation error is controlled in the order of $10^{-7}$ or less. Equation predicts $Q\approx k_{F\uparrow}+k_{F \downarrow}=0.435
\pi/a$. Figure \[fig:cartoon\](b) shows the pairing correlation function in real space $C_{ij}=\langle S^{\dag}_i P^{\dag}_i P_j
S_j\rangle$ as a function of $x=|i-j|$ for a chain of $N=60$ sites with open boundary condition, where the indices $i$ and $j$ are real space positions. Since the system only has algebraic order, $C(x)$ decays with $x$ according to a power law. On top of this, however, there is also an obvious oscillation. A curve fit with formula $C(x)=a\cos(qx+b)/x^{\eta}+c$, shown in Fig. \[fig:cartoon\](b), yields a period of $q=0.438 \pi/a$, which is in good agreement with the wave number given by Eq. before. The Fourier transform of the pair correlation function C\_q=[1N]{}\_[i,j]{} e\^[iq(i-j)]{} C\_[ij]{} \[eq:moleculedmft\] is peaked at $q=0.426\pi/a$ (to be plotted in Sec. \[sec:molecule\]). These features of the pair correlation function are the signature of the existence of the $2k_F$ CMM pairing in our system [@feiguin:220508; @zhao:063605].
Meanfield analysis for quasi 1d case
====================================
Now we move on to the quasi-1D system where a weak transverse hopping $t^\perp\ll t^\parallel$ is added. We carry out a mean-field analysis of Hamiltonian $H_{sp}$ by introducing the $s$-$p$ pairing order parameter $$\Delta_{\mathbf{r}}=U_{sp}\langle S_{\mathbf{r}} P_{\mathbf{r}}\rangle,
\label{eq:selfcons}$$ where $\langle...\rangle$ means the ground-state expectation value. Two different trial ground states are investigated, the exponential wave $\Delta_{\mathbf{r}}=\Delta e^{i\mathbf{Q\cdot r}}$, which is analogous to the Fulde-Ferrell phase and the cosine wave $\Delta_{\mathbf{r}}=\Delta \cos \mathbf{Q\cdot r}$, which is analogous to the Larkin-Ovchinnikov phase. $\mathbf{Q}$ and $\Delta$ are determined self-consistently by minimization of ground-state free energy $\langle H_{sp}\rangle$. Transverse hopping introduces a small Fermi surface curvature and spoils the perfect nesting condition as in the pure 1D problem above. However, the curvature is small for weak $t^\perp$. Thus, we expect $\mathbf{Q}$ pointing almost along the parallel direction, $\mathbf{Q}=Q(1,0,0)$, in order to maximize the phase space of pairing.
The mean-field Hamiltonian for the exponential wave can be diagonalized in momentum space by standard procedure. We get the ground state energy $$\begin{aligned}
\langle
H_{sp}\rangle=\sum_{\mathbf{k},\gamma=\pm}\Theta (-\lambda_{\mathbf{k}}^{(\gamma)})\lambda_{\mathbf{k}}^{(\gamma)} +\sum_{\mathbf{k}}\xi^p_{\mathbf{k}}-\frac{N^3\Delta^2}{U_{sp}}
\label{eq:exp}\end{aligned}$$ with the self-consistent gap equation for $\Delta$ $$\begin{aligned}
1=\frac{
U_{sp}}{N^3}\sum_{\mathbf{k}}\frac{\Theta (-\lambda_{\mathbf{k}}^{(+)})-\Theta(-\lambda_{\mathbf{k}}^{(-)})} {\sqrt{4\Delta^2+(\xi_{\mathbf{k}}^s+\xi_{\mathbf{Q-k}}^p)^2}}.
\label{eq:gapexp}\end{aligned}$$ Here, $\mathbf{k}$ is lattice momentum, $N^3$ is the total number of sites, $\Theta$ is a step function, and $\lambda_{\mathbf{k}}^{(\pm)}=\frac{1}{2}[\xi_{\mathbf{k}}^s- \xi_{\mathbf{Q-k}}^p \pm
\sqrt{4\Delta^2+(\xi_{\mathbf{k}}^s+\xi_{\mathbf{Q-k}}^p)^2}]$ is the eigenenergy of the Bogoliubov quasiparticles. As evident from these formulas, the pairing occurs between an $s$-band fermion of momentum $\mathbf{k}$ and a $p$-band fermion of momentum $\mathbf{Q-k}$ with dispersion $\xi^s_{\mathbf{k}}=-2t_s^{\parallel}\cos k_x a-2t^{\perp}\cos k_y
a-2t^{\perp}\cos k_z a-\mu_s$ and $\xi^p_{\mathbf{k}}=2t_p^{\parallel}\cos k_x a-2t^{\perp}\cos k_y
a-2t^{\perp}\cos k_z a-\mu_p+\omega_b$, respectively.
The cosine wave is spatially inhomogeneous. A full mean-field analysis requires solving the Bogoliubov-de Gennes equation to determine the gap profile self-consistently. Here we are interested only in computing the free energy for the ansatz $\Delta_{\mathbf{r}}=\Delta \cos \mathbf{Q\cdot r}$ to compare with the exponential wave case. Thus, it is sufficient to numerically diagonalize the full Hamiltonian Eq. for a finite size lattice. We introduce a vector of dimension $2N$ $$\alpha^{\dag}_{k_yk_z}=(S^{\dag}_{k_x^1
k_yk_z}...S^{\dag}_{k_x^Nk_yk_z},P_{k_x^1,-k_y,-k_z}...P_{k_x^N,-k_y,-k_z}),
\label{eq:alpha}$$ where $k_x^n=2\pi n/Na$ is the discrete momentum in the $x$ direction. The components of $\alpha$ obey anticommutation relation $\{\alpha_{k_yk_z}^{\dag(m_1)},\alpha_{k_yk_z}^{(m_2)}\}=\delta_{m_1m_2}$, where $m_1,m_2$ labels the corresponding operator component of $\alpha$. The Hamiltonian takes the compact form $H_{sp}=\sum_{k_yk_z}\alpha^{\dag}_{k_yk_z}\mathcal{H}_{k_yk_z} \alpha_{k_yk_z}+\sum_{\mathbf{k}}\xi^p_{\mathbf{k}}- (1+\delta_{Q,-Q})N^3\Delta^2/2U_{sp}$. Since $\mathcal{H}_{k_yk_z}$ is real and symmetric, it can be diagonalized by an orthogonal transformation $\alpha_{k_yk_z}=\mathcal{D}_{k_yk_z}\beta_{k_yk_z}$ to yield $2N$ eigenvalues $E_{k_yk_z}^l$. The new operators $\beta_{k_yk_z}$ automatically obey the fermionic anticommutation relationship $\{\beta_{k_yk_z}^{\dag(m_1)},\beta_{k_yk_z}^{(m_2)}\}=\delta_{m_1m_2}$. We get the ground state energy, $$\begin{aligned}
\langle
H_{sp}\rangle&=&\sum_{k_y,k_z}\sum_{l=1}^{2N}E_{k_yk_z}^l \Theta(-E_{k_yk_z}^l)+\sum_{\mathbf{k}}\xi^p_{\mathbf{k}}\nonumber\\
&&-\frac{N^3\Delta^2}{2U_{sp}}(1+\delta_{-Q,Q})\,,
\label{eq:cos}\end{aligned}$$ and the gap equation, $$\Delta=\frac{2U_{sp}}{N^3(1+\delta_{-Q,Q})}\sum_{\mathbf{k}} \sum_l\mathcal{D}_{k_yk_z}^{m_1,l}\mathcal{D}_{k_yk_z}^{m_1',l} \Theta(E_{k_yk_z}^l).
\label{eq:gapcos}$$ Here, $l$ labels the eigenenergy, and $m_1$, $m_1'$ labels the matrix elements corresponding to the original $S$, $P$ operators in the gap equation.
![(Color online) The occupation of $s$ and $p$ band within the paired state for different transverse hopping $t_\perp$. Only the first quadrant of the Brillouin zone in the $k_x-k_y$ plane is shown, $k_z=\pi/a$. The black dashed lines indicate the “bare" Fermi surfaces for corresponding noninteracting fermions ($U_{sp}=0$). (a) $\langle S_{\mathbf{k}}^{\dag}S_{\mathbf{k}} \rangle$ for $t^{\perp}=0.05$; (b) $\langle S_{\mathbf{k}}^{\dag}S_{\mathbf{k}}
\rangle$ for $t^{\perp}=0.1$; (c) $\langle
P_{\mathbf{k}}^{\dag}P_{\mathbf{k}} \rangle$ for $t^{\perp}=0.05$; (d) $\langle P_{\mathbf{k}}^{\dag}P_{\mathbf{k}} \rangle$ for $t^{\perp}=0.1$.[]{data-label="fig:fs"}](fs.eps){width="240pt" height="185pt"}
The parameters used in the mean-field calculations are the same as in the 1D case with small $t^{\perp}$’s added, and we still expect that the order parameter has the momentum around $0.435\pi/a$ as before. By self-consistently solving for $Q$ and $\Delta$, in the case $t^{\perp}=0.05$, the ground state is the cosine wave phase with $Q=0.433\pi/a$ and $\Delta=0.822$. The ground state energy per site is $-2.5927$, lower than the noninteracting value $-2.5896$. When $t^{\perp}=0.1$, the ground state is also the cosine wave phase with $Q=0.433 \pi/a$ and $\Delta=0.542$. The ground state energy per site is $-2.5955$, lower than the noninteracting value $-2.5949$. These results confirm that (i) the cosine wave state has lower energy than the exponential wave state, (ii) the order parameter has the momentum close to the prediction of Eq. , and (iii) larger transverse hopping tends to destroy the $p$-orbital pair condensate since the energy gain for larger transverse hopping is much smaller than for smaller transverse hopping.
An interesting feature of the $p$-orbital pair condensate in quasi-1D is the possible existence of Fermi surfaces with gapless energy spectrum. We monitor the fermion occupation number, i.e. $\langle S_{\mathbf{k}}^{\dag}S_{\mathbf{k}} \rangle$ and $\langle P_{\mathbf{k}}^{\dag}P_{\mathbf{k}}\rangle$ for increasing transverse hopping. The results are shown in Fig. \[fig:fs\]. For small $t^{\perp}$, they take the usual BCS form and vary smoothly from 1 (red) to 0 (blue) across the bare Fermi surface (with interaction turned off), as shown in Figs. \[fig:fs\](a) and \[fig:fs\](c) for $t^\perp=0.05$. For larger transverse hopping, sharp Fermi surfaces characterized by a sudden jump in $\langle S_{\mathbf{k}}^{\dag}S_{\mathbf{k}} \rangle$ and $\langle P_{\mathbf{k}}^{\dag}P_{\mathbf{k}}\rangle$ appear. This is clearly shown in Figs. \[fig:fs\](b) and \[fig:fs\](d) for $t^\perp=0.1$ as the occupation number changes discontinuously from 1 (red) to 0 (blue). It can be understood qualitatively as follows. As $t^{\perp}$ increases, the original Fermi surfaces acquire a larger curvature in the transverse directions and the pairing condition in Eq. cannot be satisfied everywhere anymore. Therefore in some regions fermions are not paired and Fermi surfaces survive. One should also note that the calculation is based on the assumption that $t^{\perp} \ll t^{\parallel}$, which predicts that $\mathbf{Q}$ is in the parallel direction. This prediction should fail as $t^{\perp}$ increases beyond certain critical values.
Phase Diagram
=============
Now, we systematically explore the phases of our system for general band filling and spin imbalance. Since we have $s$- and $p$- bands with different bandwidths, we introduce two dimensionless quantities for the chemical potentials $\mu_s$ and $\mu_p$ $$\begin{aligned}
\tilde{\mu}_s&=&\frac{\mu_s}{2t_s}=\frac{\mu_s}{2},\nonumber\\
\tilde{\mu}_p&=&\frac{\mu_p-\omega_b}{2t_p}=\frac{\mu_p-\omega_b}{16}.
\label{eq:renormu}\end{aligned}$$ Thus, for a non-interacting system, $-1 < \tilde{\mu}_s, \tilde{\mu}_p <1$ control the filling for the $s$ and $p$-band fermions respectively. We then define the quantities $$\begin{aligned}
\mu&=&\frac{\tilde{\mu}_s+\tilde{\mu}_p}{2},\nonumber\\
h&=&\frac{\tilde{\mu}_s-\tilde{\mu}_p}{2},
\label{eq:muandh}\end{aligned}$$ as the parameters controlling the average filling and polarization in the phase diagram. The phase at $-\mu, -h$ is the same as the state at $\mu, h$, since the transformation $\mu,h \rightarrow -\mu, -h$ gives $\mu_s, \mu_p \rightarrow -\mu_s, -\mu_p$, and the mean-field Hamiltonian with $\mu_s, \mu_p$ is identical to Hamiltonian with $-\mu_s, -\mu_p$ via a particle-hole transformation up to a constant.
![(Color online) Band occupation for the four possible phases in the system. The band colored in red represents the $s$ band occupied by spin $\downarrow$ fermions and the band colored in green represents the $p$ band occupied by $\uparrow$ fermions. The spin $\uparrow$ fermions in the $s$ band are not shown since they are inert. (a) Normal phase I (N1) with one band empty and the other partially filled. (b) Normal phase II (N2) with one band fully filled and the other partially filled. (c) Commensurate $p$-orbital pair condensate (CpPC) with both bands partially filled. The occupation numbers are the same. (d) Incommensurate $p$-orbital pair condensate (IpPC) with both bands partially filled. The occupation numbers are different.[]{data-label="fig:phases"}](phases.eps){width="215pt" height="80pt"}
We have four possible phases in such a system as shown in Fig. \[fig:phases\]. As before, we ignored the inert fully filled $s$ band of spin $\uparrow$ fermions. We consider the $p$ band of spin $\uparrow$ fermions and $s$ band of spin $\downarrow$ fermions. When one of these two bands is empty and the other is filled, the pairing does not happen and we call it normal phase I (N1) as in Fig. \[fig:phases\](a). When one of these two bands is fully filled and the other is partially filled, the pairing also does not happen since the fully filled band is inert. We call it normal phase II (N2) as in Fig. \[fig:phases\](b). When both of them are partially filled, fermions near Fermi surfaces from the two bands will be paired and the system is in superfluid phases as shown in Figs. \[fig:phases\](c) and \[fig:phases\](d). In the superfluid regime, when $h$ is small, the pairing momentum prefers $Q=\pi/a$ and we call it commensurate $p$-orbital pair condensate (CpPC). It is a special case of the $p$-orbital pair condensate, where the occupation numbers of $s$-band spin $\downarrow$ fermions and $p$-band spin $\uparrow$ fermions are the same. It is similar to the conventional unpolarized pairing (BCS), where the spin $\uparrow$ fermions and spin $\downarrow$ fermions have the same population. However, in BCS pairing the CMM of the pair has the property $Q=0$, while here $Q=\pi/a$. To understand the momentum $\pi/a$ preference, note that in conventional BCS case, the two species of fermions have the same energy spectrum and the pairing is between two fermions with opposite momenta, which leads to the CMM of pair $Q=0$. Here, the structure of energy spectrum of $p$ band is different from $s$ band. The equal occupation numbers mean $k_{F\uparrow}=\pi/a-k_{F\downarrow}$, which gives rise to $Q=k_{F\uparrow}+k_{F\downarrow}=\pi/a$, as shown in Fig. \[fig:phases\](c). At last, when $h$ is large, the pairing momentum stays at a general $Q\approx k_{F\uparrow}+k_{F\downarrow}$ and the occupation number for the two species of fermions differ. We call it incommensurate $p$-orbital pair condensate (IpPC) as shown in Fig. \[fig:phases\](d).
To determine the phases, we minimize the free energy as a function of the pairing amplitude $\Delta$ and pairing momentum $Q$ by mean-field analysis using the cosine wave function as outlined in the previous section. When the minimum is realized at $\Delta=0$, it is normal phase. When $\Delta$ is finite, there are two possibilities. When $Q=\pi/a$, it is CpPC. When $Q\neq\pi/a$, it is IpPC. For the transition between superfluid and normal phase, and the transition between CpPC and IpPC, the behaviors of free energy show that the phase transitions are first order in a lattice system. Between the superfluid and normal phases, near the phase transition, $\Delta$ changes suddenly from $0$ to finite, and the free energy shows two local minima at $\Delta=0$ and $\Delta \neq 0$. Between CpPC and IpPC, the pairing momentum changes from $Q=\pi/a$ to $Q\neq\pi/a$ discontinuously, and the free energy as a function of $Q$ also has two local minima at $Q=\pi/a$ and $Q \neq \pi/a$. Thus, they are first-order phase transitions according to our mean field analysis. Therefore, we can determine the phase boundaries between normal phase and superfluid phase by monitoring $\Delta$ changing from zero to finite. We can also monitor $Q$ changing from $Q=\pi/a$ to $Q\neq \pi/a$ to determine the phase boundaries between CpPC and IpPC.
![(Color online) The phase diagram of the $p$-orbital pair condensate for $t^{\perp}=0.05$. $\mu$ and $h$ are defined in the main text. CpPC: the $s$ band of spin $\downarrow$ fermions and the $p$ band of spin $\uparrow$ fermions have the same occupation numbers. IpPC: the $s$ band of spin $\downarrow$ fermions and the $p$ band of spin $\uparrow$ fermions have different occupation numbers. N1 with the $p$ band of spin $\uparrow$ fermions empty and the $s$ band of spin $\downarrow$ fermions partially filled. N2 with the $p$ band of spin $\uparrow$ fermions partially filled and the $s$ band of spin $\downarrow$ fermions fully filled.[]{data-label="fig:pd"}](pd.eps){width="172pt" height="150pt"}
In Fig. \[fig:pd\], we present a phase diagram for $t^{\perp}=0.05$. The x’s in Fig. \[fig:pd\] show the data points for the phase boundary obtained from the numerical procedure, and by connecting them we get the phase boundaries. An illustrative physical understanding about this phase diagram is as follows. In Fig. \[fig:pd\], when chemical potential difference $h$ is small and the two bands are still partially filled to ensure the pairing, the system tends to stay in CpPC where $Q=\pi/a$. It is similar to the conventional BCS superfluid case. As $h$ becomes larger, as long as the average filling $\mu$ is not too large or small and the two bands are still both partially filled, the pairing persists despite the spin imbalance and the system is in IpPC. If $\mu$ gets more and more negative, the average filling becomes smaller and smaller, and at certain $\mu, h$, $p$ band of spin $\uparrow$ fermions will be empty and the system will become N1 without pairing. Similarly, when $\mu$ is large and positive, the average filling is very high and at certain $\mu, h$, the $s$ band of spin $\downarrow$ fermions will be fully occupied, and the system becomes N2 without pairing. The almost straight phase boundaries in Fig. \[fig:pd\] between IpPC and normal phases indicate that these phase transitions are due to the change of band occupation as empty $\leftrightarrow$ partially filled $\leftrightarrow$ fully filled. In Fig. \[fig:pd\], the phase boundary between IpPC and N1 corresponds to the critical condition that the $s$ band of spin $\downarrow$ fermions is partially filled while the $p$ band of spin $\uparrow$ fermion becomes empty, and the almost straight phase boundary corresponds to the condition that $\tilde{\mu}_p=\mu-h=-1$ (but, as before, this is only an approximate argument due to the presence of interaction). Similarly, the almost straight phase boundary between IpPC and N2 corresponds to the condition that the $s$ band of spin $\downarrow$ fermions becomes fully filled, while the $p$ band of spin $\uparrow$ fermions is partially filled, or $\tilde{\mu}_s=\mu+h=1$. All the phase transition lines in Fig. \[fig:pd\] are mean field results, and these straight lines are expected to be corrected by quantum critical fluctuations. The phase diagram shows that the $p$-orbital pair condensate happens in large parameter regimes and is closely related to the band and orbital properties in the optical lattice systems.
Signature of the $p$-orbital pair condensate in Molecule Projection Experiment {#sec:molecule}
==============================================================================
The $p$-orbital pair condensate phase can inspire important experimental signatures for finite momentum condensation of bosonic molecules in higher orbital bands. By fast sweeping the magnetic field (and thus the interaction) from the BCS region to the deep BEC region across a Feshbach resonance, the BCS pairs are projected onto Feshbach molecules, which can be further probed for example by time-of-flight images [@liu:013607]. The bosons produced effectively reside in $p$ band and are stable, since by Pauli blocking the filled $s$-band fermions will prevent the the $p$-wave bosons from decaying [@liu:013607]. Here, we use a simple model [@Diener2004; @Altman2005] to evaluate the momentum distribution of molecules after projection $$n_{\mathbf{q}}=\sum_{\mathbf{k,k'}} f^*_{\mathbf{k}}f_{\mathbf{k'}}\langle
S^{\dag}_{\mathbf{k+q}/2}P^{\dag}_{\mathbf{-k+q}/2}P_{\mathbf{-k'+q}/2} S_{\mathbf{k'+q}/2}\rangle,
\label{eq:molecule}$$ where $f_{\mathbf{k}}$ is the molecular wave function, and the correlation function can be evaluated within mean field theory [@Altman2005]. For fast sweeps, the molecular size is small compared to lattice constant and its wave function can be approximated by a delta function in real space (a constant $\sqrt{1/N}$ in momentum space). By this assumption, $n_q$ is the same quantity as $C_q$ in Eq. . Figure \[fig:pair\](a) shows the $n_q$ of $p$-wave Feshbach molecules and a peak is located at $0.433\pi/a$. Figure \[fig:pair\](b) shows $C_q$ from Eq. , based on the DMRG results shown in Fig. \[fig:cartoon\](b).The time-of-flight experiment is predicted to distribute peaks corresponding to that in Fig. \[fig:pair\]. Note that for the 1D problem (Fig. \[fig:pair\](b)), the delta-function peak is replaced by a cusp characteristic of power law due to the lack of long range order.
![(Color online) (a) The momentum distribution function $n_q$ of projected molecules for a quasi-1D system with $t^{\perp}=0.05$ (all other parameters are same as before) according to mean field theory. Here, $q=q_x$, $q_y=q_z=0$. (b) Pair correlation function $C_q$ for a 1D chain of $N=60$ sites obtained by DMRG. The peak is located at $0.433 \pi/a$ in both figures, which corresponds to the value $k_{F\uparrow}+k_{F\downarrow}=(N_s+N-N_p)\pi/Na$ for $N_s=49$ and $N_p=15$. []{data-label="fig:pair"}](proj.eps){width="226pt" height="102pt"}
We thank Chungwei Lin for helpful discussions. This work is supported by ARO Grant No. W911NF-07-1-0293.
|
---
abstract: 'The characterization of rocky, Earth-like planets is an important goal for future large ground- and space-based telescopes. In support of developing an efficient observational strategy, we have applied Bayesian statistical inference to interpret the albedo spectrum of cloudy true-Earth analogs that include a diverse spread in their atmospheric water vapor mixing ratios. We focus on detecting water-bearing worlds by characterizing their atmospheric water vapor content via the strong 0.94$\,\mu$m H$_2$O absorption feature, with several observational configurations. Water vapor is an essential signpost when assessing planetary habitability, and determining its presence is important in vetting whether planets are suitable for hosting life. We find that R=140 spectroscopy of the absorption feature combined with a same-phase green optical photometric point at $0.525-0.575\,\mu$m is capable of distinguishing worlds with less than $0.1\times$ Earth-like water vapor levels from worlds with $1\times$ Earth-like levels or greater at a signal-to-noise ratio of 5 or better with $2\sigma$ confidence. This configuration can differentiate between $0.01\times$ and $0.1\times$ Earth-like levels when the signal-to-noise ratio is 10 or better at the same confidence. However, strong constraints on the water vapor mixing ratio remained elusive with this configuration even at signal-to-noise of 15. We find that adding the same-phase optical photometric point does not significantly help characterize the H$_2$O mixing ratio, but does enable an upper limit on atmospheric ozone levels. Finally, we find that a 0.94$\,\mu$m photometric point, instead of spectroscopy, combined with the green-optical point, fails to produce meaningful information about atmospheric water content.'
author:
- 'Adam J. R. W. Smith'
- 'Y. Katherina Feng'
- 'Jonathan J. Fortney'
- 'Tyler D. Robinson'
- 'Mark S. Marley'
- 'Roxana E. Lupu'
- 'Nikole K. Lewis'
title: |
Detecting and Characterizing Water Vapor in the Atmospheres of Earth Analogs\
through Observation of the $0.94\,\mu$m Feature in Reflected Light
---
INTRODUCTION {#sec:intro}
============
In the decades since the landmark discovery of a planet orbiting another Sunlike star [@mayor_queloz_1995], the field of exoplanetary science has grown tremendously. Thousands of exoplanets have now been found, and the Transiting Exoplanet Survey Satellite (TESS) is expected to find tens of thousands more [@ricker2014transiting; @huang2018expected]. The year 2002 saw the first detection of an atmosphere on an exoplanet [@charbonneau_brown_noyes_gilliland_2002] and the field of exoplanetary atmospheres has expanded rapidly since that time [@marley2006atmospheres; @seager2010exoplanet; @crossfield2015observations; @kaltenegger2017characterize; @madhusudhan2019exoplanetary]. The future study of exoplanetary atmospheres is of major interest to the astronomical community, with several proposed flagship telescopes set to make exoplanetary atmospheric characterization a major mission objective [@mennesson2016habitable; @bolcar_aloezos_bly_collins_crooke_dressing_fantano_feinberg_france_gochar_et_al._2017; @cooray2017origins].
A goal of future exoplanet science and atmospheric studies is the discovery and characterization of an Earth analog. Such a terrestrial planet would reside in the Habitable Zone of its host star – the orbital distance where liquid water could exist at the surface of a planet [e.g., @kasting_whitmire_reynolds_1993]. An attractive pathway to characterize such a planet around a Sunlike parent star would be via direct imaging and spectroscopy of light scattered (“reflected") from the planet’s surface and atmosphere [e.g., @feng_robinson_fortney_lupu_marley_lewis_macintosh_line_2018]. As such, two flagship scale missions currently under study, LUVOIR [@bolcar_aloezos_bly_collins_crooke_dressing_fantano_feinberg_france_gochar_et_al._2017; @roberge2018large] and HabEx [@mennesson2016habitable; @gaudi2018habitable], along with a probe-class external occulter to WFIRST [@seager2018starshade], make the detection of reflected light from rocky planet atmospheres a major science goal.
When designing such a telescope, and optimizing a proposed observing strategy, it is valuable to understand what information can be gained from optical photometry and spectroscopy. In a previous paper [@feng_robinson_fortney_lupu_marley_lewis_macintosh_line_2018], we developed the first retrieval model for terrestrial planet reflection spectra, which built off our previous efforts for optical reflection spectra for giant planets [@lupu_marley_lewis_line_traub_zahnle_2016; @nayak2017atmospheric]. @feng_robinson_fortney_lupu_marley_lewis_macintosh_line_2018 investigated optical spectra of the Earth from 0.4 to $1.0\,\mu$m at a range of spectral resolutions to understand a broad range of science questions, including one’s ability to constrain the abundances of atmospheric gases, or merely detect their presence. In addition, we studied potential constraints on planetary radius, cloud parameters, and surface gravity.
In the follow-up investigation presented here, we focus on a specific potential characterization strategy that future large space telescopes may use. Potentially interesting planets, in or near the habitable zone, will likely be detected by a search of nearby stars via single-band optical photometry. It is likely that such a detection will be performed at or near the peak brightness of the host star; for a G-type star comparable to our Sun, this peak is roughly $500-600$nm. After such a planet has been detected, a “follow the water” strategy may next ask: Does the detected planet have water vapor in its atmosphere? If so, how much?
In an initial effort to provide quantitative guidance to these questions, here we build on the work of @feng_robinson_fortney_lupu_marley_lewis_macintosh_line_2018 with an eye towards probing the 0.94$\,\mu$m water absorption band, the strongest at optical wavelengths. Determining if a planet of interest has water (and how much) would be an important milestone in determining if the planet should be followed up with additional spectroscopy. The ability to constrain the presence of water vapor in an exoplanet’s atmosphere is one useful tool that may be used to guide the search for life on other planets [@schwieterman2018exoplanet]. Following our previous work, and in order to make the problem tractable at this stage, we focus on Earth analog planets, meaning current Earth atmospheric abundances, but now with a water vapor mixing ratio that varies across a factor of a thousand, with atmospheric water vapor from 0.01$\times$ that of Earth, to $10\times$ more. We investigate the relative ability of photometry and R$\,=140$ spectroscopy, at a variety of signal-to-noise ratios (SNR), across the 0.94$\,\mu$m band, to quantify a detection of atmospheric water vapor and constrain its abundance. A concentration on a single optical band is motivated by the expectation that future space telescope missions must make multiple observations over limited spectral ranges in order to assemble a spectrum [e.g. LUVOIR; @bolcar_aloezos_bly_collins_crooke_dressing_fantano_feinberg_france_gochar_et_al._2017].
In Section \[sec:methods\], we describe the methods used in this study. In Section \[sec:results\], we describe the results of the investigation. In Section \[sec:disc\], we discuss these results and draw conclusions from them in an attempt to answer the above questions. We also suggest paths for future work.
Methods {#sec:methods}
=======
Albedo Model and Simulated Data {#sec:methods:data}
-------------------------------
[llll]{}\[!\] log(H$_2$O) & Water mixing ratio & log($3\times 10^{-3}$) & \[-8, 0\]\
log(O$_2$) & Molecular Oxygen mixing ratio & log(0.21) & \*\
log(O$_3$) & Ozone mixing ratio & log($7\times 10^{-7}$) & \[-10, -1\]\
log(CH$_4$) & Methane mixing ratio & log($1.8\times 10^{-6}$) & \*\
log(CO$_2$) & Carbon Dioxide mixing ratio & log($400\times 10^{-6}$) & \*\
R$_p$ \[R$_{\Earth}$\] & Planet Radius & 1 & \[0.5,12\]\
log(P$_0$) \[bar\] & Surface Atmospheric Pressure & log(1) & \[-2, 2\]\
log(g) & Surface Gravity & log(9.8) & \[0,2\]\
log(A$_s$) & Surface Albedo & log(0.05) & \[-2,0\]\
log(p$_t$) \[bar\] & Cloud Top Pressure & log(0.6) & \[-2, 2\]\
log($\delta$p) (bar) & Cloud Thickness & log(0.1) & \[-3, 2\]\
log($\tau$) & Cloud Optical Depth & log(10) & \[-2, 2\]\
log(f$_c$) & Cloud Coverage & log(0.5) & \[-3, 0\]\
\[table:params\]
To generate model planetary albedo spectra we employ the high-resolution albedo spectra model described in @marley1999reflected, which was extensively revised in @cahoy_marley_fortney_2010. The model was later paired with an MCMC driver in @lupu_marley_lewis_line_traub_zahnle_2016 and @nayak2017atmospheric to explore the retrieval of atmospheric parameters for gas giant exoplanets at full phase, and crescent phases, respectively. The Cahoy et al. and Lupu et al. papers have extensive descriptions of the model setup. More recently, @feng_robinson_fortney_lupu_marley_lewis_macintosh_line_2018 modified the code again to treat the surfaces and atmospheres of Earth-like terrestrial exoplanets. A fuller description can be found there, as we use the same setup for our work.
The three-dimensional albedo model divides a spherical world into a “disco-ball" of plane-parallel facets. For each facet, we calculate $\mu_s$, the angle (relative to the zenith) from which downwelling stellar radiation is incident on the facet. We also calculate $\mu_o$, the scattering angle (again, relative to the zenith) required for emergent light to reach the observer. At each facet, the model atmosphere utilizes a fixed pressure level grid, and a radiative transfer calculation is performed to determine the emergent intensity at the required zenith and azimuth angles $(\mu_o,\phi_o)$ of the observer. We take $I(\tau,\mu,\phi)$ to be the wavelength-dependent intensity at optical depth $\tau$, in the direction described by zenith angle $\mu$ and azimuth angle $\phi$. Thus, the quantity we wish to find for each facet is $I(\tau=0,\mu_o,\phi_o)$. To compute this value for each facet, we follow the steps laid out in @feng_robinson_fortney_lupu_marley_lewis_macintosh_line_2018, sections 2.1 and 2.2. We implement a two-term Henyey-Greenstein (TTHG) phase function [@kattawar1975three] to treat the directly scattered radiation and Legendre polynomials to represent the azimutally-averaged diffusely scattered radiation. We have since updated the forward and backward scattering portions of the TTHG phase function as presented in @cahoy_marley_fortney_2010, which were specifically tailored for water clouds, to be consistent with the parameterization described in @kattawar1975three instead.
With $I(\tau$=0$,\mu_o,\phi_o)$ in hand for each facet, Chebychev-Gauss quadrature [@horak1950diffuse; @horak1965calculations] is used to integrate the total planetary intensity at a given wavelength. By repeating this procedure at each wavelength of interest, we are able to build up an albedo spectrum across a given wavelength range.
![The optical planet-to-star flux ratio spectrum of an Earth-like planet, with a spectral resolution of 140, as generated by the forward model used in this paper, with major features highlighted. Of interest to this work is the H$_2$O absorption feature at $0.94\,\mu$m. Note also the lack of major features from non-water sources in the vicinity of this feature.[]{data-label="fig:features"}](Grid2Feat_w0.pdf){width="3.3in"}
We focus on the H$_2$O spectral feature centered on 0.94$\mu$m. This feature is the strongest one at optical wavelengths where reflected light spectroscopy of potentially habitable planets is most efficient. Stronger features do exist at longer near-IR wavelengths, but there is much less incident flux there from solar type stars and it can be more difficult to obtain spectra due to inner working angle constraints for coronagraphic masks. Notional plans for terrestrial planet characterization in reflected light typically give priority to the detection of this band as an indicator of atmospheric water. We use tabulated H$_2$O opacities — as well as opacities for O$_2$ and O$_3$ — generated by the Line-By-Line ABsorption Coefficient model (LBLABC; developed by D. Crisp; @meadows1996ground) constructed from the HITRAN 2012 line list [@rothman2013hitran2012] [^1], using line broadening parameters appropriate for air. Since we are studying detectability of the band at relatively low spectral resolution $R\sim~140$, detailed line positions and other parameters are not of foremost importance. We note that updated H$_2$O opacities are available [@Polyansky2018], although we expect little change at these modest temperature conditions, so we have prioritized consistency with our previous work.
Water molecules exhibit three vibrational modes ($\nu_1$, $\nu_2$, and $\nu_3$). Those rovibrational transitions in which the quantum numbers change for two or more modes are called combination bands [@bernath2015spectra]. There are several combinations bands at spectral region 0.94$\mu$m, such as 2$\nu_1$+$\nu_3$, 1$\nu_1$+2$\nu_2$+1$\nu_3$. According to HITRAN [@gordon2017hitran2016], the strengths of total individual lines in some bands such as 2$\nu_1$+$\nu_3$ are much larger than other bands and therefore they have the most impact on the opacity value. Other weak bands, however, were included in computing the water opacity in order to generate the water continuum accurately.
Following the albedo model setup of @feng_robinson_fortney_lupu_marley_lewis_macintosh_line_2018, we generated spectra of rocky exoplanets using Earth-like surface and atmosphere conditions as detailed in Table \[table:params\]. The values chosen for these parameters produce realistic Earth spectra, as validated by @feng_robinson_fortney_lupu_marley_lewis_macintosh_line_2018 against the NASA Astrobiology Institute’s sophisticated 3D, line-by-line, multiple scattering Virtual Planetary Laboratory spectral Earth model tool [@robinson2011earth]. Four such spectra were generated, with atmospheric water content[^2] of $0.01$, $0.1$, $1$, and $10$ times the Earth-like value described in Table \[table:params\]. All models were generated with phase angle $\alpha=0$ – full phase – for this initial study. Although true direct-imaging missions will not obtain full-phase observations, this assumption does not impact our results, as we do not compute integration times but instead work only in S/N space. Further, we anticipate performing a future followup investigation to expand this study and retrieve phase information.
Figure \[fig:features\] shows an example albedo spectrum at spectral resolution of $R=140$, shown as a planet-to-Sun flux ratio. The most prominent features are due to Rayleigh scattering in the blue, a broad O$_3$ absorption, weak O$_2$ absorption, and water vapor features that grow in strength at redder wavelengths. Figure \[fig:opacities\] shows the absorption cross-sections of these three important molecules, weighted by the mixing ratios of molecules in our standard “Earth-like" setup. While Rayleigh scattering imparts a slope in the blue, the optically thick water cloud is a gray scatterer throughout the rest of the optical. The broad absorption due to O$_3$ gives a subtle dip in the spectrum around 0.6$\mu$m, punctuated by narrower features due to O$_2$ and H$_2$O.
![Weighted opacities represented logarithmically as a function of wavelength for O$_2$, O$_3$, and H$_2$O, the three major species discussed in this work. Each molecule has been weighted according to their relative abundance in our fiducial model. This opacity information manifests as absorption features in the contiuum albedo spectrum set by Rayleigh scattering and scattering from the grey clouds and surface. Features of interest are the wide, but shallow, O$_3$ absorption around 0.6$\mu$m, as well as the sharp O$_2$ features. The strong H$_2$O feature at 0.94$\mu$m is the primary target of this study. []{data-label="fig:opacities"}](opacaties140step1.pdf){width="3.3in"}
{width="3.3in"}
spectrum (black dotted line). The region marked “Photometry” indicates the area covered by an potential optical photometric filter with a 10% wavelength bandpass centered at 550 nm. The region marked “R=140” indicates the area covered by an R$\,=140$ spectroscopic instrument with a 15% bandpass, representing an immediate followup observation. For this figure, we also used a signal-to-noise ratio of 15 in this region, calculated at $0.88 \,\mu$m. \[fig:datapoints\]
Observation Simulation
----------------------
With high-resolution albedo spectra in hand, we next simulated observations of these objects with a coronagraph-equipped telescope. The basic idea was to generate data that may be akin to some “first” observations, including a broadband optical photometric point (for planet discovery), followed by a reconnaissance spectrum across the $0.94\,\mu$m water band. This was achieved by reducing the resolution of the simulated spectrum to produce data points with spectral resolution R$\,=140$ in a 15% bandpass centered on the strong H$_2$O absorption feature at $0.94\,\mu$m (see Figure \[fig:features\]), covering the region from $0.85-1.00\,\mu$m. We chose this range so as to include the water absorption feature along with “continuum” reflection off the clouds, just blue-ward of the water band. What we term continuum reflection is the relatively gray reflection from the optically thick water clouds, from $\sim$ 0.6 to 1.0$ \,\mu$m, punctuated only by O$_2$ and H$_2$O absorption.
We combined this medium-resolution data with an integrated $0.525\,\mu$m - $0.575\,\mu$m photometric point (termed elsewhere in this paper as a 0.55$\,\mu$m or “green” data point). We employed the noise model of @robinson2016characterizing to simulate signal-to-noise ratios (SNR) achieved by the instrument, where the signal is defined as the reflected flux ratio $F_{\mathrm p}/F_{\mathrm s}$. For each chosen value of SNR, we selected a “continuum" reference data point at $\lambda_0 = 0.88\,\mu$m, just outside of the water band, and set the uncertainty of that data point to be $\Delta F_{\mathrm p}/F_{\mathrm s}$ = $F_{\mathrm p}/F_{\mathrm s}$/SNR. Equation 6 of @robinson2016characterizing relates exposure time to background photon count rate, planet photon count rate, and signal-to-noise ratio. Although background count rate, planet photon count rate, and signal-to-noise ratio may not be constant across all wavelengths, exposure time must be a constant within a single bandpass; therefore, we may equate the value at the reference wavelength $\lambda_0$ with that at another wavelength $\lambda$ by
$$\frac{c_p(\lambda_0)+2c_b(\lambda_0)}{c_p(\lambda_0)^2} SNR(\lambda_0)^2 = \frac{c_p(\lambda)+2c_b(\lambda)}{c_p(\lambda)^2} SNR(\lambda)^2$$
We can therefore solve for the wavelength-dependent signal-to-noise ratio $SNR(\lambda)$, allowing us to extrapolate the uncertainties achieved at all data points, given a set signal-to-noise ratio at the reference data point, a model albedo spectrum, and a model of background photon counts. We use @robinson2016characterizing to inform our background photon counts, the Cahoy et al. model described above to produce albedo spectra, and values of 5, 10, and 15 for our reference data point signal-to-noise ratios. See Figure \[fig:datapoints\] for an example data set.
Following @feng_robinson_fortney_lupu_marley_lewis_macintosh_line_2018 we add these appropriate error bars to the reduced-resolution fiducial model spectrum, but do not randomize the data points. As discussed in @feng_robinson_fortney_lupu_marley_lewis_macintosh_line_2018 this is a choice of convenience, but with a purpose. The retrieval on a single noise instance could easily bias our retrieval results. The retrieval on a large number of noise instances would be most proper, but is computationally extremely expensive. From tests @feng_robinson_fortney_lupu_marley_lewis_macintosh_line_2018 demonstrated that posteriors on atmospheric quantities of interest, comparing non-randomized data and that achieve from 10 different noise instances, led to good qualitative agreement. While acknowledging that our treatment here is likely modestly optimistic compared to a more detailed treatment, our work certainly show important trends that set a basis for more comprehensive followup work.
Our adopted spectral resolution (R$=140$) is consistent with both the current HabEx and LUVOIR designs at wavelengths around the 0.94 $\mu$m water vapor spectral feature. Proposed coronagraphs for both the HabEx and LUVOIR concepts would achieve corongraph bandwidths of 10–20%, which is consistent with our adopted bandwidth (15%). We note that the primary HabEx design also includes a starshade capable of performing high-contrast imaging and spectroscopy across the full 0.45–1.0 $\mu$m range in a single pointing, which would supersede the bandpass adopted here. Finally, our study explores retrievals at different characteristic SNRs so as to avoid tying our results to a specific telescope design. Nevertheless, our SNRs can be converted to requisite integration times for the HabEx and LUVOIR concepts using available instrument models[^3][^4] or through the @robinson2016characterizing noise model.****
Retrieval {#sec:methods:retrieval}
---------
Simulations were produced by pairing our albedo and noise models, which we then treated as observational data, and a Bayesian retrieval was performed using the PyMultiNest software [@feroz_hobson_bridges_2009; @buchner2014x], following @feng_robinson_fortney_lupu_marley_lewis_macintosh_line_2018. Computational Bayesian retrieval techniques involve the comparison of model outputs to the observed data, using a variety of algorithms (such as Markov-chain Monte Carlo, or the Multi-Nested methods used here) to explore the parameter space in an efficient manner. By quantifying the “likelihood” of each set of parameters producing the observed data, we seek to understand the distribution of possible values of those parameters. By comparing these posterior distributions to the known input values for each parameter, we can determine the information content of a data set at a given signal-to-noise ratio. We have previously used these techniques in both gas giant and terrestrial planet reflection spectra [@lupu_marley_lewis_line_traub_zahnle_2016; @feng_robinson_fortney_lupu_marley_lewis_macintosh_line_2018].
The @cahoy_marley_fortney_2010 albedo model used in this project accepts as input the 13 parameters detailed in Table \[table:params\]: mixing ratios for five molecules: ozone (O$_3$), oxygen (O$_2$), methane (CH$_4$), carbon dioxide (CO$_2$), and water (H$_2$O); surface properties of atmospheric pressure ($P_0$), gravitational acceleration ($g_{\mathrm{pl}}$), and surface reflectivity ($A_{\mathrm s}$); the planetary radius ($R_{\mathrm{pl}}$); and four cloud properties: cloud top pressure ($p_{\mathrm t}$), cloud pressure thickness ($\delta p$), cloud optical depth ($\tau$), and cloud coverage fraction ($f_{\mathrm{cld}}$). By repeatedly sampling values of each retrieved parameter and comparing the albedo spectrum output to calculate a numerical likelihood, the Bayesian retrieval tool builds up a posterior probability distribution for all “free” retrieved parameters.
During this process, we chose not to retrieve for the abundances of molecular oxygen O$_2$, methane CH$_4$, and carbon dioxide CO$_2$, instead fixing them at the values presented in Table \[table:params\]. As shown in Figure \[fig:features\], there are no major features for any of these three molecules in the region of interest. Extensive testing revealed that the retrieval posteriors for the molecules in question were uninformative when they were included in free parameters. H$_2$O posteriors retrieved with and without fixing the abundances of these molecules were virtually indistinguishable, as shown in Figure \[fig:molecules\], where we compare retrievals performed with and without these gasses. This test was conduced by performing two retrievals on the same data set: one in which these molecules were allowed to be free parameters, to be retrieved by the nested sampler; and a second, in which we held them fixed while retrieving for other parameters *given* the values of the mixing ratio for these two molecules. Since including these gasses leads to increased computational time, we elected to leave them fixed at truth values during the retrieval.
Beyond that depicted in Figure \[fig:molecules\], a total of 19 retrievals were performed: A “primary” set of twelve retrievals, and an “auxiliary” set of seven. The primary set explored planets with H$_2$O mixing ratios at 0.01$\times$, 0.1$\times$, 1$\times$, and 10$\times$ the current Earth levels, with signal-to-noise ratios of 5, 10, and 15, retrieving on the parameters as described above. When adjusting the H$_2$O mixing ratios, we held the mixing ratios of the other spectrally active atmospheric constituents fixed by increasing (decreasing) the background N$_2$ gas ratio to compensate for the decreased (increased) H$_2$O presence.
Three of the auxiliary retrievals were performed on data sets were derived from a $1\times$ Earth-like model, using SNR$\,=5, 10,$ and 15, but *without* the $0.525-0.575\,\mu$m optical photometric point. These were analyzed with our retrieval framework in order to explore the value of this green data point indirectly, by examining the information contained only in the red spectroscopic observation. While in general the $0.525-0.575\,\mu$m optical photometric point is expected to be available, understanding the effect of its absence allows us to consider cases where there is concern about the validity of the optical point data for any reason; for instance, if there was a significant time delay between the photometric detection and followup spectroscopy, or that the phase angles of the two measurements may differ.
The final set of four auxiliary retrievals were performed on data sets that used 0.10 and $0.15\,\mu$m wide photometric *filters* over the $0.94\,\mu$m absorption band (as well as the optical photometric point) for model planets with $1\times$ and $10\times$ Earth-like water mixing ratios. The rationale was to understand if *photometry* could give any constraint on water vapor, instead of more time-consuming spectroscopy.
![An illustration of the impact of including molecular oxygen O$_2$, methane CH$_4$, and carbon dioxide CO$_2$ as free parameters in retrievals. These two posterior distributions of H$_2$O were retrieved from a data set based on a forward model with $1\times$ Earth-like water mixing ratio and a signal-to-noise ratio of 15. Because of the very small difference in the posteriors, we chose to fix these parameters in order to reduce retrieval times.[]{data-label="fig:molecules"}](grid2_molecules.pdf){width="3.0in"}
Results {#sec:results}
=======
Although we retrieved on many atmospheric properties in our runs, the H$_2$O mixing ratio was of primary interest. Therefore, the results of H$_2$O mixing ratio retrieval will be shown in some detail. Results for other parameters where meaningful constraints could be placed will also be discussed.
Following @feng_robinson_fortney_lupu_marley_lewis_macintosh_line_2018, we define the following 4 terms for use when discussing our results.
1. A *non-detection* describes a retrieved posterior that is flat or nearly flat across the entire prior range.
2. A *weak detection* describes a retrieved posterior that shows a peak, but has significant “tails" going to one or both ends of the prior. This includes retrievals that produce upper or lower limits on the parameter.
3. A *detection* describes a posterior that shows a localized peak without significant “tails", but that have a $1\sigma$ range greater than one order of magnitude.
4. A *constraint* describes a peaked posterior distribution, similar to a detection, with a $1\sigma$ range of less than one order of magnitude.
Our standard setup included the $0.55\,\mu$m optical point with SNR$\,=10$, and an R$\,=140$ spectrum across the $0.94\,\mu$m water band. In retrievals with H$_2$O $=0.01\times$ Earth-like values, we found that all SNR tested resulted in weak detections of H$_2$O mixing ratios. In retrievals with H$_2$O $=0.1\times$ Earth-like values, we also found that low SNR resulted in weak detections of H$_2$O. In particular, the SNR$\,=5$ retrieval in this regime resulted in a detection weak enough to be considered a non-detection. However, with SNR$\,=15$ we achieved a detection. In retrievals with H$_2$O $=1\times$ Earth-like values, we achieved a weak detection with SNR$\,=5$, a detection with SNR$\,=10$, and a constraint with SNR$\,=15$. Similarly, in retrievals with H$_2$O $=10\times$, we achieved detections with SNR$\,=5$ and $10$, and a constraint with SNR$\,=15$. These results are summarized in Figure \[fig:h2ostack\].
![H$_2$O posteriors for data sets with H$_2$O $=0.01\times$ (top left), $0.1\times$ (top right), $1\times,$ (bottom left) and $10\times$ (bottom right) Earth-like values. Each data set combines a 0.55$\mu$m photometric point (with fixed signal-to-noise ratio of 10) and R = 140 spectroscopy centered on the 0.94$\mu$m water band. For each water mixing ratio, we vary the signal-to-noise ratio of the spectrum: SNR = 5 (teal, cross-hatched), 10 (thin purple line) and 15 (thick magenta line) are shown, along with the location of the truth value (black dashed line). Water is weakly detected even for the lowest SNR and lowest mixing ratio. To claim detection, there needs to be at least 0.1$\times$ Earth water in the atmosphere and corresponding SNR = 15 data; for water content $1\times$ Earth value, the data need SNR = 10 for detection. SNR = 15 offers constraint for H$_2$O = $1\times$ and $10\times$ Earth values only. []{data-label="fig:h2ostack"}](grid2_H2Ohistograms.pdf){width="3.0in"}
In our retrievals we were only able to constrain – or, indeed, achieve a detection – on a few atmospheric parameters, including the H$_2$O mixing ratio, $P_0$ (surface pressure), $p_{\mathrm t}$ (cloud-top pressure), and $\delta$p (cloud thickness in pressure). Surface pressure moved from a weak detection to a detection at H$_2$O $=10\times$ Earth-like values and SNR$\,=10$, and at all water abundances at SNR$\,=15$. The cloud properties p$_t$ and $\delta$p moved from weak detection to detection at H$_2$O $=0.1\times$ Earth-like values and greater with SNR$\,=10$, and at all water abundances at SNR$\,=15$. The cloud properties $\tau$ and f$_{cld}$ returned non-detections in all retrievals. $A_s$ was undetected at H$_2$O $= 0.1\times$ Earth-like values and lower with SNR$\,=5$ and $10$, and weakly detected in other retrievals. R$_{pl}$ and g$_{pl}$ are weakly detected in all retrievals.
Of additional note is that we did achieve a weak detection of O$_3$ with an upper bound in even our poorest SNR retrievals. Evidently, the green photometric discovery point has some utility in placing an upper limit on ozone. In the interest of determining the value of a same-phase optical photometric point in the retrieval of H$_2$O, we also performed retrievals with SNR$\,=5,10,$ and $15$ and H$_2$O $=1\times$ Earth-like values with this point not included, shown in Figure \[fig:optival\]. Although this modestly degraded the precision and accuracy of the retrieval for all retrieved features, particularly for the lowest SNR case, it did not substantially change the shape of the posterior for H$_2$O in the SNR$\,=10$ and 15 cases. In addition, we lost the upper limit on O$_3$ and were left with a non-detection of ozone in all retrievals.
![Three retrievals were performed on a data set which did not include an optical 0.55$\mu$m photometric data point. These were done with H$_2$O $=1\times$ Earth-like values and SNR $=5, 10,$ and $15$. Top row: we show the results of those retrievals. Bottom row: we plot the results of retrievals on data with the same H$_2$O and SNR values that does include the optical photometric data point. While the posteriors of water do not change much, this shows that the inclusion of the photometric point allows us to go from a non-detection to placing an upper limit on O$_3$.[]{data-label="fig:optival"}](grid2_greenPT.pdf){width="3.3in"}
As discussed above, an option considered for H$_2$O detection, which would require the least integration time, was the use of a photometric filter centered on the H$_2$O feature at $0.94 \,\mu$m. Therefore, we performed four retrievals under this assumption. We considered Earth-like and $10 \times$ Earth-like H$_2$O concentrations, which we judged to be the most favorable cases given our retrievals with R$\,=140$ spectroscopy. We tried two different filter widths of $0.85-1.00\,\mu$m and $0.90-1.00 \,\mu$m and combined each water band photometric point with the optical point at 0.55$\mu$m to form data sets with SNR = 15. However, we retrieved non-detections on H$_2$O in all situations, as shown in Figure \[fig:specVfilter\], which shows posteriors for water and ozone. This suggests that even well-place photometric points will be of little aid in classifying rocky planets as a tool to decide on future detailed characterization. This echoes the finding of @batalha2018color in their exhaustive study of giant planet albedos.
![Retrieved H$_2$O posterior distributions for photometric data compared with an R$\,=140$ spectrum. Retrieval results for H$_2$O (left) and O$_3$ (right) with truth H$_2$O values of $1\times$ (top) and $10\times$ (bottom) Earth-like values. We compare the R$\,=140$ scenario (teal cross-hatch) used in the rest of this paper with two scenarios employing photometric filters across the $0.94\,\mu$m H$_2$O absorption feature. One scenario used a $0.15\,\mu$m filter (thin purple line) from $0.85-1.00\,\mu$m. The second scenario used a narrower, $0.10\,\mu$m filter (thick magenta line) from $0.90-1.00\,\mu$m. All posteriors shown in this figure were retrieved from SNR=15 data. The switch from spectroscopy to photometry of the water band means that we would go from constraint of H$_2$O to non-detection even at SNR=15. The switch would not impact O$_3$ inference much, with all cases retrieving upper limits for the molecule.[]{data-label="fig:specVfilter"}](grid2_spectrumVfilter.pdf){width="3.3in"}
Discussion and Conclusions {#sec:disc}
==========================
A “follow the water” strategy for terrestrial exoplanet atmospheric characterization may be a useful one for determining which worlds may be most interesting for detailed follow-up observations. Through some initial retrieval explorations that used simulated observations from a large space-based telescope, we have been able to start shedding some light on how this might be best accomplished. First, while filter photometry allows for shorter integration times to achieve a given SNR, and will likely be how the first planet detections are made, it will be of limited aid in characterizing atmospheric water abundances.
In addition, the diagnostic power of spectroscopy was so high that retrievals for the water mixing ratio, from spectroscopy, were not particularly aided by the additional of a “discovery phase” green photometric point. While the presence or absence of this data point did have an effect on the precision of our retrieval results, we found that with a signal-to-noise ratio of 10 or higher the precision of retrieved water abundance was sufficiently comparable to results that included the data point that we do not expect it would have a substantial impact on our overall conclusions. The 2$\sigma$ lower bounds without the green photometric point are $10^{-0.5}\times, 10^{-1.1}\times,$ and $10^{-2.2}\times$ Earth-like values for SNR$\,=5, 10,$ and 15, respectively. These values are essentially similar to those found for retrievals which included the green photometric point. It is worth noting, however, that this data point did allow for upper limits to be placed on ozone abundances.
![These H$_2$O posteriors are identical to those shown in Figure \[fig:h2ostack\], but sorted by signal-to-noise ratio in order to show trends and the ability to distinguish planets based on atmospheric water content. The vertical marks along the top of the image indicate the four truth values used in this study. We see that a retrieval on SNR=5 data results in substantial differences in the posterior between high ($\geq 1\times$ Earth-like) and low ($\leq 0.1\times$ Earth-like) water mixing ratio objects, while at SNR=10 and 15, additional differences begin to emerge for 0.1$\times$ objects. For the driest case, increasing SNR does not improve detection beyond an upper limit. For planets with at least $0.1\times$ Earth water mixing ratio, we can benefit from better SNR data as detection is possible. If a terrestrial planet’s atmosphere is sufficiently abundant with water ($\geq 1\times$ Earth-like), we can detect its presence with SNR = 5.[]{data-label="fig:snrstack"}](grid2_SNRHistograms.pdf){width="3.3in"}
Using an R$\,=140$ spectrum from $0.85-1.00\,\mu$m, in conjunction with the green optical point, constraints on H$_2$O abundance were only possible in high signal-to-noise ratio cases, and then only with a significant water presence. However, if constraints on abundances are not necessary for early atmospheric characterization, it may be helpful to consider the requirements to distinguish the water-bearing worlds from the dry worlds, as the presence of water vapor in any significant quantity may indicate a world of interest to astrobiological studies. In Figure \[fig:snrstack\] we have re-plotted the data from Figure \[fig:h2ostack\] to show the appearance of the H$_2$O posterior with different truth H$_2$O values, while keeping the signal-to-noise ratio constant.
Although our retrievals were not able to place strong constraints on H$_2$O mixing ratios, we did find that a low SNR$\,=5$ retrieval may have some utility to distinguish worlds with H$_2$O $\geq 1\times$ Earth-like levels from those with H$_2$O $\leq 0.1\times$ Earth-like levels. These SNR$\,=5$ retrievals yielded uninformative $2\sigma$ lower water mixing ratio limits[^5] for $0.01\times$ and $0.1\times$ Earth-like water mixing ratios, while retrievals on Earth-like and greater water mixing ratio models returned $2\sigma$ lower-bound values in excess of $0.3\times$. Additionally, with the higher SNR$\,=10$, the $0.1\times$ worlds begin to become distinguishable from $0.01\times$ worlds, as the $2\sigma$ water mixing ratio lower-bound for $0.1\times$ worlds rises to $10^{-2.5}\times$ Earth-like values, while that of $0.01\times$ worlds remains unchanged. These distinctions are possible because the retrieved posterior distributions show a strong sensitivity to H$_2$O mixing ratio. $2\sigma$ upper limits were largely uninformative in all retrievals, with useful values only becoming apparent in high SNR$\,=15$ retrievals on very low H$_2$O models. We caution that the $0.01\times$ models were not distinguishable from essentially dry worlds even at SNR$\,=15$.
As outlined above, our work builds on that of @feng_robinson_fortney_lupu_marley_lewis_macintosh_line_2018, who applied Bayesian retrieval techniques to simulated observations of model true-Earth analogs, but with different observational assumptions. Thus a comparison between these two studies may be prudent. When @feng_robinson_fortney_lupu_marley_lewis_macintosh_line_2018 studied R=140 spectroscopy of the full optical spectrum from $0.4-1.0\,\mu$m, much stronger retrieved detections were produced on atmospheric parameters that we have no handle on in this work. This is unsurprising, given our narrow wavelength range of interest. A similar story emerges when comparing H$_2$O, specifically; while neither the present work nor the @feng_robinson_fortney_lupu_marley_lewis_macintosh_line_2018 R=140 retrievals were able to make strong statements about H$_2$O mixing ratio with a signal-to-noise ratio of 5, at high signal-to-noise ratios this work fared substantially poorer in both precision and accuracy of retrieved values. We also note that while we used a SNR reference $\lambda_0=0.88\,\mu$m, which is appropriate for our study, @feng_robinson_fortney_lupu_marley_lewis_macintosh_line_2018 used instead a value of $\lambda_0=0.55\,\mu$m, which was appropriate for the wide optical wavelength range they studied.
![In this image, the median (blue), 1$\sigma$ (dark green), and 2$\sigma$ (light green) parameter sets have been used to generate albedo spectra, and have been overplotted with the truth albedo spectra (black dotted line) and the input data set (purple). The regions with no data show a large spread in possible flux ratio values. Additional spectroscopic coverage can ensure better constraint of the atmospheric state and planetary properties such as surface albedo and surface pressure.[]{data-label="fig:yarr"}](yarr_earthGrid2_r140s15w0_PMN_L.pdf){width="3.3in"}
Another way to appraise our results is to consider how well the albedo spectrum of the planet is constrained at wavelengths outside of where data are obtained, operating within our framework of assuming a rocky exoplanet and using parameters based on our retrieval results. This is shown in Figure \[fig:yarr\]. As might be expected, the retrieved atmospheric parameters yield optical spectra that tightly correspond to the fiducial model spectrum in each of the water features, but we see considerable deviation outside of these regions. In particular, with no data across the 0.77 $\,\mu$m O$_2$ A-band, we visually see little constraint on the feature depth, compared to the excellent fit with with the weaker water features to the blue and red of this O$_2$ band. A future investigation might look at data near the O$_2$ A-band, perhaps including the H$_2$O $\alpha$-band absorption feature at 0.72 $\,\mu$m within a $\sim$ 10-15% bandpass. Another path this suggests for future studies is comparing results with other, non-Earth-like models – a retrieval where one is truly blind to atmosphere type may suggest a path to differentiate water-bearing Earth-like worlds from water-bearing small sub-Neptunes.
Taking all this together, a picture begins to emerge as to the value of the combined R=140 spectroscopy and 0.525-$0.575\,\mu$m photometry data collection setup used in this study. While clearly insufficient as a means for detailed characterization, using spectroscopy on the $0.94\,\mu$m feature appears to be a useful method for quickly distinguishing between wet and dry rocky exoplanets. This distinction can then be used to guide broader, more time-intensive followup studies in a search for life-bearing exoplanets. We note, however, that a photometric band centered on this same $0.94\,\mu$m water feature provided little utility, as even at a high SNR water vapor was not detected. In addition, we found that making use of the $0.55\,\mu$m optical data point can allow one to place some constraints on O$_3$, and at high-SNR using both can allow for determination of some useful information about the planet’s surface pressure and some cloud properties.
Additional studies may be prudent for a better understanding of the limitations of this technique. Here we only examined a single spectral resolution, of R$\,=140$. Lower resolution could be explored across this relatively wide bandpass. Furthermore, the H$_2$O mixing ratios studied here were each spaced by an order of magnitude, while other properties were left at Earth-like levels. Studies with additional granularity in H$_2$O mixing ratio may provide some benefits, as would consideration of the other physical properties of the planet.
A more physically motivated “Earth” model could include additional physical effects. It may reasonably be expected that altering the water mixing ratio will impact the properties of water clouds in the planetary atmosphere. As a world with less water will, perforce, have fewer water clouds, an observation of such a world would see deeper into the atmosphere. This would, to some extent, strengthen the water vapor absorption feature, thus we expect to see some slight degeneracy between f$_{cloud}$ and log(H$_2$O). The height in the atmosphere where water clouds reside would likely change as well, although a change in water mixing ratio would alter the greenhouse effect and hence the temperature structure of the atmosphere, including condensation levels. While these effects may alter the particulars of the model results, it is not expected that the degeneracy between f$_{cloud}$ and log(H$_2$O) would be strong; further, given our retrievals’ relative insensitivity to cloud features in general, and in particular to f$_{cloud}$ such changes are unlikely to impact the broader conclusions drawn by this study.
Finally, changes to water mixing ratios may impact other parameter values in a way which is not represented here. In particular, as explored in @wordsworth2013water, changes to water mixing ratios may impact carbon dioxide mixing ratios for Earth-like planets. This was not modelled here, although as discussed, we did not find that our results were dependent on CO$_2$ mixing ratios given our focus on the optical bandpass. Clearly, temperate rocky planets can present a wide range of atmospheric states, and much work lies ahead in assessing how to characterize these potentially habitable worlds.
The authors would like to acknowledge Xi Zhang for interesting discussions on atmospheres with high water mixing ratios and Bruce Macintosh for insights on the observational context. We thanks the referee and Ehsan Gharib for helpful discussions on the clarity of the work and related opacity issues. TDR, JJF, and MSM gratefully acknowledge support from an award through the NASA Exoplanets Research Program (\#80NSSC18K0349) and Habitable Worlds Program (\#80NSSC20K0226). M.S.M. acknowledges support from GSFC Sellers Exoplanet En- vironments Collaboration (SEEC), with funding specifically by the NASA Astrophysics Divisions Internal Scientist Funding Model. This work was made possible by support from the UCSC Other Worlds Laboratory and the WFIRST Science Investigation Team program. The results reported herein benefited from collaborations and/or information exchange within NASAÕs Nexus for Exoplanet System Science (NExSS) research coordination network, sponsored by NASAÕs Science Mission Directorate. Computation for this research was performed by the UCSC Hyades supercomputer, which was supported by the National Science Foundation (award number AST-1229745).
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[^1]: HITRAN 2012 cites the following references in construction of their water vapor line list: @shirin2006spectroscopically, @barber2006high, @brown2007co2, @furtenbacher2007marvel, @lisak2008low, @tennyson2009iupac, @tennyson2010iupac, @rothman2010hitemp, @ma2011pair, @birk2012temperature, @furtenbacher2012marvel, @lodi2012line, and @tennyson2013iupac
[^2]: Changes in H$_2$O content were compensated by changes in background N$_2$ gas content. See Section \[sec:methods:retrieval\] for more detail. All other parameters were left unchanged; see Section \[sec:disc\] for a discussion of possible ramifications of this choice.
[^3]: https://habex.ipac.caltech.edu/
[^4]: https://asd.gsfc.nasa.gov/luvoir/tools/
[^5]: $10^{-3.5}\times$ and $10^{-4.3}\times$ Earth-like levels, respectively. Both of these values approach the lower limit of the sampled range.
|
---
abstract: |
In this paper, we consider the Ising-Vanniminus model on an arbitrary order Cayley tree. We generalize the results conjectured in [@Akin2016; @Akin2017] for an arbitrary order Cayley tree. We establish existence and a full classification of translation invariant Gibbs measures with memory of length 2 associated with the model on arbitrary order Cayley tree. We construct the recurrence equations corresponding generalized ANNNI model. We satisfy the Kolmogorov *consistency* condition. We propose a rigorous measure-theoretical approach to investigate the Gibbs measures with memory of length 2 for the model. We explain whether the number of branches of tree does not change the number of Gibbs measures. Also we take up with trying to determine when phase transition does occur.\
**Keywords**: Solvable lattice models, Rigorous results in statistical mechanics, Gibbs measures, Ising-Vannimenus model, phase transition.\
**PACS**: 05.70.Fh; 05.70.Ce; 75.10.Hk.
address: |
Hasan Akin\
Ceyhun Atuf Kansu Cad. Cankaya, Ankara, Turkey, akinhasan25@gmail.com
author:
- Hasan Akin
date:
-
-
title: Gibbs Measures with memory of length 2 on an arbitrary order Cayley tree
---
Introduction
============
One of the main purposes of equilibrium statistical mechanics consists in describing all limit Gibbs distributions corresponding to a given Hamiltonian [@Georgii]. One of the methods used for the description of Gibbs measures on Cayley trees is Markov random field theory and recurrent equations of this theory [@AT1; @NHSS; @GRS; @RAU; @Rozikov; @MDA; @AGUT]. The approach we use here is based on the theory of Markov random fields on trees and recurrent equations of this theory. In this paper, we discuss their relation with the recurrent equations of the theory of Markov random fields on trees for Ising model [@NHSS; @NHSS1]. In [@ART], we obtain a new set of limiting Gibbs measures for the Ising model on a Cayley tree. In [@GU2011; @UGAT; @GATUN], the authors study the phase diagram for the Ising model on a Cayley tree of arbitrary order $k$ with competing interactions. In [@GATUN], the authors characterized each phase by a particular attractor and the obtained the phase diagram by following the evolution and detecting the qualitative changements of these attractors.
$n$-dimensional integer lattice, denoted $\mathbf{Z}^{n}$, has so-called amenability property. Moreover, analytical solutions does not exist on such lattice. But investigations of phase transitions of spin models on hierarchical lattices showed that there are exact calculations of various physical quantities (see for example, [@Bax; @Rozikov]). Such studies on the hierarchical lattices begun with the development of the Migdal-Kadanoff renormalization group method where the lattices emerged as approximants of the ordinary crystal ones. On the other hand, the study of exactly solved models deserves some general interest in statistical mechanics [@Zachary].
A Cayley tree is the simplest hierarchical lattice with non-amenable graph structure [@Preston]. Also, Cayley trees still play an important role as prototypes of graphs [@Ostilli]. This means that the ratio of the number of boundary sites to the number of interior sites of the Cayley tree tends to a nonzero constant in the thermodynamic limit of a large system. Nevertheless, the Cayley tree is not a realistic lattice, however, its amazing topology makes the exact calculations of various quantities possible.
One of the most interesting problems in statistical mechanics on a lattice is the phase transition problem, i.e. deciding whether there are many different Gibbs measures associated to a given Hamiltonian [@BRZ; @BG; @Nasir; @AGUT]. Investigations of phase transitions of spin models on hierarchical lattices showed that they make the exact calculation of various physical quantities [@Sinai; @GRRR; @GHRR]. It was established the existence of the phase transition, for the model in terms of finitely correlated states, which describes ground states of the model. Up to this day many authors have studied the existence of phase transition by means of the recurrence equations corresponding to the Ising-Vanniminus model on Cayley tree of order two and three [@Akin2016; @Akin2017; @ART; @NHSS; @RAU]. Recently, Ganikhodjaev [@Nasir] has studied the existence of phase transition for Ising model on the semi-infinite Cayley tree of second order with competing interactions up to third-nearest-neighbor generation with spins belonging to the different branches of the tree. In the present paper, for a given Hamiltonian, we provide a more general construction of Gibb measures associated with the Hamiltonian. We prove the existence of translation-invariant Gibb measures associated to the model which yield the existence of the phase transition.
It is well known that the Potts model is a generalization of the Ising model, but the Potts model on a Cayley tree is not well studied, compared to the Ising model [@Rozikov]. In the last decade, many researches have investigated Gibbs measures associated with Potts model on Cayley trees [@GTA; @AS2015; @AS1; @GATTMJ; @GTACUBO; @AT1]. In [@AS2015], we studied the existence, uniqueness and non-uniqueness of the Gibbs measures associated with the Potts model on a Bethe lattice of order three with three coupling constants by using Markov random field method. In [@AS1], we have obtained the exact solution of a phase transition problem by means of Gibbs state of the same Potts model in [@AS1].
In the present paper, we are concerned with the Ising-Vanniminus model on an arbitrary order Cayley tree. We investigate translation invariant Gibbs measures associated with Ising-Vannimenus model on arbitrary order Cayley tree. We generalize the results obtained in [@Akin2016; @Akin2017]. We use the Markov random field method to describe the Gibbs measures. We satisfy the Kolmogorov *consistency* condition. We propose a rigorous measure-theoretical approach to investigate the Gibbs measures with memory of length 2 corresponding to the Ising-Vanniminus model on a Cayley tree of arbitrary order. Also we take up with trying to determine when phase transition does occur.
The outline of this paper is as follows. In Section \[PRELIMINARIES\] we give the definitions of the Cayley tree, Gibbs measures and Ising-Vannimenus model. Section \[Construction\] provides a construction of Gibbs measures on an arbitrary order Cayley tree. In Section \[even\] we establish the existence, uniqueness and non-uniqueness of the translation-invariant Gibbs measures by means of the recurrence equations for $k$-even, while in Section \[odd\] we do the same for $k$-odd. We contain in Section \[Conlussion\] concluding remarks and discussion of the consequences of the results with next problems.
PRELIMINARIES {#PRELIMINARIES}
=============
Cayley trees
------------
Cayley trees (or Bethe lattices) are simple connected undirected graphs $G = (V, E)$ ($V$ set of vertices, $E$ set of edges) with no cycles (a cycle is a closed path of different edges), i.e., they are trees [@Ostilli]. Let $\Gamma^k=(V, L,
i)$ be the uniform Cayley tree of order $k$ with a root vertex $x^{(0)}\in V$, where each vertex has $k + 1$ neighbors with $V$ as the set of vertices and the set of edges. The notation $i$ represents the incidence function corresponding to each edge $\ell\in L$, with end points $x_1,x_2\in V$. There is a distance $d(x, y)$ on $V$ the length of the minimal point from $x$ to $y$, with the assumed length of 1 for any edge (see Figure \[cayley-level2\]).
We denote the sphere of radius $n$ on $V$ by $ W_n=\{x\in V:
d(x,x^{(0)})=n \}$ and the ball of radius $n$ by $ V_n=\{x\in V:
d(x,x^{(0)})\leq n \}.$ The set of direct successors of any vertex $x\in W_n$ is denoted by $S_k(x)=\{y\in W_{n+1}:d(x,y)=1 \}.$
\[!htbp\]\[cayley-level2\] ![Two successive generations of semi-infinite Cayley tree $\Gamma^k$ of arbitrary order $k>1$ (branching ratio is finite $k$). The fixed vertex $x^{(0)}$ is the root of the lattice that emanates $k$ edges of $\Gamma^k$ ($y_j\in S(x^{(0)}),z_i^{(j)}\in
S(y_j)$).[]{data-label="cayley-level2"}](cayley-level2.eps "fig:"){width="75mm"}
Ising-Vannimenus model
----------------------
The Ising model with competing nearest-neighbors interactions is defined by the Hamiltonian $$\label{hm1}
H(\sigma)=-J\sum_{<x,y>\subset V}\sigma(x)\sigma(y),$$ where the sum runs over nearest-neighbor vertices $<x,y>$ and the spins $\sigma(x)$ and $\sigma(y)$ take values in the set $\Phi=\{-1,+1\}$.
The Hamiltonian $$\label{hm}
H(\sigma)=-J_p\sum_{>x,y<}\sigma(x)\sigma(y)-J\sum_{<x,y>}\sigma(x)\sigma(y)$$ defines the **Ising-Vannimenus model** with competing nearest-neighbors and next-nearest-neighbors, with the sum in the first term representing the ranges of all nearest-neighbors, where $J_p,J\in \mathbb{R}$ are coupling constants corresponding to prolonged next-nearest-neighbor and nearest-neighbor potentials [@Vannimenus].
Gibbs measures
--------------
A finite-dimensional distribution of measure $\mu$ in the volume $V_n$ has been defined by formula $$\label{mu1}
\mu_n(\sigma_n)=\frac{1}{Z_{n}}\exp[-\frac{1}{T}H_n(\sigma)+\sum_{x\in
W_{n}}\sigma(x)h_{x}]$$ with the associated partition function defined as $$\label{mu}
Z_n=\sum_{\sigma_n \in
\Phi^{V_n}}\exp[-\frac{1}{T}H_n(\sigma)+\sum_{x\in
W_{n}}\sigma(x)h_{x}],$$ where the spin configurations $\sigma_n$ belongs to $\Phi^{V_n}$ and $h=\{h_x\in \mathbb{R},x\in V\}$ is a collection of real numbers that define boundary condition (see [@BG; @GRRR; @GHRR]). Physically, Eq. represents the first step of the Bethe-Peierls approach [@Bethe]. Bleher [@Bleher1990] proved that the disordered Gibbs distribution in the ferromagnetic Ising model associated to the Hamiltonian on the Cayley tree is extreme for $T \geq T^{SG}_C$, where $T^{SG}_C$ is the critical temperature of the spin glass model on the Cayley tree, and it is not extreme for $T< T^{SG}_C$. Previously, researchers frequently used memory of length 1 over a Cayley tree to study Gibbs measures [@BG; @GRRR; @GHRR].
Let $S=\{1,2,...,s\}$ be a finite state space. On the infinite product space ${{S}^{\mathbb{Z}}}$, one can define the product $\sigma$-algebra, which is generated by cylinder sets $_{m}[i_1,...,i_N]$ of length $N$ based on the block $(i_1,...,i_N)$ at the place $m$. We denote by ${{\mathfrak{M}}}({{S}^{\mathbb{Z}}})$ the set of all measures on ${{S}^{\mathbb{Z}}}$. The set of all $\sigma$-invariant measures in ${{S}^{\mathbb{Z}}}$ is denoted by ${{\mathfrak{M}}_{\sigma }}({{S}^{\mathbb{Z}}})$, where $\sigma$ is the shift transformation.
[@Denker (8.1) Proposition]\[Kolmo-Cons1\] For $\mu \in {{\mathfrak{M}}_{\sigma }}({{S}^{\mathbb{Z}}})$ the following properties are valid:
1. $\sum\limits_{i\in S}{\mu {{(}_{0}}[i])=1}$;
2. $\mu (_{n}[i_{0},...,i_{k}])\geq 0$ for any block $({{i}_{0}},{{i}_{1}},...,{{i}_{k}})\in S^{k+1}$ and any $n\in
\mathbb{Z}$;
3. $\mu (_{n}[{{i}_{0}},...,{{i}_{k}}])=\sum\limits_{{{i}_{k+1}}\in S}{\mu
{{(}_{n}}[{{i}_{0}},...,{{i}_{k}},{{i}_{k+1}}])}$;
4. $\mu (_{n}[{{i}_{0}},...,{{i}_{k}}])=\sum\limits_{{{i}_{-1}}\in S}{\mu{{(}_{n}}[{{i}_{-1}},{{i}_{0}},...,{{i}_{k}}])}$.
The proof of the Proposition \[Kolmo-Cons1\] can clearly be checked for both the Bernoulli and the Markov measures on $\sigma$-algebra [@Denker]. By a special case of Kolmogoroff’s consistency theorem (see [@Denker]), these properties are sufficient to define a measure. It is well known that a Gibbs measure is a generalization of a Markov measure to any graph, therefore any Gibbs measure should satisfy the conditions in the Proposition \[Kolmo-Cons1\]. In the next sections, we will show that the Gibbs measure associated to the Ising-Vannimenus model satisfies the conditions in the Proposition \[Kolmo-Cons1\].
Let us consider increasing subsets of the set of states for one dimensional lattices [@FV] as follows: $$\mathfrak{G}_1\subset \mathfrak{G}_2\subset... \subset \mathfrak{G}_n\subset...,$$ where $\mathfrak{G}_n$ is the set of states corresponding to non-trivial correlations between $n$-successive lattice points; $\mathfrak{G}_1$ is the set of mean field states; and $\mathfrak{G}_2$ is the set of Bethe-Peierls states, the latter extending to the so-called Bethe lattices. All these states correspond in probability theory to so-called Markov chains with memory of length $n$ (see [@D1; @FV; @Zachary]).
In [@FV], by using the idea of Bayesian extension, Fannes and Verbeure defined states known as a finite-block measure or as Markov chains with memory of length $n$ on the lattices. Recently, the author has studied the Gibbs measures with memory of length 2 associated to the Ising-Vannimenus model on the Cayley tree of order two and three [@Akin2016; @Akin2017]. The construction is based on the idea in the Proposition \[Kolmo-Cons1\]. In the present paper, we are going to establish the existence of Gibbs measures associated with the Ising-Vannimenus model with memory of length 2 on the Cayley tree of *arbitrary order*.
Construction of Gibbs measures on Cayley tree {#Construction}
=============================================
In this section, we will presents the general structure of Gibbs measures with memory of length 2 associated with the Hamiltonian on an arbitrary order Cayley tree. On non-amenable graphs, Gibbs measures depend on boundary conditions [@Rozikov]. This paper considers this dependency for Cayley trees, the simplest of graphs.
An arbitrary edge $<x^{(0)},x^{(1)}>=\ell \in L$ deleted from a Cayley tree $\Gamma _{1}^{k}$ and $\Gamma _{0}^{k}$ splits into two components: semi-infinite Cayley tree $\Gamma _{1}^{k}$ and semi-infinite Cayley tree $\Gamma _{0}^{k}$. This paper considers a semi-infinite Cayley tree $\Gamma _{0}^{k}$. For a finite subset $V_n$ of the lattice, we define the finite-dimensional Gibbs probability distributions on the configuration space $\Omega^{V_n}=\{\sigma_n=\{\sigma(x)=\pm 1, x\in V_n \}\}$ at inverse temperature $\beta=\frac{1}{kT}$ by formula.
Let $x\in W_{n}$ for some $n$ and $S(x)=\{{{y}_{1}},{{y}_{2}},\cdots ,{{y}_{k}}\}$ are the direct successors of $x$, where ${{y}_{1}},{{y}_{2}},\cdots ,{{y}_{k}}\in
{{W}_{n+1}}$.
Denote ${{B}_{1}}(x)=\left(
\begin{matrix}
{{y}_{k}},\cdots ,{{y}_{2}},{{y}_{1}} \\
x \\
\end{matrix} \right)$ a unite semi-ball with a center $x$. We denote the set of all spin configurations on $V_n$ by $\Phi^{V_n}$ and the set of all configurations on unite semi-ball $B_1(x)$ by $\Phi^{B_1(x)}$. One can get that the set $\Phi^{B_1(x)}$ consists of ${{2}^{k+1}}$ configurations: $$\Phi^{B_{1}(x)}=\left\{\left(\begin{matrix}
i_{k},\cdots ,i_{2},i_{1} \\
i \\
\end{matrix} \right):i,i_{1},i_{2},\cdots ,i_{k}\in \Phi \right\}.$$ Let $$\sigma_{S}({{x}^{(0)}})=\left( \begin{matrix}
\sigma ({{y}_{k}}),\cdots ,\sigma ({{y}_{2}}),\sigma ({{y}_{1}}) \\
\sigma ({{x}^{(0)}}) \\
\end{matrix} \right)$$ be a configuration on the set ${{x}^{(0)}}\cup S({{x}^{(0)}})$ and $$\sigma_{S}({{y}_{i}})=\left( \begin{matrix}
\sigma (z_{k}^{(i)}),\cdots ,\sigma (z_{2}^{(i)}),\sigma (z_{1}^{(i)}) \\
\sigma ({{y}_{i}}) \\
\end{matrix} \right)$$ be a configuration on the set ${{y}_{i}}\cup S({{y}_{i}}),$ ${y}_{i}\in S({{x}^{(0)}})$. Let $\Omega(S)$ be the set of all such configurations.
We wish to consider a probability measure $\mu
_{\mathbf{h}}^{(n)}$ that is formally given by $$\label{Gibbs1}
\mu _{\mathbf{h}}^{(n)}(\sigma )=\frac{1}{Z_{\mathbf{h}}^{(n)}}
\exp [-\beta {{H}_{n}}(\sigma )+\sum\limits_{x\in
{{W}_{n-1}}}{\sum\limits_{y\in S(x)}{\sigma }}
(x)\prod\limits_{y\in S(x)}{\sigma
(y)h_{B_{1}(y);\sigma_{S}(y)}}],$$ where $\beta =\frac{1}{{{k}_{B}}T}$, ${{k}_{B}}$ is the Boltzmann constant and $h_{B_{1}(y);\sigma_{S}(y)}$ is a real-valued function of $y\in V$. $\sigma_n: x\in V_n\to \sigma_n(x)$ and $Z_{\mathbf{h}}^{(n)}$ corresponds to the following partition function: $$\label{partition1}
Z_{\mathbf{h}}^{(n)}=\sum\limits_{{{\sigma }_{n}}\in {{\Phi
}^{{{V}_{n}}}}}{\exp }[-\beta H({{\sigma }_{n}})+\sum\limits_{x\in
{{W}_{n-1}}}{\sum\limits_{y\in S(x)}{\sigma }}
(x)\prod\limits_{y\in S(x)}{\sigma
(y){{h}_{B_{1}(y);\sigma_{S}(y)}}}].$$ In this paper, we suppose that vector valued function $\textbf{h}:V\rightarrow \mathbb{R}^{2(k+1)}$ is defined by $$\label{consistency}
\textbf{h}:(x,y_k,y_{k-1},\ldots, y_2,y_1)\rightarrow
\textbf{h}_{B_1(x)}=(h_{B_1(x);\sigma_{S}({{x}})}:y_i\in S(x)),$$ where $h_{B_1(x);\sigma_{S}({{x}})}\in \mathbb{R}$, $x\in W_{n-1}$ and $y_i\in S(x).$ We will consider a construction of an infinite volume distribution with given finite-dimensional distributions. More exactly, we will attempt to find a probability measure $\mu$ on $\Omega$ that is compatible with given measures $\mu_{\textbf{h}}^{(n)}$, *i.e.*, $$\label{CM}
\mu(\sigma\in\Omega:
\sigma|_{V_n}=\sigma_n)=\mu^{(n)}_{\textbf{h}}(\sigma_n), \ \ \
\textrm{for all} \ \ \sigma_n\in\Omega^{V_n}, \ n\in \mathbf{N}.$$ We say that the probability distributions $\mu_{\textbf{h}}^{n}$ satisfy the Kolmogorov consistency condition if for any configuration $\sigma_{n-1}\in\Omega^{V_{n-1}}$ $$\label{comp}
\sum_{\omega\in\Omega^{W_n}}\mu^{(n)}_{\textbf{h}}(\sigma_{n-1}\vee\omega)=\mu^{(n-1)}_{\textbf{h}}(\sigma_{n-1}).$$ This condition implies the existence of a unique measure $\mu_{\textbf{h}}^{(n)}$ defined on $\Omega$ with a required condition . Such a measure $\mu_{\textbf{h}}^{(n)}$ is called a Gibbs measure with memory of length 2 associated to the model .
We note that first two conditions of Proposition \[Kolmo-Cons1\] is trivial to check for the measure in . The condition (3) of Proposition \[Kolmo-Cons1\] is the same as the condition in . Therefore, it should be proved that the Gibbs measure satisfies the condition (3) in the Proposition \[Kolmo-Cons1\].
The recurrence equations for $k$-even {#even}
=====================================
Let $k$ be the positive even integer, where $k$ is the order of the Cayley tree. It is reasonable, though, to assume that the different branches are equivalent, as is usually done for models on trees.
Let $$\sigma _{S}^{+}({{x}^{(0)}})=\left(
\begin{matrix}
\sigma ({{y}_{k}}),\cdots ,\sigma ({{y}_{2}}),\sigma ({{y}_{1}}) \\
+ \\
\end{matrix} \right)$$ be a configuration in $\Phi^{B_1(x)}$ (see Fig. \[cayley-level2\]). Let $m$ be the number of spins down, i.e., $\sigma ({{y}_{i}})=-1$ on the first level ${{W}_{1}}$, where $0\leq m\leq k$. Then $(k-m)$ is the number of spins up, i.e., $\sigma ({{y}_{i}})=+1$ on the first level ${{W}_{1}}$. Let $$\sigma _{S}^{+}({{y}_{i}})=\left( \begin{matrix}
\sigma (z_{k}^{(i)}),\cdots ,\sigma (z_{2}^{(i)}),\sigma (z_{1}^{(i)})\\
+ \\
\end{matrix} \right)$$ be a configuration in $\Omega(S)$. Let $m$ be the number of spins down, i.e., $\sigma (z_{j}^{(i)})=-1$ on the second level ${{W}_{2}}$, where $0\leq m \leq k.$ Let $$\sigma _{S}^{-}({{x}^{(0)}})=\left(
\begin{matrix}
\sigma ({{y}_{k}}),\cdots ,\sigma ({{y}_{2}}),\sigma ({{y}_{1}})\\
- \\
\end{matrix} \right)$$ be a configuration in $\Omega(S)$. Let $m$ be the number of spins down, i.e., $\sigma (y_{i})=-1$ on the first level ${{W}_{1}}$, where $0\leq m\leq k.$ Let $$\sigma _{S}^{-}({{y}_{i}})=\left(
\begin{matrix}
\sigma (z_{k}^{(i)}),\cdots ,\sigma (z_{2}^{(i)}),\sigma (z_{1}^{(i)}) \\
- \\
\end{matrix} \right)$$ be a configuration in $\Omega(S)$ (see Fig. \[cayley-level2\]). Let $m$ be the number of spins down, i.e., $\sigma (z_{j}^{(i)})=-1$ on the second level ${{W}_{2}}$, where $0 \leq m \leq k.$
For clarity, denote the configuration of the set $\Phi^{B_1(x^{(0)})}$ by $$S_{m}^{(k-m)}(\sigma ({{x}^{(0)}}))=\left( \begin{matrix}
\overbrace{++\cdots +}^{k-m}\overbrace{--\cdots -}^{m} \\
\sigma (x^{(0)}) \\
\end{matrix} \right).$$ From the consistency condition , we can use the following equation:
[@Akin2016]\[theorem1\] The measures $\mu_{\h}^{(n)}(\s)$, $n=1,2,...,$ in satisfy the compatibility condition if and only if for any $n\in \mathbf{N}$ the following equations hold: $$\begin{aligned}
\exp
(h_{B_{1}(x^{(0)});S_{0}^{k}(+)}+h_{B_{1}(x^{(0)});S_{0}^{k}(-)})&=&\frac{\left(
\sum\limits_{i=0}^{k}\left(
\begin{array}{c}
k \\
i
\end{array}
\right)(ab)^{k-2i}(-1)^{i}{{h}_{{{B}_{1}}(y_i);S_{i}^{k-i}(+)}}\right)^{k}}{{{\left(
\sum\limits_{i=0}^{k}{\left(
\begin{array}{c}
k \\
i
\end{array}
\right){{\left( \frac{a}{b}
\right)}^{k-2i}}}(-1)^{i}{{h}_{{{B}_{1}}(y_i);S_{i}^{k-i}(+)}}
\right)}^{k}}}\\
\exp
(h_{B_{1}(x^{(0)});S_{0}^{k}(+)}+h_{B_{1}(x^{(0)});S_{k}^{0}(-)})&=&\frac{\left(
\sum\limits_{i=0}^{k}\left(
\begin{array}{c}
k \\
i
\end{array}
\right)(ab)^{k-2i}(-1)^{i}{{h}_{{{B}_{1}}(y_i);S_{i}^{k-i}(+)}}\right)^{k}}{\left(
\sum\limits_{i=0}^{k}\left(
\begin{array}{c}
k \\
i
\end{array}
\right)(ab)^{2i-k}{{(-1)}^{i+1}}{{h}_{{{B}_{1}}(y_i);S_{i}^{k-i}(-)}}\right)^{k}}\\
\exp
(h_{B_{1}(x^{(0)});S_{k}^{0}(+)}+h_{B_{1}(x^{(0)});S_{k}^{0}(-)})&=&\frac{
\left( \sum\limits_{i=0}^{k}\left(\begin{array}{c}
k \\
i \\
\end{array}\right)\left(\frac{b}{a}\right)^{k-2i}{{(-1)}^{i+1}}{{h}_{{{B}_{1}}(y_i);S_{i}^{k-i}(-)}}
\right)^{k}}{\left( \sum\limits_{i=0}^{k}\left(
\begin{array}{c}
k \\
i
\end{array}
\right)(ab)^{2i-k}{{(-1)}^{i+1}}{{h}_{{{B}_{1}}(y_i);S_{i}^{k-i}(-)}}\right)^{k}}\end{aligned}$$ where $a=e^{\beta J}$ and $b=e^{\beta J_p}$.
The Theorem \[theorem1\] partially confirms the conjecture formulated in [@Akin2016]. Also, the proof of the Theorem \[theorem1\] can be done as similar to [@Akin2016].
By means of the last equalities, from and we can get that $$\begin{aligned}
\label{mu2}
&&\exp (\sigma(x^{(0)})\prod_{y\in S(x^{(0)})} \sigma
(y)h_{B_1(x^{(0)});S_m^{k-m}(\sigma (x^{(0)}))})\\\nonumber
&=&L_2\sum _{\eta \in \Phi ^{W_2}}( \exp (\beta J\sum _{y_i\in
S(x^{(0)})} \sigma (y_i)\sum _{z_j^{(i)}\in S(y_i)} \eta
(z_j^{(i)}))\\\nonumber &&\times \exp(\beta J_p\sigma
(x^{(0)})\sum _{z_j^{(i)}\in S^2(x^{(0)})} \eta (z_j^{(i)})+\sum
_{y_i\in S(x^{(0)})} \sigma (y_i)\prod _{z_j^{(i)}\in S(y_i)} \eta
(z_j^{(i)})h_{B_1(y_i);S_m^{k-m}(\sigma (y_i))}))\nonumber\end{aligned}$$ where ${{L}_{2}}=\frac{{{Z}_{1}}}{{{Z}_{2}}}. $
Consider the configuration $S_{0}^{k}(\sigma
({{x}^{(0)}})=+)=\left(
\begin{matrix}
+,\cdots ,+,+ \\
+ \\
\end{matrix} \right)
$. For the sake of simplicity, assume such that $\exp
[(-1)^{m}{{h}_{{{B}_{1}}(y_i);S_{m}^{k-m}(\sigma
(y)=+1)}}]=u_{1+m}^{{{(-1)}^{m}}}$, we have $$\begin{aligned}
\label{Eq-eden1a}
u_{1}^{'}=\exp (h_{B_{1}(x^{(0)});S_{0}^{k}(+)})=L_{2}\left(
\sum\limits_{i=0}^{k}\left(
\begin{array}{c}
k \\
i
\end{array}
\right)(ab)^{-2i+k}u_{1+i}^{{{(-1)}^{i}}} \right)^{k}.\end{aligned}$$ Now let us consider the configuration $S_{k}^{0}(\sigma
({{x}^{(0)}})=+)=\left(
\begin{matrix}
-,\cdots ,-,- \\
+ \\
\end{matrix} \right)$ and let $$\exp [{{(-1)}^{m+1}}{{h}_{{{B}_{1}}(y_i);S_{m}^{k-m}(\sigma
(y_i)=-1)}}]=u_{k+2+m}^{{{(-1)}^{m+1}}},$$ then we have $$\begin{aligned}
\label{Eq-eden1b}
u_{k+1}^{'}=\exp (h_{B_{1}(x^{(0)});S_{k}^{0}(+)})=L_{2}
\left( \sum\limits_{i=0}^{k}\left(\begin{array}{c}
k \\
i \\
\end{array}\right)\left(\frac{b}{a}\right)^{-2i+k}u_{2+i+k}^{{(-1)}^{1+i}}
\right)^{k}.\end{aligned}$$ Similarly, for the configuration $S_{0}^{k}(\sigma
({{x}^{(0)}})=-)=\left( \begin{matrix}
+,\cdots ,+,+ \\
- \\
\end{matrix} \right)$, one can obtain $$\begin{aligned}
\label{Eq-eden1c}
{{\left( u_{k+2}^{'} \right)}^{-1}}=\exp
(-h_{B_{1}(x^{(0)});S_{0}^{k}(-)})={{L}_{2}}{{\left(
\sum\limits_{i=0}^{k}{\left(
\begin{array}{c}
k \\
i
\end{array}
\right){{\left( \frac{a}{b}
\right)}^{-2i+k}}}u_{1+i}^{{{(-1)}^{i}}} \right)}^{k}}.\end{aligned}$$ Lastly, for the configuration $S_{k}^{0}(\sigma
({{x}^{(0)}})=-)=\left(
\begin{matrix}
-,\cdots ,-,- \\
- \\
\end{matrix} \right)$ we have $$\begin{aligned}
\label{Eq-eden1d}
{{\left( u_{2(k+1)}^{'} \right)}^{-1}}=\exp
(-h_{B_{1}(x^{(0)});S_{k}^{0}(-)})=L_{2}\left(
\sum\limits_{i=0}^{k}\left(
\begin{array}{c}
k \\
i
\end{array}
\right)(ab)^{2i-k}u_{2+i+k}^{(-1)^{1+i}} \right)^{k}.\end{aligned}$$ From - we immediately get that $$\begin{aligned}
\label{Eq-eden1e}
e^{(-1)^{m}h_{B_{1}(x^{(0)});S_{m}^{k-m}(+)}}&=&\left(e^{h_{B_{1}(x^{(0)});S_{0}^{k}(+)}}
\right)^{\frac{k-m}{k}}\left(e^{h_{B_{1}(x^{(0)});S_{k}^{0}(+)}}
\right)^{\frac{m}{k}}\\
e^{(-1)^{m+1}h_{B_{1}(x^{(0)});S_{m}^{k-m}(-)}}&=&\left(e^{
-h_{B_{1}(x^{(0)});S_{0}^{k}(-)}} \right)^{\frac{k-m}{k}}\left(
e^{-h_{B_{1}(x^{(0)});S_{k}^{0}(-)}}\right)^{\frac{m}{k}}.\label{Eq-eden1f}\end{aligned}$$ Through the introduction of the new variables $v_{i}=(u_{i})^{\frac{1}{k}}$ in the equations -, we derive the following recurrence system: $$\begin{aligned}
\label{recurrence1a}
v_{1}^{'}&=&\sqrt[k]{L_2}{{\left(\frac{{{(ab)}^{2}}{{v}_{1}}+{{v}_{k+1}}}{ab}\right)}^{k}},\\\label{recurrence1b}
v_{k+1}^{'}&=&\sqrt[k]{L_2}{{\left(\frac{{{a}^{2}}{{v}_{k+2}}+{{b}^{2}}{{v}_{2(k+1)}}}{ab{{v}_{k+2}}{{v}_{2(k+1)}}}\right)}^{k}},\\\label{recurrence1c}
(v_{k+2}^{'})^{-1}&=&\sqrt[k]{L_2}{{\left(\frac{{{a}^{2}}{{v}_{1}}+{{b}^{2}}{{v}_{k+1}}}{ab}\right)}^{k}},\\\label{recurrence1d}
(v_{2(k+1)}^{'})^{-1}&=&\sqrt[k]{L_2}{{\left(\frac{{{(ab)}^{2}}{{v}_{k+2}}+{{v}_{2(k+1)}}}{(ab){{v}_{k+2}}{{v}_{2(k+1)}}}\right)}^{k}}.\end{aligned}$$
Translation-invariant Gibbs measures: Even case {#TIGM}
-----------------------------------------------
In this subsection, we are going to focus on the existence of translation-invariant Gibbs measures (TIGMs) by analyzing the equations -. Note that vector-valued function $$\label{consistency1}
\textbf{h}(x) = \{h_{B_1(x);S_{m}^{k-m}(\sigma (x))}:m\in
\{0,1,2,...,k\},\sigma (x)\in \Phi \}$$ is considered as translation-invariant if $h_{B_1(x);S_{m}^{k-m}(\sigma (x))}= h_{B_1(y);S_{m}^{k-m}(\sigma
(y))}$ for all $y\in S(x)$ and $\sigma (x)=\sigma (y)$ (see for details [@Akin2016; @Rozikov]). A translation-invariant Gibbs measure is defined as a measure, $\mu_{\textbf{h}}$, corresponding to a translation-invariant function $\textbf{h}$ (see for details [@NHSS; @Rozikov]). Here we will assume that $v'_{i}=v_{i}$ for all $i\in \{1,\ldots,2(k+1)\}$.
\[even-consistency-rem\] By using the equations and , it can be shown that if the vector-valued function $\textbf{h}(x)$ given in has the following form: $$\textbf{h}(x)=(p,-\frac{(k-1)p+q}{k},\cdots,-\frac{p+(k-1)q}{k},
q,r,-\frac{(k-1)r+s}{k},\cdots,-\frac{r+(k-1)s}{k},s),$$ where $p,q,r,s\in \mathbb{R}$, then the *consistency* condition is satisfied.
Now, we want to find Gibbs measures for considered case. To do so, we introduce some notations. Define the transformation $$\begin{aligned}
\label{cavity-even}
\textbf{F}=(F_1,F_{k+1},F_{k+2},F_{2(k+1)}): \mathbf{R}^4_+
\rightarrow \mathbf{R}^4_+\end{aligned}$$ such that $$\begin{aligned}
v'_1&=&F_1(v_1,v_{k+1},v_{k+2},v_{2(k+1)}),\\
v'_{k+1}&=&F_{k+1}(v_1,v_{k+1},v_{k+2},v_{2(k+1)}),\\
v'_{k+2}&=&F_{k+2}(v_1,v_{k+1},v_{k+2},v_{2(k+1)}),\\
v'_{2(k+1)}&=&F_{2(k+1)}(v_1,v_{k+1},v_{k+2},v_{2(k+1)}).\end{aligned}$$ The fixed points of the cavity equation $\textbf{v}=\textbf{F}(\textbf{v})$ given in the equation describe the translation-invariant Gibbs measures associated to the model corresponding to the Hamiltonian , where $\textbf{v}=(v_{1}, v_{k+1},
v_{k+2},v_{2(k+1)})$ and $k$ is positive even integer. Description of the solutions of the system of equations - is rather tricky. Assume that $v_{1}^{'}=v_{k+2}^{'}$ and $v_{k+1}^{'}=v_{2(k+1)}^{'}$ that is $$\begin{aligned}
\exp
({{(-1)}^{0}}{{h}_{{{B}_{1}}({{x}^{(0)}});S_{0}^{k}(+)}})&=&\exp
(-{{h}_{{{B}_{1}}({{x}^{(0)}});S_{0}^{k}(-)}})\\
\exp
({{(-1)}^{k}}{{h}_{{{B}_{1}}({{x}^{(0)}});S_{k}^{0}(+)}})&=&\exp
(-{{h}_{{{B}_{1}}({{x}^{(0)}});S_{k}^{0}(-)}}).\end{aligned}$$
Below we will consider the following case when the system of equations - is solvable for set
$$\begin{aligned}
\label{invariantA}
A=\left\{(v_{1}^{'}, v_{k+1}^{'}, v_{k+2}^{'},v_{2(k+1)}^{'})\in
\mathbb{R}_+^4:
v_{1}^{'}=v_{k+2}^{'}=\frac{1}{v_{k+1}^{'}}=\frac{1}{v_{2(k+1)}^{'}}
\right\}.\end{aligned}$$
Divide the equation by the equation , then we have $$\begin{aligned}
\label{recurrence2a}
(v_{1}^{'})={{\left(
\frac{{{(ab)}^{2}}{{v}_{1}}+{{v}_{k+1}}}{{{a}^{2}}{{v}_{1}}+{{b}^{2}}{{v}_{k+1}}}
\right)}^{\frac{k}{2}}}.\end{aligned}$$ Similarly, divide the equation by the equation , then we get $$\begin{aligned}
\label{recurrence2b}
v_{k+1}^{'}={{\left(
\frac{{{a}^{2}}{{v}_{k+2}}+{{b}^{2}}{{v}_{2(k+1)}}}{{{(ab)}^{2}}{{v}_{k+2}}+{{v}_{2(k+1)}}}
\right)}^{\frac{k}{2}}}={{\left(
\frac{{{a}^{2}}{{v}_{1}}+{{b}^{2}}{{v}_{(k+1)}}}{{{(ab)}^{2}}{{v}_{1}}+{{v}_{(k+1)}}}
\right)}^{\frac{k}{2}}}.\end{aligned}$$ For brevity, denote $a^2=c$ and $b^2=d$. From and , if we assume as $x'=v_{1}^{'}=\frac{1}{v_{k+1}^{'}}$ ($x>0$), then we obtain the following dynamical system $f: \mathbb{R}^{+}\rightarrow
\mathbb{R}^{+}$ $$\begin{aligned}
\label{dynamiceven-1}
x'=\left(\frac{1+c d x^{2}}{d+c x^{2}}\right)^{\frac{k}{2}}=:f(x).\end{aligned}$$
\[Remark 4.2\] For $(v_1,v_{k+1},v_{k+2},v_{2(k+1)})\in A$, that is $$h_{B_1(x);S_{0}^{k}(+)}=h_{B_1(x);S_{0}^{k}(-)}=-h_{B_1(x);S_{k}^{0}(+)}=-h_{B_1(x);S_{k}^{0}(-)}$$ the equation is valid. Also, one can verify that $\textbf{F}(A)\subset A.$ In other words, the set $A$ in is an invariant set under the mapping $\textbf{F}$ in .
Let us investigate the fixed points of the function given in , i.e., $x = f(x)$. In fact, we should show that the system has at least one solution with respect to $x'$ in the domain $\mathbb{R}^{+}$. It is obvious that $f$ is bounded and thus the curve $y = f(x)$ must intersect the line $y =m x.$ Therefore, this construction provides one element of a new set of Gibbs measures with memory of length 2 associated to the model for any $x\in
\mathbb{R}^{+}$ (see [@Preston Proposition 10.7]).
\[proposition-even\] The equation (with $x \geq 0, c> 0, d> 0$) has one solution if $d<1$. If $d
> \sqrt{\frac{k+1}{k-1}}$ then there exists $\eta_1(c,d)$, $\eta_2(c,d)$ with $0<\eta_1(c,d)<\eta_2(c,d)$ such that equation has 3 solutions if $\eta_1(c,d)<m<\eta_2(c,d)$ and has 2 solutions if either $\eta_1(c,d)=m$ or $\eta_2(c,d)=m$. In fact $$\begin{aligned}
\eta_i (c,d)=\frac{1}{x_i}\left(\frac{1+c d
x^2_i}{d+cx^2_i}\right)^{k/2}.\end{aligned}$$ where $x_1,x_2$ are the solutions of equation $c^2 d
x^4-c\left(d^2k-1-d^2-k \right) x^2+d=0.$
Let us consider the equation . Taking the first and the second derivatives of the function $f$, then we have $$f'(x)=\frac{c (d^2-1)k x \left(1+c d
x^2\right)^{-1+k/2}}{\left(d+c x^2\right)^{1+k/2} }$$ and $$f''(x)=-\frac{c(d^2-1)k\left(3 c^2 d x^4+c\left(1+d^2+ k- d^2
k\right) x^2-d\right)}{\left(d+c x^2\right)^{2+\frac{k}{2}}
\left(1+c d x^2\right)^{-\frac{k}{2}+2}}.$$ If $d< 1$, i.e. $J_p<0$, then $f$ is decreasing and the equation has only a unique solution; thus we can restrict ourselves to the case $d>1$.
Let us consider equation $$\label{Eq12}
3 c^2 d x^4+c\left(1+d^2+ k- d^2 k\right) x^2-d=0.$$
It is clear that $3 c^2 d x^4+c\left(1+d^2+ k- d^2 k\right) x^2-d$ is even function. Solving such an equation w.r.t. $x$, we can find a positive root $$x^*=\frac{\sqrt{-1-d^2-k+d^2 k+\sqrt{12 d^2+\left(1+d^2+k-d^2
k\right)^2}}}{\sqrt{6cd}}.$$ $x^*$ is a unique positive root of the quartic equation . Therefore, the function $f$ is convex up, if $$x< \frac{\sqrt{-1-d^2-k+d^2 k+\sqrt{12 d^2+\left(1+d^2+k-d^2
k\right)^2}}}{\sqrt{6cd}}.$$ The function $f$ is convex down, for $$x>\frac{\sqrt{-1-d^2-k+d^2 k+\sqrt{12 d^2+\left(1+d^2+k-d^2
k\right)^2}}}{\sqrt{6cd}}.$$ It is quite easy to see that three is more than one solution if and only if there is more than one solution to $xf'(x)=f(x)$, which is the same as $$c^2 d x^4-c\left(-1-d^2-k+d^2 k\right) x^2+d=0$$ with the help of a little elementary analysis the proof is readily completed.
It is clear that the function has a unique inflection point $x^{*}$ in the region $(0,\infty)$, therefore the function has at most three fixed points in the region $(0,\infty)$. We can conclude that the increase of $k$ affects the number of fixed points by no more than 3. So, we can obtain at most 3 TIGMs associated to the model for $(v_1,v_{k+1},v_{k+2},v_{2(k+1)})\in A$ in .
Numerical Example: Even Case
----------------------------
Previously documented analysis can analytically solve these equations for some given values $J,J_p, T$ and $k$, which we will not show all of solutions here due to the complicated nature of formulas and coefficients [@Wolfram]. In order to describe the number of the fixed points of the function , we have manipulated the function and the linear function $y=x$ via Mathematica [@Wolfram]. We have obtained at most 3 positive real roots for some parameters $J$ and $J_p$ (coupling constants), temperature $T$ and even positive integer $k$.
Let us give an illustrative example. Figs. \[fig1Cregion\] (a)-(b) show that there are 3 positive fixed points of the function , if we take $J=-5.8,
J_p=3.25,T=14.358$ and $k=12,10$. Therefore, the phase transition for the model occur.
\[!htbp\]\[fig1Cregion\] ![Graphs of the function $f$ defined in for $J=-5.8, J_p=3.25,T=14.358$ and even integers $k=12,10,8,6$.[]{data-label="fig1Cregion"}](k=12.eps "fig:"){width="65mm"} \[fig1c\] ![Graphs of the function $f$ defined in for $J=-5.8, J_p=3.25,T=14.358$ and even integers $k=12,10,8,6$.[]{data-label="fig1Cregion"}](k=10.eps "fig:"){width="65mm"} ![Graphs of the function $f$ defined in for $J=-5.8, J_p=3.25,T=14.358$ and even integers $k=12,10,8,6$.[]{data-label="fig1Cregion"}](k=8.eps "fig:"){width="65mm"} ![Graphs of the function $f$ defined in for $J=-5.8, J_p=3.25,T=14.358$ and even integers $k=12,10,8,6$.[]{data-label="fig1Cregion"}](k=6.eps "fig:"){width="65mm"}
In Figure \[fig1Cregion\] (c), there exists two positive fixed point of the function for $J=-5.8,
J_p=3.25,T=14.358$ and $k=8$.
In Figure \[fig1Cregion\] (d), we are also able to find that, for the parameters $J=-5.8, J_p=3.25,T=14.358$ and $k=6$, the function has a unique positive fixed point. Therefore, the phase transition does not occur for $J=-5.8,
J_p=3.25,T=14.358$ and $k=6$.
We note that for $J=-5.8, J_p=3.25,T=14.358$ and $k=10$, the function have three positive fixed points as $x_1^{*}=0.106457, x_2^{*}=2.13383, x_3^{*}=8.30085.$ Figure \[fig1Cregion\] (b) shows that for all $x\in (x_2^{*},x_3^{*})$, $\lim\limits_{n\rightarrow \infty}f^n(x)=x_3^{*}.$ Similarly, for all $x\in (x_1^{*},x_2^{*})$, $\lim\limits_{n\rightarrow
\infty}f^n(x)=x_1^{*}.$ Therefore, the fixed points $x_1^{*}$ and $x_3^{*}$ are stable and $x_2^{*}$ is unstable.
Therefore, there is a critical temperature $T_c > 0$ such that for $T < T_c$ this system of equations has 3 positive solutions: $h_1^{*}; h_2^{*};h_3^{*}.$ We denote the Gibbs measure that corresponds to the root $h_1^{*}$ (and respectively $h_2^{*};h_3^{*}$) by $\mu^{(1)}$ (and respectively $\mu^{(2)}$,$\mu^{(3)}$).
\[exrema1\] We can conclude that the Gibbs measures $\mu_1^{*}$ and $\mu_3^{*}$ corresponding to the stable fixed points $x_1^{*}$ and $x_3^{*}$ are extreme Gibbs distributions (for details [@Iof; @NHSS]).
For $J=-5.8, J_p=3.25,T=14.358$ and $k>6$ ($k$ is even integer), the model has phase transition. For $J=-5.8,
J_p=3.25,T=14.358$ and $k=6$, the phase transition of the model does not occur.
The Recurrence equations for $k$-odd. {#odd}
=====================================
Let us derive the recurrence equations to describe the existence of the translation-invariant Gibbs measures (TIGMs) associated to the model on the Cayley tree of order $k$-odd.
From the equations , and , one get the following equations: $$\begin{aligned}
\label{odd1a}
u_{1}^{'}&=&\exp (h_{B_{1}(x^{(0)});S_{0}^{k}(+)})=L_{2}\left(\sum
_{i=0}^k (a b)^{-2 i+k} \left(
\begin{array}{c}
k \\
i
\end{array}
\right)u_{1+i}^{(-1)^i}\right)^{k}\\
(u_{k+1}^{'})^{-1}&=&\exp
(-h_{B_{1}(x^{(0)});S_{k}^{0}(+)})={{L}_{2}}{ {\left(
\sum\limits_{i=0}^{k}{\left(
\begin{array}{c}
k \\
i
\end{array}
\right){{\left( \frac{b}{a}
\right)}^{-2i+k}}}u_{2+i+k}^{{{(-1)}^{1+i}}}
\right)}^{k}}.\label{odd1b}\end{aligned}$$ For the configuration $S_{0}^{k}(\sigma ({{x}^{(0)}})=-)=\left(
\begin{matrix}
\overbrace{+,\cdots ,+,+}^{k-odd} \\
- \\
\end{matrix} \right)$, similarly to and we obtain $$\begin{aligned}
\label{odd1c}
{{\left( u_{k+2}^{'} \right)}^{-1}}=\exp
(-{{h}_{{{B}_{1}}({{x}^{(0)}});S_{0}^{k}(-)}})={{L}_{2}}{{\left(
\sum\limits_{i=0}^{k}{\left(
\begin{array}{c}
k \\
i
\end{array}
\right){{\left( \frac{a}{b}
\right)}^{-2i+k}}}u_{1+i}^{{{(-1)}^{i}}} \right)}^{k}}.\end{aligned}$$ Lastly, for the configuration $S_{k}^{0}(\sigma
({{x}^{(0)}})=-)=\left(
\begin{matrix}
\overbrace{-,\cdots ,-,- }^{k-odd} \\
- \\
\end{matrix} \right)$ we have $$\begin{aligned}
\label{odd1d}
\left( u_{2(k+1)}^{'} \right)=\exp
({{h}_{{{B}_{1}}({{x}^{(0)}});S_{k}^{0}(-)}})={{L}_{2}}{{\left(
\sum\limits_{i=0}^{k}{\left(
\begin{array}{c}
k \\
i
\end{array}
\right){{(a b)}^{2i-k}}}u_{2+i+k}^{{{(-1)}^{1+i}}} \right)}^{k}}.\end{aligned}$$ From the equations -, it is obvious that $$\begin{aligned}
\label{even-eq4}
e^{(-1)^{m}h_{B_{1}(x^{(0)});S_{m}^{k-m}(+)}}&=&\left(e^{h_{B_{1}(x^{(0)});S_{0}^{k}(+)}}
\right)^{\frac{k-m}{k}}\left(e^{-h_{B_{1}(x^{(0)});S_{k}^{0}(+)}}
\right)^{\frac{m}{k}}\\\label{even-eq5}
e^{(-1)^{m+1}h_{B_{1}(x^{(0)});S_{m}^{k-m}(-)}}&=&\left(e^{
-h_{B_{1}(x^{(0)});S_{0}^{k}(-)}} \right)^{\frac{k-m}{k}}\left(
e^{h_{B_{1}(x^{(0)});S_{k}^{0}(-)}}\right)^{\frac{m}{k}}.\end{aligned}$$ By substituting variables $u_i=v_i^{k}$ for $i=1,2,\cdots,2(k+1)$ in the recurrent equations -, after small calculations, we can express a new recurrence system in a simpler form: $$\begin{aligned}
\label{odd4}
(v'_{1})&=&\sqrt[k]{L_2}\left(\frac{1+(a
b)^{2}v_{1}v_{k+1}}{abv_{k+1}}\right)^k,\\\label{odd4a}
(v'_{k+1})^{-1}&=&\sqrt[k]{L_2}\left(\frac{b^{2}+a^{2}v_{k+2}v_{2(k+1)}}{abv_{k+2}}\right)^k,\\\label{odd4b}
(v'_{k+2})^{-1}&=&\sqrt[k]{L_2}\left(\frac{b^{2}+a^{2}v_{1}v_{k+1}}{abv_{k+1}}\right)^k,\\\label{odd4c}
(v'_{2(k+1)})&=&\sqrt[k]{L_2}\left(\frac{1+(a
b)^{2}v_{k+2}v_{2(k+1)}}{abv_{k+2}}\right)^k.\label{odd4d}\end{aligned}$$
The translation-invariant Gibbs measures: Odd case {#TIGM-odd}
--------------------------------------------------
In this subsection, we will identify the solutions of the system of nonlinear equations - to describe the translation invariant Gibbs measures associated to the model on the arbitrary odd-order Cayley tree.
\[odd-consistency-rem\] By using the equations and , it can be shown that if the vector-valued function $\textbf{h}(x)$ given in has the following form: $$\textbf{h}(x)=(p,\cdots,\frac{(-1)^m((k-m)p-mq)}{k},\cdots,
q,r,\cdots,\frac{(-1)^{m+1}(ms-(k-m)r)}{k},\cdots,s),$$ where $p,q,r,s\in \mathbb{R}$, then the *consistency* condition is satisfied.
Now, we want to find the TIGMs for considered case. To do so, we introduce some notations. Define transformation $$\begin{aligned}
\label{cavity}
\textbf{F}=(F_1, F_{k+1}, F_{k+2}, F_{2(k+1)}): \mathbf{R}^4_+
\rightarrow \mathbf{R}^4_+\end{aligned}$$ such that $$\begin{aligned}
v'_1&=&F_1(v_1,v_{k+1},v_{k+2},v_{2(k+1)}),\\
v'_{k+1}&=&F_{k+1}(v_1,v_{k+1},v_{k+2},v_{2(k+1)}),\\
v'_{k+2}&=&F_{k+2}(v_1,v_{k+1},v_{k+2},v_{2(k+1)}),\\
v'_{2(k+1)}&=&F_{2(k+1)}(v_1,v_{k+1},v_{k+2},v_{2(k+1)}).\end{aligned}$$ The fixed points of the cavity equation $\textbf{v}=\textbf{F}(\textbf{v})$ given in the Eq. describe the translation invariant Gibbs measures associated to the model , where $\textbf{v}=(v_1,v_{k+1},v_{k+2},v_{2(k+1)})$ and $k$ is any positive odd integer greater than 1.
Divide by , then we have $$\begin{aligned}
\label{odd4e}
v_{k+1}^{2k+1}v_{k+2}^{-k}=\left(\frac{1+(ab)^2v_{k+1}^{k+1}}{b^2+a^2v_{k+2}^{k+1}}\right)^k.\end{aligned}$$ Similarly, divide by , then one gets $$\begin{aligned}
\label{odd4f}
v_{k+2}^{2k+1}v_{k+1}^{-k}=\left(\frac{1+(ab)^2v_{k+2}^{k+1}}{b^2+a^2v_{k+1}^{k+1}}\right)^k.\end{aligned}$$ Multiply the equations and , we obtain $$\begin{aligned}
\label{recurrence5}
v_{k+1}^{k+1}v_{k+2}^{k+1}=\left(\frac{1+(ab)^2v_{k+1}^{k+1}}
{b^2+a^2v_{k+2}^{k+1}}\right)^k\left(\frac{1+(ab)^2v_{k+2}^{k+1}}{b^2+a^2v_{k+1}^{k+1}}\right)^k.\end{aligned}$$ Let us consider set $$\begin{aligned}
\label{invariantB}
B=\left\{(v_1,v_{k+1},v_{k+2},v_{2(k+1)})\in
\mathbb{R}_+^4:v_{1}=v_{k+1}^{k}=v_{2({k+1})}=v_{{k+2}}^{k}\right\}.\end{aligned}$$ Assume that $v_{k+1}^{k+1}=v_{k+2}^{k+1}=x$, and $a^2=c$, $b^2=d$ then we get $$\begin{aligned}
\label{recurrence5a}
x=\left(\frac{1+c d x}{d+c x}\right)^k=:g(x).\end{aligned}$$
If $(v_1,v_{k+1},v_{k+2},v_{2(k+1)})\in B$, that is $$h_{B_1(x);S_{0}^{k}(+)}=kh_{B_1(x);S_{k}^{0}(+)}=kh_{B_1(x);S_{0}^{k}(-)}=h_{B_1(x);S_{k}^{0}(-)}$$ then the equation is valid. Also, one can verify that $\textbf{F}(B)\subset B.$ That is, the set $B$ is an invariant set under the mapping $\textbf{F}$.
Now we examine how many solutions the equation $g(x) = x$ has. Thus, similarly to the Proposition \[proposition-even\], we have the following Proposition. Here, by using the procedure given in [@Preston Proposition 10.7] we will describe the number of fixed points of the function $g$ in .
\[proposition-odd\] The equation (with $x \geq 0, c > 0, d > 0$) has one solution if $d<1$. If $d
> \frac{k+1}{k-1}$ then there exists $\eta_1(c,d)$, $\eta_2(c,d)$ with $0<\eta_1(c,d)<\eta_2(c,d)$ such that equation has 3 solutions if $\eta_1(c,d)<m<\eta_2(c,d)$ and has 2 solutions if either $\eta_1(c,d)=m$ or $\eta_2(c,d)=m$. In fact $$\begin{aligned}
\eta_i (c,d)=\frac{1}{x_i}\left(\frac{1+c d
x_i}{d+cx_i}\right)^{k}.\end{aligned}$$ where $x_1,x_2$ are the solutions of quadratic equation $c^2 d
x^2-c(d^2(k-1)-(1+ k))x+d=0.$
The proof of Proposition \[proposition-odd\] can be done easily by following the procedure in Proposition \[proposition-even\].
\[!htbp\]\[fig2Cregion\] ![Graphs of the function $g$ given in for $J = -7.3, Jp = 5.1, T = 28$ and odd integers $k=11,9,7$, respectively.[]{data-label="fig2Cregion"}](k=11.eps "fig:"){width="65mm"} \[fig1c\] ![Graphs of the function $g$ given in for $J = -7.3, Jp = 5.1, T = 28$ and odd integers $k=11,9,7$, respectively.[]{data-label="fig2Cregion"}](k=9.eps "fig:"){width="65mm"} ![Graphs of the function $g$ given in for $J = -7.3, Jp = 5.1, T = 28$ and odd integers $k=11,9,7$, respectively.[]{data-label="fig2Cregion"}](k=7.eps "fig:"){width="65mm"}
Illustrative Example: Odd Case
------------------------------
We have manipulated the equation via Mathematica [@Wolfram]. We have obtained at most 3 positive real roots for some parameters $J$ and $J_p$ and temperature $T$. As an illustrative example, the Figures \[fig2Cregion\] (a)-(b) show that there are 3 positive fixed points of the function for $J = -7.3,
J_p = 5.1, T = 28$ and $k=11,9$ values. Therefore, we have demonstrated the occurrence of phase transitions. The Figures \[fig2Cregion\] (a)-(b) shows that there are three positive fixed points of the function $g$ for $J = -7.3, J_p = 5.1, T = 28$ and $k=11,9$.
In the Figure \[fig2Cregion\] (c), there exists a unique positive fixed point of the function for $J =
-7.3, J_p = 5.1, T = 28$ and $k=7$. Therefore, the phase transition does not occur for $J = -7.3, J_p = 5.1, T = 28$ and $k=7$.
We can explicitly compute the fixed points of the function for given some parameters $J, J_p, T$ and $k$. For example, for $J = -7.3, J_p = 5.1, T = 28$ and $k=9$, the function have three positive fixed points as $x_1^{*}=0.0448184, x_2^{*}=4.93008, x_3^{*}=10.8931.$ The Figure \[fig2Cregion\] (b) shows that for all $x\in (x_2^{*},x_3^{*})$, $\lim\limits_{n\rightarrow \infty}g^n(x)=x_3^{*}.$ Similarly, for all $x\in (x_1^{*},x_2^{*})$, $\lim\limits_{n\rightarrow
\infty}g^n(x)=x_1^{*}.$ Therefore, the fixed points $x_1^{*}$ and $x_3^{*}$ are stable and $x_2^{*}$ is unstable.
As concluded in Remark \[exrema1\], we can see that the Gibbs measures $\mu_1^{*}$ and $\mu_3^{*}$ corresponding to the stable fixed points $x_1^{*}$ and $x_3^{*}$ are extreme Gibbs distributions (for details [@Iof; @NHSS]).
There is a critical temperature $T_c > 0$ such that for $T <
T_c$ the system of nonlinear equations - has 3 positive solutions: $h_1^{*}; h_2^{*};h_3^{*}.$ We denote the Gibbs measure that corresponds to the root $h_1^{*}$ (and respectively $h_2^{*};h_3^{*}$) by $\mu^{(1)}$ (and respectively $\mu^{(2)}$,$\mu^{(3)}$).
Conclusions {#Conlussion}
===========
In the present paper, we have proposed a rigorous measure-theoretical approach to investigate the Gibbs measures with memory of length 2 associated with the Ising-Vanniminus model on the arbitrary order Cayley tree. We have generalized the results conjectured in [@Akin2016; @Akin2017] for an arbitrary order Cayley tree. We have used the Markov random field method to describe the Gibbs measures. We constructed the recurrence equations corresponding generalized ANNNI model. We have satisfied the Kolmogorov *consistency* condition. We have explained whether the number of branches of tree does not change the number of Gibbs measures. We have concluded that the order $k$ of the tree significantly affects the occurrence of phase transition. Also, we have seen that the role of $k$ is rather significant on the number of Gibbs measures. Exact description of the solutions of the system of recurrence equations - (and -) is rather tricky. Therefore, we were able to resolve only case (and ) for even $k$ (and odd $k$, respectively), the other cases remain open problem. Also, depending on the even and odd of $k$, the recurrence equations obtained for even branch totaly differ from odd branch.
Note that for many problems the solution on a tree is much simpler than on a regular lattice such as $d$-dimensional integer lattice and is equivalent to the standard Bethe-Peierls theory [@Katsura]. Although the Cayley tree is not a realistic lattice; however, its amazing topology makes the exact calculations of various quantities possible. Therefore, the results obtained in our paper can inspire to study the Ising and Potts models over multi-dimensional lattices or the grid ${{\mathbb{Z}}^{d}}$. After a glimpse of some applications, we believe now that new theoretical developments can be inspired by concrete problems. By considering the method used in this paper, the investigation of Gibbs measures with memory of length $n>2$ on arbitrary order Cayley tree and Cayley tree-like lattices [@UA1; @UA2; @Moraal] is planned to be the subject of forthcoming publications.
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|
---
abstract: 'Let $\mathbb{R}=(-\infty,\infty)$, and let $Q\in C^1(\mathbb{R}): \mathbb{R}\rightarrow \mathbb{R^+}=[0,\infty)$ be an even function, which is an exponent. We consider the weight $w_\rho(x)=|x|^{\rho} e^{-Q(x)}$, $\rho\geqslant 0$, $x\in \mathbb{R}$, and then we can construct the orthonormal polynomials $p_{n}(w_\rho ^2;x)$ of degree n for $w_\rho ^2(x)$. In this paper we obtain $L_p$-convergence theorems of even order Hermite-Fejér interpolation polynomials at the zeros $\left\{x_{k,n,\rho}\right\}_{k=1}^n$ of $p_{n}(w_\rho ^2;x)$.'
address:
- '$^{1}$Department of Mathematics Education, Sungkyunkwan University, Seoul 110-745, Republic of Korea.'
- '$^{2}$Department of Mathematics, Meijo University, Nagoya 468-8502, Japan.'
author:
- Hee Sun Jung$^1$ and Ryozi Sakai$^2$
title: '$L_p$-Convergence of higher order Hermite or Hermite-Fejér interpolation polynomials with exponential-type weights'
---
MSC; 41A05, 41A10\
Keywords; higher order Hermite-Fejér interpolation polynomials
Introduction
============
Let $\mathbb{R}=(-\infty,\infty)$, and let $Q\in C^1(\mathbb{R}): \mathbb{R}\rightarrow \mathbb{R^+}=[0,\infty)$ be an even function. Consider the weight $w(x)=\exp(-Q(x))$ and define for $\rho>-\frac{1}{2}$, $$\label{eq1.1}
w_\rho(x):=|x|^{\rho} w(x), \quad x\in \mathbb{R}.$$ Suppose that $\int_0^\infty x^nw_\rho^2(x)dx<\infty$ for all $n=0,1,2,\ldots$. Then, we can construct the orthonormal polynomials $p_{n,\rho}(x)=p_{n}(w_\rho ^2;x)$ of degree $n$ for $w_\rho ^2(x)$, that is, $$\int_{-\infty}^\infty p_{n,\rho}(x)p_{m,\rho}(x)w_\rho ^2(x)dx=\delta_{mn} (\textrm{Kronecker's delta}).$$ Denote $$p_{n,\rho}(x)=\gamma_{n}x^n+\ldots, \gamma_{n}=\gamma_{n,\rho}>0,$$ and the zeros of $p_{n,\rho}(x)$ by $$-\infty<x_{n,n,\rho}<x_{n-1,n,\rho}<\ldots<x_{2,n,\rho}<x_{1,n,\rho}<\infty.$$ Let $\mathcal{P}_n$ denote the class of polynomials with degree at most $n$. For $f\in C(\mathbb{R})$ we define the higher order Hermite-Fejér interpolation polynomial $L_n(\nu,f;x)$ based at the zeros $\left\{x_{k,n,\rho}\right\}_{k=1}^n$ as follows: $$L_n^{(i)}(\nu,f;x_{k,n,\rho})=\delta_{0,i}f(x_{k,n,\rho})
\quad \textrm{for} \quad k=1,2,\ldots,n \quad \textrm{and} \quad i=0,1,\ldots,\nu-1.$$ $L_n(1,f;x)$ is the Lagrange interpolation polynomial, $L_n(2,f;x)$ is the ordinary Hermite-Fejér interpolation polynomial, and $L_n(4,f;x)$ is the Krilov-Stayermann polynomial. The fundamental polynomials $h_{k,n,\rho}(\nu;x)\in \mathcal{P}_{\nu n-1}$ for the higher order Hermite-Fejér interpolation polynomial $L_n(\nu,f;x)$ are defined as follows: $$\label{eq1.2}
\begin{array}{l}
h_{k,n,\rho}(\nu;x)=l_{k,n,\rho}^{\nu}(x) \sum_{i=0}^{\nu -1}e_i(\nu,k,n)(x-x_{k,n,\rho})^i, \\
l_{k,n,\rho}(x)=\frac{p_n(w_{\rho}^2;x)}{(x-x_{k,n,\rho})p'_n(w_{\rho}^2;x_{k,n,\rho})},\\
h_{k,n,\rho}(\nu;x_{p,n,\rho})=\delta_{k,p}, \quad h_{k,n,\rho}^{(i)}(\nu;x_{p,n,\rho})=0, \\
\quad k,p=1,2,\ldots,n, i=1,2,\ldots,\nu -1.
\end{array}$$ Using them, we can write $L_n(\nu,f;x)$ as follows: $$L_n(\nu,f;x)= \sum_{k=1}^nf(x_{k,n,\rho})h_{k,n,\rho}(\nu;x).$$ Furthermore, we extend the operator $L_n(\nu,f;x)$. Let $l$ be a non-negative integer, and let $l\le \nu -1$. For $f \in C^{l}(\mathbb{R})$ we define the $(l,\nu)$-order Hermite-Fejér interpolation polynomials $L_n(l,\nu,f;x)\in \mathcal{P}_{\nu n-1}$ as follows: For each $k=1,2,\ldots,n,$ $$\begin{array}{c}
L_n(l,\nu,f;x_{k,n,\rho})=f(x_{k,n,\rho}),\\
L_n^{(j)}(l,\nu,f;x_{k,n,\rho})=f^{(j)}(x_{k,n,\rho}), \quad j=1,2,\ldots,l,\\
L_n^{(j)}(l,\nu,f;x_{k,n,\rho})=0, \quad j=l+1,l+2,\ldots,\nu -1.
\end{array}$$ Especially, $L_n(0,\nu,f;x)$ is equal to $L_n(\nu,f;x)$, and for each $P\in \mathcal{P}_{\nu n-1}$ we see $L_n(\nu-1,\nu,P;x)=P(x)$. The fundamental polynomials $h_{s,k,n,\rho}(l,\nu;x)\in \mathcal{P}_{\nu n-1}, k=1,2,\ldots,n$, of $L_n(l,\nu,f;x)$ are defined by $$\label{eq1.3}
\begin{array}{c}
h_{s,k,n,\rho}(l,\nu;x)=l_{k,n,\rho}^{\nu}(x) \sum_{i=s}^{\nu -1}e_{s,i}(\nu,k,n)(x-x_{k,n,\rho})^i,\\
h_{s,k,n,\rho}^{(j)}(l,\nu;x_{p,n,\rho})=\delta_{s,j}\delta_{k,p}, j,s=0,1,\ldots,\nu -1, p=1,2,\ldots,n.
\end{array}$$ Then we have $$L_n(l,\nu,f;x)= \sum_{k=1}^n \sum_{s=0}^lf^{(s)}(x_{k,n,\rho})h_{s,k,n,\rho}(l,\nu;x).$$ For the ordinary Hermite and Hermite-Fejér interpolation polynomial $L_n(1,2,f;x)$, $L_n(2,f;x)$ and the related approximation process, Lubinsky [@[15]] gave some interesting convergence theorems.
In [@[7]] we obtained uniform convergence theorems with respect to the interpolation polynomials $L_n(\nu,f;x)$ and $L_n(l,\nu,f;x)$ with even integer $\nu$. In this paper we will give the $L_p$-convergence theorems of $L_n(\nu,f;x)$ and $L_n(l,\nu,f;x)$ with even order $\nu$. If we consider the higher order Hermite-Fejér interpolation polynomial $L_n(\nu,f;x)$ on a finite interval, then we can see a remarkable difference between the cases of an odd number $\nu$ and of an even number $\nu$, for example, as between the Lagrange interpolation polynomial $L_n(1,f;x)$ and the Hermite-Fejér interpolation polynomial $L_n(2,f;x)$ ([@[17]]-[@[22]]). We can also see a similar phenomenon in the cases of the infinite intervals ([@[7]]-[@[14]]). In this paper, we consider the even case in $L_p$-norm. We will discuss the odd case in $L_p$-norm elsewhere.
Here, we give the class of weights which is treated in this paper. We say that $f: \mathbb{R}\rightarrow \mathbb{R^+}$ is quasi-increasing if there exists $C>0$ such that $f(x)\le Cf(y), 0<x<y$. In the following, we introduce the class of weights defined in [@[16]].
\[Definition1.1\] Let $Q: {\mathbb{R}}\rightarrow {\mathbb{R}}^+$ be a continuous even function satisfying the following properties:
$Q'(x)$ is continuous in ${\mathbb{R}}$ and $Q(0)=0$.
$Q''(x)$ exists and is positive in ${\mathbb{R}}\setminus\{0\}$.
$ \lim_{x\rightarrow \infty}Q(x)=\infty.$
The function $$T(x):=\frac{xQ'(x)}{Q(x)}, \quad x\neq 0$$ is quasi-increasing in $(0,\infty)$, with $$T(x)\geqslant \Lambda>1, \quad x\in {\mathbb{R}}^+\setminus\{0\}.$$
There exists $C_1>0$ such that $$\frac{Q''(x)}{|Q'(x)|}\le C_1\frac{|Q'(x)|}{Q(x)}, \quad a.e. \quad x\in {\mathbb{R}}\setminus\{0\}.$$ Then we say that $w=\exp(-Q)$ is in the class $\mathcal{F}(C^2)$. Besides, if there exists a compact subinterval $J(\ni 0)$ of ${\mathbb{R}}$ and $C_2>0$ such that $$\frac{Q''(x)}{|Q'(x)|}\geqslant C_2\frac{|Q'(x)|}{Q(x)}, \quad a.e. \quad x\in {\mathbb{R}}\setminus J,$$ then we say that $w=\exp(-Q)$ is in the class $\mathcal{F}(C^2+)$. If $T(x)$ is bounded, then $w$ is called a Freud-type weight, and if $T(x)$ is unbounded, then $w$ is Erdös-type weight.
Some typical examples in $\mathcal{F}(C^2+)$ are given as follows:
\[Examples1.2\]
\(1) For $\alpha>1$ and a non-negative integer $\ell$, we put $$Q(x)=Q_{\ell,\alpha}(x):=\exp_{\ell}(|x|^{\alpha})-\exp_{\ell}(0),$$ where for $\ell \geqslant 1$, $$\exp_{\ell}(x):=\exp(\exp(\exp(\cdots\exp x)\ldots)) \quad (\ell \textrm{-times})$$ and $\exp_0(x):=x$.
For $m\geqslant 0$, $\alpha\geqslant 0$ with $\alpha+m>1$, we put $$\label{eq1.4}
Q(x)=Q_{\ell,\alpha,m}(x):=|x|^m\{\exp_{\ell} (|x|^\alpha) -\alpha^*\exp_{\ell}(0)\},$$ where for $\ell>0$ we suppose $\alpha^*=0$ if $\alpha=0$; $\alpha^*=1$ otherwise. For $\ell =0$ we suppose $m>1$ and $\alpha=0$. Note that $Q_{\ell,0,m}$ is a Freud-type weight.
For $\alpha>1$, we put $$Q(x)=Q_{\alpha}(x):=(1+|x|)^{|x|^\alpha} -1.$$
To consider the higher order Hermite-Fejér interpolation polynomial we define a class of further strengthened weights for $\nu\geqslant 2$ than Definition \[Definition1.1\] as follows:
\[Definition1.3\] Let $w(x)=\exp(-Q(x))\in \mathcal{F}(C^2+)$, and let $\nu\geqslant 2$ be an integer. Assume that $Q(x)$ is a $\nu$-times continuously differentiable function on ${\mathbb{R}}$ and satisfies the following:
$Q^{(\nu+1)}(x)$ exists and $Q^{(i)}(x), 0\le i\le \nu+1,$ are positive for $x>0$.
There exist constants $C_i>0$ such that $$\left|Q^{(i+1)}(x)\right|\le C_i\left|Q^{(i)}(x)\right|\frac{|Q'(x)|}{Q(x)},
\quad x\in {\mathbb{R}}\backslash\{0\}, \quad i=1,2,\ldots \nu.$$
There exist $0\le\delta<1$ and $c_1>0$ such that $$\label{eq1.5}
Q^{(\nu+1)}(x)\le C\left(\frac{1}{x}\right)^{\delta}, \quad x\in (0,c_1].$$ Then we say that $w(x)=\exp(-Q(x))$ is in the class $\mathcal{F}_\nu(C^2+)$.
Suppose one of the following:
1. $Q'(x)/Q(x)$ is quasi-increasing on a certain positive interval $[c_2,\infty)$.
2. $Q^{(\nu+1)}(x)$ is non-decreasing on a certain positive interval $[c_2,\infty)$.
3. There exist constants $C>0$ and $0\le \delta<1$ such that $Q^{(\nu+1)}(x)\le C(1/x)^{\delta}$ on $(0,\infty)$.
Then we write $w(x)=\exp(-Q(x))\in\mathcal{\tilde{F}}_\nu(C^2+)$.
In this paper we treat the weight $w_\rho(x)=|x|^{\rho}\exp(-Q(x))$, $\rho\geqslant 0$ of type (\[eq1.1\]). For $w_\rho(x)=|x|^{\rho}\exp(-Q(x))$, $w(x)=\exp(-Q(x))\in \tilde{\mathcal{F}}_\nu(C^2+)$, we write $w_\rho\in \tilde{\mathcal{F}}_{\nu,\rho}(C^2+)$.
In the following, we give specific examples of $w \in \tilde{\mathcal{F}}_\nu(C^2+)$.
\[Example1.4\]
Let $\nu$ be a positive integer, and let $Q_{\ell,\alpha,m}$ be defined in (\[eq1.4\]).
Let $m$ and $\alpha$ be non-negative even integers with $m+\alpha>1$. Then $w(x)=\exp(-Q_{\ell,\alpha,m})\in \mathcal{F}_\nu(C^2+)$, and one has the following.
If $\ell>0$, then we see that $Q'_{\ell,\alpha,m}(x)/Q_{\ell,\alpha,m}(x)$ is quasi-increasing on a certain positive interval $(c_1,\infty)$, and $Q_{\ell,0,m}(x)$ is non-decreasing on $(0,\infty)$.
If $\ell=0$, then we see that $Q_{0,0,m}(x), m\geqslant 2$, is non-decreasing on $(0,\infty)$. Hence $w(x)=\exp(-Q_{\ell,\alpha,m})\in \tilde{\mathcal{F}}_{\nu}(C^2+)$.
Let $m+\alpha-\nu>0$. Then $w(x)=\exp(-Q_{\ell,\alpha,m})\in \mathcal{F}_\nu(C^2+)$, and one has the following.
If $\ell\geqslant 2$ and $\alpha>0$, then there exists a constant $c_1>0$ such that $Q'_{\ell,\alpha,m}(x)/Q_{\ell,\alpha,m}(x)$ is quasi-increasing on $(c_1,\infty)$.
Let $\ell=1$. If $\alpha\geqslant 1$, then there exists a constant $c_2>0$ such that $Q'_{1,\alpha,m}(x)/Q_{1,\alpha,m}(x)$ is quasi-increasing on $(c_2,\infty)$, and if $0<\alpha<1$, then $Q'_{1,\alpha,m}(x)/Q_{1,\alpha,m}(x)$ is quasi-decreasing on $(c_2,\infty)$.
Let $\ell=1$, and $0<\alpha<1$, then $Q^{(\nu+1)}_{1,\alpha,m}(x)$ is non-decreasing on a certain positive interval on $(c_2,\infty)$. Hence $w(x)=\exp(-Q_{\ell,\alpha,m})\in \tilde{\mathcal{F}}_{\nu}(C^2+)$.
\[Example1.5\] [Let $\nu$ be a positive integer and $\alpha > \nu$. Then $w(x)=\exp(-Q_{\alpha}(x))$ belongs to $\mathcal{F}_{\nu}(C^2+)$. Moreover, there exists a positive constant $c_2 > 0$ such that $Q_{\alpha}'(x)/Q_{\alpha}(x)$ is quasi-increasing on $(c_2, \infty)$. Hence $w(x)=\exp(-Q_{\alpha})\in \tilde{\mathcal{F}}_{\nu}(C^2+)$. ]{}
In Section 2, we report the $L_p$-convergence theorems. We write some lemmas to prove the theorems and then we prove them in Section 3.
In what follows we abbreviate several notations as $x_{k,n}:=x_{k,n,\rho}, h_{kn}(x):=h_{k,n,\rho}(\nu,x)$, $l_{kn}(x):=l_{k,n,\rho}(x)$, $h_{skn}(x):=h_{s,k,n,\rho}(\nu,x)$ and $p_n(x):=p_{n,\rho}(x)$ if there is no confusion. For arbitrary nonzero real valued functions $f(x)$ and $g(x)$, we write $f(x)\sim g(x)$ if there exist constants $C_1, C_2>0$ independent of $x$ such that $C_1 g(x)\le f(x)\le C_2g(x)$ for all $x$. For arbitrary positive sequences $\{c_n\}_{n=1}^\infty$ and $\{d_n\}_{=1}^\infty$ we define $c_n\sim d_n$ similarly.
Throughout this paper $C,C_1,C_2,\ldots$ denote positive constants independent of $n,x,t$ or polynomials $P_n(x)$, and the same symbol does not necessarily denote the same constant in different occurrences.
Theorems
========
We use the following notations.\
(1) Mhaskar-Rakhmanov-Saff numbers $a_x$: $$x=\frac{2}{\pi}\int_0^1\frac{a_xuQ'(a_xu)}{(1-u^2)^{1/2}}du, \quad x>0.$$ (2) $$\label{eq2.1}
\varphi_u(x)=
\left\{
\begin{array}{lr}
\frac{|x|}{u}\frac{1-\frac{|x|}{a_{2u}}}{\sqrt{1-\frac{|x|}{a_u}+\delta_u}}, & |x|\le a_u; \\
\varphi_u(a_u), & a_u<|x|,
\end{array}
\right.$$ where $$\delta_u=(uT(a_u))^{-2/3}, \quad u>0.$$ In the rest of this paper, we assume the following:
\[Assumption2.1\][Let the weight $w_\rho\in \tilde{\mathcal{F}}_{\nu,\rho}(C^2+)$, $\rho\geqslant 0$.\
(a) If $T(x)$ is bounded, then we suppose that for some $C>0$ $$Q(x)\geqslant C|x|^{2},$$ and for $\delta$ in (\[eq1.5\]), $$\label{eq2.2}
a_n\le C n^{1/(1+\nu-\delta)}.$$ (b) There exist $0\le \gamma< 1$ and $C(\gamma)>0$ such that $$\label{eq2.3}
T(a_n)\le C(\gamma)n^\gamma,$$ here, if $T(x)$ is bounded, that is, the weight $w$ is a Freud weight, then we set $\gamma=0$. We put $$\label{eq2.4}
\varepsilon_n:=
\begin{cases}
\frac{a_n}{n},& \frac{T(a_n)}{a_n}<1; \\
\frac{1}{n^{1-\gamma}},& \frac{T(a_n)}{a_n}\geqslant 1.
\end{cases}$$ ]{}
\[Lemma2.2\] [(1)]{} Let $w=\exp(-Q)\in \mathcal{F}(C^2)$, and let $T(x)$ be unbounded. Then, for any $\eta>0$ there exists $C(\eta)>0$ such that for $t\geqslant 1$, $$\label{eq2.5}
a_t\le C(\eta)t^{\eta}.$$
Let $\lambda:=C_1$ be the constant in Definition \[Definition1.1\] (e), that is, $$\frac{Q''(x)}{Q'(x)}\le \lambda\frac{Q'(x)}{Q(x)}, \quad a.e.\quad x\in \mathbb{R}\backslash\left\{0\right\}.$$ If $1<\lambda$, and if $\eta$ is defined in (\[eq2.5\]), then there exists $C(\lambda,\eta)$ such that $$T(a_t)\le C(\lambda,\eta)t^{\frac{2(\eta+\lambda-1)}{\lambda+1}},$$ and if $\lambda\le 1+\eta$, then there exists $C(\lambda,\eta)$ such that $$\label{eq2.6}
T(a_t)\le C(\lambda,\eta)t^{4\eta}, \quad t\geqslant 1.$$
\[Remark2.3\]
\(1) If $T(x)$ is unbounded, then for any $L>0$ we have $Q(x) \ge C|x|^L$([@[24] Lemma 3.2]) and (\[eq2.2\]) holds ([@[4] Theorem 1.4]).
(\[eq2.3\]) holds for $$\gamma=\frac{2(\eta+\lambda-1)}{\lambda+1}, \quad 0<\lambda<3.$$
Let us denote the MRS-number $a_n$ and the function $T(x)$ for $Q_{r,\alpha,m}$ by $a_{n,r,\alpha,m}$ and $T_{r,\alpha,m}$ respectively. Then $T_{r,\alpha,m}(a_{n,r,\alpha,m})$ satisfies (\[eq2.3\]). In fact, we obtain from [@[23] Proposition 4] and [@[16] Example 2 and (1.34)], $$a_{n,r,\alpha,m}\sim (1+\log_r^+ n)^{1/\alpha},$$ where $$\log_r^+ (x)=
\left\{
\begin{array}{ll}
\log(\log(\log(...\log x)))\,\, (r \textrm{times}),& \textrm{ if } x>\exp_r(0),\\
0,& \textrm{ otherwise },
\end{array}
\right.$$ and for every $0< \gamma< 1$, $$T_{r,\alpha,m}(a_{n,r,\alpha,m})\sim (1+\log_r^+n)(1+\log_{r-1}^+n)(1+\log^+n)<n^\gamma$$ by $$T_{r,\alpha,m}(a_{n,r,\alpha,m})=m+T_{r,\alpha,0}(a_{n,r,\alpha,m}).$$
If the weight $w=\exp(-Q)\in\mathcal{F}(C^2+)$ satisfies $$\mu=\lim\inf_{x\rightarrow \infty}\frac{Q(x)Q''(x)}{(Q'(x))^2}
=\lim\sup_{x\rightarrow \infty}\frac{Q(x)Q''(x)}{(Q'(x))^2},$$ then we say that the weight $w$ is regular. For the regular weight $w$ we have $$\mu=\lim_{x\rightarrow \infty}\frac{Q(x)Q''(x)}{(Q'(x))^2}=1,$$ therefore, we obtain (\[eq2.6\]) for any fixed $\eta>0$ (see [@[24] Corollary 5.5]). We note that all of examples in Example \[Example1.4\] are regular.
(\[eq2.5\]) means $$0<C\le\frac{n}{a_n}\left(\frac{T(a_n)}{a_n}\right)^{\nu-1}.$$
Let $$\begin{aligned}
X_n(\nu,f;x) &:=&\sum_{k=1}^nf(x_{k,n})l_{kn}^{\nu}(x)
\sum_{i=0}^{\nu -2}e_i(\nu,k,n)(x-x_{k,n})^i,\\
Y_n(\nu,f;x)&:=&\sum_{k=1}^nf(x_{k,n})l_{kn}^{\nu}(x)e_{\nu-1}(\nu,k,n)(x-x_{k,n})^{\nu-1},\\
Z_n(l,\nu,f;x)&:=&\sum_{k=1}^n \sum_{s=1}^lf^{(s)}(x_{k,n})l_{kn}^{\nu}(x)
\sum_{i=s}^{\nu-1}e_{si}(\nu,k,n)(x-x_{k,n})^i.\end{aligned}$$ Then we know $$L_n(\nu,f;x)=X_n(\nu,f;x)+Y_n(\nu,f;x),$$ and $$L_n(l,\nu,f;x)=L_n(\nu,f;x)+Z_n(l,\nu,f;x).$$ Let $$\label{eq2.7}
\Phi(x):=\frac{1}{(1+Q(x))^{2/3}T(x)}.$$ Then we see that for $0<d\le |x|$, $$\Phi(x)\sim \frac{Q(x)^{\frac{1}{3}}}{xQ'(x)}.$$ Moreover, if we define $$\label{eq2.8}
\Phi_n(x):=\max\left\{\delta_n, 1-\frac{|x|}{a_n}\right\}, \quad n=1,2,3,....,$$ then we have the following:
\[Lemma2.4\] For $x\in \mathbb{R}$ we have $$\Phi(x)\le C\Phi_n(x), n\geqslant 1.$$
We have a chain of results as follows. In this section, we let $w_\rho\in \mathcal{\tilde{F}}_{\nu,\rho}(C^2+)$, $\rho\geqslant 0$, and we suppose Assumption \[Assumption2.1\]. In addition, we let $\alpha>0, \Delta>-1$, $1<p<\infty$. Let $C_f$ be a positive constant depending only on $f$, and let $C>0$ be a constant.
\[Proposition 2.5\] Let $\nu=2,3,4,...$, and let $$\label{eq2.9}
\Delta\geqslant \frac{1}{p}-\min\left\{1,\alpha\right\}.$$ For $f\in C(\mathbb{R})$ satisfying $$\label{eq2.10}
|f(x)|(1+|x|)^\alpha \left\{\Phi^{-\frac{3}{4}}(x)w_\rho(x)\right\}^{\nu}\le C_f, \quad x\in \mathbb{R},$$ we have $$\label{eq2.11}
\left\|(1+|x|)^{-\Delta}\left\{\Phi^{\frac{3}{4}}(x)w(x)\left(|x|+\frac{a_n}{n}\right)^{\rho}\right\}^{\nu}
X_n(\nu,f;x)\right\|_{L_p(\mathbb{R})}
\le C C_f.$$
\[Proposition2.6\] Let $\nu=2,4,6,...$ and assume that (\[eq2.9\]) holds. For $f\in C(\mathbb{R})$ satisfying $$\label{eq2.12}
|f(x)|(1+|x|)^\alpha\left\{\Phi^{-\frac{3}{4}}(x)w_\rho(x)\right\}^{\nu}
\left(|Q'(x)|+\frac{1}{|x|}\right)\le C_f, \quad x\in \mathbb{R}\setminus \{0\}$$ we have $$\left\|(1+|x|)^{-\Delta}\left\{\Phi^{\frac{3}{4}}(x)w(x)\left(|x|+\frac{a_n}{n}\right)^{\rho}\right\}^{\nu}
Y_n(\nu,f;x)\right\|_{L_p(\mathbb{R})}
\le CC_f\varepsilon_n(a_n+\log n),$$ where $\varepsilon_n$ is defined by (\[eq2.4\]).
\[Proposition2.7\] Let $\nu=2,3,4,...$ and assume that (\[eq2.9\]) holds. For $f\in C^l(\mathbb{R})$, $0\le l\le \nu-1$ satisfying $$\label{eq2.13}
|f^{(s)}(x)|(1+|x|)^\alpha \left\{\Phi^{-\frac{3}{4}}(x)w_\rho(x)\right\}^{\nu}\le C_f,
\quad x\in \mathbb{R}, \,\,\, s=1,2,...,l,$$ we have $$\begin{aligned}
&&\left\|(1+|x|)^{-\Delta}\left\{\Phi^{\frac{3}{4}}(x)w(x)\left(|x|+\frac{a_n}{n}\right)^{\rho}\right\}^{\nu}
Z_n(l,\nu,f;x)\right\|_{L_p(\mathbb{R})}\\
&\le& C_f\frac{a_n^2\log n}{n}
\left\{
\begin{array}{lr}
1, & \Delta p<1; \\
\frac{\log a_n}{a_n},& \Delta p=1;\\
\frac{1}{a_n}, &\Delta p>1.
\end{array}
\right.\end{aligned}$$
\[Proposition2.8\] Let $\nu=2,3,4,...$ and assume that (\[eq2.9\]) holds. Let $P\in \mathcal{P}_{\nu n-1}$ be fixed. Then, we have $$\left\|(1+|x|)^{-\Delta}
\left\{\Phi^{\frac{3}{4}}(x)w(x)\left(|x|+\frac{a_n}{n}\right)^{\rho}\right\}^{\nu}
\left(L_n(\nu,P;x)-P(x)\right)
\right\|_{L_p(\mathbb{R})}
\rightarrow 0 \quad \textrm{as} \quad n\rightarrow \infty.$$
\[Proposition2.9\] Let $\nu=2,4,6,...$ and assume that (\[eq2.9\]) holds. Let $P\in \mathcal{P}_{\nu n-1}$ be fixed. Then, we have $$\left\|(1+|x|)^{-\Delta}\left\{\Phi^{\frac{3}{4}}(x)w(x)
\left(|x|+\frac{a_n}{n}\right)^{\rho}\right\}^{\nu}(X_n(\nu,P;x)-P(x))
\right\|_{L_p(\mathbb{R})}
\rightarrow 0 \quad \textrm{as} \quad n\rightarrow \infty.$$
\[Remark2.10\]
Let $f\in C(\mathbb{R})$ satisfy that for given $0<\eta<1$ and $\alpha\geqslant 0$, $$\label{eq2.14}
|f(x)|(1+|x|)^\alpha w^{\nu-\eta}(x)\left\{|Q'(x)|+\frac{1}{|x|}\right\} \le C_f, \quad x\in\mathbb{R}
\quad \textrm{ and } \quad
\lim_{x\rightarrow 0}\frac{f(x)}{x}\le C_f,$$ where $C_f$ is a constant depending only on $f$. Then,
(\[eq2.10\]) and (\[eq2.12\]) hold.
Let $f\in C^l(\mathbb{R})$ for a certain $0\le l\le \nu-1$. Then (\[eq2.13\]) holds.
Let (\[eq2.14\]) be satisfied, then we see $f(0)=0$.
\[Theorem2.11\] Let $\nu=2,4,6,...$ and assume that (\[eq2.9\]) holds. For $f(x)$ satisfying (\[eq2.14\]) we have $$\left\|(1+|x|)^{-\Delta}\left\{\Phi^{\frac{3}{4}}(x)w(x)\left(|x|+\frac{a_n}{n}\right)^{\rho}\right\}^{\nu}
(L_n(\nu,f;x)-f(x))\right\|_{L_p(\mathbb{R})}
\rightarrow 0 \quad \textrm{as} \quad n\rightarrow \infty.$$
\[Theorem2.12\] Let $\nu=2,4,6,...$ and assume that (\[eq2.9\]) holds. For $f(x)$ satisfying (\[eq2.14\]) and (\[eq2.13\]), we have $$\left\|(1+|x|)^{-\Delta}\left\{\Phi^{\frac{3}{4}}(x)w(x)\left(|x|+\frac{a_n}{n}\right)^{\rho}\right\}^{\nu}
(L_n(l,\nu,f;x)-f(x))\right\|_{L_p(\mathbb{R})}
\rightarrow 0 \quad \textrm{as} \quad n\rightarrow \infty.$$
\[Remark2.13\] [From the formulas (\[eq2.2\]), (\[eq2.4\]) and (\[eq2.5\]), we have $$\varepsilon_n(a_n+\log n) \to 0 \quad \textrm{ as } \quad n \to \infty,$$ and $$\frac{a_n^2\log n}{n}
\begin{cases}
1, & \Delta p<1; \\
\frac{\log a_n}{a_n},& \Delta p=1;\\
\frac{1}{a_n}, &\Delta p>1
\end{cases}
\to 0 \quad \textrm{ as } \quad n \to \infty.$$ ]{}
Lemmas and Proof of Theorems
============================
For the coefficients $e_{si}(\nu,k,n) (e_i(\nu,k,n):=e_{0i}(\nu,k,n))$ in (\[eq1.2\]) or (\[eq1.3\]) we have the following estimates.
\[Lemma3.1\] Let $w(x)=\exp(-Q(x))\in \mathcal{F}(C^2+)$. We have the following. For each $s=0,1,...,\nu-1$ and $i=s, s+1,...,\nu-1$, $$e_0(\nu,k,n)=1, \quad |e_{si}(\nu,k,n)|\le C\left\{\frac{n}{(a_{2n}^2-x_{k,n}^2)^{1/2}}\right\}^{i-s}.$$
\[Lemma3.2\] Let $w_\rho\in \tilde{\mathcal{F}}_{\nu,\rho}(C^2+)$. If $x_{kn}\neq 0$ and $|x_{k,n}|\le a_n(1+\delta_n)$, then $e_0(\nu,k,n)=1$ and for $i=1,2,\ldots, \nu-1$, $$|e_i(\nu,k,n)|\le C\left\{\frac{T(a_n)}{a_n}+|Q'(x_{k,n})|+\frac{1}{x_{k,n}}\right\}^{\langle i \rangle}
\left\{\frac{n}{a_{2n}-|x_{k,n}|}+\frac{T(a_n)}{a_n}\right\}^{i-{\langle i \rangle}},$$ where $$<i>=
\begin{cases}
1,& \textrm{ if } i \textrm{ is odd},\\
0,& \textrm{ if } i \textrm{ is even}.
\end{cases}$$ For $x_{kn}=0$, we see $e_0(\nu,k,n)=1$ and $$|e_i(\nu,k,n)|\le C\left(\frac{n}{a_n}\right)^i \quad i=1,2,\ldots, \nu-1.$$
We have the following.
\[Lemma3.3\] Let $w_\rho\in \tilde{\mathcal{F}}_{\nu,\rho}(C^2+)$ $(\rho\geqslant 0)$, $0<p\le \infty$, and $\beta\in \mathbb{R}$. Then given $r>1$, there exist $C, n_0, \alpha>0$ such that for $n\geqslant n_0$ and $P\in \mathcal{P}_n$, $$\left\|(Pw)(x)|x|^\beta\right\|_{L_p(a_{rn}\le |x|)}
\le \exp(-Cn^\alpha)\left\|(Pw)(x)|x|^\beta\right\|_{L_p(L\frac{a_n}{n}\le |x|\le a_n(1-L\eta_n))}.$$
\[Lemma3.4\] [(1)]{} Let $P\in \mathcal{P}_{\nu n -1}$. Then, we have $$\begin{aligned}
&&\left\|(1+|x|)^{-\Delta}\left\{\Phi^{\frac{3}{4}}(x)w(x)\left(|x|+\frac{a_n}{n}\right)^{\rho}\right\}^{\nu} P(x)
\right\|_{L_p(a_{2n}\le |x|)}\\
&\le& C e^{-C_1n^{\eta}}\left\|(1+|x|)^{-\Delta}\left\{\Phi^{\frac{3}{4}}(x)w(x)
\left(|x|+\frac{a_n}{n}\right)^{\rho}\right\}^{\nu}
P(x)\right\|_{L_p(|x|\le a_{2n})}.\end{aligned}$$ So, if $$\left\|(1+|x|)^{-\Delta}\left\{\Phi^{\frac{3}{4}}(x)w(x)(|x|+1)^{\rho}\right\}^{\nu} P(x)
\right\|_{L_p(\mathbb{R})}<\infty,$$ then we have $$\left\|(1+|x|)^{-\Delta}\left\{\Phi^{\frac{3}{4}}(x)w(x)\left(|x|+\frac{a_n}{n}\right)^{\rho}\right\}^{\nu} P(x)
\right\|_{L_p(a_{2n}\le |x|)}\rightarrow 0
\quad \textrm{as} \quad n\rightarrow \infty.$$
\(1) Now, we denote the Mhaskar-Rakhmanov-Saff numbers for the exponents $Q(x)$ and $\nu Q(x)$ by $a_n(Q)$ and $a_n(\nu Q)$ respectively. Then, we have $a_n(Q)=a_{\nu n}(\nu Q)$. From (\[eq2.7\]) we note that $(1+|x|)^{\Delta}\Phi(x)^{-3\nu/4}$ is quasi-increasing. Then applying Lemma \[Lemma3.3\] with $w^{\nu}_{\nu\rho}(x)=\exp(-\nu Q(x))|x|^{\nu\rho}$, we have $$\begin{aligned}
&&\left\|(1+|x|)^{-\Delta}\left\{\Phi^{\frac{3}{4}}(x)
w(x)\left(|x|+\frac{a_n}{n}\right)^{\rho}\right\}^{\nu} P(x)\right\|_{L_p(a_{2n}(Q)\le |x|)}\\
&\le&
C\left\|w^{\nu}(x)\left(|x|+\frac{a_n}{n}\right)^{\nu\rho} P(x)\right\|_{L_p(a_{2\nu n}(\nu Q)\le |x|)}\\
&\le& C e^{-C(\nu n)^{\eta}}\left\|w^{\nu}(x)\left(|x|+\frac{a_n}{n}\right)^{\nu\rho} P(x)\right\|_{L_p(|x|\le a_{2\nu n}(\nu Q))}\\
&\le& C \frac{e^{-C\nu^{\eta} n^{\eta}}}{(1+a_{2n}(Q))^{-\Delta}\Phi(a_{2n}(Q))^{3\nu/4}}\\
&& \times\left\|(1+|x|)^{-\Delta}\left\{\Phi^{\frac{3}{4}}(x)(w(x)\left(|x|+\frac{a_n}{n}\right)^{\rho}\right\}^{\nu}
P(x)\right\|_{L_p(|x|\le a_{2n}(Q))}\\
&\le& C e^{-C_1n^{\eta}}
\left\|(1+|x|)^{-\Delta}\left\{\Phi^{\frac{3}{4}}(x)w(x)\left(|x|+\frac{a_n}{n}\right)^{\rho}\right\}^{\nu} P(x)
\right\|_{L_p(|x|\le a_{2n}(Q))}\\
&&\rightarrow 0 \quad \textrm{as} \quad n\rightarrow \infty,\end{aligned}$$ because that $(1+a_{2n}(Q))^{\Delta}\Phi(a_{2n}(Q))^{-3\nu/4}$ has order $n^s$ for some $s>0$.
In the rest of this section we let $w_\rho\in \mathcal{\tilde{F}}_{\nu,\rho}(C^2+)$, $\rho\geqslant 0$, and we suppose Assumption \[Assumption2.1\]. In addition, we let $\alpha>0, \Delta>-1$, $1<p<\infty$, and $C_f$ be a positive constant depending only on $f$.
To simplify the proofs of theorems we use the results of [@[7]]. In what follows we use the following notation. Let $x_{0,n}:=x_{1,n}+\varphi_n(x_{1,n})$, and $x_{n+1,n}=-x_{0,n}$. When $|x|\le x_{0,n}$, we define $$\label{eq3.1}
x_{m,n}:=x_{m(x),n}; \quad |x-x_{m,n}|=\min_{0\le j\le n}|x-x_{j,n}|.$$ If $|x-x_{j+1,n}|=|x-x_{j,n}|$, then we set $x_{m,n}:=x_{j,n}$. If $x_{0,n}<x$, then we put $m=0$. And if $x<x_{n+1,n}$, then we put $m=n+1$. Here, we note that there exists a constant $\delta>0$ such that $|x-x_{m,n}|\le \delta\varphi_n(x_{m,n})$.
\[Lemma3.5\] Let $m:=m(x)$ be defined by (\[eq3.1\]).
If (\[eq2.10\]) holds, then, we have $$\label{eq3.2}
\left\{\Phi^{\frac{3}{4}}(x)w(x)\left(|x|+\frac{a_n}{n}\right)^{\rho}\right\}^{\nu}|X_n(\nu,f;x)|
\le C_f\sum_{j=0}^n(1+|x_{j,n}|)^{-\alpha}\sum_{i=0}^{\nu-2}\left(\frac{1}{1+|m-j|}\right)^{\nu-i}.$$
If (\[eq2.12\]) holds, then, we have $$\label{eq3.3}
\left\{\Phi^{\frac{3}{4}}(x)w(x)\left(|x|+\frac{a_n}{n}\right)^{\rho}\right\}^{\nu}|Y_n(\nu,f;x)|
\le C_f\varepsilon_n\sum_{j=0}^n(1+|x_{j,n}|)^{-\alpha}\left(\frac{1}{1+|m-j|}\right).$$
If (\[eq2.13\]) holds, then, we have $$\begin{aligned}
\label{eq3.4} \nonumber
&& \left\{\Phi^{\frac{3}{4}}(x)w(x)\left(|x|+\frac{a_n}{n}\right)^{\rho}\right\}^{\nu}|Z_n(l,\nu,f;x)|\\
&\le& C_f\frac{a_n}{n}\sum_{j=0}^n(1+|x_{j,n}|)^{-\alpha}\sum_{i=0}^{\nu-1}\left(\frac{1}{1+|m-j|}\right)^{\nu-i}.\end{aligned}$$
In [@[7] Propositions 3.7, 3.8 and 3.9] we got the similar estimations in $L_{\infty}(\mathbb{R})$-space. Now, in there we can exchange $f(x)$ with $f(x)(1+|x|)^{\alpha}$, and then we obtain the results in $L_{p}(\mathbb{R})$ by the similar methods as the proofs of [@[7] Propositions 3.7, 3.8 and 3.9] (see [@[7] p.16-p.23]).
\[Lemma3.6\] [(1)]{} [[@[7] Lemma 4.3]]{} Uniformly for $n \ge 1$, $$\label{eq3.6}
\sup_{x\in \mathbb{R}}\left|p_{n,\rho}(x)w(x)\right|
\left(|x|+\frac{a_n}{n}\right)^{\rho}\Phi^{\frac{1}{4}}(x)
\le C a_n^{-\frac{1}{2}}.$$
[@[2] Theorem 2.5(a)] Uniformly for $n \ge 1$, $$\label{eq3.5}
|p'_{n,\rho}w|(x_{j,n})\left(|x_{j,n}|+\frac{a_n}{n}\right)^{\rho}\sim \varphi_n(x_{j,n})^{-1}[a^2_n-x^2_{j,n}]^{-1/4}.$$
\[Lemma3.7\] Let $w(x)=\exp(-Q(x))\in \mathcal{F}(C^2+)$. For the zeros $x_{j,n}=x_{j,n,\rho}$, we have the following.
[[[@[2] Theorem 2.2, (b)]]{}]{} For $n\geqslant 1$ and $1\le j\le n-1$, $$x_{j,n}-x_{j+1,n}\sim \varphi_{n}(x_{j,n}),$$ and [[@[2] Lemma A.1 (A.3)]]{} $$\varphi_{n}(x_{j,n})\sim \varphi_n(x_{j+1,n}).$$
[[[@[2] Theorem 2.2, (a)]]{}]{} For the minimum positive zero $x_{[n/2],n}$ ($[n/2]$ is the largest integer $\le n/2$), we have $$x_{[n/2],n}\sim a_n n^{-1},$$ and for large enough $n$, $$1-\frac{x_{1,n}}{a_{n}} \sim \delta_{n}.$$
[[@[2] Lemma 4.7]]{} $$b_n=\frac{\gamma_{n-1}}{\gamma_n} \sim a_n \sim x_{1,n}.$$
\[Lemma3.8\] Let $w=\exp(-Q)\in \mathcal{F}(C^2+)$. Let $L>0$ be fixed. Then we have the following.
[([@[16] Lemma 3.5, (a)])]{} Uniformly for $t>0$, $$a_{Lt}\sim a_t.$$
[([@[16] Lemma 3.5, (b)])]{} Uniformly for $t>0$, $$Q^{(j)}(a_{Lt})\sim Q^{(j)}(a_t), \quad j=0,1.$$ Moreover, $$T(a_{Lt})\sim T(a_t).$$
[([@[16] Lemma 3.11 (3.52)])]{} Uniformly for $t>0$, $$\left|1-\frac{a_{Lt}}{a_t}\right|\sim \frac{1}{T(a_t)}.$$
From Lemma \[Lemma3.6\] it is sufficient that we estimate Proposition \[Proposition 2.5\], \[Proposition2.6\] or \[Proposition2.7\] for only $|x|\le a_{2n}$.
First we set $$\begin{aligned}
X_n(\nu,f;x)&=&\sum_{j=1}^nf(x_{j,n})l_{jn}^{\nu}(x)\sum_{i=0}^{\nu -2}e_i(\nu,j,n)(x-x_{j,n})^i\\
&=&:\sum_{j\,\,;|x_{j,n}|\geqslant \frac{a_n}{3}}
+\sum_{j\,\,;|x_{j,n}|< \frac{a_n}{3}}=:X_{1,n}(\nu,f;x) +X_{2,n}(\nu,f;x).\end{aligned}$$ Using (\[eq3.2\]), we see $$\begin{aligned}
&&\left|(1+|x|)^{-\Delta}\left\{\Phi^{\frac{3}{4}}(x)w(x)\left(|x|+\frac{a_n}{n}\right)^{\rho}\right\}^{\nu} X_{1,n}(\nu,f;x)\right|\\
&\le& C_f(1+|x|)^{-\Delta}\sum_{|x_{j,n}|\geqslant \frac{1}{3}a_n}(1+|x_{j,n}|)^{-\alpha}
\sum_{i=0}^{\nu-2}\left(\frac{1}{1+|m-j|}\right)^{\nu-i}\\
&\le& CC_fa_n^{-\alpha}(1+|x|)^{-\Delta}.\end{aligned}$$ Therefore, we have by (\[eq2.9\]) $$\begin{aligned}
\label{3.9} \nonumber
&&\left\|(1+|x|)^{-\Delta}\left\{\Phi^{\frac{3}{4}}(x)w(x)\left(|x|+\frac{a_n}{n}\right)^{\rho}\right\}^{\nu}
X_{1,n}(\nu,f;x)\right\|_{L_p(|x|\le a_{2n})}\\ \nonumber
&\le& C_fa_n^{-\alpha}\left\|(1+|x|)^{-\Delta}\right\|_{L_p(|x|\le a_{2n})}\\\nonumber
&\le& C_fa_n^{-\alpha}
\left\{
\begin{array}{ll}
a_n^{\frac{1}{p}-\Delta}, & \Delta p<1, \\
\log a_n, & \Delta p=1,\\
1, & \Delta p>1
\end{array}
\right.
\le C_f.\end{aligned}$$ Now, we will estimate for $X_{2,n}(\nu,f,x)$. For $|x|<1$, since we have from (\[eq3.2\]) $$\left\{\Phi^{\frac{3}{4}}(x)w(x)\left(|x|+\frac{a_n}{n}\right)^{\rho}\right\}^{\nu}|X_{2,n}(\nu,f;x)|\le C_f,$$ we see $$\label{eq3.7}
\left\|(1+|x|)^{-\Delta}\left\{\Phi^{\frac{3}{4}}(x)w(x)\left(|x|+\frac{a_n}{n}\right)^{\rho}\right\}^{\nu}
|X_{2,n}(\nu,f;x)|\right\|_{L_p(|x|\le 1)}\\
\le C_f.$$ Let $|x| \ge 1$. We divide it into two sums as follows: $$\begin{aligned}
X_{2,n}(\nu,f;x) &=:& \sum_{\substack{j;\,\,|x_{j,n}|< \frac{a_n}{3}, \\
|x-{x_{j,n}}|\geqslant \frac{|x|}{2}}}
+\sum_{\substack{j;\,\,|x_{jn}|< \frac{a_n}{3},\\ |x-{x_{j,n}}|< \frac{|x|}{2}}}
=: X_{2,n}^{[1]}(\nu,f;x) +X_{2,n}^{[2]}(\nu,f;x).\end{aligned}$$ Using (\[eq2.10\]), Lemma \[Lemma3.7\](2), (\[eq3.6\]) and (\[eq3.5\]) with $\varphi_n(x_{j,n})\sim a_n/n$, we see that $$\begin{aligned}
\label{eq3.8}
&&\left\{\Phi^{\frac{3}{4}}(x)w(x)\left(|x|+\frac{a_n}{n}\right)^{\rho}\right\}^{\nu}
\left|f(x_{j,n})l_{jn}^{\nu}(x)\right|\\ \nonumber
&\le& C_f
(1+|x_{j,n}|)^{-\alpha}
\left|\frac{\left\{\Phi^{\frac{3}{4}}(x_{j,n})\Phi^{\frac{3}{4}}(x)w(x)\left(|x|+\frac{a_n}{n}\right)^{\rho}\right\}
p_n(x)}{p'_n(x_{j,n})w(x_{j,n})(\frac{a_n}{n}+|x_{j,n}|)^{\rho}}\right|^{\nu}\\ \nonumber
&\le& CC_f (1+|x_{j,n}|)^{-\alpha}\left(\frac{a_n}{n}\right)^{\nu}\end{aligned}$$ and Lemma \[Lemma3.1\] $$\sum_{i=0}^{\nu -2}e_i(\nu,j,n)(x-x_{j,n})^i \le C\sum_{i=0}^{\nu -2}\left(\frac{n}{a_n}\right)^i|x|^{i-\nu}
\le C\left(\frac{n}{a_n}\right)^{\nu-2}|x|^{-2}.$$ Then, we have $$\begin{aligned}
&&\left|\left\{\Phi^{\frac{3}{4}}(x)w(x)\left(|x|+\frac{a_n}{n}\right)^{\rho}\right\}^{\nu}
X_{2,n}^{[1]}(\nu,f;x)\right|\\
&\le& C_f
\sum_{\substack{j\,\,;|x_{j,n}|< \frac{a_n}{3}, \\ |x-{x_{j,n}}|\geqslant \frac{|x|}{2}}}
(1+|x_{j,n}|)^{-\alpha}\left(\frac{a_n}{n}\right)^{2}
|x|^{-2}
\le CC_f\frac{a_n^2}{n}\frac{1}{x^2}.\end{aligned}$$ Therefore, we have, using $\Delta>-1$, $$\begin{aligned}
\label{eq3.9} \nonumber
&&\left\|(1+|x|)^{-\Delta}\left\{\Phi^{\frac{3}{4}}(x)w(x)\left(|x|+\frac{a_n}{n}\right)^{\rho}\right\}^{\nu}
X_{2,n}^{[1]}(\nu,f;x)\right\|_{L_p(1\le|x|\le a_{2n})}\\
&\le& C_f\frac{a_n^2}{n} \left\|(1+|x|)^{-(\Delta+2)}\right\|_{L_p(1\le|x|\le a_{2n})}
\le CC_f\frac{a_n^2}{n}\le CC_f\end{aligned}$$ (note (\[eq2.2\]) and (\[eq2.5\])). Since, for $|x-x_{j,n}| < \frac{|x|}{2}$, we see $|x_{j,n}|\sim |x|$, the inequality (\[eq3.2\]) implies $$\begin{aligned}
\label{eq3.10}
&&\left\{\Phi^{\frac{3}{4}}(x)w(x)\left(|x|+\frac{a_n}{n}\right)^{\rho}\right\}^{\nu}
X_{2,n}^{[2]}(\nu,f;x)\\ \nonumber
&\le& CC_f
\sum_{\substack{ j;\,\, |x_{j,n}|< \frac{a_n}{3},\\|x-{x_{j,n}}|< \frac{|x|}{2}}}(1+|x_{j,n}|)^{-\alpha}
\sum_{i=0}^{\nu-2}\left(\frac{1}{1+|m-j|}\right)^{\nu-i}
\le CC_f(1+|x|)^{-\alpha}.\end{aligned}$$ Therefore, we obtain by (\[eq2.9\]) $$\begin{aligned}
\label{eq3.11}
&&\quad \quad \left\|(1+|x|)^{-\Delta}\left\{\Phi^{\frac{3}{4}}(x)w(x)\left(|x|+\frac{a_n}{n}\right)^{\rho}\right\}^{\nu}
X_{2,n}^{[2]}(\nu,f;x)\right\|_{L_p(1\le |x|\le a_{2n})}\\ \nonumber
&\le& CC_f\left\|(1+|x|)^{-(\alpha+\Delta)}\right\|_{L_p(|x|\le a_{2n})}
\le CC_fa_n^{\frac{1}{p}-(\alpha+\Delta)}\le CC_f.\end{aligned}$$ Hence, from (\[3.9\]), (\[eq3.8\]), (\[eq3.9\]), (\[eq3.11\]) and Lemma \[Lemma3.4\] we conclude (\[eq2.11\])
We repeat the methods of the proof of Proposition \[Proposition 2.5\]. $$Y_n(\nu,f;x)=:\sum_{j;|x_{j,n}|\geqslant \frac{a_n}{3}}
+\sum_{j;|x_{j,n}|< \frac{a_n}{3}}=:Y_{1,n}(\nu,f;x) +Y_{2,n}(\nu,f;x).$$ Then we have by (\[eq3.3\]), (\[eq2.9\]), (\[eq2.2\]) and (\[eq2.5\]) $$\begin{aligned}
\label{3.13} \nonumber
&&\left\|(1+|x|)^{-\Delta}\left\{\Phi^{\frac{3}{4}}(x)w(x)\left(|x|+\frac{a_n}{n}\right)^{\rho}\right\}^{\nu}
Y_{1,n}(\nu,f;x)\right\|_{L_p(|x|\le a_{2n})}\\\nonumber
&\le& C_fa_n^{-\alpha}\varepsilon_n\log n
\left\|(1+|x|)^{-\Delta}\right\|_{L_p(|x|\le a_{2n})}\\\nonumber
&\le& CC_fa_n^{-\alpha}\varepsilon_n\log n
\begin{cases}
a_n^{\frac{1}{p}-\Delta},& \Delta p<1; \\
\log a_n,& \Delta p=1;\\
1, &\Delta p>1
\end{cases}\\ \nonumber
&\le& CC_f\varepsilon_n\log n
\begin{cases}
1, & \Delta p<1; \\
\frac{\log a_n}{a_n},& \Delta p=1;\\
\frac{1}{a_n}, &\Delta p>1.
\end{cases}\end{aligned}$$ Now, we estimate $Y_{2,n}(\nu,f;x)$. For $|x|\le 1$ we obtain that from (\[eq3.3\]), $$\left\|(1+|x|)^{-\Delta}\left\{\Phi^{\frac{3}{4}}(x)w(x)\left(|x|+\frac{a_n}{n}\right)^{\rho}\right\}^{\nu} Y_{2,n}(\nu,f;x)
\right\|_{L_p(|x|\le 1)}
\le CC_f\varepsilon_n \log n.$$ Let $|x|\geqslant 1$ and $$Y_{2,n}(\nu,f;x)=:\sum_{\substack{j\,\,;|x_{j,n}|< \frac{a_n}{3},\\|x-{x_{j,n}}|\geqslant \frac{|x|}{2}>0}}
+\sum_{\substack{j\,\,;|x_{j,n}|< \frac{a_n}{3},\\|x-{x_{j,n}}|< \frac{|x|}{2}}}=:Y_{2,n}^{[1]}+Y_{2,n}^{[2]}.$$ Then we have similarly to (\[eq3.10\]) and (\[eq3.11\]), $$\begin{aligned}
\nonumber
&&\left\|(1+|x|)^{-\Delta}\left\{\Phi^{\frac{3}{4}}(x)w(x)\left(|x|+\frac{a_n}{n}\right)^{\rho}\right\}^{\nu}
Y_{2,n}^{[2]}(\nu,f;x)\right\|_{L_p(1\le |x|\le a_{2n})}\\\nonumber
&\le& CC_f\varepsilon_n\log n \left\|(1+|x|)^{-(\alpha+\Delta)}\right\|_{L_p(|x|\le a_{2n})}
\le CC_fa_n^{\frac{1}{p}-(\alpha+\Delta)}\varepsilon_n\log n\le CC_f\varepsilon_n\log n\end{aligned}$$ (see (\[eq2.9\])). Finally, we estimate $Y_{2,n}^{[1]}$. Let $|x_{j,n}|< \frac{a_n}{3}$ and $|x-{x_{j,n}}|\geqslant \frac{|x|}{2}$. Then, from Lemma \[Lemma3.2\], (\[eq2.12\]), (\[eq3.6\]) and (\[eq3.5\]), we have $$\begin{aligned}
\label{eq3.12}
&&\left\{\Phi^{\frac{3}{4}}(x)w(x)\left(|x|+\frac{a_n}{n}\right)^{\rho}\right\}^{\nu}
\left|f(x_{j,n})l_{jn}^{\nu}(x)\right||x-x_{j,n}|^{\nu-1}e_{\nu-1}(\nu,k,n)\\ \nonumber
&\le& CC_f
(1+|x_{j,n}|)^{-\alpha}\left\{|Q'(x_{j,n})|+\frac{1}{|x_{j,n}|}\right\}^{-1}\\ \nonumber
&& \quad \times
\left|\frac{\left\{\Phi^{\frac{3}{4}}(x_{j,n})\Phi^{\frac{3}{4}}(x)w(x)\left(|x|+\frac{a_n}{n}\right)^{\rho}\right\}
p_n(x)}{p'_n(x_{j,n})w(x_{j,n})(\frac{a_n}{n}+|x_{j,n}|)^{\rho}}\right|^{\nu} \frac{1}{|x-x_{j,n}|}\\ \nonumber
&& \quad \times \left\{\frac{T(a_n)}{a_n}+|Q'(x_{k,n})|+\frac{1}{x_{k,n}}\right\}
\left\{\frac{n}{a_{2n}-|x_{k,n}|}\right\}^{\nu-2} \\ \nonumber
&\le& CC_f (1+|x_{j,n}|)^{-\alpha}\left\{|Q'(x_{j,n})|+\frac{1}{|x_{j,n}|}\right\}^{-1}\\ \nonumber
&& \quad \times \Phi^{\frac{3\nu}{4}}(x_{j,n})\varphi_n^{\nu}(x_{j,n})\left(1-\frac{|x_{j,n}|}{a_n}\right)^{\frac{\nu}{4}}
\frac{1}{|x-x_{j,n}|}\\ \nonumber
&& \quad \times \left\{\frac{T(a_n)}{a_n}+|Q'(x_{k,n})|+\frac{1}{x_{k,n}}\right\}
\left\{\frac{n}{a_{2n}-|x_{k,n}|}\right\}^{\nu-2}.\end{aligned}$$ On the other hand, from (\[eq2.1\]), (\[eq2.8\]) and $|x_{j,n}| < a_n/3$, we easily see $$\label{11}
\Phi_n^{\frac{3\nu}{4}}(x_{j,n})\varphi_n^{\nu}(x_{j,n})
\left(1-\frac{|x_{j,n}|}{a_n}\right)^{\frac{\nu}{4}}
\left(\frac{n}{a_{2n}-|x_{j,n}|}\right)^{\nu-2}
\le C\left(\frac{a_n}{n}\right)^2.$$ Moreover, since $|Q'(x_{j,n})|+\frac{1}{|x_{j,n}|}\geqslant C>0$ for some constant $C$, we see $$\label{eq3.14}
\left\{\frac{T(a_n)}{a_n}+|Q'(x_{j,n})|+\frac{1}{|x_{j,n}|}\right\}
\left\{|Q'(x_{j,n})|+\frac{1}{|x_{j,n}|}\right\}^{-1}
\le C\max\left\{1,\frac{T(a_n)}{a_n}\right\}$$ and since $|x-{x_{j,n}}|\geqslant \frac{|x|}{2}$, we have $$\label{22}
\frac{1}{|x-x_{j,n}|} \le \frac{2}{|x|}.$$ Thus, using (\[eq3.12\]), (\[11\]), (\[eq3.14\]) and (\[22\]), we can estimate for $Y_{2,n}^{[1]}(\nu,f;x)$ as follows: $$\begin{aligned}
&&\left\{\Phi^{\frac{3}{4}}(x)w(x)\left(|x|+\frac{a_n}{n}\right)^{\rho}\right\}^{\nu} Y_{2,n}^{[1]}(\nu,f;x)\\
&\le& CC_f\max\left\{1,\frac{T(a_n)}{a_n}\right\}\frac{1}{|x|}\left(\frac{a_n}{n}\right)^2
\sum_{\substack{j;\,\,|x_{j,n}|< \frac{a_n}{3}, \\ |x-{x_{j,n}}|\geqslant \frac{|x|}{2}}}
(1+|x_{j,n}|)^{-\alpha} \\
&\le& CC_f\max\left\{1,\frac{T(a_n)}{a_n}\right\}\frac{1}{|x|}\frac{a_n^2}{n}
\le CC_f a_n \varepsilon_n\frac{1}{|x|}.\end{aligned}$$ Therefore we have by (\[eq2.9\]) $$\begin{aligned}
&&\left\|(1+|x|)^{-\Delta}\left\{\Phi^{\frac{3}{4}}(x)w(x)\left(|x|+\frac{a_n}{n}\right)^{\rho}\right\}^{\nu}
Y_{2,n}^{[1]}(\nu,f;x)\right\|_{L_p(1\le |x|\le a_{2n})}\\
&\le& CC_fa_n \varepsilon_n
\left\|(1+|x|)^{-(\Delta+1)}\right\|_{L_p(1\le |x|\le a_{2n})}
\le CC_f a_n \varepsilon_n a_n^{\frac{1}{p}-(\Delta+1)}
\le CC_fa_n\varepsilon_n.\end{aligned}$$
Using (\[eq3.4\]), we have from (\[eq2.2\]) and (\[eq2.5\]) $$\begin{aligned}
&&\left\|(1+|x|)^{-\Delta}\left\{\Phi^{\frac{3}{4}}(x)w(x)\left(|x|+\frac{a_n}{n}\right)^{\rho}\right\}^{\nu}
Z_n(l,\nu,f;x)\right\|_{L_p(\mathbb{R})}\\
&\le& CC_f\frac{a_n\log n}{n}\left\|(1+|x|)^{-\Delta}\right\|_{L_p(|x|\le a_{2n})}\\
&\le& CC_f\frac{a_n\log n}{n}
\begin{cases}
a_n^{\frac{1}{p}-\Delta},& \Delta p<1; \\
\log a_n,& \Delta p=1;\\
1, &\Delta p>1
\end{cases}
\le CC_f\frac{a_n^2\log n}{n}
\begin{cases}
1, & \Delta p<1; \\
\frac{\log a_n}{a_n},& \Delta p=1;\\
\frac{1}{a_n}, &\Delta p>1.
\end{cases}\end{aligned}$$
Since we see for a fixed polynomial $P\in \mathcal{P}_{\nu n-1}$ $$L_n(\nu,P)-P=L_n(\nu,P)-L_n(\nu-1,\nu,P)=Z_n(\nu-1,\nu,P),$$ we can obtain from Proposition \[Proposition2.7\], $$\begin{aligned}
&&\left\|(1+|x|)^{-\Delta}\left\{\Phi^{\frac{3}{4}}(x)w(x)\left(|x|+\frac{a_n}{n}\right)^{\rho}\right\}^{\nu}
\left\{L_n(\nu,P)-P\right\}\right\|_{L_p(\mathbb{R})}\\
&\le& CC_f\frac{a_n^2\log n}{n}
\begin{cases}
1, & \Delta p<1; \\
\frac{\log a_n}{a_n},& \Delta p=1;\\
\frac{1}{a_n}, &\Delta p>1.
\end{cases}\rightarrow 0 \quad \textrm{ as } \quad n\rightarrow \infty\end{aligned}$$ (note Remark \[Remark2.13\]).
Since we see for a fixed polynomial $P\in \mathcal{P}_{\nu n-1}$ $$X_n(\nu,P)-P=L_n(\nu,P)-P+Y_n(\nu,P)$$ and $\varepsilon_n(a_n+ \log n) \to 0$ as $n \to 0$, we have the result from Proposition \[Proposition2.6\] and Proposition \[Proposition2.8\].
Let $\varepsilon >0$ be taken arbitrarily. Then there exists a polynomial $P$ such that $$\label{eq3.16}
\left\|\left(f(x)-P(x)\right)(1+|x|)^{\alpha}\left\{\Phi^{-\frac{3}{4}}(x)
w(x)\left(|x|+\frac{a_n}{n}\right)^{\rho}\right\}^{\nu-\frac{\eta}{2}}\right\|_{L_{\infty}(\mathbb{R})}<\varepsilon.$$ In fact, by (\[eq2.14\]) we see $$\lim_{|x|\to \infty} \left|f(x)w^{\nu-\eta}(x)\right| =0,$$ hence as [@[7] (4.55)] we obtain (\[eq3.16\]). Then we also have $$\label{eq3.17}
\left\|\left(f(x)-P(x)\right)(1+|x|)^{\alpha}\left\{\Phi^{-\frac{3}{4}}(x)
w_{\rho}(x)\right\}^{\nu}\right\|_{L_{\infty}(\mathbb{R})}<C\varepsilon,$$ and from $$\left\|(1+|x|)^{\Delta-\alpha}\left\{\Phi^{-\frac{3}{4}}(x)
w(x)\left(|x|+\frac{a_n}{n}\right)^{\rho}\right\}^{\nu/2}\right\|_{L_{\infty}(\mathbb{R})}< \infty,$$ with (\[eq3.16\]), we have $$\label{eq3.18}
\left\|\left(f(x)-P(x)\right)(1+|x|)^{\Delta}\left\{\Phi^{-\frac{3}{4}}(x)
w(x)\left(|x|+\frac{a_n}{n}\right)^{\rho}\right\}^{\nu}\right\|_{L_{\infty}(\mathbb{R})}<C\varepsilon.$$ Now, we see $$\left|L_n(\nu,f)-f\right| \le \left|X_n(\nu,f-P)\right|+\left|X_n(\nu,P)-P\right|
+ \left|P-f\right|+ \left|Y_n(\nu,f)\right|.$$ From Proposition \[Proposition 2.5\] with $C_{f-P}=C\varepsilon$ and (\[eq3.17\]), we know $$\label{eq3.19}
\left\|(1+|x|)^{-\Delta}\left\{\Phi^{\frac{3}{4}}(x)w(x)\left(|x|+\frac{a_n}{n}\right)^{\rho}\right\}^{\nu}
X_n(\nu,f-P)\right\|_{L_p(\mathbb{R})}
\le C\varepsilon.$$ Then, using (\[eq3.18\]), (\[eq3.19\]), Proposition \[Proposition2.9\] and Proposition \[Proposition2.6\], we conclude $$\begin{aligned}
\left\|(1+|x|)^{-\Delta}\left\{\Phi^{\frac{3}{4}}(x)w(x)\left(|x|+\frac{a_n}{n}\right)^{\rho}\right\}^{\nu}
\left(L_n(\nu,f)-f\right)\right\|_{L_p(\mathbb{R})}
\le C\varepsilon.\end{aligned}$$ Hence, we have the result.
If we split $\left|L_n(l,\nu,f)-f\right|$ as follows: $$\left|L_n(l,\nu,f)-f\right| \le \left|L_n(\nu,f)-f\right| + \left|Z_n(l,\nu,f)\right|,$$ then the result is proved from Theorem \[Theorem2.11\] and Proposition \[Proposition2.7\].
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---
abstract: 'We analyzed the spatial distribution of $28500$ photometrically selected galaxies with magnitude $23.5<{\cal R}_{\rm AB}<25.5$ and redshift $1.4<z<3.5$ in 21 fields with a total area of $0.81$ square degrees. The galaxies were divided into three subsamples, with mean redshifts $\bar z=1.7$, $2.2$, $2.9$, according to the $U_nG{\cal R}$ selection criteria of Adelberger et al. (2004) and Steidel et al. (2003). Combining the galaxies’ measured angular clustering with redshift distributions inferred from $1600$ spectroscopic redshifts, we find comoving correlation lengths at the three redshifts of $r_0 = 4.5\pm 0.6$, $4.2\pm 0.5$, and $4.0\pm 0.6 h^{-1}$ Mpc, respectively, and infer a roughly constant correlation function slope of $\gamma=1.6\pm 0.1$. We derive similar numbers from the $1600$ object spectroscopic sample itself with a new statistic, $K$, that is insensitive to many possible systematics. Galaxies that are bright in ${\cal R}$ ($\lambda_{\rm rest}\sim 1500$–$2500$Å) cluster more strongly than fainter galaxies at $z=2.9$ and $z=2.2$ but not, apparently, at $z=1.7$. Comparison to a numerical simulation that is consistent with recent WMAP observations suggests that galaxies in our samples are associated with dark matter halos of mass $10^{11.2}$–$10^{11.8} M_\odot$ ($z=2.9$), $10^{11.8}$–$10^{12.2} M_\odot$ ($z=2.2$), $10^{11.9}$–$10^{12.3} M_\odot$ ($z=1.7$), and that a small fraction of the halos contain more than one galaxy that satisfies our selection criteria. Adding recent observations of galaxy clustering at $z\sim 0$ and $z\sim 1$ to the simulation results, we conclude that the typical object in our samples will evolve into an elliptical galaxy by redshift $z=0$ and will already have an early-type spectrum by redshift $z=1$. We comment briefly on the implied relationship between galaxies in our survey and those selected with other techniques.'
author:
- 'Kurt L. Adelberger'
- 'Charles C. Steidel'
- Max Pettini
- 'Alice E. Shapley'
- 'Naveen A. Reddy & Dawn K. Erb'
title: 'THE SPATIAL CLUSTERING OF STAR-FORMING GALAXIES AT REDSHIFTS $1.4\simlt z\simlt 3.5$'
---
INTRODUCTION {#sec:intro}
============
Early investigators studied the spatial distribution of galaxies because they hoped to learn about the structure of the universe on the largest scales. Their influential work was superseded, in the end, by its competition. Problems began with the demonstration that galaxies contained only a small fraction of the matter in the universe. Galaxy formation remained too poorly understood to quell doubts about how faithfully galaxies traced underlying distribution of dark matter. Other observations improved—gravitational lensing, peculiar velocities, intergalactic absorption lines, and so on—and seemed easier to relate to matter fluctuations. Computers became fast enough to predict the evolution of the large-scale matter distribution from the the initial conditions that microwave-background missions were measuring with increasing precision. As it became clear that the simulations and observations agreed remarkably well, most researchers concluded that the large-scale structure of the universe could be understood completely as the product of gravitational instability amplifying small inflationary perturbations. Galaxies, once believed to be the primary constituent of the universe, came to be seen as small test particles swept into ever larger structures by converging dark matter flows.
The spatial distribution of galaxies remains interesting because it can teach us about galaxy formation. Galaxy formation must be closely related to larger process of gravitational structure formation, since the formation of a galaxy begins with gas streaming into a massive potential well and ends with stars drifting in the cosmic flow. Lessons from 20 years of numerical investigations into structure formation should therefore carry over to the analysis of galaxy clustering. One example is the known positive correlation between clustering strength and mass for virialized dark matter halos. Since galaxies reside in dark matter halos, their clustering strength provides an indication of the mass of the halos that contain them. The resulting mass estimate depends on the assumptions that the microwave background and other observations have given us reliable estimates of cosmological parameters and of the initial matter power-spectrum, that numerical simulations can correctly trace the evolution of the matter distribution, at least for moderate densities, and that no process can significantly separate baryons from dark matter on Mpc scales—assumptions that are at least as plausible as those behind competing techniques for mass estimation. Another example is the evolution in the clustering strength of a population of galaxies once it has formed. This is driven solely by gravity and is easy to predict from numerical simulations. Comparing the clustering of (say) galaxies in the local universe and those at high redshift can therefore suggest or rule out possible links between them. Few other methods are as useful for unifying into evolutionary sequences the galaxy populations we observe at different look-back times. An excellent review of the history of these techniques has been written by Giavalisco (2002; pp 620-624).
This paper has two aims. The first is to present measurements of the clustering of UV-selected star-forming galaxies in a redshift range $1.4<z<3.5$ that is only partially explored. Section \[sec:data\] describes the way we obtained our data, § \[sec:methods\] describes and justifies the techniques we used to estimate the spatial clustering in our galaxy samples, and § \[sec:results\] presents our estimates of the galaxy correlation function at redshifts $z\sim 1.7$, $z\sim 2.2$, and $z\sim 2.9$. The survey analyzed here is several times larger than its predecessors; the surveyed area of $0.81$ square degrees is roughly 700 times larger than the area analyzed by Arnouts et al. (2002) and 4 times larger than the areas analyzed by Giavalisco & Dickinson (2001) and Ouchi et al. (2001). The second aim is to discuss what our measurements imply about the galaxies and their descendants. In § \[sec:correspondence\] we show that the galaxies’ correlation functions are indistinguishable from those of virialized dark matter halos with mass $M\sim 10^{12} M_\odot$. In § \[sec:evolution\] we show that the galaxies, dragged by gravity for billions of years, caught in the press of structure formation, would by redshift $z=0$ have a correlation function that is indistinguishable from that of the elliptical galaxies that surround us. Our results are summarized in § \[sec:summary\].
DATA {#sec:data}
====
Observed {#sec:observeddata}
--------
The data we analyzed were drawn from our ongoing surveys of high-redshift star-forming galaxies. A brief description of the surveys follows; see Steidel et al. (2003, 2004) for further details. Deep, multi-hour $U_nG{\cal R}$ images of 21 fields scattered around the sky were obtained with various 4m-class telescopes (table \[tab:fields\]). Tens of thousands of objects were visible in these images. For the analysis of this paper we ignored all but the subset ($\sim$ 20%) with AB magnitude $$23.5\leq {\cal R}\leq 25.5
\label{eq:maglimits}$$ and AB colors satisfying the “LBG” selection criteria of Steidel et al. (2003), $$\begin{aligned}
U_n-G &\geq& G-{\cal R}+1.0\nonumber\\
G-{\cal R}&\leq& 1.2,
\label{eq:lbg}\end{aligned}$$ the “BX” selection criteria of Adelberger et al. (2004), $$\begin{aligned}
G-{\cal R}&\geq& -0.2\nonumber\\
U_n-G &\geq& G-{\cal R}+0.2\nonumber\\
G-{\cal R}&\leq& 0.2(U_n-G)+0.4\nonumber\\
U_n-G &<& G-{\cal R}+1.0,
\label{eq:bx}\end{aligned}$$ or the “BM” selection criteria of Adelberger et al. (2004), $$\begin{aligned}
G-{\cal R}&\geq& -0.2\nonumber\\
U_n-G &\geq& G-{\cal R}-0.1\nonumber\\
G-{\cal R}&\leq& 0.2(U_n-G)+0.4\nonumber\\
U_n-G &<& G-{\cal R}+0.2.
\label{eq:bm}\end{aligned}$$ In this range of ${\cal R}$ magnitudes, the colors are characteristic of galaxies at $1.4\simlt z\simlt 3.3$. (The restriction to ${\cal R}>23.5$ helps eliminate most interlopers; see Adelberger et al. 2004 and Steidel et al. 2004.) By obtaining spectra of more than $1600$ galaxies with these colors, Steidel et al. (2003, 2004) established that the mean redshift and $\pm 1\sigma$ range of galaxies in the three samples is $$\bar z\pm \sigma_z =\cases{ 2.94\pm 0.30 & LBG \cr
2.24\pm 0.37 & BX \cr
1.69\pm 0.36 & BM. \cr}
\label{eq:meanz}$$ (These values exclude galaxies with ${\cal R}<23.5$ or ${\cal R}>25.5$ as well as the handful of low-redshift “interloper” galaxies with $z<0.8$.) Redshift histograms are shown in figure \[fig:zhistos\]. We will use the term “photometric candidates” to describe the objects with $23.5\leq {\cal R}\leq 25.5$ whose colors satisfy one set of the selection criteria presented above, and the term “spectroscopic sample” to describe the subset of photometric candidates that had a spectroscopic redshift measured by Steidel et al. (2003) or Steidel et al. (2004). Although the spectroscopic sample is sizeable, it contains only a small fraction ($\simlt 10$%) of the photometric candidates (see figure \[fig:specfrac\] and table \[tab:fields\]).
[lrrrrrrr]{} 3c324& $6.6\times 6.6$ & 11/49 & 0/166 & 0/126 & 0.0035& 0.0034& 0.0042\
B20902& $6.3\times 6.5$ & 31/65 & 1/207 & 0/189 & 0.0036& 0.0035& 0.0043\
CDFa& $8.7\times 8.9$ & 34/99 & 0/336 & 0/280 & 0.0029& 0.0028& 0.0036\
CDFb& $9.0\times 9.1$ & 20/120 & 0/316 & 0/273 & 0.0028& 0.0028& 0.0035\
DSF2237a& $9.0\times 9.1$ & 39/100 & 1/367 & 0/328 & 0.0028& 0.0028& 0.0035\
DSF2237b& $8.9\times 9.1$ & 44/161 & 1/516 & 0/309 & 0.0028& 0.0028& 0.0035\
HDF& $10.4\times 14.4$& 54/187 & 128/735 & 37/587 & 0.0022& 0.0022& 0.0028\
Q0201& $8.6\times 8.7$ & 18/90 & 4/339 & 0/285 & 0.0029& 0.0029& 0.0036\
Q0256& $8.5\times 8.4$ & 45/126 & 1/346 & 0/243 & 0.0030& 0.0029& 0.0037\
Q0302& $15.6\times 15.7$& 46/824 & 0/1778 & 0/749 & 0.0018& 0.0018& 0.0023\
Q0933& $8.9\times 9.2$ & 63/192 & 0/435 & 0/273 & 0.0028& 0.0028& 0.0035\
Q1307& $16.3\times 16.0$& 16/483 & 47/1352 & 9/936 & 0.0017& 0.0017& 0.0023\
Q1422& $7.3\times 15.6$ & 96/253 & 1/728 & 0/491 & 0.0024& 0.0024& 0.0030\
Q1623& $12.0\times 22.3$& 6/462 & 189/1220 & 2/847 & 0.0016& 0.0016& 0.0022\
Q1700& $15.4\times 15.4$& 15/406 & 62/1456 & 1/948 & 0.0018& 0.0018& 0.0024\
Q2233& $9.2\times 9.2$ & 44/76 & 1/267 & 0/181 & 0.0028& 0.0028& 0.0035\
Q2343& $22.8\times 11.5$& 10/385 & 148/938 & 8/541 & 0.0016& 0.0016& 0.0022\
Q2346& $16.5\times 17.1$& 1/362 & 34/1142 & 1/754 & 0.0016& 0.0017& 0.0022\
SSA22a& $8.6\times 8.9$ & 42/151 & 10/360 & 1/253 & 0.0029& 0.0028& 0.0036\
SSA22b& $8.6\times 8.9$ & 29/73 & 5/281 & 1/308 & 0.0029& 0.0028& 0.0036\
Westphal& $14.9\times 15.1$& 172/270 & 43/724 & 20/632 & 0.0018& 0.0018& 0.0024\
Total& 2907 & 836/4934& 676/14009& 80/9533& 0.0021& 0.0021& 0.0029\
\[tab:fields\]
All redshifts were measured with the Low-Resolution Imaging Spectrograph (LRIS; Oke et al. 1995) on the Keck telescopes. The number of redshifts in each field was determined by the number of clear nights that were allocated. Photometric candidates were selected for spectroscopy more-or-less at random, but in one way the selection was far from random: spectroscopic objects in each field were constrained to fit together in a non-interfering way on one of a small number of multislit masks. This introduced artificial angular clustering to the spectroscopic samples, particularly in fields where our image’s size significantly exceeded the spectrograph’s $\sim 8'$ field-of-view. In some cases the artificial angular clustering was increased by our desire to obtain particularly dense spectroscopic sampling in some parts of an image, e.g., near a background QSO.
Table \[tab:fields\] lists the number of BM, BX, and LBG photometric candidates and spectroscopic galaxies in each field. The field-to-field variations in the number of photometric candidates per square arcminute were caused primarily by differences in exposure times, seeing, sky brightness, telescope plus instrumental throughput, and so on, from one run to the next. Recall that we used many different telescopes and cameras during our imaging survey. The expected variations in intrinsic surface density (also shown in the table, and calculated for galaxies with bias $b=1$ as described in § \[sec:angularmethods\] below) are significantly smaller.
We estimated the angular correlation functions from the lists of photometric candidates. Inferring a comoving correlation length $r_0$ from the measured angular clustering required an estimate of the objects’ redshift distribution. For this we took the measured redshift distributions of the spectroscopic samples. Since the spectroscopic samples are large—several hundred for the BX and LBG criteria, nearly 100 for BM—random fluctuations are unlikely to have given redshift distributions to them that are significantly different from those of the parent photometric samples. We were able to measure a redshift for only $\sim 80$% of the objects we observed spectroscopically, however, and it is therefore possible that various systematics (e.g., difficulties measuring spectroscopic redshifts for galaxies in certain redshift ranges) could have caused the spectroscopic and photometric samples to have somewhat different redshift distributions. Repeated observations of a subset of the initial spectroscopic failures show that these objects have the same redshift distribution as the initial successes, implying that any systematics are not severe.
Simulated {#sec:simulated_data}
---------
To help us interpret our observations, we refered at times the GIF-LCDM numerical simulation of structure formation in a cosmology with $\Omega_M=0.3$, $\Omega_\Lambda=0.7$, $h=0.7$, $\Gamma=0.21$, and $\sigma_8=0.9$. This gravity-only simulation contained $256^3$ particles with mass $1.4\times 10^{10} h^{-1} M_\odot$ in a periodic cube of comoving side-length $141.3h^{-1}$ Mpc, used a softening length of $20 h^{-1}$ comoving kpc, and was released publicly, along with its halo catalogs, by Frenk et al. astro-ph/0007362. Further details can be found in Jenkins et al. (1998) and Kauffmann et al. (1999). Since the GIF-LCDM cosmology is consistent with the Wilkinson Microwave-Anisotropy Probe results (Spergel et al. 2003), and since modeling the gravitational growth of perturbations on large ($\sim$ Mpc) scales is not numerically challenging, we will assume that the growth of structure found in this simulation closely mirrors the growth of structure in the actual universe.
For our purposes the most interesting aspect of the simulation is the spatial distribution of virialized “halos”, or overdensities with $\delta\rho/\rho\sim 200$, since these deep potential wells are the sites where galaxies can form from cooling gas. The public halo catalogs were created, by the GIF team, by running a halo-finding algorithm at various time steps in the simulation. We will say that all the halos identified by the algorithm at time-step $t=156$ (say) had $t=156$ as their time of identification. Correlation functions for halos at the time of identification were calculated directly from the public halo catalogs. In subsequent time-steps these halos were progressively displaced by gravity. Some grew; others were destroyed as they were subsumed into larger structures. Any galaxies within the halos would be likely to survive intact, however, and it is interesting to trace the expected evolution in their correlation function over time. To do this, we assumed that the galaxies in a halo would be displaced by gravity by the same amount, and in the same direction, as the halo’s most bound particle. If $\{p_i\}$ denotes the set of particles that were the most bound particle in a halo at $t_{\rm earlier}$, we assumed that the correlation function at time $t_{\rm later}$ of the galaxies that lay within halos identified at time $t_{\rm earlier}$ would be roughly equal to the correlation function of particles $\{p_i\}$ at time $t_{\rm later}$. At spatial separations that are large compared to the typical halo radius, the expected evolution of the galaxy correlation function is insensitive to the details of this procedure. These are the only spatial separations we will consider. Our focus on large spatial scales also justifies our ignoring the possibility of galaxy mergers. Although mergers can strongly affect the correlation function on small scales, on large scales the effect is more subtle. It can be understood as follows. At time $t_{\rm later}$, the galaxies identified at time $t_{\rm earlier}$ will be found in halos with a range of masses, and their large-scale correlation function will be a weighted average of the correlation functions of the halos. Since the weighting depends on typical number of descendant galaxies in halos of each mass, and since halos with different masses have different correlation functions, the large-scale correlation function of descendant galaxies will be altered by mergers if the merger frequency depends on halo mass. In practice, however, the difference in correlation functions between the more massive and less massive halos that host the descendants is not enormous, and as a result the merger frequency would have to be an implausibly strong function of halo mass to alter the descendant correlation functions on large scales in a significant way.
METHODS {#sec:methods}
=======
Two approaches will be used to estimate the clustering strength. The first approach, which is standard, relies almost exclusively on the angular positions of the galaxies. The second relies almost exclusively on the galaxies’ measured redshifts. These approaches exploit different aspects of our data and are subject to different systematics. The level of agreement between them provides an important test of our conclusions’ robustness. This section describes and justifies the two approaches. Readers interested primarily in our scientific results may wish to skip ahead to § \[sec:results\].
Angular clustering {#sec:angularmethods}
------------------
The observed clustering of galaxies on the plane of the sky is related to the galaxies’ three-dimensional correlation function in a straightforward way. Let $z$ denote a galaxy’s redshift and ${\bf \Theta}$ denote its angular position. ${\bf \Theta}$ is written in bold face because two numbers (e.g., right ascension and declination) are required to specify the galaxy’s angular position. If $P(z_1{\bf \Theta_1}|z_2{\bf \Theta_2})$ is the probability that a galaxy at known position $(z_2,{\bf \Theta_2})$ has a neighbor at position $(z_1,{\bf \Theta_1})$, then elementary identities show that the probability that a galaxy at angular position ${\bf\Theta_2}$ will have a neighbor at angular position ${\bf\Theta_1}$ is $$P({\bf \Theta_1}|{\bf \Theta_2}) = \int dz_1 dz_2 P(z_2|{\bf \Theta_2}) P(z_1{\bf \Theta_1}|z_2{\bf \Theta_2}).
\label{eq:limber0}$$ Observations indicate that the reduced correlation function is well approximated by an isotropic power law, $$P(z_1{\bf \Theta_1}|z_2{\bf \Theta_2}) \simeq P(z_1{\bf \Theta_1})\bigl[1+(r_{12}/r_0)^{-\gamma}\bigr],$$ where $r_0$ and $\gamma$ parametrize the shape of the power law and $r_{12}$ is the distance between $(z_1,{\bf \Theta_1})$ and $(z_2,{\bf \Theta_2})$. This implies, in the circumstances of interest to us, that the reduced angular correlation function will also be a power law, $$P({\bf \Theta_1}|{\bf \Theta_2})=P({\mathbf \Theta_1})(1+A\theta_{12}^{-\beta})$$ with $\beta\equiv\gamma-1$. If the angular separation $\theta_{12}\equiv |{\bf\Theta_1}-{\bf\Theta_2}|$ between the galaxies is small, $\theta_{12}\ll 1$, and if the comoving correlation length $r_0$ does not change significantly from the front to the back of the survey, then $$\begin{aligned}
A &=& r_0^\gamma B\biggl(\frac{1}{2},\frac{\gamma-1}{2}\biggr) \int_0^{\infty}dz\,N^2(z)f^{1-\gamma}g^{-1}\nonumber\\
& &\quad\quad\quad\quad\quad\quad\quad\quad\quad \Bigg/\Biggl[\int_0^{\infty}dz'\,N(z')\Biggr]^2
\label{eq:Alimber}\end{aligned}$$ (see, e.g., Totsuji & Kihara 1969) where $N(z)$ is the survey selection function, $B$ is the beta function in the convention of Press et al. (1992), $g(z)\equiv c/H(z)$ is the change in comoving distance with redshift, $f(z)\equiv (1+z)D_A(z)$ is the change in comoving distance with angle, and $D_A(z)$ is the angular diameter distance. This follows from the relationship $\int_{-\infty}^{\infty}dz\, [r_0^2/(R^2+z^2)]^{\gamma/2} = r_0^\gamma R^{1-\gamma} B(1/2,(\gamma-1)/2)$.
Our first approach to estimating the three dimensional clustering strength will be to measure the parameters $A$ and $\beta$ of the reduced angular correlation function $\omega(\theta)\equiv A\theta^{-\beta}$, then infer values for $r_0$ and $\gamma$ using the relationships above. Our estimates of $\omega(\theta)$ in different angular bins will be based on the Landy-Szalay (1993) estimator $$\omega_{LS}(\theta) \equiv \frac{DD(\theta) - 2DR(\theta) + RR(\theta)}{RR(\theta)}
\label{eq:landyszalay}$$ where $DD(\theta)$ is the observed number of unique galaxy pairs with separation $\theta-\Delta\theta/2<\theta<\theta+\Delta\theta/2$, $DR(\theta)$ is the number of unique pairs with separation in the same range between the observed galaxy catalog and a galaxy catalog with random angular positions, and $RR(\theta)$ is the number of unique pairs in the random catalog with separations in the same range. In practice we reduce the noise in the random pair counts by creating random catalogs with many times more objects than the data catalogs ($n_{\rm rand}/n_{\rm data}\sim 100$), calculating $DR$ and $RR$, then multiplying $DR$ and $RR$ by $(n_{\rm data}-1)/n_{\rm rand}$ and $n_{\rm data}(n_{\rm data}-1)/[n_{\rm rand}(n_{\rm rand}-1)]$, respectively.
### Integral constraint {#sec:integralconstraint}
Unless fluctuations on the size of our typical field-of-view are negligible, the number of detected galaxies in any field will be somewhat higher or lower than in a fair sample of the universe, and the number of galaxies in the field’s ideal random catalog would therefore be lower or higher than the observed number. As a result the values $DR$ and $RR$ that we calculate with our approach will be incorrect to some degree. In a single field this can make the clustering appear stronger or weaker than it truly is, but when many fields are averaged it tends to make the observed clustering appear artificially weak. This can be shown as follows. Assume that the observed mean density in a field differs from the global average by the unknown factor $1+\delta$, i.e., $\rho_{\rm obs} = \bar\rho (1+\delta)$, and let $DR$ and $RR$ be the pair counts calculated from scaling the random catalogs to the observed density. One might guess that the estimator of equation \[eq:landyszalay\] ought to have $DR$ and $RR$ replaced by values corrected to the true mean density, i.e., by $DR/(1+\delta)$ and $RR/(1+\delta)^2$, and indeed Hamilton (1993, §3) has shown that the estimator $$\omega_{\rm ideal} \equiv \frac{DD - 2DR/(1+\delta) + RR/(1+\delta)^2}{RR/(1+\delta)^2}
\label{eq:hamideal}$$ is equal to the true angular correlation function on average, $$\langle \omega_{\rm ideal}(\theta) \rangle = \omega(\theta).$$ This equation does not help us directly, since we do not know $\delta$ and cannot calculate $\omega_{\rm ideal}$, but it does show that the estimator $\omega_{LS}(\theta)$ must be biased: $$\langle \omega_{LS}(\theta) \rangle = \omega(\theta) - \sigma^2\frac{DD}{RR} \simeq \omega(\theta) - \sigma^2
\label{eq:actualestimator}$$ where $\sigma^2\equiv{\rm Var}(\delta)$ and the approximation assumes the weak clustering ($\omega\ll 1$) limit. It is therefore customary to estimate $\omega(\theta)$ by adding a constant ${\cal I}\equiv \omega(\theta)-\omega_{LS}(\theta)$ to the calculated values $\omega_{LS}$. The constant ${\cal I}$ depends on the unknown values of $\delta$ in the observed field or fields and cannot be calculated exactly. If $\sigma^2\ll 1$, so that field-to-field fluctuations are in the linear regime and have a nearly Gaussian distribution, and if our data are drawn from $n$ independent fields with measured pair counts $DD_i$, $DR_i$, and $RR_i$, then the value of ${\cal I}$ appropriate to a given angular bin in our data set will have a variance of $$\begin{aligned}
{\rm Var}({\cal I}) &=& \frac{1}{RR_{\rm tot}^2}\sum_{i=1}^n \Bigl[(4\sigma_i^2+2\sigma_i^4)DD_i^2-8\sigma_i^2DD_iDR_i\nonumber\\
& & \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad+4\sigma^2DR_i^2\Bigr] \\
&\simeq& \frac{1}{RR_{\rm tot}^2}\sum_{i=1}^n 2\sigma_i^4 DD_i^2
\label{eq:var_i}\end{aligned}$$ around its expectation value $$\begin{aligned}
\langle{\cal I}\rangle &=& \frac{1}{RR_{\rm tot}}\sum_{i=1}^n\sigma_i^2DD_i.
\label{eq:exp_i}\end{aligned}$$ Here $RR_{\rm tot}\equiv \sum_{i=1}^n RR_i$ is the sum over all fields of the random pair counts in the chosen angular bin. In practice $\langle{\cal I}\rangle$ and ${\rm Var}({\cal I})$ depend very weakly on which bin is chosen. Below we will take $\langle{\cal I}\rangle$ at $100''$ as our best guess at the correction ${\cal I}$. When it matters we will discuss the effect of the uncertainty in ${\cal I}$.
We use two approaches to estimate the size of the uncertainty $\sigma_i$ in the mean galaxy density of the $i$th field. Since $$\sigma^2 \equiv \frac{1}{\Omega^2}\int_\Omega d\Omega_1\,d\Omega_2\,\omega(\theta_{12}),
\label{eq:integralconstraint}$$ it can be estimated numerically as $$\sigma^2 \simeq \sigma^2_{\rm num}\equiv \frac{\sum_i RR\,\omega(\theta_i)}{\sum_i RR}
\label{eq:rocheeales}$$ if $\omega(\theta_i)$ is known (Infante 1994; Roche & Eales 1999). Unfortunately the iterative approach suggested by equations \[eq:actualestimator\] and \[eq:rocheeales\] can be unstable, at least when the correlation function slope is allowed to vary: a large value of $\omega_{LS}$ will imply a large correction $\sigma^2$, which implies an even larger $\omega_{LS}$ and even larger correction, and so on. The instability is undoubtedly worse for large images, where the estimate of the integral constraint correction for one iteration is completely dominated by the assumed correction from the previous. We were unable to use equation \[eq:rocheeales\] as anything other than a consistency check.
A more robust estimate of $\sigma^2$ follows from theoretical considerations. Since matter fluctuations will still be in the linear regime on the large scales of our observations, the relative variance of mass from one surveyed volume to the next can be estimated from the linear cold-dark matter power-spectrum $P_L(k)$ (Bardeen et al. 1986; we adopt the parameters $\Gamma=0.2$, $\sigma_8=0.9$, $n=1$) with Parseval’s relationship $$\sigma^2_{\rm CDM} = \frac{1}{(2\pi)^3}\int d^3k P_L(|{\mathbf k}|) |W_k^2({\mathbf k})|$$ where $W_k$ is the Fourier transform of a survey volume. The shape of the observed volume in any one of our fields is reasonably approximated in the radial direction by a Gaussian with comoving width (rms) $l_z$ and in the transverse directions by a rectangle with comoving dimensions $l_x\times l_y$. In this case $$W_k = \exp\Bigl[-\frac{k_z^2l_z^2}{2}\Bigr] \frac{\sin(k_xl_x/2)}{k_xl_x/2}\frac{\sin(k_yl_y/2)}{k_yl_y/2}.$$ The implied value of $\sigma^2_{\rm CDM}$ for each sample in each field is shown in table \[tab:fields\]; the values assume the powerspectrum normalization required for the r.m.s. fluctuation as a function of redshift in spheres of comoving radius $8h^{-1}$ Mpc to obey $\sigma_8(z)=\sigma_8(0)D_L(z)$ where $\sigma_8(0)=0.9$ and $D_L(z)$ is the linear growth factor to redshift $z$. The desired corrections $\sigma^2$ are then given by $$\sigma^2\simeq b^2 \sigma^2_{\rm CDM}
\label{eq:ic_cdm}$$ where $b$, the galaxy bias, is calculated from the ratio of galaxy to matter fluctuations in spheres of comoving radius $8h^{-1}$ Mpc: $$b = \frac{\sigma_{8,{\rm g}}}{\sigma_{8}(z)}.
\label{eq:bias_from_sigma}$$ Here the galaxy variance $$\sigma^2_{8,{\rm g}} = \frac{72 (r_0/8 h^{-1}\,\,{\rm Mpc})^\gamma}{(3-\gamma)(4-\gamma)(6-\gamma)2^\gamma}
\label{eq:s2_from_r0}$$ (Peebles 1980 eq. 59.3) can be derived from the fit to the galaxy correlation function. This approach also requires an iterative solution, since the correction $\sigma^2$ to $\omega(\theta)$ depends on $\omega(\theta)$, but the advantage is that the assumed size of large-scale fluctuations is anchored in reality by our requirement that the slope of the correlation function match other observations on very large scales.
### Uncertainties in the selection functions {#sec:selfn_uncertainties}
The discussion so far assumes that we will know the precise shape of the selection function $N(z)$. In fact this is not true, and uncertainty in the true shape of our selection function is a source of error in the derived values of $r_0$.[^1] A larger width for the selection function means that projection effects are stronger, and therefore implies a larger value of $r_0$ for given angular clustering (see equation \[eq:Alimber\]). If the selection function is a Gaussian with mean $\mu$ and standard deviation $\sigma_{\rm sel}$, and if the weak redshift variations of $f$ and $g$ can be ignored, then the constant $A$ in equation \[eq:Alimber\] is proportional to $\sigma_{\rm sel}^{-1}$, and the implied value of $r_0$ is proportional to $\sigma_{\rm sel}^{1/\gamma}$. Measuring $n$ redshifts drawn from this selection function determines $\mu$ to a precision $\sigma_{\rm sel}/n^{1/2}$ and $\sigma_{\rm sel}^2$ to a relative precision $2^{1/2}/(n-1)^{1/2}$. Excluding interlopers with $z<1$, we have measured roughly 800, 700, and 80 redshifts for galaxies in the LBG, BX, and BM samples, and the selection function width is $\sigma_{\rm sel}\sim 0.3$ for each. The relative uncertainty in $\sigma_{\rm sel}$ is therefore approximately $\sim 3$% for the LBG and BX samples and $\sim 9$% for the BM sample, which implies $\sim 2$% uncertainty in $r_0$ for the LBG and BX samples and $\sim 5$% uncertainty in the BM sample.
Variations of the selection function from one field to the next (owing, for example, to differences in the depth of the data or to systematic errors in our photometric zero points) are another source of concern, especially at the redshifts $z\sim 2$ where galaxies’ colors are insensitive to redshift and small color errors mimic large redshift differences. Suppose for simplicity that all fields have the same number of photometric candidates, let the rms width of the selection function in the $i$th field be written $\sigma_i=(1+\epsilon_i)s$, where $s$ is the mean width among all fields, and let the mean redshift of the selection function be written $\mu+s\delta_i$ where $\mu$ is the mean redshift among all fields. Then the rms width of the total selection function $$[{\rm Var}(z)]^{1/2} = s[1+{\rm Var}(\delta)+{\rm Var}(\epsilon)]^{1/2},$$ exceeds the value $s$ that should be used in determining $A$. Since we will (by necessity) use the total selection function in estimating $r_0$, our estimates will be biased high. Systematic errors in our zero points are unlikely to be larger than $\Delta m=0.05$, and variations in photometric depth will at most change our characteristic color uncertainties from $\sim 0.1$ to $\sim 0.2$ magnitudes. The measured variations in galaxy redshift with $U_nG{\cal R}$ color (see, e.g., Adelberger et al. 2004) imply (a) that zero point errors with $\Delta m=0.05$ will shift the mean redshifts of galaxies that satisfy the LBG, BX, and BM selection criteria by $\Delta z \simeq 0.01$, $0.13$, and $0.11$, respectively, and (b) that increasing the photometric uncertainty from $\sigma_m=0.1$ to $\sigma_m=0.2$ will increase the widths of the LBG, BX, and BM selection functions by $\sim 10$, $20$, and $20$%. The upper limits on ${\rm Var}(\delta)$ and ${\rm Var}(\epsilon)$ are therefore $0.002$ ($0.2$) and $0.01$ ($0.04$), respectively, for the LBG (BX,BM) sample. The required reduction in $r_0$ is negligible for the LBG sample but could be as large as $\sim 7$% for the other two. We will account for uncertainties in the selection function by decreasing the best-guess value of $r_0$ for the BM and BX samples by 3.5% and increasing the uncertainty in quadrature by $0.035r_0$.
### Contaminants
As figure \[fig:zhistos\] shows, some fraction of the objects in the BX and BM samples will be low redshift interlopers. We correct for the resulting dilution in the clustering strength by using the full selection function, starting at $z=0$, in our estimate of $r_0$ from equation \[eq:Alimber\]. This is the optimal correction only if the interlopers have the same comoving correlation length $r_0\sim 4h^{-1}$ Mpc (see below) as the galaxies in the primary samples. This should be nearly true, since Budavári et al. (2003) estimate $r_0=4.51\pm 0.19 h^{-1}$ Mpc for the blue star-forming galaxy population at $z\sim 0.2$ from which our interlopers are drawn. In any case, since the correction itself is small—eliminating the tail with $z<1$ from $N(z)$ alters the inferred values of $r_0$ for the BM and BX samples by only $\sim 10$%—errors in it should not have an appreciable effect on our estimates of $r_0$.
Redshift clustering {#sec:redshiftmethod}
-------------------
We face three significant obstacles in trying to estimate the clustering strength from the spectroscopic catalogs.
\(1) The objects in a given field that were selected for spectroscopy were not distributed randomly across the field, but were instead constrained to lie on one of a small number of multislit masks. Since only a small fraction of the galaxies were observed spectroscopically in the typical field, the finite size of the masks coupled with the need to avoid spectroscopic conflicts produced significant artificial clustering in the angular positions of sources in the spectroscopic catalog. The effect was worsened in some fields by our decision to obtain particularly dense spectroscopy near background QSOs. (2) Because galaxies’ $U_nG{\cal R}$ colors change slowly with redshift near $z\sim 2$, the expected redshift distribution $N(z)$ of our BM and BX color-selected samples depends sensitively on the quality of the photometry. The larger color errors from noisy photometry will lead to a broader $N(z)$, while relatively small systematic shifts in the photometric zero points can significantly alter the mean of $N(z)$. Adelberger et al. (2004) and § \[sec:selfn\_uncertainties\] of this paper discuss this point in more detail, but the upshot is that we cannot estimate the selection function $N(z)$ with great precision for the BX and BM samples. (3) Peculiar velocities and redshift uncertainties render imprecise our estimate of each galaxy’s position in the $z$ direction. This limits the accuracy of our estimate of the distance from one galaxy to its neighbors, complicating our efforts to measure the correlation function on small spatial scales.
Effects (1)–(3) are usually compensated with the aid of detailed simulations. Although this approach should work in principle, in practice it is hard for outsiders to evaluate whether the simulations were flawed. The remainder of this section describes the alternate approach that we adopt. It is based on analyzing observable quantities that are not affected by systematics (1)–(3).
The spurious angular clustering signal can be eliminated if we take the angular positions of spectroscopic galaxies as given and estimate the clustering strength solely from their redshifts. Let $Z$ be the comoving distance to redshift $z$, and let ${\mathbf R}\equiv (1+z)D_A(z){\bf\Theta}$ be the transverse comoving separation implied by the angular separation ${\bf\Theta}$ between a galaxy and some reference position, e.g., the center of the observed field. According to elementary probability identities, if we know that one galaxy has position $({\mathbf R_2},z_2)$, then the probability that a second galaxy at transverse position ${\mathbf R_1}$ has radial position $Z_1$ is $$P(Z_1|{\mathbf R_1}{\mathbf R_2}Z_2) = \frac{P(Z_1{\mathbf R_1}|Z_2{\mathbf R_2})}{\int_0^\infty dZ_1' P(Z_1'{\mathbf R_1}|Z_2{\mathbf R_2})}
\label{eq:pzgivenrrz}$$ and the expected distribution of radial separations $Z_{12}\equiv Z_1-Z_2$ for galaxies with transverse separation $R_{12}\equiv |{\mathbf R_1}-{\mathbf R_2}|$ is $$\begin{aligned}
P(Z_{12}|R_{12}) &=& \int_0^{\infty}dZ_2\,P(Z_2|R_{12})P(Z_{12}|Z_2R_{12})\\
&\simeq& \bigl[1+\xi(R_{12},Z_{12})\bigr]\times\nonumber\\
& &\quad \int_0^{\infty} dZ_2\,\frac{P(Z_2)P(Z_2+Z_{12})}{1+P(Z_2)r_0^\gamma R_{12}^{1-\gamma}\beta(\gamma)}
\label{eq:pzgivenr}\end{aligned}$$ where we have used results from the previous section and adopted the shorthand $\beta(\gamma)$ for the beta-function given above, $\beta(\gamma)\equiv B(1/2,(\gamma-1)/2)$. (Equations \[eq:pzgivenrrz\] through \[eq:pzgivenr\] assume that the quantity $(1+z)D_A(z)$ is constant with redshift, an approximation that is valid for the small separations $|Z_{12}|\simlt 40h^{-1}$ comoving Mpc between the galaxy pairs we will use in this analysis.)
Equation \[eq:pzgivenr\] shows that the observable quantity $P(Z_{12}|R_{12})$ is sensitive to the clustering strength but independent of angular variations in the spectroscopic sampling density. Unfortunately the correlation function $\xi$ can be estimated from $P(Z_{12}|R_{12})$ with equation \[eq:pzgivenr\] only if we have a reasonably accurate estimate of the selection function shape $P(Z)$. This can be seen more clearly by Taylor-expanding the integral in equation \[eq:pzgivenr\] around $Z_{12}=0$ and approximating the selection function as a Gaussian with standard deviation $\sigma_{\rm sel}$ that is centered many standard deviations from $z=0$. One finds $$\begin{aligned}
P(Z_{12}|R_{12})&\simeq&\Bigl[1+\xi(R_{12},Z_{12})\Bigr]\times\nonumber\\
& &\quad\quad\Bigl[\frac{A_0}{\sigma_{\rm sel}} + \frac{A_2}{2\sigma_{\rm sel}^3}Z_{12}^2+\frac{A_4}{24\sigma_{\rm sel}^5}Z_{12}^4\Bigr]
\label{eq:pzgivenrapprox}\end{aligned}$$ where $$A_n\equiv\int_{-\infty}^{\infty}du\,e^{-u^2}f_n(u)/g(u,R_{12}),\nonumber$$ $f_0=1$, $f_2=u^2-1$, $f_4=u^4-6u^2+3$, $g\equiv 2\pi+(2\pi)^{1/2}a(R_{12})e^{-u^2/2}/\sigma_{\rm sel}$ and $a\equiv r_0^\gamma R_{12}^{1-\gamma}\beta(\gamma)$. The coefficients $A_n$ all have similar sizes since the integrals are dominated by contributions from $|u|\simlt 1$ where the integrands are of the same order.
In the angular clustering case above, inaccuracies in the adopted width $\sigma_{\rm sel}$ of the selection function affected the inferred amplitude of the correlation function but not its shape. Here they affect both. Moreover errors in $\sigma_{\rm sel}$ are multiplied not by $\xi$, but by $1+\xi$, which implies that they can easily dominate the true clustering signal when $\xi\ll 1$. Equation \[eq:pzgivenrapprox\] shows that one must be careful estimating the strength of redshift clustering when the shape of the selection function is poorly known.
Our solution exploits the fact that $r_0\ll\sigma_{\rm sel}$ for our survey, which implies $Z_{12}\ll\sigma_{\rm sel}$ for all separations where $\xi$ is large enough to measure. As long as $Z_{12}\ll\sigma_{\rm sel}$, the terms proportional to $Z_{12}^2$ and $Z_{12}^4$ can be neglected in equation \[eq:pzgivenrapprox\], and $P(Z_{12}|R_{12})$ will be very nearly equal to $C(R_{12})[1+\xi(R_{12},Z_{12})]$, with $C$ a function that does not depend on $Z_{12}$. The function $C$ does depend on the unknown selection function, but it can be eliminated by taking ratios of pair counts in a manner we discuss below. Ratios of $P(Z_{12}|R_{12})$ at fixed $R_{12}$ and different $Z_{12}$ will therefore be the basis of our estimate of the clustering strength in the spectroscopic sample; they are nearly immune to systematics from the irregular spectroscopic sampling and from the unknown selection function shape.
The final complication is the significant uncertainty $\sigma_Z$ in each object’s radial position $Z$ from peculiar velocities and redshift uncertainties. This uncertainty can be treated in various ways. We will follow the standard approach and estimate the value of the correlation function only within bins whose radial size $\Delta Z$ is large compared to $\sigma_Z$.
We are now ready to present the estimator that we adopt. Letting $N(a_1,a_2,R)$ denote the observed number of galaxy pairs with transverse separation $R$ and redshift separation $a_1\leq |Z_{12}|<a_2$, the discussion of the preceding paragraphs shows that the expected total number of pairs with radial separation $a_1\leq |Z_{12}|<a_2$ (and any transverse separation) is $$\langle N_{\rm tot}(a_1,a_2)\rangle = 2 (a_2-a_1) \sum_{i>j}^{\rm pairs} C(R_{ij}) [1+\bar\xi_{a_1,a_2}(R_{ij})]
\label{eq:expntot}$$ where $$\bar\xi_{a_1,a_2}(R_{ij}) \equiv \frac{1}{a_2-a_1}\int_{a_1}^{a_2} dZ\, \xi(R_{ij},Z).$$ As long as $N_{\rm tot}$ is large enough that $$\Biggl\langle\frac{N_{\rm tot}(b_1,b_2)}{N_{\rm tot}(a_1,a_2)+N_{\rm tot}(b_1,b_2)}\Biggr\rangle \simeq \frac{\langle N_{\rm tot}(b_1,b_2)\rangle}{\langle N_{\rm tot}(a_1,a_2)+N_{\rm tot}(b_1,b_2)\rangle},$$ the ratio of pair counts $$K_{a_1,a_2}^{b_1,b_2} \equiv \frac{N_{\rm tot}(b_1,b_2)}{N_{\rm tot}(a_1,a_2)+N_{\rm tot}(b_1,b_2)}
\label{eq:def_k}$$ will have expectation value $$\begin{aligned}
\label{eq:exp_k}
\langle K_{a_1,a_2}^{b_1,b_2}\rangle&\simeq&\frac{N_{\rm exp}(b_1,b_2)}{N_{\rm exp}(a_1,a_2)+N_{\rm exp}(b_1,b_2)}\\
&\simeq&\frac{N'_{\rm exp}(b_1,b_2)}{N'_{\rm exp}(a_1,a_2)+N'_{\rm exp}(b_1,b_2)}
\label{eq:exp_kb}\end{aligned}$$ regardless of angular selection effects, of uncertainties in the selection function,[^2] of peculiar velocities, and of redshift measurement errors, provided $a_2\ll\sigma_{\rm sel}$, $b_2\ll\sigma_{\rm sel}$, $a_2-a_1\gg \sigma_Z$, $b_2-b_1\gg \sigma_Z$, the selection function does not have strong features on scales smaller than $\sigma_{\rm sel}$, and $(1+z)D_A(z)$ varies slowly with $z$. Here $N_{\rm exp}(a_1,a_2)\equiv\langle N_{\rm tot}(a_1,a_2)\rangle$ is given by equation \[eq:expntot\] and $$N'_{\rm exp}(a_1,a_2) = 2 (a_2-a_1) \sum_{i>j}^{\rm pairs} [1+\bar\xi_{a_1,a_2}(R_{ij})].\nonumber$$ The second approximate equality in equation \[eq:exp\_kb\] exploits the fact that $C(R_{ij})$ is a very weak function of $R_{ij}$ in realistic situations. We estimate the correlation function from the spectroscopic sample by finding the parameters required to match the observed ratio $K_{a_1,a_2}^{b_1,b_2}$. In principle $K_{a_1,a_2}^{b_1,b_2}$ could be calculated separately for pairs in different bins of transverse separation $R_{ij}$, producing an estimate of the function $K_{a_1,a_2}^{b_1,b_2}(R)$ and allowing one to estimate both $r_0$ and $\gamma$ from the data. In practice a much larger sample is needed to fit for both $r_0$ and $\gamma$, so we hold $\gamma$ fixed and estimate $r_0$ only. Fortunately, as we will see, the best fit value of $r_0$ hardly changes as $\gamma$ is varied across the range allowed by the galaxies’ angular clustering.
The dependence of this estimator on the clustering strength is easy to understand intuitively. If the galaxies were unclustered ($\xi(r)=0$), we would observe the same number of pairs at every separation and $K_{a_1,a_2}^{b_1,b_2}$ would be equal, on average, to the ratio of the bin sizes $\eta\equiv(b_2-b_1)/(b_2-b_1+a_2-a_1)$. Correlation functions that peak near $r=0$ will produce more pairs in bins at smaller separations, driving $K_{a_1,a_2}^{b_1,b_2}$ away from $\eta$. The difference between $K_{a_1,a_2}^{b_1,b_2}$ and $\eta$ is sensitive to the strength of the clustering, and therefore can be used to estimate it. Adelberger (2005) uses Monte Carlo simulations to analyze the behavior of $K_{a_1,a_2}^{b_1,b_2}$ in more detail.
RESULTS {#sec:results}
=======
Angular {#sec:angularresults}
-------
Figure \[fig:lsraw\] shows the raw (integral-constraint correction ${\cal I}=0$) values of the Landy-Szalay estimator $\omega_{LS}$ (equation \[eq:landyszalay\]) as a function of angular separation for galaxies in the three samples. We limited these data, and our subsequent fits, to angular separations $\theta<200''$, since at larger scales the weak angular-clustering signal could be swamped by various low-level systematics. The uncertainty $\sigma_i$ in each bin was taken to be the larger of $(DD_i)^{1/2}/(RR_i)$ (Peebles 1980, §48) and the observed standard deviation of the mean of $\omega_{LS}(\theta_i)$ among the different fields in the survey. Typically the two were comparable. Numerical $\chi^2$ minimization produced the power-law fits shown with dashed lines. The correlation function parameters implied by the LBG fit, $r_0=3.35\pm 0.20 h^{-1}$ comoving Mpc, $\gamma=1.74\pm 0.1$, agree well with the estimates of Giavalisco & Dickinson (2001) which also assumed ${\cal I}=0$. It is clear, however, that these parameters cannot be correct. Substituting them into equations \[eq:s2\_from\_r0\], \[eq:bias\_from\_sigma\], and \[eq:ic\_cdm\] shows that a significant correction ${\cal I}$ should have been applied to account for fluctuations on scales larger than the field-of-view. (Porciani & Giavalisco 2002 reached a similar conclusion, and derived a result for LBGs that agrees well with the integral-constraint-corrected result we present below.)
Figure \[fig:wthet\_converg\] shows how our best-fit estimates of $r_0$ and $\gamma$ change as the correction ${\cal I}$ is applied. In our first iteration, described above, we assumed ${\cal I}=0$ and calculated the correlation function $\omega_1(\theta)$. For the second iteration we assumed the value of ${\cal I}$ implied by $\omega_1$ (equations \[eq:exp\_i\], \[eq:s2\_from\_r0\], \[eq:bias\_from\_sigma\], and \[eq:ic\_cdm\]) and estimated $\omega_2(\theta)$. For the third iteration we calculated ${\cal I}$ from $\omega_2(\theta)$. The process continued in this way until convergence. It settled on the same final parameters if we initially assumed a value for ${\cal I}$ that was too large. As figure \[fig:wthet\_converg\] shows, the applied integral constraint corrections were comparable for each of the three samples. This is because the increase in ${\cal I}$ implied by the longer correlation lengths at lower redshifts happened to be cancelled by a decrease in ${\cal I}$ that resulted from the lower-redshift samples’ greater comoving depths.
To check the plausibility of our adopted values for ${\cal I}$, we inserted into equation \[eq:rocheeales\] the best power-law fits to $\omega(\theta)$ from each sample’s final iteration. The equation returned $0.008$, $0.006$, and $0.009$ as the empirical estimates of ${\cal I}$ for BM, BX, and LBG samples. These values differ somewhat from the ones we adopt (figure \[fig:wthet\_converg\]), because the empirical and CDM approaches (see \[sec:integralconstraint\]) make different assumptions about the behavior of $\omega(\theta)$ on the scales $\theta>200''$ where we cannot measure it, but they are consistent within their large $1\sigma$ uncertainties and small changes ($<1\sigma$) to the best-fit values of $\gamma$ would make them agree perfectly. Readers may also be reassured to recall that our estimates of $r_0$ and $\gamma$ agree well with those of Porciani & Giavalisco (2002), who corrected for the integral constraint in a completely different way.
We estimated the random uncertainty in $r_0$ and $\gamma$ in two ways. First, we analyzed many alternate realizations of our $\omega(\theta)$ measurements that were generated under the assumption that the uncertainties were uncorrelated. To create a single alternate realization, we added to each measured value $\omega(\theta_i)$ a Gaussian random deviate with standard deviation equal to its uncertainty $\sigma_i$. After creating numerous alternate realizations, we calculated and tabulated the values of $r_0$ and $\gamma$ implied by each. Our $1\sigma$ confidence interval on $r_0$ was defined as the range that contained 68.3% of the measured values of $r_0$ among the alternate data sets. The $\gamma$ confidence interval is defined in the same way. We found $r_0= 4.0\pm 0.2 h^{-1}$ Mpc, $\gamma=1.57 \pm 0.07$ (LBG), $r_0= 4.3\pm 0.2 h^{-1}$ Mpc, $\gamma=1.59 \pm 0.04$ (BX), and $r_0= 4.7\pm 0.2 h^{-1}$ Mpc, $\gamma=1.55 \pm 0.06$ (BM). These numbers assume uncorrelated error bars and neglect the uncertainty in our selection functions. The uncertainty in ${\cal I}$ is also neglected, since each alternate realization had the same integral constraint correction.
Our second approach was to extract random subcatalogs from our full galaxy catalog, estimate $r_0$ and $\gamma$ for each with the iterative solution for $\omega(\theta)$ described above, measure how the r.m.s. dispersion in best-fit parameter values depended on the number of sources in the subcatalog, and extrapolate to the full catalog size. In fact we created our random subcatalogs in pairs, with both subcatalogs in a pair containing a random fraction $f\leq 0.5$ of the sources in the full catalog and no sources in common between them, and estimated the uncertainty in $r_0$ at a given value of $f$ as $2^{-1/2}$ times the r.m.s. difference in $r_0$ among pair members. This prevented us from underestimating the random uncertainty in $r_0$ as $f\to 0.5$, when random subcatalogs could otherwise contain nearly the same galaxies. With this approach we estimate $r_0= 4.0\pm 0.5 h^{-1}$ Mpc, $\gamma=1.57 \pm 0.12$ (LBG), $r_0= 4.3\pm 0.3 h^{-1}$ Mpc, $\gamma=1.59 \pm 0.05$ (BX), and $r_0= 4.7\pm 0.5 h^{-1}$ Mpc, $\gamma=1.55 \pm 0.07$ (BM). These numbers neglect the uncertainty in our selection function and do not fully account for the uncertainty in ${\cal I}$. They do not assume uncorrelated error bars, however, and we will therefore assume that they are more accurate than the numbers from the preceding paragraph.
The uncertainty in ${\cal I}$ is not negligible. According to equation \[eq:var\_i\] the $1\sigma$ uncertainty in ${\cal I}$ is $\sim 35$–$50$% as large as ${\cal I}$ itself for our 21 fields. As ${\cal I}$ varies over its $1\sigma$ allowed range, the best-fit parameters $r_0$ and $\gamma$ change by roughly as much as the uncertainties quoted above. Adding these changes in quadrature to the random uncertainties above, and making the minor corrections for the selection function uncertainties discussed in § \[sec:angularmethods\], we arrive at the following estimates: $r_0= 4.0\pm 0.6 h^{-1}$ Mpc, $\gamma=1.57 \pm 0.14$ (LBG), $r_0= 4.2\pm 0.5 h^{-1}$ Mpc, $\gamma=1.59 \pm 0.08$ (BX), and $r_0= 4.5\pm 0.6 h^{-1}$ Mpc, $\gamma=1.55 \pm 0.10$ (BM).
Other investigators (e.g., Giavalisco & Dickinson 2001; Foucaud et al. 2003) have claimed that at redshift $z\sim 3$ bright galaxies cluster more strongly than faint galaxies. Our data support this conclusion. Figure \[fig:clust\_seg\] shows that in the BX and LBG samples the correlation lengths of galaxies with $23.5<{\cal R}<24.75$ ($r_0\sim 5.0h^{-1}$ Mpc) exceed those of galaxies with $24.75<{\cal R}<25.5$ ($r_0\sim 3.7h^{-1}$ Mpc) by a significant amount. If we split the BX and LBG samples into two halves at random, rather than by apparent magnitude, the difference in correlation lengths between the two halves is this large only about $5$% (BX) to $24$% (LBG) of the time. The situation is less clear for the BM sample at $z\sim 1.7$, where the uncertainties are larger owing to the poor determination of $N(z)$ from the small number of measured redshifts, but the data do not seem to suggest stronger clustering for brighter galaxies. On the one hand it makes sense that UV-brightness should become less associated with strong clustering as redshift decreases, since UV-bright galaxies are known to be weakly clustered at $z=0$ and $z=1$. On the other, the overall clustering of the BM sample is still quite strong, stronger than one expects for typical collapsed objects at $z\sim 1.7$ (see below), and so it seems that the numerous objects too faint to satisfy our selection criteria must be less clustered than the bright objects in our sample. We will wait for additional spectroscopic observations of BM galaxies before commenting further.
The dependence of clustering strength on luminosity can produce a false impression of a change in $r_0$ with redshift, since lower redshift samples will tend to reach fainter absolute luminosities. Truncating the samples at a fixed absolute luminosity does not seem a good solution to us, however, since the bright end of the UV luminosity function rises rapidly towards higher redshifts (e.g., Adelberger & Steidel 2000) and one would therefore be comparing rare objects at lower redshifts to common objects at higher redshifts. A better approach is to compare galaxy samples of roughly the same comoving number density. Since selection with a constant apparent magnitude limit ${\cal R}<25.5$ happens to produce similar comoving number densities for the three samples (see equations \[eq:nlbg\] and \[eq:nbxbm\] and the related discussion), we will continue to use the constant apparent magnitude limits of equation \[eq:maglimits\] for our samples in the remainder of the paper. Readers should be aware that the reported value of each sample’s correlation length is somewhat arbitrary for this reason. It reflects the characteristics of the sample as defined here, not of the general galaxy population at high redshift.
Redshift
--------
For our estimator of the redshift clustering strength we took $K^{0,20}_{20,40}$, the ratio of the number of galaxy pairs with comoving radial separation $0<|Z|/(h^{-1}{\rm Mpc})<20$ to those with comoving radial separation $0<|Z|/(h^{-1}{\rm Mpc})<40$. Since $20 h^{-1}$ Mpc is significantly larger than the uncertainty in each galaxy’s radial position ($\sigma_Z\simeq 300\,{\rm km}{\rm s}^{-1}(1+z)/H(z)\sim 3h^{-1}$ comoving Mpc), and since $40 h^{-1}$ Mpc is significantly smaller than the selection functions’ widths ($\sigma_{\rm sel}>200 h^{-1}$ comoving Mpc), the expected value of $K^{0,20}_{20,40}$ should be given by equation \[eq:exp\_k\]. We limited our analysis to pairs with transverse separations $\theta_{ij}<300''$, equivalent to $R_{ij}\simlt 5.9h^{-1}$ comoving Mpc at $z=2.5$, to reduce the sensitivity of our results to any deviations of the correlation function from a $\gamma=1.55$ power-law on large scales.
Only the BX and LBG spectroscopic samples were large enough to allow meaningful measurements of $K^{0,20}_{20,40}$. For $\gamma=1.55$, the right-hand side of equation \[eq:exp\_k\] is equal to the observed ratio $K^{0,20}_{20,40}$ when $r_0=4.6h^{-1}$ (LBG) or $4.5h^{-1}$ (BX) comoving Mpc. The values change by roughly $\pm 2$% as $\gamma$ is varied from 1.45 to 1.65. When analyzed with this technique, mock galaxy catalogs from the GIF simulation (§ \[sec:simulated\_data\]) with sizes similar to our observed catalogs show a $\pm 1\sigma$ dispersion in $r_0$ around the true mean of $0.6 h^{-1}$ (LBG) and $0.9 h^{-1}$ (BX) comoving Mpc, so we adopt $4.6\pm 0.6 h^{-1}$ and $4.5\pm 0.9 h^{-1}$ Mpc as the best fit values to $r_0$ for our spectroscopic catalogs. The results do not change significantly if we eliminate pairs with $\theta_{ij}<60''$ from the analysis, showing that we have measured genuine large-scale clustering and not merely the clumping of objects within individual halos.
Figure \[fig:kclust\] presents the result in a more graphical way. We divided our lists of galaxy pairs into bins according to transverse separation $R_{ij}$, then calculated $K^{0,20}_{20,40}$ separately for each bin. Points with error bars show the values we found. The solid lines show the values predicted by the $\gamma=1.55$ correlation function described in the preceding paragraph. The plot shows that the derived correlation function parameters provide a reasonable fit to the clustering of the galaxies in the spectroscopic sample.
Summary
-------
We presented two independent estimates of the correlation function for each of our galaxy samples. The estimates were consistent with each other, but the first, based on the galaxies’ angular clustering, had somewhat smaller uncertainties. This resulted from the larger size of the photometric sample, and was accentuated by the serious systematics in the spectroscopic sample that made us throw much of our data away. We will adopt the angular clustering results for the remainder of the paper.
IMPLICATIONS
============
Correspondence to halos {#sec:correspondence}
-----------------------
On small scales, smaller than roughly the typical radius $r_{\rm vir}$ of a virialized halo, the spatial clustering of galaxies is difficult to predict or interpret. It depends on the ease with which nearby galaxies merge with each other, on the ability of a galaxy to maintain its star-formation rate as it orbits within a larger potential well, on the possible impact of a galaxy’s feed-back on its surroundings, and so on. On larger scales these baryonic complications have little effect and the correlation function of galaxies should be virtually identical to the correlation function of the halos that host them. To see that this is true, consider the galaxy correlation function in a simplified situation where every galaxy is associated with a single halo and the probability that a galaxy lies a distance $r$ from its halo’s center is $f(r)$. In this case the galaxy distribution will be a Poisson realization of the continuous density field $\rho_g({\mathbf r})$ that is created when the discrete halo distribution is convolved by $f$, and the galaxy correlation function $\xi_g$ will be equal to the correlation function of $\rho_g$ (see, e.g., Peebles 1980, eq. 33.6). Since the halo powerspectrum is $P_h({\mathbf k}) = 1/n + \int d^3r \xi_h \exp(i{\mathbf k}\cdot{\mathbf r})$ (Peebles 1980, eq. 41.5), where $\xi_h$ is the halo correlation function and $n$ is the halo number density, since the powerspectrum of $\rho_g$ is equal to $|\tilde f({\mathbf k})|^2P_h({\mathbf k})$, where $\tilde f$ is the Fourier transform of $f$, and since the powerspectrum and correlation function are Fourier-transform pairs, the galaxy and halo correlation functions will be related through $$\xi_g({\mathbf r}) = F({\mathbf r})/n + F({\mathbf r})\otimes \xi_h({\mathbf r})
\label{eq:xigxih}$$ where $\otimes$ denotes convolution and $F\equiv f\otimes f$. Now $f(r)=0$ when $r$ is greater than some maximum separation $r_{\rm max}\sim r_{\rm vir}$; galaxies cannot be located arbitrarily far from the center of their halo. The first term will therefore be zero for $r\gg 2r_{\rm vir}$. The second term will be almost exactly equal to $\xi_h(r)$ at the same large separations, because plausible correlation functions do not not change significantly from $r$ to $r+2r_{\rm vir}$ when $r_{\rm vir}\ll r$. This shows that $\xi_g(r)\simeq \xi_h(r)$ for large $r$. Although it was derived for a simplistic model, the result is more general. As long as galaxies are associated in some way with dark matter halos, and as long as there is some maximum separation $r_{\rm max}\sim r_{\rm vir}$ between each galaxy and the center of its halo, galaxies will have the same correlation function as the halos that host them for $r\gg r_{\rm vir}$. We will accordingly focus our attention solely on separations $r\simgt 1h^{-1}$ comoving Mpc that are many times larger than the typical virial radius.
Readers who are skeptical of the claimed similarity between galaxy and halo correlation functions at $r\gg r_{\rm vir}$ may wish to consider figure \[fig:xi\_halo\_0z3\], which compares observed galaxy correlation functions at redshifts $z=0.2$, $1.0$, $3.0$ to the halo correlation functions in the GIF-LCDM simulation outputs at the same redshifts (see § \[sec:simulated\_data\]). The agreement is excellent.
The implied association of galaxies with massive potential wells is hardly surprising. The interesting result is the characteristic mass of the virialized halos that contain the galaxies. This can be estimated since more massive halos cluster more strongly. Figure \[fig:xi\_halo\] compares the correlation functions at $1\simlt r/(h^{-1}{\rm comoving\ Mpc}) \simlt 10$ for galaxies in the BM, BX, and LBG samples to the correlation functions of virialized dark matter halos above various mass thresholds in the GIF-LCDM simulation (see § \[sec:simulated\_data\]). The agreement is best if galaxies in the BM, BX, and LBG samples are associated with halos of mass $M\sim 10^{12.1} M_\odot$, $10^{12.0} M_\odot$, and $10^{11.5} M_\odot$, respectively.
Uncertainties in our measured correlation functions lead to uncertainties in the estimated characteristic masses. The size of these uncertainties can be gauged most cleanly by comparing our measured angular correlation function to the angular correlation functions of halos above different mass thresholds. We calculated halo angular correlation functions numerically from equation \[eq:limber0\] after substituting in the observed redshift distributions for the different samples (fig. \[fig:zhistos\]) and the GIF-LCDM halo correlation functions at the mean redshift of each sample. Typical results are shown in figure \[fig:wthet\_halo\]. The halo angular correlation functions fall significantly below the data on small scales, since (by definition) a halo cannot have another halo as a neighbor within the radius $r_{\rm vir}$. As argued above, however, it is the larger scales $r\simgt 1h^{-1}$ comoving Mpc (i.e., $\theta\simgt 1'$) that are relevant for comparing to galaxies, and here the agreement is good. The uncertainty in the galaxies’ implied mass scale is dominated by the uncertainty in the integral constraint correction ${\cal I}$ that was applied to the data, since this moves all points up or down together. Rough $1\sigma$ limits on the threshold masses of the hosting halos are $$\begin{aligned}
\label{eq:mass_scales}
10^{11.9} &\simlt M/M_\odot\simlt& 10^{12.3}\quad\quad{\rm BM}\nonumber\\
10^{11.8}&\simlt M/M_\odot\simlt& 10^{12.2}\quad\quad{\rm BX}\\
10^{11.2}&\simlt M/M_\odot\simlt& 10^{11.8}\quad\quad{\rm LBG}.\nonumber\end{aligned}$$ Readers unfamiliar with the idea of threshold masses may wish to see footnote 6 in §\[sec:summary\], below, for further discussion of how to interpret them.
One implication of figure \[fig:wthet\_halo\] is that some halos are occupied by more than one of the galaxies in our samples; the data on small scales are strongly inconsistent with the correlation functions that assumed one object per halo. (This was first pointed out by Wechsler et al. (2001) and later denied by Porciani & Giavalisco (2002). Our analysis agrees far better with that of Wechsler et al.) The fraction of LBGs that reside in the same halo as another LBG can be calculated as follows. If there were never more than one LBG per halo, the expected number of LBGs within $1'$ of a randomly chosen LBG would be $N_1={\cal N}(1+\bar\omega_1)$ where ${\cal N}$ is the surface density of LBGs, $\bar\omega_1\equiv \int_0^{1'}\theta d\theta\,\omega_h(\theta)/\int_0^{1'}\theta d\theta$, and $\omega_h$ is halo angular correlation function that fits our data best for $\theta>1'$. The actual number of LBGs within $1'$, $N_{\rm true}$, is given by the same equation with the halo correlation function $\omega_h$ replaced by the galaxy correlation function $\omega_{\rm true}$. The expected number of additional LBGs in a halo that is known to contain one LBG, $f_{\rm LBG}$, is equal to the difference between $N_{\rm true}$ and $N_1$. Numerically integrating the angular correlation functions for observed galaxies and simulated halos, and multiplying by the LBG surface density from table \[tab:fields\], we estimate $f_{\rm LBG}\sim 0.05$. The numbers for the other samples, estimated with a similar approach, are $f_{\rm BX}\sim 0.25$ and $f_{\rm BM}\sim 0.25$. [*Some*]{} galaxies must share the same halo to explain the data, but the required number is small. Note that we are referring solely to galaxies that satisfy our color and magnitude selection criteria; we obviously cannot say anything about the spatial distribution of galaxies that are not in our samples.
Having established a rough characteristic mass for the halos that contain our galaxies, we can compare the galaxy and halo number densities and estimate what fraction of the most massive halos at redshifts $1.5\simlt z\simlt 3.5$ do not contain a galaxy that is detectable with our techniques. According to Adelberger & Steidel (2000) the comoving number density of LBGs brighter than ${\cal R}=25.5$ is $$n_{\rm LBG}=(4\pm 2)\times 10^{-3} h^{3} {\rm Mpc}^{-3}
\label{eq:nlbg}$$ for $\Omega_M=0.3$, $\Omega_\Lambda=0.7$. Combining the BM and BX completeness coefficients in table 3 of Adelberger et al. (2004) with the surface densities in table \[tab:fields\] of this paper, we estimate $$\begin{aligned}
n_{\rm BX}&=&(6\pm 3)\times 10^{-3} h^{3} {\rm Mpc}^{-3}\nonumber\\
n_{\rm BM}&=&(5\pm 2.5)\times 10^{-3} h^{3} {\rm Mpc}^{-3}
\label{eq:nbxbm}\end{aligned}$$ where the assigned uncertainties of 50% are approximate guesses that will be refined later with Monte Carlo simulations. Number densities for the population of halos that contain the galaxies can be estimated from the GIF-LCDM simulation given the range of halo masses (equation \[eq:mass\_scales\]) that are compatible with the galaxies’ clustering strength. Figure \[fig:nhalo\] shows that the number density of galaxies in the BM, BX, and LBG samples is comparable to the number density of halos that can host them. As we will discuss in § \[sec:summary\], below, this implies that the duty cycle of star-formation in the galaxies must be of order unity and shows that our surveys cannot be severely incomplete. Similar arguments have been made by Adelberger et al. (1998), Giavalisco & Dickinson (2000), Martini & Weinberg (2001), and others.
Evolution {#sec:evolution}
---------
The spatial distribution of a population of galaxies evolves in an easily predictable way as it responds to the gravitational pull of dark matter. We used a simple approach to estimate this evolution from the LCDM-GIF simulation. After connecting the observed galaxies to halos with a range of masses (equation \[eq:mass\_scales\]), we measured the evolution in the clustering of those halos in the simulation and assumed that the galaxies’ clustering would evolve in the same way. See § \[sec:simulated\_data\]. Figure \[fig:r0\_vs\_z\] shows the implied change in correlation length $r_0$ with time for the galaxies in our samples. By $z\sim 2$ galaxies in the LBG sample will have a correlation length similar to measured correlation lengths of galaxies in the BM and BX samples. By $z\sim 1$ their correlation length will be similar to the correlation length of early-type (i.e., “absorption line”) galaxies in the sample of Coil et al. (2003). By $z\sim 0.2$ their correlation length will be equal to the observed correlation length of ellipticals (Budavári et al. 2003). The evolution of $r_0$ for galaxies in the BM and BX samples is similar.
Figures \[fig:z1gals\] and \[fig:z0gals\] present a more detailed view of the possible relationships between the descendants of galaxies in our samples and various galaxy populations at lower redshift. Figure \[fig:z1gals\] shows that at $z=1$ the descendants’ clustering strength will significantly exceed that of average galaxies in optical magnitude-limited surveys. Since these surveys are dominated by star-forming (“emission-line”) galaxies, we can conclude that the typical descendant is no longer forming stars by $z\sim 1$.[^3] A similar point was made by Adelberger (2000) and Coil et al. (2004). The stronger clustering of redder and brighter sub-populations at $z\sim 1$ is more compatible with the descendants’ expected clustering, but the match is best for the sub-population with early-type spectra. This is especially true for descendants of the brightest galaxies in the BX and LBG samples. Although the observed number density of early-type galaxies at redshift $z\sim 1$, roughly $(7\pm 1.5)\times 10^{-3} h^3 {\rm Mpc}^{-3}$ (Chen et al. 2003), is consistent with the idea that most had a BM/BX/LBG galaxy as a progenitor, we cannot rule out the idea that some had multiple merged BM/BX/LBG progenitors and others had none.
Figure \[fig:z0gals\] is an analogous plot for redshift $z\sim 0.2$. Correlation lengths for various populations in the Sloan Digital Sky Survey (SDSS) were taken from Budavári et al. (2003). Number densities were calculated assuming a surveyed volume of $10^8 h^{-3}$ Mpc$^3$ with an uncertainty of $\sim 15$% (T. Budavári 2004, private communication). The expected $z=0.2$ clustering strength of typical LBG descendants (darker shaded box) agrees best with the clustering of galaxies with early-type SEDs in the Budavári et al. (2003) sample. The galaxies in our high-redshift samples are roughly as numerous as these early-type galaxies, though the possibility of merging prevents us from estimating the number density of their BM/BX/LBG descendants at $z\sim 0.2$. The descendants of brighter LBGs will have a correlation length closer to that of bright ellipticals, though there are probably not enough descendants to account for the entire bright elliptical population.
SUMMARY & DISCUSSION {#sec:summary}
====================
The first part of the paper was concerned with measuring the spatial clustering of large samples of star-forming galaxies at redshifts $z\sim 1.7$, $2.2$, and $2.9$. We fit a three-dimensional correlation of the form $\xi(r)=(r/r_0)^{-\gamma}$ to the galaxies’ angular clustering with standard techniques and to the galaxies’ redshift clustering with a new estimator. The new estimator, $K$, is insensitive to many of the possible systematic biases in our spectroscopic surveys. We reached consistent conclusions about the correlation function with the two approaches, but adopted the angular results since their random uncertainties were somewhat smaller. As given in § \[sec:angularresults\], the best-fit correlation-function parameters from the angular clustering are $$r_0/(h^{-1}{\rm Mpc}),\gamma=\cases{ 4.0\pm 0.6, 1.57\pm 0.14& LBG \cr
4.2\pm 0.5, 1.59\pm 0.08& BX \cr
4.5\pm 0.6, 1.55\pm 0.10& BM \cr}
\label{eq:correlationlengths}$$ where BM, BX, and LBG are the names we have given the $U_nG{\cal R}$ color-selection criteria that we used to find galaxies at $z\sim 1.7$, $2.2$, and $2.9$ (§ \[sec:observeddata\]). The quoted $1\sigma$ errors include random uncertainties, uncertainties in the integral constraint corrections, and uncertainties in the shapes of the selection functions.
Since the value of $r_0$ depends on the apparent-magnitude limit of the samples, at least in the two higher-redshift bins (figure \[fig:clust\_seg\]), the reported values of $r_0$ are somewhat arbitrary. We chose to limit each sample to a fixed range of apparent magnitudes, $23.5<{\cal R}<25.5$, on the grounds that this resulted in a similar comoving density in each redshift bin. Different magnitude limits would have resulted in different correlation lengths. Readers should be aware that the numbers we give are appropriate to the samples as we have defined them, not to the general galaxy population at high redshifts.
The second part of the paper was based on the proposition that WMAP (Spergel et al. 2003) and other experiments have given us reliable measurements of the cosmological parameters and of the shape of the dark matter power-spectrum. This implies that we know what sorts of virialized dark-matter halos existed at different epochs in the past and how their spatial distribution evolved over time. Since galaxies reside within dark-matter halos, they will have the same correlation function as the halos on large scales (equation \[eq:xigxih\]). The galaxies’ clustering should therefore tell us what sort of halo they reside within. We found a good match (figures \[fig:xi\_halo\] and \[fig:wthet\_halo\]) between the correlation functions of the galaxies and of halos with threshold masses ranging from $10^{11.5}M_\odot$ (LBG) to $10^{12.1}M_\odot$ (BM).[^4] Equation \[eq:mass\_scales\] gives rough $1\sigma$ limits on the halos’ total masses. Similar masses for Lyman-break galaxies have been derived with the same approach by Jing & Suto (1998), Adelberger et al. (1998), Giavalisco & Dickinson (2000), Porciani & Giavalisco (2002), and others. Although the estimated masses were derived solely from the galaxy clustering, they seem reasonable on other grounds. They cannot be much higher. The number density of halos would be lower than the number density of LBGs, for example, if the halo mass were greater than $10^{11.8}M_\odot$. Such large halo masses would be possible only if significantly more than one LBG resided in the typical halo, and that is something that our observations rule out (§ \[sec:correspondence\]). Nor can they be much lower. The halos that contain LBGs would not have enough baryons to form the median LBG stellar mass of roughly $10^{10}M_\odot$ (Shapley et al. 2001[^5]) unless their total mass were greater than about $10^{11}M_\odot$.
We should mention in passing that our best-fit halo masses seem to imply that only a small fraction of the baryons in the halos are associated with the observed galaxies. For example, the best-fit mass threshold of $10^{11.5}M_\odot$ for LBGs corresponds to a median total mass of $10^{11.86}M_\odot$ and median baryonic mass of $1.2\times 10^{11}M_\odot$ (for $\Omega_b/\Omega_M\simeq 0.17$, Spergel et al. 2003), roughly ten times larger than the observed stellar masses of LBGs. Since the $10^8$ supernovae that explode during the assembly of the typical LBG’s stellar mass will eject roughly $10^8 M_\odot$ of metals (e.g., Woosley & Weaver 1995), enough to enrich at most $1.3\times 10^{10} M_\odot$ of gas to LBGs’ typical metallicities of $0.4Z_\odot$ (Pettini et al. 2002), their observed interstellar gas cannot contain a large fraction of the remaining baryons. These baryons need not be associated with other objects in the halo, however. They may be locked in dim stars that formed in previous episodes of star-formation (e.g., Papovich, Dickinson, & Ferguson 2001), or may have been heated by various processes to undetectably high temperatures. The latter is presumably the case for nearby galaxies, whose ratios of mass in stars and gas to total mass are usually also smaller than the WMAP value $\Omega_b/\Omega_M\simeq 0.17$. The Milky Way, for example, has a total mass of $10^{12}M_\odot$ (Zaritsky 1999; Wilkinson & Evans 1999) and a mass in gas and stars of only $\sim 8\times 10^{10} M_\odot$ (K. Freeman 2004, private communication), yet few would assert that its missing baryons belong to another galaxy in its halo.
After establishing plausible total masses for the halos associated with the galaxies, we considered some of the implications. Our arguments were not new (see, e.g., Moustakas & Somerville 2002, Martini & Weinberg 2001, Adelberger et al. 1998). They seemed worth revisiting only because our knowledge of the cosmogony, of the local universe, and of high-redshift galaxies has improved so much in the last few years. We began by estimating the completeness of our surveys from a comparison of the galaxies’ number densities to the number densities of halos with similar clustering strength (figure \[fig:nhalo\]). Similar number densities would imply that almost all of the most massive halos contained a galaxy that satisfied our selection criteria; a much lower galaxy number density would imply that most of the galaxies in massive halos are missed by our survey. Defining $\eta$ as the ratio of galaxy to halo number density, we found rough $1\sigma$ limits of $0.2<\eta_{\rm LBG}<1$, $0.6<\eta_{\rm BX}<3$, and $0.5<\eta_{\rm BM}<2.5$. These limits were derived from the clustering at radii $r\simgt 1h^{-1}$ comoving Mpc. The clustering on smaller scales, sensitive to the possible presence of more than one galaxy in a halo, implies that the upper limits on $\eta_{\rm BX}$ and $\eta_{\rm BM}$ should be revised downwards to $\sim 1.25$. The data appear consistent with the claim of Franx et al. (2003) that our selection criteria find roughly half of the most massive galaxies at $z\sim 2$. A completeness of order $50$% seems plausible to us for other reasons as well. Shapley et al. (2001) estimate a lifetime for the typical LBG of $\sim 3\times 10^8$ yr, for example, which implies that the typical LBG will be bright enough for us to detect for only about half of the time that elapsed between the survey selection limits of $z\sim 3.4$ and $z\sim 2.6$.
We considered next the way the clustering of the galaxies would evolve (figure \[fig:r0\_vs\_z\]). Analysis of the GIF-LCDM simulation suggested that the correlation length of LBG descendants would be similar by $z\sim 2.2$ to the correlation length of galaxies in the BX sample and by $z\sim 1.7$ to the correlation length of galaxies in the BM sample. The spatial clustering is therefore consistent with the idea that we are seeing the same population at all three redshifts, though the selection criteria’s $\sim 50$% incompleteness leaves room for the populations to be distinct and the difference in stellar masses between the LBG ($10^{10}M_\odot$) and BX ($2\times 10^{10} M_\odot$; Steidel et al. 2005, in preparation) samples may not be consistent with continuous star-formation at observed rates through the elapsed time. Turning our attention to lower redshifts, we found that at $z\sim 1$ our descendants’ clustering would most closely match the observed clustering of galaxies that are red and bright and have early-type spectra (figure \[fig:z1gals\]). By $z\sim 0.2$ the estimated clustering of the descendants suggested elliptical galaxies as the most likely counterparts (figure \[fig:z0gals\]). The correspondence is especially hard to dispute for the descendants of the brightest and most strongly clustered galaxies in the high-redshift samples.
One conclusion seems difficult to escape: the descendants of the galaxies in our samples must have significantly larger stellar masses than their high-redshift forebears. Only $\sim 25$% of the total stellar mass in the local universe is found in galaxies with stellar masses smaller than $2\times 10^{10}M_\odot$ (Kauffmann et al. 2003), similar to the values in our high-redshift samples, and these faint galaxies are too weakly clustered to have descended from the galaxies we find at $1.4<z<3.5$. Only elliptical galaxies have a spatial distribution consistent with our expectations for the descendants, and the characteristic stellar mass of ellipticals is $10^{11}M_\odot$ (Padmanabhan et al. 2004). The increase in stellar mass from $z\sim 2$ to $z\sim 0$ could have been produced by ongoing star formation or by mergers. Our results at redshift $z\sim 1$ may favor the latter, but in any case our findings are strongly inconsistent with traditional notions of monolithic collapse.
It would be unfair to close without mentioning one population of high-redshift galaxies that we have ignored completely. These are the bright ($K\simlt 20$) near-infrared-selected galaxies. Their reported star-formation rates ($\sim 200M_\odot$/yr), correlation lengths ($>9 h^{-1}$ Mpc, Daddi et al. 2004) and stellar masses ($2\times 10^{11}M_\odot$; van Dokkum et al. 2004) are extraordinary, far larger than the corresponding values for typical galaxies in our samples.[^6] Galaxies with similarly extreme properties are not a negligible component of the high redshift universe. The shape of the $850\mu$m background implies that up to a third (Cowie, Barger, & Kneib 2002) of all stars could have formed in objects with star-formation rates greater than $\sim 200 M_\odot$; halos at $z\sim 3$ with the large masses $M\simgt 10^{12.7}M_\odot$ implied by $r_0\sim 8$–$9h^{-1}$ Mpc contain in total almost $20$% as many baryons as the more numerous and smaller halos with $M\sim 10^{11.5} M_\odot$ that contain LBGs; objects with stellar masses $M_\ast>2\times 10^{11}M_\odot$ contain nearly $5$% of all stars in the local universe (Kauffmann et al. 2003) and $20$% of the stars in local elliptical galaxies (Padmanabhan et al. 2004). No treatment will be entirely complete if it neglects galaxies similar to those found in near-IR surveys. The galaxies we studied are neither the most massive, nor the most rapidly star-forming, nor the most clustered galaxies in the high redshift universe, but it is precisely this that makes them plausible progenitors for the early-type galaxies that surround us today.
We are grateful to the Virgo consortium for its public release of the GIF simulation data and to G. Kauffmann for bringing the data to our attention. A. Coil and T. Budavári responded helpfully to our questions about their data. M. Giavalisco, the referee, gave us an insightful report. KLA, AES, and NAR were supported by fellowships from the Carnegie Institute of Washington, the Miller Foundation, and the National Science Foundation. DKE and CCS were supported by grant AST 03-07263 from the National Science Foundation.
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[^1]: The dependence of the integral constraint correction on $r_0$ means that errors in $N(z)$ alter the inferred value of $\gamma$ as well. We will neglect this small effect.
[^2]: More correctly, $\langle K\rangle$ is independent of uncertainties in the selection function width. The expectation value $\langle K\rangle$ [*will*]{} be affected by errors in the mean redshift of the selection function if these errors are large enough to significantly alter the mapping of redshifts and angles to distances.
[^3]: In principle the descendants could have emission-line spectra if they made up a small part of an otherwise weakly clustered population, but this possibility seems to be ruled out by the fact that the number densities of BM/BX/LBG galaxies are similar to the number density of galaxies in the $z\sim 1$ emission-line sample. (Our estimate of the number density in the emission-line sample, shown in figure \[fig:nhalo\], is from A. Coil, private communication.)
[^4]: When we say (for example) that the galaxies have the same correlation function as halos with threshold mass $M=10^{11.5}M_\odot$, we mean the subset of halos with mass $M>10^{11.5}M_\odot$. The median mass of this subset is $M=10^{11.86}M_\odot$ in the GIF-LCDM simulation at $z=2.97$. Halo subsets can be defined with schemes more elaborate than our simple mass threshold (e.g., Kauffmann et al. 1999, Bullock, Wechsler, & Somerville 2002), but the differences between the possible schemes are too small to affect our analysis when the subsets they produce are constrained to have the same clustering on large scales.
[^5]: This mass is for a Baldry-Glazebrook (2003; eq. 3) IMF, and is therefore $1.82$ times lower than the value Shapley et al. calculated for an IMF with a Salpeter slope between $0.1$ and $100M_\odot$. Their assumed Salpeter IMF is probably unrealistic since the IMF in the solar neighborhood is known to flatten near $\sim 1M_\odot$ and eventually turn over at lower masses. See Leitherer (1998) or Renzini (2004) for further discussion.
[^6]: These values are not larger, however, than those for galaxies in our samples with similar $K$ magnitudes. See, e.g., Shapley et al. 2004 and Adelberger et al. 2005
|
---
abstract: 'In previous work ([@gklr19]), we noted that the known cases of hyper-Kähler manifolds satisfy a natural condition on the LLV decomposition of the cohomology; informally, the Verbitsky component is the dominant representation in the LLV decomposition. Assuming this condition holds for all hyper-Kähler manifolds, we obtain an upper bound for the second Betti number in terms of the dimension.'
address:
- 'Stony Brook University, Department of Mathematics, Stony Brook, NY 11794-3651'
- 'Stony Brook University, Department of Mathematics, Stony Brook, NY 11794-3651'
author:
- 'Yoon-Joo Kim'
- Radu Laza
bibliography:
- 'Betti.HK.bib'
title: 'A conjectural bound on the second Betti number for hyper-Kähler manifolds'
---
[^1]
Introduction
============
A fundamental open question in the theory of compact hyper-Kähler manifold is the boundedness question: [*are there finitely many diffeomorphism types of hyper-Kählers in a given dimension?*]{} In accordance with the Torelli principle, Huybrechts [@huy03 Thm 4.3] proved that there are finitely many diffeomorphism types of hyper-Kähler manifolds once the dimension and the (unnormalized) Beauville–Bogomolov lattice $(H^2(X, {\mathbb{Z}}), q_X)$ are fixed. Thus, bounding the hyper-Kähler manifolds is equivalent to bounding the second Betti number $b_2 = b_2(X)$, and then the Beauville–Bogomolov form (e.g., the discriminant). In dimension $2$, a compact hyper-Kähler manifold is always a K3 surface, thus $b_2 = 22$. In dimension $4$, Beauville and Guan [@guan01] gave a sharp bound $b_2\le 23$ (in fact, Guan showed $3 \le b_2 \le 8$ or $b_2 = 23$). Beyond these, we are not aware of any other boundedness result on $b_2$ (see Remark \[rem\_saw\]). The purpose of this note is to give a conjectural bound on $b_2(X)$ for an arbitrary compact hyper-Kähler manifold $X$ of dimension $2n$. Our bound depends on a natural conjectural condition satisfied by the Looijenga–Lunts–Verbitsky (LLV) decomposition of the cohomology $H^*(X)$ for hyper-Kähler manifolds $X$.
To state our results, let us recall that Verbitsky [@ver95] and Looijenga–Lunts [@ll97] noted that the cohomology $H^*(X)$ of a hyper-Kähler manifold admits a natural action by the Lie algebra $\mathfrak g = \mathfrak{so}(b_2 + 2)$, generalizing the usual hard Lefschetz theorem. As a $\mathfrak g$-module, the cohomology of a hyper-Kähler manifold $X$ decomposes as $$\label{eq:llv_decomp}
H^* (X) = \sideset{}{_{\mu}} {\bigoplus} V_{\mu}^{\oplus m_{\mu}},$$ where $V_{\mu}$ indicates an irreducible $\mathfrak g$-module of highest weight $\mu=(\mu_0, \cdots, \mu_r)$, with $r= \left \lfloor \frac{b_2(X)}{2} \right \rfloor=\mathrm{rk}\ \mathfrak g-1$. We refer to $\mathfrak g$ as the [*LLV algebra*]{} of $X$, and to as the [*LLV decomposition*]{} of $H^*(X)$ (see [@gklr19] for further discussion). Motivated by the behavior of the LLV decomposition in the known cases of hyper-Kähler manifolds [@gklr19], we have made the following conjecture.
Let $X$ be a compact hyper-Kähler manifold of dimension $2n$. Then the weights $\mu = (\mu_0, \cdots, \mu_r)$ occurring in the LLV decomposition of $H^*(X)$ satisfy $$\label{eq:conj}
\mu_0 + \cdots + \mu_{r-1} + |\mu_r| \le n.$$
The conjecture holds for all currently known examples of compact hyper-Kähler manifolds (cf. [@gklr19 Thm 1.14]). Furthermore, the equality in always holds for [*the Verbitsky component*]{} $V_{(n)}$, the irreducible component corresponding to the subring generated by $H^2(X)$, which shows that is sharp. Beyond the evidence given by the validity of in the known cases, we have some partial arguments of motivic nature (and depending on standard conjectures) showing that at least is plausible. This will be discussed elsewhere.
The purpose of this note is to show that conjecture implies a general bound on $b_2(X)$.
Let $X$ be a compact hyper-Kähler manifold of dimension $2n$. If the condition holds for $X$, then $$\label{bound_b2}
b_2(X) \le \begin{cases}
\frac{21 + \sqrt{96n + 433}}{2} & \mbox{if } H^*_{\operatorname{odd}} (X) = 0 \\
2k + 1 & \mbox{if } H^k (X) \neq 0 \mbox{ for some odd } k
\end{cases} .$$
A slightly weaker version of is $$b_2(X) \le \max \left\{ \tfrac{21 + \sqrt{96n + 433}}{2}, 4n-1 \right\},$$ which reads explicitly
$n$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $\ge 8$
-------------- ------ ------ ------ ------ ------ ------ ------ ---------
$b_2(X) \le$ $22$ $23$ $23$ $24$ $25$ $26$ $27$ $4n-1$
.
In low dimensions, our bounds agree with the known results and seem fairly sharp. For instance, we know $\mathrm{K3}^{[n]}$ type hyper-Kähler manifolds have $b_2 = 23$ and $\mathrm{OG10}$ manifolds have $b_2 = 24$. These examples almost reach our maximum bound of $b_2$ for low dimensions. Similarly, $\mathrm{Kum}_n$ type hyper-Kähler manifolds have $H^3 (X) \neq 0$ and $b_2 = 7$, showing also that the second inequality in is sharp.
\[rem\_saw\] Sawon [@saw15] has previously obtained the same bound for $n = 3, 4$, and moreover predicted the general formula when $H^*_{\operatorname{odd}} (X) = 0$. However, the results in loc. cit. were based on the assumption $\mu_2 = \cdots = \mu_r = 0$ for the highest weights $\mu$ appearing in the LLV decomposition . This is a too strong assumption, which is rarely satisfied (e.g., [@gklr19 Cor 3.2, 3.6]).
A few words about the proof of our conjectural bound. First, in [@gklr19 (3.31)], we have already noticed that the condition has consequences on the odd cohomology (specifically, if $b_2 \ge 4n$ then there should be no odd cohomology). A slight generalization of the argument in loc. cit. then gives the second inequality in . The main content of this note is the control of the even cohomology under the assumption . Essentially, our argument is a representation theoretic refinement of Beauville’s argument that $b_2 \le 23$ for hyper-Kähler fourfolds. Namely, the starting point is the Salamon’s relation [@sal96], a linear relation satisfied by the Betti numbers of hyper-Kähler manifolds. Inspired by the shape of it, we define a numerical function $s(W)$ for a $\mathfrak g$-module $W$ and verify its basic properties, most importantly $s ( W_1 \oplus W_2 ) \le \max \{ s(W_1), s(W_2) \}$. In this setting, Salamon’s relation reads $s(H^*(X)) = \frac{n}{3}$. Now the punchline is an explicit formula for $s(V_{\mu})$ for irreducible $\mathfrak g$-modules $V_{\mu}$ (), which is obtained by applying the Weyl character formula. Combining it with , we conclude $$\frac{n}{3} = s(H^*(X)) \le s(V_{(n)}) = \frac{8n (b_2+n)}{(b_2+1) (b_2+2)} ,$$ which in turn gives the first inequality in .
Cohomology of compact hyper-Kähler manifolds {#sec:backgrounds}
============================================
We briefly review some relevant results on the cohomology of hyper-Kähler manifolds. Let $X$ be a compact hyper-Kähler manifold of dimension $2n$ and $H^* (X) = H^* (X, {\mathbb{C}})$. Let $\mathfrak g \subset {\mathfrak{gl}}(H^* (X))$ be the Lie algebra generated by all the Lefschetz and dual Lefschetz operators associated to elements in $H^2 (X)$ (cf. [@ver95], [@ll97]). We call this the Looijenga–Lunts–Verbitsky (LLV) algebra of $X$. Let $$(V, q) = (H^2 (X), q_{X}) \oplus U$$ be the Mukai completion of $H^2 (X)$ equipped with the Beauville–Bogomolov form, and set $r = \left \lfloor \frac{b_2(X)}{2} \right \rfloor$. Then $\mathfrak g$ is isomorphic to the special orthogonal Lie algebra ${\mathfrak{so}}(V, q) \cong {\mathfrak{so}}(b_2 + 2, {\mathbb{C}})$ of rank $r+1$. The cohomology $H^*(X)$ of a hyper-Kähler manifold $X$ admits a $\mathfrak g$-module structure, generalizing the hard Lefschetz theorem. We refer to the $\mathfrak g$-module irreducible decomposition of $H^* (X)$ as the [*LLV decomposition*]{} of the cohomology (see [@gklr19 Thm 1.1] for some examples). Finally, the subring of $H^* (X)$ generated by $H^2 (X)$ becomes an irreducible $\mathfrak g$-module component of $H^* (X)$. We refer to it as [*the Verbitsky component*]{} and denote it by $V_{(n)}$. It always occurs with multiplicity $1$ in the LLV decomposition.
Fix a Cartan and a Borel subalgebra of $\mathfrak g$. Representation theory of ${\mathfrak{so}}(V, q)$ depends on the parity of $\dim V = b_2 + 2$. If $b_2 = 2r$ is even, then we can fix a suitable basis $\varepsilon_0, \cdots, \varepsilon_r$ of the dual Cartan subalgebra such that the associated weights of the standard module $V$ are $\pm \varepsilon_0, \cdots, \pm \varepsilon_r$. Similarly, if $b_2 = 2r+1$ is odd, then we can choose $\varepsilon_i$ such that $V$ has the associated weights $0, \pm \varepsilon_0, \cdots, \pm \varepsilon_r$. (Note that the index of the basis starts from $0$.) Any dominant integral weight $\mu$ can be expressed in this basis as $$\mu = (\mu_0, \cdots, \mu_r) = \sum_{i=0}^r \mu_i \varepsilon_i .$$ Here if $b_2 = 2r$ is even, then $\mu_i$ satisfy the condition $\mu_0 \ge \cdots \ge \mu_{r-1} \ge |\mu_r| \ge 0$ and all $\mu_i$ are either mutually integers or half-integers. If $b_2 = 2r+1$ is odd, then $\mu_0 \ge \cdots \ge \mu_r \ge 0$ and all $\mu_i$ are again either mutually integers or half-integers. It will be important whether $\mu_i$ are integers or half-integers, so we define:
Let $\mu = (\mu_0, \cdots, \mu_r)$ be a dominant integral weight of $\mathfrak g = {\mathfrak{so}}(V, q)$.
1. If all $\mu_i$ are integers, we say $\mu$ is *even*. If all $\mu_i$ are half-integers (i.e., $\mu_i \in \frac{1}{2} {\mathbb{Z}}\setminus {\mathbb{Z}}$), we say $\mu$ is *odd*.
2. An irreducible $\mathfrak g$-module $V_{\mu}$ of highest weight $\mu$ is called even (resp. odd) if $\mu$ is even (resp. odd).
3. A $\mathfrak g$-module $W$ is called even (resp. odd) if all of its irreducible components $V_{\mu}$ are even (resp. odd).
By [@gklr19 Prop 2.35], the even (odd) cohomology $H^*_{\operatorname{even}} (X)$ is always an even (resp. odd) $\mathfrak g$-module.
Let $h$ be the degree operator on $H^* (X)$, the operator acting as multiplication by $k$ on $H^{2n+k} (X)$. For a suitable choice of a Cartan and a Borel subalgebra, we can assume $h = \varepsilon_0^{\vee}$ (e.g., [@gklr19 (2.28)]). By definition, the degree decomposition of the cohomology $$H^* (X) = \bigoplus_{k=-2n}^{2n} H^{2n + k} (X)$$ is the $h$-eigenspace decomposition. In general, an arbitrary $\mathfrak g$-module admits the $h$-eigenspace decomposition $$\label{eq:h_eigendecomp}
W = \bigoplus_{k \in {\mathbb{Z}}} W_k ,$$ where $W_k$ denotes the eigenspace of $W$ with eigenvalue $k$, which is always an integer because $h = \varepsilon_0^{\vee}$.
Consider the LLV decomposition of the cohomology $$H^* (X) = \sideset{}{_{\mu}} {\bigoplus} V_{\mu}^{\oplus m_{\mu}} . \tag{\ref{eq:llv_decomp} (restated)}$$ If $V_{\mu}$ is contained in the odd cohomology, then $\mu$ is odd by the above discussion. Hence all $\mu_i$ are half-integers, and in particular we have $\mu_i \ge \frac{1}{2}$ (possibly except for the last $|\mu_r| \ge \frac{1}{2}$, if $b_2$ is even). If we specifically assume $H^k (X) \neq 0$ for odd $k < 2n$, then there exists at least one irreducible component $V_{\mu}$ with $(V_{\mu})_{k-2n} \neq 0$. This means $h = \varepsilon_0^{\vee}$ acts on some part of $V_{\mu}$ by $k-2n$, so $V_{\mu}$ has an associated weight $\theta = \theta_0 \varepsilon_0 + \cdots + \theta_r \varepsilon_r$ with $\theta_0 = \frac{k}{2} - n$. This forces $\mu_0 \ge n - \frac{k}{2}$. Summarizing, we have $$\mu_0 \ge n - \frac{k}{2}, \qquad \mu_1, \cdots, \mu_{r-1}, |\mu_r| \ge \frac{1}{2} ,$$ which gives the following.
\[cor:odd\_weight\_bound\] Let $X$ be a compact hyper-Kähler manifold of dimension $2n$. Assume $H^k (X) \neq 0$ for some odd integer $k < 2n$. Then there exists a weight $\mu$ in with $\mu_0 + \cdots + \mu_{r-1} + |\mu_r| \ge n - \frac{k}{2} + \frac{r}{2}$.
Finally, let us recall the Salamon’s relation. Let $b_k = b_k(X)$ the $k$-th Betti number of $X$. Salamon [@sal96] proved that the Betti numbers of hyper-Kähler manifolds $X$ satisfy a linear relation: $$\sum_{k=1}^{2n} (-1)^k (6k^2 - 2n) b_{2n + k} = n b_{2n} .$$ One can manipulate the identity into the following form $$\label{eq:salamon_old}
\sum_{k=-2n}^{2n} (-1)^k k^2 b_{2n + k} = \frac{n}{3} e(X),$$ where $e(X) = \sum_{k=-2n}^{2n} (-1)^k b_{2n+k}$ is the topological Euler characteristic of $X$.
Proof of Main Theorem
=====================
Inspired by Salamon’s relation , we define a constant $s(W)$ associated to an arbitrary $\mathfrak g$-module $W$.
Let $W$ be a $\mathfrak g$-module and $W = \bigoplus_k W_k$ its $h$-eigenspace decomposition in . Assume $\sum_k (-1)^k \dim W_k\neq 0$ (N.B. This is automatic if $W$ is either even or odd). Then we define a constant $s(W)\in {\mathbb{Q}}$ associated to $W$ by $$s(W) = \frac{\sum_{k \in {\mathbb{Z}}} (-1)^k k^2 \dim W_k}{\sum_{k \in {\mathbb{Z}}} (-1)^k \dim W_k} .$$ In particular, if $e(X) \neq 0$, Salamon’s relation reads $$\label{eq:salamon_new}
s(H^*(X)) = \frac{n}{3} .$$
The case of odd cohomology will be easily handled by Corollary \[cor:odd\_weight\_bound\]. Thus, we can focus on the case of vanishing odd cohomology (in particular, $e(X) \neq 0$). The main content then is to bound the value $s(H^* (X))$ in terms of $b_2$ and the LLV decomposition . Once this is done, assuming our conjecture , Salamon’s relation leads to the desired inequality between $b_2$ and $n$. Let us start from some straightforward properties of the constant $s(W)$.
\[prop:s\_prop1\] Let $\{ W_i \}_{i \in I}$ be a finite set of $\mathfrak g$-modules with well-defined $s(W_i)$.
1. If all $W_i$ are simultaneously even or odd, then $\min_i \{ s(W_i) \} \le s \left( \bigoplus_i W_i \right) \le \max_i \{ s(W_i) \}$.
2. $s \left( \bigotimes_i W_i \right) = \sum_i s(W_i)$.
It is enough to prove the proposition for two $\mathfrak g$-modules $W$ and $W'$. Assume without loss of generality $s(W) \le s(W')$, and let us consider the case when $W$ and $W'$ are even (the odd case is similar). In this case, all eigenvalues $k$ of $W$ are even, so we have $$\sum_k k^2 \dim W_k = s(W) \dim W, \qquad \sum_k k^2 \dim W'_k = s(W') \dim W' .$$ Adding the two equalities and using $s(W) \le s(W')$ gives us the first item.
For the second item, we compute $$\begin{aligned}
\sum_k (-1)^k k^2 \dim (W \otimes W')_k &= \sum_k (-1)^k k^2 \left( \sum_{i+j = k} \dim W_i \dim W'_j \right) \\
&= \sum_{i, j} (-1)^{i+j} (i^2 + 2ij + j^2) \dim W_i \dim W'_j \\
&= \left( \sum_i (-1)^i i^2 \dim W_i \right) e(W') + \left( \sum_j (-1)^j j^2 \dim W'_j \right) e(W) \\
& \qquad + 2 \left( \sum_i (-1)^i i \dim W_i \right) \left( \sum_j (-1)^j j \dim W'_j \right) .
\end{aligned}$$ Here we used the notation $e(W) = \sum_i (-1)^i \dim W_i$ and $e(W') = \sum_j (-1)^j \dim W'_j$ for simplicity. Notice that $\sum_i (-1)^i i \dim W_i = 0$, since by Weyl symmetry we always have $\dim W_i = \dim W_{-i}$. This proves $\sum_k (-1)^k k^2 \dim (W \otimes W')_k = (\sum_i (-1)^i i^2 \dim W_i) e(W') + (\sum_j (-1)^j j^2 \dim W'_j) e(W)$. Dividing both hand sides by $e(W \otimes W') = e(W) e(W')$ gives us the result.
In fact, we can associate to an arbitrary $\mathfrak g$-module $W$ the following formal power series $$S(W) = \sum_k (-1)^k \dim W_k \cdot \exp (kt) \ \ \in \ \ {\mathbb{Q}}[[t]] .$$ One can easily show $$S(W \oplus W') = S(W) + S(W') , \quad S(W \otimes W') = S(W)\cdot S(W') ,$$ so that $S$ defines a *ring homomorphism* from the representation ring $K(\mathfrak g)$ of $\mathfrak g$ $$S : K(\mathfrak g) \to {\mathbb{Q}}[[t]] .$$ By Weyl symmetry, we have $\dim W_k = \dim W_{-k}$, giving that all the odd degree terms of $S(W)$ vanish. Thus, we can write $$S(W) = s_0 + s_2 t^2 + s_4 t^4 + \cdots \ \ \in \ \ {\mathbb{Q}}[[t]] ,\qquad s_i = \frac{1}{i!} \sum_k (-1)^k k^i \dim W_k .$$ From this perspective, our constant $s(W)$ is the ratio between the first two coefficients $$s(W)=\frac{2s_2}{s_0}$$ of the formal power series $S(W)$.
A more interesting result is the explicit computation of $s(W)$ for irreducible $\mathfrak g$-modules $W = V_{\mu}$.
\[thm:generalized\_weyl\] Let $V_{\mu}$ be an irreducible $\mathfrak g$-module of highest weight $\mu$. If $\mu_r \ge 0$, then $$s(V_{\mu}) = 8 \cdot \frac{\left( \sum_{i=0}^r \mu_i \right) b_2 + \left( \sum_{i=0}^r (\mu_i - i)^2 - i^2 \right)}{(b_2+1)(b_2+2)} .$$ If $b_2$ is even and $\mu_r < 0$, then $s(V_{\mu}) = s(V_{\mu'})$ where $\mu' = (\mu_0, \cdots, \mu_{r-1}, -\mu_r)$.
We postpone the proof of to the following section. For now let us conclude the proof of our Main Theorem using this result. First, we note the following consequence of .
\[cor:s\_prop2\]
1. $s(V_{(m)}) = \frac{8m (b_2+m)} {(b_2+1)(b_2+2)}$ for $m \in {\mathbb{Z}}_{\ge 0}$.
2. If $\mu$ is even, then $s(V_{\mu}) \le s(V_{(m)})$ for $m = \mu_0 + \cdots + \mu_{r-1} + |\mu_r|$.
3. $s(V_{(m)}) \le s(V_{(n)})$ for $m \le n$.
The first item is immediate from letting $\mu = m \varepsilon_0$ in . The third item follows from it directly. For the second item, since $s(V_{\mu}) = s(V_{\mu'})$, we can assume without loss of generality $\mu_r \ge 0$. Let us temporarily define a function $A(\mu)$ of a dominant integral weight $\mu$ by $$A(\mu) = \sum_{i = 0}^r (\mu_i - i)^2 .$$ The second item is equivalent to the inequality $A(\mu) \le A(m \varepsilon_0)$. For it, one first proves an inequality $$\label{eq:induction}
(\mu_i - i)^2 + (\mu_j - j)^2 < (\mu_i + 1 - i)^2 + (\mu_j - 1 - j)^2 \qquad \mbox{for } \ 0 \le i < j \le r ,$$ which easily follows from $\mu_i \ge \mu_j$. The desired $A(\mu) \le A(m \varepsilon_0)$ follows from inductively applying the inequality to modify the dominant integral weight $\mu$ until it reaches $m \varepsilon_0$.
\[Proof of Main Theorem\] Assume $H^k (X) \neq 0$ for some odd integer $k$. By , there exists at least one component $V_{\mu} \subset H^*_{\operatorname{odd}} (X)$ with $\mu_0 + \cdots + \mu_{r-1} + |\mu_r| \ge n - \frac{k}{2} + \frac{r}{2}$. Thus, under the condition , we get $r \le k$ and hence $b_2 = (2r \mbox{ or } 2r+1) \le 2k+1$.
Now assume $H^*_{\operatorname{odd}} (X) = 0$. Among the irreducible components $V_{\mu}$ of the LLV decomposition , we always have the Verbitsky component $V_{(n)}$. Thus, if we assume the condition holds for $X$, then combining with gives us $$s(H^* (X)) \le \max \{ s(V_{\mu}) : \mu \mbox{ appearing in \eqref{eq:llv_decomp}} \} = s(V_{(n)}) = \frac{8n (b_2 + n)} {(b_2 + 1)(b_2 + 2)} .$$ On the other hand, we have the Salamon’s relation $s(H^* (X)) = \frac{n}{3}$ in . We conclude $$s(H^*(X)) = \frac{n}{3} \le \frac{8n (b_2 + n)} {(b_2 + 1)(b_2 + 2)} ,$$ giving the desired bound on $b_2$ in Main Theorem.
Computation of $s(W)$ for irreducible $\mathfrak g$-modules
===========================================================
In this section, we prove by using standard representation theoretic methods.
Let us first fix the notation. Here, we simply write $b = b_2(X)$. Let $(V, q)$ be a quadratic space of dimension $b+2$ and $\mathfrak g = {\mathfrak{so}}(V, q)$ be the associated simple Lie algebra of type $\mathrm B_{r+1}$ / $\mathrm D_{r+1}$. We fix a Cartan and Borel subalgebra of $\mathfrak g$ so that the positive and simple roots are well defined. We also use the following notation:
- $\mathfrak W$ is the Weyl group of $\mathfrak g$;
- $R_+$ is the set of positive roots of $\mathfrak g$;
- For $w \in \mathfrak W$, $\ell (w)$ is the length of $w$. That is, $\ell (w)$ is the minimum length of the decomposition of $w$ into a product of simple reflexions $w = s_{\alpha_1} \cdots s_{\alpha_{\ell}}$ where all $\alpha_i$ are simple roots of $\mathfrak g$;
- $\rho$ is the half sum of all the positive roots $$\label{eq:rho}
\rho = \frac{1}{2} \sum_{\alpha \in R_+} \alpha .$$
Throughout, we consider an irreducible representation $W=V_\mu$ of highest weight $\mu$.
Our proof is inspired by the proof of Weyl dimension formula (following [@kirillov §8.5]). The Weyl dimension formula is a closed formula computing $\dim V_{\mu}$, which can be derived from the Weyl character formula.
\[Weyl character formula\] \[thm:weyl\_character\_formula\] The formal character of the irreducible $\mathfrak g$-module $V_{\mu}$ of highest weight $\mu$ can be computed from a formal power series expansion of the rational function $$\operatorname{ch}(V_{\mu}) = \frac{\sum_{w \in \mathfrak W} (-1)^{\ell (w)} e^{w . (\mu + \rho)}}{\prod_{\alpha \in R_+} (e^{\alpha / 2} - e^{- \alpha / 2})} .$$
Due to its importance to our proof of , let us review first the proof of Weyl dimension formula. To start, we recall the formal character $\operatorname{ch}(V_{\mu})$ lives in the representation ring ${\mathbb{Z}}[\Lambda]^{\mathfrak W} \subset {\mathbb{Q}}[\Lambda]$ of $\mathfrak g$, where $\Lambda$ is the weight lattice of $\mathfrak g$. We introduce a ring homomorphism “projection to $\rho$-direction” $$\mathrm{pr}_{\rho} : {\mathbb{Q}}[\Lambda] \to {\mathbb{Q}}[q^{\pm 1}] , \qquad e^{\theta} \mapsto q^{4(\rho, \theta)} ,$$ where $(,)$ is the Killing form of $\mathfrak g$ and $\rho$ is the one defined in . We then define a polynomial $f(q) = f_{\mu}(q) \in {\mathbb{Q}}[q^{\pm 1}]$ as the formal character $\operatorname{ch}(V_{\mu})$ applied by $\mathrm{pr}_{\rho}$ $$\label{eq:f}
f(q) = \sum_{\theta} \dim V_{\mu}(\theta) \: q^{4(\rho, \theta)} .$$ Clearly, the dimension of $V_{\mu}$ is encoded in $f$ by $$\label{eq:dimension}
\dim V_{\mu} = f(1) .$$ If we apply $\mathrm{pr}_{\rho}$ to the Weyl character formula above, then using Weyl denominator identity (e.g., [@kirillov Thm 8.39]), the Weyl character formula is translated into $$\label{eq:weyl_character}
f(q) = \prod_{\alpha \in R_+} \frac{q^{2(\mu + \rho, \alpha)} - q^{-2(\mu + \rho, \alpha)}} {q^{2(\rho, \alpha)} - q^{-2(\rho, \alpha)}} .$$ The Weyl dimension formula is obtained by computing $f(1) = \lim_{q \to 1} f(q)$ with the aid of .
Now let us begin the proof of . First, notice that for irreducible modules $V_{\mu}$, we can ignore the sign terms $(-1)^k$ in the definition of $s(V_{\mu})$ (i.e., $V_\mu$ is either even or odd). Thus, we have $$s(V_{\mu}) = \frac{\sum_k k^2 \dim (V_{\mu})_k}{\dim V_{\mu}} .$$ The following lemma expresses $s(V_{\mu})$ in terms of $f(q)$, imitating above.
\[lem:step1\] Let $f = f(q)$ be as in . Then $s(V_{\mu}) = \frac{6}{b(b+1)(b+2)} (\log f)'' (1)$.
Consider the derivates of $f$ $$f'(q) = \sum_{\theta} 4(\rho, \theta) \dim V_{\mu} (\theta) \: q^{4(\rho, \theta) - 1}, \qquad f'' (q) = \sum_{\theta} 4(\rho, \theta) (4(\rho, \theta) - 1) \dim V_{\mu} (\theta) \: q^{4(\rho, \theta) - 2} .$$ The Weyl symmetry gives us $\dim V_{\mu} (\theta) = \dim V_{\mu} (-\theta)$. From it, we obtain $f'(1) = 0$ and $$f'' (1) = 16 \sum_{\theta} (\rho, \theta)^2 \dim V_{\mu} (\theta) .$$ Let us now specialize the discussion to $\mathfrak g = {\mathfrak{so}}(V, q)$ and use the precise value of $\rho$. We have $$\begin{aligned}
\label{eq:rho_explicit}
\rho = \begin{cases}
r \varepsilon_0 + (r-1) \varepsilon_1 + \cdots + \varepsilon_{r-1} & \mbox{when } \ b = 2r \mbox{ is even} \\
(r + \tfrac{1}{2}) \varepsilon_0 + (r - \tfrac{1}{2}) \varepsilon_1 + \cdots + \tfrac{1}{2} \varepsilon_r \quad & \mbox{when } \ b = 2r + 1 \mbox{ is odd}
\end{cases} .
\end{aligned}$$
Assume $b = 2r$ is even. Letting $\theta = \sum_{i=0}^r \theta_i \varepsilon_i$, we have $(\rho, \theta) = \sum_{i=0}^r (r-i)\theta_i$. This gives us $$f''(1) = 16 \sum_{\theta} \left( \sum_{i=0}^r (r-i)^2 \theta_i^2 + 2 \sum_{0 \le i < j \le r} (r-i)(r-j) \theta_i \theta_j \right) \dim V_{\mu} (\theta) .$$ Again by Weyl symmetry, we have $\dim V_{\mu} (\theta) = \dim V_{\mu} (w.\theta)$ for any $w \in \mathfrak W$. Therefore we deduce
- $\sum_{\theta} \theta_i^2 \dim V_{\mu} (\theta) = \sum_{\theta} \theta_j^2 \dim V_{\mu} (\theta)$; and
- $\sum_{\theta} \theta_i \theta_j \dim V_{\mu} (\theta) = 0$ for $i \neq j$.
This finally leads us to the identity $$\begin{aligned}
f''(1) &= 16 \cdot \left( r^2 + (r-1)^2 + \cdots + 1^2 \right) \sum_{\theta} \theta_0^2 \dim V_{\mu} (\theta) \\
&= 16 \cdot \frac{r(r+1)(2r+1)}{6} \sum_k \left( \frac{k}{2} \right)^2 \dim (V_{\mu})_k
= \frac{b(b+1)(b+2)}{6} \sum_k k^2 \dim (V_{\mu})_k .
\end{aligned}$$ Combining it with $f(1) = \dim V_{\mu}$ and $f'(1) = 0$, we have $(\log f)''(1) = \frac{f''(1)}{f(1)} = \frac{b(b+1)(b+2)}{6} s(V_{\mu})$, as claimed.
Next, assume $b = 2r+1$ is odd. Similar argument gives us the computation $$f''(1) = \frac{(r+1)(2r+1)(2r+3)}{3} \sum_k k^2 \dim (V_{\mu})_k = \frac{b(b+1)(b+2)}{6} \sum_k k^2 \dim (V_{\mu})_k .$$ Hence the same result follows, regardless of the parity of $b$.
The next step is to use the Weyl character formula and compute the value $(\log f)''(1)$.
\[lem:step2\] Let $f = f(q)$ be as in . If $\mu_r \ge 0$, then $(\log f)''(1) = \frac{4}{3} b \left[ \left( \sum_{i=0}^r \mu_i \right) b + \left( \sum_{i=0}^r \mu_i^2 - 2i\mu_i \right) \right]$.
From the $q$-polynomial version of the Weyl character formula , we derive $$\log f(q) = \sum_{\alpha \in R_+} \log \left( \frac{q^{2(\mu + \rho, \alpha)} - q^{-2(\mu + \rho, \alpha)}} {q - 1} \right) - \log \left( \frac{q^{2(\rho, \alpha)} - q^{-2(\rho, \alpha)}} {q - 1} \right) .$$ Notice that $(\log f)''(1)$ is twice the coefficient of the term $(q-1)^2$ in the Taylor series of $\log f$. For a general *positive integer* $a$, the Taylor series expansion of $\log \left( \frac{q^a - q^{-a}}{q-1} \right)$ at $q = 1$ is $$\log \left( \frac{q^a - q^{-a}} {q-1} \right) = \log (2a) - \frac{1}{2} (q-1) + \frac{1}{24} (4a^2 + 5) (q-1)^2 + \cdots ,$$ which has a degree $2$ coefficient $\frac{1}{24} (4a^2 + 5)$. Since $2(\mu+\rho, \alpha)$ and $2(\rho, \alpha)$ are both positive integers for any $\alpha \in R_+$ (N.B. Here we used the fact $\mu_r \ge 0$), we conclude $$\begin{aligned}
(\log f)''(1) &= 2 \sum_{\alpha \in R_+} \frac{1}{24} (16 (\mu + \rho, \alpha)^2 + 5) - \frac{1}{24} (16 (\rho, \alpha)^2 + 5) \\
&= \frac{4}{3} \sum_{\alpha \in R_+} (\mu + \rho, \alpha)^2 - (\rho, \alpha)^2 .
\end{aligned}$$
Recalling , let us get into an explicit computation for $\mathfrak g = {\mathfrak{so}}(V, q)$. Assume $b = 2r$ is even. The positive roots are $R_+ = \{ \varepsilon_i \pm \varepsilon_j : 0 \le i < j \le r \}$. We get $$\begin{aligned}
&\sum_{\alpha \in R_+} (\mu + \rho, \alpha)^2 - (\rho, \alpha)^2 \\
=& \: \sum_{0 \le i < j \le r} (\mu_i - \mu_j)^2 + 2 (\mu_i - \mu_j) (j-i) + (\mu_i + \mu_j)^2 + 2 (\mu_i + \mu_j) (2r - i - j) \\
=& \: \sum_{0 \le i < j \le r} \left[ 2(\mu_i^2 + \mu_j^2) + 4r (\mu_i + \mu_j) - (i\mu_i + j\mu_j) \right] \\
=& \: 2r \left[ 2r \sum_{i=0}^r \mu_i + \sum_{i=0}^r (\mu_i^2 - 2i \mu_i) \right]
= b \left[ \left( \sum_{i=0}^r \mu_i \right) b + \left( \sum_{i=0}^r \mu_i^2 - 2i \mu_i \right) \right] .
\end{aligned}$$ This proves the result in this case.
Similarly for $b = 2r+1$ odd, the positive roots are $R_+ = \{ \varepsilon_i : 0 \le i \le r \} \cup \{ \varepsilon_i \pm \varepsilon_j : 0 \le i < j \le r \}$, giving: $$\begin{aligned}
&\sum_{\alpha \in R_+} (\mu + \rho, \alpha)^2 - (\rho, \alpha)^2 \\
=& \: \sum_{0 \le i < j \le r} (\mu_i - \mu_j)^2 + 2 (\mu_i - \mu_j) (j-i) + (\mu_i + \mu_j)^2 + 2 (\mu_i + \mu_j) (2r + 1 - i - j) \\
& \: + \sum_{i=0}^r \mu_i^2 + 2 \mu_i (r + \tfrac{1}{2} - i) \\
=& \: \sum_{0 \le i < j \le r} \left[ 2(\mu_i^2 + \mu_j^2) + (4r + 2) (\mu_i + \mu_j) - (i\mu_i + j\mu_j) \right] + \sum_{i=0}^r \mu_i^2 + (2r+1) \mu_i - 2i\mu_i \\
=& \: (2r + 1) \left[ (2r + 1) \sum_{i=0}^r \mu_i + \sum_{i=0}^r (\mu_i^2 - 2i \mu_i) \right]
= b \left[ \left( \sum_{i=0}^r \mu_i \right) b + \left( \sum_{i=0}^r \mu_i^2 - 2i \mu_i \right) \right] .
\end{aligned}$$ This completes the proof of the lemma.
Combining and \[lem:step2\], the theorem follows for the case $\mu_r \ge 0$. Now assume $b_2 = 2r$ is even and $\mu_r < 0$. In this case, $\rho$ does not have the $\varepsilon_r$-coordinate by . Hence, the Weyl character formula () implies that the weights associated to $V_{\mu}$ and $V_{\mu'}$ are bijective via the action $(\theta_0, \cdots, \theta_{r-1}, \theta_r) \mapsto (\theta_0, \cdots, \theta_{r-1}, -\theta_r)$. By definition, the constant $s(W)$ captures only the $h$-eigenspaces, i.e., only the $\varepsilon_0$-coordinates of the weights associated to $W$. This means $s(V_{\mu}) = s(V_{\mu'})$.
[^1]: The two authors were partially supported by NSF grant DMS-1802128.
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---
abstract: 'We give a new proof of the Skeletal Lemma, which is the main technical tool in our paper on Hamilton cycles in line graphs \[T. Kaiser and P. Vrána, Hamilton cycles in 5-connected line graphs, European J. Combin. 33 (2012), 924–947\]. It generalises results on disjoint spanning trees in graphs to the context of 3-hypergraphs. The lemma is proved in a slightly stronger version that is more suitable for applications. The proof is simplified and formulated in a more accessible way.'
author:
- |
Tomáš Kaiser$^{\:1}$\
Petr Vrána$^{\:1}$
title: '**Quasigraphs and skeletal partitions**'
---
Introduction {#sec:introduction}
============
The main tool used in our work on Hamilton cycles in line graphs [@KV-hamilton] is a result called ‘Skeletal Lemma’ [@KV-hamilton Lemma 17]. It deals with quasigraphs in 3-hypergraphs (see below for definitions) and is related to Tutte’s and Nash-Williams’ characterisation of graphs with two disjoint spanning trees.
In our recent paper [@KV-ess-9], we need to use the lemma in a slightly stronger form that unfortunately does not follow from the formulation given in the paper. Instead of pointing out the necessary modifications to the long and complicated proof, we decided to use this opportunity to rewrite the proof completely, trying to formulate it in a way as conceptually simple as we can. That is the purpose of the present paper which is a companion paper to [@KV-ess-9]. In addition, the present paper aims to give the full proof in detail, even in parts where the argument in [@KV-hamilton] is somewhat sketchy.
The structure of the paper is as follows. In Section \[sec:quasi\], we review the basic notions related to quasigraphs, the structures forming a central concept of our proof. In Section \[sec:anti-conn\], we develop the basic properties of the notion of connectivity and especially anticonnectivity of a quasigraph on a set of vertices. This allows us to define, for any quasigraph, a sequence of successively more and more refined partitions of the vertex set that serves as a measure of ‘quality’ of the quasigraph. This is done in Section \[sec:sequence\]. Section \[sec:skeletal\] gives the proof of the main result, a stronger version of the Skeletal Lemma (Theorem \[t:enhancing\]). Finally, in Section \[sec:bad\], we infer the result we need for the above mentioned application in [@KV-ess-9] (Theorem \[t:no-bad\]).
Quasigraphs {#sec:quasi}
===========
A *$3$-hypergraph* is a hypergraph whose hyperedges have size $2$ or $3$. Throughout this paper, let $H$ be a 3-hypergraph. A *quasigraph* in $H$ is a mapping $\pi$ that assigns to each hyperedge $e$ of $H$ either a subset of $e$ of size 2, or the empty set. The hyperedges $e$ with $\pi(e)\neq \emptyset$ are said to be *used* by $\pi$. (See Figure \[fig:quasi\] for an illustration.) The number of hyperedges used by $\pi$ is denoted by ${\|\pi\|}$.
Given a quasigraph $\pi$ in $H$, we let $\pi^*$ denote the graph on $V(H)$, obtained by considering the pairs $\pi(e')$ as edges whenever $\pi(e')\neq\emptyset$ ($e'\in E(H)$). If $\pi^*$ is a forest, then $\pi$ is *acyclic*. If $\pi^*$ is the union of a cycle and a set of isolated vertices, then $\pi$ is a *quasicycle*. A 3-hypergraph $H$ is *acyclic* if there exists no quasicycle in $H$.
If $e$ is a hyperedge of $H$, then we define $\pi-e$ as the quasigraph which satisfies $(\pi-e)(e) = \emptyset$, and coincides with $\pi$ on all hyperedges other than $e$. If $e$ is a hyperedge not used by $\pi$, and if $u,v\in e$, then $\pi+(uv)_e$ is the quasigraph that coincides with $\pi$ except that its value on $e$ is $uv$ rather than $\emptyset$.
The *complement* $\overline\pi$ of $\pi$ is the subhypergraph of $H$ (on the same vertex set) consisting of the hyperedges not used by $\pi$.
Let ${\mathcal P}$ be a partition of $V(H)$. We say that ${\mathcal P}$ is *nontrivial* if ${\mathcal P}\neq{\left\{{V(H)}\right\}}$. If $X\subseteq V(H)$, then the partition ${\mathcal P}[X]$ of $X$ *induced* by ${\mathcal P}$ has all nonempty intersections $P\cap X$, where $P\in{\mathcal P}$, as its classes.
If $e\in E(H)$, then $e/{\mathcal P}$ is defined as the set of all classes of ${\mathcal P}$ intersected by $e$. If there is more than one such class, then $e$ is said to be *${\mathcal P}$-crossing*. The hypergraph $H/{\mathcal P}$ has vertex set ${\mathcal P}$ and its hyperedges are all the sets of the form $e/{\mathcal P}$, where $e$ is a ${\mathcal P}$-crossing hyperedge of $H$. Thus, $H/{\mathcal P}$ is a 3-hypergraph. A quasigraph $\pi/{\mathcal P}$ in this hypergraph is defined by setting, for every ${\mathcal P}$-crossing hyperedge $e$ of $H$, $$(\pi/{\mathcal P})(e/{\mathcal P}) =
\begin{cases}
\pi(e)/{\mathcal P}& \text{if $\pi(e)$ is ${\mathcal P}$-crossing,}\\
\emptyset & \text{otherwise.}
\end{cases}$$
We extend the above notation and write, e.g., $uv/{\mathcal P}$ for the set of classes of ${\mathcal P}$ intersecting ${\left\{{u,v}\right\}}$, where $u,v\in V(H)$.
We often consider quasigraphs $\gamma$ in the complement of the quasigraph $\pi/{\mathcal P}$ (typically, such a $\gamma$ is a quasicycle). Note that $\gamma$ assigns a value to each hyperedge $e/{\mathcal P}$ such that $\pi(e)$ does not cross ${\mathcal P}$ (including $\pi(e)=\emptyset$). The situation is illustrated in Figure \[fig:complement\].
(Anti)connectivity {#sec:anti-conn}
==================
In this section, we define and explore the notions of components and anticomponents of a quasigraph (on a set of vertices) that are completely essential for our arguments.
Recall that $H$ denotes a 3-hypergraph. Let $\pi$ be a quasigraph in $H$ and $X\subseteq V(H)$. We say that $\pi$ is *connected on $X$* if the induced subgraph of $\pi^*$ on $X$ is connected. The *components* of $\pi$ on $X$ are defined as the vertex sets of the connected components of the induced subgraph of $\pi^*$ on $X$.
We say that $\pi$ is *anticonnected on $X$* if for each nontrivial partition ${\mathcal R}$ of $X$, there is a hyperedge $f$ of $H$ intersecting at least two classes of ${\mathcal R}$, and such that $\pi(f)$ is a subset of one of the classes of ${\mathcal R}$ (possibly $\pi(f)=\emptyset$). If we need to refer to the hypergraph $H$, we say that $\pi$ is *anticonnected on $X$ in $H$*.
The above notions are illustrated in Figure \[fig:anti\].
7
\[l:union\] Let $\pi$ be a quasigraph in (a $3$-hypergraph) $H$ and $X,Y$ subsets of $V(H)$ such that $\pi$ is anticonnected on $X$ and $Y$. Then $\pi$ is anticonnected on $X\cup Y$ whenever one of the following holds:
1. $X$ and $Y$ intersect, or
2. there is a hyperedge $h$ of $H$ intersecting both $X$ and $Y$, such that $\pi(h)$ is a subset of $X$ or $Y$ (possibly $\pi(h)=\emptyset$).
Let ${\mathcal R}$ be a nontrivial partition of $X\cup Y$. We find for ${\mathcal R}$ the hyperedge whose existence is required by the definition of anticonnectedness.
Suppose first that ${\mathcal R}[X]$ is nontrivial. Since $\pi$ is anticonnected on $X$, there is a hyperedge $f$ of $H$ such that $f$ intersects at least two classes of ${\mathcal R}[X]$ and one of them contains $\pi(f)$. Thus, $f$ intersects at least two classes of ${\mathcal R}$ and one of them contains $\pi(f)$.
We can thus assume, by symmetry, that both ${\mathcal R}[X]$ and ${\mathcal R}[Y]$ are trivial. This implies that ${\mathcal R}={\left\{{X,Y}\right\}}$, so $X$ and $Y$ are disjoint. In this case, the hyperedge $h$ from (ii) has the required property.
By Lemma \[l:union\], the maximal sets $Y\subseteq X$ such that $\pi$ is anticonnected on $Y$ partition $X$. We call them the *anticomponents* of $\pi$ on $X$.
\[l:no-change\] Let $\pi$ and $\rho$ be quasigraphs in $H$ and $Y$ be a subset of $V(H)$ such that $\pi(e) = \rho(e)$ for every hyperedge of $H$ with ${\left|{e\cap Y}\right|} \geq 2$. Then $\pi$ is anticonnected on $Y$ if and only $\rho$ is anticonnected on $Y$.
Suppose that $\pi$ is anticonnected on $Y$ and let ${\mathcal R}$ be a nontrivial partition of $Y$. Consider a hyperedge $f$ of $H$ such that $f$ intersects two classes of ${\mathcal R}$ and one of them contains $\pi(f)$. By the assumption, $\rho(f) = \pi(f)$, so the same holds for $\rho$ in place of $\pi$. Since ${\mathcal R}$ is arbitrary, $\rho$ is anticonnected on $Y$. The lemma follows by symmetry.
We prove several further lemmas that describe some of the basic properties of (anti)connectivity of quasigraphs.
\[l:H-e\] Let $\pi$ be a quasigraph in $H$, $X\subseteq V(H)$ and $e$ a hyperedge of $H$ with ${\left|{e\cap X}\right|}\leq 1$. If $\pi$ is anticonnected on $X$ in $H$, then $\pi$ is anticonnected on $X$ in $H-e$.
Let ${\mathcal R}$ be a nontrivial partition of $X$. Since $\pi$ is anticonnected on $X$ in $H$, there is a hyperedge $f$ of $H$ intersecting at least two classes of ${\mathcal R}$, one of which contains $\pi(f)$. The hyperedge $f$ is distinct from $e$ as ${\left|{e\cap X}\right|} \leq 1$. Thus, $f \in E(H-e)$. Since ${\mathcal R}$ is arbitrary, $\pi$ is anticonnected on $X$ in $H-e$.
\[l:sub-add\] Let $\pi$ be a quasigraph in $H$ and $Y\subseteq X$ subsets of $V(H)$. Suppose that $e$ is a hyperedge of $H$ not used by $\pi$ and containing vertices $u,v\in Y$. Define $\rho$ as the quasigraph $\pi
+ (uv)_e$. The following holds:
1. if $\pi$ is anticonnected on $X$ and $\rho$ is anticonnected on $Y$, then $\rho$ is anticonnected on $X$,
2. if $\pi$ is connected on $X$, then so is $\rho$.
We prove (i). Consider an arbitrary partition ${\mathcal R}$ of $X$. We aim to show that there is a hyperedge $f$ of $H$ such that $f$ intersects two classes of ${\mathcal R}$ and $\rho(f)$ is contained in one of them. This is certainly true if ${\mathcal R}[Y]$ is nontrivial, since $\rho$ is assumed to be anticonnected on $Y$. Thus, we may assume that $Y$ is contained in a class of ${\mathcal R}$.
Since $\pi$ is anticonnected on $X$, there is a hyperedge $h$ of $H$ such that $h$ intersects two classes of ${\mathcal R}$ and $\pi(h)$ is contained in one of them. We set $f := h$. If $h\neq e$, this choice works because $\rho(h) = \pi(h)$. If $h=e$, then $\rho(h)$ is contained in $Y$ and therefore in a class of ${\mathcal R}$. This concludes the proof of (i).
Part (ii) follows directly from the fact that $\pi^*$ is a subgraph of $\rho^*$, and therefore the induced subgraph of $\pi^*$ on $X$ is a subgraph of the induced subgraph of $\rho^*$ on $X$.
\[l:sub-remove\] Let $\pi$ be a quasigraph in $H$ and $Y\subseteq X$ subsets of $V(H)$. Suppose that $e$ is a hyperedge of $H$ with $\pi(e)\subseteq
Y$. Define $\sigma$ as the quasigraph $\pi-e$. It holds that
1. if $\pi$ is anticonnected on $X$, then so is $\sigma$,
2. if $\pi$ is connected on $X$ and $\sigma$ is connected on $Y$, then $\sigma$ is connected on $X$.
We prove (i). Suppose, for contradiction, that $\sigma$ is not anticonnected on $X$. By the definition, there is a partition ${\mathcal S}$ of $X$ such that for all hyperedges $f$ of $H$ intersecting at least two classes of ${\mathcal S}$, $\sigma(f)$ intersects two classes of ${\mathcal S}$ as well. On the other hand, since $\pi$ is anticonnected on $X$ and $\pi=\sigma$ except for the value at $e$, it must be that the hyperedge $e$ intersects two classes of ${\mathcal S}$ (and $\pi(e)$ is contained in one class). Since $\sigma(e)=\emptyset$, we obtain a contradiction.
Next, we prove (ii). Note that $\sigma^*$ is a subgraph of $\pi^*$. Since $\sigma$ is connected on $Y$, so is $\pi$. We show that $\sigma$ is connected on $X$. Let $\pi^*_X$ be the induced subgraph of $\pi^*$ on $X$, and let $\pi(e) = {\left\{{u,v}\right\}}$. We need to prove that any two vertices in $X$ are joined by a walk in the induced subgraph of $\sigma^*$ on $X$, which equals $\pi^*_X-uv$. This is easy from the fact that $\pi^*_X$ is connected, and that the edge $uv$ may be replaced in any walk by a path from $u$ to $v$ in the induced subgraph of $\sigma^*$ on $Y$ (which is connected).
Let us now define two notions that will play a role when we introduce the sequence of a quasigraph in Section \[sec:sequence\]. Suppose that $X\subseteq V(H)$ such that the quasigraph $\pi$ is both connected and anticonnected on $X$. Let $e$ be a hyperedge with ${\left|{e\cap X}\right|} = 2$.
We say that $e$ is an *$X$-bridge* (with respect to $\pi$) if $e$ is used by $\pi$, $\pi(e) \subseteq X$, and $\pi-e$ is not connected on $X$ in $H-e$. Similarly, $e$ is an *$X$-antibridge* (with respect to $\pi$) if $e$ is not used by $\pi$ and $\pi$ is not anticonnected on $X$ in $H-e$.
8
The plane sequence of a quasigraph {#sec:sequence}
==================================
Let $\pi$ be a quasigraph in a $3$-hypergraph $H$. In [@KV-hamilton], we associate with $\pi$ a sequence of partitions of $V(H)$. In the present paper, we proceed similarly, but for technical reasons, we need to extend the original definition to involve a two-dimensional analogue of a sequence. A *plane sequence* is a family $({\mathcal P}_{i,j})_{i,j\geq 0}$ of partitions of $V(H)$.
It will be convenient to consider the lexicographic order $\leq$ on pairs of nonnegative integers: $(i,j) \leq (i',j')$ if either $i < i'$, or $i = i'$ and $j \leq j'$. This is extended in the natural way to the set $${{\mathcal T}}= {\left\{{(i,j)}:\,{0\leq i < \infty, 0\leq j\leq\infty}\right\}}\cup{\left\{{(\infty,\infty)}\right\}}.$$ For instance, $(1,\infty) < (2,0) < (\infty,\infty)$. This is a well-ordering on the set ${{\mathcal T}}$, which allows us to perform induction over ${{\mathcal T}}$.
The *(plane) sequence of $\pi$*, denoted by ${\tilde{\mathbb{P}}^{\pi}}$, consists of partitions ${{\mathcal P}^{\pi}_{i,j}}$ of $V(H)$, where $(i,j)\in{{\mathcal T}}$. We let ${{\mathcal P}^{\pi}_{0,0}}$ be the trivial partition ${\left\{{V(H)}\right\}}$. If $j \geq 1$ and ${{\mathcal P}^{\pi}_{i,j-1}}$ is defined, then we let $${{\mathcal P}^{\pi}_{i,j}} =
\begin{cases}
{\left\{{K}:\,{K\text{ is a component of $\pi$ on some
$X\in{{\mathcal P}^{\pi}_{i,j-1}}$}}\right\}} & \text{if $j$ is
odd,}\\
{\left\{{K}:\,{K\text{ is an anticomponent of $\pi$ on some
$X\in{{\mathcal P}^{\pi}_{i,j-1}}$}}\right\}} & \text{if $j$ is even.}
\end{cases}$$ See Figure \[fig:seq\] for an example.
9
So far, this yields the partitions ${{\mathcal P}^{\pi}_{0,0}},{{\mathcal P}^{\pi}_{0,1}},\dots$. We first notice that since $H$ is finite, there is some $j_0$ such that $${{\mathcal P}^{\pi}_{0,j_0}} = {{\mathcal P}^{\pi}_{0,j_0 + 1}} = \dots,$$ and we set ${{\mathcal P}^{\pi}_{0,\infty}}$ equal to ${{\mathcal P}^{\pi}_{0,j_0}}$. We will use an analogous definition to construct ${{\mathcal P}^{\pi}_{i,\infty}}$ for $i > 0$ when ${{\mathcal P}^{\pi}_{i,0}},{{\mathcal P}^{\pi}_{i,1}},\dots$ will have been defined. (See Figure \[fig:layout\] for a schematic illustration.)
[![The order of partitions in the construction of a plane sequence of a quasigraph.[]{data-label="fig:layout"}](skeletal-figs.pdf "fig:")]{}
By the construction, ${{\mathcal P}^{\pi}_{0,\infty}}$ has the property that $\pi$ is both connected and anticonnected on each of its classes. We call any such partition of $V(H)$ *$\pi$-solid*.
The definition of the plane sequence of $\pi$ will be completed once we define ${{\mathcal P}^{\pi}_{i,0}}$ for all $i\geq 1$. Thus, let $i\geq 1$ be fixed, and suppose that the partition ${\mathcal P}:={{\mathcal P}^{\pi}_{i-1,\infty}}$ is already defined.
Let $A,B\in{\mathcal P}$. The *exposure step* of the pair $AB$ is the least $(s,t)$ (with respect to the ordering defined above) such that $A$ and $B$ are contained (as subsets) in different classes of ${{\mathcal P}^{\pi}_{s,t}}$. Similarly, for a pair of vertices $u,v$ of $H$ contained in different classes of ${\mathcal P}$, the exposure step of the pair $uv$ is the least $(s,t)$ such that $u$ and $v$ are contained in different classes of ${{\mathcal P}^{\pi}_{s,t}}$.
Suppose that $\gamma$ is a quasicycle in $\overline{\pi/{\mathcal P}}$. The *exposure step* of $\gamma$ is the least exposure step of $\gamma(e/{\mathcal P})$, where $e$ ranges over all hyperedges of $H$ such that $e/{\mathcal P}$ is used by $\gamma$. If the exposure step of $\gamma$ is $(s,t)$, we also say that $\gamma$ is *exposed* at (step) $(s,t)$. We say that a hyperedge $e$ of $H$ is a *leading hyperedge* of $\gamma$ if $e/{\mathcal P}$ is used by $\gamma$ and the exposure step of $\gamma(e/{\mathcal P})$ equals that of $\gamma$.
The terms ‘exposure step’ and ‘leading hyperedge’ are defined similarly for a cycle in the graph $(\pi/{\mathcal P})^*$, by viewing it as a quasicycle in $H/{\mathcal P}$. Later in this section, we will generalise these notions to the situation where the plane sequence has already been completely defined.
We extend the notions of $X$-bridge and $X$-antibridge defined in Section \[sec:anti-conn\] as follows: given a $\pi$-solid partition ${\mathcal R}$ of $V(H)$, $e$ is an *${\mathcal R}$-bridge* (*${\mathcal R}$-antibridge*) if there is $X\in{\mathcal R}$ such that $e$ is an $X$-bridge ($X$-antibridge, respectively).
We say that a hyperedge $e$ of $H$ crossing ${\mathcal R}$ is *redundant* (with respect to $\pi$ and ${\mathcal R}$) if $e$ is not used by $\pi$ and $e$ is not an ${\mathcal R}$-antibridge. Note that a hyperedge $e$ unused by $\pi$ is redundant if ${\left|{e}\right|} = 2$, or more generally, if each of its vertices is in a different class of ${\mathcal R}$.
Furthermore, a hyperedge $e$ of $H$ is *weakly redundant* (with respect to $\pi$ and ${\mathcal R}$) if either it is redundant, or it is used by $\pi$ and is not an ${\mathcal R}$-bridge.
We are now ready to define the partition ${{\mathcal P}^{\pi}_{i,0}}$. We will say that it is obtained from ${{\mathcal P}^{\pi}_{i-1,\infty}}$ by the *limit step $(i-1,\infty)$*. At the same time, we will define the *decisive hyperedge at $(i-1,\infty)$*, $d^\pi_{i-1}$, for the current limit step. This will be a hyperedge of $H$; for technical reasons, we also allow two extra values, ${\text{\textsc{stop}}}$ and ${\text{\textsc{terminate}}}$.
If the complement of $\pi/{\mathcal P}$ in $H/{\mathcal P}$ is acyclic, we define ${{\mathcal P}^{\pi}_{i,0}} = {\mathcal P}$ and say that $\pi$ *terminates at $(i-1,\infty)$*. We set $d^\pi_{i-1} = {\text{\textsc{terminate}}}$.
Otherwise, let $L$ be the set of such hyperedges $f$ of $H$ for which there exists a quasicycle $\gamma$ in $\overline{\pi/{\mathcal P}}$ such that $f$ is a leading hyperedge of $\gamma$. We define ${{\mathcal P}^{\pi}_{i,0}} = {\mathcal P}$ if $L$ contains a weakly redundant hyperedge $f$ (with respect to $\pi$ and ${\mathcal P}$). In this case, we say that $\pi$ *stops at $(i-1,\infty)$*; we set $d^\pi_{i-1} = {\text{\textsc{stop}}}$.
If no weakly redundant hyperedge exists in $L$, choose the maximum hyperedge $e$ in $L$ according to a fixed linear ordering $\leq_E$ of all hyperedges in $H$. (For the purposes of this and the following section, the choice of $\leq_E$ is not important; it will be discussed in more detail in Section \[sec:bad\]). Set $d^\pi_{i-1}$ equal to $e$. We say that $\pi$ *continues* at $(i-1,\infty)$. Moreover, in this case, any quasicycle in $\overline{\pi/{\mathcal P}}$ whose leading hyperedge is $e$ will be referred to as a *decisive quasicycle at $(i-1,\infty)$*.
Since $e$ is not weakly redundant, we can distinguish the following two cases for the definition of ${{\mathcal P}^{\pi}_{i,0}}$:
- if $e$ is a ${\mathcal P}$-antibridge, then the classes of ${{\mathcal P}^{\pi}_{i,0}}$ are all the anticomponents of $\pi$ on $X$ in $H-e$, where $X$ ranges over all classes of ${\mathcal P}$,
- if $e$ is a ${\mathcal P}$-bridge, then the classes of ${{\mathcal P}^{\pi}_{i,0}}$ are all the components of $\pi-e$ on $X$ in $H$, where $X$ ranges over all classes of ${\mathcal P}$.
The subsequent partitions ${{\mathcal P}^{\pi}_{i,1}},{{\mathcal P}^{\pi}_{i,2}},\dots$ are then defined as described above, and the partition ${{\mathcal P}^{\pi}_{i,\infty}}$ is defined analogously to ${{\mathcal P}^{\pi}_{0,\infty}}$. Iterating, we obtain the whole plane sequence ${\tilde{\mathbb{P}}^{\pi}}$ and the partitions ${{\mathcal P}^{\pi}_{0,\infty}},{{\mathcal P}^{\pi}_{1,\infty}},\dots$. By the finiteness of $H$, there is some $i_0$ such that ${{\mathcal P}^{\pi}_{i_0,\infty}} = {{\mathcal P}^{\pi}_{i_0+1,\infty}}$, and we define ${{\mathcal P}^{\pi}_{\infty,\infty}}$ as ${{\mathcal P}^{\pi}_{i_0,\infty}}$.
Observe that in the cases where $\pi$ stops or terminates at $(i-1,\infty)$ and we define ${{\mathcal P}^{\pi}_{i,0}}$ as ${{\mathcal P}^{\pi}_{i-1,\infty}}$, this partition will in fact equal ${{\mathcal P}^{\pi}_{\infty,\infty}}$ since none of the subsequent steps in the construction of the plane sequence of $\pi$ will lead to any modifications.
Now that the sequence of a quasigraph $\pi$ has been completely defined, let us revisit the definitions of the terms ‘exposure step’ and ‘leading hyperedge’. Although these are defined relative to a partition ${{\mathcal P}^{\pi}_{i-1,\infty}}$ (for some $i\geq 1$), this only affects the scope of the definitions: for instance, if vertices $u,v$ are contained in different classes of ${{\mathcal P}^{\pi}_{\ell,\infty}}$, where $\ell\geq i$, then the exposure step of $uv$ is the same whether we use ${{\mathcal P}^{\pi}_{i-1,\infty}}$ or ${{\mathcal P}^{\pi}_{\ell,\infty}}$ for the definition.
In particular, if we let ${\mathcal Q}={{\mathcal P}^{\pi}_{\infty,\infty}}$, then it makes sense to speak of leading hyperedges of any quasicycle in $\overline{\pi/{\mathcal Q}}$ or the exposure step of a pair of vertices contained in different classes of ${\mathcal Q}$.
We now define a partial order on the set of all quasigraphs in $H$ that is crucial for our argument. First, we define the *signature* ${\mathbb S^{\pi}}$ of a quasigraph $\pi$ as the sequence $${\mathbb S^{\pi}} =
({{\mathcal P}^{\pi}_{0,0}},{{\mathcal P}^{\pi}_{0,1}},\dots,{{\mathcal P}^{\pi}_{0,\infty}},d^\pi_0,{{\mathcal P}^{\pi}_{1,0}},\dots,
{{\mathcal P}^{\pi}_{1,\infty}},d^\pi_1,\dots,{{\mathcal P}^{\pi}_{\ell,\infty}},d^\pi_\ell),$$ where $\ell$ is minimum such that $d^\pi_\ell\in{\left\{{{\text{\textsc{terminate}}},{\text{\textsc{stop}}}}\right\}}$.
We derive from this an order ${\sqsubseteq}$ on quasigraphs in $H$, setting $\pi {\sqsubseteq}\rho$ if ${\mathbb S^{\pi}}$ is smaller than or equal to ${\mathbb S^{\rho}}$ in the lexicographic order on the set of signatures of quasigraphs.
It will be convenient to define several related notions to facilitate the comparison of quasigraphs. Let $(i,j)\in{{\mathcal T}}$. We define the *$(i,j)$-prefix* ${\mathbb S^{\pi}}_{(i,j)}$ of ${\mathbb S^{\pi}}$ as follows:
- if $j < \infty$, then ${\mathbb S^{\pi}}_{(i,j)}$ is the initial segment of ${\mathbb S^{\pi}}$ ending with (and including) ${{\mathcal P}^{\pi}_{i,j}}$,
- if $j = \infty$, then ${\mathbb S^{\pi}}_{(i,j)}$ is the initial segment of ${\mathbb S^{\pi}}$ ending with (and including) $d^\pi_i$.
We let $\pi{\sqsubseteq_{(i,j)}} \rho$ if ${\mathbb S^{\pi}}_{(i,j)}$ is lexicographically smaller or equal to ${\mathbb S^{\rho}}_{(i,j)}$. Furthermore, we define $$\begin{aligned}
\pi {\equiv}\rho &\text{\quad if\quad} \pi{\sqsubseteq}\rho \text{ and
}\rho{\sqsubseteq}\pi,\\
\pi {{\equiv}_{(i,j)}} \rho &\text{\quad if\quad} \pi{\sqsubseteq_{(i,j)}} \rho \text{ and
}\rho{\sqsubseteq_{(i,j)}} \pi.\end{aligned}$$
Lastly, the notation $\pi{\sqsubset}\rho$ means $\pi{\sqsubseteq}\rho$ and $\pi\not{\equiv}\rho$.
The main result: a variant of the Skeletal Lemma {#sec:skeletal}
================================================
In this section, we are finally in a position to state and prove the main result of this paper that is essentially a more specific version of [@KV-hamilton Lemma 17]. Before we state it, we need one more definition.
Let $H$ be a $3$-hypergraph and $\pi$ an acyclic quasigraph in $H$. A partition ${\mathcal P}$ of $V(H)$ is *$\pi$-skeletal* if both of the following conditions hold:
(1) for each $X\in{\mathcal P}$, $\pi$ is both connected on $X$ and anticonnected on $X$ (i.e., ${\mathcal P}$ is $\pi$-solid),
(2) the complement of $\pi/{\mathcal P}$ in $H/{\mathcal P}$ is acyclic.
\[t:enhancing\] Let $\pi$ be a quasigraph in a 3-hypergraph $H$. If ${{\mathcal P}^{\pi}_{\infty,\infty}}$ is not $\pi$-skeletal or $\pi$ is not acyclic, then there is a quasigraph $\rho$ in $H$ such that either $\rho \sqsupset \pi$, or $\rho{\equiv}\pi$ and ${\|\rho\|} < {\|\pi\|}$.
An obvious corollary of Theorem \[t:enhancing\] (which will be further strengthened in Section \[sec:bad\]) is the following:
\[cor:main\] For any $3$-hypergraph $H$, there exists an acyclic quasigraph $\pi$ such that ${{\mathcal P}^{\pi}_{\infty,\infty}}$ is $\pi$-skeletal.
Before proving Theorem \[t:enhancing\], we need to establish the following crucial lemma. The situation is illustrated in Figure \[fig:qc-addition\].
[![The situation in Lemma \[l:qc-addition\]: the quasigraph $\pi$ (bold) and the partition ${\mathcal Q}$ (dark gray). Only some of the hyperedges and vertices are shown; in particular, ${\mathcal Q}$ is assumed to be $\pi$-solid.[]{data-label="fig:qc-addition"}](skeletal-figs.pdf "fig:")]{}
\[l:qc-addition\] Let $\pi$ be a quasigraph in a 3-hypergraph $H$ and $X\subseteq
V(H)$ such that $\pi$ is anticonnected on $X$. Suppose that ${\mathcal Q}$ is a $\pi$-solid partition of $V(H)$ refining ${\left\{{X,V(H)-X}\right\}}$ and that $\gamma$ is a quasicycle in $\overline{\pi/{\mathcal Q}}$ all of whose vertices are subsets of $X$ (as classes of ${\mathcal Q}$).
If $\gamma$ has a redundant leading hyperedge $e$ (with respect to $\pi$ and ${\mathcal Q}$), then there are vertices $u,v\in e$ such that each of $u$ and $v$ is contained in a different class in $\gamma(e)$, and the quasigraph $\pi + (uv)_e$ is anticonnected on $X$.
Let the vertices of $\gamma^*$ be $Q_1,\dots,Q_k\in{\mathcal Q}$ in order, such that $\gamma(e) = {\left\{{Q_k,Q_1}\right\}}$.
\[cl:S\] The quasigraph $\pi$ is anticonnected on $Q_1\cup\dots\cup Q_k$ in $H-e$.
Since $e$ is redundant, it is not a ${\mathcal Q}$-antibridge, so $\pi$ is anticonnected on each $Q_i$ in $H-e$, where $i=1,\dots,k$. We prove, by induction on $j$, that $\pi$ is anticonnected on $Q_1\cup\dots\cup Q_j$ in $H-e$, where $1\leq j\leq k$. The case $j=1$ is clear. Supposing that $j > 1$ and the statement is valid for $j-1$, we prove it for $j$. Consider two consecutive vertices $Q_{j-1}, Q_j$ of $\gamma^*$ and the edge $f$ of $\gamma^*$ joining them. Since $\gamma$ is a quasigraph in $\overline{\pi/{\mathcal Q}}$, $f$ corresponds to a hyperedge $h\neq e$ of $H$ intersecting both $Q_{j-1}$ and $Q_j$, and such that $\pi(h)$ is contained in $Q_{j-1}$ or $Q_j$ (including the case $\pi(h)=\emptyset$). By the induction hypothesis and Lemma \[l:union\], $\pi$ is anticonnected on $(Q_1\cup\dots\cup Q_{j-1})\cup Q_j$ in $H-e$.
Let $u\in Q_k\cap e$ and $v\in Q_1\cap e$. Observe that $u$ and $v$ are contained in different classes of $\gamma(e)$ as stated in the lemma. By Claim \[cl:S\], $u$ and $v$ are contained in the same anticomponent of $\pi$ on $X$ in $H-e$. It follows that $u$ and $v$ are contained in the same anticomponent $A$ of $\rho := \pi+(uv)_e$ on $X$ in $H$. In fact, the following holds:
\[cl:rho-anti\] The quasigraph $\rho$ is anticonnected on $X$ in $H$.
Suppose the contrary and consider a partition ${\mathcal R}$ of $X$ such that for each hyperedge $f$ of $H$ crossing ${\mathcal R}$, $\rho(f)$ also crosses ${\mathcal R}$. Then $e$ must cross ${\mathcal R}$, since otherwise ${\mathcal R}$ would demonstrate that $\pi$ is not anticonnected on $X$, contrary to the assumption of the lemma. Thus, $\rho(e)$ crosses ${\mathcal R}$, so $u$ and $v$ are contained in distinct classes of ${\mathcal R}$. The partition ${\mathcal R}[A]$ of $A$ (the anticomponent of $\rho$ defined above) induced by ${\mathcal R}$ is therefore nontrivial. Since $\rho$ is anticonnected on $A$, there is a hyperedge $h$ of $H$ such that $h$ crosses ${\mathcal R}[A]$ and $\rho(h)$ is contained in a class of ${\mathcal R}[A]$. But then $h$ crosses ${\mathcal R}$ while $\rho(h)$ does not, a contradiction with the choice of ${\mathcal R}$ which proves the claim.
We have shown that the present choice of $u$ and $v$ satisfies all requirements of the lemma. This concludes the proof.
\[l:qc\] Suppose that ${\mathcal R}$ is a partition of $V(H)$ and $X\in{\mathcal R}$. If $e$ is a hyperedge of $H$ with $e\cap X\supseteq{\left\{{u,v}\right\}}$ and $\tau$ is the quasigraph $\pi + (uv)_e$, then $\overline{\pi/{\mathcal R}} = \overline{\tau/{\mathcal R}}$.
The hypergraph $\overline{\pi/{\mathcal R}}$ consists of hyperedges $f/{\mathcal R}$ such that $f$ crosses ${\mathcal R}$ and $f/{\mathcal R}$ is not used by $\pi/{\mathcal R}$. For $f\neq e$ we clearly have $f\in\overline{\pi/{\mathcal R}}$ if and only if $f\in\overline{\tau/{\mathcal R}}$. As for the hyperedge $e$, if it does not cross ${\mathcal R}$, there is no corresponding hyperedge in either of $\overline{\pi/{\mathcal R}}$ and $\overline{\tau/{\mathcal R}}$. If $e$ crosses ${\mathcal R}$, then $e/{\mathcal R}$ is not used by $\pi/{\mathcal R}$ (since $e$ is not used by $\pi$) nor by $\tau/{\mathcal R}$ (since $u,v\in X\in{\mathcal R}$). Thus, $e/{\mathcal R}$ is a hyperedge of both $\overline{\pi/{\mathcal R}}$ and $\overline{\tau/{\mathcal R}}$. We conclude that $\overline{\pi/{\mathcal R}}=\overline{\tau/{\mathcal R}}$.
Given $i,j\in{{\mathcal T}}$ with $i,j < \infty$ and $(i,j)\neq (0,0)$, the *predecessor* of the partition ${{\mathcal P}^{\pi}_{i,j}}$ is the partition ${{\mathcal P}^{\pi}_{i,j-1}}$ if $j > 0$, and ${{\mathcal P}^{\pi}_{i-1,\infty}}$ if $j = 0$ and $i > 0$. The predecessors of the other partitions in the sequence for $\pi$ are undefined.
\[obs:exposed\] Let ${\mathcal Q}$ be a partition of $V(H)$. If a quasicycle $\gamma$ in $\overline{\pi/{\mathcal Q}}$ is exposed at $(i,j)$ with respect to $\pi$, then $i,j < \infty$ and $(i,j)\neq (0,0)$; in particular, the predecessor of ${{\mathcal P}^{\pi}_{i,j}}$ exists. In addition, if $j = 1$ and $i\geq 1$, then $d^\pi_{i-1}$ is a hyperedge not used by $\pi$.
Clearly, $(i,j) \neq (0,0)$ since ${{\mathcal P}^{\pi}_{0,0}} = {\left\{{V(H)}\right\}}$. By the definition of the sequence of $\pi$, for any $r\geq 0$, ${{\mathcal P}^{\pi}_{r,\infty}}$ is equal to one (actually, infinitely many) of the partitions ${{\mathcal P}^{\pi}_{r,s}}$, where $0\leq s < \infty$, and hence it cannot be the exposing partition for $\gamma$. A similar argument applies to ${{\mathcal P}^{\pi}_{\infty,\infty}}$. As for the last statement, suppose that the exposing partition is ${{\mathcal P}^{\pi}_{i,1}}$. Clearly, $d^\pi_{i-1}\notin{\left\{{{\text{\textsc{terminate}}},{\text{\textsc{stop}}}}\right\}}$, so it is a hyperedge. If it were used by $\pi$, it would be a ${{\mathcal P}^{\pi}_{i-1,\infty}}$-bridge, and the classes of ${{\mathcal P}^{\pi}_{i,0}}$ would be the components of $\pi-d^\pi_{i-1}$ on the classes of ${{\mathcal P}^{\pi}_{i-1,\infty}}$. By the definition of the sequence of $\pi$, ${{\mathcal P}^{\pi}_{i,1}} = {{\mathcal P}^{\pi}_{i,0}}$ and so ${{\mathcal P}^{\pi}_{i,1}}$ cannot be the exposing partition for $\gamma$.
\[obs:anti\] Suppose that $\pi$ is a quasigraph in $H$, $X\subseteq V(H)$ and $e$ is a hyperedge such that $e\cap X = {\left\{{u,v}\right\}}$. If $\pi+(uv)_e$ is anticonnected on $X$, then $e$ is not an $X$-antibridge with respect to $\pi$.
Assume the contrary. Then there is a nontrivial partition ${\mathcal R}$ of $X$ such that for each hyperedge $f$ of $H-e$, $\pi(f)$ crosses ${\mathcal R}$ whenever $f$ does. Since there is no such partition for $\pi+(uv)_e$ in $H$, it must be that $e$ crosses ${\mathcal R}$ but $uv$ does not. That is impossible since $e\cap X = {\left\{{u,v}\right\}}$.
\[l:stable\] Let $\pi$ be a quasigraph in $H$. Let $e$ be a hyperedge not used by $\pi$ such that vertices $u,v\in e$ are contained in different classes of a partition ${{\mathcal P}^{\pi}_{i,j}}$ (where $i,j\geq 0$), but both of them are contained in the same class $X$ of its predecessor. If the quasigraph $\pi + (uv)_e$ is anticonnected on $X$, then $\pi + (uv)_e \sqsupset \pi$.
Let $\rho = \pi + (uv)_e$. Suppose that $\rho\not\sqsupset\pi$. We begin by proving the following claim: $$\label{eq:stable} \text{$\pi {\sqsubseteq_{(s,t)}}
\rho$ for all $(s,t) \leq (i,j)$.}$$
We proceed by induction on $(s,t)$, assuming the claim for all smaller pairs in ${{\mathcal T}}$. The claim holds for $(s,t) = (0,0)$; assume therefore that $(s,t) > (0,0)$. Suppose first that $0 < t < \infty$ and the statement holds for $(s,t-1)$. If $t$ is odd, then any class $A$ of ${{\mathcal P}^{\pi}_{s,t}}$ is a component of $\pi$ on a class of ${{\mathcal P}^{\pi}_{s,t-1}}$. We have either $A\supseteq X$, or $A\cap X = \emptyset$. In both cases, $\rho$ is clearly connected on $A$ (by Lemma \[l:sub-add\](ii) in the former case). Thus, ${{\mathcal P}^{\pi}_{s,t}} \leq {{\mathcal P}^{\rho}_{s,t}}$ and $\pi{\sqsubseteq_{(s,t)}} \rho$.
Next, if $t$ is even (and nonzero), we proceed similarly: if $A$ is an anticomponent of $\pi$ on a class of ${{\mathcal P}^{\pi}_{s,t-1}}$ and $A\supseteq X$, then $\rho$ is anticonnected on $A$ by Lemma \[l:sub-add\](i), while if $A\cap X = \emptyset$, the same is true for trivial reasons. Thus, again, $\pi{\sqsubseteq_{(s,t)}} \rho$.
The next case is $t = \infty$. By the induction hypothesis, ${{\mathcal P}^{\pi}_{s,\infty}} \leq {{\mathcal P}^{\rho}_{s,\infty}}$; without loss of generality, we may assume that the partitions are equal. Let ${\mathcal S}= {{\mathcal P}^{\pi}_{s,\infty}}$. We need to show that $d^\pi_s \leq_E d^\rho_s$. By Lemma \[l:qc\], $\overline{\pi/{\mathcal S}} = \overline{\rho/{\mathcal S}}$. In particular, the two hypergraphs have the same quasicycles, and these quasicycles have the same leading hyperedges. It follows that $d^\pi_s \leq_E d^\rho_s$ or $d^\pi_s = {\text{\textsc{stop}}}$ — but the latter does not hold since $\pi$ cannot stop (or terminate) at $(s,\infty)$ as ${{\mathcal P}^{\pi}_{i,j}}$ differs from its predecessor and $(s,\infty) < (i,j)$.
It remains to consider the case $t=0$. Here, we have $s > 0$ and the induction hypothesis implies that $\pi{\sqsubseteq_{(s-1,\infty)}}\rho$. Let us assume that $\pi{{\equiv}_{(s-1,\infty)}}\rho$ and denote ${{\mathcal P}^{\pi}_{s-1,\infty}}$ by ${\mathcal S}$. We want to show that ${{\mathcal P}^{\pi}_{s,0}} \leq {{\mathcal P}^{\rho}_{s,0}}$.
Let $f := d^\pi_{s-1}$. For the same reason as above, $\pi$ continues at $(s-1,\infty)$, so $f\notin{\left\{{{\text{\textsc{terminate}}},{\text{\textsc{stop}}}}\right\}}$. This means that $f$ is an ${\mathcal S}$-bridge or an ${\mathcal S}$-antibridge with respect to $\pi$. In addition, $d^\rho_{s-1} = f$ by the assumption that $\pi{{\equiv}_{(s-1,\infty)}}\rho$. Hence, $f$ is an ${\mathcal S}$-bridge or an ${\mathcal S}$-antibridge with respect to $\rho$ as well.
Consider the set $X$ from the statement of the lemma. Since the anticomponents of $\pi-f$ and $\rho-f$ on $X$ are clearly the same if ${\left|{f\cap X}\right|}\leq 1$, we may assume that ${\left|{f\cap X}\right|}\geq 2$ — indeed, since $f$ crosses ${\mathcal S}$, we must have equality. Let $S$ be the class of ${\mathcal S}$ containing $X$.
We distinguish three cases:
(a) $f = e$,
(b) $f\neq e$ and $f$ is not used by $\pi$,
(c) $f\neq e$ and $f$ is used by $\pi$.
In case (a), $f$ ($=e$) is not used by $\pi$, so it is an $S$-antibridge with respect to $\pi$. Since it intersects $S$ in two vertices, these two vertices are $u$ and $v$, and each of them is contained in a different anticomponent of $\pi$ on $S$ in $H-f$. Moreover, it must be that $i = s$, $j = 0$ and $X = S$ (since $u,v$ are contained in distinct classes of ${{\mathcal P}^{\pi}_{s,0}}$ but in one class $S$ of its predecessor). However, an assumption of the lemma is that $\rho$ is anticonnected on $X$, a contradiction with Observation \[obs:anti\]. In other words, case (a) cannot occur.
In case (b), $f$ is also an $S$-antibridge with respect to $\pi$ (and $\rho$). To prove that ${{\mathcal P}^{\pi}_{s,0}} \leq {{\mathcal P}^{\rho}_{s,0}}$, it is enough to show that $\rho$ is anticonnected on each anticomponent of $\pi$ on $S$ in $H-f$. Let $A$ be such an anticomponent. We have ${\left|{e\cap A}\right|}=2$, otherwise it is easy to see that $f$ could not be an $S$-antibridge with respect to $\pi$. It follows that $X\subseteq A$. By Lemma \[l:sub-add\](i), $\rho$ is anticonnected on $A$ as claimed. The discussion of case (b) is complete.
Lastly, in case (c), $f$ is an $S$-bridge with respect to $\pi$ and $\rho$. The quasigraph $\rho-f$ is clearly connected on each component of $\pi-f$ on $S$ in $H$, so ${{\mathcal P}^{\pi}_{s,0}} \leq {{\mathcal P}^{\rho}_{s,0}}$.
To summarise, each of the cases (a)–(c) leads either to a contradiction, or to the sought conclusion ${{\mathcal P}^{\pi}_{s,0}} \leq {{\mathcal P}^{\rho}_{s,0}}$. This concludes the proof of .
It remains to show that $\rho\sqsupset\pi$. Since $j < \infty$, there are three cases to distinguish based on the value of $j$. If $j$ is odd, then the classes of ${{\mathcal P}^{\pi}_{i,j}}\setminus{{\mathcal P}^{\pi}_{i,j-1}}$ are the components of $\pi$ on $X$. Since $u$ and $v$ are in different classes of ${{\mathcal P}^{\pi}_{i,j}}$, the replacement of $\pi$ with $\rho$ has the effect of adding the edge $uv$ to $\pi^*$, joining the two components into one. Therefore, ${{\mathcal P}^{\rho}_{i,j}} > {{\mathcal P}^{\pi}_{i,j}}$, and by , $\rho\sqsupset\pi$.
If $j$ is even and $j > 0$, the classes of ${{\mathcal P}^{\pi}_{i,j}}$ are the anticomponents of $\pi$ on $X$. Let $A_1$ and $A_2$ be such anticomponents containing $u$ and $v$, respectively. By Lemma \[l:union\], $\pi$ is anticonnected on $A_1$ and $A_2$ since $e$ intersects both of of these sets and is not used by $\pi$. This contradiction means that the present case is not possible.
Finally, if $j=0$, $uv$ is exposed at $(i,0)$ with respect to $\pi$. Let $f = d^\pi_{i-1}$ be the corresponding decisive hyperedge. Since $\pi$ continues at $(i-1,\infty)$, $f$ is an $X$-bridge or an $X$-antibridge with respect to $\pi$. This shows that $f\neq e$ because $e$ is not an $X$-antibridge by Observation \[obs:anti\], and is not an $X$-bridge because it is not used by $\pi$. Furthermore, similarly to the preceding case, it cannot be that $f$ is an $X$-antibridge, for then $e$ would intersect two anticomponents of $\pi$ on $X$ in $H-f$, which is impossible by Lemma \[l:union\].
Thus, $f$ is an $X$-bridge with respect to $\pi$, and the classes of ${{\mathcal P}^{\pi}_{i,0}}$ are the components of $\pi-f$ on $X$ in $H$. Since $u,v$ are in different components, the quasigraph $\rho-f$ is connected on $X$ in $H$, and consequently $\rho$ stops at ${{\mathcal P}^{\rho}_{i-1,\infty}}$ and $\rho\sqsupset\pi$. This concludes the proof.
\[l:stable-cycle\] Let $\pi$ be a quasigraph in $H$. Let $e$ be a hyperedge of $H$ used by $\pi$ and $X\in V(H)$ such that $\pi(e)\subseteq X$, $\pi-e$ is connected on $X$, and one of the following conditions is satisfied:
(a) the vertices of $\pi(e)$ are contained in different classes of ${{\mathcal P}^{\pi}_{i,j}}$ ($0\leq i,j < \infty$) and $X$ is a class of its predecessor,
(b) $X\in{{\mathcal P}^{\pi}_{\infty,\infty}}$ and ${{\mathcal P}^{\pi}_{\infty,\infty}}$ is $\pi$-skeletal.
Then $\pi-e {\sqsupseteq}\pi$. In addition, if (a) is satisfied, then $\pi-e {\sqsupset}\pi$.
Suppose that condition (a) is satisfied. We prove, by induction on $(s,t)$, that $$\label{eq:1}
\pi {\sqsubseteq_{(s,t)}} \pi-e \text{ for all $(s,t) \leq (i,j)$}.$$ The statement holds if $(s,t)=(0,0)$. Consider $(s,t) > (0,0)$. If $0 < t < \infty$, then we may suppose that $\pi {{\equiv}_{(s,t-1)}} \pi-e$; by Lemma \[l:sub-remove\], ${{\mathcal P}^{\pi}_{s,t}} \leq {{\mathcal P}^{\pi-e}_{s,t}}$ and therefore $\pi {\sqsubseteq_{(s,t)}} \pi-e$ as desired.
Suppose that $t = \infty$. Let ${\mathcal P}= {{\mathcal P}^{\pi}_{s,\infty}}$. Without loss of generality, ${\mathcal P}= {{\mathcal P}^{\pi-e}_{s,\infty}}$. Since $X$ is a subset of a class of ${\mathcal P}$, Lemma \[l:qc\] implies that the complement of $\pi/{\mathcal P}$ is the same as the complement of $(\pi-e)/{\mathcal P}$. In particular, the quasicycles in these hypergraphs, as well as the sets of their leading hyperedges, are the same.
We state the following simple observation as a claim for easier reference later in the proof: 0
\[cl:no-wr\] No leading hyperedge of any quasicycle in the complement of $\pi/{\mathcal P}$ is weakly redundant.
Indeed, $\pi$ would have stopped at $(s,\infty)$, but condition (a) implies that ${{\mathcal P}^{\pi}_{i,j}}$ differs from its predecessor. Since $(s,\infty) < (i,j)$, this would be a contradiction.
By Claim \[cl:no-wr\], if $\pi-e$ stops at $(s,\infty)$, then $\pi{\sqsubset}\pi-e$. We may therefore assume that $\pi-e$ continues at $(s,\infty)$, in which case the decisive hyperedges at $(s,\infty)$ for $\pi$ and $\pi-e$ coincide. We conclude that $\pi{\sqsubseteq_{(s,\infty)}}\pi-e$ in this case.
It remains to consider the case $t=0$. It suffices to show that ${{\mathcal P}^{\pi}_{s,0}} \leq {{\mathcal P}^{\pi-e}_{s,0}}$ assuming that $\pi{{\equiv}_{(s-1,\infty)}}\pi-e$. Let $f := d^\pi_{s-1}$ be the decisive hyperedge at $(s-1,\infty)$ with respect to $\pi$. The assumption implies that $f = d^{\pi-e}_{s-1}$. Moreover, $f\notin{\left\{{{\text{\textsc{terminate}}},{\text{\textsc{stop}}}}\right\}}$ because ${{\mathcal P}^{\pi}_{i,j}}$ differs from its predecessor, so $\pi$ has to continue at $(s-1,\infty)$ as $(i,j) \geq (s,0)$.
If there is a class $A\in{{\mathcal P}^{\pi}_{s,0}}$ with $\pi(e)\subseteq A$, then we have $X\subseteq A$ (where $X$ is the set from the lemma). It follows from Lemma \[l:sub-remove\] that $\pi-e$ is connected on $A$ whenever $\pi$ is, and similarly for anticonnectivity. Consequently, ${{\mathcal P}^{\pi}_{s,0}} \leq {{\mathcal P}^{\pi-e}_{s,0}}$.
We may therefore assume that $\pi(e)$ intersects two classes of ${{\mathcal P}^{\pi}_{s,0}}$. By condition (a), $(i,j)=(s,0)$ and $X\in{{\mathcal P}^{\pi}_{s-1,\infty}}$. Moreover, $f$ is an $X$-bridge or $X$-antibridge with respect to $\pi$. We can see that $f\neq e$: otherwise, $e$ would necessarily be an $X$-bridge, but $\pi-e$ is assumed to be connected on $X$. Since $\pi(e)$ intersects two classes of ${{\mathcal P}^{\pi}_{s,0}}$ and it is used by $\pi$, we find that $f$ is not used by $\pi$ and the classes of ${{\mathcal P}^{\pi}_{s,0}}$ are the anticomponents of $\pi$ on $X$ in $H-f$. But then $f$ is a redundant leading hyperedge in a quasicycle in the complement of $(\pi-e)/{{\mathcal P}^{\pi}_{s-1,\infty}}$, contradicting our assumption that $\pi-e$ continues at $(s-1,\infty)$. Summing up, if $\pi(e)$ intersects two classes of ${{\mathcal P}^{\pi}_{s,0}}$, then $\pi{\sqsubset_{(s-1,\infty)}}\pi-e$. The case $t=0$ is settled.
Having proved , let us now show that $$\pi{\sqsubset_{(i,j)}}\pi-e.$$ By the assumption of the lemma, $\pi(e)$ intersects two classes of ${{\mathcal P}^{\pi}_{i,j}}$. We note that $j < \infty$ and consider two possibilities: $j = 0$ and $j > 0$ ($j$ finite). If $j = 0$, then we have seen in the above paragraph that if $\pi(e)$ intersects two classes of ${{\mathcal P}^{\pi}_{s,0}}$ (for any $s\leq i$), then $s=i$ and $\pi{\sqsubset_{(s-1,\infty)}}\pi-e$. In particular, $\pi{\sqsubset_{(i,j)}}\pi-e$.
Suppose now that $j > 0$ is finite. Without loss of generality, $\pi{{\equiv}_{(i,j-1)}}\pi-e$. Since $\pi(e)$ intersects two classes of ${{\mathcal P}^{\pi}_{i,j}}$, we see from the definition of the sequence of $\pi$ that $j$ is even. Hence, the vertices of $\pi(e)$ lie in different anticomponents of $\pi$ on $X$, and $\pi-e$ is anticonnected on the union of these anticomponents by Lemma \[l:union\]. Consequently, ${{\mathcal P}^{\pi}_{i,j}} < {{\mathcal P}^{\pi-e}_{i,j}}$ and $\pi{\sqsubset_{(i,j)}} \pi-e$. This concludes the proof for condition (a).
A very similar inductive proof to that used to prove works when condition (b) is satisfied. The main difference is that in the $t = \infty$ case, Claim \[cl:no-wr\] now holds for a different reason, namely that ${{\mathcal P}^{\pi}_{\infty,\infty}}$ is $\pi$-skeletal, which means that $\pi$ never stops — consequently, there is no weakly redundant leading hyperedge of a quasicycle in $\overline{\pi/{{\mathcal P}^{\pi}_{s-1,\infty}}}$.
Another difference is that the case discussed in the last paragraph of the proof of cannot occur if condition (b) is satisfied, which makes the proof for condition (b) somewhat shorter.
We can now proceed to the proof of Theorem \[t:enhancing\].
Let ${\mathcal Q}={{\mathcal P}^{\pi}_{\infty,\infty}}$. We distinguish the following cases.
[${\mathcal Q}$ is not $\pi$-skeletal.]{} By the construction, it is clear that ${\mathcal Q}$ is $\pi$-solid. Thus, $\overline{\pi/{\mathcal Q}}$ contains a quasicycle. Consider the least $s$ such that ${{\mathcal P}^{\pi}_{s,\infty}} = {\mathcal Q}$. Since $\pi$ stops at ${{\mathcal P}^{\pi}_{s,\infty}}$, there is a quasicycle $\gamma$ in $\overline{\pi/{\mathcal Q}}$ and a leading hyperedge $e$ of $\gamma$ such that $e$ is weakly redundant. Let the exposure step for $\gamma$ be $(i,j)$, where $0\leq i,j < \infty$. We put ${\mathcal P}= {{\mathcal P}^{\pi}_{i,j}}$, and let $X$ be the class of the predecessor of ${{\mathcal P}^{\pi}_{i,j}}$ containing both vertices of $\gamma(e)$. Furthermore, let $Q_1,Q_2\in{\mathcal Q}$ be the two vertices of $\gamma(e)$, and let $P_i\in{\mathcal P}$ be such that $P_i\supseteq Q_i$ ($i = 1,2$).
[$e$ is not used by $\pi$.]{} First, $j$ is odd or $j = 0$; otherwise, $P_1,P_2$ would be anticomponents of $\pi$ on $X$, but Lemma \[l:union\] shows that $\pi$ is anticonnected on $P_1\cup P_2$, which would be a contradiction.
Consider first the case that $j$ is odd, so $P_1$ and $P_2$ are components of $\pi$ on $X$. Moreover, suppose for now that $j > 1$. By the construction of the sequence for $\pi$, $\pi$ is anticonnected on $X$. Applying Lemma \[l:qc-addition\] (with the current values of $X$, $Q$, $\gamma$ and $e$), we obtain vertices $u,v$ such that $u\in Q_1$, $v\in Q_2$ and $\pi + (uv)_e$ is anticonnected on $X$. Lemma \[l:stable\] then implies that $\pi + (u_1u_2)_e \sqsupset \pi$ and we are done.
Suppose that $j = 1$. If $i=0$ or the decisive hyperedge $d^\pi_{i-1}$ at $(i-1,\infty)$ is not used by $\pi$, then the above argument works, since in this case $\pi$ is anticonnected on $X$. The other case ($i > 0$ and $d^\pi_{i-1}$ is used by $\pi$) is excluded by Observation \[obs:exposed\]. This settles the case $j=1$ and more broadly the case that $j$ is odd.
It remains to consider the possibility that $j = 0$. Clearly, $i > 0$ as ${{\mathcal P}^{\pi}_{0,0}} = {\left\{{V(H)}\right\}}$. The predecessor of ${{\mathcal P}^{\pi}_{i,0}}$ is ${{\mathcal P}^{\pi}_{i-1,\infty}}$, which is $\pi$-solid. Thus, $\pi$ is anticonnected on $X$ and the argument used for odd $j>1$ applies.
[$e$ is used by $\pi$.]{} Since $e$ is a leading hyperedge of a quasicycle in $\overline{\pi/{\mathcal Q}}$, $\pi(e)$ is a subset of some $Q\in{\mathcal Q}$. Since $e$ is weakly redundant with respect to ${\mathcal Q}$, it is not a $Q$-bridge — in other words, $\pi-e$ is connected on $Q$. Lemma \[l:stable-cycle\] implies that $\pi-e \sqsupseteq \pi$. Since $\pi-e$ uses fewer hyperedges than $\pi$, it has the desired properties.
[$\pi$ is not acyclic.]{} Suppose that there is a cycle $C$ in the graph $\pi^*$. That means that either $\pi/{\mathcal Q}$ is not acyclic, or there is a cycle in the induced subgraph of $\pi^*$ on some $Q\in{\mathcal Q}$.
[The quasigraph $\pi/{\mathcal Q}$ is not acyclic.]{} Let $C$ be a cycle in $(\pi/{\mathcal Q})^*$ and suppose its exposure step is $(i,j)$, where $0 \leq i,j < \infty$. Let $e$ be a leading hyperedge of $C$. Note that each of the vertices of $\pi(e)$ is in a different class of the partition ${{\mathcal P}^{\pi}_{i,j}}$, but both are contained in the same class $X$ of its predecessor.
Since $e$ is contained in the cycle $C$ all of whose vertices are subsets of $X$, $\pi-e$ is connected on $X$. Thus, the assumptions of Lemma \[l:stable-cycle\] are satisfied (we use the current values of $e$ and $X$). It follows that $\pi-e\sqsupseteq \pi$; since ${\|\pi-e\|} < {\|\pi\|}$, $\pi-e$ has the desired properties.
[There is $Q\in{\mathcal Q}$ such that the induced subgraph of $\pi^*$ on $Q$ contains a cycle.]{} Let $C$ be a cycle in the induced subgraph of $\pi^*$ on $Q$, and let $e$ be a hyperedge such that $\pi(e)$ is an edge of $C$. Clearly, $\pi-e$ is connected on $Q$. By Lemma \[l:stable-cycle\], $\pi-e\sqsupseteq\pi$. Since $\pi-e$ uses fewer hyperedges than $\pi$ does, we are done.
Removing bad leaves {#sec:bad}
===================
For the purposes of the application of the Skeletal Lemma in [@KV-ess-9], we have to prove the lemma in a stronger form (Theorem \[t:enhancing\]) allowing us to deal with a certain configuration that is problematic for the analysis in [@KV-ess-9], namely a ‘bad leaf’ in a quasigraph. As a result, we will be able to exclude this configuration in Theorem \[t:no-bad\] (at the cost of some local modifications to the hypergraph).
Let $H$ be a 3-hypergraph and let $\pi$ be an acyclic quasigraph in $H$. In each component of the graph $\pi^*$, we choose an arbitrary root and orient all the edges of $\pi^*$ toward the root. A hyperedge $e$ of $H$ is *associated with* a vertex $u$ if it is used by $\pi$ and $u$ is the tail of $\pi(e)$ in the resulting oriented graph. Thus, every vertex has at most one associated hyperedge, and conversely, each hyperedge is associated with at most one vertex.
A vertex $u$ of $H$ is a *bad leaf* for $\pi$ if all of the following hold:
1. $u$ is a leaf of $\pi^*$,
2. $u$ is incident with exactly three hyperedges, exactly one of which has size 3 (say, $e$), and
3. $e$ is associated with $u$.
12
To eliminate a bad leaf $u$, we use a *switch* operation illustrated in Figure \[fig:bad\](b). Suppose that $u$ is incident with hyperedges $ua$, $ub$ and $ucd$, where $ucd$ is associated with $u$, and $\pi(ucd) = uc$. We remove from $H$ the hyperedges $ua$, $ub$ and $ucd$ and add the hyperedges $uab$, $uc$ and $ud$; the resulting hypergraph is denoted by $H^{(u)}$. We say that a hypergraph $\tilde
H$ is *related* to $H$ if it can be obtained from $H$ by a finite series of switch operations.
With $\pi$ as above, a quasigraph $\pi^{(u)}$ in $H^{(u)}$ is obtained by setting $\pi^{(u)}(uc) = uc$, and leaving both $ud$ and $uab$ unused. Observe that $(\pi^{(u)})^* = \pi^*$, and $\pi^{(u)}$ has fewer bad leaves than $\pi$.
A problem we have to address is that a partition ${\mathcal P}$ which is $\pi$-skeletal in $H$ need no longer be $\pi^{(u)}$-skeletal in $H^{(u)}$, since the switch may create an unwanted cycle in $\overline{\pi^{(u)}/{\mathcal P}}$. This is illustrated in Figure \[fig:switch-cycle\]. The following paragraphs describe the steps taken to resolve this problem. First, we extend the order $\sqsubseteq$ defined on quasigraphs in $H$ to the set of all quasigraphs in hypergraphs related to $H$. Since all such hypergraphs have the same vertex set, we can readily compare the partitions of their vertex sets.
We have to be more careful, however, in the definition of the sequence of $\pi$, where a linear ordering $\leq_E$ of hyperedges of $H$ is used: this ordering should involve all hyperedges of hypergraphs related to $H$. We define $\leq_E$ as follows. We fix a linear ordering $\leq$ of $V(H)$. On the set of 3-hyperedges of hypergraphs related to $H$, $\leq_E$ is the associated lexicographic ordering, and the same holds for the set of 2-hyperedges of hypergraphs related to $H$. Finally, we make each 2-hyperedge greater than any 3-hyperedge with respect to $\leq_E$.
This allows for a definition of the sequence of $\pi$ consistent with the switch operation. Furthermore, the definition of the ordering $\sqsubseteq$ as given in Section \[sec:sequence\] is well suited for our purpose, and remains without change.
Let us mention that although the incorporation of the decisive hyperedges in the signature of a quasigraph may have seemed unnecessary (mainly thanks to Lemma \[l:qc\]), the present section is the reason why we chose this definition. In fact, the only situation in our arguments when the comparison of the decisive hyperedges is relevant is immediately after a switch, as in Lemma \[l:stable-bad\] below.
We first prove that switching a bad leaf of $\pi$ does not affect the (anti)connectivity of $\pi$ on a set of vertices.
\[l:sub-switch\] Let $\pi$ be a quasigraph in $H$ and $X\subseteq V(H)$. Suppose that $\pi$ has a bad leaf $u$ and $\sigma$ is obtained from $\pi$ by switching at $u$. The following holds:
1. if $\pi$ is anticonnected on $X$, then so is $\sigma$,
2. if $\pi$ is connected on $X$, then so is $\sigma$.
We prove (i). Suppose that $\pi$ is anticonnected on $X$, but the quasigraph $\sigma$ (in a hypergraph $\tilde H$ related to $H$) is not. The definition implies that there is a partition ${\mathcal P}$ of $X$ such that for every hyperedge $f$ of $\tilde H$ crossing ${\mathcal P}$, $\sigma(f)$ crosses ${\mathcal P}$. At the same time, there is a hyperedge $e$ of $H$ such that $e$ crosses ${\mathcal P}$ but $\pi(e)$ does not. Clearly, $e$ must be incident with $u$ (since the other hyperedges exist both in $H$ and $\tilde H$, and the values of $\pi$ and $\sigma$ coincide).
Let the neighbours of $u$ in $\tilde H$ be labelled as in Figure \[fig:bad\]b. Let $A$ be the class of ${\mathcal P}$ containing $u$; by the above property of $\sigma$, we can easily see that $a\in A$ if $a\in X$, and similarly for $b$ and $d$. This implies that $c\in X-A$ and $e=ucd$, but then $\pi(ucd)$ crosses ${\mathcal P}$, a contradiction.
Part (ii) is immediate from the fact that $\pi^* = \sigma^*$.
\[l:stable-bad\] Let $\pi$ be a quasigraph in $H$ such that ${{\mathcal P}^{\pi}_{\infty,\infty}}$ is $\pi$-skeletal. If $\pi$ has a bad leaf $u$ and the quasigraph $\sigma$ (in a hypergraph related to $H$) is obtained from $\pi$ by a switch at $u$, then $\pi \sqsupseteq \sigma$.
We show that $$\label{eq:stable-bad}
\text{$\pi {\sqsubseteq_{(i,j)}}
\sigma$ for all $(i,j) \geq (0,0)$.}$$ We proceed by induction on $(i,j)$. The claim is trivial for $(i,j)=(0,0)$. We may assume that $\pi\not\sqsubset\sigma$ for otherwise we are done. Suppose thus that $j > 0$ and $\pi{{\equiv}_{(i,j-1)}}\sigma$. If $j$ is odd, then the classes of ${{\mathcal P}^{\pi}_{i,j}}$ are the components of $\pi$ on classes of ${{\mathcal P}^{\pi}_{i,j-1}}$. Let $X\in{{\mathcal P}^{\pi}_{i,j-1}}$ and let $A$ be a component of $\sigma$ on $X$. By Lemma \[l:sub-switch\](ii), $\sigma$ is connected on $A$. It follows that ${{\mathcal P}^{\pi}_{i,j}}\leq {{\mathcal P}^{\sigma}_{i,j}}$ and follows. An analogous argument, using Lemma \[l:sub-switch\](i), can be used for even $j > 0$.
Let us consider the case $j = \infty$. We assume without loss of generality that $\pi{{\equiv}_{(i,\infty)}}\sigma$. Since ${{\mathcal P}^{\pi}_{\infty,\infty}}$ is assumed to be $\pi$-skeletal, $\pi$ does not stop at $(i,\infty)$. Furthermore, we may assume that $\pi$ does not terminate at $(i,\infty)$, for otherwise we immediately conclude $\pi{\sqsubseteq_{(i,\infty)}}\sigma$.
Let ${\mathcal P}={{\mathcal P}^{\pi}_{i,\infty}}$ and let $\gamma$ be a quasicycle in the complement of $\pi/{\mathcal P}$ in $H/{\mathcal P}$. Define a quasigraph $\gamma'$ in the complement of $\sigma/{\mathcal P}$ in $\tilde H/{\mathcal P}$ as follows (see Figure \[fig:switch-cases\] for an illustration of several of the cases):
- if $\gamma$ uses a hyperedge $f/{\mathcal P}$, where $f$ is a hyperedge of $H$ not incident with $u$, then set $\gamma'(f/{\mathcal P}) =
\gamma(f/{\mathcal P})$,
- if $\gamma$ uses $au/{\mathcal P}$ and $bu/{\mathcal P}$, then set $\gamma'(abu/{\mathcal P}) = ab/{\mathcal P}$,
- if $\gamma$ uses $au/{\mathcal P}$ but not $bu/{\mathcal P}$, then set $\gamma'(abu/{\mathcal P}) = au/{\mathcal P}$ (and symmetrically with $au$ and $bu$ reversed),
- if $\gamma$ uses $ucd/{\mathcal P}$ (so $u$ and $d$ are in different classes of ${\mathcal P}$), then set $\gamma'(ud/{\mathcal P}) = ud/{\mathcal P}$.
A look at Figures \[fig:bad\] and \[fig:switch-cases\] shows that $\gamma'$ is a quasicycle. Thus, $\sigma$ does not terminate at $(i,\infty)$. We need to relate leading hyperedges of $\gamma$ to those of $\gamma'$.
Any leading hyperedge of $\gamma$ that is not incident with $u$ is a leading hyperedge of $\gamma'$, and vice versa. We assert that neither $au$ nor $bu$ is a leading hyperedge of $\gamma$. If they were, they would be redundant (since their size is $2$) and ${{\mathcal P}^{\pi}_{\infty,\infty}}$ would not be $\pi$-skeletal, contrary to the assumption. Finally, if $ucd$ is a leading hyperedge of $\gamma$, then $ud$ is a leading hyperedge of $\gamma'$. Note that $ud >_E ucd$.
It follows that if $\sigma$ does not stop at $(i,\infty)$, then $d^\pi_i \leq_E d^\sigma_i$. On the other hand, if it does stop, then the same inequality holds by the definition of $\leq_E$. In both cases, we have $\pi{\sqsubseteq_{(i,\infty)}}\sigma$.
The last possibility left to consider is $j=0$. We need to show that ${{\mathcal P}^{\pi}_{i,0}} \leq{{\mathcal P}^{\sigma}_{i,0}}$ under the assumption that $\pi{{\equiv}_{(i-1,\infty)}}\sigma$. Let ${\mathcal R}= {{\mathcal P}^{\pi}_{i-1,\infty}}$ and let $f := d^\pi_{i-1} = d^\sigma_{i-1}$. Since ${{\mathcal P}^{\pi}_{\infty,\infty}}$ is $\pi$-skeletal, $f\neq{\text{\textsc{stop}}}$. If $f = {\text{\textsc{terminate}}}$, then $\pi{\equiv}\sigma$. We may thus assume that $f$ is a hyperedge, and in that case it is not incident with $u$ (since $H$ and $\tilde H$ have no common hyperedge incident with $u$). Thus, $\sigma-f$ is obtained from $\pi-f$ by a switch at $u$. Similarly, if $f$ is not used by $\pi$, then $\sigma$ is obtained from $\pi$ by a switch at $u$, in the hypergraph $H-f$.
If $f$ is an $X$-antibridge with respect to $\pi$ for some $X\in{\mathcal R}$, we may use Lemma \[l:sub-switch\] in the hypergraph $H-f$. We find that $\sigma$ is anticonnected on each anticomponent of $\pi$ on $X$ in $H-f$, and hence $\pi{\sqsubseteq_{(i,0)}} \sigma$. On the other hand, if $f$ is an $X$-bridge with respect to $\pi$, then Lemma \[l:sub-switch\] implies that $\sigma-f$ is connected on each component of $\pi-f$ on $X$ in $H$, and $\pi{\sqsubseteq_{(i,0)}} \sigma$ again. This proves and the lemma follows.
Let us now state the result we need to use in [@KV-ess-9].
\[t:no-bad\] Let $H$ be a $3$-hypergraph. There exists a hypergraph $\tilde H$ related to $H$ and an acyclic quasigraph $\sigma$ in $\tilde H$ such that $\sigma$ has no bad leaves and $V(\tilde H)$ admits a $\sigma$-skeletal partition ${\mathcal S}$.
Let $\sigma$ be a quasigraph in a hypergraph $\tilde H$ related to $H$ chosen as follows:
(1) $\sigma$ is $\sqsubseteq$-maximal in the set of all quasigraphs in 3-hypergraphs related to $H$,
(2) subject to (1), $\sigma$ uses as few hyperedges as possible,
(3) subject to (1) and (2), the number of bad leaves is as small as possible.
We define ${\mathcal S}:= {{\mathcal P}^{\sigma}_{\infty,\infty}}$, where the partition is obtained via the plane sequence with respect to $\tilde H$. Note that $\sigma$ is acyclic and ${\mathcal S}$ is $\sigma$-skeletal by Theorem \[t:enhancing\] and the choice of $\sigma$.
It remain to prove that $\sigma$ has no bad leaves. Suppose to the contrary that there is a bad leaf $u$ for $\sigma$. By Lemma \[l:stable-bad\], ${{\sigma}^{({u})}} \sqsupseteq \sigma$. Furthermore, ${{\sigma}^{({u})}}$ uses the same number of hyperedges as $\sigma$, and has one bad leaf fewer, a contradiction with the choice of $\sigma$.
[99]{}
T. Kaiser and P. Vrána, The hamiltonicity of essentially 9-connected line graphs, submitted for publication.
T. Kaiser and P. Vrána, Hamilton cycles in 5-connected line graphs, European J. Combin. 33 (2012), 924–947.
|
---
abstract: 'We present a systematic investigation of the effect of H, B, C, and N interstitials on the electronic, lattice and magnetic properties of La(Fe,Si)$_{13}$ using density functional theory. The parent LaSiFe$_{12}$ alloy has a shallow, double-well free energy function that is the basis of first order itinerant electron metamagnetism. On increasing the dopant concentration, the resulting lattice expansion causes an initial increase in magnetisation for all interstitials that is only maintained at higher levels of doping in the case of hydrogen. Strong *s-p* band hybridisation occurs at high B,C and N concentrations. We thus find that the electronic effects of hydrogen doping are much less pronounced than those of other interstitials, and result in the double-well structure of the free energy function being least sensitive to the amount of hydrogen. This microscopic picture accounts for the change in the metamagnetic transition from first order to second order on doping with B,C, and N interstitials, as observed experimentally.'
author:
- 'Z. Gercsi'
- 'K.G. Sandeman'
- 'A. Fujita'
title: 'Electronic structure and metamagnetic transition of interstitially doped LaSiFe$_{12}$'
---
Introduction
============
The magnetocaloric effect (MCE) is the temperature change of a substance subjected to a change in applied magnetic field. The discovery of the effect can be attributed to Weiss and Piccard’s observation of the magnetization of nickel close to its Curie point in 1917, [@WeissPicc]after a recent re-examination of the original literature by Smith.[@Anders] The adiabatic demagnetization of paramagnetic salts was shown by Giauque and MacDougall in 1933[@GiaMac1933] following initial proposals by both Debye and Giauque in the previous decade.[@Debye; @Giauque] The study of room temperature MCEs associated with a magnetic phase transition was revived in 1997 by Pecharsky and Gschneidner who observed a ‘giant’ entropy change of $\sim$14 JK$^{-1}$kg$^{-1}$ in a 0-2 Tesla field change in Gd$_{5}$Si$_{2}$Ge$_{2}$.[@Gd5(Si2Ge2)] While experiments up to that point had postulated the possibility of room temperature refrigeration using, for example, a second order Curie transition such as that found in Gd, [@brown_1976] it was the effects associated with the first order transition seen in Gd$_{5}$Si$_{2}$Ge$_{2}$ that initiated widespread research interest in the MCE. Today, a large set of magnetic materials show ‘large’ or ‘giant’ magnetocaloric effects.[@SandemanScripta] However, a good refrigerant material also needs to fulfil auxiliary requirements such as tuneable thermal conductivity, durability and elemental abundance and so the number of material systems that are close to commercialisation is relatively small. This situation provides motivation for the use of theoretical models that may aid the understanding and prediction of magnetocaloric effects.
Density functional theory (DFT) is a valuable tool with which to describe the changes in matter at the electronic level that may lead to a large MCE. Elemental Gd has a ferromagnetic (FM) ordering temperature around room temperature that makes it an ideal candidate magnetocaloric material. DFT calculations based on thermally induced spin fluctuations in a disordered local moment picture showed that the magnetic order in Gd is linked to the $c/a$ ratio and atomic unit cell volume. [@Gd-Nature] Such magneto-elastic coupling is useful for generating a large MCE. However, the cost of heavy rare-earth Gd inhibits its use in everyday applications as refrigerant. In Gd$_{5}$(Si$_{2}$Ge$_{2}$), DFT calculations indicated breaking and reforming of Si-Ge bonds between layers within the unit cell, affecting both the location of the Fermi level and the effective magnetic exchange coupling, increasing the latter to the level where a first order magneto-structural transition is observed. [@Gd5(Si2Ge2)]
Manganites and manganese silicides have also been the subject of DFT studies. In manganites, the broad variety of crystallographic, magnetic and electronic phases are attributed to the strong interplay between spin, charge, orbital and lattice degrees of freedom that often couples to external magnetic fields and results in measurable MCE. For a qualitative description of these correlated physical quantities, state-of-the-art hybrid exchange density functionals such can be applied.[@Manganites_review] In the case of manganese based metallic silicides, ground state and finite temperature DFT models have been used to model and predict new Mn-based metamagnets. [@Zsolt2; @Zsolt1; @TDFT_Staunton] Those calculations used accurate structural data obtained from high resolution neutron diffraction on CoMnSi, a noncollinear antiferromagnet (AFM) that exhibits giant magneto-elastic coupling. [@Alex_PRL]
Experimentally, the most intensively studied MCE materials are based on either Fe$_{2}$P or La(Fe,Si)$_{13}$. Both have been the subject of some modelling studies. In Fe$_{2}$P, iron has two inequivalent crystallographic sites and the low moment site (3$f$) has a metamagnetic transition[@Yamada] at the Curie temperature, 212 K. The Curie point can be tuned through room temperature by partial replacement of Fe by Mn as well as P by, for example, Si.[@Dung1] The so-called mixed magnetism of this material has been investigated by a number of DFT studies that have identified the mechanism of magneto-elastic coupling and the change of electron density across the Curie transition.[@Dung1; @Erna1; @Erna2; @Fe2P_ZG1; @Liu1]
In this article, we perform a DFT study of compounds based on LaFe$_{13-x}$M$_{x}$ (M=Si, Al), a cubic NaZn$_{13}$-type material which was first synthesized by Kripyakevich et al.[@Krypiakewytsch] To date, much of the compositional tuning that is used to adjust the magnetocaloric effect and its temperature range is the result of empirical work rather than theory-led prediction.
LaFe$_{13-x}$M$_{x}$ exhibits a large MCE associated with a paramagnetic to ferromagnetic transition on cooling at a temperature, $T_{C}$, between 180 and 250 K. A magnetic field-dependent itinerant-electron metamagnetic (IEM) transition above $T_{C}$, can be shifted towards room temperature by Si addition. However, on increasing the Si content above $x>1.8$, a change in the nature of the FM phase transition from first-order to second-order takes place that results in a considerable reduction of the useful MCE. A first-principles calculation by Wang et al.[@Wang] indicated that hybridization between the Fe-$d$ and Si-$p$ states is linked to the reduction of Fe magnetic moment as well as to the smearing of the first-order type transition for alloys with high Si-content.
The partial replacement of the transition metal element Fe by Co or Mn has been explored in an attempt to preserve the first order nature of the transition around the Curie temperature, $T_{C}$, although both elements cause significant weakening of the field-induced IEM transition[@Liu-Co]. Similarly, interstitial doping of $s$-block or $p$-block elements was also pursued experimentally in order to raise the IEM to room temperature. These empirical studies found that the preparation of single phase compositions is limited to low interstitial concentrations and that only hydrogen is capable of the increase of magnetic transition to room temperature without the diminution of useful isothermal entropy change.[@IEM2] Theoretical calculations by Kuz’min and Richter[@Kuzmin_DFT] on LaFe$_{12}$Si, without interstitial substitution, found that the free energy, as a function of magnetization ($F(M)$) has several shallow minima and maxima, to which they attributed the reduced hysteresis and improved magnetocaloric performance of La(Fe,Si)$_{13}$. Recently, Fujita and Yako [@FujitaScripta] extended this approach and further detailed the dependence of such an energy plot on both the lattice size and the degree of Fe/Si substitution.
We note that no systematic investigation of the effect of interstitial $s$-block or $p$-block elements on the electronic, lattice and magnetic properties of La(Fe,Si)$_{13}$ has been carried out to date. The work presented here, using a theoretical approach based on DFT, attempts to describe the effect of the size of four different dopants and their valence electrons to understand how interstitials can influence the magnetocaloric performance of these alloys. We describe our theoretical methods in section \[sec\_methods\] before presenting our results and discussion in section \[sec\_res-dis\]. Conclusions are drawn in section \[sec\_conc\].
Methods {#sec_methods}
=======
Computational models used \[sub:Theoretical-model\]
---------------------------------------------------
Our computational approach is divided into two, complementary parts. In the first part, we investigate the effect of interstitial doping on the equilibrium unit cell volume using the projector augmented wave (PAW) method[@VASP] as implemented in the Vienna ab-initio simulation package (VASP). The VASP code with Perdew-Burke-Ernzerhof (PBE) parameterization[@PBE] is employed, where site-based magnetic moments were calculated using the Vosko-Wilk-Nusair interpolation[@Vosko] within the general gradient approximation (GGA) for the exchange-correlation potential.
La(Fe,Si)$_{13}$ has 8 formula units per conventional cell. The La atoms occupy the 8$a$ sites ($\frac{1}{4}$,$\frac{1}{4}$,$\frac{1}{4}$), while diffraction studies show that Fe and Si atoms can occupy both 8$b$ $(000)$ and the 96$i$ $(0yz)$ crystallographic positions.[@LaFe-Si1; @LaFe-Si2; @LaFe-Si3; @LaFe-Si4] In order to keep the computational requirements at a feasible level, we follow the approach previously adopted by Kuz’min and Richter,[@Kuzmin_DFT] limiting our investigations to an atomically ordered version of LaSiFe$_{12}$Z$_{x}$, where the 8$b$ sites are occupied solely by silicon whilst iron is located exclusively on the 96$i$ sites. In such a case, the cell that forms the basis of the calculations contains 2 La, 26 Fe and 2 Si atoms. Furthermore, interstitial elements (Z) H, B, C, and N were considered to occupy the 24$d$ crystallographic site only. Using this model, we may vary the concentration of interstitials, $x_{Z}$ in a step size of 0.5 from $x=0$ to 3 in LaSiFe$_{12}$Z$_{x}$.
Full structural relaxation was carried out for both collinear ferromagnetic (FM) and non-magnetic (NM) states in the case of parent LaFe$_{12}$Si alloy, while only the lattice parameter $a$ was relaxed (without relaxation of the internal atomic positions) for the materials doped with $s$- or $p$-block interstitials. A 7 $\times$ 7 $\times$ 7 $k$-point grid was used to discretize the first Brillouin zone and the energy convergence criterion was set to $5\times10^{-7}$ eV during the energy minimization process. The spin-orbit interaction was turned off during the calculations. Finally, data presented in Fig. \[fig:PDOS\] was calculated on a dense 19 $\times$ 19 $\times$ 19 grid of $k$-points for high accuracy.
In the second part of this study, we have taken a fixed spin moment (FSM) approach within the tight-binding theorem using linear muffin tin orbitals (TB-LMTO) as implemented in Stuttgart TB-LMTO code.[@LMTO; @Stuttgartcode; @FSM] This method requires carefully adjusted overlapping Wigner-Seitz (WS) atomic spheres included in the calculations to complete the basis and to provide an accurate description of the electron density throughout the entire unit cell. Consequently, the structural parameters of the relaxed lattice are inherently dependent on the volume occupied by the WS spheres and/or empty spheres. For this reason, we used VASP code (see above) for relaxation. Nevertheless, the TB-LMTO approach allows us to evaluate the total energy difference between FM and NM states, $\Delta F(M)$ as a function of fixed spin moment $M$ as well as the corresponding density of states (DOS) and band dispersions. H, B, C, and N atoms were considered to fully occupy the 24$d$ crystallographic site, LaSiFe$_{12}$Z$_{3}$, for the study. A dense mesh with 48$\times$48$\times$48 $k$-points (for the DOS calculations) or with 12$\times$12$\times$12 $k$-points (for the FSM calculations) was used.
Results and Disscussion {#sec_res-dis}
=======================
Effect of dopants on the lattice expansion and magnetic properties
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![\[fig:a/a0\] Magnetic moment/formula unit (a) and relative lattice expansion ($\frac{a}{a_{0}}$) (b) as a function of dopant concentration in LaSiFe$_{12}$Z$_{x}$. The lattice parameter $a_{0}$ corresponds to the fully relaxed, interstitials free FM structure. ](Fig1){width="8.5cm"}
Fig. \[fig:a/a0\] shows the calculated lattice parameter in the FM state in LaSiFe$_{12}$Z$_{x}$ as a function of interstitial doping. Our calculations obtain a relaxed structure that differs by only about 0.1% from the experimentally reported value. This remarkable agreement validates our choice of exchange correlation, GGA. The lattice expansion increases monotonically with dopant concentration at a rate that depends strongly on the size of the interstitial element. The empirical atomic radius of hydrogen (25 pm) is much smaller than that of the boron (85 pm), carbon (70 pm), or nitrogen (65 pm). The trend in calculated lattice expansion in Fig. \[fig:a/a0\] correlates well with the relative atomic size of the interstitial, showing the predominant influence of the latter on the size of the unit cell. At full doping ($x=3$), we here find a relative lattice expansion of 0.4% for hydrogen and a considerably higher value of 1.7% for carbon**,** which match with the experimental values H and C, respectively. **[@Jia2_Hdoping; @ZhangCdope]** The values of $\frac{\triangle a}{a}$ are 1.8% and 1.25% for Z=B and N respectively at full doping, but these are yet to be confirmed experimentally.
There are only limited experimental data available on the relative effects of lattice expansion of the dopants studied here, particularly as full occupation of the 24$d$ site ($x=3$) by any of the dopants has not been achieved in practice. In terms of valence electron number, however, a different sequence exists: H$(s{}^{1})=$ B$(p{}^{1})<$ C$(p{}^{2})<$ N$(p{}^{3}$). A closer look at Fig. \[fig:a/a0\] reveals a non-monotonic behaviour in the lattice expansion as a function of doping, especially in the case of nitrogen. Indeed, additional charges significantly alter the electronic structure (apart from H) in ways that go beyond the simple picture of chemical pressure effects, as we discuss later. The contribution of additional valence electrons is also reflected in the calculated magnetic moment ($M$). $M$ rises initially (Fig. \[fig:a/a0\], top) for each dopant but a monotonic increase up to $x=2.5$ is only seen in the case of hydrogen. These observations imply a mechanism for hybrid band formation and band broadening for any dopant with a larger atomic radius than that of hydrogen.
In order to depict the changes in the electronic structure that are brought about by the interstitial elements, we next examine the partial electronic density of states (PDOS). Fig. \[fig:PDOS\] shows the PDOS of the parent alloy together with the fully hydrogenated and fully nitrogenated materials ($x=3$). In the parent alloy (bottom), the PDOS is dominated by Fe (red line) around the Fermi level (E$_{F}$) with typical spin-split states. The spin-up ($\uparrow$) states are mostly occupied, while the unoccupied states are dominated by spin-down ($\downarrow$) states separated by about 2.5eV in energy. Furthermore, the filled bands at the lower end of energy range (-9.5 eV) relate mostly to silicon 3$s$ states which are overlapped with $p$-states of both La and Fe. A large energy gap appears from -9.5 eV up to about -6.5eV, where a high population of 3$p$ states of Si (black) is located (-6.5 to -4.5 eV). In this latter energy range, there is negligible contribution from Fe $d$-states. Most of the aforementioned features in the electronic structure are preserved in fully hydrogenated LaSiFe$_{12}$H$_{3}$ (middle of Fig. \[fig:PDOS\]). The main difference in the PDOS compared to the parent alloy is the development of additional states in the gap around -7.5 eV related to the hydrogen interstitials. Small additional peaks also appear around -5 eV, where they overlap with the $p$ states of Si.
In strong contrast to hydrogenation, fully nitrogenated LaSiFe$_{12}$N$_{3}$ exhibits a large overlap of N $p$ states with Fe $d$ states in the -7.5 to -4 eV energy interval (top of Fig. \[fig:PDOS\]). These peaks indicate $p-d$ hybridization, and as a result, increased covalency in bond formation. Another important consequence of nitrogenation is the appearance of states in the vicinity of the Fermi level. The existence of a “double peak” feature just below and above E$_{F}$ for LaSiFe$_{12}$, created by Fe $d$ states in the minority DOS, is altered only a little by hydrogenation. On the other hand, nitrogenation fills this valley at E$_{F}$, which results in the strong alteration of the magnetic properties and ultimately leads to the disappearance of IEM transition. We address the latter behavior in detail in the next section.
![\[fig:PDOS\] Partial density of states (PDOS) of the parent LaSiFe$_{12}$ alloy (bottom) in comparison with LaSiFe$_{12}$H$_{3}$ (middle) and LaSiFe$_{12}$N$_{3}$ (top). ](Fig2){width="8.5cm"}
The free energy landscape
--------------------------
We now turn our interest to the results of our second computational approach, fixed spin moment calculations using TB-LMTO. Our aim is to visualize the energy difference between FM and NM states in the parent alloy and the doped materials. The purpose of our analysis is to identify the main factors that lead to the field-induced isothermal entropy change of LaSiFe$_{12}$Z$_{3}$ around the magnetic transition being lower than that of LaSiFe$_{12}$, as found experimentally for interstitials other than hydrogen.
Fig. \[fig:dFvsM\] compares the free energy curves $F(M)$ calculated by the FSM method for LaSiFe$_{12}$Z$_{3}$, where Z = H, N, B and empty sphere (Es), respectively, together with those of the parent alloy. For direct comparison, we set the non-magnetic energy state as F(0) for each individual composition. Fig. \[fig:dFvsM\](a) shows that the parent LaSiFe$_{12}$ has a very shallow magnetic energy landscape, in accordance with the predictions by Kuz’min and Richter, who used the full-potential local-orbital (FPLO) method.[@Kuzmin_DFT] They noted the advantage of such a potential energy landscape in permitting a low hysteresis, first order metamagnetic transition (IEM). The main difference in our calculation of the parent compound is that we find only two minima rather than the multiple minima that were predicted in their work. The field-induced magnetisation of hydrogenated La-Fe-Si under pressure previously was seen to exhibit multiple steps, confirming Kuz’min and Richter’s predictions. It may be interesting to investigate this property in the undoped compound to eliminate the possibility of pressure-induced hydrogen segregation. Spontaneous hydrogen segregation is known to occur in materials with a smaller hydrogen content than the empirical maximum. [@Barcza2011; @Krautz2012; @Zimm2013; @Baumfeld2014]
![\[fig:dFvsM\] Free energy of the FM state (relative to the free energy of the non-magnetic state) as a function of magnetization $M$, calculated by the fixed spin method (FSM) for LaSiFe$_{12}$Z and for LaSiFe$_{12}$Z$_{3}$, ( Z = H, N, B, empty sphere (Es)). (b) The derivative, $d(\Delta F)/dM$. Local minima in the free energy occur where $d(\Delta F)/dM=0$. **** ](Fig3){width="8.5cm"}
The derivative of $\Delta F(M)$ is also shown in Fig. \[fig:dFvsM\]; local minima in the free energy function are where $d\Delta f/dM=0$. We may conclude that full doping of any of the four interstitials studied removes the shallow double-well potential, resulting in only a single well. This corresponds to the disappearance of the first order metamagnetic transition, as found experimentally for B, C, and N doping. We note that boron addition is detrimental to the total magnetisation as the minimum in $\Delta F(M)$ occurs at a lower value of $M$ than for any other dopant. Of all the interstitials studied, hydrogen alters the double well picture the least. The shallow landscape of $\Delta F(M)$ takes on a concave curvature in the range of $M=7-9$$\mu_{B}$ at full doping. The intrinsically small energy barrier for the parent compound and the hydrogen-doped material makes these compositions particularly sensitive to external parameters such as magnetic field, pressure and temperature and renders the first order IEM transition quasi-reversible.
![\[fig:FSMvsDOS\] Up- and down DOS for LaSiFe$_{12}$ (i, j), LaSiFe$_{12}$Es$_{3}$ (g, h), LaSiFe$_{12}$H$_{3}$ (e, f), LaSiFe$_{12}$N$_{3}$ (c, d) and LaSiFe$_{12}$B$_{3}$ (a, b) at M=7$\mu_{B}$ (left column) and 13$\mu_{B}$ (right column) as calculated by FSM.](Fig4){width="8.5cm"}
Fig. \[fig:dFvsM\]b also reveals the sensitive nature of the metamagnetic states to the lattice parameters. An increase of about 0.5% in the lattice parameter upon H-doping generates a ground state with a high-spin configuration. In order to separate the changes in the magnetic state caused by the volume expansion (chemical pressure) due to the inclusion of large interstitials such as nitrogen from those caused by additional valence electrons, we also carried out calculations with empty spheres (Es) included at the 24$d$ crystallographic site. For a direct comparison, the same lattice constant was adopted for both Z = N and Es.
![\[fig:FSMvsBdisp\] Up- and down spin band dispersions for LaSiFe$_{12}$ (a,b), LaSiFe$_{12}$H$_{3}$ (c,d) and for LaSiFe$_{12}$N$_{3}$ (e,f) and LaSiFe$_{12}$B$_{3}$ (g,h) at M=13$\mu_{B}$ as calculated by FSM. ](Fig5){width="8.5cm"}
To further examine the appearance of a metastable low-spin state at around $M=7\mu_{B}$ and the emergence of an energy barrier near $M=13\mu_{B}$, DOS calculations were performed with fixed spin moments at these two values of magnetisation for each system. The results are shown in Fig. \[fig:FSMvsDOS\]. For both X=H and Es, the Fermi level is located in a deep valley for both the minority and majority-spin DOS at M=7$\mu_{B}$, similar to that found in the parent alloy. This well-defined valley in the DOS is destroyed for X = B and N. By contrast, the electronic features found in the DOS for M=13$\mu_{B}$ differ significantly; high peaks in the DOS appears at E$_{F}$ for both minority and majority-spin states for the undoped as well as for Z=H and Es alloys, whilst only a moderate peak in height for Z=N and B are observed. It is thus apparent that the valleys and peaks around E$_{F}$ can be attributed to the low-spin state and the energy barrier in $\Delta F(M)$.
In Fig. \[fig:FSMvsBdisp\], we show the electronic band structure calculated for $M=13\mu_{B}$, in order to provide an explanation for the above peaks in the electronic DOS. We note that a 0.01 eV offset was added to the energy level of $p$ bands of N for clarity in the figure. We may make several qualitative observations. First, the flat and quasi-degenerate t$_{2g}$ and e$_{g}$ bands are widely observed toward representative $k$-directions along the Brillouin zone in the parent alloy. The additional charge supplied by the H atoms shifts the position of the Fermi level upwards but the character of these 3$d$ bands is conserved. Second, in the empty sphere configuration (Z=Es), where the significantly larger lattice structure of the nitrogenated alloy is adopted without any addition of charge, strong narrowing of the bandwidth is found but the dispersion of each band is once again preserved (not shown).
Third, the situation is very different for both Z=N and B, as the band structure is strongly altered at full doping. For Z=B, strong $p-d$ band mixing occurs between -0.3 and -0.6 eV for the majority spin states in all $k$-directions. In addition, broad bands originating from the $p$-states of boron appear at the $\Gamma$ point about -0.3 eV for the minority spins. The formation of these hybrid $p-d$ bands ultimately results in the vanishing of the well-formed peak and valley structures of the DOS around E$_{F}$ as shown in Fig. \[fig:FSMvsDOS\] and in Fig. \[fig:PDOS\]. Finally, for nitrogen doping most of the flat bands **** appear just above E$_{F}$, apart from some minor ones around -0.5 eV in the W direction for the minority spins. Here, the $p-d$ mixing occurs mainly between bands in the K and W directions. Bands of $e_{g}$ character show especially pronounced mixing with $p$-electrons. The number of flat bands around E$_{F}$ is decreased as compared to the undoped system and as a result the DOS has uneven features with small peaks that are detrimental to the IEM. transition.
Conclusions {#sec_conc}
===========
We have investigated the effect of selected $s$- and $p$-block interstitial elements on the electronic, lattice and magnetic properties of La(Fe,Si)$_{13}$ using DFT. Our calculations find that a good correlation between the expansion of the unit cell and the size of the dopant. Fixed spin moment calculations yield a double well structure in the free energy of LaFe$_{12}$Si as a function of magnetisation, which may be seen as the basis of the itinerant electron metamagnetic transition. Significantly, hydrogenation alters the electronic and magnetic structure of LaSiFe$_{12}$ to a much smaller degree than B, C and N dopants. This means that the first order IEM is much more robust to H insertion than to interstitial B, C, or N.
An analysis of the projected electronic DOS reveals that the dominant electronic states related to hydrogen insertion appear at around -8 to -7 eV, where very little contribution from Fe, Si and La elements is present. The additional charge of the hydrogen atoms elevates the Fermi level but the character of the bands is unaltered. The latter feature is also evident in the empty sphere configuration, where the nitrogenated lattice parameters are simulated without the N inclusions; only a narrowing of the band-width is found but the dispersion of each band remains mostly unaffected.
Consequently, hydrogen provides perhaps the only chemical pressure on the lattice that avoids significant alterations to the electronic structure of LaSiFe$_{12}$. In the case of the other dopants (B, C, N) broad bands originating from their $p$-states appear at energy levels, where the 3$d$ states of Fe are also present. The formation of these hybrid $p-d$ bands results in the disappearance of the peak and valley structures in the electronic DOS around E$_{F}$, thereby reshaping the shallow free energy landscape and ultimately destroying the first order IEM of LaSiFe$_{12}$.
The research leading to these results has received funding from the European Community’s 7th Framework Programme under Grant Agreement No. 310748 DRREAM. Computing resources provided by Darwin HPC and Camgrid facilities at The University of Cambridge and the HPC Service at Imperial College London are also gratefully acknowledged.
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---
abstract: 'The article is devoted to investigating the application of hedging strategies to online expert weight allocation under delayed feedback. As the main result we develop the General Hedging algorithm $\mathcal{G}$ based on the exponential reweighing of experts’ losses. We build the artificial probabilistic framework and use it to prove the adversarial loss bounds for the algorithm $\mathcal{G}$ in the delayed feedback setting. The designed algorithm $\mathcal{G}$ can be applied to both countable and continuous sets of experts. We also show how algorithm $\mathcal{G}$ extends classical Hedge (Multiplicative Weights) and adaptive Fixed Share algorithms to the delayed feedback and derive their regret bounds for the delayed setting by using our main result.'
address: Skolkovo Institute of Science and Technology
author:
- Alexander Korotin
- 'Vladimir V’yugin'
- Evgeny Burnaev
title: Adaptive Hedging under Delayed Feedback
---
hedging ,decision-theoretic online learning ,experts problem ,delayed feedback ,adaptive algorithms ,non-replicating algorithms ,adversarial setting.
Introduction
============
We consider the **Decision-Theoretic Online Learning** (DTOL) framework [@LiW94; @FrS97; @devroye2013prediction; @cesa2007improved; @kotlowski2018minimaxity; @devroye2013prediction] which is closely related to the paradigm of **prediction with expert advice** [@cesa-bianchi; @Vov90; @VoV98; @Vovk1999; @LiW94; @adamskiy2012putting; @BoW2002; @KorBurAggrBase2018]. A **master algorithm** at every step ${t=1,\dots,T}$ of the game has to choose the weight allocation for a given pool of expert strategies (experts). We call this problem the **experts problem**. We investigate the **adversarial case**, i.e., no assumptions are made about the nature of the data (stochastic, deterministic, etc.).
The performance of the master algorithm is measured by the **regret** over the entire game. The regret $R_T$ is the difference between the cumulative loss of the online algorithm and the loss of some given comparator. A typical comparator is the best fixed expert in the pool or the best fixed convex linear combination of experts. The goal of the algorithm is to minimize the regret, i.e., $R_{T}\rightarrow \min$.
In the classical online learning, the algorithm suffers loss of its decision at each step $t$ at the end of the same step the decision is made. In contrast to the classical scenario, we consider the **delayed feedback** learning. At each step $t$ of the game the algorithm makes a decision, and its result will be revealed only at the end of a time point $t+D_{t}$ (where $D_{t}\geq 0$ is some **delay**).
It turns out that there exists a wide range of algorithms for the non-delayed scenario ($D_{t}\equiv 0$). Almost all of them exploit the **follow-the-best-expert idea**: the better expert performed in the past, the higher relative weight is assigned to the expert. The pure **Follow the Leader** (FTL) strategy is well-known to have good performance in the stochastic setting[^1] [@kotlowski2018minimaxity], but it may be inefficient when the data is generated by an adversary (see discussion in [@shalev2012online; @de2014follow]).
**Follow the Perturbed Leader** (FTPL) algorithm [@kalai2005efficient] adds random noise to expert evaluation process. This prevents overfitting in adversarial setting. For example, exponential [@kalai2005efficient], random-walk [@devroye2013prediction], and dropout [@van2014follow] noise has been shown to achieve low expected regret for the experts problem.
**Follow the Regularized Leader**[^2] (FTRL) is a powerful algorithm from online convex optimization framework [@HazanOCO16], [@shalev2012online]. The usage of the linear loss function on a simplex allows to deal with the experts problem. The quadratic regularization leads to Online Gradient Descent (OGD) algorithm [@shalev2012online], the Entropic regularization provides Exponential Weights algorithm, also known as **Hedge** [@FrS97].
The idea of **multiplicative weight updates** (MW) of Hedge algorithm is used in many successive algorithms (MW2 [@cesa2007improved], Variation-MW [@hazan2010extracting], Optimistic-MW [@chiang2012online], AEG-Path and AMEG-Path [@steinhardt2014adaptivity] and other algorithms [@gaillard2014second; @koolen2015second]). The main goal of such algorithms is to obtain the first or the second order regret bound (e.g. in terms of best expert’s loss) or achieve improvement for easy-data. Also, some Hedge-based algorithms (AdaHedge [@erven2011adaptive], Flip-Flop [@de2014follow]) are designed to be parameter-free.
Almost all described algorithms provide $O(\sqrt{T})$ adversarial regret guarantees w.r.t. the best expert in the pool. Note that this bound is minimax optimal up to some multiplicative factor because $\Omega(\sqrt{T})$ is known to be the lower bound [@cesa-bianchi].[^3]
An important variant of the experts problem is to develop an **adaptive master algorithm**. Such an algorithm has to track the shifts (switches) of the best expert and achieve low **tracking regret** with respect to shifting sequences of experts.[^4] There are many meta-approaches such as restarts [@AdaAda16; @hazan2009efficient] or specialist experts [@freund1997using] to create adaptive algorithms from non-adaptive ones. However, the most recognizable approach is to use the **Fixed Share** extension for Hedge [@HeW98; @cesa2012mirror; @AdaAda16; @BoW2002].
When it comes to the delayed feedback setting, many of the above described **non-delayed algorithms** do not have theoretical guarantees of performance or do not even have a modification for the delayed feedback setting.
There exists a bunch of meta-algorithms that allow to produce a version for delayed feedback setting from the basic non-delayed version [@WeO2002; @Mes2009; @Mes2007; @joulani2013online]. The roots of meta-approach lie in the work [@WeO2002]. The authors studied the setting under fixed known feedback delay $D$. They proved that the optimal (non-adaptive) algorithm is to run $D+1$ independent versions of the optimal non-delayed algorithm on $D+1$ disjoint time grids ${GR_{d}=\{t\mbox{ }|\mbox{ }t\equiv d\mbox{ }(\mbox{mod } D+1)\}}$ for $1{\le d\le D+1}$. Thus, the optimal worst-case adversarial regret is $(D+1)\cdot \Omega(\sqrt{\frac{T}{D+1}})=\Omega(\sqrt{T(1+D)})$. The described meta-approach was enhanced for the unknown and dynamic feedback delay in [@joulani2013online]. Their meta-algorithm **BOLD** (Black-box Online Learning with Delays) also runs independent copies of the basic algorithm on disjoint time lines.
We call algorithms obtained by meta-approaches (such as BOLD) **replicated** algorithms. Whereas replicating is simple and in some cases is theoretically optimal, it has several obvious practical drawbacks. Firstly, it uses only part of the observed data at every step of the game. Secondly, separate replicating learning processes generated by the meta-algorithm do not even interact.
Non-adaptive algorithms based on FTRL and FTPL have several **non-replicated** adaptations for delayed feedback setting. The most straightforward ones are Delayed OGD [@LSZ2009], Delayed FTPL and FTRL [@QuanDGD15] and FTRL with Memory [@AHS2015]. For the fixed and known feedback delay $D$ their best regret bound is $O(\sqrt{T(1+D)})$, which is optimal.
**In this work,** we aim to create an adaptive non-replicated algorithm for the delayed feedback setting. We base our research on the Hedge algorithm (and its adaptive extension Fixed Share), which is the state-of-the-art basis for many existing algorithms. In order to achieve the desired goal, we develop the general probabilistic framework for Hedge-based algorithms. Using this framework, we propose the **General Hedging Algorithm** $\mathcal{G}$, prove its loss bounds both for delayed and non-delayed cases. As a corollary of the main result, we show how classical non-delayed Hedge and Fixed Share algorithms (as the cases of $\mathcal{G}$) can be extended to the delayed feedback setting and what regret bounds they have.
**The main contributions of this paper are:**
1. Developing the General Hedging algorithm $\mathcal{G}$ for the delayed feedback scenario which is applicable to both non-delayed and delayed online settings. Proving the algorithm’s loss bound (and regret bound, for the case of a countable set of experts) in a general form.
2. Developing (for a finite number of experts) non-replicated versions of basic Hedge [@FrS97] and adaptive Fixed Share [@HeW98] algorithms (as special cases of algorithm $\mathcal{G}$) for the delayed feedback scenario as well as deriving their regret bounds.
The General Hedging algorithm $\mathcal{G}$ which we develop is motivated by the paper [@adamskiy2012putting]. In that work the authors considered the special case of the prediction with experts’ advice with the **logarithmic loss function**. For the traditional non-delayed scenario ($D_{t}\equiv 0$) they developed the **Bayesian Merging Algorithm** for mixing (averaging) experts’ predictions. Their algorithm is based on the natural graphical model (similar to the one in Figure \[figure:model-general-n\] of Section \[sec-algorithm\]) implied by the probabilistic origin of the logarithmic loss function.
In contrast to [@adamskiy2012putting], we consider the decision-theoretic online learning scenario (hedging), which is more general than prediction with experts’ advice.[^5] At the same time we investigate both non-delayed and delayed feedback settings. We build the artificial probabilistic framework for arbitrary bounded losses by using the **entropithication** transform (loss exponentiation, see e.g. [@grunwald2004game; @van2015fast]), state the General Hedging algorithm $\mathcal{G}$ and prove its loss bound.
**The article is structured as follows:**
In Section \[sec-prelim\] we give preliminary notions, describe the notation and the setting of the game of the delayed feedback experts’ weights allocation.
In Section \[sec-algorithm\] we describe the developed probabilistic framework, the main algorithm $\mathcal{G}$, and formulate the main Theorem \[theorem-main-loss-bound-D\] about its loss bound. In Section \[sec-proofs\] we prove the main theorem.
In Section \[sec-examples\] we provide the examples of the application of algorithm $\mathcal{G}$: Delayed Hedge in Subsection \[hedge-delayed\], Delayed Fixed Share in Subsection \[sec-fs\].
In Section \[sec-experiments\] we conduct massive computational experiments and provide the detailed discussion of the results.
In \[sec-appendix-math\] we provide the necessary mathematical background.
Preliminaries {#sec-prelim}
=============
We use **bold** font to denote vectors (e.g. $\bm{w}\in \mathbb{R}^{M}$ for some integer $M$). In most cases, superscript is used for indexing elements of a vector (e.g. ${(w^{1},\dots,w^{N})=\bm{w}}$). Subscript is always used to indicate time (e.g. $l_{t}, R_{T}, w_{t}^{n}$).
We consider the online game of delayed hedging of a (finite or infinite) pool of experts. We use $\mathcal{N}$ to denote the pool and $n\in\mathcal{N}$ as an index of an expert. In this paper $\mathcal{N}$ is either a discrete set (e.g. ${\mathcal{N}=\{1,2,\dots, N\}}$) or a continuous subset of Euclidean space (e.g. ${\mathcal{N}=\mathbb{R}^{M}}$). By $\Delta(\mathcal{N})$ for a discrete (continuous) set $\mathcal{N}$ we denote all discrete (continuous) probability distributions on $\mathcal{N}$.
For convenience, we do all calculations in the paper assuming that $\mathcal{N}$ is a discrete countable set. All the results also hold true for the continuous $\mathcal{N}$ but sums over $n$ (e.g. ${\sum_{n\in\mathcal{N}}[\ldots]}$) should be replaced with corresponding intergrals (e.g. ${\int_{n\in\mathcal{N}}[\ldots]\cdot dn}$).
At each integer time step ${t=1,2,\dots, T}$ of the game the master (hedging) algorithm has to assign the weights $w^{n}_{t}$ to all experts ${n\in\mathcal{N}}$ so that $$\bm{w}_{t}=\{w_{t}^{n}\text{ for }n \in\mathcal{N}\}\in \Delta(\mathcal{N}).$$ At the end of the step $t+D_{t}$ (for integer $D_{t}\geq 0)$ experts reveal their losses ${\bm{l}_{t}=\{l_{t}^{n}\text{ for }n\in\mathcal{N}\}}$ at the step $t$. The loss of the algorithm’s decision of the step $t$ is $$h_{t}=\sum_{n\in\mathcal{N}}[l_{t}^{n}\cdot w_{t}^{n}]\stackrel{def}{=}\langle \bm{w}_{t}, \bm{l}_{t}\rangle,$$ i.e., the average experts’ loss w.r.t. $\bm{w}_{t}$.
The sequence $D_{1},D_{2},\dots,D_{T}$ is called the sequence of delays. For correctness, we assume that ${t+D_{t}\leq T}$ for all $t=1,2,\dots,T$. We denote the set of all time indices of the losses revealed before the end of the step $t$ by ${\mathcal{D}_{t}=\{\tau|\tau+D_{\tau}\leq t\}}$. Also, we denote ${d\mathcal{D}_{t}=\mathcal{D}_{t}\setminus \mathcal{D}_{t-1}}$.
There are many scenarios on how the sequence $D_{t}$ is chosen (randomly, adversarially) and whether it is known to the learner in advance or not (see e.g. [@WeO2002; @Mes2009; @Mes2007; @agarwal2011distributed; @joulani2013online]). Yet, we do not specify the particular scenario, and consider the game in the general form.
In this work we assume that all the losses are bounded: ${l_{t}^{n}\in [0,H]}$ for all ${t=1,\dots,T}$ and ${n\in\mathcal{N}}$. This is a common assumption in online learning (see [@shalev2012online; @HazanOCO16] or any other survey on online learning). The game setting is described by the following Protocol \[protocol\].
We use $H_{T}=\sum_{t=1}^{T}h_{t}$ and $L_{T}^{n}=\sum_{t=1}^{T}l_{t}^{n}$ to denote the cumulative (total) loss of the algorithm and expert $n\in\mathcal{N}$.
The performance of the algorithm is measured by the (cumulative) regret. The regret is the difference between the cumulative loss of the algorithm and the cumulative loss of some given comparator. A typical approach is to compete with the best expert in the pool. The cumulative regret with respect to the best expert is $$R_{T}=H_{T}-\min_{n\in\mathcal{N}}L_{T}^{n}.
\label{base-regret}$$
The goal of the algorithm is to minimize the regret, i.e., ${R_{T}\rightarrow \min}$. In order to theoretically guarantee algorithm’s performance, some upper bound is usually proved for the cumulative regret ${R_{T}\leq f(T)}$.
In the basic setting , sub-linear upper bound $f(T)$ for the regret leads to the asymptotic performance of the algorithm equal to the performance of the best expert. More precisely, we have $\lim_{T\rightarrow \infty}\frac{R_{T}}{T}=0$.
Generalized Hedging Algorithm {#sec-algorithm}
=============================
In this section we describe the generalization $\mathcal{G}$ of the classical hedging algorithm based on exponential reweighing of experts’ losses. The basic algorithm was introduced by [@FrS97].
We investigate the adversarial case, i.e., no assumptions (stochastic, functional, etc.) are made about the nature of data (experts’ losses). However, it turns out that in this case it is convenient to develop algorithms using some probabilistic framework.
Probabilistic Framework
-----------------------
Recall that $\bm{l}_{t}=\{l_{t}^{n}\text{ for }n\in\mathcal{N}\}$ is a dictionary of experts’ losses at the step $t$. The framework that we build implies that data is generated by some probabilistic model with hidden states. The graphical model is shown in Figure \[figure:model-general-n\].
![General probabilistic model for data generation process[]{data-label="figure:model-general-n"}](hedge-model-general-n.eps)
We suppose that there is some hidden sequence of experts $ n_{t}\in \mathcal{N}$ (for ${t=1,2,\dots,T}$) that generates the experts’ losses $\bm{l}_{t}$. In particular, hidden expert $n_{t}$ at the step $t$ is called **active expert**. The **conditional probability** to observe the vector $\bm{l}_{t}$ of experts’ losses at the step $t$ is
$$p(\bm{l}_{t}| n_{t})=p(l_{t}^{n_{t}}| n_{t})=\frac{e^{-\eta l_{t}^{n_{t}}}}{Z_{t}},
\label{observation-probability}$$
where $\eta>0$ is some fixed learning rate and $Z_{t}=\int_{l\in [0, H]}e^{-\eta l}dl$ is the normalizing constant. Constant $Z_{t}$ is $n_{t}$-independent. The idea of conditional probability is to assume that if at the step $t$ expert $n_{t}\in\mathcal{N}$ is active, then the loss vector $\bm{l}_{t}=(l_t^1, l_t^2,...,l_t^N)$ is not completely random, i.e. loss $l_t^{n_{t}}$ is random, while all the other components are deterministic (e.g. given by nature).[^6]
For the first active expert $n_{1}$ some known **prior distribution** is given $p(n_{1})=p_{0}(n_{1})$. The sequence $(n_{1},\dots, n_{T})$ of active experts is generated step by step. For $t\in\{1,\dots,T-1\}$ each $ n_{t+1}$ is sampled from some known distribution $p(n_{t+1}|N_{t})$, where $N_{t}=(n_{1},\dots, n_{t})$.[^7] Thus, active expert $n_{t+1}$ depends on the previous experts $N_{t}$.
For every sequence of experts $N_{t}=(n_{1},n_{2},\dots,n_{t})$ we denote the cumulative loss of the sequence by $$L_{t}^{N_{t}}=\sum_{\tau=1}^{t}l_{\tau}^{n_{\tau}}.$$ For all $t$ we define the following lists of loss vectors: $$\bm{L}_{t}=(\bm{l}_{1},\dots,\bm{l}_{t}),\qquad {\bm{L}_{\mathcal{D}_{t}}=\{\bm{l}_{\tau}\text{ for }\tau\in \mathcal{D}_{t}\}}, \qquad {\bm{L}_{d\mathcal{D}_{t}}=\{\bm{l}_{\tau}\text{ for }\tau\in d\mathcal{D}_{t}\}}.
\nonumber$$
The considered probabilistic model is: $$\begin{aligned}
p(N_{T},\bm{L}_{T})=p(N_{T})\cdot p(\bm{L}_{T}|N_{T})=
\bigg[p_{0}(n_{1})\prod_{t=2}^{T}p(n_{t}|N_{t-1})\bigg]\cdot \bigg[\prod_{t=1}^{T}p(\bm{l}_{t}| n_{t})\bigg].
\label{model-general}\end{aligned}$$
The probability $p(N_{T})$ is that of hidden states (active experts).[^8]
General Hedging Algorithm
-------------------------
The hedging algorithm \[algorithm-main\] is shown below. We denote it by $\mathcal{G}=\mathcal{G}(p)$ ($\mathcal{G}$ stands for **G**eneral), where $p$ indicates the probability distribution $p({N}_{T})$ of active experts to which the algorithm is applied.
The idea of the algorithm $\mathcal{G}$ is simple: set the weight allocation $\bm{w}_{t}$ for the current step $t$ according to the posterior probability $p(n_{t}|\bm{L}_{\mathcal{D}_{t-1}})$ of the expert $n_{t}$ computed from the underlying probabilistic model. We illustrate this idea in Figure \[figure:model-posterior\].
![The idea of General Hedge Algorithm. The weights $\bm{w}_{t}$ used at the step $t$ correspond to the posterior probability $p(n_{t}|L_{\mathcal{D}_{t-1}})$ of the hidden state at the current step[]{data-label="figure:model-posterior"}](hedge-model-posterior.eps)
Consider a finite pool $\mathcal{N}=\{1,\dots,N\}$ and set $p_{0}(n_{1})\equiv \frac{1}{N}$ for all $n_{1}\in\mathcal{N}$. Consider the non-delayed scenario ($D_{t}\equiv 0$ for all $t$). If we use ${p(n_{1}=n_{2}=\dots=n_{T})\equiv 1}$, the experts’ weights become $w_{t}^{n}\propto e^{-\eta L_{t-1}^{n}}$. The resulting algorithm $\mathcal{G}(p)$ turns to be classical non-delayed Hedge (for more detailed discussion see Subsection \[hedge-delayed\]). Also, non-delayed Fixed Share is the case of $\mathcal{G}$ for specially chosen Markovian $p(\cdot)$ (see Subsection \[sec-fs\]).
The **time and memory complexities** of the algorithm depend on the properties of the underlying distribution $p$. For Markovian models (when hidden state $n_{t}$ depends only on the previous state $n_{t-1}$ for all $t$) it is possible to provide linear in $T+\sum_{t=1}^{T}D_{t}$ schemes to compute weights (see Subsection \[sec-fs\]) which require $O(N\cdot \max_t D_{t})$ memory. For the arbitrary $p(\cdot)$ time and memory complexity may be even exponential.
Guarantees of Performance
-------------------------
The algorithm has **theoretical guarantees of performance**. We state the following main theorem.
\[theorem-main-loss-bound-D\] Let $\mathcal{N}$ be a countable (or continuous) set of experts. Let $p(\cdot)$ be a discrete (or continuous) distribution on $\mathcal{N}^{T}$. Then for the hedging algorithm $\mathcal{G}$ applied to model $p$ with learning rate $\eta>0$ the following upper bound for the total loss over the entire game holds true: $$\begin{aligned}
H_{T}\leq -\frac{1}{\eta}\ln \bigg[\mathbb{E}_{p({N}_{T})}\big[e^{-\eta L_{T}^{{N}_{T}}}\big]\bigg]+
\eta\frac{H^{2}}{8}T+\eta \big[\frac{H^{2}\cdot \sum_{t=1}^{T}D_{t}}{4}\big]T.
\label{main-loss-bound-convex}\end{aligned}$$
The proof of this theorem is given in Section \[sec-proofs\]. Note that while the algorithm may seem to be designed for the stochastic setting, we apply it to the pure adversarial case[^9] and obtain the loss guarantees. At the same time, the adversarial loss bound depends on the probability distribution $p(\cdot)$ for which the algorithm in designed.
One may wonder how Theorem \[theorem-main-loss-bound-D\] is applied to estimate the regret, for example, the regret with respect to the best expert . If the set $\mathcal{N}$ of experts is countable, then the following simple corollary holds true.
\[corollary-regret-sequence\] If the set of experts $\mathcal{N}$ is countable, then under the conditions of Theorem \[theorem-main-loss-bound-D\], the regret with respect to any sequence ${{N}_{T}^{*}=(n_{1}^{*},n_{2}^{*},\dots, n_{T}^{*})\in\mathcal{N}^{T}}$ is
$$\begin{aligned}
R_{T}({N}_{T}^{*})=H_{T}-L_{T}^{{N}_{T}^{*}}\leq
-\frac{1}{\eta}\ln p({N}_{T}^{*})+\eta\frac{H^{2}}{8}T+\eta \big[\frac{H^{2}\cdot \sum_{t=1}^{T}D_{t}}{4}\big]T.
\nonumber\end{aligned}$$
The corollary results from the following inequality for the expectation in the right part of : $$\begin{aligned}
-\frac{1}{\eta}\ln \bigg[\mathbb{E}_{p({N}_{T})}\big[e^{-\eta L_{T}^{{N}_{T}}}\big]\bigg]\leq -\frac{1}{\eta}\ln \bigg[p({N}^{*}_{T})e^{-\eta L_{T}^{{N}_{T}^{*}}}\bigg]=
L_{T}^{{N}_{T}^{*}}-\frac{1}{\eta}\ln p({N}_{T}^{*}),
\nonumber\end{aligned}$$ which leads to the desired bound.
If $\mathcal{N}$ is continuous under the conditions of Theorem \[theorem-main-loss-bound-D\], then the first term in the upper bound is represented by the integral (instead of a countable sum). It is not possible to extract a single summand as in the finite case. However, sometimes the expectation can be directly computed or estimated w.r.t. the loss of the best expert in the pool. For example, see approaches of [@VoV2001; @Kaln2007; @zhdanov2010identity] applied to Online Kernel Regression.
Examples {#sec-examples}
========
In this Section we provide the examples of useful underlying probability models $p(\cdot)$ and use them to apply the algorithm $\mathcal{G}$ to construct online expert weight allocation algorithms. We consider a finite pool of experts ${\mathcal{N}=\{1,2,\dots, N\}}$.
Basic Delayed Exponential Weights (Hedge) {#hedge-delayed}
-----------------------------------------
Consider the following underlying probability $p$. Let ${p(n_{1})=p_{0}(n_{1})}$ be some prior and ${p(n_{t}|n_{t-1})=\mathbb{I}_{[n_{t}=n_{t-1}]}}$ for ${t=2,\dots,T}$. This means that the hidden active expert does not change during the game. We denote the corresponding algorithm applied to $p$ by ${\mathcal{G}_{\text{base}}=\mathcal{G}_{\text{base}}(p_{0})}$. The corresponding graphical model is shown in Figure \[figure:model-simple-n\].
![Hedge probabilistic model for data generation process[]{data-label="figure:model-simple-n"}](hedge-model-simple-n.eps)
It is easy to see that for all $t$ the weight allocation $${w_{t}^{n}\propto p_{0}(n)\cdot e^{-\eta L_{\mathcal{D}_{t}}^{n}}}$$ is proportional to the observed losses of the expert $n\in\mathcal{N}$. If there are no delays ($D_{t}\equiv 0$ for all $t$), then the algorithm becomes classical Hedge by [@FrS97].
### Algorithm
The pseudo-code of Algorithm \[algorithm-delayed-hedge\] ($\mathcal{G}_{\text{base}}$) is shown below. In the code we assume that the operation **Output**($\ldots$) sets the weight allocation ($\bm{w}_{t}$) for the current step. Function **GetRevealedLosses**() obtains all the vectors of losses ${\bm{l}_{\tau}=(l_{\tau}^{1},\dots,l_{\tau}^{N})}$ of the steps ${\tau\in d\mathcal{D}_{t}}$ in the form of an iterable list of pairs ${(\tau, \bm{l}_{\tau})}$.
$\bm{w}\leftarrow \bm{p}_{0}$
The algorithm requires $O(N)$ **memory** and $O(NT)$ **time complexity**.
### Regret bound
According to Corollary \[corollary-regret-sequence\], the regret of the algorithm with respect to any fixed expert $n\in\mathcal{N}$ is bounded: $$\begin{aligned}
R_{T}(n)=H_{T}-L_{T}^{n}\leq
-\frac{1}{\eta}\ln p_{0}(n)+\eta\frac{H^{2}}{8}T+\eta \big[\frac{H^{2}\cdot \sum_{t=1}^{T}D_{t}}{4}\big].
\nonumber\end{aligned}$$
The typical prior is $p_{0}\equiv \frac{1}{N}$. For this basic case in the non-delayed feedback setting ($D_{t}\equiv 0$) the $\eta$ is chosen in advance (with prior knowledge of $T$) to minimize the regret. The optimal choice is $\eta\propto \frac{1}{H\sqrt{T}}$, which results in $O(\sqrt{T})$ classical regret.
However, the choice of optimal $\eta$ in the delayed setting highly depends on how the sequence of delays is generated. If the learner knows $\sum_{t=1}^{T}D_{t}$ in advance or $D_{t}$ is sampled from some distribution with known expectation $\mathbb{E}D$, the optimal choice is $$\eta\propto \frac{1}{H\sqrt{T+\sum_{t=1}^{T}D_{t}}}
\qquad
\text{or}
\qquad
\eta\propto \frac{1}{H\sqrt{T(1+\mathbb{E}D)}}
\nonumber$$ respectively. This choice results in $O(\sqrt{T+\sum_{t=1}^{T}D_{t}})$ and $O(\sqrt{T(1+\mathbb{E}D)})$ regret bounds respectively.
If the sequence of delays is chosen by an adversary, the classical choice $\eta\propto \frac{1}{H\sqrt{T}}$ results in $O[\sqrt{T}(1+\overline{D})]$ regret, where $\overline{D}=\frac{1}{T}\sum_{t=1}^{T}D_{t}$.
Adaptive Delayed Exponential Weights (Fixed Share) {#sec-fs}
--------------------------------------------------
Consider the following underlying probability $p$. Let ${p(n_{1})=p_{0}(n_{1})}$ be some prior and $${p(n_{t}|n_{t-1})=\alpha_{t}p_{0}(n_{t})+(1-\alpha_{t})\cdot \mathbb{I}_{[n_{t}=n_{t-1}]}}
\label{fixed-share-trans-prob}$$ for $t=2,\dots,T$ and sequence $0\leq \alpha_{2},\dots,\alpha_{T}\leq 1$. This means that the hidden active expert changes to random (according to prior $p_{0}$) between steps $t-1$ and $t$ with some small probability $\alpha_{t}$.
We denote the corresponding algorithm applied to $p$ by $\mathcal{G}_{\text{fs}}$. The graphical model is shown in Figure \[figure:model-markov-n\].
![Fixed Share probabilistic model for data generation process[]{data-label="figure:model-markov-n"}](hedge-model-markov-n.eps)
The sequence $\alpha_{t}$ can be arbitrary. However, the classical approach is to use $\alpha_{t}=\frac{1}{t}$ (see [@HeW98; @AdaAda16; @cesa2012mirror]), because in this special case the regret bound is better (than e.g. in the case $\alpha_{t}\equiv const$). In our case at the end of the subsection we will also use the sequence $\alpha_{t}=\frac{1}{t}$ when estimating the regret.
### Equivalence to Fixed Share in the Non-Delayed Setting
To begin with, we examine the application of algorithm $\mathcal{G}$ to the described probabilistic model $p(\cdot)$ in the non-delayed case, i.e., $D_{t}\equiv 0$ for all ${t=1,2,\dots,T}$. In the non-delayed case we have $\mathcal{D}_{t}=\{1,2,\dots,t\}$ for all $t$. Thus, for all $t$ we get $$w_{t}^{n}=p(n_{t}=n|\bm{L}_{\mathcal{D}_{t-1}})=p(n_{t}=n|\bm{L}_{t-1}).$$ We set ${\bm{u}_{t}=(u_{t}^{1},\dots, u_{t}^{N})\in\Delta(\mathcal{N})}$, $u_{t}^{n}=p(n_{t}=n|\bm{L}_{t})$ for all ${n\in\mathcal{N}}$ and ${t=1,2,\dots, T}$. We get $$\begin{aligned}
w_{t}^{n}=p(n_{t}=n|\bm{L}_{t-1})=
\nonumber
\\
\sum_{n'\in\mathcal{N}}\bigg[p(n_{t}=n|n_{t-1}=n')\cdot \underbrace{p(n_{t-1}=n'|\bm{L}_{t-1})}_{u_{t-1}^{n'}}\bigg].
\label{fs-reweighing-part}\end{aligned}$$ Combining with we see that $$\bm{w}_{t}=(1-\alpha_{t})\cdot \bm{u}_{t-1}+\alpha_{t}\cdot \bm{p}_{0}.
\label{fixed-share-first-update}$$ On the other hand, $$\begin{aligned}
u_{t}^{n}=p(n_{t}=n|\bm{L}_{t})=\frac{p(\bm{L}_{t}|n_{t}=n)\cdot p(n_{t}=n)}{p(\bm{L}_{t})}=
\nonumber
\\
\frac{p(\bm{l}_{t}|n_{t}=n)\cdot p(\bm{L}_{t-1}|n_{t}=n)\cdot p(n_{t}=n)}{p(\bm{L}_{t})}=
\nonumber
\\
\underbrace{p(n_{t}=n|\bm{L}_{t-1})}_{w_{t}^{n}}\cdot \underbrace{p(\bm{l}_{t}|n_{t}=n)}_{\exp (-\eta\cdot l_{t}^{n})}\cdot \big[\frac{p(\bm{L}_{t-1})}{p(\bm{L}_{t})}\big].
\nonumber\end{aligned}$$ Thus, $$\bm{u}_{t}\propto \bm{w}_{t}\cdot \exp(-\eta \cdot \bm{l}_{t}).
\label{fixed-share-second-update}$$ Formulas and mean that the algorithm’s decision $\bm{w}_{t}$ can be iteratively updated step by step by using the additional weight $\bm{u}_{t}$. The obtained weight updates and exactly match the updates of the Fixed Share algorithm by [@HeW98]. Thus, in the non-delayed case $\mathcal{G}(p)$ is equal to Fixed Share.
### Algorithm for the Delayed Setting
Now we examine the algorithm $\mathcal{G}(p)$ under the setting of the delayed feedback, i.e., for all ${t=1,2,\dots,T}$ delay $D_{t}$ is some non-negative integer value.
For all $t=1,2,\dots,T$ and $\tau\leq t$ we use $\mathcal{D}_{t}^{\tau}$ to denote the set of all time steps $t'\leq \tau$ such that the loss vector $\bm{l}_{t'}$ is revealed not later than the step $t$. Formally, we define $$\mathcal{D}_{t}^{\tau}=\{t'|(t'+D_{t'}\leq t)\land (t'\leq \tau)\}.$$
In the next few paragraphs we describe the efficient scheme to recompute the algorithms decision $\bm{w}_{t}$ at every step $t$.
Suppose that at the beginning of the step $t$ we keep all the probabilities $p(n_{\tau}|\bm{L}_{\mathcal{D}_{t}^{\tau}})$ for all $\tau=1,\dots,t$ and $n\in\mathcal{N}$. We denote the corresponding $N$-dimensional probability vectors by $\bm{u}_{\tau}$. We also denote $\bm{u}_{0}=\bm{p}_{0}$. Similar to calculations lead to the simple formula that allows to obtain $\bm{w}_{t}$: $$\bm{w}_{t}=(1-\alpha_{t})\cdot \bm{u}_{t-1}+\alpha_{t}\cdot \bm{p}_{0}.$$ After the decision on $\bm{w}_{t}$ is made, the algorithm obtains losses of steps ${\tau\in d\mathcal{D}_{t}}$. Thus, we need to calculate new probability vector $\bm{u}_{t}$ with coordinates $p(n_{\tau}|\bm{L}_{\mathcal{D}_{t}^{\tau}})$. Moreover, we have to update all vectors $\bm{u}_{\tau}$ for $\tau<t$ from $p(n_{\tau}|\bm{L}_{\mathcal{D}_{t-1}^{\tau}})$ to $p(n_{\tau}|\bm{L}_{\mathcal{D}_{t}^{\tau}})$.
Let $\tau_{\min}=\min\{\tau:\tau \in d\mathcal{D}_{t}\}$. Note that all $\bm{u}_{\tau}$ for $\tau <\tau_{\min}$ do not require being updated because $\mathcal{D}_{t}^{\tau}=\mathcal{D}_{t-1}^{\tau}$. Next, for $\tau=\tau_{\min},\dots,t-1, t$ we recompute the vectors $\bm{u}_{\tau}$ iteratively.
We explain how to compute $\bm{u}_{\tau}$ below (assuming that previous $\bm{u}_{\tau-1}$ is already computed). For convenience, we introduce the temporary vector variable ${\bm{v}=(v^{1},\dots, v^{N})\in\Delta(\mathcal{N})}$, where $v^{n}=p(n_{\tau}=n|\bm{L}_{\mathcal{D}_{t}^{\tau-1}})$ for all ${n\in\mathcal{N}}$. First, we express $\bm{v}$ using $\bm{u}_{\tau}$. Next, we derive the expression for $\bm{u}_{\tau}$ using $\bm{v}$.
We deduce the formula to compute $v^{n_{\tau}}$ by using $u_{\tau-1}^{n_{\tau}}$: $$\begin{aligned}
v^{n_{\tau}}=p(n_{\tau}|\bm{L}_{\mathcal{D}_{t}^{\tau-1}})=\sum_{n_{\tau-1}\in\mathcal{N}}p(n_{\tau-1}|\bm{L}_{\mathcal{D}_{t}^{\tau-1}})\cdot p(n_{\tau}|n_{\tau-1})=
\label{fixed-share-transition-decomposition}
\\
\sum_{n_{\tau-1}\in\mathcal{N}}\bigg[p(n_{\tau-1}|\bm{L}_{\mathcal{D}_{t}^{\tau-1}})\cdot \big[\alpha_{\tau}\cdot p_{0}(n_{\tau})+(1-\alpha_{\tau})\cdot \mathbb{I}_{[n_{\tau}=n_{\tau-1}]}\big]\bigg]=
\label{fixed-share-transition-usage}
\\
\underbrace{\big[\sum_{n_{\tau-1}\in\mathcal{N}}p(n_{\tau-1}|\bm{L}_{\mathcal{D}_{t}^{\tau-1}})\big]}_{\text{Sums to }1}\cdot \alpha_{\tau}\cdot p_{0}(n_{\tau})+(1-\alpha_{\tau})\cdot u_{\tau-1}^{n_{\tau}}=
\nonumber
\\
\alpha_{\tau}\cdot p_{0}(n_{\tau})+(1-\alpha_{\tau})\cdot u_{\tau-1}^{n_{\tau}}.
\label{fixed-share-transition-recomputing}\end{aligned}$$ In line we exploit the fact that the elements of $\mathcal{D}_{t}^{\tau-1}$ are strictly lower than $\tau$. In line we use the definition of the transition probability. The vector form of is $$\bm{v}=(1-\alpha_{\tau})\cdot \bm{u}_{\tau-1}+\alpha_{\tau}\cdot \bm{p}_{0}.
\nonumber$$
To derive $\bm{u}_{\tau}$ using $\bm{v}$ we consider two cases: $\tau\notin\mathcal{D}_{t}$ and $\tau\in\mathcal{D}_{t}$. In the first case ${\bm{L}_{\mathcal{D}_{t}^{\tau-1}}\equiv\bm{L}_{\mathcal{D}_{t}^{\tau}}}$, which leads to $\bm{u}_{\tau}=\bm{v}$. If $\tau\in\mathcal{D}_{t}$, we have $$\begin{aligned}
u_{\tau}^{n_{\tau}}=p(n_{\tau}|\bm{L}_{\mathcal{D}_{t}^{\tau}})\propto
p(\bm{L}_{\mathcal{D}_{t}^{\tau}}|n_{\tau})\cdot p(n_{\tau})=
\nonumber
\\
p(\bm{L}_{\mathcal{D}_{t}^{\tau-1}}|n_{\tau})\cdot p(\bm{l}_{\tau}|n_{\tau})\cdot p(n_{\tau})\propto p(n_{\tau}|\bm{L}_{\mathcal{D}_{t}^{\tau-1}})\cdot p(\bm{l}_{\tau}|n_{\tau})\propto
\nonumber
\\
p(n_{\tau}|\bm{L}_{\mathcal{D}_{t}^{\tau-1}})\cdot \exp(-\eta l_{\tau}^{n_{\tau}})=v^{n_{\tau}}\cdot \exp(-\eta l_{\tau}^{n_{\tau}}).
\label{fixed-share-update-observation}\end{aligned}$$ The vector form of expression is $$\bm{u}_{\tau}\propto \bm{v}\cdot \exp(-\eta \cdot \bm{l}_{\tau}).$$
The pseudo-code of algorithm \[algorithm-delayed-fixed-share\] ($\mathcal{G}_{\text{fs}}$) is shown below. In addition to the notations of Algorithm \[algorithm-delayed-hedge\] ($\mathcal{G}_{\text{base}}$), we assume that an extra function **GetSwitchProbability**() provides the value of the current switch probability $0\leq\alpha_{t}\leq 1$ (which may be chosen online).
$\bm{u}\leftarrow$ List($[\bm{p}_{0}]$) $\bm{\alpha}\leftarrow$ List($[1]$) $\bm{l}\leftarrow$ List(\[Null\]) $\bm{v}\leftarrow$ Null
The **List**() class corresponds to the dynamic array. We assume that it has integer index $\{0,1,\dots,|List|-1\}$, supports the append-to-right operation in $O(1)$ time. We also assume that all operations to get or set list element (by index) require $O(1)$ time. At the end of each step $t$ list $\bm{u}$ keeps the posterior probabilities described above.
The **time complexity** of the algorithm is bounded by $${O\big(N\cdot(T+\sum_{t=1}^{T}D_{t})\big)}.$$ Indeed, at the steps $t$ such that $|d\mathcal{D}_{t}|=0$ the algorithm performs $O(N)$ operations. At other steps the algorithm performs $${O\big(N\cdot(t+1-\min\{\tau:\tau \in d\mathcal{D}_{t}\})\big)}$$ operations which are bounded by $O(N\cdot D_{\tau})$ for the minimal $\tau\in d\mathcal{D}_{t}$.
The **memory complexity** of the algorithm is $O(NT)$. However, it is possible to significantly reduce the memory complexity. Note that if for some $\tau, t$ we have $\mathcal{D}_{t}^{\tau}=\{1,2,\dots,t\}$, the weights $\bm{u}_{0},\dots,\bm{u}_{\tau-1}$ will never be used or recomputed after the step $t$. Thus, they become useless, and it is meaningful to keep only elements $\bm{u}_{t'}$ with $t'\geq \tau$ (same for lists $\bm{l}$ and $\bm{\alpha}$). The reduction will result in $O(N\cdot \max D_{t})$ memory complexity. We did not include the explained trick in the pseudo-code of Algorithm \[algorithm-delayed-fixed-share\] in order to keep it simple.
Regret Bound
------------
We use $\alpha_{t}=\frac{1}{t}$. We combine Corollary \[corollary-regret-sequence\] with Lemma \[fixed-share-prob-bound\] and obtain the regret bound for the algorithm with respect to any switching sequence ${{N}_{T}=(n_{1},n_{2},\dots,n_{T})}$: $$\begin{aligned}
R_{T}({N}_{T})\leq H_{T}-L_{T}^{{N}_{T}}\leq
\nonumber
\\
(K+1)\cdot \frac{\ln N+\ln T}{\eta}+\eta\frac{H^{2}}{8}T+
\eta \big[\frac{H^{2}\cdot \sum_{t=1}^{T}D_{t}}{4}\big]T,
\label{fixed-share-regret-bound}\end{aligned}$$ where $K=|\{t:\,n_{t}\neq n_{t-1}\}|$ is the number of expert’s switches in ${N}_{T}$.
Similar to the non-adaptive case, the algorithm requires choosing optimal learning rate $\eta$ in order to minimize the regret bound. The optimal $\eta$ should be chosen with respect to $T$ and $\sum_{t=1}^{T}D_{t}$.[^10] The following discussion is similar to the one at the end of the previous subsection \[hedge-delayed\].
If the learner knows $\sum_{t=1}^{T}D_{t}$ beforehand or $D_{t}$ is sampled from some distribution with known expectation $\mathbb{E}D$, the choice of $$\eta\propto \frac{1}{H}\sqrt{\frac{\ln T}{T+\sum_{t=1}^{T}D_{t}}}\
\quad
\text{or}
\quad
\eta\propto \frac{1}{H}\sqrt{\frac{\ln T}{T(1+\mathbb{E}D)}}
\nonumber$$ respectively results in $${O\big((K+2)\cdot \sqrt{(T+\sum_{t=1}^{T}D_{t})\cdot \ln T}\big)}$$ and $${O\big((K+2)\cdot \sqrt{T(1+\mathbb{E}D)\cdot \ln T}\big)}$$ (expected) regret bound with respect to any sequence with no more than $K$ expert switches.
If the sequence of delays is chosen by an adversary and unknown to the learner, then classical choice $\eta\propto \frac{\sqrt{\ln T}}{H\sqrt{T}}$ results in ${O[(K+2)\sqrt{T\ln T}(1+\overline{D})]}$ regret, where ${\overline{D}=\frac{1}{T}\sum_{t=1}^{T}D_{t}}$.
Proof of Performance {#sec-proofs}
====================
In this section we prove Theorem \[theorem-main-loss-bound-D\]. The proof is complicated, and we split it into two sequential parts. Firstly, we prove the bound for the non-delayed case in Subsection \[sec-proof-1\], i.e., $\{D_{t}\}_{t=1}^{T}=(0,\dots, 0)$. Secondly, we obtain the bound for arbitrary sequence of delays $\{D_{t}\}_{t=1}^{T}$ in Subsection \[sec-proof-2\].
Bound for Non-delayed Setting {#sec-proof-1}
-----------------------------
We set $D_{t}\equiv 0$ for all $t$ and deal with the bound for algorithm $\mathcal{G}$ in this case. Note that $\mathcal{D}_{t}=\{1,2,\dots,t\}$ for all $t=1,2,\dots,T$ and $\bm{L}_{\mathcal{D}_{t}}=\bm{L}_{t}$.
Recall that $w_{t}^{n_{t}}=p(n_{t}|\bm{L}_{\mathcal{D}_{t-1}})=p(n_{t}|\bm{L}_{t-1})$. Define the **mixloss** at the step $t$: $$\begin{aligned}
m_{t}=-\frac{1}{\eta}\ln\big[\sum_{n_{t}\in\mathcal{N}}e^{-\eta l_{t}^{n_{t}}}\cdot w_{t}^{ n_{t}}\big]=
-\frac{1}{\eta}\ln\big[\sum_{n_{t}\in\mathcal{N}}e^{-\eta l_{t}^{n_{t}}}\cdot p( n_{t}|\bm{L}_{t-1})\big]=
\label{mixloss-definition}
\\
-\frac{1}{\eta}\ln\big[\sum_{n_{t}\in\mathcal{N}}Z_{t}\cdot p(\bm{l}_{t}|n_{t})\cdot p(n_{t}|\bm{L}_{t-1})\big]=
-\frac{1}{\eta}\ln Z_{t}-\frac{1}{\eta}\ln p(\bm{l}_{t}|\bm{L}_{t-1}).
\nonumber\end{aligned}$$ Define the **cumulative mixloss** $M_{T}$ over the entire game: $$\begin{aligned}
M_{T}=\sum_{t=1}^{T}m_{t}=\frac{1}{\eta}\ln \prod_{t=1}^{T}Z_{t}-\frac{1}{\eta}\ln\prod_{t=1}^{T}p(\bm{l}_{t}|\bm{L}_{t-1})=
\nonumber
\\
\frac{1}{\eta}\ln \prod_{t=1}^{T}Z_{t}-\frac{1}{\eta}\ln p(\bm{L}_{T})=\frac{1}{\eta}\ln \bigg[\prod_{t=1}^{T}Z_{t}-\frac{1}{\eta}\ln \sum_{N_{T}\in\mathcal{N}^{T}}p(N_{T})p(\bm{L}_{T}|N_{T})\bigg]=
\nonumber
\\
\frac{1}{\eta}\ln \prod_{t=1}^{T}Z_{t}-\frac{1}{\eta}\ln \bigg[\sum_{N_{T}\in\mathcal{N}^{T}}\big[p(N_{T})\prod_{t=1}^{T}p(\bm{l}_{t}|n_{t})\big]\bigg]=
\nonumber
\\
-\frac{1}{\eta}\ln\bigg[ \sum_{N_{T}\in\mathcal{N}^{T}}\big[p(N_{T})\prod_{t=1}^{T}\frac{p(\bm{l}_{t}|n_{t})}{Z_{t}}\big]\bigg]=
-\frac{1}{\eta}\ln \bigg[\mathbb{E}_{p({N}_{T})}\big[e^{-\eta L_{T}^{{N}_{T}}}\big]\bigg].
\nonumber\end{aligned}$$ For all $t=1,\dots,T$ we apply Hoeffding’s inequality to a random variable $$X_{t}=l_{t}^{n_{t}}\in [0,H],$$ where $n_{t}\sim p(n_{t}|\bm{L}_{t-1})=w_{t}^{n_{t}}$: $$\ln \sum_{n=1}^{N}w_{t}^{n}e^{-\eta l_{t}^{n_{t}}}\leq -\eta \langle \bm{w}_{t},\bm{l}_{t}\rangle+\eta^{2}\frac{H^{2}}{8},$$ which is equal to $$h_{t}\leq m_{t}+\eta \frac{H^2}{8}.
\label{single-loss-bound}$$ We sum for $t=1,2,\dots,T$ and obtain $$\begin{aligned}
H_{T}\leq M_{T}+\eta \frac{H^2}{8}T=
-\frac{1}{\eta}\ln \bigg[\mathbb{E}_{p({N}_{T})}\big[e^{-\eta L_{T}^{{N}_{T}}}\big]\bigg]+\eta \frac{H^2}{8}T,
\label{non-delayed-loss-bound}\end{aligned}$$ which finishes the proof.
Bound for Delayed Setting {#sec-proof-2}
-------------------------
In this section we consider the case of arbitrary sequence of delays $\{D_{t}\}_{t=1}^{T}$.
We use the superscript $(\ldots)^{\mathcal{D}}$ to denote the variables obtained by algorithm $\mathcal{G}$ (for example, weights $\bm{w}_{t}^{\mathcal{D}}$, etc.) with the sequence of delays $\{D_{t}\}_{t=1}^{T}$. Our main idea is to prove that the weights $\bm{w}_{t}^{\mathcal{D}}$ are approximately equal to the weights $\bm{w}_{t}^{0}$ obtained by the algorithm in the game with the same experts but with no delays, i.e., $\{D_{t}\}_{t=1}^{T}=(0,\dots, 0)$. Thus, the losses $h_{t}^{\mathcal{D}}$ and $h_{t}^{0}$ will be approximately equal.
We divide this part of the proof of the theorem into two steps:
**Step 1. Proof for a simple probability distribution $p$**
To begin with, we consider the case of a simple Hidden Markov Model $p(\cdot)$. Let ${p(n_{1})=p_{0}(n_{1})}$ and ${p(n_{t+1}|n_{t})=\mathbb{I}_{[n_{t+1}=n_{t}]}}$ for all ${t=1,\dots,T-1}$. The corresponding algorithm is $\mathcal{G}_{\text{base}}=\mathcal{G}(p)$.
We compare the losses $H_{T}^{0}$ and $H_{T}^{\mathcal{D}}$ of algorithm $\mathcal{G}_{\text{base}}$ applied to the same data with no delays and with the given sequence of delays $\{D_{t}\}_{t=1}^{T}$ respectively. $$\begin{aligned}
|H_{T}^{0}-H_{T}^{\mathcal{D}}|=|\sum_{t=1}^{T}h_{t}^{0}-\sum_{t=1}^{T}h_{t}^{\mathcal{D}}|\leq \sum_{t=1}^{T}|h_{t}^{0}-h_{t}^{\mathcal{D}}|=
\nonumber
\\
\sum_{t=1}^{T}|\langle \bm{w}_{t}^{0}, \bm{l}_{t}\rangle - \langle \bm{w}_{t}^{\mathcal{D}}, \bm{l}_{t}\rangle |= \sum_{t=1}^{T}|\langle \bm{w}_{t}^{0}- \bm{w}_{t}^{\mathcal{D}}, \bm{l}_{t}\rangle | \leq
\nonumber
\\
H\cdot \sum_{t=1}^{T}\max_{\mathcal{N}'\subset \mathcal{N}}\bigg[\sum_{n\in \mathcal{N}'}\big[(w_{t}^{n})^{0}-(w_{t}^{n})^{\mathcal{D}}\big]\bigg]
\label{unfinished-loss-diff-bound}\end{aligned}$$
Note that ${(w_{t}^{n})^{0}\propto e^{-\eta L_{t-1}^{n}}}$ and ${(w_{t}^{n})^{\mathcal{D}}\propto e^{-\eta L_{\mathcal{D}_{t-1}}^{n}}}$ for all $t$. This means that ${(w_{t}^{n})^{0}\propto [(w_{t}^{n})^{\mathcal{D}}\cdot a^{n}]}$, where $$-\frac{1}{\eta}\ln (a^{n})=\sum_{\tau=1}^{t-1}l_{\tau}^{n}-\sum_{\tau\in\mathcal{D}_{t-1}}l_{\tau}^{n}\in \big[0, [(t-1)-|\mathcal{D}_{t-1}|]\cdot H\big].$$ Thus, according to Lemma \[lemma-change-bound\], we obtain the bound $$\max_{\mathcal{N}'\subset \mathcal{N}}\bigg[\sum_{n\in \mathcal{N}'}\big[(w_{t}^{n})^{0}-(w_{t}^{n})^{\mathcal{D}}\big]\bigg]\leq \eta H\frac{t-1-|\mathcal{D}_{t-1}|}{4}$$ for all $t$. Combining it with and Lemma \[lemma-delay-sum\] we obtain: $$|H_{T}^{0}-H_{T}^{\mathcal{D}}|\leq \eta\frac{H^{2}}{4}\big[\frac{T(T-1)}{2}-\sum_{t=1}^{T-1}|\mathcal{D}_{t}|\big]=\eta\frac{H^{2}}{4}\sum_{t=1}^{T}D_{t}.$$ The final step is to combine current result with the loss bound for the non-delayed case: $$\begin{aligned}
H_{T}^{\mathcal{D}}\leq H_{T}^{0}+|H_{T}^{0}-H_{T}^{\mathcal{D}}|\leq
\nonumber
\\
-\frac{1}{\eta}\ln \bigg[\mathbb{E}_{p({N}_{T})}\big[e^{-\eta L_{T}^{{N}_{T}}}\big]\bigg]+\eta\frac{H^{2}}{8}T+\eta\frac{H^{2}}{4}\sum_{t=1}^{T}D_{t},
\nonumber\end{aligned}$$ and finish the proof of the bound for algorithm $\mathcal{G}_{\text{base}}$.
**Step 2. Proof for an arbitrary probability distribution $p$**
Now we consider the case of an arbitrary probability distribution $p$. From the given set of experts $\mathcal{N}$ we create a new super set $\mathcal{S}=\mathcal{N}^{T}$ of **super experts** $s\in\mathcal{S}$ ($\mathcal{S}$ for **S**uper). Each super expert $s$ corresponds to some sequence ${{N}_{T}=(n_{1},\dots,n_{T})\in \mathcal{N}^{T}}$ of basic experts $n\in\mathcal{N}$ of length $T$. We denote the $t$-th component of super expert $s$ by $n_{t}(s)$. We denote the full sequence of experts corresponding to $s$ by ${N}_{T}(s)$. We do not use subscript in order not to overburden the notation. The loss of super expert $s\in\mathcal{S}$ at the step $t$ is $l_{t}^{n_{t}(s)}$, where $l_{t}^{n}$ (for $n\in\mathcal{N}$) are the losses of basic experts. We use $E(\bm{L}_{\mathcal{D}_t})$ and $E(\bm{l}_{t})$ to denote all the super experts’ losses at the steps $\mathcal{D}_{t}$ and $t$ respectively ($E$ for **E**nhanced).
We define the probability model for hidden super experts. In order not to confuse the reader with notation, we use capital $P$ (instead of regular $p$) to denote all probabilities related to super experts. Let $P(s_{1})=P_{0}(s_{1})=p({N}_{T}(s_{1}))$ and $$P(s_{t+1}|s_{t})=[s_{t+1}=s_{t}].$$ The described probability distribution corresponds to algorithm $\mathcal{G}_{\text{base}}(P)$ for super experts $s\in\mathcal{S}$ and initial distribution $P_{0}$. We have $s_{1}=s_{2}=\dots=s_{T}$ w.p. 1.
The main idea is to show that the losses of algorithm $\mathcal{G}_{\text{base}}$ are equal to the losses of algorithm $\mathcal{G}(p)$. In order to prove this, we show that for all $t$ the sum of the weights $$\widehat{w}_{t}^{n}\stackrel{\mbox{def}}{=}\sum_{s|n_{t}(s)=n}w_{t}^{s}$$ in algorithm $\mathcal{G}_{\text{base}}(P)$ is equal to $w_{t}^{n}$ in algorithm $\mathcal{G}(p)$. This sum corresponds to the weight that is allocated to the base expert $n\in\mathcal{N}$ as a part of the super experts’ weight allocation for step $t$. We perform several calculations:
$$\begin{aligned}
\widehat{w}_{t}^{n}=\sum_{s|n_{t}(s)=n}w_{t}^{s}=
\nonumber
\\
\sum_{s|n_{t}(s)=n}P\big(s|E(\bm{L}_{\mathcal{D}_{t-1}})\big)=
\sum_{s|n_{t}(s)=n}\frac{P\big(E(\bm{L}_{\mathcal{D}_{t-1}}|s)\big)P(s)}{P\big(E(\bm{L}_{\mathcal{D}_{t-1}})\big)}=
\nonumber\end{aligned}$$
Now note that $P(s)=P_{0}(s)=p\big({N}_{T}(s)\big),$ and $$\begin{aligned}
P\big(E(\bm{L}_{\mathcal{D}_{t-1}})|s)=\prod_{\tau\in\mathcal{D}_{t-1}}P(E(\bm{l}_{\tau})|s)=
\nonumber
\\
\prod_{\tau\in\mathcal{D}_{t-1}}p\big(\bm{l}_{\tau}|n_{\tau}(s)\big)=
p\big(\bm{L}_{\mathcal{D}_{t-1}}|{N}_{T}(s)\big)=
p\big(\bm{L}_{\mathcal{D}_{t-1}}|{N}_{\mathcal{D}_{t-1}}(s)\big).
\nonumber\end{aligned}$$ Thus, we continue computations: $$\begin{aligned}
\sum_{s|n_{t}(s)=n}w_{t}^{s}=
\sum_{s|n_{t}(s)=n}\frac{p\big(\bm{L}_{\mathcal{D}_{t-1}}|{N}_{\mathcal{D}_{t-1}}(s)\big)\cdot p\big({N}_{T}(s)\big)}{P\big(E(\bm{L}_{\mathcal{D}_{t-1}})\big)}=
\nonumber
\\
\sum_{{N}_{T}|n_{t}=n}\frac{p(\bm{L}_{\mathcal{D}_{t-1}}|{N}_{\mathcal{D}_{t-1}})\cdot p({N}_{T})}{P\big(E(\bm{L}_{\mathcal{D}_{t-1}})\big)}=
\nonumber
\\
\sum_{{N}_{T}|n_{t}=n}\frac{p(\bm{L}_{\mathcal{D}_{t-1}}|{N}_{\mathcal{D}_{t-1}})\cdot p({N}_{\mathcal{D}_{t-1}})\cdot p({N}_{T}|{N}_{\mathcal{D}_{t-1}})}{P\big(E(\bm{L}_{\mathcal{D}_{t-1}})\big)}=
\nonumber
\\
\sum_{{N}_{T}|n_{t}=n}\frac{p(\bm{L}_{\mathcal{D}_{t-1}})\cdot p({N}_{\mathcal{D}_{t-1}}|\bm{L}_{\mathcal{D}_{t-1}})\cdot p({N}_{T}|{N}_{\mathcal{D}_{t-1}})}{P\big(E(\bm{L}_{\mathcal{D}_{t-1}})\big)}=
\nonumber
\\
\frac{p(\bm{L}_{\mathcal{D}_{t-1}})}{P\big(E(\bm{L}_{\mathcal{D}_{t-1}})\big)}\sum_{{N}_{T}|n_{t}=n}p({N}_{T}|\bm{L}_{\mathcal{D}_{t-1}})=
\nonumber
\\
\frac{p(\bm{L}_{\mathcal{D}_{t-1}})}{P\big(E(\bm{L}_{\mathcal{D}_{t-1}})\big)}p(n_{t}=n|\bm{L}_{\mathcal{D}_{t-1}})=\frac{p(\bm{L}_{\mathcal{D}_{t-1}})}{P\big(E(\bm{L}_{\mathcal{D}_{t-1}})\big)}w_{t}^{n}
\nonumber\end{aligned}$$ Let us show that the value of $n$-independent normalizing constant $\frac{p(\bm{L}_{\mathcal{D}_{t-1}})}{P\big(E(\bm{L}_{\mathcal{D}_{t-1}})\big)}$ is equal to $1$. Indeed,
$$\begin{aligned}
1=\sum_{s\in\mathcal{S}}w_{t}^{s}=
\sum_{n\in\mathcal{N}}\widehat{w}_{t}^{n}=\frac{p(\bm{L}_{\mathcal{D}_{t-1}})}{P\big(E(\bm{L}_{\mathcal{D}_{t-1}})\big)}\sum_{n\in\mathcal{N}}w_{t}^{n}=\frac{p(\bm{L}_{\mathcal{D}_{t-1}})}{P\big(E(\bm{L}_{\mathcal{D}_{t-1}})\big)}.
\nonumber\end{aligned}$$
We conclude that $\widehat{w}_{t}^{n}=w_{t}^{n}$ for all $t=1,\dots T$ and $n\in\mathcal{N}$. Thus, we proved that algorithms $\mathcal{G}_{\text{base}}(P)$ and $\mathcal{G}(p)$ have exactly the same losses. Let $H_{T}$ be the cumulative loss of these algorithms. Then, by using part 1 of the proof of the theorem we conclude:
$$\begin{aligned}
H_{T}\leq
-\frac{1}{\eta}\ln \bigg[\mathbb{E}_{P(s)}\big[e^{-\eta L_{T}^{s}}\big]\bigg]+\eta\frac{H^{2}}{8}+\eta\frac{H^{2}}{4}\sum_{t=1}^{T}D_{t}=
\nonumber
\\
-\frac{1}{\eta}\ln \bigg[\mathbb{E}_{p({N}_{T})}\big[e^{-\eta L_{T}^{{N}_{T}}}\big]\bigg]+\eta\frac{H^{2}}{8}+\eta\frac{H^{2}}{4}\sum_{t=1}^{T}D_{t}
\nonumber\end{aligned}$$
and finish the proof.
Experiments {#sec-experiments}
===========
We empirically compare developed non-replicating algorithm \[algorithm-delayed-hedge\] ($\mathcal{G}_{\text{base}}$) and algorithm \[algorithm-delayed-fixed-share\] ($\mathcal{G}_{\text{fs}}$) with their analogous replicated ones obtained from non-delayed Hedge and Fixed Share by using meta-algorithm BOLD [@joulani2013online].
To begin with, we recall the main idea of replicating meta-algorithm BOLD. For the sequence of the delays $\{D_{t}\}_{t=1}^{T}$ meta-algorithm BOLD splits the time line into disjoint subsequences. Each subsequence $\{t_{1}<\dots<t_{S}\}$ satisfies ${t_{s}+D_{t_{s}}<t_{s+1}}$, so it is possible to run an independent copy of some non-delayed algorithm $\mathcal{A}$ on the subsequence. For simplicity we assume that all the delays $D_{t}$ are known to the BOLD beforehand. Thus, the meta-algorithm can choose the optimal learning rate for each copy of $\mathcal{A}$ depending on the length of the corresponding subsequence. For more details about algorithm **BOLD** please refer to the original paper [@joulani2013online].
We use BOLD$(\mathcal{H}_{\text{base}})$ and BOLD$(\mathcal{H}_{\text{fs}})$ to denote replicated Hedge and Fixed Share respectively.
We conduct the experiments on the **artificial data**. The artificial data is widely used to illustrate the performance of the Hedge-like algorithms (see [@HeW98; @BoW2002; @erven2011adaptive; @de2014follow]).
To generate the data we use schemes similar to the ones from [@erven2011adaptive]. In all our experiments we set $N=4$ experts and use binary losses, i.e. $\{0, 1\}$. Thus, we set $H=1$. The length of the game is $T=10 000$.
We sample $l_{t}^{n}\sim \text{Bernoulli}(q^{n})$, i.i.d. random variables for all ${n=1,2,3,4}$ and ${t=1,2,\dots T}$. We use two variants of $\bm{q}$: the first one is $${\bm{q}_{1}=[q^{1}, q^{2}, q^{3}, q^{4}]=[0.35, 0.4, 0.45, 0.5]},$$ when all the experts suffer approximately similar losses; the second one, $$\bm{q}_{2}=[q^{1}, q^{2}, q^{3}, q^{4}]=[0.2, 0.4, 0.5, 0.7],$$ when experts differ a lot.
The sequence of delays is random. Each $D_{t}$ is sampled from Poisson distribution with known to the learner mean $\lambda$, i.e. $D_{t}\sim\text{Poisson}(\lambda)$.
Note that all the computational results are **averaged** on ${R=250}$ random realizations of data (losses, delays) for all considered parameters ($\bm{q}, \lambda$).
Experiments with Hedge {#sec-exp-hedge}
----------------------
In this subsection we compare non-replicating algorithm $\mathcal{G}_{\text{base}}$ and replicating algorithm BOLD$(\mathcal{H}_{\text{base}})$.
For each copy of $\mathcal{H}_{\text{base}}$ started by BOLD on the subsequence of length $S$ we use its optimal learning rate $$\eta^{*}=\operatorname*{arg\,min}_{\eta>0}\big[\frac{\ln N}{\eta}+\eta\frac{H^{2}S}{8}\big]=\frac{2}{H}\sqrt{\frac{2\ln N}{S}}.
\label{learning-rate-subprocess}$$
Note that BOLD($\mathcal{H}_{\text{base}}$) runs roughly $\approx[1+\mathbb{E}D]$ copies of $\mathcal{H}_{\text{base}}$, each of length $\approx \frac{T}{[\mathbb{E}D+1]}$ with learning rate[^11] $$\eta^{\mathcal{H}_{\text{base}}}\approx \frac{2}{H}\sqrt{\frac{2\ln N}{T}(1+\mathbb{E}D)}.$$ Thus, in order to equalize the learning speed of $\mathcal{G}_{\text{base}}$ and BOLD($\mathcal{H}_{\text{base}}$), it is fair to assign $[1+\mathbb{E}D]$ times lower learning rate $$\eta^{\mathcal{G}_{\text{base}}}=[1+\mathbb{E}D]^{-1}\cdot \eta^{\mathcal{H}_{\text{base}}}=\frac{2}{H}\sqrt{\frac{2\ln N}{T(1+\mathbb{E}D)}}
\label{learning-rate-main}$$ to algorithm $\mathcal{G}_{\text{base}}$. The usage of such $\eta$ leads to $O(\sqrt{T(1+\mathbb{E}D)})$ regret bound (see Subsection \[hedge-delayed\]).
For integer values of $\lambda=\mathbb{E}D\in [0, 250]$, we compare the total regret $R_{T}$ of $\mathcal{G}_{\text{base}}$ and BOLD($\mathcal{H}_{\text{base}}$) with respect to the best expert. The resulting empirical dependence is shown in Figures \[figure:hedge-similar\], \[figure:hedge-distant\] for losses generated with the use of $\bm{q}_{1}$ and $\bm{q}_{2}$ respectively.
We discuss the results in Section \[sec-experiments-discussion\] below.
Experiments with Fixed Share
----------------------------
In this subsection we compare non-replicating algorithm $\mathcal{G}_{\text{fs}}$ and replicating algorithm BOLD$(\mathcal{H}_{\text{fs}})$.
We set $K=10$ switches and generate datasets which have $K$ switches of the best expert. To create such a dataset, we randomly select $K$ time steps ${t_{1}<t_{2}<\dots<T_{K}}$. On each $k$-th segment $[t_{k}+1,t_{k+1}]$ (for $k=0,1,\dots, K$ and $t_{0}=0, t_{K+1}=T$) we fix random permutation $\sigma_{k}$ on the set of $N$ elements and sample the losses of expert $n=1,2,3,4$ from Bernoulli$(q^{\sigma_{k}(n)})$. Thus, we obtain the sequence of losses which has up to $K$ switches of the best expert. We also assume that the learner does not know $K$ in advance.
In order not to overburden the reader, we use the same learning rates as in the previous subsection. For every copy of $\mathcal{H}_{\text{fs}}$ generated by BOLD, the learning rate is defined by . For the $\mathcal{G}_{\text{fs}}$ the learning rate is given by .
We discuss the results in Section \[sec-experiments-discussion\] below.
Discussion {#sec-experiments-discussion}
----------
In all Figures \[figure:hedge-similar\], \[figure:hedge-distant\], \[figure:fixed-share-similar\], \[figure:fixed-share-distant\] we see that the non-replicated algorithms outperform their corresponding replicating opponents.
For Hedge algorithm from Figures \[figure:hedge-similar\], \[figure:hedge-distant\] we also conclude that with the increase of the expected delay $\mathbb{E}D$ the gap between performance of non-replicating Hedge ($\mathcal{G}_{\text{base}}$) and replicating BOLD$(\mathcal{H}_{base})$ increases. Indeed, the bigger the expected delay is, the more infrequent the separate learning processes generated by BOLD become and the less data they see. Nevertheless, while each base copy of $\mathcal{H}_{\text{base}}$ runs on $\approx \frac{1}{1+\mathbb{E}D}$ times less data than the non-replicated $\mathcal{G}_{\text{base}}$, it uses $\approx(1+\mathbb{E}D)$ times higher learning rate, which should balance the learning speed with the non-replicated $\mathcal{G}_{\text{base}}$.
Note that Hedge is equal to Online Mirror Descent (OMD) with Entropic Regularization (see e.g. [@shalev2012online]). OMD runs Online Gradient Descent (OGD) $$\bm{x}_{t}\leftarrow \bm{x}_{t-1}-\eta\cdot \bm{l}_{t}$$ in the mirrored space $\mathbb{R}^{N}$ and after each gradient step transforms the mirrored weight $\bm{x}_{t}$ into primal weight $$\bm{w}_{t+1}=\text{SoftMax}(\bm{x}_{t})=\bigg[\frac{e^{x^{1}_{t}}}{\sum_{n=1}^{N}e^{x^{n}_{t}}},\dots,\frac{e^{x^{N}_{t}}}{\sum_{n=1}^{N}e^{x^{n}_{t}}}\bigg],$$ so that $\bm{w}_{t+1}\in \Delta(\mathcal{N})$ is the decision of the algorithm on weight allocation.
In the case of i.i.d. experts losses, the mirrored estimates of $\bm{x}_{t}$ for both $\mathcal{G}_{\text{base}}$ on $t$ observations and $\mathcal{H}_{\text{base}}$ on $\approx\frac{1}{1+\mathbb{E}D}t$ observations have the same expectation. Indeed, $$\begin{aligned}
\mathbb{E}(\bm{x}_{t}^{\mathcal{G}_{\text{base}}})=\mathbb{E}\sum_{\tau=1}^{t}\big[\eta^{\mathcal{G}_{\text{base}}}\cdot \bm{l}_{\tau}\big]=\eta^{\mathcal{G}_{\text{base}}}\cdot\sum_{\tau=1}^{t}\mathbb{E} \bm{l}_{\tau}=\eta^{\mathcal{G}_{\text{base}}}\cdot\big( t\cdot\mathbb{E}\bm{l}\big)=
\label{mirrored-expectation-base}
\\
\big(\frac{\eta^{\mathcal{H}_{\text{base}}}}{1+\mathbb{E}D})\cdot \big( t\cdot\mathbb{E}\bm{l}\big)=\big(\eta^{\mathcal{H}_{\text{base}}})\cdot \underbrace{\big( \frac{t}{1+\mathbb{E}D}\cdot\mathbb{E}\bm{l}\big)}_{\approx \sum\limits_{\tau\in \text{SP}(t)}\mathbb{E}\bm{l}_{\tau}}\approx
\label{mirrored-expectation-intermediate}
\\
\sum\limits_{\tau\in \text{SP}(t)}\big[\eta^{\mathcal{H}_{\text{base}}}\cdot\mathbb{E}\bm{l}_{\tau}\big]=\mathbb{E}(\bm{x}_{t}^{\text{BOLD}}),
\label{mirrored-expectation-bold}\end{aligned}$$ where we use $\text{SP}(t)$ to denote the set of all time steps $\tau\leq t$ included in the **s**eparate learning **p**rocess (generated by BOLD) that is used at the step $t$. In the transition between lines and we use definition of the learning rates. In line we note that the size of the set $\text{SP}(t)$ is $\approx \frac{t}{1+\mathbb{E}D}$.
Same as in -, we compare the co-variance matrices of the estimates of the mirrored estimates of $\bm{x}_{t}$ obtained by $\mathcal{G}_{\text{base}}$ and BOLD. Again, using the i.i.d. assumption we derive
$$\begin{aligned}
\mathbb{V}(\bm{x}_{t}^{\mathcal{G}_{\text{base}}})=
\mathbb{V}\sum_{\tau=1}^{t}\big[\eta^{\mathcal{G}_{\text{base}}}\cdot \bm{l}_{\tau}\big]=
\big(\eta^{\mathcal{G}_{\text{base}}}\big)^{2}\cdot\sum_{\tau=1}^{t}\mathbb{V} \bm{l}_{\tau}=
\nonumber
\\
\big(\eta^{\mathcal{G}_{\text{base}}}\big)^{2}\cdot\big( t\cdot\mathbb{V}\bm{l}\big)=
\big(\frac{\eta^{\mathcal{H}_{\text{base}}}}{1+\mathbb{E}D}\big)^{2}\cdot \big( t\cdot\mathbb{V}\bm{l}\big)=
\nonumber
\\
\frac{\big(\eta^{\mathcal{H}_{\text{base}}})^{2}}{1+\mathbb{E}D}\cdot \underbrace{\big( \frac{t}{1+\mathbb{E}D}\cdot\mathbb{V}\bm{l}\big)}_{\approx \sum\limits_{\tau\in \text{SP}(t)}\mathbb{V}\bm{l}_{\tau}}\approx
\frac{1}{1+\mathbb{E}D}\cdot \sum\limits_{\tau\in \text{SP}(t)}\big[\eta^{\mathcal{H}_{\text{base}}}\cdot\mathbb{V}\bm{l}_{\tau}\big]=
\nonumber
\\
(1+\mathbb{E}D)^{-1}\cdot \mathbb{V}(\bm{x}_{t}^{\text{BOLD}}).
\nonumber\end{aligned}$$
Note that all the described co-variance matrices are diagonal because we consider the case when the losses of different experts are independent.
We see that while the expectation of the estimates of the mirrored weight $\bm{x}_{t}$ is equal for both non-replicated $\mathcal{G}_{\text{base}}$ and replicated BOLD$(\mathcal{H}_{\text{base}})$, the variance differs $1+\mathbb{E}D$ times. In particular, this means that the distribution of mirrored weights $\bm{x}_{t}$ for these two algorithms differs. The mirrored weight of $\mathcal{G}_{\text{base}}$ is more robust than the corresponding weight of a copy of $\mathcal{H}_{\text{base}}$. As we see from the experiments, these robustness of mirrored weight $\bm{x}_{t}$ also leads to robustness of the primal weights $\bm{w}_{t+1}$ and results in better performance.
If the data does not behave like stochastic, e.g. is maximally adversarial, the above argument obviously does not work, and the replicated algorithms may outperform their non-replicated analogues.
We also note another important advantage of the non-replicating algorithms. They are more **interpretable** than their replicated analogues. The weights obtained by non-replicated algorithms are smooth (thus, more interpretable), whereas the weights of replicated algorithms are smooth only inside every domain of the independent learning subprocess.
To illustrate this, we plot the weight evolution of experts obtained by $\mathcal{G}_{\text{fs}}$ and BOLD$(\mathcal{H}_{\text{fs}})$ in a single experiment with $\mathbb{E}D=40$ and $K=10$ experts’ switches with experts’ losses generated using $\bm{q}_{2}$. The weight evolution on time interval $(4200, 4300)$ is shown in Figures \[figure:evolution-non-replicated\] and \[figure:evolution-replicated\]. One may clearly see that the experts’ weights of replicated algorithm in Figure \[figure:evolution-replicated\] look like uninterpretable noise (because the weights of separate learning processes significantly differ).
We also attach the plot of the full weight evolution of the non-replicated algorithm in Figure \[figure:evolution-non-replicated-all\].
![Evolution of weights of **non-replicated** algorithm during the full game.[]{data-label="figure:evolution-non-replicated-all"}](evolution-non-replicated-all){width="98.00000%"}
To conclude, it seems that the replicated algorithms outperform non-replicated ones on the stochastic-like data. It would be interesting to obtain some concrete **empirical condition** on adversarial data under which the non-replicated algorithms perform better than their replicated analogues. This problem serves as the challenge for our further research.
Conclusion
==========
In the article we developed the general hedging algorithm $\mathcal{G}$ (based on classical Hedge) for the delayed feedback experts’ weight allocation (see Section \[sec-algorithm\], Algorithm \[algorithm-main\]). The developed algorithm is applicable both to hedging countable and continuous sets of experts. Thanks to our main result (Theorem \[theorem-main-loss-bound-D\]), we can bound its loss or regret with respect to the switching sequence of experts.
We described two examples of applications of algorithm $\mathcal{G}$ for delayed feedback setting. Algorithm \[algorithm-delayed-hedge\] ($\mathcal{G}_{\text{base}}$, Subsection \[hedge-delayed\]) is an extension of the classical Hedge for the delayed feedback. Algorithm \[algorithm-delayed-fixed-share\] ($\mathcal{G}_{\text{fs}}$, Subsection \[sec-fs\]) is the adaptation of classical Fixed Share. Both algorithms are non-replicated, which means that they use all the observed data to make the decision (in contrast to existing meta-approaches to delayed feedback setting).
It seems that the general probabilistic model which we described can be enhanced even more. First of all, it is reasonable to consider dynamic time-dependent learning rates $\eta_{t}$ for different time steps $t$.[^12] This may rid the learner from choosing the learning rate beforehand. Secondly, it is possible to consider different observation probabilities (or potential, see [@cesa-bianchi]). The different choice may allow to obtain the generalized versions and loss bound of Theorem \[theorem-main-loss-bound-D\] for many other algorithms based on multiplicative weights (e.g. MW2 [@cesa2007improved]). The described statements serve as the challenge for our further research.
Acknowledgements {#acknowledgements .unnumbered}
================
The research was partially supported by the Russian Foundation for Basic Research grant 16-29-09649 ofi m.
Math Tools {#sec-appendix-math}
==========
In this appendix we describe the math tools that we use in out article. We start with the well-known Hoeffding’s & Pinsker’s inequalities and then state and prove the important Lemmas (used in the proof of our main Theorem \[theorem-main-loss-bound-D\]).
**Hoeffding’s inequality.** [*L*et $X\in[a,b]\subset \mathbb{R}$ be a random variable. Then, $$\ln \mathbb{E}e^{sX}\leq s\mathbb{E}X+s^{2}\frac{(b-a)^2}{8}
\label{hoeffding}$$ for all $s\in\mathbb{R}$.]{}
**Pinsker’s inequality.** Let $p(x)$ and $q(x)$ be probabilities (or densities) of $x\in X$ for two discrete (continuous) distributions over discrete (continuous) set $X\subset \mathbb{R}^{N}$. Then $$\max_{X'\subset X}|p(X')-q(X')|\leq \sqrt{\frac{1}{2}KL(p||q)},
\label{pinsker}$$ where $KL(p||q)$ is Kullback–Leibler divergence between $p$ and $q$.
The following technical Lemma plays an important role in the proof of Theorem \[theorem-main-loss-bound-D\] (Section \[sec-proofs\]).
\[lemma-change-bound\] Let $X\subset \mathbb{R}^{N}$ be a countable (or continuous) set. Let $p(x)$ and $q(x)$ denote probabilities (or densities) of two random variables with values in $X$. Let $a:X\rightarrow \mathbb{R}$ be a measurable function such that for all $x\in X$ we have $-\frac{1}{\eta}\ln a(x)\in [0, C]$. Then if $q(x)\propto p(x)\cdot a(x)$, the following holds true[^13] $$\sum_{x: p(x)\geq q(x)}[p(x)-q(x)]\leq \frac{\eta C}{4}.
\label{change-bound}$$
Apply Pinsker’s inequality \[pinsker\] for $p(\cdot)$ and $q(x)$ and obtain $$\sum_{x: p(x)\geq q(x)}[p(x)-q(x)]\leq \sqrt{\frac{1}{2}KL(p||q)}.
\label{pinsker-to-lemma}$$ Note that $q(x)=\frac{p(x)a(x)}{\sum_{x'\in X}p(x')a(x')}$. We compute the divergence $$\begin{aligned}
\nonumber
KL(p||q)=\sum_{x\in X}p(x)\ln \frac{p(x)}{q(x)}=
\sum_{x\in X}p(x)\ln \frac{\sum_{x'\in X}p(x')a(x')}{a(x)}=
\\
\ln \sum_{x\in X}p(x)a(x) -\sum_{x\in X}p(x)\ln a(x)=
\nonumber
\\
\eta \bigg[\sum_{x\in X}p(x)l^{x}-\frac{1}{\eta}\ln \sum_{x\in X}p(x)e^{-\eta l^{x}}\bigg]\leq
\eta \cdot \frac{\eta C^{2}}{8}=\frac{\eta^{2}C^{2}}{8},
\label{mix-gap-in-pinsker}\end{aligned}$$ where in we denote $l^{x}=-\frac{1}{\eta}\ln a(x)\in [0, C]$ (for $x\in X$) and use Hoeffding’s inequality for variable which is equal to $l^{x}$ w.p. $p(x)$. To finish, we obtain the bound by combining with the upper bound .
Let $T>0$ be an integer and $\{D_{t}\}_{t=1}^{T}$ be the sequence of integer delays such that $t+D_{t}\leq T$. Let $\mathcal{D}_{t}=\{\tau|\tau+D_{\tau}\leq t\}$. Then $$\sum_{t=1}^{T-1}|\mathcal{D}_{t}|+\sum_{t=1}^{T-1}D_{t}=\frac{T(T-1)}{2}.
\label{delay-sum-eq}$$\[lemma-delay-sum\]
Note that all $\mathcal{D}_{\tau}$ for $\tau\geq t+D_{t}$ contain $t$. Thus, $$\sum_{t=1}^{T}|\mathcal{D}_{t}|=\sum_{t=1}^{T}\big[T+1-(t+D_{t})\big].$$ Since $|\mathcal{D}_{T}|=T$ and $D_{T}=0,$, the obtained expression is equivalent to desired equality .
\[fixed-share-prob-bound\] Let ${N}_{T}$ be the sequence of experts $(n_{1},n_{2},\dots,n_{T})\in \mathcal{N}^{T}$, where $\mathcal{N}=\{1,2,\dots,N\}$. Let $p(\cdot)$ be the probabilistic model used in Fixed Share with prior $p_{0}\equiv \frac{1}{N}$ and switch probabilities $\alpha_{t}=\frac{1}{t}$ for all $t=2,\ldots,T$. Then $$-\ln p({N}_{T})\leq |d{N}_{T}+1|\cdot (\ln N+\ln T),$$ where $|d{N}_{T}|=|\{t:\,n_{t}\neq n_{t-1}\}|$ is the number of expert switches in $N_{T}$.
Simple calculations $$\begin{aligned}
-\ln p({N}_{T})=
\nonumber
\\
-\ln\frac{1}{N}-\sum_{t\in d{N}_{T}}\ln \frac{\alpha_{t}}{N}-\sum_{t\notin d{N}_{T}}\ln(1-\alpha_{t}+\frac{\alpha_{t}}{N})\leq
\nonumber
\\
-|d{N}_{T}+1|\cdot \ln\frac{1}{N}-\sum_{t\in d{N}_{T}}\ln \alpha_{t}-\sum_{t\notin d{N}_{T}}\ln(1-\alpha_{t})\leq\nonumber
\\
-|d{N}_{T}+1|\cdot \ln\frac{1}{N}-|d{N}_{T}|\cdot \ln \frac{1}{T}-\sum_{t=2}^{T}\ln\frac{t-1}{t}=
\nonumber
\\
|d{N}_{T}+1|\cdot (\ln N+\ln T)
\nonumber\end{aligned}$$ prove the lemma.
[^1]: Pure Follow the Leader strategy is known to be the minimax in the simplest stochastic setting (experts’ losses are i.i.d. between experts and time steps).
[^2]: Equivalently, Online Mirror Descent, see [@mcmahan2011follow].
[^3]: More precisely, the lower bound is $\Omega(\sqrt{T\ln N})$, where $N$ is the number of experts in the finite pool.
[^4]: Sometimes in online learning the term **adaptive** means that the algorithm dynamically changes its learning rate during the game. Please do not get confused.
[^5]: Hedging scenario assumes that the learner has access only to losses of experts while in prediction with experts’ advice the learner knows experts’ predictions and observes true outcomes (the losses are computed by using the known loss function). Prediction with experts’ advice can be reduced to Hedging by forgetting about the expert’s predictions and using only the computed losses of the experts.
[^6]: Another definition of conditional probability is also possible. All the elements $(l_t^1, l_t^2,...,l_t^N)$ can be considered as independent random variables. If expert $n_{t}\in\mathcal{N}$ is active, the probability of observing $l_t^{n_{t}}$ is equal to the current right part of equation . All the other losses are i.i.d. uniform variables on $[0, H]$. For the case of finite $\mathcal{N}$ the formula is replaced by $$p(\bm{l}_{t}| n_{t})=p(l_{t}^{n_{t}}| n_{t})\times\bigg[\prod_{n\neq n_{t}}p(l_{t}^{n}|n_{t})\bigg]=\frac{e^{-\eta l_{t}^{n_{t}}}}{Z_{t}}\times \frac{1}{H^{N-1}},
\label{observation-probability-hard}$$ i.e. has an additional denominating factor of $H^{N-1}$. However, for the infinite number of experts this approach requires a more detailed specification of probabilities in terms of measures, because the denominator becomes infinite. In Section \[sec-proofs\] we will see that the exact value of the normalization constant $Z_{t}$ is important neither for the algorithm, nor for its regret bound. Thus, for convenience it is reasonable to consider the model .
[^7]: In case $p( n_{t+1}|N_{t})=p(n_{t+1}| n_{t})$, we obtain a traditional Hidden Markov Process: the hidden state at step $t+1$ depends only on the previous hidden state at step $t$.
[^8]: The form $p({N}_{T})=p_{0}(n_{1})\prod_{t=2}^{t}p(n_{t}|{N}_{t-1})$ is used only for convenience and association with online scenario. It does not impose any restrictions on the type of probability distribution. In fact, $p(N_{T})$ may be any distribution on $\mathcal{N}^{T}$ of any form.
[^9]: The only assumption is that the losses are bounded, i.e. $l_{t}^{n}\in[0,H]$ for all $t=1,\dots,T$ and $n\in\mathcal{N}$.
[^10]: It is also possible to minimize the bound w.r.t. particular number of switches $K$.
[^11]: In the case $D_{t}\equiv D=\mathbb{E}D$ for all $t$, all approximations become equalities.
[^12]: The usual choice of dynamic learning rate in the non-delayed setting is $\eta_{t}\propto \frac{1}{\sqrt{t}}$.
[^13]: In the continuous case the sum should be replaced by the integral.
|
---
address:
- |
$^1$School of Physics and Astronomy, University of Minnesota,\
Minneapolis, MN 55455-0149
- |
$^{2}$ Department of Physics, University of Maryland,\
College Park, MD 20742-4111
author:
- 'Anand Bhattacharya$^{1}$, Igor Žutić$^{2}$, Oriol T. Valls$^{1}$ and A.M. Goldman$^{1}$'
title: 'Comment on “Is The Nonlinear Meissner Effect Unobservable?”'
---
In a recent Letter [@lxw] by Li, Hirschfeld and Wölfle (LHW) on nonlocal effects in unconventional superconductors, it was suggested that these effects might explain the null result for the nonlinear Meissner effect (NLME) in our experiments [@buan; @preprint] on optimally doped YBa$%
_{2}$Cu$_{3}$O$_{6.95}$ (YBCO) single crystals for which an appreciable signal is predicted by theory [@ys; @zv].
We have no objection to the main part of the LHW letter, which deals with a detailed calculation of the nonlocal effects[@kl]. However, the remarks made about our experimental results do not directly follow from these detailed calculations but are critically dependent on a qualitative argument which fails to work for YBCO.
The qualitative argument relies on treating YBCO as a “weakly 3-D” system. This leads LHW to the conclusion that nonlocal effects will wipe out the NLME (for the geometry of our experiments) for fields below about $0.8\sim
1H_{c1}$. However, the estimate of $H_{c1}$ from the “weakly 3-D” argument is[@lxw] $\Phi _{0}/(2\pi \lambda _{0}\lambda _{0c})$ (where $\Phi _{0}$ is the flux quantum, and $\lambda _{0}$ and $\lambda _{0c}$ are the[* *]{} zero temperature penetration depths for currents flowing in the [*a-b*]{} plane and along the [*c-*]{}axis respectively), which leads to a value of $%
H_{c1}$ of twenty Gauss or less. This is over an order of magnitude below the experimental value of the field at which first flux penetration occurs which is[@preprint] 300 Gauss or more[@buan]. In our experiments, the samples used have typical dimensions of 1.5mm x 1.5mm x 50$\mu $m ([*a* ]{}x [*b*]{} x [*c*]{}). The magnetic field is applied in the [*a-b*]{} plane and given the sample geometry, most of the screening current flows in the [*a-b*]{} plane with components along the nodal directions, with the exception of the return currents that flow near the edges in the[* c*]{}-axis direction. Thus, we expect the field of first flux entry to be closer to $\Phi _{0}/(2\pi \lambda _{0}^{2}),$ which is borne out by experiment. For currents in the [*a-b*]{} plane, the nonlocal effects are very small, much smaller than the NLME at fields of 300G. For the return currents, the nonlocal effects may be relevant, [*but*]{} these do not contribute in any way to NLME, as they have no components in the nodal directions. Thus, for our experimental geometry, the nonlocal contributions are irrelevant.
The same conclusion can be reached even more starkly by starting from the estimate of the characteristic nonlocal energy, given in LHW as $%
E_{nl}=\xi_{0c}\Delta_0/\lambda_0$. Quasiparticle effects will be ineffective, due to nonlocality, for quasiparticles within an angle of less than $\phi_{nl}$ from a node, where $\phi_{nl}$ is determined from the condition $\Delta(\phi_{nl})/E_{nl}\sim 1$. This yields $\phi_{nl}\sim 0.001$ implying that the NLME requires an applied field $H>H_m$ with $H_m/H_0\sim
0.001$ where $H_0$ is[@zv] the characteristic field scale of the NLME. Since $H_0$ is about[@zv; @preprint] 8000 gauss, we have that $H_m$ is about ten gauss, in rough agreement with the argument in the previous paragraph.
To summarize: we find it quite plausible that the nonlocal effects indeed render the NLME unobservable at fields below ten or twenty Gauss. Since the experiments are performed at fields over one order of magnitude larger, however, with the sample remaining in the Meissner state, it is obvious that the explanation for our negative result must lie elsewhere. In our opinion[@preprint] the presence of at least a few percent component of imaginary $s$ or $d_{xy}$ character in the gap remains the most likely explanation.
M.R. Li, P.J. Hirschfeld and P. Wölfle, Phys. Rev. Lett.[** 81**]{}, 5640 (1998)
J.Buan et al. , Phys. Rev. Lett. [**72**]{}, 2632 (1994).
A. Bhattacharya, I. Žutić, Oriol.T. Valls, Allen M. Goldman, Ulrich Welp, Boyd Veal, cond-mat/9812234.
S.K.Yip and J.A.Sauls, Phys. Rev. Lett. [**69**]{}, 2264 (1992); D.Xu, S.K.Yip and J.A.Sauls, Phys. Rev. B [**51**]{} (22), 16233 (1995).
I. Zutic and Oriol T. Valls, Phys Rev [**B56**]{}, 1279, (1997); Phys Rev [**B58**]{}, 8738, (1998).
I. Kosztin, A.J. Leggett, Phys. Rev. Lett. [**79**]{}, 135 (1997).
|
---
abstract: 'We study stochastic differential equations with jumps with no diffusion part. We provide some basic stochastic characterizations of solutions of the corresponding non-local partial differential equations and prove the Harnack inequality for a class of these operators. We also establish key connections between the recurrence properties of these jump processes and the non-local partial differential operator. One of the key results is the regularity of solutions of the Dirichlet problem for a class of operators with weakly Hölder continuous kernels.'
address:
- 'Department of Electrical and Computer Engineering, The University of Texas at Austin, 1 University Station, Austin, TX 78712'
- 'Department of Electrical and Computer Engineering, The University of Texas at Austin, 1 University Station, Austin, TX 78712'
- 'Department of Mathematics, The University of Texas at Austin, 1 University Station, Austin, TX 78712'
author:
- Ari Arapostathis
- Anup Biswas
- Luis Caffarelli
title: |
On a class of stochastic differential equations\
with jumps and its properties
---
[^1]
------------------------------------------------------------------------
------------------------------------------------------------------------
Introduction
============
Stochastic differential equations (SDEs) with jumps have received wide attention in stochastic analysis as well as in the theory of differential equations. Unlike continuous diffusion processes, SDEs with jumps have long range interactions and therefore the generators of such processes are non-local in nature. These processes arise in various applications, for instance, in mathematical finance and control [@cont-tankov; @soner] and image processing [@gilboa-osher]. There have been various studies on such processes from a stochastic analysis viewpoint concentrating on existence, uniqueness, and stability properties of the solution of the stochastic differential equation [@abels-kassmann; @bass04; @chen-kim-song; @chen-wang; @kurtz; @komatsu], as well as from a differential equation viewpoint focusing on the existence and regularity of viscosity solutions [@barles-chas-imbert; @barles-chas-imbert-11; @caffarelli-silvestre-regu]. One of our objectives in this paper is to establish stochastic representations of solutions of SDEs with jumps via the associated integro-differential operator.
Let us consider a Markov process $X$ in ${\mathbb{R}^{d}}$ with generator ${{\mathcal{A}}}$. Let $D$ be a smooth bounded domain in ${\mathbb{R}^{d}}$. We denote the first exit time of the process $X$ from $D$ by $\tau(D)=\inf\{t\ge 0: X_{t}\notin D\}$. One can formally say that $$\label{ee1.1}
u(x)\;{:=}\;\operatorname{\mathbb{E}}_{x}\biggl[\int_{0}^{\tau(D)}f(X_{s})\,{\mathrm{d}}{s}\biggr]$$ satisfies the following equation $$\label{ee1.2}
{{\mathcal{A}}}u\;=\;-f \quad \text{in}~D, \quad u\;=\;0\quad \text{in}~D^{c}\,,$$ where $\operatorname{\mathbb{E}}_{x}$ denotes the expectation operator on the canonical space of the process starting at $x$ when $t=0$. An important question is when can we actually identify the solution of as the right hand side of . When ${{\mathcal{A}}}=\Delta + b$, i.e., $X$ is a drifted Brownian notion, one can use the regularity of the solution and Itô’s formula to establish . Clearly then, one standard method to obtain a representation of the mean first exit time from $D$ is to find a classical solution of for non-local operators. This is related to the work in [@bass-2009] where estimates on classical solutions are obtained when $D={\mathbb{R}^{d}}$. The author in [@bass-2009] also raises questions concerning the existence and regularity of solutions to the Dirichlet problem for non-local operators. We provide a partial answer to these questions in Theorem \[T6.1\].
One of the main results of this paper is the existence of a classical solution of for a fairly general class of non-local operators. We focus on operators of the form $$\label{ee1.3}
{\mathcal{I}}f(x)\;=\; b(x)\cdot {\nabla}f(x)
+\int_{{\mathbb{R}^{d}}}{\mathfrak{d}}f(x;z)\,\uppi(x,z)\,{\mathrm{d}}{z}\,,$$ where $$\label{ee1.4}
{\mathfrak{d}}f(x;z)\;{:=}\; f(x+z)-f(x)-{\bm1_{\{{\lvertz\rvert}\le 1\}}}{\nabla}{f}(x)\cdot z\,,$$ with $\bm1_{A}$ denoting the indicator function of a set $A$. The kernel $\uppi$ satisfies the usual integrability conditions. When $\uppi(x,z) = \frac{k(x,z)}{{\lvertz\rvert}^{d+\alpha}}$, with $\alpha\in (1,2)$, and $b$, $k$ and $f$ are locally Hölder in $x$ with exponent $\beta$, and $k(x,\cdot)-k(x,0)$ satisfies the integrability condition in , we show in Theorem \[T3.1\] that $u$ defined by is the unique solution of in $C^{2s+\beta}_{\mathrm{loc}}(D)\cap C({\mathbb{R}^{d}})$. This result can be extended to include non-zero boundary conditions provided that the boundary data is regular enough. The proof is based on various regularity results concerning the Dirichlet problem, that are of independent interest and can be found in Section \[S6\]. For the case $k\equiv1$, with continuous $f$ and $g$, we characterize the solution of $$\label{ee1.5}
{\mathcal{I}}u\;=\;-f \quad \text{in}~D, \quad u\;=\;g\quad \text{in}~D^{c}$$ in the viscosity framework. Theorem \[T3.2\], which appears later in Section \[sec-pde\], asserts that $$u(x)\;=\;\operatorname{\mathbb{E}}_{x}\biggl[\int_{0}^{\tau(D)}f(X_{s})\,{\mathrm{d}}{s}
+g(X_{\tau(D)})\biggr]\,,
\quad x\in{\mathbb{R}^{d}}\,,$$ is the unique viscosity solution to . One of the hurdles in establishing this lies in showing that $\operatorname{\mathbb{E}}_{x}[\tau(\Bar{D})]=0$ whenever $x\in\partial D$. When $X$ is a drifted Brownian motion, this can be easily deduced from the fact that Brownian motion has infinitely many zeros in every finite interval. But similar crossing properties are not known for $\alpha$-stable processes. We also have to restrict ourselves to the regime $\alpha\in (1,2)$, so that the jump process ‘dominates’ the drift, and this allows us to establish that $\operatorname{\mathbb{E}}_{x}[\tau(\Bar{D})]=0$ whenever $x\in\partial D$. The proof technique uses an estimate of the first exit time of an $\alpha$-stable process from a cone [@hernandez]. These auxiliary results can be found in Section \[sec-pde\].
Recall that a function $h$ is said to be *harmonic* with respect to $X$ in $D$ if $h(X_{t\wedge\tau(D)})$ is a martingale. One of the important properties of nonnegative harmonic functions for nondegenerate continuous diffusions is the Harnack inequality, which plays a crucial role in various regularity and stability estimates. The work in [@bass-levin] proves the Harnack inequality for a class of pure jump processes, and this is further generalized in [@bass-kassmann] for non-symmetric kernels that may have variable order. A parabolic Harnack inequality is obtained in [@Barlow-Bass-Chen-Kassmann] for symmetric jump processes associated with the Dirichlet form with a symmetric kernel. In [@song-vondracek-04] sufficient conditions on Markov processes to satisfy the Harnack inequality are identified. Let us also mention the work in [@ari-ghosh-04; @foondun; @song-vondracek-05] where a Harnack inequality is established for jump processes with a non-degenerate diffusion part. Recently, [@kassmann-mimica] proves a Harnack type estimate for harmonic functions that are not necessarily nonnegative in all of ${\mathbb{R}^{d}}$.
In this paper we prove a Harnack inequality for harmonic functions relative to the operator ${\mathcal{I}}$ in when $k$ and $b$ are locally bounded and measurable, and either $k(x,z) = k(x,-z)$, or ${\lvert\uppi(x,z)-\uppi(x,0)\rvert}$ is a lower order kernel (Theorem \[T4.1\]). The method of the proof is based on verifying the sufficient conditions in [@song-vondracek-04]. Later we use this Harnack estimate to obtain certain stability results for the process. Let us also mention that the estimates obtained in Section \[sec-pde\] and Section \[sec-harnack\] may also be used to establish Hölder continuity for harmonic functions by following a similar method as in [@bass-kassmann-holder]. However we don’t pursue this here.
In Section \[sec-stability\] we discuss the ergodic properties of the process such as positive recurrence, invariant probability measures, etc. We provide a sufficient condition for positive recurrence and the existence of an invariant probability measure. This is done via imposing a *Lyapunov stability* condition on the generator. Following Has$'$minskiĭ’s method, we establish the existence of a unique invariant probability measure for a fairly large class of processes. We also show that one may obtain a positive recurrent process by using a non-symmetric kernel and no drift (see Theorem \[T5.3\]). In this case, the non-symmetric part of the kernel plays the role of the drift. Let us mention here that in [@wee] the author provides sufficient conditions for stability for a class of jump diffusions and this is accomplished by constructing suitable Lyapunov type functions. However, the class of kernels considered in [@wee] satisfies a different set of hypotheses than those assumed in this paper, and in a certain way lies in the complement of the class of Lévy kernels that we consider. Stability of $1$-dimensional processes is discussed in [@Wang-08] under the assumption of Lebesgue-irreducibility. Lastly, we want to point out one of the interesting results of this paper, which is the characterization of the mean hitting time of a bounded domain as a viscosity solution of the exterior Dirichlet problem (Theorem \[T5.4\]). This is established for the class of operators with weakly Hölder continuous kernels in Definition \[D-hkernel\].
The organization of the paper is as follows. In Section \[S1.1\] we introduce the notation used in the paper. In Section \[S2\] we introduce the model and assumptions. Section \[sec-pde\] establishes stochastic representations of viscosity solutions. In Section \[sec-harnack\] we show the Harnack inequality. Section \[sec-stability\] establishes the connections between the recurrence properties of the process and solutions of the non-local equations. Finally, Section \[S6\] is devoted to the proof of the regularity of solutions to the Dirichlet problem for weakly Hölder continuous kernels. These results are used in Section \[sec-stability\].
Notation {#S1.1}
--------
The standard norm in the $d$-dimensional Euclidean space ${\mathbb{R}^{d}}$ is denoted by ${\lvert\,\cdot\,\rvert}$, and we let ${\mathbb{R}^{d}}_{*}{:=}{\mathbb{R}^{d}}\setminus\{0\}$. The set of non-negative real numbers is denoted by ${\mathbb{R}}_{+}$, ${\mathbb{N}}$ stands for the set of natural numbers, and $\bm1_{A}$ denotes the indicator function of a set $A$. For vectors $a, b\in{\mathbb{R}^{d}}$, we denote the scalar product by $a\cdot b$. We denote the maximum (minimum) of two real numbers $a$ and $b$ by $a\vee b$ ($a\wedge b$). We let $a^{+}{:=}a\vee 0$ and $a^-{:=}(-a)\vee 0$. By $\lfloor a\rfloor$ ($\lceil a\rceil$) we denote the largest (least) integer less than (greater than) or equal to the real number $a$. For $x\in{\mathbb{R}^{d}}$ and $r\ge 0$, we denote by $B_{r}(x)$ the open ball of radius $r$ around $x$ in ${\mathbb{R}^{d}}$, while $B_{r}$ without an argument denotes the ball of radius $r$ around the origin. Also in the interest of simplifying the notation we use $B\equiv B_{1}$, i.e., the unit ball centered at $0$.
Given a metric space ${{\mathcal{S}}}$, we denote by ${{\mathcal{B}}}({{\mathcal{S}}})$ and $B_{b}({{\mathcal{S}}})$ the Borel $\sigma$-algebra of ${{\mathcal{S}}}$ and the set of bounded Borel measurable functions on ${{\mathcal{S}}}$, respectively. The set of Borel probability measures on ${{\mathcal{S}}}$ is denoted by ${{\mathcal{P}}}({{\mathcal{S}}})$, $\parallel \cdot\parallel_{\mathrm{TV}}$ denotes the total variation norm on ${{\mathcal{P}}}({{\mathcal{S}}})$, and $\delta_{x}$ the Dirac mass at $x$. For any function $f:{{\mathcal{S}}}\to {\mathbb{R}^{d}}$ we define ${\lVertf\rVert}_{\infty}{:=}\sup_{x\in{{\mathcal{S}}}}\,{\lvertf(x)\rvert}$.
The closure and the boundary of a set $A\subset{\mathbb{R}^{d}}$ are denoted by $\Bar{A}$ and $\partial{A}$, respectively, and ${\lvertA\rvert}$ denotes the Lebesgue measure of $A$. We also define $$\tau(A)\;{:=}\;\inf\;\{s\ge 0 : X_{s}\notin A\}\,.$$ Therefore $\tau(A)$ denotes the first exit time of the process $X$ from $A$. For $R>0$, we often use the abbreviated notation $\tau_{R}{:=}\tau(B_{R})$.
We introduce the following notation for spaces of real-valued functions on a set $A\subset{\mathbb{R}^{d}}$. The space $L^{p}(A)$, $p\in[1,\infty)$, stands for the Banach space of (equivalence classes) of measurable functions $f$ satisfying $\int_{A} {\lvertf(x)\rvert}^{p}\,{\mathrm{d}}{x}<\infty$, and $L^{\infty}(A)$ is the Banach space of functions that are essentially bounded in $A$. For an integer $k\ge 0$, the space $C^{k}(A)$ ($C^{\infty}(A)$) refers to the class of all functions whose partial derivatives up to order $k$ (of any order) exist and are continuous, $C_{c}^{k}(A)$ is the space of functions in $C^{k}(A)$ with compact support, and $C_{b}^{k}(A)$ is the subspace of $C^{k}(A)$ consisting of those functions whose derivatives up to order $k$ are bounded. Also, the space $C^{k,r}(A)$, $r\in(0,1]$, is the class of all functions whose partial derivatives up to order $k$ are Hölder continuous of order $r$. For simplicity we write $C^{0,r}(A)=C^{r}(A)$. For any $\gamma> 0$, $C^\gamma(A)$ denotes the space $C^{\lfloor\gamma\rfloor,\gamma-\lfloor\gamma\rfloor}(A)$, under the convention $C^{k,0}(A) = C^{k}(A)$.
In general if $\mathcal{X}$ is a space of real-valued functions on a domain $D$, $\mathcal{X}_{\mathrm{loc}}$ consists of all functions $f$ such that $f\varphi\in\mathcal{X}$ for every $\varphi\in C_{c}^{\infty}(D)$.
For a nonnegative multiindex $\beta=(\beta_{1},\dotsc,\beta_{d})$, we let ${\lvert\beta\rvert}{:=}\beta_{1}+\dotsb+\beta_{d}$ and $D^{\beta}{:=}\partial_{1}^{\beta_{1}}\dotsb
\partial_{d}^{\beta_{d}}$, where $\partial_{i} {:=}\frac{\partial}{\partial x_{i}}$.
Let $D$ be a bounded domain with a $C^{2}$ boundary. Define $d_{x} \;{:=}\; \text{dist}(x, \partial D)$ and $d_{xy}{:=}\min(d_{x}, d_{y})$. For $u\in C(D)$ and $r\in{\mathbb{R}}$, we introduce the weighted norm $$\begin{aligned}
{[\kern-0.45ex[\kern0.1ex u\kern0.1ex]\kern-0.45ex]}^{(r)}_{0;D}&\;{:=}\;
\sup_{x\in D}\; d_{x}^{r}\,{\lvertu(x)\rvert}\,,\\
\intertext{and, for $k\in{\mathbb{N}}$ and $\delta\in(0,1]$, the seminorms}
{[\kern-0.45ex[\kern0.1ex u\kern0.1ex]\kern-0.45ex]}^{(r)}_{k;D}&\;{:=}\; \sup_{{\lvert\beta\rvert}=k}\;
\sup_{x\in D}\; d_{x}^{k+r}{\bigl\lvertD^{\beta}u(x)\bigr\rvert}\\
{[\kern-0.45ex[\kern0.1ex u\kern0.1ex]\kern-0.45ex]}^{(r)}_{k,\delta;D}&\;{:=}\; \sup_{{\lvert\beta\rvert}=k}\;
\sup_{x,y\in D}\; \biggl(d_{xy}^{k+\delta+r}\,
\frac{{\bigl\lvertD^{\beta}u(x)-D^{\beta}u(y)\bigr\rvert}}{{\lvertx-y\rvert}^{\delta}}\biggr)\,.\end{aligned}$$ For $r\in{\mathbb{R}}$ and $\gamma\ge0$, with $\gamma+r\ge0$, we define the space $$\mathscr{C}^{(r)}_{\gamma}(D)\;{:=}\;
\bigl\{u\in C^{\gamma}(D)\cap C({\mathbb{R}}^{d})\,\colon
u(x)=0~\text{for}\,x\in D^{c},
~ {\bm\lvert\kern-0.50ex\bm\lvert\kern0.1ex u
\kern0.1ex\bm\rvert\kern-0.50ex\bm\rvert}^{(r)}_{\gamma;D}<\infty\bigr\}\,,$$ where $${\bm\lvert\kern-0.50ex\bm\lvert\kern0.1ex u
\kern0.1ex\bm\rvert\kern-0.50ex\bm\rvert}^{(r)}_{\gamma;D}\;{:=}\;
\sum_{k=0}^{\lceil\gamma\rceil-1}
{[\kern-0.45ex[\kern0.1ex u\kern0.1ex]\kern-0.45ex]}^{(r)}_{k,D}
+ {[\kern-0.45ex[\kern0.1ex u\kern0.1ex]\kern-0.45ex]}^{(r)}_{\lceil\gamma\rceil-1,\,\gamma+1-\lceil\gamma\rceil;\,D}\,,$$ under the convention ${\bm\lvert\kern-0.50ex\bm\lvert\kern0.1ex u
\kern0.1ex\bm\rvert\kern-0.50ex\bm\rvert}^{(r)}_{0;D}={[\kern-0.45ex[\kern0.1ex u\kern0.1ex]\kern-0.45ex]}^{(r)}_{0;D}$. We also use the notation ${\bm\lvert\kern-0.50ex\bm\lvert\kern0.1ex u
\kern0.1ex\bm\rvert\kern-0.50ex\bm\rvert}^{(r)}_{k,\delta;D}
={\bm\lvert\kern-0.50ex\bm\lvert\kern0.1ex u
\kern0.1ex\bm\rvert\kern-0.50ex\bm\rvert}^{(r)}_{k+\delta;D}$ for $\delta\in(0,1]$. It is straightforward to verify that ${\bm\lvert\kern-0.50ex\bm\lvert\kern0.1ex u
\kern0.1ex\bm\rvert\kern-0.50ex\bm\rvert}^{(r)}_{\gamma;D}$ is a norm, under which $\mathscr{C}^{(r)}_{\gamma}(D)$ is a Banach space.
If the distance functions $d_x$ or $d_{xy}$ are not included in the above definitions, we denote the corresponding seminorms by $[\,\cdot\,]_{k;D}$ or $[\,\cdot\,]_{k,\delta;D}$ and define $${\lVertu\rVert}_{C^{k,\delta}(D)} \;{:=}\;
\sum_{\ell=0}^{k} [u]_{\ell;D} + [ u]_{k,\delta;D}\,.$$ Thus, ${\lVertu\rVert}_{C^{\gamma}(D)}$ is well defined for any $\gamma>0$, by the identification $C^{\gamma}(D)=C^{\lfloor\gamma\rfloor,\gamma-\lfloor\gamma\rfloor}(A)$.
We recall the well known interpolation inequalities [@GilTru Lemma 6.32, p. 30]. Let $u\in C^{2,\beta}(D)$. Then for any $\varepsilon$ there exists a constant $C=C(\varepsilon,j,k,r)$ such that $$\begin{gathered}
{[\kern-0.45ex[\kern0.1ex u\kern0.1ex]\kern-0.45ex]}^{(0)}_{j,\gamma;D}\;\le\; C\,{\bm\lvert\kern-0.50ex\bm\lvert\kern0.1ex u
\kern0.1ex\bm\rvert\kern-0.50ex\bm\rvert}^{(0)}_{0;D}
+ \varepsilon\,[\![u]\!]^{(0)}_{k,\beta;D}
\\[5pt]
{\bm\lvert\kern-0.50ex\bm\lvert\kern0.1ex u
\kern0.1ex\bm\rvert\kern-0.50ex\bm\rvert}^{(0)}_{j,\gamma;D}\;\le\; C\,{\bm\lvert\kern-0.50ex\bm\lvert\kern0.1ex u
\kern0.1ex\bm\rvert\kern-0.50ex\bm\rvert}^{(0)}_{0;D}
+ \varepsilon\,{[\kern-0.45ex[\kern0.1ex u\kern0.1ex]\kern-0.45ex]}^{(0)}_{k,\beta;D}
\end{gathered}
\qquad
j=0,1,2,\qquad 0\le\beta,\,\gamma\le 1\,,\quad
j+\gamma < k + \beta\,.$$
Preliminaries {#S2}
=============
Let $\alpha\in(1,2)$. Let $b:{\mathbb{R}^{d}}\to{\mathbb{R}^{d}}$ and $\uppi:{\mathbb{R}^{d}}\times{\mathbb{R}^{d}}\to {\mathbb{R}}$ be two given measurable functions where $\uppi$ is nonnegative. We define the non-local operator ${\mathcal{I}}$ as follows: $$\label{ee2.1}
{\mathcal{I}}f(x)\;{:=}\;
b(x)\cdot{\nabla}f(x) +\int_{{\mathbb{R}^{d}}}{\mathfrak{d}}f(x;z)\,
\uppi(x,z)\,{\mathrm{d}}{z}\,,$$ with ${\mathfrak{d}}f$ as in . We always assume that $$\int_{{\mathbb{R}^{d}}}({\lvertz\rvert}^{2}\wedge 1)\,\uppi(x,z)\,{\mathrm{d}}{z}\;<\;\infty
\qquad \forall\,x\in{\mathbb{R}^{d}}\,.$$ Note that is well-defined for any $f\in C^{2}_{b}({\mathbb{R}^{d}})$. Let $\Omega={{\mathcal{D}}}([0, \infty), {\mathbb{R}^{d}})$ denote the space of all right continuous functions mapping $[0,\infty)$ to ${\mathbb{R}^{d}}$, having finite left limits (cádlág). Define $X_{t}=\omega(t)$ for $\omega\in\Omega$ and let $\{{{\mathcal{F}}}_{t}\}$ be the right-continuous filtration generated by the process $\{X_{t}\}$. In this paper we always assume that given any initial distribution $\nu_{0}$ there exists a strong Markov process $(X, \operatorname{\mathbb{P}}_{\nu_{0}})$ that satisfies the martingale problem corresponding to ${\mathcal{I}}$, i.e., $\operatorname{\mathbb{P}}_{\nu_{0}}(X_{0}\in A)=\nu_{0}(A)$ for all $A\in{{\mathcal{B}}}({\mathbb{R}^{d}})$ and for any $f\in C^{2}_{b}({\mathbb{R}^{d}})$, $$f(X_{t})-f(X_{0})-\int_{0}^{t}{\mathcal{I}}f(X_{s})\,{\mathrm{d}}{s}$$ is a martingale with respect to the filtration $\{{{\mathcal{F}}}_{t}\}$. We denote the law of the process by $\operatorname{\mathbb{P}}_{x}$ when $\nu_{0}=\delta_{x}$. Sufficient conditions on $b$ and $\uppi$ to ensure the existence of such processes are available in the literature. Unfortunately, the available sufficient conditions do not cover a wide class of processes. We refer the reader to [@bass04] for the available results in this direction, as well as to [@applebaum; @chen-kim-song; @chen-wang; @komatsu; @kurtz]. When $b\equiv 0$, well-posedness of the martingale problem is obtained under some regularity assumptions on $\uppi$ in [@abels-kassmann].
Let us mention once more that our goal here is not to study the existence of a solution to the martingale problem. Therefore, we do not assume any regularity conditions on the coefficients, unless otherwise stated. Before we proceed to state our assumptions and results, we recall the Lévy-system formula, the proof of which is a straightforward adaptation of the proof for a purely non-local operator and can be found in [@bass-levin Proposition 2.3 and Remark 2.4] (see also [@chen-kim-song; @foondun]).
\[levy-system\] If $A$ and $B$ are disjoint Borel sets in ${{\mathcal{B}}}({\mathbb{R}^{d}})$, then for any $x\in{\mathbb{R}^{d}}$, $$\sum_{s\le t}\bm1_{\{X_{s-}\in A,\, X_{s}\in B\}}
-\int_{0}^{t}\int_{B} \bm1_{\{X_{s}\in A\}}
\uppi(X_{s}, z-X_{s})\,{\mathrm{d}}{z}\,{\mathrm{d}}{s}$$ is a $\operatorname{\mathbb{P}}_{x}$-martingale.
Probabilistic representations of solutions of non-local PDE {#sec-pde}
===========================================================
The aim in this section is to give a rigorous mathematical justification of the connections between stochastic differential equations with jumps and viscosity solutions to associated non-local differential equations.
Recall the generator in where $f$ is in $C^{2}_{b}({\mathbb{R}}^{d})$. We also recall the definition of a viscosity solution [@barles-chas-imbert; @caffarelli-silvestre-regu].
\[D-visc\] Let $D$ be a domain with $C^{2}$ boundary. A function $u:{\mathbb{R}^{d}}\to{\mathbb{R}}$ which is upper (lower) semi-continuous on $\Bar{D}$ is said to be a sub-solution (super-solution) to $$\begin{aligned}
{\mathcal{I}}u& \;=\; -f \quad \text{in}~D\,, \\[3pt]
u& \;=\; g \quad \text{in}~D^{c}\,,\end{aligned}$$ where ${\mathcal{I}}$ is given by , if for any $x\in\Bar{D}$ and a function $\varphi\in C^{2}({\mathbb{R}^{d}})$ such that $\varphi(x)=u(x)$ and $\varphi(z)> u(z)$ $\bigl(\varphi(z)< u(z)\bigr)$ on ${\mathbb{R}^{d}}\setminus\{x\}$, it holds that $${\mathcal{I}}\varphi(x)\ge -f(x) \quad\bigl({\mathcal{I}}\varphi(x)\le -f(x) \bigr)\,,
\quad\text{if}~x\in D\,,$$ while, if $x\in\partial D$, then $$\max\;({\mathcal{I}}\varphi(x)+f(x), g(x)-u(x))\ge 0
\quad\bigl(\min\;({\mathcal{I}}\varphi(x)+f(x), g(x)-u(x))\le 0\bigr)\,.$$ A function $u$ is said to be a viscosity solution if it is both a sub- and a super-solution.
In Definition \[D-visc\] we may assume that $\varphi$ is bounded, provided $u$ is bounded. Otherwise, we may modify the function $\varphi$ by replacing it with $u$ outside a small ball around $x$. It is evident that every classical solution is also a viscosity solution.
Let $f$ and $g$ be two continuous functions on ${\mathbb{R}^{d}}$, with $g$ bounded. Given a bounded domain $D$, we let $$\label{E3.1}
u(x)\;=\;\operatorname{\mathbb{E}}_{x}\biggl[\int_{0}^{\tau(D)}f(X_{s})\,{\mathrm{d}}{s}+g(X_{\tau(D)})\biggr]
\quad \text{for}~x\in{\mathbb{R}^{d}}\,,$$ where $\operatorname{\mathbb{E}}_{x}$ denotes the expectation operator relative to $\operatorname{\mathbb{P}}_{x}$. In this section we characterize $u$ as a solution of a non-local differential equation. As usual, we say that $b$ is locally bounded, if for any compact set $K$, $\sup_{x\in K}\;{\lvertb(x)\rvert}<\infty$.
Three lemmas concerning operators with measurable kernels
---------------------------------------------------------
\[L3.1\] Let $D$ be a bounded domain. Suppose $X$ is a strong Markov process associated with ${\mathcal{I}}$ in , with $b$ locally bounded, and that the integrability conditions $$\label{ee3.2}
\sup_{x\in K}\;\int_{\{{\lvertz\rvert}>1\}}{\lvertz\rvert}\,\uppi(x,z)\,{\mathrm{d}}{z}\;<\;\infty\,,
\quad\text{and}\quad \inf_{x\in K}\;\int_{{\mathbb{R}^{d}}}{\lvertz\rvert}^{2}\uppi(x,z)\,{\mathrm{d}}{z}
\;=\;\infty$$ hold for any compact set $K$. Then $\sup_{x\in D}\operatorname{\mathbb{E}}_{x}[(\tau(D))^{m}]<\infty$, for any positive integer $m$.
Without loss of generality we assume that $0\in D$. Otherwise we inflate the domain to include $0$. Let $\Bar d=\operatorname{diam}(D)$ and $M_{D}=\sup_{x\in D}\,{\lvertb(x)\rvert}$. Recall that $B_{R}$ denotes the ball of radius $R$ around the origin. We choose $R>1\vee 2(\Bar d\vee M_{D})$, and large enough so as to satisfy the inequality $$\inf_{x\in D}\;\int_{B_{R}}{\lvertz\rvert}^{2}\,\uppi(x,z)\,{\mathrm{d}}{z}
\;>\; 1+ 2\Bar d M_{D}+2\Bar d\sup_{x\in D}\;
\int_{\{1< {\lvertz\rvert}\le R\}} {\lvertz\rvert}\, \uppi(x,z)\,{\mathrm{d}}{z}\,.$$ We let $f\in C^{2}_{b}({\mathbb{R}^{d}})$ be a radially increasing function such that $f(x)={\lvertx\rvert}^{2}$ for ${\lvertx\rvert}\le 2R$ and $f(x)=8R^{2}$ for ${\lvertx\rvert}\ge 2R+1$. Then, for any $x\in D$, we have $$\begin{aligned}
{\mathcal{I}}f (x) &\;=\;b(x)\cdot{\nabla}f(x)
+\int_{{\mathbb{R}^{d}}}{\mathfrak{d}}f(x;z)\,\uppi(x,z)\,{\mathrm{d}}{z}\\
&\;\ge\; -2\Bar d M_{D}
+ \int_{B_{R}}\bigl(f(x+z)-f(x)-{\nabla}f(x)\cdot z\bigr)\,\uppi(x,z)\,{\mathrm{d}}{z}\\
&\mspace{50mu} +\int_{\{1< {\lvertz\rvert}\le R\}}{\nabla}f(x)\cdot z\;\uppi(x,z)\,{\mathrm{d}}{z}
+\int_{B_{R}^{c}}\bigl(f(x+z)-f(x)\bigr)\,\uppi(x,z)\,{\mathrm{d}}{z}\,.\end{aligned}$$ Also, for any ${\lvertz\rvert}\ge R$, it holds that ${\lvertx+z\rvert}\ge \Bar d\ge {\lvertx\rvert}$. Therefore $f(x+z)\ge f(x)$. Hence $$\begin{aligned}
{\mathcal{I}}f(x) &\;\ge\; -2\Bar d M_{D} +\int_{\{1< {\lvertz\rvert}\le R\}}{\nabla}f(x)\cdot z\;
\uppi(x,z)\,{\mathrm{d}}{z}\\
&\mspace{200mu}+ \int_{B_{R}}\bigl(f(x+z)-f(x)-{\nabla}f(x)\cdot z\bigr)\,
\uppi(x,z)\,{\mathrm{d}}{z}\\
&\;\ge\; -2\Bar d M_{D} -2\Bar d\int_{\{1<{\lvertz\rvert}\le R\}} {\lvertz\rvert}\,
\uppi(x,z)\,{\mathrm{d}}{z}
+\int_{B_{R}}{\lvertz\rvert}^{2}\,\uppi(x,z)\,{\mathrm{d}}{z}\\
&\;\ge\; 1\,.\end{aligned}$$ Thus $$\begin{aligned}
\operatorname{\mathbb{E}}_{x}[f(X_{\tau(D)\wedge t})]-f(x)
&\;=\; \operatorname{\mathbb{E}}_{x}\biggl[\int_{0}^{\tau(D)\wedge t}{\mathcal{I}}f(X_{s})\,{\mathrm{d}}{s}\biggr]\\[5pt]
&\;\ge\; \operatorname{\mathbb{E}}_{x}[\tau(D)\wedge t]\qquad\forall\,x\in D\,.\end{aligned}$$ Letting $t\to\infty$ we obtain $\operatorname{\mathbb{E}}_{x}[\tau(D)]\le 8R^{2}$. Since $x\in D$ is arbitrary, this shows that $$\sup_{x\in D}\;\operatorname{\mathbb{E}}_{x}[\tau(D)]\le 8R^{2}\,.$$
We continue by using the method of induction. We have proved the result for $m=1$. Assume that it is true for $m$, i.e., $M_{m}{:=}\sup_{x\in D}\operatorname{\mathbb{E}}_{x}[(\tau(D))^{m}]<\infty$. Let $h(x)= M_{m}f(x)$ where $f$ is defined above. Then from the calculations above we obtain $$\label{ee3.3}
\operatorname{\mathbb{E}}_{x}[h(X_{\tau(D)\wedge t})]-h(x)\;\ge\; \operatorname{\mathbb{E}}_{x}[M_{m}(\tau(D)\wedge t)]
\qquad\forall\,x\in D\,.$$ Denoting $\tau(D)$ by $\tau$ we have $$\begin{aligned}
\operatorname{\mathbb{E}}_{x}[\tau^{m+1}]
&\;=\;\operatorname{\mathbb{E}}_{x}\biggl[\int_{0}^{\infty} (m+1)(\tau-t)^{m}\,
\bm1_{\{t<\tau\}}\,{\mathrm{d}}{t}\biggr]
\\[5pt]
&\;=\; \operatorname{\mathbb{E}}_{x}\biggl[\int_{0}^{\infty} (m+1)
\operatorname{\mathbb{E}}_{x}\bigl[(\tau-t)^{m}\,
\bm1_{\{t<\tau\}}\bigm|{{\mathcal{F}}}_{t\wedge\tau}\bigr]\,{\mathrm{d}}{t}\biggr]
\\[5pt]
&\;=\; \operatorname{\mathbb{E}}_{x}\biggl[\int_{0}^{\infty} (m+1) \bm1_{\{t\wedge \tau<\tau\}}
\operatorname{\mathbb{E}}_{X_{t\wedge \tau}}[\tau^{m}]\,{\mathrm{d}}{t}\biggr]
\\[5pt]
&\;\le\; \sup_{x\in D}\;\operatorname{\mathbb{E}}_{x}[\tau^{m}]\,
\operatorname{\mathbb{E}}_{x}\biggl[\int_{0}^{\infty} (m+1) \bm1_{\{t\wedge \tau<\tau\}}\,{\mathrm{d}}{t}\biggr]
\\[5pt]
&\;\le\; M_{m}(m+1)\,\operatorname{\mathbb{E}}_{x}[\tau]\,,\end{aligned}$$ and in view of , the proof is complete.
Boundedness of solutions to the Dirichlet problem on bounded domains and with zero boundary data is asserted in the following lemma.
\[L3.2\] Let $b$ and $f$ be locally bounded functions and $D$ a bounded domain. Suppose $\uppi$ satisfies . Then there exists a constant $C$, depending on $\operatorname{diam}(D)$, $\sup_{x\in D}\,{\lvertb(x)\rvert}$ and $\uppi$, such that any viscosity solution $u$ to the equation $$\begin{aligned}
{\mathcal{I}}u &\;=\;f \quad \text{in}~D\,,\\
u&\;=\;0 \quad \text{in}~D^{c}\,,\end{aligned}$$ satisfies ${\lVertu\rVert}_{\infty}\le C\, \sup_{x\in D}\,{\lvertf(x)\rvert}$.
As shown in the proof of Lemma \[L3.1\], there exists a nonnegative, radially nondecreasing function $\xi\in C^{2}_{b}({\mathbb{R}^{d}})$ satisfying ${\mathcal{I}}\xi(x)>\sup_{x\in D}\,{\lvertf(x)\rvert}$ for all $x\in \bar{D}$. Let $M>0$ be the smallest number such that $M-\xi$ touches $u$ from above at least at one point. We claim that $M\le {\lVert\xi\rVert}_{\infty}$. If not, then $M-\xi(x)>0$ for all $x\in D^{c}$. Therefore $M-\xi$ touches $u$ in $D$ from above. Hence by the definition of a viscosity solution we have ${\mathcal{I}}(M-\xi(x))\ge f(x)$, or equivalently, ${\mathcal{I}}\xi(x)\le - f(x)$, where $x\in D$ is a point of contact from above. But this contradicts the definition of $\xi$. Thus $M\le {\lVert\xi\rVert}_{\infty}$. Also by the definition of $M$ we have $$\sup_{x\in D}\;u(x)\;\le\; \sup_{x\in D}\;(M-\xi(x))\;\le\; M
\;\le\; {\lVert\xi\rVert}_{\infty}\,.$$ The result then follows by applying the same argument to $-u$.
\[D3.2\] Let ${\mathfrak{L}_{\alpha}}$ denote the class of operators ${\mathcal{I}}$ of the form $$\label{gen-op}
{\mathcal{I}}f(x)\;{:=}\;b(x)\cdot{\nabla}f(x)+\int_{{\mathbb{R}^{d}}}
{\mathfrak{d}}f(x;z)\,
\frac{k(x,z)}{{\lvertz\rvert}^{d+\alpha}}\,{\mathrm{d}}{z}\,,\quad f\in C^{2}_{b}({\mathbb{R}^{d}})\,,$$ with $b:{\mathbb{R}^{d}}\to{\mathbb{R}^{d}}$ and $k:{\mathbb{R}^{d}}\times{\mathbb{R}^{d}}\to(0,\infty)$ Borel measurable and locally bounded, and $\alpha\in(1,2)$. We also assume that $x\mapsto\sup_{z\in{\mathbb{R}^{d}}}\,k^{-1}(x,z)$ is locally bounded. The subclass of ${\mathfrak{L}_{\alpha}}$ consisting of those ${\mathcal{I}}$ satisfying $k(x,z)=k(x,-z)$ is denoted by ${\mathfrak{L}^{\mathsf{sym}}_{\alpha}}$.
Consider the following *growth condition*: There exists a constant $K_{0}$ such that $$\label{E3.5}
x\cdot b(x)\,\vee\,{\lvertx\rvert}\,k(x,z)\;\le\; K_{0}\,(1+{\lvertx\rvert}^{2})\qquad
\forall\, x, z\in{\mathbb{R}^{d}}\,.$$ It turns out that under , the Markov process associated with ${\mathcal{I}}$ does not have finite explosion time, as the following lemma shows.
\[L3.3\] Let ${\mathcal{I}}\in{\mathfrak{L}_{\alpha}}$ and suppose that for some constant $K_{0}>0$, the data satisfies the growth condition in . Let $X$ be a Markov process associated with ${\mathcal{I}}$. Then $$\operatorname{\mathbb{P}}_{x}\,\biggl(\sup_{s\in[0, T]}\,{\lvertX_{s}\rvert}<\infty\biggr)\;=\;1
\qquad\forall\, T\,>\,0\,.$$
Let $\delta\in(0,\alpha-1)$, and $f\in C^{2}({\mathbb{R}^{d}})$ be a non-decreasing, radial function satisfying $$f(x)\;=\;\bigl(1+{\lvertx\rvert}^{\delta}\bigr)\quad\text{for}~{\lvertx\rvert}\ge 1\,,
\quad\text{and}\quad
f(x)\;\ge\; 1\quad\text{for}~{\lvertx\rvert}< 1\,.$$ We claim that $$\label{ee3.6}
{\biggl\lvert\int_{{\mathbb{R}^{d}}}{\mathfrak{d}}f(x;z)\,\frac{k(x,z)}{{\lvertz\rvert}^{d+\alpha}}\,{\mathrm{d}}{z}\biggr\rvert}
\;\le\; \kappa_{0}\,(1+{\lvertx\rvert}^{\delta})\qquad\forall x\in{\mathbb{R}^{d}}\,,$$ for some constant $\kappa_{0}$. To prove first note that since the second partial derivatives of $f$ are bounded over ${\mathbb{R}^{d}}$, it follows that ${\Bigl\lvert\int_{{\lvertz\rvert}\le 1}{\mathfrak{d}}f(x;z)\,\frac{k(x,z)}{{\lvertz\rvert}^{d+\alpha}}\,{\mathrm{d}}{z}\Bigr\rvert}$ is bounded by some constant. It is easy to verify that, provided $z\ne 0$, then $$\label{ee3.7}
\begin{aligned}
{\bigl\lvert{\lvertx+z\rvert}^{\delta} - {\lvertx\rvert}^{\delta}\bigr\rvert}
&\;\le\; 2\delta {\lvertz\rvert}\,{\lvertx\rvert}^{\delta-1}\,,
&\qquad\text{if}~ {\lvertx\rvert}\ge2{\lvertz\rvert}\,,\\[5pt]
{\bigl\lvert{\lvertx+z\rvert}^{\delta} - {\lvertx\rvert}^{\delta}\bigr\rvert}
&\;\le\; 8 {\lvertz\rvert}^{\delta}\,, &\qquad\text{if}~ {\lvertx\rvert}<2{\lvertz\rvert}\,,
\end{aligned}$$ for some constant $\kappa$. By the hypothesis in , for some constant $c$, we have $$\label{ee3.8}
k(x,z)\;\le\; c\,(1+{\lvertx\rvert})\qquad\forall x\in{\mathbb{R}^{d}}\,.$$ Combining – we obtain, for ${\lvertx\rvert}>1$, $$\begin{aligned}
{\biggl\lvert\int_{{\lvertz\rvert}>1}{\mathfrak{d}}f(x;z)\,\frac{k(x,z)}{{\lvertz\rvert}^{d+\alpha}}\,{\mathrm{d}}{z}\biggr\rvert}
&\;\le\;
\int_{1<{\lvertz\rvert}\le \frac{{\lvertx\rvert}}{2}}
2\delta\,c\,(1+{\lvertx\rvert})\,{\lvertx\rvert}^{\delta-1}\,{\lvertz\rvert}\,
\frac{1}{{\lvertz\rvert}^{d+\alpha}}\,{\mathrm{d}}{z}\nonumber\\[5pt]
&\mspace{100mu}
+\int_{{\lvertz\rvert}>\frac{{\lvertx\rvert}}{2}}
8c\,(1+{\lvertx\rvert})\,{\lvertz\rvert}^{\delta}\,
\frac{1}{{\lvertz\rvert}^{d+\alpha}}\,{\mathrm{d}}{z}\nonumber\\[5pt]
&\;\le\;
\kappa(d)\biggl(\frac{2\,\delta\,c}{\alpha-1}\,
(1+{\lvertx\rvert})\,{\lvertx\rvert}^{\delta-1}
+\frac{2^{3+\alpha-\delta}c}{\alpha-\delta}\,(1+{\lvertx\rvert})\,{\lvertx\rvert}^{\delta-\alpha}
\biggr)\end{aligned}$$ for some constant $\kappa(d)$, thus establishing .
By and the assumption on the growth of $b$ in , we obtain $${\lvert{\mathcal{I}}f(x)\rvert}\;\le\; K_{1}\; f(x)\qquad\forall x\in{\mathbb{R}^{d}}\,,$$ for some constant $K_{1}$. Then, by Dynkin’s formula, we have, $$\begin{aligned}
\operatorname{\mathbb{E}}_{x}\bigl[f(X_{t\wedge\tau_{n}})\bigr]&\;=\;
f(x) + \operatorname{\mathbb{E}}_{x}\,\biggl[
\int_{0}^{t\wedge\tau_{n}}{\mathcal{I}}f(X_{s})\,{\mathrm{d}}{s}\biggr]\nonumber\\[5pt]
&\;\le\; f(x) + K_{1}\,
\operatorname{\mathbb{E}}_{x}\,\biggl[\int_{0}^{t\wedge\tau_{n}} f(X_{s})\,{\mathrm{d}}{s}\biggr]\nonumber\\[5pt]
&\;\le\; f(x) + K_{1}\,
\int_{0}^{t} \operatorname{\mathbb{E}}_{x}\,\bigl[f(X_{s\wedge\tau_{n}})\bigr]\,{\mathrm{d}}{s}\,,\end{aligned}$$ where in the second inequality we use the property that $f$ is radial and non-decreasing. Hence, by the Gronwall inequality, we have $$\label{ee3.9}
\operatorname{\mathbb{E}}_{x}\bigl[f(X_{t\wedge\tau_{n}})\bigr]\;\le\; f(x)\,{\mathrm{e}}^{K_{1} t}\qquad
\forall\,t>0\,,\quad \forall\,n\in{\mathbb{N}}\,.$$ Since $\operatorname{\mathbb{E}}_{x}\bigl[f(X_{t\wedge\tau_{n}})\bigr]\,\ge\,f(n)\,\operatorname{\mathbb{P}}_{x}(\tau_{n}\le t)$, we obtain by that $$\begin{aligned}
\operatorname{\mathbb{P}}_{x}\,\biggl(\sup_{s\in[0, T]}\,{\lvertX_{s}\rvert}\ge n\biggr)&\;=\;
\operatorname{\mathbb{P}}_{x}(\tau_{n}\le T)\\[5pt]
&\;\le\; \frac{f(x)}{1+n^{\delta}}\,{\mathrm{e}}^{K_{1} T}\qquad
\forall\,T>0\,,\quad \forall\,n\in{\mathbb{N}}\,,\end{aligned}$$ from which the conclusion of the lemma follows.
A class of operators with weakly Hölder continuous kernels {#S3.2}
----------------------------------------------------------
We introduce a class of kernels whose numerators $k(x,z)$ are locally Hölder continuous in $x$, and $z\mapsto k(x,z)$ is bounded, locally in $x$. We call such kernels $\uppi$ weakly Hölder continuous since they have the property that for any $f$ satisfying $\int_{{\mathbb{R}^{d}}} \frac{f(z)}{{\lvertz\rvert}^{d+\alpha}}\,{\mathrm{d}}{z}<\infty$ the map $x\mapsto \int_{{\mathbb{R}^{d}}} f(z)\frac{k(x,z)}{{\lvertz\rvert}^{d+\alpha}}\,{\mathrm{d}}{z}$ is locally Hölder continuous.
\[D-hkernel\] Let $\lambda: [0,\infty) \to (0,\infty)$ be a nondecreasing function that plays the role of a parameter. For a bounded domain $D$ define $\lambda_{D}{:=}\sup\,\{\lambda(R)\,\colon
D\subset B_{R+1}\}$. Let $\mathfrak{I}_{\alpha}(\beta,\theta,\lambda)$, where $\beta\in(0,1]$, $\theta\in(0,1)$, denote the class of operators ${\mathcal{I}}$ as in that satisfy, on each bounded domain $D$, the following properties:
- $\alpha\in (1,2)$.
- $b$ is locally Hölder continuous with exponent $\beta$, and satisfies $${\lvertb(x)\rvert}\;\le\;\lambda_{D}\,,\quad\text{and}\quad
{\lvertb(x)-b(y)\rvert}\;\le\; \lambda_{D}\,{\lvertx-y\rvert}^{\beta}
\qquad \forall\, x,\,y\in D\,.$$
- The map $k(x,z)$ is continuous in $x$ and measurable in $z$ and satisfies $$\begin{split}
{\lvertk(x,z)-k(y,z)\rvert}\;\le\; \lambda_{D}\,{\lvertx-y\rvert}^{\beta}
\qquad \forall\, x,\,y\in D\,,\quad \forall\,z\in{\mathbb{R}^{d}}\\[5pt]
\lambda^{-1}_{D}\;\le\; k(x,z)\;\le\;\lambda_{D}
\qquad \forall\, x\in D\,,\quad \forall\,z\in{\mathbb{R}^{d}}\,.
\end{split}$$
- For any $x\in D$, we have $$\label{ee3.10}
{\biggl\lvert\int_{{\mathbb{R}^{d}}} \bigl({\lvertz\rvert}^{\alpha-\theta}\wedge1\bigr)\,\,
\frac{{\lvertk(x,z)-k(x,0)\rvert}}{{\lvertz\rvert}^{d+\alpha}}\,{\mathrm{d}}{z}\biggr\rvert}
\;\le\;\lambda_{D}\,.$$
It is evident that if ${\lvertk(x,z)-k(x,0)\rvert}\le \Tilde{\lambda}_{D}{\lvertz\rvert}^{\theta'}$ for some $\theta'>\theta$, then property (d) of Definition \[D-hkernel\] is satisfied .
We may view ${\mathcal{I}}$ as the sum of the translation invariant operator ${\mathcal{I}}_{0}$ defined by $${\mathcal{I}}_{0}\,u(x)\;{:=}\;b(x)\cdot{\nabla}u(x)
+ \int_{{\mathbb{R}^{d}}} {\mathfrak{d}}u(x;z)\, \frac{k(x,0)}{{\lvertz\rvert}^{d+2s}}\,{\mathrm{d}}{z}\,,$$ which is uniformly elliptic on every bounded domain, and a perturbation that takes the form $$\widetilde{\mathcal{I}}\,u(x)\;{:=}\;
\int_{{\mathbb{R}^{d}}} {\mathfrak{d}}u(x;z)\, \frac{k(x,z)-k(x,0)}{{\lvertz\rvert}^{d+2s}}\,{\mathrm{d}}{z}\,.$$ We are not assuming that the numerator $k$ is symmetric, as in the approximation techniques in [@caffarelli-silvestre-approx; @Kriventsov; @Bjorland-12]. Moreover, these operators are not addressed in [@ChangLara-Davila] due to the presence of the drift term.
For operators in the class $\mathfrak{I}_{\alpha}(\beta,\theta,\lambda)$, we have the following regularity result concerning solutions to the Dirichlet problem.
\[T3.1\] Let ${\mathcal{I}}\in\mathfrak{I}_{\alpha}(\beta,\theta,\lambda)$, $D$ be a bounded domain with $C^{2}$ boundary, and $f\in C^{\beta}(\Bar{D})$. We assume that neither $\beta$, nor $2s+\beta$ are integers, and that either $\beta<s$, or that $\beta\ge s$ and $${\lvertk(x,z)-k(x,0)\rvert}\;\le\; \Tilde{\lambda}_{D}\, {\lvertz\rvert}^{\theta}\qquad
\forall x\in D\,, \;\forall z\in {\mathbb{R}^{d}}\,,$$ for some positive constant $\Tilde{\lambda}_{D}$. Let $\operatorname{\mathbb{E}}_{x}$ denote the expectation operator corresponding to the Markov process $X$ with generator given by ${\mathcal{I}}$. Then $u(x){:=}\operatorname{\mathbb{E}}_{x}\bigl[\int_{0}^{\tau(D)}f(X_{s})\,{\mathrm{d}}{s}\bigr]$ is the unique solution in $C^{\alpha+\beta}(D)\cap C(\Bar{D})$ to the equation $$\begin{aligned}
{\mathcal{I}}u &\;=\;-f \quad \text{in}~D\,,\\
u&\;=\;0 \quad \text{in}~D^{c}\,.\end{aligned}$$
For $\varepsilon>0$, we denote by $D_{\varepsilon}$ the $\varepsilon$-neighborhood of $D$, i.e., $$\label{ee3.11}
D_{\varepsilon}\;{:=}\;\{z\in {\mathbb{R}^{d}}:\,\operatorname{dist}(z, D)< \varepsilon\}\,.$$ Note that for $\varepsilon$ small enough, $D_{\varepsilon}$ has a $C^{2}$ boundary. Then by Theorem \[T6.1\] there exists $u_\varepsilon\in C^{\alpha+\beta}(D)\cap C(\Bar{D})$ satisfying $$\begin{aligned}
{\mathcal{I}}u_\varepsilon &\;=\;-f \quad \text{in}~D_{\varepsilon},\\
u_\varepsilon&\;=\;0 \quad \text{in}~D_{\varepsilon}^{c}\,.\end{aligned}$$ In the preceding equation $f$ stands for the Lipschitz extension of $f$. We also have the estimate (recall the definition of ${\bm\lvert\kern-0.50ex\bm\lvert\kern0.1ex \,\cdot\,
\kern0.1ex\bm\rvert\kern-0.50ex\bm\rvert}^{(r)}_{\beta;D}$ in Section \[S1.1\]) $${\bm\lvert\kern-0.50ex\bm\lvert\kern0.1ex u_\varepsilon
\kern0.1ex\bm\rvert\kern-0.50ex\bm\rvert}^{(-r)}_{\alpha+\beta;D_{\varepsilon}}
\;\le\; C_{0} \, {\lVertf\rVert}_{C^{\beta}(\Bar{D}_\varepsilon)}\,,$$ with $r$ some fixed constant in $\bigl(0,\frac{\alpha}{2}\bigr)$. As can be seen from the Lemma \[L3.2\] and the proof of Theorem \[T6.1\], we may select a constant $C_{0}$, that does not depend on $\varepsilon$, for $\varepsilon$ small enough. Since $u_\varepsilon=0$ in $D^{c}_\varepsilon$, it follows that $${\lVertu_\varepsilon\rVert}_{C^{r}({\mathbb{R}}^{d})}\;\le\;
c_{1}\,{\bm\lvert\kern-0.50ex\bm\lvert\kern0.1ex u_\varepsilon
\kern0.1ex\bm\rvert\kern-0.50ex\bm\rvert}^{(-r)}_{\alpha+\beta;D_{\varepsilon}}$$ for some constant $c_{1}$, independent of $\varepsilon$, for all small enough $\varepsilon$. Hence $u_\varepsilon\to u$ as $\varepsilon\to 0$, along some subsequence, and $u\in C^{\alpha+\beta}(D)\cap C(\Bar{D})$ by Theorem \[T6.1\]. By Itô’s formula, we obtain $$u_\varepsilon(x)\;=\; \operatorname{\mathbb{E}}_{x}\bigl[u_\varepsilon(X_{\tau(D)})\bigr]
+\operatorname{\mathbb{E}}_{x}\biggl[\int_{0}^{\tau(D)}f(X_{s})\,{\mathrm{d}}{s}\biggr]\,.$$ Letting $\varepsilon\searrow 0$, we obtain the result. Uniqueness follows from Theorem \[T6.1\].
Theorem \[T3.1\] can be extended to account for non-zero boundary conditions, provided the boundary data is regular enough, say in $C^3({\mathbb{R}}^{d})\cap C_{b}({\mathbb{R}^{d}})$.
Some results concerning the fractional Laplacian with drift
-----------------------------------------------------------
In the rest of this section we consider a smaller class of operators, but the data of the Dirichlet problem is only continuous. We focus on stochastic differential equations driven by a symmetric $\alpha$-stable process. More precisely, we consider a process $X$ satisfying $$\label{ee3.12}
dX_{t}\;=\;b(X_{t})\,{\mathrm{d}}{t}+{\mathrm{d}}{L}_{t}\,,$$ where $L_{t}$ is a symmetric $\alpha$-stable process with generator given by $$-(-\Delta)^{\nicefrac{\alpha}{2}}f(x)
\;=\;c(d,\alpha) \int_{{\mathbb{R}^{d}}}{\mathfrak{d}}f(x;z)\,\frac{1}{{\lvertz\rvert}^{d+\alpha}}\,{\mathrm{d}}{z}\,,
\quad f\in C^{2}_{b}({\mathbb{R}^{d}})\,,$$ with $\alpha\in (1,2)$, and $c(d,\alpha)$ a normalizing constant. Then the solution of is also a solution to the martingale problem for $\overline{\mathcal{I}}$ given by $$\overline{\mathcal{I}}f(x) \;{:=}\; -(-\Delta)^{\nicefrac{\alpha}{2}}f(x)
+ b(x)\cdot {\nabla}{f}(x) \,,\quad\alpha\in (1,2)\,.$$ The following condition is in effect for the rest of this section unless mentioned otherwise.
\[cond1\] There exists a positive constant $M$ such that $$\begin{aligned}
{\lvertb(x)-b(y)\rvert} \;\le\; & M {\lvertx-y\rvert}\qquad\forall\,x, y\in{\mathbb{R}}^{d}\,, \\[3pt]
{\lVertb\rVert}_{\infty} \;\le\; & M\,.\end{aligned}$$
Under Condition \[cond1\], equation has a unique adapted strong cádlág solution for any initial condition $X_{0}=x\in{\mathbb{R}}^{d}$, which is a Feller process [@applebaum]. We need the following assertion whose proof is standard, and therefore omitted.
\[L3.4\] Assume Condition \[cond1\] holds and $T>0$. Let $x_{n}\to x$ as $n\to \infty$ and $X^{n}$, $X$ denote the solutions to with initial data $X^{n}_{0}=x_{n}$, $X_{0}=x$, respectively. Then $$\lim_{n\to\infty}\;
\operatorname{\mathbb{E}}\Biggl[\sup_{s\in[0, T]}\,{\lvertX^{n}_{s}-X_{s}\rvert}^{2}\Biggr]\;=\;0\,.$$
The rest of the section is devoted to the proof of the following result.
\[T3.2\] Let $D\subset{\mathbb{R}^{d}}$ be a bounded domain with $C^{1}$ boundary, $f\in C_b(D)$, and $g\in C_{b}(D^{c})$. The function $u(\cdot)$ defined in is continuous and bounded, and is the unique viscosity solution to the equation $$\label{ee3.13}
\begin{split}
\overline{\mathcal{I}}u& \;=\;-f \quad \text{in}~D\,,\\
u& \;=\; g \quad \text{in}~D^{c}\,.
\end{split}$$
The proof of Theorem \[T3.2\] relies on several lemmas which follow. The following lemma is a careful modification of [@song-vondracek-05 Lemma 2.1].
\[L3.5\] Let $D$ be a given bounded domain. There exits a constant $\kappa_{1}>0$ such that for any $x\in D$ and $r\in (0, 1)$ $$\operatorname{\mathbb{P}}_{x}\biggl(\sup_{0\le s\le t}\;{\lvertX_{s}-x\rvert}>r\biggr)\;\le\;
\kappa_{1} t\, r^{-\alpha}\qquad \forall\,x\in D\,,$$ where $X$ satisfies , and $X_{0}=x$.
Let $f\in C^{2}_{b}({\mathbb{R}^{d}})$ be such that $f(x)={\lvertx\rvert}^{2}$ for ${\lvertx\rvert}\le \frac{1}{2}$, and $f(x)=1$ for ${\lvertx\rvert}\ge 1$. Let $c_{1}$ be a constant such that $$\begin{aligned}
{\lVert{\nabla}f\rVert}_{\infty}&\;\le\; c_{1}\,,\\[5pt]
{\bigl\lvertf(x+z)-f(x)-{\nabla}f(x)\cdot z\bigr\rvert}&\;\le\; c_{1} {\lvertz\rvert}^{2}\qquad
\forall\, x,z\in{\mathbb{R}^{d}}\,.\end{aligned}$$ Define $f_{r}(y)=f(\frac{y-x}{r})$ where $x$ is a point in $D$. For $y\in \Bar{B}_{r}(x)$, we obtain $$\begin{aligned}
\biggl|\int_{{\mathbb{R}^{d}}} {\mathfrak{d}}f_{r}(y;z)\,
\frac{1}{{\lvertz\rvert}^{d+\alpha}}\,{\mathrm{d}}{z}\biggr|
&\;\le\; \biggl|\int_{{\lvertz\rvert}\le r}
\bigl(f_{r}(y+z)-f_{r}(y)-{\nabla}f_{r}(y)\cdot z\bigr)
\frac{1}{{\lvertz\rvert}^{d+\alpha}}\,{\mathrm{d}}{z}\biggr|\\[5pt]
&\mspace{150mu}+\biggl| \int_{{\lvertz\rvert}>r}\bigl(f_{r}(y+z)-f_{r}(y)\bigr)
\frac{1}{{\lvertz\rvert}^{d+\alpha}}\,{\mathrm{d}}{z}\biggr| \\[5pt]
& \;\le\; c_{1}\, \frac{1}{r^{2}}\int_{{\lvertz\rvert}\le r}{\lvertz\rvert}^{2-d-\alpha}\,{\mathrm{d}}{z}
+ 2\int_{{\lvertz\rvert}>r}{\lvertz\rvert}^{-d-\alpha}\,{\mathrm{d}}{z} \\[5pt]
&\;\le\; \frac{c_{2}}{r^{\alpha}}\end{aligned}$$ for some constant $c_{2}$. Since $\alpha>1$, we have $${\bigl\lvert\overline{\mathcal{I}}f_{r}(y)\bigr\rvert}\;\le\; \frac{c_{3}}{r^{\alpha}}
\qquad \forall y\in \Bar{B}_{r}(x)\,,$$ where $c_{3}$ is a positive constant depending on $c_{2}$ and $M$. Therefore using Itô’s formula we obtain $$\frac{c_{3}}{r^{\alpha}}\, \operatorname{\mathbb{E}}_{x}\bigl[\tau(\Bar{B}_{r}(x))\wedge t\bigr]
\;\ge\; \operatorname{\mathbb{E}}_{x}\bigl[f_{r}(X_{\tau(\Bar{B}_{r}(x))\wedge t})\bigr]\,.$$ Since $f_{r}=1$ on $B^{c}_{r}(x)$, we have $\operatorname{\mathbb{P}}_{x}(\tau(\Bar{B}_{r}(x))\le t)\le c_{3}r^{-\alpha}t$. This completes the proof.
\[R3.2\] It is clear from the proof of Lemma \[L3.5\] that the result also holds for operators ${\mathcal{I}}\in{\mathfrak{L}_{\alpha}}$. However, in this case, the constant $\kappa_{1}$ depends also on the local bounds of $k$ and $b$.
We define the following process $$\label{ee3.14}
Y_{t}\;{:=}\;x+L_{t}\,.$$ In other words, $Y$ is a symmetric $\alpha$-stable Lévy process starting at $x$. It is straightforward to verify using the martingale property that for any measurable function $f:{\mathbb{R}^{d}}\to {\mathbb{R}}$, we have $$\label{ee3.15}
\operatorname{\mathbb{E}}_{x}[f(Y_t)]\;=\;\operatorname{\mathbb{E}}_{\frac{x}{a}}[f(aY_{a^{-\alpha}t})]\,.$$
We recall the following theorem from [@hernandez Theorem 1].
\[hernan\] Let $\theta\in (0, \pi)$. Let $G$ be a closed cone in ${\mathbb{R}^{d}}$, $d\ge 2$, of angle $\theta$ with vertex at $0$. For $d=1$ we let the cone to be the closed half line. Define $$\eta(G)\;=\;\inf\;\{t\ge 0\,\colon Y_{t}\notin G\}\,.$$ Then there exists a constant $p_\alpha(\theta)>0$ such that $$\begin{aligned}
\operatorname{\mathbb{E}}_{x}[(\eta(G))^{p}]
&\;<\;\infty \qquad \text{for}~ p<p_\alpha(\theta)\,,\\[5pt]
\operatorname{\mathbb{E}}_{x}[(\eta(G))^{p}]
&\;=\;\infty \qquad \text{for}~ p>p_\alpha(\theta)\,,\end{aligned}$$ for all $x\in G\setminus\{0\}$.
The result in [@hernandez] is proven for open cones. The statement in Theorem \[hernan\] follows from the fact that every closed cone is contained in an open cone except for the vertex of the cone and with probability $1$ the exit location from an open cone is not the vertex. The following result is also obtained in [@chen-kim-song] for $d\ge 2$, using estimates of the transition density. Our proof technique is different, so we present it here.
\[L3.6\] Under the process $X$ defined in , for any bounded domain $D$, satisfying the exterior cone condition, and $x\in\partial D$, it holds that $\operatorname{\mathbb{P}}_{x}(\tau(\Bar{D})>0)=0$.
Let $x_{0}\in\partial D$ be a fixed point. We consider an open cone $G$ in the complement of $\Bar{D}$ at a distance $r$ from the boundary $\partial D$, such that the distance between the cone and the boundary equals the length of the linear segment connecting $x_{0}$ with the vertex of the cone at $x_{r}$. In fact, we may choose an angle $\theta$ and an axis for the cone that can be kept fixed for all $r$ small enough and the above mentioned property holds. It is quite clear that this can be done for some truncated cone. So first we assume that the full cone $G$ with angle $\theta$ and vertex $x_{r}$ lies in $D^{c}$. Let $\eta(G^{c})$ denote the first hitting time of $G$. Since a translation of coordinates does not affect the first hitting time, we may assume that $x_{r}=0$. Then from Theorem \[hernan\], there exists $p\in(0,p_\alpha(\theta))$, satisfying $$\begin{aligned}
\operatorname{\mathbb{E}}_{x_{0}}\bigl[\bigl(\eta(G^{c})\bigr)^{p}\bigr]\;=\;{\lvertx_{0}\rvert}^{\alpha p}\,
\operatorname{\mathbb{E}}_{\frac{x_{0}}{{\lvertx_{0}\rvert}}}\bigl[\bigl(\eta(G^{c})\bigr)^{p}\bigr]
\;<\; \infty\,,\end{aligned}$$ where we used the property . Since $G$ is open, by upper semi-continuity we have $$\sup\;\bigl\{\operatorname{\mathbb{E}}_{z}\bigl[\bigl(\eta(G^{c})\bigr)^{p}\bigr]\,
\colon z\in G^{c},\, {\lvertz\rvert}=1\bigr\}\;<\;\infty\,.$$ Therefore we can find a constant $\kappa>0$ not depending on $r$ (for $r$ small) such that $$\label{ee3.16}
\operatorname{\mathbb{E}}_{x_{0}}\bigl[\bigl(\eta(G^{c})\bigr)^{p}\bigr]
\;\le\;\kappa |r|^{\alpha p}\,.$$ Let $\alpha'\in (1,\alpha)$. Then by , for any $\varepsilon>0$, we may choose $r$ small enough so that $$\label{ee3.17}
\operatorname{\mathbb{P}}_{x_{0}}\bigl(\eta(G^{c})>r^{\alpha'}\bigr)\;\le\;
\kappa r^{(\alpha-\alpha')p}\;<\;\varepsilon\,.$$ Using Condition \[cond1\] and , , we have $$\label{ee3.18}
\sup_{s\in[0, r^{\alpha'}]}\;{\lvertX_{s}-Y_{s}\rvert}\;\le\; M r^{\alpha'}\,,$$ with probability $1$. Hence on $\{\eta(G^{c})\le r^{\alpha'}\}$ we have $|Y_{\eta(G^{c})}-X_{\eta(G^{c})}|\le M r^{\alpha'}$ by . But $Y_{\eta(G^{c})}\in G$ and $\operatorname{dist}(x_{0}, G)=r$. Since $Mr^{\alpha'}< r$ for $r$ small enough we have $X_{\eta(G^{c})}\in (\Bar{D})^{c}$ on $\{\eta(G^{c})\le r^{\alpha'}\}$. Therefore from we obtain $$\operatorname{\mathbb{P}}_{x_{0}}\bigl(\tau(\Bar{D})>r^{\alpha'}\bigr)\;<\;\varepsilon$$ for all $r$ small enough. This concludes the proof for the case when we can fit a whole cone in $D^{c}$ near $x_{0}$. For any other scenario we can modify the domain locally around $x_{0}$ and deduce that the first exit time from the new domain is $0$. We use Lemma \[L3.5\] to assert that with high probability the paths spend $r^{\alpha}$ amount of time in a ball of radius of order $r$. Combining these two facts concludes the proof.
The result of Lemma \[L3.6\] still holds if $X$ satisfies in a weak-sense for some locally bounded measurable drift $b$ (see also [@chen-wang]).
The following corollary follows from Lemma \[L3.6\].
\[C3.1\] Under the process $X$ defined in , for any bounded domain $D$ with $C^{1}$ boundary, $\operatorname{\mathbb{P}}_{x}\bigl(\tau(D)=\tau(\Bar{D})\bigr)=1$ for all $x\in\Bar{D}$.
\[L3.7\] Under the process $X$ defined in , for any bounded domain $D$ with $C^{1}$ boundary, and $x\in D$, we have $$\begin{aligned}
\operatorname{\mathbb{P}}_{x}\bigl(X_{\tau(D)-}\in\partial D,\, X_{\tau(D)}\in \Bar{D}^{c}\bigr)
&\;=\;0\,,\\[5pt]
\operatorname{\mathbb{P}}_{x}\bigl(X_{s-}\in\partial D,\, X_{s}\in D,\,
X_{t}\in\Bar{D}~\text{for all}~t\in[0,s]\bigr)
&\;=\;0\,.\end{aligned}$$
We only prove the first equality, as the proof for the second one follows along the same lines. Condition \[cond1\] implies that $X_{t}$ has a density for every $t>0$ [@bogdan-jakubowski]. We let $$\Hat{D}_{R}\;{:=}\;\{z\in D^{c}:\,\operatorname{dist}(z, D)\ge R\}\,.$$ It is enough to prove that $\operatorname{\mathbb{P}}_{x}\bigl(X_{\tau(D)-}\in\partial D,\, X_{\tau(D)}\in \Hat{D}_{R}\bigr)=0$ for every $R>0$. For any $t>0$, we obtain by Proposition \[levy-system\] that $$\begin{aligned}
\operatorname{\mathbb{P}}_{x}\bigl(X_{t\wedge\tau(D)-}\in\partial D,\, X_{t\wedge\tau(D)}
\in \Hat{D}_{R}\bigr)
&\;\le\;
\operatorname{\mathbb{E}}_{x}\Biggl[\sum_{s\le t}
\bm1_{\{X_{s-}\in\partial D,\, X_{s}\in \Hat{D}_{R}\}}\biggr]
\\[5pt]
&\;=\; c(d,\alpha)\;\operatorname{\mathbb{E}}_{x}\biggl[\int_{0}^{t} \bm1_{\{X_{s}\in\partial D\}}
\int_{\Hat{D}_{R}}\frac{1}{{\lvertX_{s}-z\rvert}^{d+\alpha}}\,{\mathrm{d}}{z}\,{\mathrm{d}}{s}\biggr]
\\[5pt]
&\;\le\; \frac{\kappa}{R^{\alpha}}\,\operatorname{\mathbb{E}}_{x}
\biggl[\int_{0}^{t} \bm1_{\{X_{s}\in\partial D\}}\,{\mathrm{d}}{s}\biggr]\end{aligned}$$ for some constant $\kappa$. But the term on the right hand side of the above inequality is $0$ by the fact the $X_{s}$ has density. Hence $\operatorname{\mathbb{P}}_{x}\bigl(X_{t\wedge\tau(D)-}\in\partial D,\,
X_{t\wedge\tau(D)}\in \Hat{D}_{R}\bigr)=0$ for any $t>0$. Letting $t\to \infty$ completes the proof.
Uniqueness follows by the comparison principle in [@hector Corollary 2.9]. Since $f$ and $g$ are bounded, it follows from Lemma \[L3.1\] that $u$ is bounded. Also in view of Lemma \[L3.6\], $u(x)=g(x)$ for $x\in \overline{D^{c}}$. First we show that $u$ is continuous in $D$. Let $x_{n}\to x$ in $D$ as $n\to\infty$. To simplify the notation, we let $\tau^{n}$ denote the first exit time from $D$ for the process $X^{n}$ that starts at $x_{n}$. Similarly, $\tau$ corresponds to the process $X$ that starts at $x$. From Lemma \[L3.4\] we obtain $$\operatorname{\mathbb{E}}\biggl[\sup_{s\in [0, T+1]}\;{\lvertX^{n}_{s}-X_{s}\rvert}^{2}\biggr]
\;\xrightarrow[n\to\infty]{}\; 0\,.$$ Passing to a subsequence we may assume that $$\label{ee3.19}
\sup_{s\in [0, T+1]}\;{\lvertX^{n}_{s}-X_{s}\rvert}\;\to\; 0
\qquad\text{as $n\to\infty$, a.s.}$$ Recall the definition of $D_{\varepsilon}$ in . It is evident that, for any $\varepsilon>0$, implies that $$\liminf_{n\to\infty}\; \tau^{n}\wedge T\;\le\; \tau(D_{\varepsilon})\wedge T\,.$$ Since $\tau(D_{\varepsilon})
\;\xrightarrow[\varepsilon\searrow0]{}\;\tau(\Bar{D})$ a.s., we obtain $$\label{ee3.20}
\liminf_{n\to\infty}\; \tau^{n}\wedge T\;\le\; \tau(\Bar{D})\wedge T\,.$$ On the other hand, $\tau(\Bar{D})\,=\,\tau(D)$ a.s. by Corollary \[C3.1\], and thus we obtain from that $$\liminf_{n\to\infty}\; \tau^{n}\wedge T\;\le\; \tau\wedge T
\quad\text{a.s.}$$ The reverse inequality, i.e., $$\liminf_{n\to\infty}\; \tau^{n}\wedge T\;\ge\; \tau\wedge T
\quad\text{a.s.}\,,$$ is evident from . Hence we have $$\label{ee3.21}
\lim_{n\to\infty}\; \tau^{n}\wedge T\;=\;\tau\wedge T \,,$$ with probability $1$. It then follows by and that $$\operatorname{\mathbb{E}}\biggl[\int_{0}^{\tau^{n}\wedge T} f(X^{n}_{s})\,{\mathrm{d}}{s} -
\int_{0}^{\tau\wedge T} f(X_{s})\,{\mathrm{d}}{s}\biggr]\;\xrightarrow[n\to\infty]{}\;0
\qquad \forall\,T>0\,.$$ By Lemma \[L3.1\] we can take limits as $T\to\infty$ to obtain, $$\label{ee3.22}
\operatorname{\mathbb{E}}\, \biggl[\int_{0}^{\tau^{n}} f(X^{n}_{s})\,{\mathrm{d}}{s} -
\int_{0}^{\tau} f(X_{s})\,{\mathrm{d}}{s}\biggr]\;\xrightarrow[n\to\infty]{}\;0\,.$$ Since $g$ is bounded, by Lemma \[L3.1\] we obtain $$\operatorname{\mathbb{E}}\,\bigl[\bm1_{\{\tau\ge T\}}\,{\bigl\lvertg(X^{n}_{\tau^{n}})-g(X_{\tau})\bigr\rvert}
\bigr]\;\le\;
2\,{\lVertg\rVert}_{\infty}\,\operatorname{\mathbb{P}}(\tau\ge T)\;\xrightarrow[T\to\infty]{}\;0\,.$$ From now on we consider a continuous extension of $g$ on ${\mathbb{R}^{d}}$, also denoted by $g$. We use the triangle inequality $$\begin{aligned}
\label{ee3.23}
\operatorname{\mathbb{E}}\,\bigl[\bm1_{\{\tau< T\}}\,{\bigl\lvertg(X^{n}_{\tau^{n}})-g(X_{\tau})\bigr\rvert}\bigr]
\;&\le\;
\operatorname{\mathbb{E}}\,\bigl[\bm1_{\{\tau< T\}}\,{\bigl\lvertg(X^{n}_{\tau^{n}})-g(X_{\tau^{n}})\bigr\rvert}\bigr]
+\operatorname{\mathbb{E}}\,\bigl[\bm1_{\{\tau< T\}}\,{\bigl\lvertg(X_{\tau^{n}})-g(X_{\tau})\bigr\rvert}\,\bigr]
\nonumber\\[5pt]
&\le\;
\operatorname{\mathbb{E}}\,\bigl[\bm1_{\{\tau^{n}< T+\varepsilon\}}\bm1_{\{\tau< T\}}\,
{\bigl\lvertg(X^{n}_{\tau^{n}})-g(X_{\tau^{n}})\bigr\rvert}\bigr]\nonumber\\[5pt]
&\mspace{10mu}+
2{\lVertg\rVert}_{\infty}\,\operatorname{\mathbb{P}}(\tau^{n}\ge\tau+\varepsilon)
+
\operatorname{\mathbb{E}}\,\bigl[\bm1_{\{\tau< T\}}\,
{\bigl\lvertg(X_{\tau^{n}})-g(X_{\tau})\bigr\rvert}\bigr]\,.\end{aligned}$$ The second term on the right hand side of tends to $0$ as $n\to\infty$, by . The first term is dominated by $$\operatorname{\mathbb{E}}\,\biggl[\biggl(\sup_{0\le t\le T+\varepsilon}\;
{\bigl\lvertg(X^{n}_{t})-g(X_{t})\bigr\rvert}\biggr)\,
\bm1_{\{\tau^{n}< T+\varepsilon\}}\bm1_{\{\tau< T\}}\biggr]\,,$$ so it also tends to $0$ as $n\to\infty$, by , and the continuity and boundedness of $g$. For the third term, we write $$\begin{gathered}
\label{ee3.24}
\operatorname{\mathbb{E}}\,\bigl[\bm1_{\{\tau< T\}}\,
{\bigl\lvertg(X_{\tau^{n}})-g(X_{\tau})\bigr\rvert}\bigr]\;\le\;
\operatorname{\mathbb{E}}\,\bigl[\bm1_{\{\tau< T\}}\,
{\bigl\lvertg(X_{\tau^{n}\wedge T})-g(X_{\tau\wedge T})\bigr\rvert}\bigr]\\[5pt]
+\operatorname{\mathbb{E}}\,\bigl[\bm1_{\{\tau< T\}}\,
{\bigl\lvertg(X_{\tau^{n}})-g(X_{\tau^{n}\wedge T})\bigr\rvert}\bigr]\,.\end{gathered}$$ The first term on the right hand side of tends to $0$, as $n\to\infty$, by , Lemma \[L3.7\], and the continuity and boundedness of $g$. The second term also tends to $0$ as $T\to\infty$, uniformly in $n$, by Lemma \[L3.1\]. Combining the above, we obtain $$\label{ee3.25}
\operatorname{\mathbb{E}}\,\bigl[{\bigl\lvertg(X^{n}_{\tau^{n}})-g(X_{\tau})\bigr\rvert}\bigr]
\;\xrightarrow[T\to\infty]{}\;0\,.$$ By , and , it follows that $u(x_{n})\to u(x)$, as $n\to \infty$, which shows that $u$ is continuous.
Next we show that $u$ is a viscosity solution to . By the strong Markov property of $X$, for any $t\ge 0$, we have $$\label{ee3.26}
u(x)\;=\;
\operatorname{\mathbb{E}}_{x}\biggl[\int_{0}^{\tau(D)\wedge t}f(X_{s})\,{\mathrm{d}}{s}
+u(X_{\tau(D)\wedge t})\biggr]\,.$$ Let $\varphi\in C^{2}_{b}({\mathbb{R}^{d}})$ be such that $\varphi(x)=u(x)$ and $\varphi(z)>u(z)$ for all $z\in{\mathbb{R}^{d}}\setminus\{x\}$. Then by and Itô’s formula we have $$\begin{aligned}
\operatorname{\mathbb{E}}_{x}\biggl[\int_{0}^{\tau(D)\wedge t}{\mathcal{I}}\varphi(X_{s})\,{\mathrm{d}}{s}\biggr]
& \;=\;\operatorname{\mathbb{E}}_{x}\bigl[\varphi(X_{\tau(D)\wedge t})\bigr]-\varphi(x)
\\[5pt]
&\;\ge\; \operatorname{\mathbb{E}}_{x}\bigl[u(X_{\tau(D)\wedge t})\bigr]-u(x)\\[5pt]
&\;=\;-\,\operatorname{\mathbb{E}}_{x}\biggl[\int_{0}^{\tau(D)\wedge t}f(X_{s})\,{\mathrm{d}}{s}\biggr]\,.\end{aligned}$$ Dividing both sides by $t$ and letting $t\to 0$ we obtain ${\mathcal{I}}\varphi(x)\ge -f(x)$ and thus $u$ is a sub-solution. Similarly we can show that $u$ is super-solution and so is a viscosity solution.
The following theorem proves the stability of the viscosity solutions over a convergent sequence of domains.
Let $D_{n}$, $D$ be a collection of $C^{1}$ domains such that $D_{n}\to D$ in the Hausdorff topology, as $n\to\infty$. Let $f, \, g\in C_b({\mathbb{R}^{d}})$. Then $u_{n}\to u$, as $n\to\infty$, where $u_{n}$ and $u$ are the viscosity solutions of in $D_{n}$ and $D$, respectively.
We only need to establish that for any $T>0$, $\tau(D_{n})\wedge T\to\tau(D)\wedge T$ with probability $1$, and that $X_{\tau(D_{n})}\to X_{\tau(D)}$ on $\{\tau(D)<T\}$, as $n\to\infty$. This can be shown following the same argument as in the proof of Theorem \[T3.2\].
The Harnack property for operators containing a drift term {#sec-harnack}
==========================================================
In this section, we prove a Harnack inequality for harmonic functions. The classes of operators considered are summarized in the following definition.
\[D4.1\] With $\lambda$ as in Definition \[D-hkernel\], we let ${\mathfrak{L}_{\alpha}}(\lambda)$, denote the class of operators ${\mathcal{I}}\in{\mathfrak{L}_{\alpha}}$ satisfying $${\lvertb(x)\rvert}\;\le\;\lambda_{D}\,,\quad\text{and}\quad
\lambda^{-1}_{D}\;\le\; k(x,z)\;\le\; \lambda_{D} \qquad\forall\, x\in D\,,~
z\in{\mathbb{R}^{d}}\,,$$ for a bounded domain $D$. As in Definition \[D3.2\], the subclass of ${\mathfrak{L}_{\alpha}}(\lambda)$ consisting of those ${\mathcal{I}}$ satisfying $k(x,z)=k(x, -z)$ is denoted by ${\mathfrak{L}^{\mathsf{sym}}_{\alpha}}(\lambda)$. Also by $\mathfrak{I}_{\alpha}(\theta,\lambda)$ we denote the subset of ${\mathfrak{L}_{\alpha}}(\lambda)$ satisfying $${\biggl\lvert\int_{{\mathbb{R}^{d}}} \bigl({\lvertz\rvert}^{\alpha-\theta}\wedge1\bigr)\,\,
\frac{{\lvertk(x,z)-k(x,0)\rvert}}{{\lvertz\rvert}^{d+\alpha}}\,{\mathrm{d}}{z}\biggr\rvert}
\;\le\;\lambda_{D}\qquad \forall x\in D\,,$$ for any bounded domain $D$.
A measurable function $h:{\mathbb{R}^{d}}\to{\mathbb{R}}$ is said to be *harmonic* with respect to ${\mathcal{I}}$ in a domain $D$ if for any bounded subdomain $G\subset D$, it satisfies $$h(x)\;=\;\operatorname{\mathbb{E}}_{x}[h(X_{\tau(G)})] \qquad \forall\, x\in G\,,$$ where $(X, \operatorname{\mathbb{P}}_{x})$ is a strong Markov process associated with ${\mathcal{I}}$.
\[T4.1\] Let $D$ be a bounded domain of ${\mathbb{R}^{d}}$ and $K\subset D$ be compact. Then there exists a constant $C_{H}$ depending on $K$, $D$ and $\lambda$, such that any bounded, nonnegative function which is harmonic in $D$ with respect to an operator ${\mathcal{I}}\in{\mathfrak{L}^{\mathsf{sym}}_{\alpha}}(\lambda)\cup\mathfrak{I}_{\alpha}(\theta,\lambda)$, $\theta \in (0, 1)$, satisfies $$h(x)\;\le\; C_{H}\, h(y)\qquad \text{for all}~ x, y\in K\,.$$
We prove Theorem \[T4.1\] by verifying the conditions in [@song-vondracek-04] where a Harnack inequality is established for a general class of Markov processes. We accomplish this through Lemmas \[L4.1\]–\[L4.3\] which follow. Let us also mention that some of the proof techniques are standard but we still add them for clarity. In fact, the Harnack property with non-symmetric kernel is also discussed in [@song-vondracek-04] under some regularity condition on $k(\cdot,\cdot)$ and under the assumption of the existence of a harmonic measure. Our proof of Lemma \[L4.1\](b) which follows holds under very general conditions, and does not rely on the existence of a harmonic measure. In Lemmas \[L4.1\]–\[L4.3\], $(X, \operatorname{\mathbb{P}}_{x})$ is a strong Markov process associated with ${\mathcal{I}}\in{\mathfrak{L}^{\mathsf{sym}}_{\alpha}}(\lambda)\cup\mathfrak{I}_{\alpha}(\theta,\lambda)$, and $D$ is a bounded domain.
\[L4.1\] Let $D$ be a bounded domain. There exist positive constants $\kappa_{2}$ and $r_{0}$ such that for any $x\in D$ and $r\in (0,r_{0})$,
- $\inf_{z\in B_{\frac{r}{2}}(x)}\operatorname{\mathbb{E}}_{z}[\tau(B_{r}(x))]
\ge \kappa^{-1}_{2}r^{\alpha}$,
- $\sup_{z\in B_{r}(x)}\operatorname{\mathbb{E}}_{z}[\tau(B_{r}(x))]\le \kappa_{2}\, r^{\alpha}$.
By Lemma \[L3.5\] and Remark \[R3.2\] there exists a constant $\kappa_{1}$ such that $$\label{ee4.1}
\operatorname{\mathbb{P}}_{x}(\tau(B_{r}(x))\le t)\;\le\; \kappa_{1} t r^{-\alpha}\,,$$ for all $t\ge 0$, and all $x\in D_{2}{:=}\{y: \,\operatorname{dist}(y, D)< 2\}$. We choose $t=\frac{r^{\alpha}}{2\kappa_{1}}$. Then for $z\in B_{\frac{r}{2}}(x)$, we obtain by that $$\begin{aligned}
\operatorname{\mathbb{E}}_{z}[\tau(B_{r}(x))]&\;\ge\; \operatorname{\mathbb{E}}_{z}[\tau(B_\frac{r}{2}(z))]\\[5pt]
&\;\ge\; \frac{r^{\alpha}}{2\kappa_{1}}\,
\operatorname{\mathbb{P}}_{z}\Bigl(\tau(B_\frac{r}{2}(z))>
\frac{r^{\alpha}}{2\kappa_{1}}\Bigr)
\\[5pt]
&\;\ge\; \frac{r^{\alpha}}{4\kappa_{1}}\,.\end{aligned}$$ This proves the part (a).
To prove part (b) we consider a radially non-decreasing function $f\in C^{2}_{b}({\mathbb{R}^{d}})$, which is convex in $B_{4}$, and satisfies $$f(x+z)-f(x)-z\cdot{\nabla}f(x)\;\ge\; c_{1}{\lvertz\rvert}^{2}
\qquad \text{for}~{\lvertx\rvert}\le 1\,,\;{\lvertz\rvert}\le 3\,,$$ for some positive constant $c_{1}$. For an arbitrary point $x_{0}\in D$, define $g_{r}(x){:=}f(\frac{x-x_{0}}{r})$. Then for $x\in B_{r}(x_{0})$ and ${\mathcal{I}}\in{\mathfrak{L}^{\mathsf{sym}}_{\alpha}}(\lambda)$ we have $$\begin{aligned}
\int_{{\mathbb{R}^{d}}}{\mathfrak{d}}g_{r}(x;z)\,\frac{k(x,z)}{{\lvertz\rvert}^{\alpha+d}}\,{\mathrm{d}}{z}
&\;=\;\int_{{\lvertz\rvert}\le 3r}\bigl(g_{r}(x+z)-g_{r}(x)-
z\cdot{\nabla}g_{r}(x)\bigr)
\frac{k(x,z)}{{\lvertz\rvert}^{\alpha+d}}\,{\mathrm{d}}{z}\\
&\mspace{200mu}+
\int_{{\lvertz\rvert}> 3r}\bigl(g_{r}(x+z)-g_{r}(x)\bigr)
\frac{k(x,z)}{{\lvertz\rvert}^{\alpha+d}}\,{\mathrm{d}}{z}
\\[5pt]
&\;\ge\; \frac{c_{1}}{r^{2}}\,\lambda^{-1}_{D}
\int_{{\lvertz\rvert}\le 3r} {\lvertz\rvert}^{2-d-\alpha}\,{\mathrm{d}}{z}\\[5pt]
&\;=\; c_{2}\,\frac{3^{2-\alpha}}{2-\alpha}\, \lambda^{-1}_{D}\,r^{-\alpha}\end{aligned}$$ for some constant $c_{2}>0$, where in the first equality we use the fact that $k(x,z)=k(x, -z)$, and for the second inequality we use the property that $g(x+z)\ge g(x)$ for ${\lvertz\rvert}\ge 3r$. It follows that we may choose $r_{0}$ small enough such that $${\mathcal{I}}g_{r}(x)\;\ge\; c_{3} r^{-\alpha}
\qquad\text{for all}~r\in (0, r_{0})\,,~x\in B_{r}(x_{0})\,,~\text{and}~
x_{0}\in D\,,$$ with $c_{3}{:=}\frac{c_{2}}{2}\,\frac{3^{2-\alpha}}{2-\alpha}\,\lambda^{-1}_{D}$.
To obtain a similar estimate for ${\mathcal{I}}\in\mathfrak{I}_{\alpha}(\theta,\lambda)$ we fix some $\theta_1\in(0, \theta\wedge(\alpha-1))$. Let $\Hat{k}(x,z){:=}k(x,z)-k(x,0)$. We have $$\begin{aligned}
\int_{{\mathbb{R}^{d}}}{\mathfrak{d}}g_{r}(x;z)\,\frac{k(x,z)}{{\lvertz\rvert}^{\alpha+d}}\,{\mathrm{d}}{z}
&\;=\;\int_{{\lvertz\rvert}\le 3r}{\mathfrak{d}}g_{r}(x;z)
\frac{k(x,z)}{{\lvertz\rvert}^{\alpha+d}}\,{\mathrm{d}}{z}
-\int_{3r<{\lvertz\rvert}<1}z\cdot{\nabla}g_{r}(x)\frac{k(x,z)-k(x,0)}{{\lvertz\rvert}^{d+\alpha}}\,
{\mathrm{d}}{z}\\[5pt]
&\mspace{150mu} +\int_{{\lvertz\rvert}> 3r}\bigl(g_{r}(x+z)-g_{r}(x)\bigr)
\frac{k(x,z)}{{\lvertz\rvert}^{\alpha+d}}\,{\mathrm{d}}{z}\\[5pt]
&\ge\; \frac{c_{1}}{\lambda_{D}\,r^{2}}
\int_{{\lvertz\rvert}\le 3r} {\lvertz\rvert}^{2-d-\alpha}\,{\mathrm{d}}{z}
- \frac{{\lVert{\nabla}f\rVert}_{\infty}}{r}\int_{3r<{\lvertz\rvert}<1}{\lvertz\rvert}
\frac{{\lvert\Hat{k}(x,z)\rvert}}{{\lvertz\rvert}^{d+\alpha}}\, {\mathrm{d}}{z}
\\[5pt]
&\ge\; c_{2}\,\frac{3^{2-\alpha}}{(2-\alpha)\,\lambda_{D}\,r^{\alpha}}
-\frac{{\lVert{\nabla}f\rVert}_{\infty}}{r}\int_{3r<{\lvertz\rvert}<1}
{\lvertz\rvert}^{\alpha-\theta_{1}}(3r)^{-\alpha+\theta_{1}+1}
\frac{{\lvert\Hat{k}(x,z)\rvert}}{{\lvertz\rvert}^{d+\alpha}}\, {\mathrm{d}}{z}\\[5pt]
&\ge\; c_{2}\,\frac{3^{2-\alpha}}{(2-\alpha)\,\lambda_{D}\,r^{\alpha}}
-\frac{{\lVert{\nabla}f\rVert}_{\infty}}{r}\int_{3r<{\lvertz\rvert}<1}
{\lvertz\rvert}^{\alpha-\theta}(3r)^{-\alpha+\theta_{1}+1}
\frac{{\lvert\Hat{k}(x,z)\rvert}}{{\lvertz\rvert}^{d+\alpha}}\, {\mathrm{d}}{z}\\[5pt]
&\ge\; c_{2}\,\frac{3^{2-\alpha}}{(2-\alpha)\,\lambda_{D}\,r^{\alpha}}
- \kappa(d) 3^{\alpha-\theta_{1}+1}r^{-\alpha+\theta_{1}}\,\lambda_{D}\,
{\lVert{\nabla}f\rVert}_{\infty}
\\[5pt]
&\ge\; c_{4}\,r^{-\alpha} \qquad \forall\, x\in B_r(x_{0})\,,\end{aligned}$$ for some constant $c_{4}>0$ and $r$ small, where in the third inequality we used the fact that $\theta_{1}<\alpha-1$. Thus by Itô’s formula we obtain $$\operatorname{\mathbb{E}}_{x}\bigl[\tau(B_{r}(x_{0}))\bigr]\;\le\; c_{4}^{-1}
r^{\alpha}{\lVertf\rVert}_{\infty} \qquad \forall\, x\in B_r(x_{0})\,.$$ This completes the proof.
\[L4.2\] There exists a constant $\kappa_{3}>0$ such that for any $r\in(0, 1)$, $x\in D$ and $A\subset B_{r}(x)$ we have $$\operatorname{\mathbb{P}}_{z}\bigl(\tau(A^{c})
\,<\,\tau(B_{3r}(x))\bigr)\;\ge\; \kappa_{3}\,\frac{|A|}{|B_{r}(x)|}
\qquad \forall\, z\in B_{2r}(x)\,.$$
Let $\Hat\tau{:=}\tau(B_{3r}(x))$. Suppose $\operatorname{\mathbb{P}}_{z}(\tau(A^{c})<\Hat\tau)<\nicefrac{1}{4}$ for some $z\in B_{2r}(x)$. Otherwise there is nothing to prove as $\frac{|A|}{|B_{r}(x)|}\le 1$. By Lemma \[L3.5\] and Remark \[R3.2\] there exists $t>0$ such that $\operatorname{\mathbb{P}}_{y}(\Hat\tau\le t r^{\alpha})\le \nicefrac{1}{4}$ for all $y\in B_{2r}(x)$. Hence using the Lévy-system formula we obtain $$\begin{aligned}
\label{ee4.2}
\operatorname{\mathbb{P}}_{y}(\tau(A^{c})<\Hat\tau)&\;\ge\;
\operatorname{\mathbb{E}}_{y}\Biggl[\sum_{s\le \tau(A^{c})\wedge\Hat\tau\wedge t r^{\alpha}}
\bm1_{\{X_{s-}\neq X_{s}, X_{s}\in A\}}\Biggr]\nonumber\\[5pt]
&\;=\;\operatorname{\mathbb{E}}_{y}\biggl[\int_{0}^{\tau(A^{c})\wedge\Hat\tau\wedge t r^{\alpha}}
\int_{A}
\frac{k(X_{s}, z-X_{s})}{|z-X_{s}|^{d+\alpha}}\,{\mathrm{d}}{z}\,{\mathrm{d}}{s}\biggr]
\nonumber\\[5pt]
& \;\ge\; \operatorname{\mathbb{E}}_{y}\biggl[\int_{0}^{\tau(A^{c})\wedge\Hat\tau\wedge t r^{\alpha}}
\int_{A}\frac{\lambda^{-1}_{D}}{(4r)^{d+\alpha}}\,{\mathrm{d}}{z}\,{\mathrm{d}}{s}\biggr]
\nonumber\\[5pt]
&\;\ge\; \kappa'_{3}\, r^{-\alpha}\frac{|A|}{|B_{r}(x)|}
\operatorname{\mathbb{E}}_{y}[\tau(A^{c})\wedge\Hat\tau\wedge t r^{\alpha}]\end{aligned}$$ for some constant $\kappa'_{3}>0$, where in the third inequality we use the fact that $|X_{s}-z|\le 4r$ for $s< \Hat\tau$, $z\in A$. On the other hand, we have $$\begin{aligned}
\label{ee4.3}
\operatorname{\mathbb{E}}_{y}[\tau(A^{c})\wedge\Hat\tau\wedge t r^{\alpha}]&\;\ge\; t\, r^{\alpha}\,
\operatorname{\mathbb{P}}_{y}(\tau(A^{c})\ge\Hat\tau\ge t r^{\alpha})
\nonumber\\[5pt]
&\;=\; t\, r^{\alpha}\,\bigl[1-\operatorname{\mathbb{P}}_{y}(\tau(A^{c})<\Hat\tau)
-\operatorname{\mathbb{P}}_{y}(\Hat\tau<t r^{\alpha})\bigr]\nonumber\\[5pt]
&\;\ge\; \frac{t}{2}\,r^{\alpha}\,.\end{aligned}$$ Therefore combining –, we obtain $\operatorname{\mathbb{P}}_{z}(\tau(A^{c})<\Hat\tau)\ge
\frac{t\kappa'_{3}}{2}\frac{|A|}{|B_{r}(x)|}$.
\[L4.3\] There exists positive constants $\kappa_{i}$, $i=4, 5$, such that if $x\in D$, $r\in (0, 1)$, $z\in B_{r}(x)$, and $H$ is a bounded nonnegative function with support in $B_{2r}^{c}(x)$, then $$\operatorname{\mathbb{E}}_{z}\bigl[H(X_{\tau(B_{r}(x)})\bigr]\;\le\; \kappa_{4}\,
\operatorname{\mathbb{E}}_{z}\bigl[\tau(B_{r}(x)\bigr]
\int_{{\mathbb{R}^{d}}} H(y)\frac{k(x, y-x)}{|y-x|^{d+\alpha}}\,{\mathrm{d}}{y}\,,$$ and $$\operatorname{\mathbb{E}}_{z}\bigl[H(X_{\tau(B_{r}(x)})\bigr]\;\ge\; \kappa_{5}\,
\operatorname{\mathbb{E}}_{z}\bigl[\tau(B_{r}(x)\bigr]
\int_{{\mathbb{R}^{d}}} H(y)\frac{k(x, y-x)}{|y-x|^{d+\alpha}}\,{\mathrm{d}}{y}\,.$$
The proof follows using the same argument as in [@song-vondracek-04 Lemma 3.5].
By Lemmas \[L4.1\], \[L4.2\] and \[L4.3\], the hypotheses (A1)–(A3) in [@song-vondracek-04] are satisfied. Hence the proof follows from [@song-vondracek-04 Theorem 2.4].
Positive recurrence and invariant probability measures {#sec-stability}
======================================================
In this section we study the recurrence properties for a Markov process with generator ${\mathcal{I}}\in{\mathfrak{L}_{\alpha}}$ (see Definition \[D3.2\] and \[D4.1\]). Many of the results of this section are based on the assumption of the existence of a *Lyapunov function*.
We say that the operator ${\mathcal{I}}$ of the form satisfies the Lyapunov stability condition if there exists a ${{\mathcal{V}}}\in C^{2}({\mathbb{R}^{d}})$ such that $\inf_{x\in{\mathbb{R}^{d}}}{{\mathcal{V}}}(x)>-\infty$, and for some compact set ${{\mathcal{K}}}\subset{\mathbb{R}^{d}}$ and $\varepsilon>0$, we have $$\label{E5.01}
{\mathcal{I}}\,{{\mathcal{V}}}(x)\;\le\; -\varepsilon \qquad\forall\, x\in {{\mathcal{K}}}^{c}.$$
It is straightforward to verify that if ${{\mathcal{V}}}$ satisfies for ${\mathcal{I}}\in{\mathfrak{L}_{\alpha}}$, then $$\label{E5.02}
\int_{{\lvertz\rvert}\ge 1}|{{\mathcal{V}}}(z)|\frac{1}{{\lvertz\rvert}^{d+\alpha}}\,{\mathrm{d}}{z}<\infty\,.$$
If there exists a constant $\gamma\in (1,\alpha)$ such that $$\frac{b(x)\cdot x}{{\lvertx\rvert}^{2-\gamma}\,\sup_{z\in{\mathbb{R}^{d}}} k(x,z)\vee 1}\;
\xrightarrow[{\lvertx\rvert}\to\infty]{}\;-\infty\,,$$ then the operator ${\mathcal{I}}$ satisfies the Lyapunov stability condition.
Consider a nonnegative function $f\in C^{2}({\mathbb{R}^{d}})$ such that $f(x)={\lvertx\rvert}^\gamma$ for ${\lvertx\rvert}\ge 1$, and let $\Bar{k}(x) {:=}\sup_{z\in{\mathbb{R}^{d}}} k(x,z)$. Since the second derivatives of $f$ are bounded in ${\mathbb{R}^{d}}$, and $k$ is also bounded, it follows that $${\biggl\lvert\int_{{\lvertz\rvert}\le1}{\mathfrak{d}}f(x;z)\,\frac{k(x,z)}{{\lvertz\rvert}^{d+\alpha}}\,{\mathrm{d}}{z}\biggr\rvert}
\;\le\; \kappa_{1}\,\Bar{k}(x)$$ for some constant $\kappa_{1}$ which depends on the bound of the trace of the Hessian of $f$. Following the same steps as in the proof of , and using the fact that $k$ is bounded in ${\mathbb{R}^{d}}\times{\mathbb{R}^{d}}$, we obtain $$\label{E5.03}
{\biggl\lvert\int_{{\lvertz\rvert}>1} ({\lvertx+z\rvert}^\gamma -{\lvertx\rvert}^\gamma)\,
\frac{k(x,z)}{{\lvertz\rvert}^{d+\alpha}}\,{\mathrm{d}}{z}\biggr\rvert}
\;\le\; \kappa_{2}\,\Bar{k}(x)\,
(1+{\lvertx\rvert}^{\gamma-\alpha})\qquad \text{if~} {\lvertx\rvert}>1\,,$$ for some constant $\kappa_{2}>0$. Since also, $$\label{E5.04}
{\biggl\lvert\int_{{\mathbb{R}^{d}}}\bm1_{B_{1}}(x+z)\,
\frac{k(x,z)}{{\lvertz\rvert}^{d+\alpha}}\,{\mathrm{d}}{z}\biggr\rvert}
\;\le\; \kappa_{3}\,\Bar{k}(x)\,({\lvertx\rvert}-1)^{-\alpha}\qquad\text{for~} {\lvertx\rvert}>2\,,$$ for some constant $\kappa_{3}$, it follows by the above that $$\label{E5.05}
{\biggl\lvert\int_{{\mathbb{R}^{d}}}{\mathfrak{d}}f(x;z)\,\frac{k(x,z)}{{\lvertz\rvert}^{d+\alpha}}\,{\mathrm{d}}{z}\biggr\rvert}
\;\le\; \kappa_{4}\,\Bar{k}(x)\,(1+{\lvertx\rvert}^{\gamma-\alpha})\qquad\forall x\in{\mathbb{R}^{d}}\,,$$ for some constant $\kappa_{4}$. Therefore by the hypothesis and , it follows that ${\mathcal{I}}f(x)\to-\infty$ as ${\lvertx\rvert}\to\infty$.
\[L5.1\] Let $X$ be the Markov process associated with a generator ${\mathcal{I}}\in{\mathfrak{L}_{\alpha}}(\lambda)$, and suppose that ${\mathcal{I}}$ satisfies the Lyapunov stability hypothesis and the growth condition in . Then for any $x\in {{\mathcal{K}}}^{c}$ we have $$\operatorname{\mathbb{E}}_{x}[\tau({{\mathcal{K}}}^{c})]
\;\le\;\frac{2}{\varepsilon}\,\bigl({{\mathcal{V}}}(x)+(\inf{{\mathcal{V}}})^-\bigr)\,.$$
Let $R_{0}>0$ be such that ${{\mathcal{K}}}\subset B_{R_{0}}$. We choose a cut-off function $\chi$ which equals $1$ on $B_{R_{1}}$, with $R_{1}>2R_{0}$, vanishes outside of $B_{R_{1}+1}$, and ${\lVert\chi\rVert}_{\infty}=1$. Then $f{:=}\chi{{\mathcal{V}}}$ is in $C^{2}_{b}({\mathbb{R}^{d}})$. Clearly if ${\lvertx\rvert}\;\le\; R_{0}$ and ${\lvertx+z\rvert}\;\ge\; R_{1}$, then ${\lvertz\rvert}\;>\; R_{0}$, and thus ${\lvertx+z\rvert}\;\le\; 2{\lvertz\rvert}$. Therefore, for large enough $R_{1}$, we obtain $$\begin{aligned}
{\biggl\lvert\int_{{\mathbb{R}^{d}}}\bigl(f(x+z)-{{\mathcal{V}}}(x+z)\bigr)
\frac{k(x,z)}{{\lvertz\rvert}^{d+\alpha}}\,{\mathrm{d}}{z}\biggr\rvert}
&\;\le\; 2\int_{\{{\lvertx+z\rvert}\ge R_{1}\}}|{{\mathcal{V}}}(x+z)|\,
\frac{k(x,z)}{{\lvertz\rvert}^{d+\alpha}}\,{\mathrm{d}}{z}
\\[5pt]
&\;\le\; 2^{d+\alpha+1}\lambda_{B_{R_{0}}}
\int_{\{{\lvertx+z\rvert}\ge R_{1}\}}|{{\mathcal{V}}}(x+z)|\,\frac{1}{|x+z|^{d+\alpha}}\,{\mathrm{d}}{z}
\\[5pt]
&\;\le\; \frac{\varepsilon}{2}\qquad \forall\, x\in B_{R_{0}}\,.\end{aligned}$$ Hence, for all $R_{1}$ large enough, we have $${\mathcal{I}}f(x)\;\le\;-\frac{\varepsilon}{2}
\qquad \forall x\in B_{R_{0}}\setminus {{\mathcal{K}}}\,.$$ Let $\widetilde\tau_{R}=\tau({{\mathcal{K}}}^{c})\wedge\tau(B_{R})$. Then applying Itô’s formula we obtain $$\operatorname{\mathbb{E}}_{x}\bigl[{{\mathcal{V}}}(X_{\widetilde\tau_{R_{0}}})\bigr]-{{\mathcal{V}}}(x)\;\le\;
-\frac{\varepsilon}{2}\,\operatorname{\mathbb{E}}_{x}[\widetilde\tau_{R_{0}}]\qquad\forall\,
x\in B_{R_{0}}\setminus {{\mathcal{K}}}\,,$$ implying that $$\label{E5.06}
\operatorname{\mathbb{E}}_{x}[\widetilde\tau_{R_{0}}]\;\le\;
\frac{2}{\varepsilon}\,\bigl({{\mathcal{V}}}(x)+(\inf{{\mathcal{V}}})^-\bigr)\,.$$ By the growth condition and Lemma \[L3.3\], $\tau(B_{R})\to\infty$ as $R\to\infty$ with probability $1$. Hence the result follows by applying Fatou’s lemma to .
Existence of invariant probability measures
-------------------------------------------
Recall that a Markov process is said be to positive recurrent if for any compact set $G$ with positive Lebesgue measure it holds that $\operatorname{\mathbb{E}}_{x}[\tau(G^{c})]<\infty$ for any $x\in{\mathbb{R}^{d}}$. We have the following theorem.
\[T5.1\] If ${\mathcal{I}}\in{\mathfrak{L}_{\alpha}}(\lambda)$ satisfies the Lyapunov stability hypothesis, and the growth condition in , then the associated Markov process is positive recurrent.
First we note that if the Lyapunov condition is satisfied for some compact set ${{\mathcal{K}}}$, then it is also satisfied for any compact set containing ${{\mathcal{K}}}$. Hence we may assume that ${{\mathcal{K}}}$ is a closed ball centered at origin. Let $D$ be an open ball with center at origin and containing ${{\mathcal{K}}}$. We define $$\Hat\tau_{1}\;{:=}\;\inf\;\{t\ge 0\,:\, X_{t}\notin D\}\,,\qquad
\Hat\tau_{2}\;{:=}\;\inf\;\{t>\tau\,:\, X_{t}\in {{\mathcal{K}}}\}\,.$$ Therefore for $X_{0}=x\in {{\mathcal{K}}}$, $\Hat\tau_{2}$ denotes the first return time to ${{\mathcal{K}}}$ after hitting $D^{c}$. First we prove that $$\label{E5.07}
\sup_{x\in {{\mathcal{K}}}}\;\operatorname{\mathbb{E}}_{x}[\Hat\tau_{2}]\;<\;\infty\,.$$ By Lemma \[L5.1\] we have $\operatorname{\mathbb{E}}_{x}[\tau({{\mathcal{K}}}^{c})]\le\frac{2}{\varepsilon}[{{\mathcal{V}}}(x)+(\inf{{\mathcal{V}}})^-]$ for $x\in {{\mathcal{K}}}^{c}$. By Lemma \[L3.1\] we have $\sup_{x\in {{\mathcal{K}}}}\operatorname{\mathbb{E}}_{x}[\Hat\tau_{1}]<\infty$. Let ${\mathscr{P}}_{\Hat\tau_{1}}(x,\cdot\,)$ denote the exit distribution of the process $X$ starting from $x\in {{\mathcal{K}}}$. In order to prove it suffices to show that $$\sup_{x\in {{\mathcal{K}}}}\;\int_{D^{c}}
\bigl({{\mathcal{V}}}(y)+(\inf{{\mathcal{V}}})^-\bigr)\,{\mathscr{P}}_{\Hat\tau_{1}}(x,{\mathrm{d}}{y})\;<\;\infty\,,$$ and since ${{\mathcal{V}}}$ is locally bounded it is enough that $$\label{E5.08}
\sup_{x\in {{\mathcal{K}}}}\;\int_{B_{R}^{c}}\bigl({{\mathcal{V}}}(y)+(\inf{{\mathcal{V}}})^-\bigr)\,
{\mathscr{P}}_{\Hat\tau_{1}}(x,{\mathrm{d}}{y})\;<\;\infty$$ for some ball $B_{R}$. To accomplish this we choose $R$ large enough so that $$\frac{|x-z|}{{\lvertz\rvert}}\;>\;\frac{1}{2}\qquad\text{for}~ {\lvertz\rvert}\ge R\,,\; x\in D\,.$$ Then, for any Borel set $A\subset B_{R}^{c}$, by Proposition \[levy-system\] we have that $$\begin{aligned}
\operatorname{\mathbb{P}}_{x}(X_{\Hat\tau_{1}\wedge t}\in A)
&\;=\;\operatorname{\mathbb{E}}_{x}\Biggl[\sum_{s\le \Hat\tau_{1}\wedge t}\bm1_{\{X_{s-}\in
D,\, X_{s}\in A\}}\Biggr]
\\[5pt]
&\;=\;\operatorname{\mathbb{E}}_{x}\biggl[\int_{0}^{\Hat\tau_{1}\wedge t}\bm1_{\{X_{s}\in D\}}
\int_{A}\frac{k(X_{s}, z-X_{s})}{|X_{s}-z|^{d+\alpha}}\,{\mathrm{d}}{z}\,{\mathrm{d}}{s}\biggr]
\\[5pt]
&\;\le\; 2^{d+\alpha}\lambda_{D}\,
\operatorname{\mathbb{E}}_{x}\biggl[\int_{0}^{\Hat\tau_{1}\wedge t}\int_{A}
\frac{1}{{\lvertz\rvert}^{d+\alpha}}\,{\mathrm{d}}{z}\,{\mathrm{d}}{s}\biggr]
\\[5pt]
&\;=\; 2^{d+\alpha}\lambda_{D}\,\operatorname{\mathbb{E}}_{x}[\Hat\tau_{1}\wedge t]\,\mu(A)\,,\end{aligned}$$ where $\mu$ is the $\sigma$-finite measure on ${\mathbb{R}^{d}}_*$ with density $\frac{1}{{\lvertz\rvert}^{d+\alpha}}$. Thus letting $t\to\infty$ we obtain $${\mathscr{P}}_{\Hat\tau_{1}}(x, A)\;\le\;
2^{d+\alpha}\lambda_{D}\,\biggl(\sup_{x\in {{\mathcal{K}}}}\;\operatorname{\mathbb{E}}_{x}[\Hat\tau_{1}]\biggr)
\,\mu(A)\,.$$ Therefore, using a standard approximation argument, we deduce that for any nonnegative function $g$ it holds that $$\int_{B_{R}^{c}}g(y)\,{\mathscr{P}}_{\Hat\tau_{1}}(x, {\mathrm{d}}{y})\;\le\;
\Tilde\kappa \int_{B_{R}^{c}}g(y)\mu({\mathrm{d}}{y})$$ for some constant $\Tilde\kappa$. This proves since ${{\mathcal{V}}}$ is integrable on $B_{R}^{c}$ with respect to $\mu$ and $\mu(B_{R}^{c})<\infty$.
Next we prove that the Markov process is positive recurrent. We need to show that for any compact set $G$ with positive Lebesgue measure, $\operatorname{\mathbb{E}}_{x}[\tau(G^{c})]<\infty$ for any $x\in{\mathbb{R}^{d}}$. Given a compact $G$ and $x\in G^{c}$ we choose a closed ball ${{\mathcal{K}}}$, which satisfies the Lyapunov condition relative to ${{\mathcal{V}}}$, and such that $G\cup\{x\}\subset {{\mathcal{K}}}$. Let $D$ be an open ball containing ${{\mathcal{K}}}$. We define a sequence of stopping times $\{\Hat\tau_{k}\,,\; k=0,1,\dotsc\}$ as follows: $$\begin{aligned}
\Hat\tau_{0} &\;=\;0\\[3pt]
\Hat\tau_{2n+1}&\;=\;\inf\{t>\Hat\tau_{2n}:\,X_{t}\notin D\}\,,\\[3pt]
\Hat\tau_{2n+2}&\;=\;\inf\{t>\Hat\tau_{2n+1}:\,X_{t}\in {{\mathcal{K}}}\}\,,
\quad n=0,1,\dotsc.\end{aligned}$$ Using the strong Markov property and , we obtain $\operatorname{\mathbb{E}}_{x}[\Hat\tau_{n}]<\infty$ for all $n\in{\mathbb{N}}$. From Lemma \[L3.5\] there exist positive constants $t$ and $r$ such that $$\sup_{x\in{{\mathcal{K}}}}\;\operatorname{\mathbb{P}}_{x}(\tau(D)<t)\;\le\;
\sup_{x\in{{\mathcal{K}}}}\;\operatorname{\mathbb{P}}_{x} (\tau(B_r(x))< t)\;\le\; \frac{1}{4}\,.$$ Therefore, using a similar argument as in Lemma \[L4.2\], we can find a constant $\delta>0$ such that $$\inf_{x\in {{\mathcal{K}}}}\;\operatorname{\mathbb{P}}_{x}(\tau(G^{c})<\tau(D))\;>\;\delta\,.$$ Hence $$p\;{:=}\;\sup_{x\in {{\mathcal{K}}}}\;\operatorname{\mathbb{P}}_{x}(\tau(D)<\tau(G^{c}))
\;\le\; 1-\delta\;<\;1\,.$$ Thus by the strong Markov property we obtain $$\operatorname{\mathbb{P}}_{x}(\tau(G^{c})>\Hat\tau_{2n})\;\le\;
p\,\operatorname{\mathbb{P}}_{x}(\tau(G^{c})>\Hat\tau_{2n-2})\;\le\;\dotsb\;\le\; p^{n}
\qquad \forall\,x\in {{\mathcal{K}}}\,.$$ This implies $\operatorname{\mathbb{P}}_{x}(\tau(G^{c})<\infty)=1$. Hence, for $x\in {{\mathcal{K}}}$, we obtain $$\begin{aligned}
\operatorname{\mathbb{E}}_{x}[\tau(G^{c})]&\;\le\; \sum_{n=1}^{\infty}
\operatorname{\mathbb{E}}_{x}\bigl[\Hat\tau_{2n}
\bm1_{\{\Hat\tau_{2n-2}<\tau(G^{c})\le\Hat\tau_{2n}\}}\bigr]\\[5pt]
&\;=\; \sum_{n=1}^{\infty}\sum_{l=1}^{n}
\operatorname{\mathbb{E}}_{x}\bigl[(\Hat\tau_{2l}- \Hat\tau_{2l-2})
\bm1_{\{\Hat\tau_{2n-2}<\tau(G^{c})\le\Hat\tau_{2n}\}}\bigr]\\[5pt]
&\;=\; \sum_{l=1}^{\infty}\sum_{n=l}^{\infty}
\operatorname{\mathbb{E}}_{x}\bigl[(\Hat\tau_{2l}- \Hat\tau_{2l-2})
\bm1_{\{\Hat\tau_{2n-2}<\tau(G^{c})\le\Hat\tau_{2n}\}}\bigr]
\\[5pt]
&\;=\; \sum_{l=1}^{\infty}\operatorname{\mathbb{E}}_{x}\bigl[(\Hat\tau_{2l}- \Hat\tau_{2l-2})
\bm1_{\{\Hat\tau_{2l-2}<\tau(G^{c})\}}\bigr]\\[5pt]
&\;\le\; \sum_{l=1}^{\infty} p^{l-1}\sup_{x\in {{\mathcal{K}}}}\;\operatorname{\mathbb{E}}_{x}[\Hat\tau_{2}]\\[5pt]
&\;=\;\frac{1}{1-p}\,\sup_{x\in {{\mathcal{K}}}}\;\operatorname{\mathbb{E}}_{x}[\Hat\tau_{2}]\;<\;\infty\,.\end{aligned}$$ Since also $\operatorname{\mathbb{E}}_{x}[\tau({{\mathcal{K}}}^{c})]<\infty$ for all $x\in{\mathbb{R}^{d}}$, this completes the proof.
\[T5.2\] Let $X$ be a Markov process associated with a generator ${\mathcal{I}}\in{\mathfrak{L}^{\mathsf{sym}}_{\alpha}}(\lambda)\cup\mathfrak{I}_{\alpha}(\theta,\lambda)$, and suppose that the Lyapunov stability hypothesis and the growth condition in hold. Then $X$ has an invariant probability measure.
The proof is based on Has$'$minskiĭ’s construction. Let ${{\mathcal{K}}}$, $D$, $\Hat\tau_{1}$, and $\Hat\tau_{2}$ be as in the proof of Theorem \[T5.1\]. Let $\Hat X$ be a Markov process on ${{\mathcal{K}}}$ with transition kernel given by $$\Hat\operatorname{\mathbb{P}}_{x}({\mathrm{d}}{y})\;=\;\operatorname{\mathbb{P}}_{x}(X_{\Hat\tau_{2}}\in {\mathrm{d}}{y})\,.$$ Let $f$ be any bounded, nonnegative measurable function on $D$. Define $Q_{f}(x)=\operatorname{\mathbb{E}}_{x}[f(X_{\Hat\tau_{2}})]$. We claim that $Q_{f}$ is harmonic in $D$. Indeed if we define $\Tilde f(x)=\operatorname{\mathbb{E}}_{x}[f(X_{\tau({{\mathcal{K}}}^{c})})]$ for $x\in D^{c}$, then by the strong Markov property we obtain $Q_{f}(x)=\operatorname{\mathbb{E}}_{x}[\Tilde f(X_{\Hat\tau_{1}})]$, and the claim follows. By Theorem \[T4.1\] there exists a positive constant $C_{H}$, independent of $f$, satisfying $$\label{E5.09}
Q_{f}(x)\;\le\; C_{H} Q_{f}(y)\qquad \forall\,x, y\in {{\mathcal{K}}}\,.$$ We note that $Q_{\bm1_{{{\mathcal{K}}}}}\equiv 1$. Let $Q(x, A){:=}Q_{\bm1_{A}}(x)$, for $A\subset {{\mathcal{K}}}$. For any pair of probability measures $\mu$ and $\mu'$ on ${{\mathcal{K}}}$, we claim that $$\label{E5.10}
{\biggl\lVert\int_{{{\mathcal{K}}}}\bigl(\mu({\mathrm{d}}{x})-\mu'({\mathrm{d}}{x})\bigr)
Q(x,\cdot\,)\biggr\rVert}_{\mathrm{TV}}\;\le\;
\frac{C_{H}-1}{C_{H}}\,{\lVert\mu-\mu'\rVert}_{\mathrm{TV}}\,.$$ This implies that the map $\mu\to\int_{{{\mathcal{K}}}}Q(x,\cdot\,)\mu({\mathrm{d}}{x})$ is a contraction and hence it has a unique fixed point $\Hat\mu$ satisfying $\Hat\mu(A)=\int_{{{\mathcal{K}}}}Q(x, A)\Hat\mu({\mathrm{d}}{x})$ for any Borel set $A\subset {{\mathcal{K}}}$. In fact, $\Hat\mu$ is the invariant probability measure of the Markov chain $\Hat X$. Next we prove . Given any two probability measure $\mu$, $\mu'$ on ${{\mathcal{K}}}$, we can find subsets $F$ and $G$ of ${{\mathcal{K}}}$ such that $$\begin{aligned}
{\biggl\lVert\int_{{{\mathcal{K}}}}\bigl(\mu({\mathrm{d}}{x})-\mu'({\mathrm{d}}{x})\bigr)
Q(x,\cdot\,)\biggr\rVert}_{\mathrm{TV}}&\;=\;
2\int_{{{\mathcal{K}}}}\bigl(\mu({\mathrm{d}}{x})-\mu'({\mathrm{d}}{x})\bigr)\,Q(x,F)\,,
\\[5pt]
{\lVert\mu-\mu'\rVert}_{\mathrm{TV}} & \;=\; 2(\mu-\mu')(G)\,.\end{aligned}$$ In fact, the restriction of $(\mu-\mu')$ to $G$ is a nonnegative measure and its restriction to $G^{c}$ it is non-positive measure. By , we have $$\label{E5.11}
\inf_{x\in G^{c}}\;Q(x, F)\;\ge\;\sup_{x\in G}\;Q(x, F)$$ Hence, using , we obtain $$\begin{aligned}
{\biggl\lVert\int_{{{\mathcal{K}}}}\bigl(\mu({\mathrm{d}}{x})-\mu'({\mathrm{d}}{x})\bigr)Q(x,\cdot\,)\biggr\rVert}_{\mathrm{TV}}
&\;=\;
2\int_{G}\bigl(\mu({\mathrm{d}}{x})-\mu'({\mathrm{d}}{x})\bigr)Q(x, F)
+2\int_{G^{c}}\bigl(\mu({\mathrm{d}}{x})-\mu'({\mathrm{d}}{x})\bigr)Q(x,F)
\\[5pt]
&\;\le\; 2(\mu-\mu')(G)\,\sup_{x\in G}\;Q(x, F)
+ 2(\mu-\mu')(G^{c})\,\inf_{x\in G^{c}}\;Q(x, F)\\[5pt]
&\;\le\; 2(\mu-\mu')(G)\,\sup_{x\in G}\;Q(x, F)
- \frac{2}{C_{H}}(\mu-\mu')(G)\,\sup_{x\in G}\;Q(x, F)\\[5pt]
&\;\le\; \bigl(1-C_{H}^{-1}\bigr){\lVert\mu-\mu'\rVert}_{\mathrm{TV}} \,.\end{aligned}$$ This proves .
We define a probability measure $\nu$ on ${\mathbb{R}^{d}}$ as follows. $$\int_{{\mathbb{R}^{d}}}f(x)\,\nu({\mathrm{d}}{x})
\;=\;\frac{\int_{{{\mathcal{K}}}}\operatorname{\mathbb{E}}_{x}\bigl[\int_{0}^{\Hat\tau_{2}}f(X_{s})\,{\mathrm{d}}{s}\bigr]
\Hat\mu({\mathrm{d}}{x})}
{\int_{{{\mathcal{K}}}}\operatorname{\mathbb{E}}_{x}[\Hat\tau_{2}]\Hat\mu({\mathrm{d}}{x})}\,, \qquad f\in C_{b}({\mathbb{R}^{d}})\,.$$ It is straight forward to verify that $\nu$ is an invariant probability measure of $X$ (see for example, [@ari-bor-ghosh Theorem 2.6.9]).
If $k(\cdot,\cdot)=1$ and the drift $b$ belongs to certain Kato class, in particular bounded, (see [@bogdan-jakubowski]) then the transition probability has a continuous density, and therefore any invariant probability measure has a continuous density. Since any two distinct ergodic measures are mutually singular, this implies the uniqueness of the invariant probability measure. As shown later in Proposition \[P5.3\] open sets have strictly positive mass under any invariant measure.
The following result is fairly standard.
Let ${\mathcal{I}}\in{\mathfrak{L}_{\alpha}}$, and ${{\mathcal{V}}}\in C^{2}({\mathbb{R}^{d}})$ be a nonnegative function satisfying satisfying ${{\mathcal{V}}}(x)\to\infty$ as ${\lvertx\rvert}\to\infty$, and ${\mathcal{I}}\,{{\mathcal{V}}}\le 0$ outside some compact set ${{\mathcal{K}}}$. Let $\nu$ be an invariant probability measure of the Markov process associated with the generator ${\mathcal{I}}$. Then $$\int_{{\mathbb{R}^{d}}}|{\mathcal{I}}\,{{\mathcal{V}}}(x)|\,\nu({\mathrm{d}}{x})\;\le\; 2\int_{{{\mathcal{K}}}}|{\mathcal{I}}\,{{\mathcal{V}}}(x)|\,\nu({\mathrm{d}}{x})\,.$$
Let $\varphi_{n}:{\mathbb{R}}_{+}\to{\mathbb{R}}_{+}$ be a smooth non-decreasing, concave, function such that $$\varphi_{n}(x)\;=\;\begin{cases}
x & \text{for}~x\le n\,,\\
n+\nicefrac{1}{2} & \text{for}~x\ge n+1\,.
\end{cases}$$ Due to concavity we have $\varphi_{n}(x)\le {\lvertx\rvert}$ for all $x\in{\mathbb{R}}_{+}$. Then ${{\mathcal{V}}}_{n}(x){:=}\varphi_{n}({{\mathcal{V}}}(x))$ is in $C^{2}_{b}({\mathbb{R}^{d}})$ and it also follows that ${\mathcal{I}}\,{{\mathcal{V}}}_{n}(x)\to{\mathcal{I}}\,{{\mathcal{V}}}(x)$ as $n\to\infty$. Since $\nu$ is an invariant probability measure, it holds that $$\label{E5.12}
\int_{{\mathbb{R}^{d}}}{\mathcal{I}}\,{{\mathcal{V}}}_{n}(x)\,\nu({\mathrm{d}}{x})\;=\;0\,.$$ By concavity, $\varphi_{n}(y)\le \varphi_{n}(x)+(y-x)\cdot\varphi_{n}'(x)$ for all $x,y\in{\mathbb{R}}_{+}$. Hence $$\begin{aligned}
{\mathcal{I}}\,{{\mathcal{V}}}_{n}(x)
&\;=\; \int_{{\mathbb{R}^{d}}}{\mathfrak{d}}{{\mathcal{V}}}_{n}(x;z)\,
\frac{k(x,z)}{{\lvertz\rvert}^{d+\alpha}}\,{\mathrm{d}}{z}
+ \varphi_{n}'({{\mathcal{V}}}(x))\,b(x)\cdot{\nabla}{{\mathcal{V}}}(x)
\\[5pt]
&\;\le\; \int_{{\mathbb{R}^{d}}}\varphi_{n}'({{\mathcal{V}}}(x))\,{\mathfrak{d}}{{\mathcal{V}}}(x;z)\,
\frac{k(x,z)}{{\lvertz\rvert}^{d+\alpha}}\,{\mathrm{d}}{z}
+ \varphi_{n}'({{\mathcal{V}}}(x))\,b(x)\cdot{\nabla}{{\mathcal{V}}}(x)
\\[5pt]
&\;=\;\varphi_{n}'({{\mathcal{V}}}(x))\,{\mathcal{I}}\,{{\mathcal{V}}}(x)\,,\end{aligned}$$ which is negative for $x\in {{\mathcal{K}}}^{c}$. Therefore using we obtain $$\begin{aligned}
\label{E5.13}
\int_{{\mathbb{R}^{d}}}|{\mathcal{I}}\,{{\mathcal{V}}}_{n}(x)|\,\nu({\mathrm{d}}{x})
&\;=\;\int_{{{\mathcal{K}}}}|{\mathcal{I}}\,{{\mathcal{V}}}_{n}(x)|\,\nu({\mathrm{d}}{x})
-\int_{{{\mathcal{K}}}^{c}}{\mathcal{I}}\,{{\mathcal{V}}}_{n}(x)\,\nu({\mathrm{d}}{x})
\nonumber\\[5pt]
&\;=\;\int_{{{\mathcal{K}}}}|{\mathcal{I}}\,{{\mathcal{V}}}_{n}(x)|\,\nu({\mathrm{d}}{x})
+\int_{{{\mathcal{K}}}}{\mathcal{I}}\,{{\mathcal{V}}}_{n}(x)\,\nu({\mathrm{d}}{x})
\nonumber\\[5pt]
&\;\le\; 2\int_{{{\mathcal{K}}}}|{\mathcal{I}}\,{{\mathcal{V}}}_{n}(x)|\,\nu({\mathrm{d}}{x})\,.\end{aligned}$$ On the other hand, with $A_{n}{:=}\{y\in{\mathbb{R}^{d}}\,\colon {{\mathcal{V}}}(y)\ge n\}$, and provided ${{\mathcal{V}}}(x)<n$, we have $$\begin{aligned}
{\lvert{\mathcal{I}}\,{{\mathcal{V}}}_{n}(x)\rvert} &\;\le\; {\lvert{\mathcal{I}}\,{{\mathcal{V}}}(x)\rvert}
+\int_{x+z\in A_{n}} {\lvert{{\mathcal{V}}}(x+z)-{{\mathcal{V}}}_{n}(x+z)\rvert}\,
\frac{k(x,z)}{{\lvertz\rvert}^{d+\alpha}}\,{\mathrm{d}}{z}\\[5pt]
&\;\le\; {\lvert{\mathcal{I}}\,{{\mathcal{V}}}(x)\rvert}
+\int_{x+z\in A_{n}} {\lvert{{\mathcal{V}}}(x+z)\rvert}\,
\frac{k(x,z)}{{\lvertz\rvert}^{d+\alpha}}\,{\mathrm{d}}{z}\,.\end{aligned}$$ This together with imply that there exists a constant $\kappa$ such that $${\lvert{\mathcal{I}}\,{{\mathcal{V}}}_{n}(x)\rvert} \;\le\; \kappa+{\lvert{\mathcal{I}}\,{{\mathcal{V}}}(x)\rvert}
\qquad \forall\,x\in{{\mathcal{K}}}\,,$$ and all large enough $n$. Therefore, letting $n\to\infty$ and using Fatou’s lemma for the term on the left hand side of , and the dominated convergence theorem for the term on the right hand side, we obtain the result.
A class of operators with variable order kernels
------------------------------------------------
It is quite evident from Theorem \[T5.2\] that the Harnack inequality plays a crucial role in the analysis. Therefore one might wish to establish positive recurrence for an operator with a variable order kernel, and deploy the Harnack inequality from [@bass-kassmann] to prove a similar result as in Theorem \[T5.2\].
\[T5.3\] Let $\uppi:{\mathbb{R}^{d}}\times{\mathbb{R}^{d}}\to{\mathbb{R}^{d}}$ be a nonnegative measurable function satisfying the following properties, for $1<\alpha'<\alpha<2$:
- There exists a constant $c_{1}>0$ such that $\bm1_{\{{\lvertz\rvert}>1\}}\uppi(x,z)\le \frac{c_{1}}{{\lvertz\rvert}^{d+\alpha'}}$ for all $x\in{\mathbb{R}^{d}}$;
- There exists a constant $c_{2}>0$ such that $$\uppi(x,z-x)\;\le\; c_{2}\, \uppi(y,y-z)\,,\quad \text{whenever}
\quad {\lvertz-x\rvert}\wedge{\lvertz-y\rvert}\ge 1\,,\;{\lvertx-y\rvert}\le 1\,;$$
- For each $R>0$ there exists $q_{R}>0$ such that $$\frac{q_{R}^{-1}}{{\lvertz\rvert}^{d+\alpha'}}\;\le\; \uppi(x,z)
\;\le\; \frac{q_{R}}{{\lvertz\rvert}^{d+\alpha}}\qquad
\forall x\in{\mathbb{R}^{d}}\,,\;\forall z\in B_{R}\,;$$
- For each $R>0$ there exists $R_{1}>0$, $\sigma\in(1,2)$, and $\kappa_{\sigma}=\kappa_{\sigma}(R,R_{1})>0$ such that $$\frac{\kappa_{\sigma}^{-1}}{{\lvertz\rvert}^{d+\sigma}}\;\le\;
\uppi(x,z)\;\le\; \frac{\kappa_{\sigma}}{{\lvertz\rvert}^{d+\sigma}}\qquad
\forall x\in B_{R}\,,\;\forall z\in B_{R_{1}}^{c};$$
- There exists ${{\mathcal{V}}}\in C^{2}({\mathbb{R}^{d}})$ that is bounded from below in ${\mathbb{R}^{d}}$, a compact set ${{\mathcal{K}}}\subset{\mathbb{R}^{d}}$ and a constant $\varepsilon>0$, such that $$\int_{{\mathbb{R}^{d}}}{\mathfrak{d}}{{\mathcal{V}}}(x;z)\,
\uppi(x,z)\,{\mathrm{d}}{z}\;<\;-\varepsilon\quad \forall x\in {{\mathcal{K}}}^{c}\,.$$
Then the Markov process associated with the above kernel has an invariant probability measure.
The first three assumptions guarantee the Harnack property for associated harmonic functions [@bass-kassmann]. Then the conclusion of Theorem \[T5.3\] follows by using an argument similar to the one used in the proof of Theorem \[T5.2\].
Next we present an example of a kernel $\uppi$ that satisfies the conditions in Theorem \[T5.3\]. We accomplish this by adding a non-symmetric bump function to a symmetric kernel.
Let $\varphi:{\mathbb{R}^{d}}\to[0,1]$ be a smooth function such that $$\varphi(x)\;=\;\begin{cases}
1 & \text{for}~{\lvertx\rvert}\le \frac{1}{2}\,,\\[5pt]
0 & \text{for}~{\lvertx\rvert}\ge 1\,.
\end{cases}$$ Define for $1< \alpha'<\beta'< \alpha<2$, $$\gamma(x,z)\;{:=}\;\varphi\biggl(2\frac{x+z}{1+{\lvertx\rvert}}\biggr)(1-\varphi(4x))
(\alpha'-\beta')\,,$$ and let $$\begin{aligned}
\widetilde\uppi(x,z)&\;{:=}\;\frac{1}{{\lvertz\rvert}^{d+\beta'+\gamma(x,z)}}\,,\\
\uppi(x,z)&\;{:=}\;\frac{1}{{\lvertz\rvert}^{d+\alpha}}+\widetilde\uppi(x,z)\,.\end{aligned}$$ We prove that $\uppi$ satisfies the conditions of Theorem \[T5.3\]. Let us also mention that there exists a unique solution to the martingale problem corresponding to the kernel $\uppi$ [@Komatsu-73; @komatsu]. We only show that conditions (b) and (e) hold. It is straightforward to verify (a), (c) and (d).
Note that $\alpha'-\beta'\le \gamma(x,z)\le 0$ for all $x,z$. Let $x,\, y,\, z\in{\mathbb{R}^{d}}$ such that $|x-z|\wedge|y-z|\ge 1$ and ${\lvertx-y\rvert}\le 1$. Then ${\lvertz-y\rvert}\le 1+{\lvertz-x\rvert}$. By a simple calculation we obtain $$\begin{aligned}
\widetilde\uppi(x,z-x) & \;\le\;
\biggl(1+\frac{1}{{\lvertz-x\rvert}}\biggr)^{d+\beta'+\gamma(x,z-x)}
\frac{1}{{\lvertz-y\rvert}^{d+\beta'+\gamma(x,z-x)}}\\[5pt]
&\;\le\; 2^{d+\beta'}\frac{1}{{\lvertz-y\rvert}^{d+\beta'+\gamma(y, z-y)}}
{\lvertz-y\rvert}^{-\beta'(x,z-x)
+\gamma(y, z-y)}\,.\end{aligned}$$ Hence it is enough to show that $$\label{E5.14}
{\lvertz-y\rvert}^{-\gamma(x,z-x)+\gamma(y, z-y)}\;<\;\varrho$$ for some constant $\varrho$ which does not depend on $x$, $y$ and $z$. Note that if ${\lvertx\rvert}\le 2$, which implies that ${\lverty\rvert}\le 3$, then for ${\lvertz\rvert}\ge 4$ we have $\gamma(x,z-x)=0=\gamma(y, z-y)$. Therefore for ${\lvertx\rvert}\le 2$, it holds that $$\label{E5.15}
{\lvertz-y\rvert}^{-\gamma(x,z-x)+\gamma(y, z-y)}\;\le\; 7^{\beta'-\alpha'}\,.$$ Suppose that ${\lvertx\rvert}\ge 2$. Then ${\lverty\rvert}\ge 1$. Since we only need to consider the case where $\gamma(x,z-x)\neq \gamma(y, z-y)$ we restrict our attention to $z\in{\mathbb{R}^{d}}$ such that ${\lvertz\rvert}\le 2(1+{\lvertx\rvert})$. We obtain $$\begin{aligned}
\label{E5.16}
\log({\lvertz-y\rvert})(-\gamma(x,z-x)+\gamma(y, z-y))&\;\le\;
\log\bigl(3(1+{\lvertx\rvert})\bigr)\,{\lVert\varphi'\rVert}_{\infty}\,
\frac{2{\lvertz\rvert}(\beta'-\alpha')}{(1+{\lvertx\rvert})(1+{\lverty\rvert})}\nonumber\\[5pt]
&\;\le\; \log\bigl(3(1+{\lvertx\rvert})\bigr)\,{\lVert\varphi'\rVert}_{\infty}\,
\frac{4(1+{\lvertx\rvert})(\beta'-\alpha')}{(1+{\lvertx\rvert}){\lvertx\rvert}}\,.\end{aligned}$$ Since the term on the right hand side of is bounded in ${\mathbb{R}^{d}}$, the bound in follows by –.
Next we prove the Lyapunov property. We fix a constant $\eta\in(\alpha',\beta')$, and choose some function ${{\mathcal{V}}}\in C^{2}({\mathbb{R}^{d}})$ such that ${{\mathcal{V}}}(x) = {\lvertx\rvert}^{\eta}$ for ${\lvertx\rvert}>1$. Since $\widetilde\uppi(x,z)\,\le\,\frac{1}{{\lvertz\rvert}^{d+\alpha'}}$ for all $x\in{\mathbb{R}^{d}}$ and $z\in{\mathbb{R}^{d}}_{*}$, it follows that $$x\;\mapsto\;
{\biggl\lvert\int_{{\lvertz\rvert}\le 1} {\mathfrak{d}}{{\mathcal{V}}}(x;z)\,\widetilde\uppi(x,z)\,{\mathrm{d}}{z}\biggr\rvert}$$ is bounded by some constant on ${\mathbb{R}^{d}}$. By , $${\biggl\lvert\int_{{\mathbb{R}^{d}}}{\mathfrak{d}}{{\mathcal{V}}}(x;z)\,\uppi(x,z)\,{\mathrm{d}}{z}\biggr\rvert}
\;\le\; c_{0}\,(1+{\lvertx\rvert}^{\eta-\alpha})\qquad\forall x\in{\mathbb{R}^{d}}\,,$$ for some constant $c_{0}$. Therefore, in view of , it is enough to show that for ${\lvertx\rvert}\ge 4$, there exist positive constants $c_{1}$ and $c_{2}$ such that $$\label{E5.17}
\int_{{\lvertz\rvert}>1}
\bigl({\lvertx+z\rvert}^\eta-{\lvertx\rvert}^\eta\bigr)\,\widetilde\uppi(x,z)\,{\mathrm{d}}{z}\;\le\;
c_{1}-c_{2}{\lvertx\rvert}^{\eta-\alpha'}\,.$$ By the definition of $\gamma$ it holds that $$\label{E5.18}
\widetilde\uppi(x,z)\;=\;\frac{1}{{\lvertz\rvert}^{d+\beta'}}\,,\qquad
\text{if~}{\lvertx+z\rvert}\,\ge\,\frac{3}{4}\,{\lvertx\rvert}\,,~\text{and~}{\lvertx\rvert}\,\ge\,2\,,$$ while $$\label{E5.19}
\widetilde\uppi(x,z)\;=\;\frac{1}{{\lvertz\rvert}^{d+\alpha'}}\,,\qquad
\text{if~}{\lvertx+z\rvert}\,\le\,\frac{{\lvertx\rvert}}{4}\,.$$ Suppose that ${\lvertx\rvert}>2$. Since ${\lvertx+z\rvert}\,\le\,\frac{{\lvertx\rvert}}{4}$ implies that $\frac{3}{4}{\lvertx\rvert}\,\le\,{\lvertz\rvert}\le\frac{5}{4}{\lvertx\rvert}$, we obtain by that $$\begin{aligned}
\label{E5.20}
\int_{{\lvertx+z\rvert}\le\frac{{\lvertx\rvert}}{4},\,{\lvertz\rvert}>1}
\bigl({\lvertx+z\rvert}^\eta-{\lvertx\rvert}^\eta\bigr)\,\widetilde\uppi(x,z)\,{\mathrm{d}}{z}
&\;\le\;
-\int_{{\lvertx+z\rvert}\le\frac{{\lvertx\rvert}}{4}}
\left(1-\tfrac{1}{4^\eta}\right){\lvertx\rvert}^\eta\,\left(\tfrac{4}{5}\right)^{d+\alpha'}
\frac{1}{{\lvertx\rvert}^{d+\alpha'}}\,{\mathrm{d}}{z}\nonumber\\[5pt]
&\;\le\;
-\left(1-\tfrac{1}{4^\eta}\right)\,\left(\tfrac{4}{5}\right)^{d+\alpha'}\,
{\lvertx\rvert}^{\eta-\alpha'}\int_{{\lvertx+z\rvert}\le\frac{{\lvertx\rvert}}{4}}
\frac{{\mathrm{d}}{z}}{{\lvertx\rvert}^{d}}\nonumber\\[5pt]
&\;\le\;
-m_{1}\,{\lvertx\rvert}^{\eta-\alpha'}\,,\qquad\text{if~} {\lvertx\rvert}>2\,,\end{aligned}$$ for some constant $m_{1}>0$, where we use the fact that the integral in the second inequality is independent of $x$ due to rotational invariance. Also, ${\lvertx+z\rvert}\le \frac{3}{4}{\lvertx\rvert}$ implies $\frac{1}{4}{\lvertx\rvert}\le {\lvertz\rvert}\le \frac{7}{4}{\lvertx\rvert}$, and in a similar manner, using , we obtain $$\begin{aligned}
\label{E5.21}
\int_{{\lvertx+z\rvert}\le\frac{3{\lvertx\rvert}}{4},\,{\lvertz\rvert}>1}
\bigl({\lvertx+z\rvert}^\eta-{\lvertx\rvert}^\eta\bigr)\,\frac{1}{{\lvertx\rvert}^{d+\beta'}}\,{\mathrm{d}}{z}
&\;\ge\;
-\int_{\frac{1}{4}{\lvertx\rvert}\le {\lvertz\rvert}\le \frac{7}{4}{\lvertx\rvert}}
{\lvertx\rvert}^\eta\,4^{d+\beta'}
\frac{1}{{\lvertx\rvert}^{d+\beta'}}\,{\mathrm{d}}{z}\nonumber\\[5pt]
&\;\ge\;
-m_{2}\,{\lvertx\rvert}^{\eta-\beta'}\,,\qquad\text{if~} {\lvertx\rvert}>2\,,\end{aligned}$$ for some constant $m_{2}>0$. Let $A_1{:=}\bigl\{z\, : \frac{1}{4}{\lvertx\rvert}\le {\lvertx+z\rvert}\le
\frac{3}{4}{\lvertx\rvert}\bigr\}$. Since $\eta$ is positive, we have $$\int_{\{{\lvertz\rvert}\geq 1\}\cap A_1}
\bigl({\lvertx+z\rvert}^\eta-{\lvertx\rvert}^\eta\bigr)\widetilde\uppi(x,z)\, {\mathrm{d}}{z}
\;\le\;0\,.$$ Thus, combining this observation with and , we obtain $$\begin{aligned}
\label{E5.22}
\int_{{\lvertx+z\rvert}>\frac{{\lvertx\rvert}}{4},\,{\lvertz\rvert}>1}
\bigl({\lvertx+z\rvert}^\eta-{\lvertx\rvert}^\eta\bigr)\,\widetilde\uppi(x,z)\,{\mathrm{d}}{z}
&\;\le\;
\int_{{\lvertx+z\rvert}>\frac{3}{4}{\lvertx\rvert},\,{\lvertz\rvert}>1}
\bigl({\lvertx+z\rvert}^\eta-{\lvertx\rvert}^\eta\bigr)\,\frac{1}{{\lvertz\rvert}^{d+\beta'}}\,{\mathrm{d}}{z}
\nonumber\\[5pt]
&\;=\;
\int_{{\lvertz\rvert}>1}
\bigl({\lvertx+z\rvert}^\eta-{\lvertx\rvert}^\eta\bigr)\,\frac{1}{{\lvertz\rvert}^{d+\beta'}}\,{\mathrm{d}}{z}
\nonumber\\[5pt]
&\mspace{50mu}-
\int_{{\lvertx+z\rvert}\le\frac{3{\lvertx\rvert}}{4},\,{\lvertz\rvert}>1}
\bigl({\lvertx+z\rvert}^\eta-{\lvertx\rvert}^\eta\bigr)\,\frac{1}{{\lvertx\rvert}^{d+\beta'}}\,{\mathrm{d}}{z}
\nonumber\\[5pt]
&\;\le\;
m_{3}\,(1+{\lvertx\rvert}^{\eta-\beta'})\end{aligned}$$ for some constant $m_{3}>0$. Combining and , we obtain $$\label{E5.23}
\int_{{\lvertz\rvert}>1}
\bigl({\lvertx+z\rvert}^\eta-{\lvertx\rvert}^\eta\bigr)\,\widetilde\uppi(x,z)\,{\mathrm{d}}{z}
\;\le\;
m_{3}\,(1+{\lvertx\rvert}^{\eta-\beta'})
-m_{1}\,{\lvertx\rvert}^{\eta-\alpha'}
\,,\qquad\text{if~} {\lvertx\rvert}>2\,.$$ Therefore, follows by , and the Lyapunov property holds.
\[P5.3\] Let $D$ be any bounded open set in ${\mathbb{R}^{d}}$ and $X$ be a Markov process associated with either ${\mathcal{I}}\in{\mathfrak{L}_{\alpha}}$, or a generator with kernel $\uppi$ as in Theorem \[T5.3\]. Suppose that for any compact set $K$ and any open set $G$, it holds that $sup_{x\in K}\operatorname{\mathbb{P}}_{x}(\tau(G^{c})>T)\to 0$ as $T\to \infty$. Then for any invariant probability measure $\nu$ of $X$ we have $\nu(D)>0$.
We argue by contradiction. Suppose $\nu(D)=0$. Let $x_{0}\in D$ and $r\in(0,1)$ be such that $B_{2r}(x_{0})\subset D$. By Lemma \[L3.5\] and Remark \[R3.2\] (see also [@bass-kassmann Proposition 3.1]), we have $$\sup_{x\in B_{r}(x_{0})}\;\operatorname{\mathbb{P}}_{x}\bigl(\tau(B_{r}(x))\le t\bigr)\;\le\;
\kappa\, t\,,\quad t>0\,,$$ for some constant $\kappa$ which depends on $r$. Therefore there exists $t_{0}>0$ such that $$\inf_{x\in B_{r}(x_{0})}\;\operatorname{\mathbb{P}}_{x}\bigl(\tau(B_{r}(x))\;\ge\;
t_{0}\bigr)\;\ge\; \frac{1}{2}\,.$$ Let $K$ be a compact set satisfying $\nu(K)>\frac{1}{2}$. By the hypothesis there exists $T_{0}>0$ such that $\sup_{x\in K}\operatorname{\mathbb{P}}_{x}(\tau(B^{c}_{r}(x_{0})>T)\le\nicefrac{1}{2}$ for all $T\ge T_{0}$. Hence $$\begin{aligned}
0\;=\;\nu(D)&\;\ge\; \frac{1}{T_{0}+t_{0}}
\int_{0}^{T_{0}+t_{0}}\int_{{\mathbb{R}^{d}}}\nu({\mathrm{d}}{x})P(t,x; B_{2r}(x_{0}))\,{\mathrm{d}}{t}
\\[5pt]
&\;=\;\frac{1}{T_{0}+t_{0}}\int_{{\mathbb{R}^{d}}}\nu({\mathrm{d}}{x})\,
\operatorname{\mathbb{E}}_{x}\Biggl[\int_{0}^{T_{0}+t_{0}}\bm1_{\{B_{2r}(x_{0})\}}(X_{s})\,{\mathrm{d}}{t}\Biggr]
\\[5pt]
&\;\ge\;\frac{1}{T_{0}+t_{0}}\int_{K}\nu({\mathrm{d}}{x})\,
\operatorname{\mathbb{E}}_{x}\Biggl[\bm1_{\{\tau(B^{c}_{r}(x_{0}))\le T_{0}\}}
\operatorname{\mathbb{E}}_{X_{\tau(B^{c}_{r}(x_{0}))}}
\biggl[\bm1_{\{\tau(B_{2r}(x_{0}))\ge t_{0}\}}\\[5pt]
&\mspace{400mu}
\int_{\tau(B_{r}(x_{0}))}^{T_{0}+t_{0}}
\bm1_{\{B_{2r}(x_{0})\}}(X_{s})\,{\mathrm{d}}{t}\biggr]\Biggr]
\\[5pt]
&\;\ge\;\frac{1}{T_{0}+t_{0}}\nu(K)
\inf_{x\in K}\;\operatorname{\mathbb{P}}_{x}\bigl(\tau(B^{c}_{r}(x_{0}))\le T_{0}\bigr)
\inf_{x\in B_{r}(x_{0})}\;\operatorname{\mathbb{P}}_{x}\bigl(\tau(B_{2r}(x_{0}))\ge t_{0}\bigr)\,t_{0}
\\[5pt]
&\;\ge\;\frac{1}{T_{0}+t_{0}}\frac{\nu(K)}{2}\,
\inf_{x\in B_{r}(x_{0})}\;\operatorname{\mathbb{P}}_{x}\bigl(\tau(B_{r}(x))\ge t_{0}\bigr)\,t_{0}
\\[5pt]
&\;\ge\; \frac{t_{0}}{T_{0}+t_{0}}\frac{\nu(K)}{4}\;>\;0\,.\end{aligned}$$ But this is a contradiction. Hence $\nu(D)>0$.
Mean recurrence times for weakly Hölder continuous kernels
----------------------------------------------------------
This section is devoted to the characterization of the mean hitting time of bounded open sets for Markov processes with generators studied in Section \[S3.2\]. The results hold for any bounded domain $D$ with $C^{2}$ boundary, but for simplicity we state them for the unit ball centered at $0$. As introduced earlier, we use the notation $B\equiv B_{1}$.
For nondegenerate continuous diffusions, it is well known that if some bounded domain $D$ is positive recurrent with respect to some point $x\in \Bar{D}^{c}$, then the process is positive recurrent and its generator satisfies the Lyapunov stability hypothesis in [@ari-bor-ghosh Lemma 3.3.4]). In Theorem \[T5.4\] we show that the same property holds for the class of operators $\mathfrak{I}_{\alpha}(\beta,\theta,\lambda)$.
\[T5.4\] Let ${\mathcal{I}}\in\mathfrak{I}_{\alpha}(\beta,\theta,\lambda)$. We assume that ${\mathcal{I}}$ satisfies the growth condition in . Moreover, we assume that $\operatorname{\mathbb{E}}_{x}[\tau(B^{c})]<\infty$ for some $x$ in $\Bar{B}^{c}$. Then $u(x){:=}\operatorname{\mathbb{E}}_{x}[\tau(B^{c})]$ is a viscosity solution to $$\begin{aligned}
{\mathcal{I}}u &\;=\;-1 \quad \text{in}~\Bar{B}^{c}\,,\\
u& \;=\;0 \quad \text{in}~\Bar{B}\,.\end{aligned}$$
In order to prove Theorem \[T5.4\] we need the following two lemmas.
\[L5.2\] Let ${\mathcal{I}}\in\mathfrak{I}_{\alpha}(\beta,\theta,\lambda)$, and $G$ a bounded open set containing $\Bar{B}$. Then there exist positive constants $r_{0}$ and $M_{0}$ depending only on $G$ such that $$\int_{\Bar{B}^{c}(x)}\operatorname{\mathbb{E}}_{z}[\tau(B^{c})]\,\frac{1}{{\lvertz\rvert}^{d+\alpha}}\,{\mathrm{d}}{z}
\;<\;\frac{M_{0}}{r^{\alpha}}\,\operatorname{\mathbb{E}}_{x}[\tau(B^{c})]$$ for every $r<\operatorname{dist}(x, B)\wedge r_{0}$, and for all $x\in G\setminus\Bar{B}$, such that $\operatorname{\mathbb{E}}_{x}[\tau(B^{c})]<\infty$.
Let $\Breve\tau{:=}\tau(B^{c})$, and $\Hat\tau_{r}{:=}\tau\bigl(B_{r}(x)\bigr)$. We select $r_{0}$ as in Lemma \[L4.1\], and without loss of generality we assume $r_{0}\le 1$. We have $$\label{E5.24}
\operatorname{\mathbb{E}}_{x}\Bigl[\bm1_{\{\Hat\tau_{r}<\Breve\tau\}}
\operatorname{\mathbb{E}}_{X_{\Hat\tau_{r}}}[\Breve\tau]\Bigr]
\;\le\; \operatorname{\mathbb{E}}_{x}[\Breve\tau]\,.$$ By Definition \[D-hkernel\] we have $$k(y,z)\;\ge\; \lambda^{-1}_{G}\;>\;0
\qquad \forall y\in B_{r_{0}}(x)\,.$$ Let $A\subset \Bar{B}^{c}_{r}(x)\cap \Bar{B}^{c}$ be any Borel set. Using Proposition \[levy-system\], we have $$\begin{aligned}
\operatorname{\mathbb{P}}_{x}(X_{\Hat\tau_{r}\wedge t}\in A)
&\;=\;\operatorname{\mathbb{E}}_{x}\left[\sum_{s\le \Hat\tau_{r}\wedge t}
\bm1_{\{X_{s-}\in
B_{r}(x),\, X_{s}\in A\}}\right]
\\[5pt]
&\;=\;\operatorname{\mathbb{E}}_{x}\biggl[\int_{0}^{\Hat\tau_{r}\wedge t}
\bm1_{\{X_{s}\in B_{r}(x) \}}
\int_{A}\uppi(X_{s}, z-X_{s})\,{\mathrm{d}}{z}\,{\mathrm{d}}{s}\biggr]
\\[5pt]
&\;\ge\; \lambda^{-1}_{G}\,
\operatorname{\mathbb{E}}_{x}\biggl[\int_{0}^{\Hat\tau_{r}\wedge t}
\int_{A}\frac{1}{{\lvertz\rvert}^{d+\alpha}}\,{\mathrm{d}}{z}\,{\mathrm{d}}{s}\biggr]
\\[5pt]
&\;\ge\; \lambda^{-1}_{G}\,
\operatorname{\mathbb{E}}_{x}[\Hat\tau_{r}\wedge t]
\int_{A}\frac{1}{{\lvertz\rvert}^{d+\alpha}}\,{\mathrm{d}}{z}\,.\end{aligned}$$ Letting $t\to\infty$, we obtain $$\label{E5.25}
\operatorname{\mathbb{P}}_{x}(X_{\Hat\tau_{r}}\in A)\;\ge\;
\lambda^{-1}_{G}\,
\operatorname{\mathbb{E}}_{x}[\Hat\tau_{r}]
\int_{A}\frac{1}{{\lvertz\rvert}^{d+\alpha}}\,{\mathrm{d}}{z}\,.$$ By Lemma \[L4.1\] it holds that $\operatorname{\mathbb{E}}_{x}[\Hat\tau_{r}]>
\kappa_{1}\, r^{\alpha}$ for some positive constant $\kappa_{1}$ which depends on $G$. Hence combining and we obtain $$\begin{aligned}
\lambda^{-1}_{G}\,\kappa_{1}\, r^{\alpha}\,
\int_{\Bar{B}^{c}(x)}\operatorname{\mathbb{E}}_{z}[\Breve\tau]\,\frac{1}{{\lvertz\rvert}^{d+\alpha}}\,{\mathrm{d}}{z}
&\;\le\;
\operatorname{\mathbb{E}}_{x}\bigl[\bm1_{\{X_{\Hat\tau_{r}}\in \Bar{B}^{c}\}}
\operatorname{\mathbb{E}}_{X_{\Hat\tau_{r}}}[\Breve\tau]\bigr]\\
&\;\le\; \operatorname{\mathbb{E}}_{x}[\Breve\tau]\,,\end{aligned}$$ where the first inequality follows by the standard approximation technique using step functions. This completes the proof.
Lemma \[L5.2\] of course implies that if $\operatorname{\mathbb{E}}_{x}[\tau(B^{c})]<\infty$ at some point $x\in\Bar{B}^{c}$ then $\operatorname{\mathbb{E}}_{x}[\tau(B^{c})]$ is finite a.e.-$x$. We can express the bound in Lemma \[L5.2\] without reference to Lemma \[L4.1\] as $$\int_{\Bar{B}^{c}(x)}\operatorname{\mathbb{E}}_{z}[\tau(B^{c})]\,\frac{1}{{\lvertz\rvert}^{d+\alpha}}\,{\mathrm{d}}{z}
\;\le\;
\lambda_{G}\,\frac{\operatorname{\mathbb{E}}_{x}[\tau(B^{c})]}
{\operatorname{\mathbb{E}}_{x}[\Bar{B}^{c}]}\,.$$ Now let $x^{\prime}$ be any point such that $\operatorname{dist}(x^{\prime},x)\wedge\operatorname{dist}(x^{\prime},B)= 2r$. We obtain $$\frac{\omega(r)}{{\lvert2r\rvert}^{d+\alpha}}\;\inf_{z\in B_{r}(x^{\prime})}\;
\operatorname{\mathbb{E}}_{z}[\tau(B^{c})]
\;\le\; \frac{M_{0}}{r^{\alpha}}\,
\operatorname{\mathbb{E}}_{x}[\tau(B^{c})]\,.$$ Therefore for some $y\in B_{r}(x^{\prime})$, we have $\operatorname{\mathbb{E}}_{y}[\tau(B^{c})]< C_{1}\, \operatorname{\mathbb{E}}_{x}[\tau(B^{c})]$. Applying Lemma \[L5.2\] once more we obtain $$\int_{{\mathbb{R}^{d}}}\operatorname{\mathbb{E}}_{x+z}[\tau(B^{c})]\,\frac{1}{(1+{\lvertz\rvert})^{d+\alpha}}\,{\mathrm{d}}{z}
\;\le\;
C_{0} \operatorname{\mathbb{E}}_{x}[\tau(B^{c})]\,,$$ with the constant $C_{0}$ depending only on $\operatorname{dist}(x,B)$ and the parameter $\lambda$, i.e., the local bounds on $k$. We introduce the following notation.
We say that $v\in L^{1}({\mathbb{R}^{d}},s)$ if $$\int_{{\mathbb{R}^{d}}}\frac{{\lvertv(z)\rvert}}{(1+{\lvertz\rvert})^{d+\alpha}}\,{\mathrm{d}}{z}\;<\;\infty\,.$$
Thus we have the following.
If $\operatorname{\mathbb{E}}_{x_{0}}[\tau(B^{c})]<\infty$ for some $x_{0}\in\Bar{B}^{c}$, then the function $u(x){:=}\operatorname{\mathbb{E}}_{x}[\tau(B^{c})]$ is in $L^{1}({\mathbb{R}^{d}},s)$.
In what follows, without loss of generality we assume that $\beta<s$. Then, by Theorem \[T3.1\], $u_{n}(x){:=}\operatorname{\mathbb{E}}_{x}\bigl[\tau(B_{n}\cap \Bar{B}^{c})\bigr]$ is the unique solution in $C^{\alpha+\beta}(B_{n}\setminus \Bar{B})\cap C(\Bar{B}_{n}\setminus B)$ of $$\label{E5.26}
\begin{split}
{\mathcal{I}}u_{n} &\;=\;-1 \quad \text{in}\quad B_{n}\cap \Bar{B}^{c}\,,\\[5pt]
u_{n}&\;=\;0 \quad \text{in}\quad B_{n}^{c}\cup B\,.
\end{split}$$ The following lemma provides a uniform barrier on the solutions $u_{n}$ near $B$.
\[L5.3\] Let ${\mathcal{I}}\in\mathfrak{I}_{\alpha}(\beta,\theta,\lambda)$, and $$\widetilde\tau_{n}{:=}\tau(B_{n}\cap \Bar{B}^{c})\,,\quad n\in{\mathbb{N}}\,.$$ Then, provided that $\sup_{x\in F}\,\operatorname{\mathbb{E}}_{x}[\tau(B^{c})]<\infty$ for all compact sets $F\subset\Bar{B}^{c}$, there exists a continuous, nonnegative radial function $\varphi$ that vanishes on $B$, and satisfies, for some $\eta>0$, $$\operatorname{\mathbb{E}}_{x}[\widetilde\tau_{n}]\le \varphi(x)\qquad
\forall\,x\in B_{1+\eta}\setminus B\,,\quad\forall\,n>1\,.$$
The proof relies on the construction of barrier. Let $\Hat{k}(x,z)=k(x,z)-k(x,0)$. By Lemma \[L6.2\], for $q\in (\nicefrac{\alpha-1}{2}, \nicefrac{\alpha}{2})$, there exists a constant $c_{0}>0$ such that for $\varphi_{q}(x){:=}[(1-{\lvertx\rvert})^{+}]^{q}$ we have $$(-\Delta)^{\nicefrac{\alpha}{2}}\varphi_{q}(x)
\;>\; c_{0}\,(1-{\lvertx\rvert})^{q-\alpha}\qquad \forall\, x\in B\,.$$ We recall the Kelvin transform from [@RosOton-Serra]. Define $\Hat{\varphi}(x)={\lvertx\rvert}^{\alpha-d}\varphi_{q}(x^*)$ where $x^*{:=}\frac{x}{{\lvertx\rvert}^{2}}$. Then by [@RosOton-Serra Proposition A.1] there exists a positive constant $c_{1}$ such that $$(-\Delta)^{\nicefrac{\alpha}{2}}\Hat{\varphi}(x)
\;>\;c_{1}\,({\lvertx\rvert}-1)^{q-\alpha}\qquad \forall\, x\in B_{2}\setminus\Bar{B}\,.$$ We restrict $\Hat{\varphi}$ outside a large compact set so that it is bounded on ${\mathbb{R}^{d}}$. By $\widehat{\mathcal{I}}$ we denote the operator $$\widehat{\mathcal{I}}f(x)\;=\;b(x)\cdot{\nabla}{f}(x) +
\int_{{\mathbb{R}^{d}}}{\mathfrak{d}}f(x;z)\,
\frac{\Hat{k}(x,z)}{{\lvertz\rvert}^{d+\alpha}}\,{\mathrm{d}}{z}\,.$$ It is clear that ${\lvert{\nabla}\Hat\varphi(x)\rvert}\le c_{2} ({\lvertx\rvert}-1)^{q-1}$ for all ${\lvertx\rvert}\in (1,2)$, for some constant $c_{2}$. Also, using the fact that $\Hat{\varphi}$ is Hölder continuous of exponent $q$ and we obtain $${\biggl\lvert\int_{{\mathbb{R}^{d}}}{\mathfrak{d}}\Hat\varphi(x;z)\,
\frac{\Hat{k}(x,z)}{{\lvertz\rvert}^{d+\alpha}}\,{\mathrm{d}}{z}\biggr\rvert}\;\le\; c_{3}
({\lvertx\rvert}-1)^{q+\theta-\alpha}\qquad
\forall\,x\in B_{2}\setminus\Bar{B}\,,$$ for some constant $c_{3}$. Hence $$\begin{aligned}
{\bigl\lvert\widehat{\mathcal{I}}\Hat\varphi (x)\bigr\rvert} \;\le\; c_{4}\,
\bigl({\lvertx\rvert}-1\bigr)^{(q-1)\wedge(q+\theta-\alpha)}\,,
\quad \text{for}~ x\in B_{2}\setminus\Bar{B}\,,\end{aligned}$$ for some constant $c_{4}$. Since $\theta>0$, $\alpha>1$, and ${\mathcal{I}}= \widehat{\mathcal{I}}- k(x,0) (-\Delta)^{\nicefrac{\alpha}{2}}$, it follows that we can find $\eta$ small enough such that $${\mathcal{I}}\Hat\varphi(x)\;<\; -4\,, \quad \text{for}
~ x\in B_{1+\eta}\setminus\Bar{B}\,.$$ Let $K$ be a compact set containing $B_{1+\eta}$. We define $$\Tilde{\varphi}(x)\;=\;\Hat{\varphi}(x)\,\bm1_{K}(x)+\operatorname{\mathbb{E}}_{x}[\tau(B^{c})]
\,\bm1_{K^{c}}(x)\,.$$ Since the hypotheses of Lemma \[L5.2\] are met, we conclude that $\bm1_{K^{c}}(x)\operatorname{\mathbb{E}}_{x}[\tau(B^{c})]$ is integrable with respect to the kernel $\uppi$. For $x\in B_{1+\eta}\setminus\Bar{B}$, we obtain $$\begin{aligned}
{\mathcal{I}}\Tilde{\varphi}(x)&\;<\; -4+
\int_{{\mathbb{R}^{d}}}\bigl(\operatorname{\mathbb{E}}_{x+z}[\tau(B^{c})]\,
-\Hat{\varphi}(x+z)\bigr)\,\bm1_{K^{c}}(x+z)\,
\uppi(x,z)\,{\mathrm{d}}{z}\\[5pt]
&\;=\; -4
+ \int_{K^{c}}\operatorname{\mathbb{E}}_{z}[\tau(B^{c})]\,
\frac{\uppi(x,z-x)}{\uppi(x,z)}\uppi(x,z)\,{\mathrm{d}}{z}
- \int_{{\mathbb{R}^{d}}}\Hat{\varphi}(x+z)\,\bm1_{K^{c}}(x+z)\,
\uppi(x,z)\,{\mathrm{d}}{z}\,.\end{aligned}$$ Since the kernel is comparable to ${\lvertz\rvert}^{-d-\alpha}$ on any compact set, we may choose $K$ large enough and use Lemma \[L5.2\] to obtain $${\mathcal{I}}\Tilde{\varphi}(x)\;<\; -2
\qquad \forall\,x\in B_{1+\eta}\setminus\Bar{B}\,.$$ Let $$\psi(x)\;{:=}\;\biggl(1\,\vee\sup_{z\in K\setminus B_{1+\eta}}
\operatorname{\mathbb{E}}_{z}[\tau(B^{c})]\biggr)\,\biggl(1\,\vee\sup_{z\in K\setminus B_{1+\eta}}
\frac{1}{\Tilde\varphi(z)}\biggr)\,\Tilde\varphi(x)\,.$$ Then, ${\mathcal{I}}\psi<-2$ on $B_{1+\eta}\setminus\Bar{B}$, while $\psi\ge u_{n}$ on $B_{1+\eta}^{c}\cup B$. Therefore, by the comparison principle, $u_{n}\le \psi$ on $B_{1+\eta}\setminus\Bar{B}$ for all $n\in{\mathbb{N}}$ and the proof is complete.
Consider the sequence of solutions $\{u_{n}\}$ defined in . First we note that $u_{n}(x)\le \operatorname{\mathbb{E}}_{x}[\tau(B^{c})]$ for all $x$. Clearly $u_{n+1}-u_{n}$ is bounded, nonnegative and harmonic in $B_{n}\setminus \Bar{B}$. By Theorem \[T4.1\] the operator ${\mathcal{I}}$ has the Harnack property. Therefore $$\sup_{x\in F}\;\sum_{n\ge 1}\bigl(u_{n+1}(x)-u_{n}(x)\bigr)\;<\;\infty$$ for any compact subset $F$ in $\Bar{B}^{c}$. Hence Lemma \[L3.3\] combined with Fatou’s lemma implies that $\sup_{x\in F}\,\operatorname{\mathbb{E}}_{x}[\tau(B^{c})]<\infty$ for any compact set $F\subset\Bar{B}^{c}$.
We write $$u_{n} \;=\; u_{1}+\sum_{m=1}^{n-1}\bigl(u_{m+1}(x)-u_{m}(x)\bigr)\,,$$ and use the Harnack property once more to conclude that $u_{n}\nearrow u$ uniformly over compact subsets of $\Bar{B}^{c}$. Since $u\le \varphi$ in a neighborhood of $\partial B$ by Lemma \[L5.3\], and $\varphi$ vanishes on $\partial B$, it follows that $u\in C({\mathbb{R}^{d}})$. That $u$ is a viscosity solution follows from the fact that $u_{n}\to u$ uniformly over compacta as $n\to\infty$ and Lemma \[L5.2\].
The Dirichlet problem for weakly Hölder continuous kernels {#S6}
==========================================================
This section is devoted to the study of the Dirichlet problem $$\label{ea6.1}
\begin{split}
{\mathcal{I}}u(x)
&\;=\;f(x) \quad \text{in}~D\,,\\
u & \;=\; 0 \quad \text{in}~D^{c}\,,
\end{split}$$ where ${\mathcal{I}}\in\mathfrak{I}_{\alpha}(\beta,\theta,\lambda)$, $f$ is Hölder continuous with exponent $\beta$, and $D$ is a bounded open set with a $C^{2}$ boundary. In this section, it is convenient to use $s\equiv\frac{\alpha}{2}$ as the parameter reflecting the order of the kernel. Throughout this section, we assume $s>\nicefrac{1}{2}$.
Recall the definition of weighted Hölder norms in Section \[S1.1\]. We start with the following lemma.
\[L6.1\] Let $D$ be a $C^{2}$ bounded domain in ${\mathbb{R}^{d}}$, and $r\in(0,s]$. Suppose $k:{\mathbb{R}^{d}}\times{\mathbb{R}^{d}}\to {\mathbb{R}}$ and the constants $\beta\in(0,1)$, $\theta\in\bigl(0,(2s-1)\wedge\beta\bigr)$, and $\lambda_{D}>0$ satisfy parts and of Definition \[D-hkernel\]. We define $$\begin{split}
\Tilde{k}(x,z)&\;{:=}\; c(d,2s)\,
\biggl(\frac{k(x,z)}{k(x,0)}-1\biggr)\,,\\[5pt]
{{\mathcal{H}}}[v](x)&\;{:=}\;\int_{{\mathbb{R}^{d}}}{\mathfrak{d}}v(x;z)\,
\frac{\Tilde{k}(x,z)}{{\lvertz\rvert}^{d+2s}}\,{\mathrm{d}}{z}\,,
\end{split}$$ where $c(d,2s)=c(d,\alpha)$ is the normalization constant of the fractional Laplacian.
Suppose that either of the following assumptions hold:
- $\beta\le r$.
- $\beta\in(r,1)$ and $\frac{\Tilde{k}(x,z)}{{\lvertz\rvert}^{\theta}}$ is bounded on $(x,z)\in D\times {\mathbb{R}^{d}}$, or, equivalently, it satisfies $$\label{ea6.2}
{\lvertk(x,z)-k(x,0)\rvert}\;\le\; \Tilde{\lambda}_{D}\, {\lvertz\rvert}^{\theta}\qquad
\forall x\in D\,, \;\forall z\in {\mathbb{R}^{d}}\,,$$ for some positive constant $\Tilde{\lambda}_{D}$.
Then, if $v\in\mathscr{C}_{2s-\theta}^{(-r)}(D)$, we have $${\bigl[\kern-0.75ex\bigl[\kern0.1ex {{\mathcal{H}}}[v]\kern0.1ex\bigr]\kern-0.75ex\bigr]}^{(2s-r-\theta)}_{0;D}\;\le\; M_{0}\,
{\bm\lvert\kern-0.50ex\bm\lvert\kern0.1ex v
\kern0.1ex\bm\rvert\kern-0.50ex\bm\rvert}^{(-r)}_{2s-\theta;D}\,,$$ and if $v\in\mathscr{C}_{2s+\beta-\theta}^{(-r)}(D)$, it holds that ${{\mathcal{H}}}[v]\in\mathscr{C}_{\beta}^{(2s-r-\theta)}(D)$, and $$\label{ea6.3}
{\bigl\lvert\kern-0.72ex\bigl\rvert\kern-0.55ex
\bigl\lvert\kern-0.72ex\bigl\rvert\kern0.1ex {{\mathcal{H}}}[v]
\kern0.1ex\bigl\lvert\kern-0.72ex\bigl\rvert\kern-0.55ex
\bigl\lvert\kern-0.72ex\bigl\rvert}^{(2s-r-\theta)}_{\beta;D}\;\le\;
M_{1}\, {\bm\lvert\kern-0.50ex\bm\lvert\kern0.1ex v
\kern0.1ex\bm\rvert\kern-0.50ex\bm\rvert}^{(-r)}_{2s+\beta-\theta;D}$$ for some constants $M_{0}$ and $M_{1}$ which depend only on $d$, $s$, $\beta$, $r$, and $D$.
Moreover, over a set of parameters of the form $\{(r,\beta)\,\colon r\in(\varepsilon,1),\, \beta\in(0,1)\}$, constants $M_{0}$ and $M_{1}$ can be selected which do not depend on $\beta$ or $r$, but only on $\varepsilon>0$.
Let $x\in D$, and define $R=\frac{d_{x}}{4}$. We suppose that $R<1$. It is clear that $\Tilde{k}$ satisfies , and that it is Hölder continuous. Abusing the notation, we’ll use the same symbol $\lambda_{D}$ as a constant in the estimates. We have, $$\label{E6.4}
{\bigl\lvert{\mathfrak{d}}{v}(x;z)\bigr\rvert} \;\le\;
{\lvertz\rvert}^{2s-\theta}\, R^{r+\theta-2s}\,
{[\kern-0.45ex[\kern0.1ex v\kern0.1ex]\kern-0.45ex]}^{(-r)}_{2s-\theta;D}\qquad \forall\, z\in B_{R}\,.$$ Also, since ${\lvertz\rvert}\ge R$ on $B^{c}_{R}$, we obtain $$\begin{aligned}
\label{ea6.5}
{\bigl\lvert{\mathfrak{d}}{v}(x;z)\bigr\rvert} &\;\le\;
\Bigl({\lvertz\rvert}^{r}\, {[\kern-0.45ex[\kern0.1ex v\kern0.1ex]\kern-0.45ex]}^{(-r)}_{r;D}
+{\lvertz\rvert}\,R^{r-1}\,{[\kern-0.45ex[\kern0.1ex v\kern0.1ex]\kern-0.45ex]}^{(-r)}_{1;D}\Bigr)\,{\bm1_{\{{\lvertz\rvert}\le 1\}}}+ 2\, {\lVertv\rVert}_{C(D)}\,\bm 1_{\{{\lvertz\rvert}>1\}}\nonumber\\[5pt]
&\;\le\;
\bigl({\lvertz\rvert}\wedge1\bigr)^{2s-\theta}\, R^{r+\theta-2s}
\Bigl({[\kern-0.45ex[\kern0.1ex v\kern0.1ex]\kern-0.45ex]}^{(-r)}_{r;D} +{[\kern-0.45ex[\kern0.1ex v\kern0.1ex]\kern-0.45ex]}^{(-r)}_{1;D}\Bigr)
+ 2\, {\lVertv\rVert}_{C(D)}\,\bm 1_{\{{\lvertz\rvert}>1\}}\end{aligned}$$ for all $z\in B^{c}_{R}$. Integrating, using , and –, as well as the Hölder interpolation inequalities, we obtain $${\lvert{{\mathcal{H}}}[v](x)\rvert}\;\le\;c_{1}\, (4\,d_{x})^{r+\theta-2s}\,
{\bm\lvert\kern-0.50ex\bm\lvert\kern0.1ex v
\kern0.1ex\bm\rvert\kern-0.50ex\bm\rvert}^{(-r)}_{2s-\theta;D}\qquad\forall x\in D\,,$$ for some constant $c_{1}$. Therefore, for some constant $M_{0}$, we have $$\label{ea6.6}
{\bigl[\kern-0.75ex\bigl[\kern0.1ex {{\mathcal{H}}}[v]\kern0.1ex\bigr]\kern-0.75ex\bigr]}^{(2s-r-\theta)}_{0;D}\;\le\; M_{0}\,
{\bm\lvert\kern-0.50ex\bm\lvert\kern0.1ex v
\kern0.1ex\bm\rvert\kern-0.50ex\bm\rvert}^{(-r)}_{2s-\theta;D}\,.$$
Next consider two points $x\,,y\in D$. If ${\lvertx-y\rvert}\ge4 d_{xy}$, then provides a suitable estimate. Indeed, if $x,\,y\in D$ are such $4 d_{xy} \le {\lvertx-y\rvert}$, then, for any $r$ we have $$\begin{aligned}
d_{xy}^{2s-r-\theta}\,d_{xy}^{\beta}\,
\frac{{\lvert{{\mathcal{H}}}[v](x)-{{\mathcal{H}}}[v](y)\rvert}}{{\lvertx-y\rvert}^{\beta}}
&\;\le\; \frac{1}{4^{\beta}}\,d_{xy}^{2s-r-\theta}\,
{\lvert{{\mathcal{H}}}[v](x)-{{\mathcal{H}}}[v](y)\rvert}\\[5pt]
&\;\le\;\frac{1}{4^{\beta}}\,d_{x}^{2s-r-\theta}\,{\lvert{{\mathcal{H}}}[v](x)\rvert}
+\frac{1}{4^{\beta}}\,d_{y}^{2s-r-\theta}\,{\lvert{{\mathcal{H}}}[v](y)\rvert}\\[5pt]
&\;\le\;\frac{2 M_{0}}{4^{\beta}}\,{\bm\lvert\kern-0.50ex\bm\lvert\kern0.1ex v
\kern0.1ex\bm\rvert\kern-0.50ex\bm\rvert}^{(-r)}_{2s-\theta;D}\,.\end{aligned}$$
So it suffices to consider the case ${\lvertx-y\rvert}< 4 d_{xy}$. Therefore, we may suppose that $x$ is as above and that $y\in B_{R}(x)$. Then $d_{xy}\le 4R$. With ${{\widetilde{\uppi}}}(x,z){:=}\frac{\Tilde{k}(x,z)}{{\lvertz\rvert}^{d+2s}}$, we write $$\begin{aligned}
F(x,y;z)&\;{:=}\;{\mathfrak{d}}{v}(x;z)\,{{\widetilde{\uppi}}}(x,z)-{\mathfrak{d}}{v}(y;z)\,{{\widetilde{\uppi}}}(y,z)\\[5pt]
&\;=\;F_{1}(x,y;z)+F_{2}(x,y;z)\,,\end{aligned}$$ with $$\begin{aligned}
F_{1}(x,y;z)&\;{:=}\;\Bigl({\mathfrak{d}}{v}(x;z)+{\mathfrak{d}}{v}(y;z)\Bigr)\,
\frac{{{\widetilde{\uppi}}}(x,z)-{{\widetilde{\uppi}}}(y,z)}{2}\,,\\[5pt]
F_{2}(x,y;z)&\;{:=}\;\Bigl({\mathfrak{d}}{v}(x;z)-{\mathfrak{d}}{v}(y;z)\Bigr)\,
\frac{{{\widetilde{\uppi}}}(x,z)+{{\widetilde{\uppi}}}(y,z)}{2}\,.\end{aligned}$$ We modify the estimate in , and write $$\begin{aligned}
{\bigl\lvert{\mathfrak{d}}{v}(x;z)+{\mathfrak{d}}{v}(y;z)\bigr\rvert} &\;\le\;
2\,{\lvertz\rvert}^{\gamma_{0}}\,
R^{r-\gamma_{0}}\,{[\kern-0.45ex[\kern0.1ex v\kern0.1ex]\kern-0.45ex]}^{(-r)}_{\gamma_{0};D}\,,
\qquad\text{if~} z\in B_{R}\,,\\[5pt]
\intertext{with $\gamma_{0}=(2s+\beta-\theta)\wedge (s+1)$, and}
{\bigl\lvert{\mathfrak{d}}{v}(x;z)+{\mathfrak{d}}{v}(y;z)\bigr\rvert} &\;\le\;
2\,\Bigl({\lvertz\rvert}^{r}\, {[\kern-0.45ex[\kern0.1ex v\kern0.1ex]\kern-0.45ex]}^{(-r)}_{r;D}
+{\lvertz\rvert}\,R^{r-1}\,{[\kern-0.45ex[\kern0.1ex v\kern0.1ex]\kern-0.45ex]}^{(-r)}_{1;D}\Bigr)\,{\bm1_{\{{\lvertz\rvert}\le 1\}}}+ 4 {\lVertv\rVert}_{C(D)}\,\bm 1_{\{{\lvertz\rvert}>1\}}\,,\end{aligned}$$ if $z\in B^{c}_{R}$. We use the Hölder continuity of $x\mapsto\Tilde k(x,\,\cdot\,)$ to obtain $$\int_{{\mathbb{R}^{d}}} F_{1}(x,y;z)\,{\mathrm{d}}{z}
\;\le\; c_{2}\,R^{r-2s}\, {\lvertx-y\rvert}^{\beta}\,
{\bm\lvert\kern-0.50ex\bm\lvert\kern0.1ex v
\kern0.1ex\bm\rvert\kern-0.50ex\bm\rvert}^{(-r)}_{\gamma_{0};D}$$ for some constant $c_{2}$. We write this as $$\begin{aligned}
\label{ea6.7}
R^{2s-r-\theta}\,R^{\beta}\,
\frac{\int_{{\mathbb{R}^{d}}} F_{1}(x,y;z)\,{\mathrm{d}}{z}}{{\lvertx-y\rvert}^{\beta}}
&\;\le\; R^{2s-r-\beta}\,R^{\beta}\,
\frac{\int_{{\mathbb{R}^{d}}} F_{1}(x,y;z)\,{\mathrm{d}}{z}}{{\lvertx-y\rvert}^{\beta}}\nonumber\\[5pt]
&\;\le\; c_{2}\,{\bm\lvert\kern-0.50ex\bm\lvert\kern0.1ex v
\kern0.1ex\bm\rvert\kern-0.50ex\bm\rvert}^{(-r)}_{\gamma_{0};D}\,.\end{aligned}$$ For $F_{2}$, we use $${\mathfrak{d}}{v}(x;z) \;=\; z\cdot \int_{0}^{1}
\bigl({\nabla}v(x+t z) - {\nabla}v(x)\bigr)\,{\mathrm{d}}{t}\,,$$ combined with the following fact: If $f\in C^{\gamma}(B)$ for $\gamma\in(0,1]$ and $x$, $y$, $x+z$, $y+z$ are points in $B$ and $\delta\in(0,\gamma)$, then adopting the notation $\varDelta f_{x}(z) {:=}f(x+z)-f(x)$, we obtain by Young’s inequality, that $$\begin{aligned}
\frac{{\lvert\varDelta f_{x}(z) - \varDelta f_{y}(z)\rvert}}
{{\lvertz\rvert}^{\gamma-\delta}{\lvertx-y\rvert}^{\delta}}
&\;\le\;
\frac{\gamma-\delta}{\gamma}\,
\frac{{\lvert\varDelta f_{x}(z)\rvert} + {\lvert\varDelta f_{y}(z)\rvert}}
{{\lvertz\rvert}^{\gamma}}
+\frac{\delta}{\gamma}\,
\frac{{\lvert\varDelta f_{x+z}(y-x)\rvert} + {\lvert\varDelta f_{x}(y-x)\rvert}}
{{\lvertx-y\rvert}^{\gamma}}\\[5pt]
&\;\le\; 2 [f]_{\gamma; B}\,.\end{aligned}$$ The same inequality also holds for $\gamma\in(1,2)$ and $\delta\in(\gamma-1, 1)$. For this we use $$\begin{aligned}
\frac{{\lvert\varDelta f_{x}(z) - \varDelta f_{y}(z)\rvert}}
{{\lvertz\rvert}^{\gamma-\delta}{\lvertx-y\rvert}^{\delta}}
&\;\le\;
\frac{1-\delta}{2-\gamma}\,
\frac{{\lvertz\rvert}\,{\Bigl\lvert\int_{0}^{1}
\bigl({\nabla}f(x+t z) - {\nabla}f(y+t z)\bigr)\,{\mathrm{d}}{t}\Bigr\rvert}}
{{\lvertx-y\rvert}^{\gamma-1}\,{\lvertz\rvert}}\nonumber\\[5pt]
&+\frac{1+\delta-\gamma}{2-\gamma}\,
\frac{{\lvertx-y\rvert}\,{\Bigl\lvert\int_{0}^{1}
\bigl({\nabla}f(y+z+t (x-y)) - {\nabla}f(y+t (x-y))\bigr)\,{\mathrm{d}}{t}\Bigr\rvert}}
{{\lvertz\rvert}^{\gamma-1}\,{\lvertx-y\rvert}}\end{aligned}$$ Therefore, in either of the cases (i) or (ii) we obtain, $${\bigl\lvert{\nabla}v(x+t z) - {\nabla}v(x) -
{\nabla}v(y+t z) + {\nabla}v(y)\bigr\rvert}
\;\le\; 2{\lvertt z\rvert}^{2s-\theta-1}{\lvertx-y\rvert}^{\beta}
\, [{\nabla}v]_{2s-\theta -1+ \beta;\,B_{2R}(x)}$$ for $t\in[0,1]$, and $$\label{ea6.8}
{\bigl\lvert{\mathfrak{d}}{v}(x;z)-{\mathfrak{d}}{v}(y;z)\bigr\rvert} \;\le\;
\frac{2}{2s-\theta}\;
{\lvertz\rvert}^{2s-\theta}\,{\lvertx-y\rvert}^{\beta}\,
R^{r+\theta-\beta-2s}\,{[\kern-0.45ex[\kern0.1ex v\kern0.1ex]\kern-0.45ex]}^{(-r)}_{2s+\beta-\theta;D}
\qquad\forall\,z\in B_{R}\,.$$ Concerning the integration on $B^{c}_{R}$, we use $$\begin{aligned}
\label{ea6.9}
{\bigl\lvertv(x)-v(y)&- z\cdot\bigl({\nabla}v(x)-{\nabla}v(y)\bigr){\bm1_{\{{\lvertz\rvert}\le 1\}}}\bigr\rvert}\nonumber\\[5pt]
&\;\le\;
{\lvertx-y\rvert}^{\beta\vee r}\,d_{xy}^{r-\beta\vee r}\,
{[\kern-0.45ex[\kern0.1ex v\kern0.1ex]\kern-0.45ex]}^{(-r)}_{\beta\vee r;D}
+\bigl({\lvertz\rvert}\wedge1\bigr)\,{\lvertx-y\rvert}^{\beta}\,d_{xy}^{r-\beta-1}\,
{[\kern-0.45ex[\kern0.1ex v\kern0.1ex]\kern-0.45ex]}^{(-r)}_{1+\beta;D}\nonumber\\[5pt]
&\;\le\;
c_{3}\,\bigl({\lvertz\rvert}\wedge1\bigr)^{2s-\theta}\,{\lvertx-y\rvert}^{\beta}\,
R^{r+\theta-\beta-2s}\,
{\bm\lvert\kern-0.50ex\bm\lvert\kern0.1ex v
\kern0.1ex\bm\rvert\kern-0.50ex\bm\rvert}^{(-r)}_{1+\beta;D}\qquad \forall\, z\in B^{c}_{R}\,,\end{aligned}$$ for some constant $c_{3}$, and $$\label{ea6.10}
{\lvertv(x+z)-v(y+z)\rvert}\;\le\;
{\lvertx-y\rvert}^{\beta\vee r}\,
(d_{x+z}\wedge d_{y+z})^{r-\beta\vee r}\,
{[\kern-0.45ex[\kern0.1ex v\kern0.1ex]\kern-0.45ex]}^{(-r)}_{\beta\vee r;D}
\qquad\forall\,z\in B^{c}_{R}\,.$$ Integrating the terms on the right hand side of – is straightforward. Doing so, and using the fact that $1+\beta<2s+\beta-\theta$, one obtains the desired estimate.
Concerning the integral of ${\lvertv(x+z)-v(y+z)\rvert}$ on $B^{c}_{R}$, we distinguish between the cases (i) and (ii). Let ${{\widetilde{\uppi}}}(z){:=}\frac{{\lvert{{\widetilde{\uppi}}}(x,z)+{{\widetilde{\uppi}}}(y,z)\rvert}}{2}$. In case (i) we have $$\begin{aligned}
\label{ea6.11}
\int_{B^{c}_{R}} {\lvertv(x+z)-&v(y+z)\rvert}\,{{\widetilde{\uppi}}}(z)\,{\mathrm{d}}{z}
\nonumber\\[5pt]
&\;\le\;
{\lvertx-y\rvert}^{r}\, {[\kern-0.45ex[\kern0.1ex v\kern0.1ex]\kern-0.45ex]}^{(-r)}_{r;D}
\int_{B^{c}_{R}} \,{{\widetilde{\uppi}}}(z)\,{\mathrm{d}}{z} \nonumber\\[5pt]
&\;\le\;
{\lvertx-y\rvert}^{\beta}\,R^{r-\beta}\,R^{\theta-2s}\,
{[\kern-0.45ex[\kern0.1ex v\kern0.1ex]\kern-0.45ex]}^{(-r)}_{r;D}
\int_{{\mathbb{R}^{d}}}
\bigl({\lvertz\rvert}\wedge\operatorname{diam}(D)\bigr)^{2s-\theta}\,{{\widetilde{\uppi}}}(z)\,{\mathrm{d}}{z}\,,\end{aligned}$$ where we use the fact that ${\lvertz\rvert}>R$ on $B^{c}_{R}$. In case (ii) the integral is estimated over disjoint sets. We define $${\mathcal{Z}}_{xy}(a)\;{:=}\;\{z\in{\mathbb{R}^{d}}\,\colon d_{x+z}\wedge d_{y+z} < a\}
\qquad\text{for~}a\in(0,R)\,.$$ Since $d_{x+z}\wedge d_{y+z} \in[R,\operatorname{diam}(D)]$ for $x\in{\mathcal{Z}}^{c}_{xy}(R)$, integration is straightforward, after replacing $(d_{x+z}\wedge d_{y+z})^{r-\beta}$ in with $R^{r-\beta}$. Thus, similarly to , we obtain $$\begin{aligned}
\label{ea6.12}
\int_{B^{c}_{R}\,\cap\,{\mathcal{Z}}^{c}_{xy}(R)}& {\lvertv(x+z)-v(y+z)\rvert}\,{{\widetilde{\uppi}}}(z)\,{\mathrm{d}}{z}
\nonumber\\[5pt]
&\;\le\;
{\lvertx-y\rvert}^{\beta}\, R^{r-\beta}\, {[\kern-0.45ex[\kern0.1ex v\kern0.1ex]\kern-0.45ex]}^{(-r)}_{\beta;D}
\int_{B^{c}_{R}\,\cap\,{\mathcal{Z}}^{c}_{xy}(R)} \,{{\widetilde{\uppi}}}(z)\,{\mathrm{d}}{z} \nonumber\\[5pt]
&\;\le\;
{\lvertx-y\rvert}^{\beta}\, R^{r+\theta-\beta-2s}\,
{[\kern-0.45ex[\kern0.1ex v\kern0.1ex]\kern-0.45ex]}^{(-r)}_{\beta;D}
\int_{{\mathbb{R}^{d}}}
\bigl({\lvertz\rvert}\wedge\operatorname{diam}(D)\bigr)^{2s-\theta}\,{{\widetilde{\uppi}}}(z)\,{\mathrm{d}}{z}\,.\end{aligned}$$ Since ${\mathcal{Z}}_{xy}(R)\subset B^{c}_{R}$, it remains to compute the integral on ${\mathcal{Z}}_{xy}(R)$. Recall the definition of $D_{\varepsilon}$ in . We also define for $\varepsilon>0$, $$\widetilde{D}(\varepsilon)\;=\;\{z\in D:\,\operatorname{dist}(z,\partial D)\ge \varepsilon\}\,.$$ In other words $\widetilde{D}(\varepsilon) = (D^{c})^{c}_{\varepsilon}$. We’ll make use of the following simple fact: There exists a constant $C_{0}$, such that for all $x\in D$ and positive constants $R$ and $\varepsilon$ which satisfy $0<\varepsilon\le R$ and $d_{x}\ge 3R$, it holds that $$\label{ea6.13}
\int_{x+z\,\in D_{\varepsilon}\setminus \widetilde{D}(\varepsilon)}
\frac{{\mathrm{d}}{z}}{{\lvertz\rvert}^{d}}
\;\le\; \frac{C_{0}\, \varepsilon}{R}\,.$$ Observe that the support of ${\lvertv(x+z)-v(y+z)\rvert}$ in ${\mathcal{Z}}_{xy}(R)$ is contained in the disjoint union of the sets $$\begin{aligned}
\widetilde{{\mathcal{Z}}}_{xy}(R)&\;{:=}\;\bigl\{z\in {\mathcal{Z}}_{xy}(R)\,\colon
d_{x+z}\wedge d_{y+z} > 0\bigr\}\,,\\
\intertext{and}
\widehat{{\mathcal{Z}}}_{xy}&\;{:=}\;\bigl\{z\in {\mathbb{R}}^{d}\,\colon x+z\in
D_{{\lvertx-y\rvert}}\setminus D
~\text{or}~y+z\in D_{{\lvertx-y\rvert}}\setminus D\bigr\}\,.\end{aligned}$$ We also have the bound ${\lvertv(x+z)-v(y+z)\rvert}\le {\lvertx-y\rvert}^{r}{[\kern-0.45ex[\kern0.1ex v\kern0.1ex]\kern-0.45ex]}^{(-r)}_{r;D}$ for $z\in\widehat{{\mathcal{Z}}}_{xy}$. Therefore, using , we obtain $$\begin{aligned}
\label{ea6.14}
\int_{\widehat{{\mathcal{Z}}}_{xy}} {\lvertv(x+z)-v(y+z)\rvert}\,{{\widetilde{\uppi}}}(z)\,{\mathrm{d}}{z}
&\;\le\;
{\lvertx-y\rvert}^{r}{[\kern-0.45ex[\kern0.1ex v\kern0.1ex]\kern-0.45ex]}^{(-r)}_{r;D}\,R^{\theta-2s}\,
\int_{\widehat{{\mathcal{Z}}}_{xy}}{\lvertz\rvert}^{2s-\theta}\,{{\widetilde{\uppi}}}(z)\,{\mathrm{d}}{z} \nonumber\\[5pt]
&\;\le\;
{\lvertx-y\rvert}^{r}{[\kern-0.45ex[\kern0.1ex v\kern0.1ex]\kern-0.45ex]}^{(-r)}_{r;D}\,R^{\theta-2s}\,
\int_{\widehat{{\mathcal{Z}}}_{xy}}
\frac{{\mathrm{d}}{z}}{{\lvertz\rvert}^{d}} \nonumber\\[5pt]
&\;\le\;
2\,\Tilde{\lambda}_{D}\,C_{0}\,{\lvertx-y\rvert}^{r+1}
{[\kern-0.45ex[\kern0.1ex v\kern0.1ex]\kern-0.45ex]}^{(-r)}_{r;D}\,R^{\theta-2s}\,R^{-1}\nonumber\\[5pt]
&\;\le\;
2\,\Tilde{\lambda}_{D}\,C_{0}\,{\lvertx-y\rvert}^{\beta}\,R^{r+\theta-\beta-2s}\,
{[\kern-0.45ex[\kern0.1ex v\kern0.1ex]\kern-0.45ex]}^{(-r)}_{r;D}\,.\end{aligned}$$ In order to evaluate the integral over $\widetilde{{\mathcal{Z}}}_{xy}(R)$, we define $$G(z) \;{:=}\; \frac{{\lvertv(x+z)-v(y+z)\rvert}}{{\lvertx-y\rvert}^{\beta}\,
{[\kern-0.45ex[\kern0.1ex v\kern0.1ex]\kern-0.45ex]}^{(-r)}_{\beta;D}}\,.$$ By we have $$\bigl\{z\in \widetilde{{\mathcal{Z}}}_{xy}(R)\,\colon G(z)> h\bigr\}
\;\subset\;
\bigl\{z\in {\mathbb{R}^{d}}\,\colon x+z\in
\widetilde{D}^{c}\bigl(h^{\frac{-1}{\beta-r}}\bigr)\bigr\}\,\cup\,
\bigl\{z\in {\mathbb{R}^{d}}\,\colon y+z\in
\widetilde{D}^{c}\bigl(h^{\frac{-1}{\beta-r}}\bigr)\bigr\}\,.$$ Therefore, by , we obtain $$\begin{aligned}
{{\widetilde{\uppi}}}\bigl(\bigl\{z\in \widetilde{{\mathcal{Z}}}_{xy}(R)\,\colon G(z)> h\bigr\}\bigr)
&\;\le\; 2 R^{\theta-2s}\,
\int_{\widehat{{\mathcal{Z}}}_{xy}}{\lvertz\rvert}^{2s-\theta}\,{{\widetilde{\uppi}}}(z)\,{\mathrm{d}}{z} \nonumber\\[5pt]
&\;\le\;
2\,\Tilde{\lambda}_{D}\,C_{0}\,R^{\theta-2s-1}
h^{\frac{-1}{\beta-r}}\,.\end{aligned}$$ It follows that $$\begin{aligned}
\label{ea6.15}
\int_{\widetilde{{\mathcal{Z}}}_{xy}(R)} G(z)\,{{\widetilde{\uppi}}}(z)\,{\mathrm{d}}{z}
&\;=\; \int_{0}^{\infty}
{{\widetilde{\uppi}}}\bigl(\bigl\{z\in \widetilde{{\mathcal{Z}}}_{xy}(R)\,\colon G(z)> h\bigr\}\bigr)
\,{\mathrm{d}}{h}\nonumber\\[5pt]
&\;\le\;
2\,\Tilde{\lambda}_{D}\,C_{0}\,R^{\theta-2s-1}\,
\int_{R^{r-\beta}}^{\infty}h^{\frac{-1}{\beta-r}}\,{\mathrm{d}}{h}
\nonumber\\[5pt]
&\;\le\;
\frac{2\,(\beta-r)}{1+r-\beta}\,\Tilde{\lambda}_{D}\,C_{0}\,R^{\theta-2s-1}\,
R^{1+r-\beta}\,.\end{aligned}$$
Thus, combining – with in case (i), or with , and in case (ii), and using the Hölder interpolation inequalities, we obtain $$\label{ea6.16}
R^{2s-r-\theta}\,R^{\beta}\,
\frac{\int_{{\mathbb{R}^{d}}} F_{2}(x,y;z)\,{\mathrm{d}}{z}}{{\lvertx-y\rvert}^{\beta}}
\;\le\; c_{4}\,{[\kern-0.45ex[\kern0.1ex v\kern0.1ex]\kern-0.45ex]}^{(-r)}_{2s+\beta-\theta;D}$$ for some constant $c_{4}$.
Therefore, by , and we obtain , and the proof is complete.
It is evident from the proof of Lemma \[L6.1\] that the assumption in may be replaced by the following: There exists a constant ${M}_{D}$, such that for all $x\in D$ and positive constants $R$ and $\varepsilon$ which satisfy $0<\varepsilon\le R$ and $d_{x}\ge 3R$, it holds that $$\int_{x+z\,\in D_{\varepsilon}\setminus \widetilde{D}(\varepsilon)}
\frac{\Tilde{k}(x,z)}{{\lvertz\rvert}^{d-\theta}}\,{\mathrm{d}}{z}
\;\le\; M_{D}\,\frac{\varepsilon}{R}\,.$$ The same applies to Theorems \[T3.1\] and \[T6.1\].
In order to proceed, we need certain properties of solutions of $(-\Delta)^{s} = f$ in a bounded domain $D$, and $u=0$ on $D^{c}$, with $f$ not necessarily in $L^{\infty}(D)$. We start with exhibiting a suitable supersolution.
\[L6.2\] For any $q\in \bigl(s-\nicefrac{1}{2}, s\bigr)$ there exists a constant $c_{0}\; > \; 0$ and a radial continuous function $\varphi$ such that $$\begin{cases}
(-\Delta)^{s}\varphi(x) \;\ge\; d_{x}^{q-2s}\,,
& \text{in}~B_{4}\setminus \Bar{B}_{1}\,,\\
\varphi \;=\; 0 & \text{in}~B_{1}\,,\\
0\;\le\; \varphi\;\le\; c_{0}({\lvertx\rvert}-1)^{q}
& \text{in} \quad B_{4}\setminus B_{1}\,,\\
1\;\le\; \varphi\;\le\; c_{0} & \text{in} \quad {\mathbb{R}}^{d}\setminus B_{4}\,.
\end{cases}$$
In view of the Kelvin transform [@RosOton-Serra Proposition A.1] it is enough to prove the following: for $q\in \bigl(s-\nicefrac{1}{2}, s\bigr)$, and with $\psi(x) \;{:=}\; [(1-{\lvertx\rvert})^{+}]^{q}$, we have $$\label{ea6.17}
(-\Delta)^{s}\psi(x)\;\ge\; c_{1}\,(1-{\lvertx\rvert})^{q-2s}\,,
\quad \text{for all}~ x\in B_{1}\,,$$ for some positive constant $c_{1}$. To prove we let $x_{0}\in B_{1}$. Due to the rotational symmetry we may assume $x_{0}=r e_{1}$ for some $r\in (0,1)$. Denote $z=(z_{1},\dotsc, z_{d})$. Then $$\begin{aligned}
-(-\Delta)^{s}\psi(x_{0}) \;&=\; c(d,\alpha)
\int_{{\mathbb{R}^{d}}}\bigl(\psi(x_{0}+z)-\psi(x_{0})\bigr)\frac{1}{{\lvertz\rvert}^{d+2s}}\,{\mathrm{d}}{z}
\\[5pt]
& = \; c(d,\alpha)\int_{{\mathbb{R}^{d}}}
\Bigl(\bigl[(1-|re_{1}+z|)^{+}\bigr]^{q}-(1-r)^{q}\Bigr)
\frac{1}{{\lvertz\rvert}^{d+2s}}\,{\mathrm{d}}{z}\\[5pt]
& \le \; c(d,\alpha)\int_{{\mathbb{R}^{d}}}
\Bigl(\bigl[(1-|re_{1}+z_{1}|)^{+}\bigr]^{q}-(1-r)^{q}\Bigr)
\frac{1}{{\lvertz\rvert}^{d+2s}}\,{\mathrm{d}}{z}\\[5pt]
& = \; c_{2}\int_{{\mathbb{R}}}\Bigl(\bigl[(1-|r+z|)^{+}\bigr]^{q}-(1-r)^{q}\Bigr)
\frac{1}{{\lvertz\rvert}^{1+2s}}\,{\mathrm{d}}{z}\\[5pt]
& \le \; c_{2}\int_{{\mathbb{R}}}\Bigl(\bigl[(1-r-z)^{+}\bigr]^{q}-(1-r)^{q}\Bigr)
\frac{1}{{\lvertz\rvert}^{1+2s}}\,{\mathrm{d}}{z}\\[5pt]
& = \; c_{2}(1-r)^{q-2s}\int_{{\mathbb{R}}}\Bigl(\bigl[(1-z)^{+}\bigr]^{q}-1\Bigr)
\frac{1}{{\lvertz\rvert}^{1+2s}}\,{\mathrm{d}}{z}\end{aligned}$$ for some constant $c_{2}$, where in the first inequality we use the fact that $(1-{\lvertz\rvert})^{+}\le (1-|z_{1}|)^{+}$ and in the second inequality we use $1-{\lvertz\rvert}\le 1-z$. Define $$\begin{aligned}
A(q) & \;{:=}\; \int_{{\mathbb{R}}}\Bigl([(1-z)^{+}]^{q}-1\Bigr)\frac{1}{{\lvertz\rvert}^{1+2s}}\,{\mathrm{d}}{z}
\;=\;\int_{0}^{\infty}\frac{z^{q}-1}{{\lvert1-z\rvert}^{1+2s}}\,{\mathrm{d}}{z}
-\int_{-\infty}^{0}\frac{1}{{\lvert1-z\rvert}^{1+2s}}\,{\mathrm{d}}{z}\,,\\[5pt]
B(q) &\;{:=}\; \int_{0}^{\infty}\frac{z^{q}-1}{{\lvert1-z\rvert}^{1+2s}}\,{\mathrm{d}}{z}\,.\end{aligned}$$ We need to show that $A(q)<0$ for $q$ close to $s$. It is known that $A(s)=0$ [@RosOton-Serra Proposition 3.1]. Therefore it is enough to show that $B(q)$ is strictly increasing for $q\in \bigl(s-\nicefrac{1}{2}, s\bigr)$. We have $$\begin{aligned}
B(q) \;&=\;\int_{0}^{1}\frac{z^{q}-1}{{\lvert1-z\rvert}^{1+2s}}\,{\mathrm{d}}{z}\, + \,
\int_{1}^{\infty}\frac{z^{q}-1}{{\lvert1-z\rvert}^{1+2s}}\,{\mathrm{d}}{z}\\[5pt]
&=\; \int_{0}^{1}\frac{(z^{q}-1)(1-z^{2s-1-q})}{{\lvert1-z\rvert}^{1+2s}}\,{\mathrm{d}}{z}\,.\end{aligned}$$ Therefore, for $q\in \bigl(s-\nicefrac{1}{2}, s\bigr)$, we obtain $$\frac{dB(q)}{dq}\; =\;
\int_{0}^{1}\frac{(z^{q}-z^{2s-1-q})\log{z}}{{\lvert1-z\rvert}^{1+2s}}\,{\mathrm{d}}{z}\; >\; 0\,,$$ where we use the fact that $\log{z}\le 0$ in $[0, 1]$. This completes the proof.
\[L6.3\] Let $f$ be a continuous function in $D$ satisfying $\sup_{x\in D}\,d_{x}^{\delta} |f(x)|<\infty$ for some $\delta<s$. Then there exists a viscosity solution $u\in C({\mathbb{R}}^{d})$ to $$\begin{split}
(-\Delta)^{s} u(x) & \;=\;-f(x) \quad \text{in}~D\,,\\[5pt]
u& \;=\; 0 \quad \text{in}~D^{c}\,.
\end{split}$$ Also, for every $q< s$ we have
$$\begin{aligned}
{\lvertu(x)\rvert}&\;\le\; C_{1}\, {[\kern-0.45ex[\kern0.1ex f\kern0.1ex]\kern-0.45ex]}^{(\delta)}_{0;D}\,d^{q}_{x}
\qquad\forall\,x\in\Bar{D}\,,\label{ea6.18a}\\[5pt]
{\lVertu\rVert}_{C^{q}(\Bar{D})}&\;\le\; C_{1}\,\sup_{x\in D}\;
d_{x}^{\delta}\,{\lvertf(x)\rvert}\,,\label{ea6.18b}\end{aligned}$$
for some constant $C_{1}$ that depends only on $s$, $\delta$, $q$ and the domain $D$. Moreover, since $u=0$ on $D^{c}$, it follows that the Hölder norm of $u$ on ${\mathbb{R}}^{d}$ is bounded by the same constant.
Existence of a continuous viscosity solution follows from Lemma \[L6.2\] and Perron’s method, since we can always choose $q$ close enough to $s$ in Lemma \[L6.2\] so as to satisfy $2s-q>\delta$, and obtain a bound on the solution $u$. From the barrier there exists a compact set $K_{1}\subset D$ such that $$\label{ea6.19}
|u(x)|\;\le\; \kappa_{1} \biggl(\sup_{x\in K_{1}}\;|u(x)|
+{[\kern-0.45ex[\kern0.1ex f\kern0.1ex]\kern-0.45ex]}^{(\delta)}_{0;D}\biggr)\,d_{x}^{q}
\qquad \forall \, x\in K_{1}^{c}\,,$$ where the constant $\kappa_{1}$ depends only on $K_{1}$ and $D$. Also, using the same argument as in Lemma \[L3.2\], we can show that for any compact $K_{2}\subset D$, there exists a constant $\kappa_{2}$, depending on $D$, and satisfying $$\label{ea6.20}
\sup_{x\in K_{2}}\;|u(x)|\;\le\; \kappa_{2}\biggl(\sup_{x\in K_{2}}\;|f(x)|
+\sup_{x\in D\setminus K_{2}}\;|u(x)|\biggr)\,.$$ We choose $K_{2}$ and $K_{1}\subset K_{2}$ such that $\sup_{x\in K^{c}_{2}\cap D}\, |d_{x}^{q}|<\frac{1}{2\kappa_{1}\kappa_{2}}$. Then from – we obtain $$\label{ea6.21}
\sup_{x\in K_{2}}\;|u(x)|\;\le\; \kappa_{3}\;{[\kern-0.45ex[\kern0.1ex f\kern0.1ex]\kern-0.45ex]}^{(\delta)}_{0;D}$$ for some constant $\kappa_{3}$. Hence the bound in follows by combining and .
The estimate in is easily obtained by following the argument in the proof of [@RosOton-Serra Proposition 1.1].
Our main result in this section is the following.
\[T6.1\] Let ${\mathcal{I}}\in\mathfrak{I}_{2s}(\beta,\theta,\lambda)$, $f$ be locally Hölder continuous with exponent $\beta$, and $D$ be a bounded domain with a $C^{2}$ boundary. We assume that neither $\beta$, nor $2s+\beta$ are integers, and that either $\beta<s$, or that $\beta\ge s$ and $${\lvertk(x,z)-k(x,0)\rvert}\;\le\; \Tilde{\lambda}_{D}\, {\lvertz\rvert}^{\theta}\qquad
\forall x\in D\,, \;\forall z\in {\mathbb{R}^{d}}\,,$$ for some positive constant $\Tilde{\lambda}_{D}$. Then the Dirichlet problem in has a unique solution in $C^{2s+\beta}_{\mathrm{loc}}(D)\cap C(\Bar{D})$. Moreover, for any $r<s$, we have the estimate $${\bm\lvert\kern-0.50ex\bm\lvert\kern0.1ex u
\kern0.1ex\bm\rvert\kern-0.50ex\bm\rvert}^{(-r)}_{2s+\beta;D}
\;\le\; C_{0} \, {\lVertf\rVert}_{C^{\beta}(\Bar{D})}$$ for some constant $C_{0}$ that depends only on $d$, $\beta$, $r$, $s$ and the domain $D$.
Consider the case $\beta\ge s$. We write as $$\label{ea6.22}
\begin{split}
(-\Delta)^{s} u(x) & \;=\;\mathcal{T}[u](x)\;{:=}\;\frac{c(d,2s)}{k(x,0)}\,
\bigl(-f(x) + b(x)\cdot{\nabla}{u}(x)\bigr)+ {{\mathcal{H}}}[u](x)
\quad \text{in}~D\,,\\[5pt]
u& \;=\; 0 \quad \text{in}~D^{c}\,,
\end{split}$$ and we apply the Leray–Schauder fixed point theorem. Also, without loss of generality, we assume $\theta<2s-1$. We choose any $r\in(0,s)$ which satisfies $$r\;>\;\Bigl(s -\frac{\theta}{2}\Bigr)
\vee\Bigl(1-s+\frac{\theta}{2}\Bigr)\,,$$ and let $v\in\mathscr{C}_{2s+\beta-\theta}^{(-r)}(D)$. Then ${{\mathcal{H}}}[v]\in
\mathscr{C}_{\beta}^{(2s-r-\theta)}(D)$ by Lemma \[L6.1\]. Since ${\nabla}V\in \mathscr{C}_{2s+\beta-\theta-1}^{(1-r)}(D)$ and $(1-r)\wedge (2s-r-\theta) < s$ by hypothesis, then applying Lemma \[L6.3\] we conclude that there exists a solution $u$ to $(-\Delta)^{s}u = \mathcal{T}[v]$ on $D$, with $u=0$ on $D^{c}$, such that $u\in\mathscr{C}_{0}^{(-q)}(D)$ for any $q<s$.
Next we obtain some estimates that are needed in order to apply the Leray–Schauder fixed point theorem. By Lemma \[L6.1\] we obtain $${\bigl\lvert\kern-0.72ex\bigl\rvert\kern-0.55ex
\bigl\lvert\kern-0.72ex\bigl\rvert\kern0.1ex {{\mathcal{H}}}[v]
\kern0.1ex\bigl\lvert\kern-0.72ex\bigl\rvert\kern-0.55ex
\bigl\lvert\kern-0.72ex\bigl\rvert}^{(2s-r-\nicefrac{\theta}{2})}_{0;D}
\;=\; {\bigl\lvert\kern-0.72ex\bigl\rvert\kern-0.55ex
\bigl\lvert\kern-0.72ex\bigl\rvert\kern0.1ex {{\mathcal{H}}}[v]
\kern0.1ex\bigl\lvert\kern-0.72ex\bigl\rvert\kern-0.55ex
\bigl\lvert\kern-0.72ex\bigl\rvert}^{(2s-(r-\nicefrac{\theta}{2})-\theta)}_{0;D}
\;\le\;
\kappa_{1}\, {\bm\lvert\kern-0.50ex\bm\lvert\kern0.1ex v
\kern0.1ex\bm\rvert\kern-0.50ex\bm\rvert}^{(-r+\nicefrac{\theta}{2})}_{2s-\theta;D}\,,$$ and similarly, $$\label{ea6.23}
{\bigl\lvert\kern-0.72ex\bigl\rvert\kern-0.55ex
\bigl\lvert\kern-0.72ex\bigl\rvert\kern0.1ex {{\mathcal{H}}}[v]
\kern0.1ex\bigl\lvert\kern-0.72ex\bigl\rvert\kern-0.55ex
\bigl\lvert\kern-0.72ex\bigl\rvert}^{(2s-r-\nicefrac{\theta}{2})}_{\beta;D}\;\le\;
\kappa_{1}\, {\bm\lvert\kern-0.50ex\bm\lvert\kern0.1ex v
\kern0.1ex\bm\rvert\kern-0.50ex\bm\rvert}^{(-r+\nicefrac{\theta}{2})}_{2s+\beta-\theta;D}\,,$$ for some constant $\kappa_{1}$ which does not depend on $\theta$ or $r$. Thus, since by hypothesis $2s-r-\nicefrac{\theta}{2}<s$ and $1-r+\nicefrac{\theta}{2}<s$, we obtain by Lemma \[L6.3\] that $$\label{ea6.24}
{\lVertu\rVert}_{C^{r}({\mathbb{R}^{d}})}\;\le\; \kappa'_{1}\,
\Bigl({\lVertf\rVert}_{C(\Bar{D})}+{\bm\lvert\kern-0.50ex\bm\lvert\kern0.1ex {\nabla}v
\kern0.1ex\bm\rvert\kern-0.50ex\bm\rvert}^{(1-r+\nicefrac{\theta}{2})}_{0;D}
+{\bm\lvert\kern-0.50ex\bm\lvert\kern0.1ex v
\kern0.1ex\bm\rvert\kern-0.50ex\bm\rvert}^{(-r+\nicefrac{\theta}{2})}_{2s-\theta;D}\Bigr)\,$$ for some constant $\kappa'_{1}$. Also, by Lemma 2.10 in [@RosOton-Serra], there exists a constant $\kappa_{2}$, depending only on $\beta$, $s$, $r$ and $d$, such that $$\label{ea6.25}
{\bm\lvert\kern-0.50ex\bm\lvert\kern0.1ex u
\kern0.1ex\bm\rvert\kern-0.50ex\bm\rvert}^{(-r)}_{2s+\beta;D}\;\le\; \kappa_{2}\,\Bigl({\lVertu\rVert}_{C^{r}({\mathbb{R}^{d}})}
+{\bigl\lvert\kern-0.72ex\bigl\rvert\kern-0.55ex
\bigl\lvert\kern-0.72ex\bigl\rvert\kern0.1ex \mathcal{T}[v]
\kern0.1ex\bigl\lvert\kern-0.72ex\bigl\rvert\kern-0.55ex
\bigl\lvert\kern-0.72ex\bigl\rvert}^{(2s-r)}_{\beta;D}\Bigr)\,.$$ It follows by – that $v\mapsto u$ is a continuous map from $\mathscr{C}^{(-r)}_{2s+\beta-\theta}$ to itself. Moreover, since $\mathscr{C}^{(-r)}_{2s+\beta}(D)$ is precompact in $\mathscr{C}^{(-r)}_{2s+\beta-\theta}(D)$, it follows that $v\mapsto u$ is compact.
Next we obtain a bound for ${\bm\lvert\kern-0.50ex\bm\lvert\kern0.1ex u
\kern0.1ex\bm\rvert\kern-0.50ex\bm\rvert}^{(-r)}_{2s+\beta;D}$. By we have $$\begin{aligned}
{\bigl\lvert\kern-0.72ex\bigl\rvert\kern-0.55ex
\bigl\lvert\kern-0.72ex\bigl\rvert\kern0.1ex {{\mathcal{H}}}[v]
\kern0.1ex\bigl\lvert\kern-0.72ex\bigl\rvert\kern-0.55ex
\bigl\lvert\kern-0.72ex\bigl\rvert}^{(2s-r)}_{\beta;D}
&\;\le\;\bigl(\operatorname{diam}(D)\bigr)^{\nicefrac{\theta}{2}}
{\bigl\lvert\kern-0.72ex\bigl\rvert\kern-0.55ex
\bigl\lvert\kern-0.72ex\bigl\rvert\kern0.1ex {{\mathcal{H}}}[v]
\kern0.1ex\bigl\lvert\kern-0.72ex\bigl\rvert\kern-0.55ex
\bigl\lvert\kern-0.72ex\bigl\rvert}^{(2s-r-\nicefrac{\theta}{2})}_{\beta;D}\nonumber\\[5pt]
&\;\le\; \kappa_{1}\,\bigl(\operatorname{diam}(D)\bigr)^{\nicefrac{\theta}{2}}\,
{\bm\lvert\kern-0.50ex\bm\lvert\kern0.1ex v
\kern0.1ex\bm\rvert\kern-0.50ex\bm\rvert}^{(-r+\nicefrac{\theta}{2})}_{2s+\beta-\theta;D}\Bigr)\,.\end{aligned}$$ Therefore, since also $2s-r>1-r+\nicefrac{\theta}{2}$, we obtain $$\label{ea6.26}
{\bigl\lvert\kern-0.72ex\bigl\rvert\kern-0.55ex
\bigl\lvert\kern-0.72ex\bigl\rvert\kern0.1ex \mathcal{T}[v]
\kern0.1ex\bigl\lvert\kern-0.72ex\bigl\rvert\kern-0.55ex
\bigl\lvert\kern-0.72ex\bigl\rvert}^{(2s-r)}_{\beta;D}
\;\le\; \kappa_{3}\,\Bigl({\lVertf\rVert}_{C^{\beta}(\Bar{D})}
+ {[\kern-0.45ex[\kern0.1ex v\kern0.1ex]\kern-0.45ex]}^{(-r+\nicefrac{\theta}{2})}_{1;D}
+{\bm\lvert\kern-0.50ex\bm\lvert\kern0.1ex v
\kern0.1ex\bm\rvert\kern-0.50ex\bm\rvert}^{(-r+\nicefrac{\theta}{2})}_{2s+\beta-\theta;D}\Bigr)$$ for some constant $\kappa_{3}$. By the Hölder interpolation inequalities, for any $\varepsilon>0$, there exists $\widetilde{C}(\varepsilon)>0$ such that $$\label{ea6.27}
{[\kern-0.45ex[\kern0.1ex v\kern0.1ex]\kern-0.45ex]}^{(-r+\nicefrac{\theta}{2})}_{1;D} +
{\bm\lvert\kern-0.50ex\bm\lvert\kern0.1ex v
\kern0.1ex\bm\rvert\kern-0.50ex\bm\rvert}^{(-r+\nicefrac{\theta}{2})}_{2s+\beta-\theta;D}\;\le\;
\widetilde{C}(\varepsilon)\,{[\kern-0.45ex[\kern0.1ex v\kern0.1ex]\kern-0.45ex]}^{(-r+\nicefrac{\theta}{2})}_{0;D}
+\varepsilon\,{\bm\lvert\kern-0.50ex\bm\lvert\kern0.1ex v
\kern0.1ex\bm\rvert\kern-0.50ex\bm\rvert}^{(-r+\nicefrac{\theta}{2})}_{2s+\beta;D}\,.$$ Combining , , and , and then using and the inequality $${[\kern-0.45ex[\kern0.1ex v\kern0.1ex]\kern-0.45ex]}^{(-r+\nicefrac{\theta}{2})}_{2s+\beta;D}\;\le\;
\bigl(\operatorname{diam}(D)\bigr)^{\nicefrac{\theta}{2}}
{\bm\lvert\kern-0.50ex\bm\lvert\kern0.1ex v
\kern0.1ex\bm\rvert\kern-0.50ex\bm\rvert}^{(-r)}_{2s+\beta;D}$$ we obtain $$\label{ea6.28}
{\bm\lvert\kern-0.50ex\bm\lvert\kern0.1ex u
\kern0.1ex\bm\rvert\kern-0.50ex\bm\rvert}^{(-r)}_{2s+\beta;D}\;\le\; \kappa_{4}(\varepsilon)\,
\Bigl({\lVertf\rVert}_{C^{\beta}(\Bar{D})}
+{\bm\lvert\kern-0.50ex\bm\lvert\kern0.1ex v
\kern0.1ex\bm\rvert\kern-0.50ex\bm\rvert}^{(-r+\nicefrac{\theta}{2})}_{0;D}\Bigr)
+ \varepsilon\,{\bm\lvert\kern-0.50ex\bm\lvert\kern0.1ex v
\kern0.1ex\bm\rvert\kern-0.50ex\bm\rvert}^{(-r)}_{2s+\beta;D}\,.$$ In order to apply the Leray–Schauder fixed point theorem, it suffices to show that the set of solutions $u\in \mathscr{C}^{(-r)}_{2s+\beta}(D)$ of $(-\Delta)^{s} u(x) \;=\;\xi\,\mathcal{T}[u](x)$, for $\xi\in[0,1]$, with $u=0$ on $D^{c}$, is bounded in $\mathscr{C}^{(-r)}_{2s+\beta}(D)$. However, from the above calculations, any such solution $u$ satisfies with $v\equiv u$. Moreover by Lemma \[L3.2\], $$\label{ea6.29}
\sup_{x\in D}\; {\lvertu(x)\rvert} \;\le\; \kappa_{5}\; \sup_{x\in D}\; {\lvertf(x)\rvert}$$ for some constant $\kappa_{5}$. We also have that $$\begin{aligned}
\label{ea6.30}
{\bm\lvert\kern-0.50ex\bm\lvert\kern0.1ex u
\kern0.1ex\bm\rvert\kern-0.50ex\bm\rvert}^{(-r+\nicefrac{\theta}{2})}_{0;D}&
\;\le\; \varepsilon^{-r+\nicefrac{\theta}{2}}\,
\sup_{x\in D,\,d_{x}\ge\varepsilon}\; {\lvertu(x)\rvert}
+ \varepsilon^{\nicefrac{\theta}{2}}\,
\sup_{x\in D,\,d_{x}<\varepsilon}\; d_{x}^{-r}\,{\lvertu(x)\rvert}\nonumber\\[5pt]
&\;\le\;\varepsilon^{-r+\nicefrac{\theta}{2}}\,\sup_{x\in D}\; {\lvertu(x)\rvert}
+ \varepsilon^{\nicefrac{\theta}{2}}\,
{\bm\lvert\kern-0.50ex\bm\lvert\kern0.1ex u
\kern0.1ex\bm\rvert\kern-0.50ex\bm\rvert}^{(-r)}_{0;D}\,.\end{aligned}$$ Choosing $\varepsilon>0$ small enough, and using – on the right hand side of with $v\equiv u$, we obtain $$\label{ea6.31}
{\bm\lvert\kern-0.50ex\bm\lvert\kern0.1ex u
\kern0.1ex\bm\rvert\kern-0.50ex\bm\rvert}^{(-r)}_{2s+\beta;D}\;\le\; \kappa_{6}\,
{\lVertf\rVert}_{C^{\beta}(\Bar{D})}$$ for some constant $\kappa_{6}$. Hence by the Leray–Schauder fixed point theorem the map $v\mapsto u$ given by has a fixed point $u\in\mathscr{C}^{(-r)}_{2s+\beta}(D)$, i.e., $$(-\Delta)^{s} u(x)=\mathcal{T}[u](x)\,.$$ Hence, this is a solution to . Uniqueness is obvious as $u$ is a classical solution. The bound in then applies and the proof is complete. The proof in the case $\beta<s$ is completely analogous.
Optimal regularity up to the boundary can be obtained under additional hypotheses. The following result is a modest extension of the results in [@RosOton-Serra Proposition 1.1].
Let ${\mathcal{I}}\in\mathfrak{I}_{2s}(\beta,\theta,\lambda)$ with $\theta>s$, $f$ be locally Hölder continuous with exponent $\beta$, and $D$ be a bounded domain with a $C^{2}$ boundary. Suppose in addition that $b=0$ and that $k$ is symmetric, i.e., $k(x,z)=k(x,-z)$. Then the solution of the Dirichlet problem in is in $C^{s}({\mathbb{R}^{d}})$. Moreover, for any $\beta<s$ we have $u\in \mathscr{C}_{2s+\beta}^{(-s)}(D)$.
By Theorem \[T6.1\], the Dirichlet problem in has a unique solution in $C^{2s+\rho}_{\mathrm{loc}}(D)\cap C(\Bar{D})$, for any $\rho<\beta\wedge s$. Moreover, for any $r<s$, we have the estimate $${\bm\lvert\kern-0.50ex\bm\lvert\kern0.1ex u
\kern0.1ex\bm\rvert\kern-0.50ex\bm\rvert}^{(-r)}_{2s+\rho;D}
\;\le\; C_{0} \, {\lVertf\rVert}_{C^{\beta}(\Bar{D})}\,.$$ Fix $r=2s-\theta$. Then $$\int_{R<{\lvertz\rvert}<1}
{\lvertz\rvert}^{r}\,\frac{\Tilde{k}(x,z)}{{\lvertz\rvert}^{d+2s}}\,
{\mathrm{d}}{z} \;=\; \int_{R<{\lvertz\rvert}<1}
{\lvertz\rvert}^{2s-\theta}\,\frac{\Tilde{k}(x,z)}{{\lvertz\rvert}^{d+2s}}\,{\mathrm{d}}{z}
\;\le\;\lambda_{D}\,.$$ By and the symmetry of the kernel, it follows that $${\biggl\lvert\int_{R<{\lvertz\rvert}} {\mathfrak{d}}{u}(x;z)\,
\frac{\Tilde{k}(x,z)}{{\lvertz\rvert}^{d+2s}}\biggr\rvert}\,
{\mathrm{d}}{z} \;\le\; \kappa_{1}
\Bigl({[\kern-0.45ex[\kern0.1ex u\kern0.1ex]\kern-0.45ex]}^{(-r)}_{r;D}
+ {\lVertu\rVert}_{C(\Bar{D}}\Bigr)\qquad \forall x\in D\,,$$ for some constant $\kappa_{1}$. Combining this with the estimate in Lemma \[L6.1\] we obtain $$\begin{aligned}
{\bigl[\kern-0.75ex\bigl[\kern0.1ex {{\mathcal{H}}}[u]\kern0.1ex\bigr]\kern-0.75ex\bigr]}^{(0)}_{0;D}
\;\le\;M_{0}\,{\bm\lvert\kern-0.50ex\bm\lvert\kern0.1ex u
\kern0.1ex\bm\rvert\kern-0.50ex\bm\rvert}^{(-r)}_{r;D}
\;<\;\infty\,,\end{aligned}$$ implying that ${{\mathcal{H}}}[u]\in L^{\infty}(D)$. It then follows by [@RosOton-Serra Proposition 1.1] that $u\in C^{s}({\mathbb{R}^{d}})$, and that for some constant $C$ depending only on $s$, we have $$\begin{aligned}
{\lVertu\rVert}_{C^{s}({\mathbb{R}^{d}})}
&\;\le\;
C\,{\bigl\lVert\mathcal{T}[u]\bigr\rVert}_{L^{\infty}(D)}\\[5pt]
&\;\le\; C\,\lambda_{D}^{-1}\,c(d,2s)
\Bigl({\lVertf\rVert}_{L^{\infty}(D)}+{\lVert{{\mathcal{H}}}[u]\rVert}_{L^{\infty}(D)}\Bigr)\\[5pt]
&\;\le\;
C\,\lambda_{D}^{-1}\,c(d,2s)
\Bigl({\lVertf\rVert}_{L^{\infty}(D)}+M_{0}\,{\bm\lvert\kern-0.50ex\bm\lvert\kern0.1ex u
\kern0.1ex\bm\rvert\kern-0.50ex\bm\rvert}^{(-r)}_{r;\,D}
\Bigr)\,.\end{aligned}$$ Using the Hölder interpolation inequalities we obtain from the preceding estimate that $${\lVertu\rVert}_{C^{s}({\mathbb{R}^{d}})}
\;\le\; \Tilde{C}\,{\lVertf\rVert}_{L^{\infty}(D)}$$ for some constant $\Tilde{C}$ depending only on $s$, $\theta$, and $\lambda_{D}$.
Applying Lemma \[L6.1\] once more, we conclude that ${{\mathcal{H}}}[u]\in\mathscr{C}_{\beta'}^{(s)}(D)$ for any $\beta'\leq r$, and that $${\bigl\lvert\kern-0.72ex\bigl\rvert\kern-0.55ex
\bigl\lvert\kern-0.72ex\bigl\rvert\kern0.1ex {{\mathcal{H}}}[u]
\kern0.1ex\bigl\lvert\kern-0.72ex\bigl\rvert\kern-0.55ex
\bigl\lvert\kern-0.72ex\bigl\rvert}^{(s)}_{\beta';\,D}\;\le\;
M_{1}\, {\bm\lvert\kern-0.50ex\bm\lvert\kern0.1ex u
\kern0.1ex\bm\rvert\kern-0.50ex\bm\rvert}^{(-r)}_{2s+\beta'-\theta ;\,D}\,.$$ Hence, applying [@RosOton-Serra Proposition 1.4], we obtain $$\begin{aligned}
{\bm\lvert\kern-0.50ex\bm\lvert\kern0.1ex u
\kern0.1ex\bm\rvert\kern-0.50ex\bm\rvert}^{(-s)}_{2s+\beta';\,D}
&\;\le\; C_{1}\Bigl({\lVertu\rVert}_{C^{s}({\mathbb{R}^{d}})}
+{\bigl\lvert\kern-0.72ex\bigl\rvert\kern-0.55ex
\bigl\lvert\kern-0.72ex\bigl\rvert\kern0.1ex \mathcal{T}[u]
\kern0.1ex\bigl\lvert\kern-0.72ex\bigl\rvert\kern-0.55ex
\bigl\lvert\kern-0.72ex\bigl\rvert}^{(s)}_{\beta';\,D}\Bigr)\end{aligned}$$ for some constant $C_{1}$, and we can repeat this procedure to reach $u\in\mathscr{C}_{2s+\beta}^{(-s)}(D)$.
**Acknowledgments.** We thank Dennis Kriventsov for helping us in clarifying some points of his paper [@Kriventsov] and for suggesting us the paper [@RosOton-Serra]. We also thank Héctor Chang Lara and Gonzalo D[á]{}vila for their help. The work of Anup Biswas was supported in part by an award from the Simons Foundation (No. 197982 to The University of Texas at Austin) and in part by the Office of Naval Research through the Electric Ship Research and Development Consortium. The work of Ari Arapostathis was supported in part by the Office of Naval Research through the Electric Ship Research and Development Consortium. This research of Luis Caffarelli is supported by an award from NSF.
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[^1]: Corresponding Author: Ari Arapostathis, 1616 Guadalupe St., UTA 7.502, Austin, TX 78701, ari@ece.utexas.edu, (512) 471-3265.
|
---
abstract: |
In a quite pragmatic sense [oops]{} is the fastest general way of solving one task after another, always optimally exploiting solutions to earlier tasks when possible. It can be used for increasingly hard problems of optimization or prediction. Suppose there is only one task and a bias in form of a probability distribution $P$ on programs for a universal computer. In the $i$-th phase $(i=1,2,3, \ldots)$ of asymptotically optimal [*non*]{}incremental universal search (Levin, 1973, 1984) we test all programs $p$ with runtime $\leq 2^iP(p)$ until the task is solved. Now suppose there is a sequence of tasks, e.g., the $n$-th task is to find a shorter path through a maze than the best found so far. To reduce the search time for new tasks, previous [*incremental*]{} extensions of universal search tried to modify $P$ through experience with earlier tasks—but in a heuristic and non-general and suboptimal way prone to overfitting. [Oops]{}, however, does it right.
Tested self-delimiting program prefixes (beginnings of code that may continue) are immediately executed while being generated. They grow by one instruction whenever they request this. The storage for the first found program computing a solution to the current task becomes non-writeable. Programs tested during search for solutions to later task may copy non-writeable code into separate modifiable storage, to edit it and execute the modified result. Prefixes may also recompute the probability distribution on their suffixes in arbitrary computable ways. To solve the $n$-th task we sacrifice half the total search time for testing (via universal search) programs that have the most recent successful program as a prefix. The other half remains for testing fresh programs starting at the address right above the top non-writeable address. When we are searching for a universal solver for all tasks in the sequence we have to time-share the second half (but not the first!) among all tasks $1..n$. For realistic limited computers we need efficient backtracking in program space to reset storage contents modified by tested programs. We introduce a recursive procedure for doing this in time-optimal fashion.
[Oops]{} can solve tasks unsolvable by traditional reinforcement learners and AI planners, such as [*Towers of Hanoi*]{} with 30 disks (minimal solution size $> 10^9$). In our experiments OOPS demonstrates incremental learning by reusing previous solutions to discover a prefix that temporarily rewrites the distribution on its suffixes, such that universal search is accelerated by a factor of 1000. This illustrates how [oops]{} can benefit from self-improvement and metasearching, that is, searching for faster search procedures.
We mention several [oops]{} variants and outline [oops]{}-based reinforcement learners. Since [oops]{} will scale to larger problems in essentially unbeatable fashion, we also examine its physical limitations.
author:
- |
Jürgen Schmidhuber juergen@idsia.ch - www.idsia.ch/ juergen\
IDSIA, Galleria 2, 6928 Manno-Lugano, Switzerland
bibliography:
- 'bib.bib'
nocite: '[@Schmidhuber:02ijfcs]'
title: Optimal Ordered Problem Solver
---
\[section\] \[section\] \[section\]
[oops]{}, bias-optimality, incremental optimal universal search, efficient planning & backtracking in program space, metasearching & metalearning, self-improvement
[*Based on arXiv:cs.AI/0207097 v1 (TR-IDSIA-12-02 version 1.0, July 2002) [@Schmidhuber:02oops; @Schmidhuber:02nips]. All sections are illustrated by Figures \[storage\] and \[searchtree\] at the end of this paper. Frequently used symbols are collected in reference Table \[general\] (general [oops]{}-related symbols) and Table \[specific\] (less important implementation-specific symbols, explained in the appendix, Section \[language\]).* ]{}
Introduction {#intro}
============
We train children and most machine learning systems on sequences of harder and harder tasks. This makes sense since new problems often are more easily solved by reusing or adapting solutions to previous problems.
Often new tasks depend on solutions for earlier tasks. For example, given an NP-hard optimization problem, the $n$-th task in a sequence of tasks may be to find an approximation to the unknown optimal solution such that the new approximation is at least 1 % better (according to some measurable performance criterion) than the best found so far.
Alternatively we may want to find a strategy for solving [*all*]{} tasks in a given sequence of more and more complex tasks. For example, we might want to teach our learner a program that computes [fac]{}$(n)= 1 \times 2 \times \ldots n$ for any given positive integer $n$. Naturally, the $n$-th task in the “training sequence” will be to compute [fac]{}$(n)$.
In general we would like our learner to continually profit from useful information conveyed by solutions to earlier tasks. To do this in an optimal fashion, the learner may also have to improve the way it exploits earlier solutions. Is there a general yet time-optimal way of achieving such a feat? Indeed, there is. The Optimal Ordered Problem Solver ([oops]{}) is a simple, general, theoretically sound way of solving one task after another, efficiently searching the space of programs that compute solution candidates, including programs that organize and manage and adapt and reuse earlier acquired knowledge.
Overview
--------
Section \[survey\] will survey previous relevant work on general optimal search algorithms. Section \[oops\] will use the framework of universal computers to explain [oops]{} and how it benefits from incrementally extracting useful knowledge hidden in training sequences. The remainder of the paper is devoted to “Realistic” [oops]{} which uses a recursive procedure for time-optimal planning and backtracking in program space to perform efficient storage management (Section \[realistic\]) on realistic, limited computers. Appendix \[language\] describes an pilot implementation of Realistic [oops]{} based on a stack-based universal programming language inspired by [Forth]{} [@Forth:70], with initial primitives for defining and calling recursive functions, iterative loops, arithmetic operations, domain-specific behavior, and even for rewriting the search procedure itself. Experiments in Section \[experiments\] use the language of Appendix \[language\] to solve 60 tasks in a row: we first teach [oops]{} something about recursion, by training it to construct samples of the simple context free language $\{ 1^k2^k \}$ ($k$ 1’s followed by $k$ 2’s), for $k$ up to 30. This takes roughly 0.3 days on a standard personal computer (PC). Thereafter, within a few additional days, [oops]{} demonstrates the benefits of incremental knowledge transfer: it exploits certain properties of its previously discovered universal $1^k2^k$-solver to greatly accelerate the search for a universal solver for all $k$ disk [*Towers of Hanoi*]{} problems, solving all instances up to $k=30$ (solution size $2^k-1$). Previous, less general reinforcement learners and [*non*]{}learning AI planners tend to fail for much smaller instances.
Survey of Universal Search and Suboptimal Incremental Extensions {#survey}
================================================================
Let us start by briefly reviewing general, asymptotically optimal search methods by [@Levin:73; @Levin:84] and [@Hutter:01fast]. These methods are [*non*]{}incremental in the sense that they do not attempt to accelerate the search for solutions to new problems through experience with previous problems. We will point out drawbacks of existing heuristic extensions for [*incremental*]{} search. The remainder of the paper will describe [oops]{} which overcomes these drawbacks.
Bias-Optimality {#bias}
---------------
For the purposes of this paper, a problem $r$ is defined by a recursive procedure $f_r$ that takes as an input any potential solution (a finite symbol string $y \in Y$, where $Y$ represents a search space of solution candidates) and outputs 1 if $y$ is a solution to $r$, and 0 otherwise. Typically the goal is to find as quickly as possible some $y$ that solves $r$.
Define a probability distribution $P$ on a finite or infinite set of programs for a given computer. $P$ represents the searcher’s initial bias (e.g., $P$ could be based on program length, or on a probabilistic syntax diagram). A [*bias-optimal*]{} searcher will not spend more time on any solution candidate than it deserves, namely, not more than the candidate’s probability times the total search time:
\[[Bias-Optimal Searchers]{}\] \[bias-optimal\] Given is a problem class $\cal R$, a search space $\cal C$ of solution candidates (where any problem $r \in \cal R$ should have a solution in $\cal C$), a task-dependent bias in form of conditional probability distributions $P(q \mid r)$ on the candidates $q \in \cal C$, and a predefined procedure that creates and tests any given $q$ on any $r \in \cal R$ within time $t(q,r)$ (typically unknown in advance). A searcher is [*$n$-bias-optimal*]{} ($n \geq 1$) if for any maximal total search time $T_{max} > 0$ it is guaranteed to solve any problem $r \in \cal R$ if it has a solution $p \in \cal C$ satisfying $t(p,r) \leq P(p \mid r)~T_{max}/n$. It is [*bias-optimal*]{} if $n=1$.
This definition makes intuitive sense: the most probable candidates should get the lion’s share of the total search time, in a way that precisely reflects the initial bias.
Near-Bias-Optimal Nonincremental Universal Search {#levin}
-------------------------------------------------
The following straight-forward method (sometimes referred to as [*Levin Search*]{} or [Lsearch]{}) is near-bias-optimal. For simplicity, we notationally suppress conditional dependencies on the current problem. Compare [@Levin:73; @Levin:84; @Solomonoff:86; @Schmidhuber:97bias; @LiVitanyi:97; @Hutter:01fast] (Levin also attributes similar ideas to Allender):
\[lsearch\] Set current time limit T=1. [While]{} problem not solved [do:]{}
> Test all programs $q$ such that $t(q)$, the maximal time spent on creating and running and testing $q$, satisfies $t(q) < P(q)~T$. Set $T := 2 T.$
Note that [Lsearch]{} has the optimal order of computational complexity: Given some problem class, if some unknown optimal program $p$ requires $f(k)$ steps to solve a problem instance of size $k$, then [Lsearch]{} will need at most $O(P(p) f(k)) = O(f(k))$ steps — the constant factor $P(p)$ may be huge but does not depend on $k$.
The near-bias-optimality of [Lsearch]{} is hardly affected by the fact that for each value of $T$ we repeat certain computations for the previous value. Roughly half the total search time is still spent on $T$’s maximal value (ignoring hardware-specific overhead for parallelization and nonessential speed-ups due to halting programs if there are any). Note also that the time for testing is properly taken into account here: any result whose validity is hard to test is automatically penalized.
Universal [Lsearch]{} provides inspiration for nonuniversal but very practical methods which are optimal with respect to a limited search space, while suffering only from very small slowdown factors. For example, designers of planning procedures often just face a binary choice between two options such as depth-first and breadth-first search. The latter is often preferrable, but its greater demand for storage may eventually require to move data from on-chip memory to disk. This can slow down the search by a factor of 10,000 or more. A straightforward solution in the spirit of [Lsearch]{} is to start with a 50 % bias towards either technique, and use both depth-first and breadth-first search in parallel — this will cause a slowdown factor of at most 2 with respect to the best of the two options (ignoring a bit of overhead for parallelization). Such methods have presumably been used long before Levin’s 1973 paper. [@Wiering:96levin] and [@Schmidhuber:97bias] used rather general but nonuniversal variants of [Lsearch]{} to solve machine learning toy problems unsolvable by traditional methods. Probabilistic alternatives based on [*probabilistically chosen maximal program runtimes*]{} in [*Speed-Prior*]{} style [@Schmidhuber:00v2; @Schmidhuber:02colt] also outperformed traditional methods on toy problems [@Schmidhuber:95kol; @Schmidhuber:97nn].
Asymptotically Fastest Nonincremental Problem Solver {#hutter}
----------------------------------------------------
Recently my postdoc [@Hutter:01fast] developed a more complex asymptotically optimal search algorithm for [*all*]{} well-defined problems. [Hsearch]{} (or [*Hutter Search*]{}) cleverly allocates part of the total search time to searching the space of proofs for provably correct candidate programs with provable upper runtime bounds; at any given time it focuses resources on those programs with the currently best proven time bounds. Unexpectedly, [Hsearch]{} manages to reduce the constant slowdown factor to a value smaller than $5$. In fact, it can be made smaller than $1 + \epsilon$, where $\epsilon$ is an arbitrary positive constant (M. Hutter, personal communication, 2002).
Unfortunately, however, [Hsearch]{} is not yet the final word in computer science, since the search in proof space introduces an unknown [*additive*]{} problem class-specific constant slowdown, which again may be huge. While additive constants generally are preferrable to multiplicative ones, both types may make universal search methods practically infeasible—in the real world constants do matter. For example, the last to cross the finish line in the Olympic 100 m dash may be only a constant factor slower than the winner, but this will not comfort him. And since constants beyond $2^{500}$ do not even make sense within this universe, both [Lsearch]{} and [Hsearch]{} may be viewed as academic exercises demonstrating that the $O()$ notation can sometimes be practically irrelevant despite its wide use in theoretical computer science.
Previous Work on Incremental Extensions of Universal Search {#incremental}
-----------------------------------------------------------
[Hsearch]{} and [Lsearch]{} (Sections \[levin\], \[hutter\]) neglect one potential source of speed-up: they are nonincremental in the sense that they do not attempt to minimize their constant slowdowns by exploiting experience collected in previous searches for solutions to earlier tasks. They simply ignore the constants — from an asymptotic point of view, incremental search does not buy anything.
A heuristic attempt [@Schmidhuber:97bias] to greatly reduce the constants through experience was called [*Adaptive*]{} [Lsearch]{} or [Als]{} — compare related ideas by [@Solomonoff:86; @Solomonoff:89]. Essentially [Als]{} works as follows: whenever [Lsearch]{} finds a program $q$ that computes a solution for the current problem, $q$’s probability $P(q)$ is substantially increased using a “learning rate,” while probabilities of alternative programs decrease appropriately. Subsequent [Lsearch]{}es for new problems then use the adjusted $P$, etc. [@Schmidhuber:97bias] and [@Wiering:96levin] used a nonuniversal variant of this approach to solve reinforcement learning (RL) tasks in partially observable environments unsolvable by traditional RL algorithms.
Each [Lsearch]{} invoked by [Als]{} is bias-optimal with respect to the most recent adjustment of $P$. On the other hand, the rather arbitrary $P$-modifications themselves are not necessarily optimal. They might lead to [*overfitting*]{} in the following sense: modifications of $P$ after the discovery of a solution to problem 1 could actually be harmful and slow down the search for a solution to problem 2, etc. This may provoke a loss of near-bias-optimality with respect to the initial bias during exposure to subsequent tasks. Furthermore, [Als]{} has a fixed prewired method for changing $P$ and cannot improve this method by experience. The main contribution of this paper is to overcome all such drawbacks in a principled way.
Other Work on Incremental Learning {#other}
----------------------------------
Since the early attempts of [@Newell:63] at building a “General Problem Solver” —see also [@SOAR:93]—much work has been done to develop mostly heuristic machine learning algorithms that solve new problems based on experience with previous problems, by incrementally shifting the inductive bias in the sense of [@Utgoff:86]. Many pointers to [*learning by chunking, learning by macros, hierarchical learning, learning by analogy,*]{} etc. can be found in the book by [@Mitchell:97]. Relatively recent general attempts include program evolvers such as [Adate]{} [@Olsson:95] and simpler heuristics such as [*Genetic Programming (GP)*]{} [@Cramer:85; @Banzhaf:98]. Unlike logic-based program synthesizers [@Green:69; @Waldinger:69; @Deville:94], program evolvers use biology-inspired concepts of [*Evolutionary Computation*]{} [@Rechenberg:71; @Schwefel:74] or [*Genetic Algorithms*]{} [@Holland:75] to evolve better and better computer programs. Most existing GP implementations, however, do not even allow for programs with loops and recursion, thus ignoring a main motivation for search in program space. They either have very limited search spaces (where solution candidate runtime is not even an issue), or are far from bias-optimal, or both. Similarly, traditional reinforcement learners [@Kaelbling:96] are neither general nor close to being bias-optimal.
A first step to make GP-like methods bias-optimal would be to allocate runtime to tested programs in proportion to the probabilities of the mutations or “crossover operations” that generated them. Even then there would still be room for improvement, however, since GP has quite limited ways of making new programs from previous ones—it does not learn better program-making strategies.
This brings us to several previous publications on [*learning to learn*]{} or [*metalearning*]{} [@Schmidhuber:87], where the goal is to learn better learning algorithms through self-improvement without human intervention—compare the human-assisted self-improver by [@Lenat:83]. We introduced the concept of incremental search for improved, probabilistically generated code that modifies the probability distribution on the possible code continuations: [*incremental self-improvers*]{} [@Schmidhuber:97ssa] use the [*success-story algorithm*]{} SSA to undo those self-generated probability modifications that in the long run do not contribute to increasing the learner’s cumulative reward per time interval. An earlier meta-GP algorithm [@Schmidhuber:87] was designed to learn better GP-like strategies; [@Schmidhuber:87] also combined principles of reinforcement learning economies [@Holland:85] with a “self-referential” metalearning approach. A gradient-based metalearning technique [@Schmidhuber:93selfreficann] for [*continuous*]{} program spaces of differentiable recurrent neural networks (RNNs) was also designed to favor better learning algorithms; compare the remarkable recent success of the related but technically improved RNN-based metalearner by [@Hochreiter:01meta].
The algorithms above generally are not near-bias-optimal though. The method discussed in this paper, however, combines optimal search and incremental self-improvement, and will be [*$n$-bias-optimal*]{}, where $n$ is a small and practically acceptable number, such as 8.
OOPS on Universal Computers {#oops}
===========================
An informed reader familiar with concepts such as universal computers [@Turing:36] and self-delimiting programs [@Levin:74; @Chaitin:75] will probably understand the simple basic principles of [oops]{} by just reading the abstract. For the others, Subsection \[basics\] will start the formal description of [oops]{} by introducing notation and explaining program sets that are prefix codes. Subsection \[principles\] will provide [oops]{} pseudocode and point out its essential properties and a few essential differences to previous work. The remainder of the paper is about practical implementations of the basic principles on realistic computers with limited storage.
\[general\]
[**Symbol**]{} [**Description**]{}
-------------------- ----------------------------------------------------------------------------------------- -- -- -- -- --
$Q$ variable set of instructions or tokens
$Q_i$ $i$-th possible token (an integer)
$n_Q$ current number of tokens
$Q^*$ set of strings over alphabet $Q$, containing the search space of programs
$q$ total current code $\in Q^*$
$q_n$ $n$-th token of code $q$
$q^n$ $n$-th frozen program $\in Q^*$, where total code $q$ starts with $q^1q^2 \ldots$
$qp$ [*$q$-pointer*]{} to the highest address of code $q=q_{1:qp}$
$a_{last}$ start address of a program (prefix) solving all tasks so far
$a_{frozen}$ top frozen address, can only grow, $1 \leq a_{last} \leq a_{frozen} \leq qp$
$q_{1:a_{frozen}}$ current code bias
$R$ variable set of tasks, ordered in cyclic fashion; each task has a computation tape
$S$ set of possible tape symbols (here: integers)
$S^*$ set of strings over alphabet $S$, defining possible states stored on tapes
$s^i$ an element of $S^*$
$s(r)$ variable state of task $r \in R$, stored on tape $r$
$s_i(r)$ $i$-th component of $s(r)$
$l(s)$ length of any string $s$
$z(i)(r)$ equal to $q_i$ if $0 < i \leq l(q)$ or equal to $s_{-i}(r)$ if $-l(s(r)) \leq i \leq 0$
$ip(r)$ current instruction pointer of task $r$, encoded on tape $r$ within state $s(r)$
$p(r)$ variable probability distribution on $Q$, encoded on tape $r$ as part of $s(r)$
$p_i(r)$ current history-dependent probability of selecting $Q_i$ if $ip(r)=qp+1$
Formal Setup and Notation {#basics}
-------------------------
Unless stated otherwise or obvious, to simplify notation, throughout the paper newly introduced variables are assumed to be integer-valued and to cover the range implicit in the context. Given some finite or countably infinite alphabet $Q=\{Q_1,Q_2, \ldots \}$, let $Q^*$ denote the set of finite sequences or strings over $Q$, where $\lambda$ is the empty string. Then let $q,q^1,q^2, \ldots \in Q^*$ be (possibly variable) strings. $l(q)$ denotes the number of symbols in string $q$, where $l(\lambda) = 0$; $q_n$ is the $n$-th symbol of string $q$; $q_{m:n}= \lambda$ if $m>n$ and $q_m q_{m+1} \ldots q_n$ otherwise (where $q_0 := q_{0:0} := \lambda$). $q^1q^2$ is the concatenation of $q^1$ and $q^2$ (e.g., if $q^1=abc$ and $q^2=dac$ then $q^1q^2 = abcdac$).
Consider countable alphabets $S$ and $Q$. Strings $s,s^1,s^2, \ldots \in S^*$ represent possible internal [*states*]{} of a computer; strings $q,q^1,q^2, \ldots \in Q^*$ represent [*token sequences*]{} or [*code*]{} or [*programs*]{} for manipulating states. We focus on $S$ being the set of integers and $Q := \{ 1, 2, \ldots, n_Q \}$ representing a set of $n_Q$ instructions of some universal programming language [@Goedel:31; @Turing:36]. (The first universal programming language due to [@Goedel:31] was based on integers as well, but ours will be more practical.) $Q$ and $n_Q$ may be variable: new tokens may be defined by combining previous tokens, just as traditional programming languages allow for the declaration of new tokens representing new procedures. Since $Q^* \subset S^*$, substrings within states may also encode programs.
$R$ is a set of currently unsolved tasks. Let the variable $s(r) \in S^*$ denote the current state of task $r \in R$, with $i$-th component $s_i(r)$ on a [*computation tape*]{} $r$ (a separate tape holding a separate state for each task, initialized with task-specific inputs represented by the initial state). Since subsequences on tapes may also represent executable code, for convenience we combine current code $q$ and any given current state $s(r)$ in a single [*address space*]{}, introducing negative and positive addresses ranging from $-l(s(r))$ to $l(q)+1$, defining the content of address $i$ as $z(i)(r) := q_i$ if $0 < i \leq l(q)$ and $z(i)(r) := s_{-i}(r)$ if $-l(s(r)) \leq i \leq 0$. All dynamic task-specific data will be represented at nonpositive addresses (one code, many tasks). In particular, the current instruction pointer [*ip(r)*]{} $:= z(a_{ip}(r))(r)$ of task $r$ (where $ip(r) \in {-l(s(r)), \ldots, l(q)+1})$ can be found at (possibly variable) address $a_{ip}(r) \leq 0$. Furthermore, $s(r)$ also encodes a modifiable probability distribution $p(r) = \{ p_1(r), p_2(r), \ldots, p_{n_Q}(r) \}$ $(\sum_i p_i(r) = 1)$ on $Q$.
Code is executed in a way inspired by self-delimiting binary programs [@Levin:74; @Chaitin:75] studied in the theory of Kolmogorov complexity and algorithmic probability [@Solomonoff:64; @Kolmogorov:65]. Section \[try\] will present details of a practically useful variant of this approach. Code execution is time-shared sequentially among all current tasks. Whenever any $ip(r)$ has been initialized or changed such that its new value points to a valid address $\geq -l(s(r))$ but $\leq l(q)$, and this address contains some executable token $Q_i$, then $Q_i$ will define task $r$’s next instruction to be executed. The execution may change $s(r)$ including $ip(r)$. Whenever the time-sharing process works on task $r$ and $ip(r)$ points to the smallest positive currently unused address $l(q)+1$, $q$ will grow by one token (so $l(q)$ will increase by 1), and the current value of $p_i(r)$ will define the current probability of selecting $Q_i$ as the next token, to be stored at new address $l(q)$ and to be executed immediately. That is, executed program beginnings or [*prefixes*]{} define the probabilities of their possible suffixes. (Programs will be interrupted through errors or halt instructions or when all current tasks are solved or when certain search time limits are reached—see Section \[principles\].)
To summarize and exemplify: programs are grown incrementally, token by token; their prefixes are immediately executed while being created; this may modify some task-specific internal state or memory, and may transfer control back to previously selected tokens (e.g., loops). To add a new token to some program prefix, we first have to wait until the execution of the prefix so far [*explicitly requests*]{} such a prolongation, by setting an appropriate signal in the internal state. Prefixes that cease to request any further tokens are called self-delimiting programs or simply programs (programs are their own prefixes). So our procedure yields [*task-specific prefix codes*]{} on program space: with any given task, programs that halt because they have found a solution or encountered some error cannot request any more tokens. Given a single task and the current task-specific inputs, no program can be the prefix of another one. On a different task, however, the same program may continue to request additional tokens.
$a_{frozen} \geq 0 $ is a variable address that can increase but not decrease. Once chosen, the [*code bias*]{} $q_{0:a_{frozen}}$ will remain unchangeable forever — it is a (possibly empty) sequence of programs $q^1q^2 \ldots$, some of them prewired by the user, others [*frozen*]{} after previous successful searches for solutions to previous task sets (possibly completely unrelated to the current task set $R$).
To allow for programs that exploit previous solutions, the instruction set $Q$ should contain instructions for invoking or calling code found below $a_{frozen}$, for copying such code into some $s(r)$, and for editing the copies and executing the results. Examples of such instructions will be given in the appendix (Section \[language\]).
Basic Principles of OOPS {#principles}
------------------------
Given a sequence of tasks, we solve one task after another in the given order. The solver of the $n$-th task $(n \geq 1)$ will be a program $q^i$ $(i \leq n)$ stored such that it occupies successive addresses somewhere between 1 and $l(q)$. The solver of the $1$st task will start at address 1. The solver of the $n$-th task $(n>1)$ will either start at the same address as the solver of the $n-1$-th task, or right after its end address. To find a universal solver for all tasks in a given task sequence, do:
\[oopsalgbasics\]
[**FOR**]{} task index $n=1,2,\ldots$ [**DO:**]{}
[**1.**]{} Initialize current time limit $T := 2$.
[**2.**]{} Spend at most $T/2$ on a variant of [Lsearch]{} that searches for a program solving task $n$ and starting at the start address $a_{last}$ of the most recent successful code (1 if there is none). That is, the problem-solving program either must be equal to $q_{a_{last}:a_{frozen}}$ or must have $q_{a_{last}:a_{frozen}}$ as a prefix. If solution found, go to [**5.**]{}
[**3.**]{} Spend at most $T/2$ on [Lsearch]{} for a fresh program that starts at the first writeable address and solves [*all*]{} tasks $1..n$. If solution found, go to [**5.**]{}
[**4.**]{} Set $T := 2 T$, and go to [**2.**]{}
[**5.**]{} Let the top non-writeable address $a_{frozen}$ point to the end of the just discovered problem-solving program.
Essential Properties of OOPS {#properties}
----------------------------
The following observations highlight important aspects of [oops]{} and point out in which sense [oops]{} is optimal.
A program starting at $a_{last}$ and solving task $n$ will also solve all tasks up to $n$.
[*Proof*]{} (exploits the nature of self-delimiting programs): Obvious for $n=1$. For $n> 1$: By induction, the code between $a_{last}$ and $a_{frozen}$, which cannot be altered any more, already solves all tasks up to $n-1$. During its application to task $n$ it cannot request any additional tokens that could harm its performance on these previous tasks. So those of its prolongations that solve task $n$ will also solve tasks $1, \ldots, n-1$.
$a_{last}$ does not increase if task $n$ can be more quickly solved by testing prolongations of $q_{a_{last}:a_{frozen}}$ on task $n$, than by testing fresh programs starting above $a_{frozen}$ on all tasks up to $n$.
Once we have found an optimal solver for all tasks in the sequence, at most half of the total future time will be wasted on searching for faster alternatives.
Unlike the learning rate-based bias shifts of [Als]{} (Section \[incremental\]), those of [oops]{} do not reduce the probabilities of programs that were meaningful and executable [*before*]{} the addition of any new $q^i$.
But consider formerly meaningless program prefixes trying to access code for earlier solutions when there weren’t any: such prefixes may suddenly become prolongable and successful, once some solutions to earlier tasks have been stored. That is, unlike with [Als]{} the acceleration potential of [oops]{} is not bought at the risk of an unknown slowdown due to nonoptimal changes of the underlying probability distribution through a heuristically chosen learning rate. As new tasks come along, [oops]{} remains near-bias-optimal with respect to the initial bias, while still being able to profit in from subsequent code bias shifts in an optimal way.
\[fastest\] Given the initial bias and subsequent code bias shifts due to $q^1, q^2, \ldots, $ no bias-optimal searcher with the same initial bias will solve the current task set substantially faster than [oops]{}.
Ignoring hardware-specific overhead (e.g., for measuring time and switching between tasks), [oops]{} will lose at most a factor 2 through allocating half the search time to prolongations of $q_{a_{last}:a_{frozen}}$, and another factor 2 through the incremental doubling of time limits in [Lsearch]{} (necessary because we do not know in advance the final time limit).
\[startaddress\] If the current task (say, task $n$) can already be solved by some previously frozen program $q^k$, then the probability of a solver for task $n$ is at least equal to the probability of the most probable program computing the start address $a(q^k)$ of $q^k$ and setting instruction pointer $ip(n):=a(q^k)$.
As we solve more and more tasks, thus collecting and freezing more and more $q^i$, it will generally become harder and harder to identify and address and copy-edit useful code segments within the earlier solutions.
As a consequence we expect that much of the knowledge embodied by certain $q^j$ actually will be about how to access and copy-edit and otherwise use programs $q^i$ ($i<j$) previously stored below $q^j$.
Tested program prefixes may rewrite the probability distribution on their suffixes in computable ways (based on previously frozen $q^i$), thus temporarily redefining the search space structure of [Lsearch]{}, essentially rewriting the search procedure. If this type of metasearching for faster search algorithms is useful to accelerate the search for a solution to the current problem, then [oops]{} will automatically exploit this.
Since there is no fundamental difference between domain-specific problem-solving programs and programs that manipulate probability distributions and rewrite the search procedure itself, we collapse both learning and metalearning in the same time-optimal framework.
If the overall goal is just to solve one task after another, as opposed to finding a universal solver for all tasks, it suffices to test only on task $n$ in step [**3.**]{}
For example, in an optimization context the $n$-th task usually is not to find a solver for all tasks in the sequence, but just to find an approximation to some unknown optimal solution such that the new approximation is better than the best found so far.
Summary
-------
[Lsearch]{} is about optimal time-sharing, given one problem. [Oops]{} is about optimal time-sharing, given a sequence of problems. The basic principles of [Lsearch]{} can be explained in one line: time-share all program tests such that each program gets a constant fraction of the total search time. Those of [oops]{} require just a few more lines: use self-delimiting programs and freeze those that were successful; given a new task, spend a fixed fraction of the total search time on programs starting with the most recently frozen prefix (test only on the new task, never on previous tasks); spend the rest of the time on fresh programs (when looking for a universal solver, test them on all previous tasks).
[Oops]{} spends part of the total search time for a new problem on programs that exploit previous solutions in computable ways. If the new problem can be solved faster by copy-editing/invoking previous code than by solving the new problem from scratch, then [oops]{} will find this out. If not, then at least it will not suffer from the previous solutions.
If [oops]{} is so simple indeed, then why does the paper not end here but has 31 additional pages? The answer is: to describe the additional efforts required to make OOPS work on realistic limited computers, as opposed to universal machines.
OOPS on Realistic Computers {#realistic}
===========================
Unlike the Turing machines originally used to describe [Lsearch]{} and [Hsearch]{}, realistic computers have limited storage. So we need to efficiently reset storage modifications computed by the numerous programs [oops]{} is testing. Furthermore, our programs typically will be composed from more complex primitive instructions than those of typical Turing machines. In what follows we will address such issues in detail.
Multitasking & Prefix Tracking By Recursive Procedure [**“Try”**]{} {#try}
-------------------------------------------------------------------
[Hsearch]{} and [Lsearch]{} assume potentially infinite storage. Hence they may largely ignore questions of storage management. In any practical system, however, we have to efficiently reuse limited storage. Therefore, in both subsearches of Method \[oopsalgbasics\] (steps [**2**]{} and [**3**]{}), Realistic [oops]{} evaluates alternative prefix continuations by a practical, token-oriented backtracking procedure that can deal with several tasks in parallel, given some [*code bias*]{} in the form of previously found code.
The novel recursive method [**Try**]{} below essentially conducts a depth-first search in program space, where the branches of the search tree are program prefixes (each modifying a bunch of task-specific states), and backtracking (partial resets of partially solved task sets and modifications of internal states and continuation probabilities) is triggered once the sum of the runtimes of the current prefix on all current tasks exceeds the current time limit multiplied by the prefix probability (the product of the history-dependent probabilities of the previously selected prefix components in $Q$). This ensures near-[*bias-optimality*]{} (Def. \[bias-optimal\]), given some initial probabilistic bias on program space $ \subseteq Q^*$.
Given task set $R$, the current goal is to solve all tasks $r \in R$, by a single program that either appropriately uses or extends the current code $q_{0:a_{frozen}}$ (no additional freezing will take place before all tasks in $R$ are solved).
### Overview of “Try” {#tryoverview}
We assume an initial set of user-defined primitive behaviors reflecting prior knowledge and assumptions of the user. Primitives may be assembler-like instructions or time-consuming software, such as, say, theorem provers, or matrix operators for neural network-like parallel architectures, or trajectory generators for robot simulations, or state update procedures for multiagent systems, etc. Each primitive is represented by a token $\in Q$. It is essential that those primitives whose runtimes are not known in advance can be interrupted by [oops]{} at any time.
The searcher’s initial bias is also affected by initial, user-defined, task-dependent probability distributions on the finite or infinite search space of possible self-delimiting program prefixes. In the simplest case we start with a maximum entropy distribution on the tokens, and define prefix probabilities as the products of the probabilities of their tokens. But prefix continuation probabilities may also depend on previous tokens in context sensitive fashion defined by a probabilistic syntax diagram. In fact, we even permit that any executed prefix assigns a task-dependent, self-computed probability distribution to its own possible suffixes (compare Section \[basics\]).
Consider the left-hand side of Figure \[storage\]. All instruction pointers $ip(r)$ of all current tasks $r$ are initialized by some address, typically below the topmost code address, thus accessing the code bias common to all tasks, and/or using task-specific code fragments written into tapes. All tasks keep executing their instructions in parallel until interrupted or all tasks are solved, or until some task’s instruction pointer points to the yet unused address right after the topmost code address. The latter case is interpreted as a request for code prolongation through a new token, where each token has a probability according to the present task’s current state-encoded distribution on the possible next tokens. The deterministic method [**Try**]{} systematically examines all possible code extensions in a depth-first fashion (probabilities of prefixes are just used to order them for runtime allocation). Interrupts and backtracking to previously selected tokens (with yet untested alternatives) and the corresponding partial resets of states and task sets take place whenever one of the tasks encounters an error, or the product of the task-dependent probabilities of the currently selected tokens multiplied by the [*sum*]{} of the runtimes on all tasks exceeds a given total search time limit $T$.
To allow for efficient backtracking, [**Try**]{} tracks effects of tested program prefixes, such as task-specific state modifications (including probability distribution changes) and partially solved task sets, to reset conditions for subsequent tests of alternative, yet untested prefix continuations in an optimally efficient fashion (at most as expensive as the prefix tests themselves).
Since programs are created online while they are being executed, [**Try**]{} will never create impossible programs that halt before all their tokens are read. No program that halts on a given task can be the prefix of another program halting on the same task. It is important to see, however, that in our setup a given prefix that has solved one task (to be removed from the current task set) may continue to demand tokens as it tries to solve other tasks.
### Details of “Try:” Bias-Optimal Depth-First Planning in Program Space {#trydetails}
To allow us to efficiently undo state changes, we use global Boolean variables $mark_i(r)$ (initially [False]{}) for all possible state components $s_i(r)$. We initialize time $t_0 := 0;$ probability $P := 1$; [*q-pointer*]{} $qp := a_{frozen}$ and state $s(r)$ — including $ip(r)$ and $p(r)$ — with task-specific information for all task names $r$ in a so-called [*ring*]{} $R_0$ of tasks, where the expression [*“ring”*]{} indicates that the tasks are ordered in cyclic fashion; $\mid R \mid$ denotes the number of tasks in ring $R$. Given a global search time limit $T$, we [**Try**]{} to solve all tasks in $R_0$, by using existing code in $q= q_{1:qp}$ and / or by discovering an appropriate prolongation of $q$:
—————————————————————————————–
[*($r_0 \in R_0$; returns [True]{} or [False]{}; may have the side effect of increasing $a_{frozen}$ and thus prolonging the frozen code $q_{1:a_{frozen}}$):*]{}
[**1.**]{} Make an empty stack $\cal S$; set local variables $r := r_0; R := R_0; t:= t_0;$ [*Done*]{}$:=$ [False]{}.
[While]{} there are unsolved tasks ($\mid R \mid > 0$) [and]{} there is enough time left ($t \leq PT$) [and]{} instruction pointer valid ($-l(s(r)) \leq ip(r) \leq qp$) [and]{} instruction valid ($1 \leq z(ip(r))(r) \leq n_Q$) [and]{} no halt condition is encountered [*(e.g., error such as division by 0, or robot bumps into obstacle; evaluate conditions in the above order until first satisfied, if any)*]{} [Do:]{}
> Interpret / execute token $z(ip(r))(r)$ according to the rules of the given programming language, continually increasing $t$ by the consumed time. This may modify $s(r)$ including instruction pointer $ip(r)$ and distribution $p(r)$, but not code $q$. Whenever the execution changes some state component $s_i(r)$ whose $mark_i(r)=$ [False]{}, set $mark_i(r) :=$ [True]{} and save the previous value $\hat{s}_i(r)$ by pushing the triple $(i, r, \hat{s}_i(r))$ onto $\cal S$. Remove $r$ from $R$ if solved. [If]{} $\mid R \mid > 0$, [**set $r$ equal to the next task in ring $R$**]{} [*(like in the round-robin method of standard operating systems).*]{} [Else]{} set [*Done*]{} $:=$ [True]{}; $a_{frozen} := qp$ [*(all tasks solved; new code frozen, if any).*]{}
[**2.**]{} Use $\cal S$ to efficiently reset only the modified $mark_i(k)$ to [False]{} [*(the global mark variables will be needed again in step [**3**]{}),*]{} but do not pop $\cal S$ yet.
[**3.**]{} [If]{} $ip(r)=qp+1$ ([**i.e., if there is an online request for prolongation of the current prefix through a new token**]{}): [While]{} [*Done*]{} $=$ [False]{} and there is some yet untested token $Z \in Q$ (untried since $t_0$ as value for $q_{qp+1}$) [Do:]{}
> Set $q_{qp+1}:=Z$ and [*Done*]{} $:=$ [**Try ($qp+1, r, R, t, P * p(r)(Z)$)**]{}, where $p(r)(Z)$ is $Z$’s probability according to current distribution $p(r)$.
[**4.**]{} Use $\cal S$ to efficiently restore only those $s_i(k)$ changed since $t_0$, thus restoring all tapes to their states at the beginning of the current invocation of [**Try**]{}. This will also restore instruction pointer $ip(r_0)$ and original search distribution $p(r_0)$. Return the value of [*Done*]{}.
—————————————————————————————–
A successful [**Try**]{} will solve all tasks, possibly increasing $a_{frozen}$ and prolonging total code $q$. In any case [**Try**]{} will completely restore all states of all tasks. It never wastes time on recomputing previously computed results of prefixes, or on restoring unmodified state components and marks, or on already solved tasks — tracking / undoing effects of prefixes essentially does not cost more than their execution. So the $n$ in Def. \[bias-optimal\] of [*$n$-bias-optimality*]{} is not greatly affected by the undoing procedure: we lose at most a factor 2, ignoring hardware-specific overhead such as the costs of single [*push*]{} and [*pop*]{} operations on a given computer, or the costs of measuring time, etc.
Since the distributions $p(r)$ are modifiable, we speak of self-generated continuation probabilities. As the variable suffix $q' := q_{a_{frozen}+1 : qp}$ of the total code $q = q_{1:qp}$ is growing, its probability can be readily updated: $$\label{prob}
P(q' \mid s^0)= \prod_{i=a_{frozen}+1}^{qp} P^i(q_i \mid s^i),$$ where $s^0$ is an initial state, and $P^i(q_i \mid s^i)$ is the probability of $q_i$, given the state $s^i$ of the task $r$ whose variable distribution $p(r)$ (as a part of $s^i$) was used to determine the probability of token $q_i$ at the moment it was selected. So we allow the probability of $q_{qp+1}$ to depend on $q_{0:qp}$ and intial state $s^0$ in a fairly arbitrary computable fashion. Note that unlike the traditional Turing machine-based setup by [@Levin:74] and [@Chaitin:75] (always yielding binary programs $q$ with probability $2^{-l(q)}$) this framework of self-generated continuation probabilities allows for token selection probabilities close to 1.0, that is, even long programs may have high probability.
[*Example.*]{} In many programming languages the probability of token “(”, given a previous token “[While]{}”, equals 1. Having observed the “(” there is not a lot of new code to execute yet — in such cases the rules of the programming language will typically demand another increment of instruction pointer [*ip(r)*]{}, which will lead to the request of another token through subsequent increment of the topmost code address. However, once we have observed a complete expression of the form “[While]{} (condition) [Do]{} (action),” it may take a long time until the conditional loop — interpreted via $ip(r)$ — is exited and the top address is incremented again, thus asking for a new token.
The [*round robin*]{} [**Try**]{} variant above keeps circling through all unsolved tasks, executing one instruction at a time. Alternative [**Try**]{} variants could also sequentially work on each task until it is solved, then try to prolong the resulting $q$ on the next task, and so on, appropriately restoring previous tasks once it turns out that the current task cannot be solved through prolongation of the prefix solving the earlier tasks. One potential advantage of [*round robin*]{} [**Try**]{} is that it will quickly discover whether the currently studied prefix causes an error for at least one task, in which case it can be discarded immediately.
[**Nonrecursive C-Code**]{}. An efficient iterative (nonrecursive) version of [**Try**]{} for a broad variety of initial programming languages was implemented in C. Instead of local stacks $\cal S$, a single global stack is used to save and restore old contents of modified cells of all tapes / tasks.
Realistic OOPS for Finding Universal Solvers {#unisolve}
--------------------------------------------
Recall that the instruction set $Q$ should contain instructions for invoking or calling code found below $a_{frozen}$, for copying such code into $s(r)$, and for editing the copies and executing the results (examples in Appendix \[language\]).
Now suppose there is an ordered sequence of tasks $r_1, r_2, \ldots$. Inductively suppose we have solved the first $n$ tasks through programs stored below address $a_{frozen}$, and that the most recently discovered program starting at address $a_{last} \leq a_{frozen}$ actually solves all of them, possibly using information conveyed by earlier programs $q^1,q^2,\ldots$. To find a program solving the first $n+1$ tasks, Realistic [oops]{} invokes [**Try**]{} as follows (using set notation for task rings, where the tasks are ordered in cyclic fashion—compare basic Method \[oopsalgbasics\]):
—————————————————————————————–
\[oopsalg\] Initialize current time limit $T := 2$ and $q$-pointer $qp := a_{frozen}$ [*(top frozen address)*]{}.
[**1.**]{} Set instruction pointer $ip(r_{n+1}) := a_{last}$ [*(start address of code solving all tasks up to $n$)*]{}.
> [If]{} [**Try ($qp, r_{n+1}, \{ r_{n+1} \}, 0, \frac{1}{2}$)**]{} then exit.
>
> [*(This means that half the search time is assigned to the most recent $q_{a_{last}:a_{frozen}}$ and all possible prolongations thereof).*]{}
[**2.**]{} [If]{} it is possible to initialize all $n+1$ tasks within time $T$:
> Set local variable $a := a_{frozen} + 1$ [*(first unused address)*]{}; for all $r \in \{ r_1, r_2, \ldots, r_{n+1} \}$ set $ip(r) := a$. [If]{} [**Try ($qp, r_{n+1}, \{ r_1, r_2, \ldots, r_{n+1} \}, 0, \frac{1}{2}$)**]{} set $a_{last} := a$ and exit.
>
> [*(This means that half the time is assigned to all new programs with fresh starts).*]{}
[**3.**]{} Set $T := 2 T$, and go to [**1.**]{}
—————————————————————————————–
Therefore, given tasks $r_1, r_2, \ldots,$ first initialize $a_{last}$; then for $i := 1, 2, \ldots $ invoke Realistic [oops]{}$(i)$ to find programs starting at (possibly increasing) address $a_{last}$, each solving all tasks so far, possibly eventually discovering a universal solver for all tasks in the sequence.
As address $a_{last}$ increases for the $n$-th time, $q^n$ is defined as the program starting at $a_{last}$’s old value and ending right before its new value. Program $q^m$ ($m>i,j$) may exploit $q^i$ by calling it as a subprogram, or by copying $q^i$ into some state $s(r)$, then editing it there, e.g., by inserting parts of another $q^j$ somewhere, then executing the edited variant.
Near-Bias-Optimality of Realistic OOPS {#near-optimal}
--------------------------------------
[oops]{} for realistic computers is not only asymptotically optimal in the sense of [@Levin:73] (see Method \[lsearch\]), but also near bias-optimal (compare Def. \[bias-optimal\], Observation \[fastest\]). To see this, consider a program $p$ solving the current task set within $k$ steps, given current code bias $q_{0:a_{frozen}}$ and $a_{last}$. Denote $p$’s probability by $P(p)$ (compare Eq. (\[prob\]) and Method \[oopsalg\]; for simplicity we omit the obvious conditions). A bias-optimal solver would find a solution within at most $k/P(p)$ steps. We observe that [oops]{} will find a solution within at most $2^3k/P(p)$ steps, ignoring a bit of hardware-specific overhead (for marking changed tape components, measuring time, switching between tasks, etc, compare Section \[try\]): At most a factor 2 might be lost through allocating half the search time to prolongations of the most recent code, another factor 2 for the incremental doubling of $T$ (necessary because we do not know in advance the best value of $T$), and another factor 2 for [**Try**]{}’s resets of states and tasks. So the method is essentially [*8-bias-optimal*]{} (ignoring hardware issues) with respect to the current task. If we do [*not*]{} want to ignore hardware issues: on currently widely used computers we can realistically expect to suffer from slowdown factors smaller than acceptable values such as, say, 100.
The advantages of [oops]{} materialize when $P(p) >> P(p')$, where $p'$ is among the most probable fast solvers of the current task set that do [*not*]{} use previously found code. Ideally, $p$ is already identical to the most recently frozen code. Alternatively, $p$ may be rather short and thus likely because it uses information conveyed by earlier found programs stored below $a_{frozen}$. For example, $p$ may call an earlier stored $q^i$ as a subprogram. Or maybe $p$ is a short and fast program that copies a large $q^i$ into state $s(r_j)$, then modifies the copy just a little bit to obtain $\bar{q}^i$, then successfully applies $\bar{q}^i$ to $r_j$. Clearly, if $p'$ is not many times faster than $p$, then [oops]{} will in general suffer from a much smaller constant slowdown factor than [*non*]{}incremental asymptotically optimal search, precisely reflecting the extent to which solutions to successive tasks do share useful mutual information, given the set of primitives for copy-editing them.
Given an optimal problem solver, problem $r$, current code bias $q_{0:a_{frozen}}$, the most recent start address $a_{last}$, and information about the starts and ends of previously frozen programs $q^1,q^2,\ldots,q^k$, the total search time $T(r,q^1,q^2,\ldots,q^k,a_{last},a_{frozen})$ for solving $r$ can be used to define the degree of bias $$B(r,q^1,q^2,\ldots,q^k,a_{last},a_{frozen}) :=
1 / T(r,q^1,q^2,\ldots,q^k,a_{last},a_{frozen}).$$ Compare the concept of [*conceptual jump size*]{} [@Solomonoff:86; @Solomonoff:89].
Realistic OOPS Variants for Optimization etc.
---------------------------------------------
Sometimes we are not searching for a universal solver, but just intend to solve the most recent task $r_{n+1}$. E.g., for problems of fitness function maximization or optimization, the $n$-th task typically is just to find a program than outperforms the most recently found program. In such cases we should use a reduced variant of [oops]{} which replaces step [**2**]{} of Method \[oopsalg\] by:
> [**2.**]{} Set $a := a_{frozen} + 1$; set $ip(r_{n+1}) := a$. [If]{} [**Try ($qp, r_{n+1}, \{ r_{n+1} \}, 0, \frac{1}{2}$)**]{}, then set $a_{last} := a$ and exit.
Note that the reduced variant still spends significant time on testing earlier solutions: the probability of any prefix that computes the address of some previously frozen program $p$ and then calls $p$ determines a lower bound on the fraction of total search time spent on $p$-like programs. Compare Observation \[startaddress\].
Similar [oops]{} variants will also assign [*prewired*]{} fractions of the total time to the second most recent program and its prolongations, the third most recent program and its prolongations, etc. Other [oops]{} variants will find a program that solves, say, just the $m$ most recent tasks, where $m$ is an integer constant, etc. Yet other [oops]{} variants will assign more (or less) than half of the total time to the most recent code and prolongations thereof. We may also consider probabilistic [oops]{} variants in [*Speed-Prior*]{} style [@Schmidhuber:00v2; @Schmidhuber:02colt].
One not necessarily useful idea: Suppose the number of tasks to be solved by a single program is known in advance. Now we might think of an [OOPS]{} variant that works on all tasks in parallel, again spending half the search time on programs starting at $a_{last}$, half on programs starting at $a_{frozen}+1$; whenever one of the tasks is solved by a prolongation of $q_{a_{last}:a_{frozen}}$ (usually we cannot know in advance which task), we remove it from the current task ring and freeze the code generated so far, thus increasing $a_{frozen}$ (in contrast to [**Try**]{} which does not freeze programs before the entire current task set is solved). If it turns out, however, that not all tasks can be solved by a program starting at $a_{last}$, we have to start from scratch by searching only among programs starting at $a_{frozen}+1$. Unfortunately, in general we cannot guarantee that this approach of [*early freezing*]{} will converge.
Illustrated Informal Recipe for OOPS Initialization {#recipe}
---------------------------------------------------
Given some application, before we can switch on [oops]{} we have to specify our initial bias.
1. Given a problem sequence, collect primitives that embody the prior knowledge. Make sure one can interrupt any primitive at any time, and that one can undo the effects of (partially) executing it.
[ *For example, if the task is path planning in a robot simulation, one of the primitives might be a program that stretches the virtual robot’s arm until its touch sensors encounter an obstacle. Other primitives may include various traditional AI path planners [@Russell:94], artificial neural networks [@Werbos:74; @Rumelhart:86; @Bishop:95] or support vector machines [@Vapnik:92] for classifying sensory data written into temporary internal storage, as well as instructions for repeating the most recent action until some sensory condition is met, etc.* ]{}
2. Insert additional prior bias by defining the rules of an initial probabilistic programming language for combining primitives into complex sequential programs.
[ *For example, a probabilistic syntax diagram may specify high probability for executing the robot’s stretch-arm primitive, given some classification of a sensory input that was written into temporary, task-specific memory by some previously invoked classifier primitive.* ]{}
3. To complete the bias initialization, add primitives for addressing / calling / copying & editing previously frozen programs, and for temporarily modifying the probabilistic rules of the language (that is, these rules should be represented in modifiable task-specific memory as well). Extend the initial rules of the language to accommodate the additional primitives.
[ *For example, there may be a primitive that counts the frequency of certain primitive combinations in previously frozen programs, and temporarily increases the probability of the most frequent ones. Another primitive may conduct a more sophisticated but also more time-consuming Bayesian analysis, and write its result into task-specific storage such that it can be read by subsequent primitives. Primitives for editing code may invoke variants of Evolutionary Computation [@Rechenberg:71; @Schwefel:74], Genetic Algorithms [@Holland:75], Genetic Programming [@Cramer:85; @Banzhaf:98], Ant Colony Optimization [@Gambardella:00; @Dorigo:99], etc.* ]{}
4. Use [oops]{}, which invokes [**Try**]{}, to bias-optimally spend your limited computation time on solving your problem sequence.
The experiments (Section \[experiments\]) will use assembler-like primitives that are much simpler (and thus in a certain sense less biased) than those mentioned in the robot example above. They will suffice, however, to illustrate the basic principles.
Example Initial Programming Language {#seed}
------------------------------------
The efficient search and backtracking mechanism described in Section \[try\] is designed for a broad variety of possible programming languages, possibly list-oriented such as LISP, or based on matrix operations for recurrent neural network-like parallel architectures. Many other alternatives are possible.
A given language is represented by $Q$, the set of initial tokens. Each token corresponds to a primitive instruction. Primitive instructions are computer programs that manipulate tape contents, to be composed by [oops]{} such that more complex programs result. In principle, the “primitives” themselves could be large and time-consuming software, such as, say, traditional AI planners, or theorem provers, or multiagent update procedures, or learning algorithms for neural networks represented on tapes. Compare Section \[recipe\].
For each instruction there is a unique number between 1 and $n_Q$, such that all such numbers are associated with exactly one instruction. Initial knowledge or bias is introduced by writing appropriate primitives and adding them to $Q$. Step [**1**]{} of procedure [**Try**]{} (see Section \[try\]) translates any instruction number back into the corresponding executable code (in our particular implementation: a pointer to a $C$-function). If the presently executed instruction does not directly affect instruction pointer $ip(r)$, e.g., through a conditional jump, or the call of a function, or the return from a function call, then $ip(r)$ is simply incremented.
Given some choice of programming language / initial primitives, [**we typically have to write a new interpreter from scratch,**]{} instead of using an existing one. Why? Because procedure [**Try**]{} (Section \[try\]) needs total control over all (usually hidden and inaccessible) aspects of storage management, including garbage collection etc. Otherwise the storage clean-up in the wake of executed and tested prefixes could become suboptimal.
For the experiments (Section \[experiments\]) we wrote (in $C$) an interpreter for an example, stack-based, universal programming language inspired by [Forth]{} [@Forth:70], whose disciples praise its beauty and the compactness of its programs.
The appendix (Section \[language\]) describes the details. Data structures on tapes (Section \[data\]) can be manipulated by primitive instructions listed in Sections \[stacks\], \[control\], \[self\]. Section \[user\] shows how the user may compose complex programs from primitive ones, and insert them into total code $q$. Once the user has declared his programs, $n_Q$ will remain fixed.
Limitations and Possible Extensions of OOPS {#miscellaneous}
===========================================
In what follows we will discuss to which extent “no free lunch theorems” are relevant to [oops]{} (Section \[nfl\]), which are the essential limitations of [oops]{} (Section \[limits\]), and how to use [oops]{} for reinforcement learning (Section \[rl\]).
How Often Can we Expect to Profit from Earlier Tasks? {#nfl}
-----------------------------------------------------
How likely is it that any learner can indeed profit from earlier solutions? At first naive glance this seems unlikely, since it has been well-known for many decades that most possible pairs of symbol strings (such as problem-solving programs) do not share any algorithmic information; e.g., [@LiVitanyi:97]. Why not? Most possible combinations of strings $x,y$ are algorithmically incompressible, that is, the shortest algorithm computing $y$, given $x$, has the size of the shortest algorithm computing $y$, given nothing (typically a bit more than $l(y)$ symbols), [**which means that $x$ usually does not tell us anything about $y$.**]{} Papers in evolutionary computation often mention [*no free lunch theorems*]{} [@Wolpert:97] which are variations of this ancient insight of theoretical computer science.
Such at first glance discouraging theorems, however, have a quite limited scope: they refer to the very special case of problems sampled from i.i.d. uniform distributions on finite problem spaces. But of course there are infinitely many distributions besides the uniform one. In fact, the uniform one is not only extremely unnatural from any computational perspective — although most objects are random, computing random objects is much harder than computing nonrandom ones — but does not even make sense as we increase data set size and let it go to infinity: [*There is no such thing as a uniform distribution on infinitely many things,*]{} such as the integers.
Typically, successive real world problems are [**not**]{} sampled from uniform distributions. Instead they tend to be closely related. In particular, teachers usually provide sequences of more and more complex tasks with very similar solutions, and in optimization the next task typically is just to outstrip the best approximative solution found so far, given some basic setup that does not change from one task to the next. Problem sequences that humans consider to be [*interesting*]{} are [*atypical*]{} when compared to [*arbitrary*]{} sequences of well-defined problems [@Schmidhuber:97nn]. In fact, it is no exaggeration to claim that almost the entire field of computer science is focused on comparatively few atypical problem sets with exploitable regularities. For all [*interesting*]{} problems the consideration of previous work is justified, to the extent that [*interestingness*]{} implies relatedness to what’s already known [@Schmidhuber:02predictable]. Obviously, [oops]{}-like procedures are advantageous only where such relatedness does exist. In any case, however, they will at least not do much harm.
Fundamental Limitations of OOPS {#limits}
-------------------------------
An appropriate task sequence may help [oops]{} to reduce the slowdown factor of plain [Lsearch]{} through experience. Given a single task, however, [oops]{} does [**not**]{} by itself invent an appropriate series of easier subtasks whose solutions should be frozen first. Of course, since both [Lsearch]{} and [oops]{} may search in general algorithm space, some of the programs they execute may be viewed as self-generated subgoal-definers and subtask solvers. But with a single given task there is no incentive to [*freeze*]{} intermediate solutions [*before*]{} the original task is solved. The potential speed-up of [oops]{} [*does*]{} stem from exploiting external information encoded within an ordered task sequence. This motivates its very name.
Given some final task, a badly chosen training sequence of intermediate tasks may cost more search time than required for solving just the final task by itself, without any intermediate tasks.
[Oops]{} is designed for resetable environments. In [*non*]{}resetable environments it loses its theoretical foundation, and becomes a heuristic method. For example, it is possible to use [oops]{} for designing optimal trajectories of robot arms in virtual [*simulations.*]{} But once we are working with a real physical robot there may be no guarantee that we will be able to precisely reset it as required by backtracking procedure [**Try**]{}.
[Oops]{} neglects one source of potential speed-up relevant for reinforcement learning [@Kaelbling:96]: it does not predict future tasks from previous ones, and does not spend a fraction of its time on solving predicted tasks. Such issues will be addressed in the next subsection.
Outline of OOPS-based Reinforcement Learning (OOPS-RL) {#rl}
------------------------------------------------------
At any given time, a reinforcement learner [@Kaelbling:96] will try to find a [*policy*]{} (a strategy for future decision making) that maximizes its expected future reward. In many traditional reinforcement learning (RL) applications, the policy that works best in a given set of training trials will also be optimal in future test trials [@Schmidhuber:01direct]. Sometimes, however, it won’t. To see the difference between searching (the topic of the previous sections) and reinforcement learning (RL), consider an agent and two boxes. In the $n$-th trial the agent may open and collect the content of exactly one box. The left box will contain $100n$ Swiss Francs, the right box $2^n$ Swiss Francs, but the agent does not know this in advance. During the first 9 trials the optimal policy is [*“open left box.”*]{} This is what a good searcher should find, given the outcomes of the first 9 trials. But this policy will be suboptimal in trial 10. A good reinforcement learner, however, should extract the underlying regularity in the reward generation process and predict the future reward, picking the right box in trial 10, without having seen it yet.
The first general, asymptotically optimal reinforcement learner is the recent AIXI model [@Hutter:01aixi; @Hutter:02selfopt]. It is valid for a very broad class of environments whose reactions to action sequences (control signals) are sampled from arbitrary computable probability distributions. This means that AIXI is far more general than traditional RL approaches. However, while AIXI clarifies the theoretical limits of RL, it is not practically feasible, just like [Hsearch]{} is not. From a pragmatic point of view, what we are really interested in is a reinforcement learner that makes optimal use of given, limited computational resources. In what follows, we will outline how to use [oops]{}-like bias-optimal methods as components of universal yet feasible reinforcement learners.
We need two [oops]{} modules. The first is called the predictor or world model. The second is an action searcher using the world model. The life of the entire system should consist of a sequence of [*cycles*]{} 1, 2, ... At each cycle, a limited amount of computation time will be available to each module. For simplicity we assume that during each cyle the system may take exactly one action. Generalizations to actions consuming several cycles are straight-forward though. At any given cycle, the system executes the following procedure:
1. For a time interval fixed in advance, the predictor is first trained in bias-optimal fashion to find a better world model, that is, a program that predicts the inputs from the environment (including the rewards, if there are any), given a history of previous observations and actions. So the $n$-th task ($n=1,2,\ldots$) of the first [oops]{} module is to find (if possible) a better predictor than the best found so far.
2. Once the current cycle’s time for predictor improvement is used up, the current world model (prediction program) found by the first [oops]{} module will be used by the second module, again in bias-optimal fashion, to search for a future action sequence that maximizes the predicted cumulative reward (up to some time limit). That is, the $n$-th task ($n=1,2,\ldots$) of the second [oops]{} module will be to find a control program that computes a control sequence of actions, to be fed into the program representing the current world model (whose input predictions are successively fed back to itself in the obvious manner), such that this control sequence leads to higher predicted reward than the one generated by the best control program found so far.
3. Once the current cycle’s time for control program search is used up, we will execute the current action of the best control program found in step 2. Now we are ready for the next cycle.
The approach is reminiscent of an earlier, heuristic, non-bias-optimal RL approach based on two adaptive recurrent neural networks, one representing the world model, the other one a controller that uses the world model to extract a policy for maximizing expected reward [@Schmidhuber:91nips]. The method was inspired by previous combinations of [*non*]{}recurrent, [*reactive*]{} world models and controllers [@Werbos:87specifications; @NguyenWidrow:89; @JordanRumelhart:90].
At any given time, until which temporal horizon should the predictor try to predict? In the AIXI case, the proper way of treating the temporal horizon is not to discount it exponentially, as done in most traditional work on reinforcement learning, but to let the future horizon grow in proportion to the learner’s lifetime so far [@Hutter:02selfopt]. It remains to be seen whether this insight carries over to [oops]{}-based RL. In particular, is it possible to prove that variants of OOPS-RL as above are a near-bias-optimal way of spending a given amount of computation time on RL problems? Or should we instead combine [oops]{} and Hutter’s time-bounded AIXI$(t,l)$ model? We observe that certain important problems are still open.
Experiments
===========
Experiments can tell us something about the usefulness of a particular initial bias such as the one incorporated by a particular programming language with particular initial instructions. In what follows we will describe illustrative problems and results obtained using the [Forth]{}-inspired language specified in the appendix (Section \[language\]). The latter should be consulted for the details of the instructions appearing in programs found by [oops]{}.
While explaining the learning system’s setup, we will also try to identify several more or less hidden sources of initial bias.
On Task-Specific Initialization {#init}
-------------------------------
Besides the $61$ initial primitive instructions from Sections \[stacks\], \[control\], \[self\] (appendix), the only user-defined (complex) tokens are those declared in Section \[user\] (except for the last one, [tailrec]{}). That is, we have a total of $61+7=68$ initial non-task-specific primitives.
Given any task, we add task-specific instructions. In the present experiments, we do [*not*]{} provide a [*probabilistic syntax diagram*]{} defining conditional probabilities of certain tokens, given previous tokens. Instead we simply start with a maximum entropy distribution on the $n_Q >68$ tokens $Q_i$, initializing all probabilities $p_i= \frac{1}{n_Q}$, setting all $p[curp][i]:=1$ and $sum[curp]:=n_Q$ (compare Section \[data\]).
Note that the instruction numbers themselves significantly affect the initial bias. Some instruction numbers, in particular the small ones, are computable by very short programs, others are not. In general, programs consisting of many instructions that are not so easily computable, given the initial arithmetic instructions (Section \[stacks\]), tend to be less probable. Similarly, as the number of frozen programs grows, those with higher addresses in general become harder to access, that is, the address computation may require longer subprograms.
For the experiments we [*insert substantial prior bias*]{} by assigning the lowest (easily computable) instruction numbers to the task-specific instructions, and by [**boosting**]{} (see instruction [*boostq*]{} in Section \[self\]) the appropriate [*“small number pushers”*]{} (such as [*c1, c2, c3*]{}; compare Section \[stacks\]) that push onto data stack [*ds*]{} the numbers of the task-specific instructions, such that they become executable as part of code on [*ds*]{}. We also boost the simple arithmetic instructions [*by2*]{} (multiply top stack element by 2) and [*dec*]{} (decrement top stack element), such that the system can easily create other integers from the probable ones. For example, without these boosts the code sequence [*(c3 by2 by2 dec)*]{} (which returns integer 11) would be much less likely. Finally we express our initial belief in the occasional future usefulness of previously useful instructions, by also boosting [*boostq*]{} itself.
The following numbers represent maximal values enforced in the experiments: state size: $l(s)=3000$; absolute tape cell contents $s_i(r)$: $10^9$; number of self-made functions: $100$, of self-made search patterns or probability distributions per tape: $20$; callstack pointer: $maxcp=100$; data stack pointers: $maxdp=maxDp=200$.
Towers of Hanoi: the Problem {#hanoi}
----------------------------
Given are $n$ disks of $n$ different sizes, stacked in decreasing size on the first of three pegs. One may move some peg’s top disk to the top of another peg, one disk at a time, but never a larger disk onto a smaller. The goal is to transfer all disks to the third peg. Remarkably, the fastest way of solving this famous problem requires $2^n - 1$ moves $(n \geq 0)$.
The problem is of the [*reward-only-at-goal*]{} type — given some instance of size $n$, there is no intermediate reward for achieving instance-specific subgoals.
The exponential growth of minimal solution size is what makes the problem interesting: Brute force methods searching in raw solution space will quickly fail as $n$ increases. But the rapidly growing solutions do have something in common, namely, the short algorithm that generates them. Smart searchers will exploit such algorithmic regularities. Once we are searching in general algorithm space, however, it is essential to efficiently allocate time to algorithm tests. This is what [oops]{} does, in near-bias-optimal incremental fashion.
Untrained humans find it hard to solve instances $n>6$. [@Anderson:86] applied traditional reinforcement learning methods and was able to solve instances up to $n=3$, solvable within at most 7 moves. [@Langley:85] used learning production systems and was able to solve instances up to $n=5$, solvable within at most 31 moves. ([*Side note:*]{} [@Baum:99] also applied an alternative reinforcement learner based on the artificial economy by [@Holland:85] to a simpler 3 peg blocks world problem where any disk may be placed on any other; thus the required number of moves grows only linearly with the number of disks, not exponentially; [@Kwee:01market] were able to replicate their results for $n$ up to 5.) Traditional AI planning procedures—compare chapter V by [@Russell:94] and [@Jana:97]—do not learn but systematically explore all possible move combinations, using only absolutely necessary task-specific primitives (while [oops]{} will later use more than 70 general instructions, most of them unnecessary). On current personal computers AI planners tend to fail to solve Hanoi problem instances with $n > 15$ due to the exploding search space (Jana Koehler, IBM Research, personal communication, 2002). [oops]{}, however, searches program space instead of raw solution space. Therefore, in principle it should be able to solve arbitrary instances by discovering the problem’s elegant recursive solution—given $n$ and three pegs $S,A,D$ (source peg, auxiliary peg, destination peg), define the following procedure:
> *[hanoi]{}(S,A,D,n): [If]{} $n=0$ exit; [Else Do]{}:*
>
> call [hanoi]{}(S, D, A, n-1); move top disk from S to D; call [hanoi]{}(A, S, D, n-1).
Task Representation and Domain-Specific Primitives {#setup}
--------------------------------------------------
The $n$-th problem is to solve all Hanoi instances up to instance $n$. Following our general rule, we represent the dynamic environment for task $n$ on the $n$-th task tape, allocating $n+1$ addresses for each peg, to store the order and the sizes of its current disks, and a pointer to its top disk (0 if there isn’t one).
We represent pegs $S, A, D$ by numbers 1, 2, 3, respectively. That is, given an instance of size $n$, we push onto data stack [*ds*]{} the values $1, 2, 3, n$. By doing so we insert [*substantial, nontrivial prior knowledge*]{} about the fact that it is useful to represent each peg by a symbol, and to know the problem size in advance. The task is completely defined by $n$; the other 3 values are just useful for the following primitive instructions added to the programming language of Section \[language\]: Instruction [*mvdsk()*]{} assumes that $S, A, D$ are represented by the first three elements on data stack [*ds*]{} above the current base pointer $cs[cp].base$ (Section \[data\]). It operates in the obvious fashion by moving a disk from peg $S$ to peg $D$. Instruction [*xSA()*]{} exchanges the representations of $S$ and $A$, [*xAD()*]{} those of $A$ and $D$ (combinations may create arbitrary peg patterns). Illegal moves cause the current program prefix to halt. Overall success is easily verifiable since our objective is achieved once the first two pegs are empty.
Incremental Learning: First Solve Simpler Context Free Language Tasks {#1n2n}
---------------------------------------------------------------------
Despite the near-bias-optimality of [oops]{}, within reasonable time (a week) on a personal computer, the system with 71 initial instructions was not able to solve instances involving more than 3 disks. What does this mean? [*Since search time of an optimal searcher is a natural measure of initial bias,*]{} it just means that the already nonnegligible bias towards our task set was still too weak.
This actually gives us an opportunity to demonstrate that [oops]{} can indeed benefit from its incremental learning abilities. Unlike Levin’s and Hutter’s [*non*]{}incremental methods it always tries to profit from experience with previous tasks. Therefore, to properly illustrate its behavior, we need an example where it [*does*]{} profit. In what follows, we will first train it on additional, easier tasks, to teach it something about recursion, hoping that the resulting code bias shifts will help to solve the Hanoi tasks as well.
For this purpose we use a seemingly unrelated problem class based on the context free language $\{1^n2^n\}$: given input $n$ on the data stack [*ds*]{}, the goal is to place symbols on the auxiliary stack [*Ds*]{} such that the $2n$ topmost elements are $n$ 1’s followed by $n$ 2’s. Again there is no intermediate reward for achieving instance-specific subgoals.
After every executed instruction we test whether the objective has been achieved. By definition, the time cost per test (measured in unit time steps; Section \[primitives\]) equals the number of considered elements of [*Ds*]{}. Here we have an example of a test that may become more expensive with instance size.
We add two more instructions to the initial programming language: instruction [*1toD()*]{} pushes 1 onto [*Ds*]{}, instruction [*2toD()*]{} pushes 2. Now we have a total of five task-specific instructions (including those for Hanoi), with instruction numbers 1, 2, 3, 4, 5, for [*1toD*]{}, [*2toD*]{}, [*mvdsk*]{}, [*xSA*]{}, [*xAD*]{}, respectively, which gives a total of 73 initial instructions.
So we first boost (Section \[self\]) the “small number pushers” [*c1, c2*]{} (Section \[stacks\]) for the first training phase where the $n$-th task $(n=1, \ldots, 30)$ is to solve all $1^n2^n$ problem instances up to $n$. Then we undo the $1^n2^n$-specific boosts of [*c1, c2*]{} and boost instead the Hanoi-specific instruction number pushers $c3, c4, c5$ for the subsequent training phase where the $n$-th task (again $n=1, \ldots, 30$) is to solve all Hanoi instances up to $n$.
C-Code
------
All of the above was implemented by a dozen pages of code written in C, mostly comments and documentation: Multitasking and storage management through an iterative variant of [*round robin*]{} [**Try**]{} (Section \[try\]); interpreter and 62 basic instructions (Section \[language\]); simple user interface for complex declarations (Section \[user\]); applications to $1^n2^n$-problems (Section \[1n2n\]) and Hanoi problems (Section \[hanoi\]). The current nonoptimized implementation considers between one and two million discrete unit time steps per second on an off-the-shelf PC (1.5 GHz).
Experimental Results for Both Task Sets
---------------------------------------
Within roughly 0.3 days, [oops]{} found and froze code solving all thirty $1^n2^n$-tasks. Thereafter, within 2-3 additional days, it also found a universal Hanoi solver. The latter does not call the $1^n2^n$ solver as a subprogram (which would not make sense at all), but it does profit from experience: it begins with a rather short prefix that reshapes the distribution on the possible suffixes, an thus the search space, by temporally increasing the probabilities of certain instructions of the earlier found $1^n2^n$ solver. This in turn happens to increase the probability of finding a Hanoi-solving suffix. It is instructive to study the sequence of intermediate solutions. In what follows we will transform integer sequences discovered by [oops]{} back into readable programs (compare instruction details in Section \[language\]).
1. For the $1^n2^n$-problem, within 480,000 time steps (less than a second), [oops]{} found nongeneral but working code for $n=1$: [*(defnp 2toD).*]{}
2. At time $10^7$ (roughly 10 [*s*]{}) it had solved the 2nd instance by simply prolonging the previous code, using the old, unchanged start address $a_{last}$: [*(defnp 2toD grt c2 c2 endnp).*]{} So this code solves the first two instances.
3. At time $10^8$ (roughly 1 [*min*]{}) it had solved the 3rd instance, again through prolongation:
[*(defnp 2toD grt c2 c2 endnp boostq delD delD bsf 2toD).*]{}
Here instruction [*boostq*]{} greatly boosted the probabilities of the subsequent instructions.
4. At time $2.85*10^9$ (less than 1 hour) it had solved the 4th instance through prolongation:
[*(defnp 2toD grt c2 c2 endnp boostq delD delD bsf 2toD fromD delD delD delD fromD bsf by2 bsf).*]{}
5. At time $3*10^9$ (a few minutes later) it had solved the 5th instance through prolongation:
[*(defnp 2toD grt c2 c2 endnp boostq delD delD bsf 2toD fromD delD delD delD fromD bsf by2 bsf by2 fromD delD delD fromD cpnb bsf).*]{}
The code found so far was lengthy and unelegant. But it does solve the first 5 instances.
6. Finally, at time $30,665,044,953$ (roughly 0.3 days), [oops]{} had created and tested a new, elegant, recursive program (no prolongation of the previous one) with a new increased start address $a_{last}$, solving all instances up to 6: [*(defnp c1 calltp c2 endnp).*]{}
That is, it was cheaper to solve all instances up to 6 by discovering and applying this new program to all instances so far, than just prolonging the old code on instance 6 only.
7. The program above turns out to be a near-optimal universal $1^n2^n$-problem solver. On the stack, it constructs a 1-argument procedure that returns nothing if its input argument is 0, otherwise calls the instruction [*1toD*]{} whose code is 1, then calls itself with a decremented input argument, then calls [*2toD*]{} whose code is 2, then returns.
That is, all remaining $1^n2^n$-tasks can profit from the solver of instance 6. Reusing this current program $q_{a_{last}:a_{frozen}}$ again and again, within very few additional time steps (roughly 20 milliseconds), by time $30,665,064,543$, [oops]{} had also solved the remaining 24 $1^n2^n$-tasks up to $n=30$.
8. Then [oops]{} switched to the Hanoi problem. Almost immediately (less than 1 [*ms*]{} later), at time $30,665,064,995$, it had found the trivial code for $n=1$: [*(mvdsk).*]{}
9. Much later, by time $260 *10^9$ (more than 1 day), it had found fresh yet somewhat bizarre code (new start address $a_{last}$) for $n=1,2$: [*(c4 c3 cpn c4 by2 c3 by2 exec).*]{}
The long search time so far indicates that the Hanoi-specific bias still is not very high.
10. Finally, by time $541 *10^9$ (roughly 3 days), it had found fresh code (new $a_{last}$) for $n=1,2,3$:
[*(c3 dec boostq defnp c4 calltp c3 c5 calltp endnp).*]{}
11. The latter turns out to be a near-optimal universal Hanoi solver, and greatly profits from the code bias embodied by the earlier found $1^n2^n$-solver (see analysis in Section \[discussion\] below). Therefore, by time $679*10^9$, [oops]{} had solved the remaining 27 tasks for $n$ up to 30, reusing the same program $q_{a_{last}:a_{frozen}}$ again and again.
The entire 4-day search for solutions to all 60 tasks tested 93,994,568,009 prefixes corresponding to 345,450,362,522 instructions costing 678,634,413,962 time steps. [*Recall once more that search time of an optimal solver is a natural measure of initial bias.*]{} Clearly, most tested prefixes are short — they either halt or get interrupted soon. Still, some programs do run for a long time; for example, the run of the self-discovered universal Hanoi solver working on instance 30 consumed 33 billion steps, which is already 5 % of the total time. The stack used by the iterative equivalent of procedure [**Try**]{} for storage management (Section \[try\]) never held more than 20,000 elements though.
Analysis of the Results {#discussion}
-----------------------
The final 10-token Hanoi solution demonstrates the benefits of incremental learning: it greatly profits from rewriting the search procedure with the help of information conveyed by the earlier recursive solution to the $1^n2^n$-problem. How?
The prefix [*(c3 dec boostq)*]{} (probability 0.003) prepares the foundations: Instruction [*c3*]{} pushes 3; [*dec*]{} decrements this; [*boostq*]{} takes the result 2 as an argument (interpreted as an address) and thus boosts the probabilities of all components of the 2nd frozen program, which happens to be the previously found universal $1^n2^n$-solver. This causes an online bias shift on the space of possible suffixes: it greatly increases the probability that [*defnp, calltp, endnp,*]{} will appear in the remainder of the online-generated program. These instructions in turn are helpful for building (on the data stack [*ds*]{}) the double-recursive procedure generated by the suffix [*(defnp c4 calltp c3 c5 calltp endnp),*]{} which essentially constructs (on data stack $ds$) a 4-argument procedure that returns nothing if its input argument is 0, otherwise decrements the top input argument, calls the instruction [*xAD*]{} whose code is 4, then calls itself on a copy of the top 4 arguments, then calls [*mvdsk*]{} whose code is 5, then calls [*xSA*]{} whose code is 3, then calls itself on another copy of the top 4 arguments, then makes yet another (unnecessary) argument copy, then returns (compare the standard Hanoi solution).
The total probability of the final solution, given the previous codes, is calculated as follows: since $n_Q=73$, given the boosts of [*c3, c4, c5, by2, dec, boostq*]{}, we have probability $( \frac{1+73}{7 * 73})^3$ for the prefix [*(c3 dec boostq)*]{}; since this prefix further boosts [*defnp, c1, calltp, c2, endnp,*]{} we have probability $( \frac{1+73}{12 * 73})^7$ for the suffix [*(defnp c4 calltp c3 c5 calltp endnp)*]{}. That is, the probability of the complete 10-symbol code is $9.3 * 10^{-11}$. On the other hand, the probability of the essential Hanoi-specific suffix [*(defnp c4 calltp c3 c5 calltp endnp)*]{}, given just the initial boosts, is only $( \frac{1+73}{7 * 73})^3 ( \frac{1}{7 * 73})^4 = 4.5 * 10^{-14}$, which explains why it was not quickly found without the help of the solution to an easier problem set. (Without any initial boosts its probability would actually have been similar: $(\frac{1}{73})^7=9 * 10^{-14}$.) This would correspond to a search time of several years, as opposed to a few days.
So in this particular setup the simple recursion for the $1^n2^n$-problem indeed provided useful incremental training for the more complex Hanoi recursion, speeding up the search by a factor of 1000 or so.
On the other hand, the search for the universal solver for all $1^n2^n$-problems (first found with instance $n=6$) did not at all profit from solutions to earlier solved tasks (although instances $n>6$ did profit).
Note that the time spent by the final 10-token Hanoi solver on increasing the probabilities of certain instructions and on constructing executable code on the data stack (less than 50 time steps) quickly becomes negligible as the Hanoi instance size grows. In this particular application, most time is spent on executing the code, not on constructing it.
Once the universal Hanoi solver was discovered, why did the solution of the remaining Hanoi tasks substantially increase the total time (by roughly 25 %)? Because the sheer [*runtime*]{} of the discovered, frozen, near-optimal program on the remaining tasks was already comparable to the previously consumed [*search time*]{} for this program, due to the very nature of the Hanoi task: Recall that a solution for $n=30$ takes more than a billion [*mvdsk*]{} operations, and that for each [*mvdsk*]{} several other instructions need to be executed as well. Note that experiments with traditional reinforcement learners [@Kaelbling:96] rarely involve problems whose solution sizes exceed a few thousand steps.
Note also that we could continue to solve Hanoi tasks up to $n>40$. The execution time required to solve such instances with an optimal solver greatly exceeds the search time required for finding the solver itself. There it does not matter much whether [oops]{} already starts with a prewired Hanoi solver, or first has to discover one, since the initial search time for the solver becomes negligible anyway.
Of course, different initial bias can yield dramatically different results. For example, using hindsight we could set to zero the probabilities of all 73 initial instructions (most are unnecessary for the 30 Hanoi tasks) except for the 7 instructions used by the Hanoi-solving suffix, then make those 7 instructions equally likely, and thus obtain a comparatively high Hanoi solver probability of $(\frac{1}{7})^7 = 1.2 * 10^{-6}$. This would allow for finding the solution to the 10 disk Hanoi problem within less than an hour, without having to learn easier tasks first (at the expense of obtaining a nonuniversal initial programming language). The point of this experimental section, however, is [**not**]{} to find the most reasonable initial bias for particular problems, but to illustrate the basic functionality of the first general, near-bias-optimal, incremental learner.
[**Future research**]{} may focus on devising particularly compact, particularly reasonable sets of initial codes with particularly broad practical applicability. It may turn out that the most useful initial languages are not traditional programming languages similar to the [Forth]{}-like one from Section \[language\], but instead based on a handful of primitive instructions for massively parallel cellular automata [@Ulam:50; @Neumann:66; @Zuse:69; @Wolfram:84], or on a few nonlinear operations on matrix-like data structures such as those used in recurrent neural network research [@Werbos:74; @Rumelhart:86; @Bishop:95]. For example, we could use the principles of [oops]{} to create a non-gradient-based, near-bias-optimal variant of the successful recurrent network metalearner by [@Hochreiter:01meta]. It should also be of interest to study probabilistic [*Speed Prior*]{}-based [oops]{} variants [@Schmidhuber:02colt] and to devise applications of [oops]{}-like methods as components of universal reinforcement learners (Section \[rl\]). In ongoing work, we are applying [oops]{} to the problem of optimal trajectory planning for robotics in a realistic physics simulation. This involves the interesting trade-off between comparatively fast program-composing primitives or [*“thinking primitives”*]{} and time-consuming [*“action primitives”*]{}, such as [*stretch-arm-until-touch-sensor-input*]{} (compare Section \[recipe\]).
Physical Limitations of OOPS {#outlook}
----------------------------
Due to its generality and its optimality properties, [oops]{} should scale to large problems in an essentially unbeatable fashion, thus raising the question: Which are its physical limitations? To give a very preliminary answer, we first observe that with each decade computers become roughly 1000 times faster by cost, reflecting Moore’s empirical law first formulated in 1965. Within a few decades [*non*]{}reversible computation will encounter fundamental heating problems associated with high density computing [@Bennett:82]. Remarkably, however, [oops]{} can be naturally implemented using the [*reversible*]{} computing strategies [@Fredkin:82], since it completely resets all state modifications due to the programs it tests. But even when we naively extrapolate Moore’s law, within the next century [oops]{} will hit the limit of [@Bremermann:82]: approximately $10^{51}$ operations per second on $10^{32}$ bits for the “ultimate laptop” [@Lloyd:00] with 1 kg of mass and 1 liter of volume. Clearly, the Bremermann limit constrains the maximal [*conceptual jump size*]{} [@Solomonoff:86; @Solomonoff:89] from one problem to the next. For example, given some prior code bias derived from solutions to previous problems, within 1 minute, a sun-sized [oops]{} (roughly $2 \times 10^{30} kg$) might be able to solve an additional problem that requires finding an additional 200 bit program with, say, $10^{20}$ steps runtime. But within the next centuries, [oops]{} will fail on new problems that require additional 300 bit programs of this type, since the speed of light greatly limits the acquisition of additional mass, through a function quadratic in time.
Still, even the comparatively modest hardware speed-up factor $10^9$ expected for the next 30 years appears quite promising for [oops]{}-like systems. For example, with the 73 token language used in the experiments (Section \[experiments\]), we could learn from scratch (within a day or so) to solve the 20 disk Hanoi problem ($>10^6$ moves), without any need for boosting task-specific instructions, or for incremental search through instances $< 20$, or for additional training sequences of easier tasks. Comparable speed-ups will be achievable much earlier by distributing [oops]{} across large computer networks or by using supercomputers—on the fastest current machines our 60 tasks (Section \[experiments\]) should be solvable within a few seconds as opposed to 4 days.
Acknowledgments {#acknowledgments .unnumbered}
---------------
Thanks to Ray Solomonoff, Marcus Hutter, Sepp Hochreiter, Bram Bakker, Alex Graves, Douglas Eck, Viktor Zhumatiy, Giovanni Pettinaro, Andrea Rizzoli, Monaldo Mastrolilli, Ivo Kwee, and several unknown NIPS referees, for useful discussions or helpful comments on drafts or short versions of this paper, to Jana Koehler for sharing her insights concerning AI planning procedures, and to Philip J. Koopman Jr. for granting permission to reprint the quote in Section \[control\]. Hutter’s frequently mentioned work was funded through the author’s SNF grant 2000-061847 “Unification of universal inductive inference and sequential decision theory.”
Example Programming Language {#language}
============================
[Oops]{} can be seeded with a wide variety of programming languages. For the experiments, we wrote an interpreter for a stack-based universal programming language inspired by [Forth]{} [@Forth:70]. We provide initial instructions for defining and calling recursive functions, iterative loops, arithmetic operations, and domain-specific behavior. Optimal metasearching for better search algorithms is enabled through bias-shifting instructions that can modify the conditional probabilities of future search options in currently running self-delimiting programs. Sections \[data\], explains the basic data structures; Sections \[stacks\], \[control\], \[self\] define basic primitive instructions; Section \[user\] shows how to compose complex programs from primitive ones, and explains how the user may insert them into total code $q$.
Data Structures on Tapes {#data}
------------------------
\[specific\]
[**Symbol**]{} [**Description**]{}
----------------- ---------------------------------------------------------------------------- -- -- -- -- --
[*ds*]{} data stack holding arguments of functions, possibly also edited code
[*dp*]{} stack pointer of [*ds*]{}
[*Ds*]{} auxiliary data stack
[*Dp*]{} stack pointer of [*Ds*]{}
[*cs*]{} call stack or runtime stack to handle function calls
[*cp*]{} stack pointer of [*cs*]{}
$cs[cp].ip$ current function call’s instruction pointer $ip(r) := cs[cp](r).ip$
$cs[cp].base$ current base pointer into [*ds*]{} right below the current input arguments
$cs[cp].out$ number of return values expected on top of [*ds*]{} above $cs[cp].base$
[*fns*]{} stack of currently available self-made functions
[*fnp*]{} stack pointer of [*fns*]{}
$fns[fnp].code$ start address of code of most recent self-made function
$fns[fnp].in$ number of input arguments of most recent self-made function
$fns[fnp].out$ number of return values of most recent self-made function
[*pats*]{} stack of search patterns (probability distributions on $Q$)
[*patp*]{} stack pointer of [*pats*]{}
[*curp*]{} pointer to current search pattern in [*pats*]{}, $0 \leq curp \leq patp$
$p[curp][i]$ $i$-th numerator of current search pattern
$sum[curp]$ denominator; the current probability of $Q_i$ is $p[curp][i] / sum[curp]$
Each tape $r$ contains various stack-like data structures represented as sequences of integers. For any stack $Xs(r)$ introduced below (here $X$ stands for a character string reminiscent of the stack type) there is a (frequently not even mentioned) stack pointer $Xp(r)$; $0 \leq Xp(r) \leq maxXp$, located at address $a_{Xp}$, and initialized by 0. The $n$-th element of $Xs(r)$ is denoted $Xs[n](r)$. For simplicity we will often omit tape indices $r$. Each tape has:
1. A data stack [*ds(r)*]{} (or [*ds*]{} for short, omitting the task index) for storing function arguments. (The corresponding stack pointer is $dp: 0 \leq dp \leq maxdp$).
2. An auxiliary data stack [*Ds*]{}.
3. A runtime stack or [*callstack*]{} $cs$ for handling (possibly recursive) functions. Callstack pointer $cp$ is initialized by 0 for the “main” program. The $k$-th callstack entry ($k = 0, \ldots, cp$) contains 3 variables: an instruction pointer $cs[k](r).ip$ (or simply $cs[k].ip$, omitting task index $r$) initialized by the start address of the code of some procedure $f$, a pointer $cs[k].base$ pointing into [*ds*]{} right below the values which are considered input arguments of $f$, and the number $cs[k].out$ of return values $ds[cs[k].base+1], \ldots, ds[dp]$ expected on top of [*ds*]{} once $f$ has returned. $cs[cp]$ refers to the topmost entry containing the current instruction pointer $ip(r):=cs[cp](r).ip$.
4. A stack [*fns*]{} of entries describing self-made functions. The entry for function [*fn*]{} contains 3 integer variables: the start address of [*fn*]{}’s code, the number of input arguments expected by [*fn*]{} on top of [*ds*]{}, and the number of output values to be returned.
5. A stack [*pats*]{} of search patterns. $pats[i]$ stands for a probability distribution on search options (next instruction candidates). It is represented by $n_Q+1$ integers $p[i][n]$ ($1 \leq n \leq n_Q$) and [*sum\[i\]*]{} (for efficiency reasons). Once $ip(r)$ hits the current search address $l(q)+1$, the history-dependent probability of the $n$-th possible next instruction $Q_n$ (a candidate value for $q_{ip(r)}$) is given by $p[curp][n] / sum[curp]$, where $curp$ is another tape-represented variable ($0 \leq curp \leq patp$) indicating the current search pattern.
6. A binary [*quoteflag*]{} determining whether the instructions pointed to by [*ip*]{} will get executed or just [*quoted*]{}, that is, pushed onto [*ds*]{}.
7. A variable holding the index $r$ of this tape’s task.
8. A stack of integer arrays, each having a name, an address, and a size (not used in this paper, but implemented and mentioned for the sake of completeness).
9. Additional problem-specific dynamic data structures for problem-specific data, e.g., to represent changes of the environment. An example environment for the [*Towers of Hanoi*]{} problem is described in Section \[experiments\].
Primitive Instructions {#primitives}
----------------------
Most of the 61 tokens below do not appear in the solutions found by [oops]{} in the experiments (Section \[experiments\]). Still, we list all of them for completeness’ sake, and to provide at least one example way of seeding [oops]{} with an initial set of behaviors.
In the following subsections, any instruction of the form [*inst ($x_1, \ldots, x_n$)*]{} expects its $n$ arguments on top of data stack [*ds*]{}, and replaces them by its return values, adjusting [*dp*]{} accordingly — the form [*inst()*]{} is used for instructions without arguments.
Illegal use of any instruction will cause the currently considered program prefix to halt. In particular, it is illegal to set variables (such as stack pointers or instruction pointers) to values outside their prewired given ranges, or to pop empty stacks, or to divide by zero, or to call a nonexistent function, etc.
Since CPU time measurements on our PCs turned out to be unreliable, we defined our own, rather realistic time scales. By definition, most instructions listed below cost exactly 1 unit time step. Some, however, consume more time: Instructions making copies of strings with length $n$ (such as [*cpn(n)*]{}) cost $n$ time steps; so do instructions (such as [*find(x)*]{}) accessing an [*a priori*]{} unknown number $n$ of tape cells; so do instructions (such as [*boostq(k)*]{}) modifying the probabilities of an [*a priori*]{} unknown number $n$ of instructions.
### Basic Data Stack-Related Instructions {#stacks}
1. [Arithmetic.]{} [*c0(),c1(), c2(), ..., c5()*]{} return constants 0, 1, 2, 3, 4, 5, respectively; [*inc(x)*]{} returns $x+1$; [*dec(x)*]{} returns $x-1$; [*by2(x)*]{} returns $2x$; [*add(x,y)*]{} returns $x+y$; [*sub(x,y)*]{} returns $x-y$; [*mul(x,y)*]{} returns $x*y$; [*div(x,y)*]{} returns the smallest integer $\leq x/y$; [*powr(x,y)*]{} returns $x^y$ (and costs $y$ unit time steps).
2. [Boolean.]{} Operand [*eq(x,y)*]{} returns 1 if $x=y$, otherwise 0. Analogously for [*geq(x,y)*]{} (greater or equal), [*grt(x,y)*]{} (greater). Operand [*and(x,y)*]{} returns 1 if $x>0$ and $y>0$, otherwise 0. Analogously for [*or(x,y)*]{}. Operand [*not(x)*]{} returns 1 if $x \leq 0$, otherwise 0.
3. [Simple Stack Manipulators.]{} [*del()*]{} decrements [*dp*]{}; [*clear()*]{} sets $dp :=0$; [*dp2ds()*]{} returns [*dp*]{}; [*setdp(x)*]{} sets $dp :=x$; [*ip2ds()*]{} returns $cs[cp].ip$; [*base()*]{} returns $cs[cp].base$; [*fromD()*]{} returns $Ds[Dp]$; [*toD()*]{} pushes $ds[dp]$ onto [*Ds*]{}; [*delD()*]{} decrements [*Dp*]{}; [*topf()*]{} returns the integer name of the most recent self-made function; [*intopf()*]{} and [*outopf()*]{} return its number of requested inputs and outputs, respectively; [*popf()*]{} decrements [*fnp*]{}, returning its old value; [*xmn(m,n)*]{} exchanges the $m$-th and the $n$-th elements of [*ds*]{}, measured from the stack’s top; [*ex()*]{} works like [*xmn(1,2)*]{}; [*xmnb(m,n)*]{} exchanges the $m$-th and the $n$-th elements [*above*]{} the current base $ds[cs[cp].base]$; [*outn(n)*]{} returns $ds[dp-n+1]$; [*outb(n)*]{} returns $ds[cs[cp].base + n]$ (the $n$-th element above the base pointer); [*inn(n)*]{} copies $ds[dp]$ onto $ds[dp-n+1]$; [*innb(n)*]{} copies $ds[dp]$ onto $ds[cs[cp].base + n]$.
4. [Pushing Code.]{} Instruction [*getq(n)*]{} pushes onto [*ds*]{} the sequence beginning at the start address of the $n$-th frozen program (either user-defined or frozen by [oops]{}) and ending with the program’s final token. [*insq(n,a)*]{} inserts the $n$-th frozen program above $ds[cs[cp].base+a]$, then increases [*dp*]{} by the program size. Useful for copying previously frozen code into modifiable memory, to later edit the copy.
5. [Editing Strings on Stack.]{} Instruction [*cpn(n)*]{} copies the n topmost [*ds*]{} entries onto the top of [*ds*]{}, increasing [*dp*]{} by $n$; [*up()*]{} works like [*cpn(1)*]{}; [*cpnb(n)*]{} copies $n$ [*ds*]{} entries above $ds[cs[cp].base]$ onto the top of [*ds*]{}, increasing [*dp*]{} by $n$; [*mvn(a,b,n)*]{} copies the $n$ [*ds*]{} entries starting with $ds[cs[cp].base+a]$ to $ds[cs[cp].base+b]$ and following cells, appropriately increasing [*dp*]{} if necessary; [*ins(a,b,n)*]{} inserts the $n$ [*ds*]{} entries above $ds[cs[cp].base+a]$ after $ds[cs[cp].base+b]$, appropriately increasing [*dp*]{}; [*deln(a,n)*]{} deletes the $n$ [*ds*]{} entries above $ds[cs[cp].base+a]$, appropriately decreasing [*dp*]{}; [*find(x)*]{} returns the stack index of the topmost entry in [*ds*]{} matching $x$; [*findb(x)*]{} the index of the first [*ds*]{} entry above base $ds[cs[cp].base]$ matching $x$. Many of the above instructions can be used to edit stack contents that may later be interpreted as executable code.
### Control-Related Instructions {#control}
Each call of callable code $f$ increments $cp$ and results in a new topmost callstack entry. Functions to make and execute functions include:
1. Instruction [*def(m,n)*]{} defines a new integer function name (1 if it is the first, otherwise the most recent name plus 1) and increments [*fnp*]{}. In the new [*fns*]{} entry we associate with the name: $m$ and $n$, the function’s expected numbers of input arguments and return values, and the function’s start address $cs[cp].ip + 1$ (right after the address of the currently interpreted token [*def*]{}).
2. Instruction [*dof(f)*]{} calls $f$: it views $f$ as a function name, looks up $f$’s address and input number $m$ and output number $n$, increments $cp$, lets $cs[cp].base$ point right below the $m$ topmost elements (arguments) in [*ds*]{} (if $m < 0$ then $cs[cp].base=cs[cp-1].base$, that is, all [*ds*]{} contents corresponding to the previous instance are viewed as arguments), sets $cs[cp].out := n$, and sets $cs[cp].ip$ equal to $f$’s address, thus calling $f$.
3. [*ret()*]{} causes the current function call to return; the sequence of the $n=cs[cp].out$ topmost values on [*ds*]{} is copied down such that it starts in [*ds*]{} right above $ds[cs[cp].base]$, thus replacing the former input arguments; [*dp*]{} is adjusted accordingly, and $cp$ decremented, thus transferring control to the [*ip*]{} of the previous callstack entry (no copying or [*dp*]{} change takes place if $n<0$ — then we effectively return the entire stack contents above $ds[cs[cp].base]$). Instruction [*rt0(x)*]{} calls [*ret()*]{} if $x \leq 0$ (conditional return).
4. [*oldq(n)*]{} calls the $n$-th frozen program (either user-defined or frozen by [oops]{}) stored in $q$ below $a_{frozen}$, assuming (somewhat arbitrarily) zero inputs and outputs.
5. Instruction [*jmp1(val, n)*]{} sets $cs[cp].ip$ equal to $n$ provided that $val$ exceeds zero (conditional jump, useful for iterative loops); [*pip(x)*]{} sets $cs[cp].ip :=x$ (also useful for defining iterative loops by manipulating the instruction pointer); [*bsjmp(n)*]{} sets current instruction pointer $cs[cp].ip$ equal to the [*address*]{} of $ds[cs[cp].base+n]$, thus interpreting stack contents above $ds[cs[cp].base+n]$ as code to be executed.
6. [*bsf(n)*]{} uses $cs$ in the usual way to [*call*]{} the code starting at $ds[cs[cp].base+n]$ (as usual, once the code is executed, we will return to the address of the next instruction right after [*bsf*]{}); [*exec(n)*]{} interprets $n$ as the number of an instruction and executes it.
7. [*qot()*]{} flips a binary flag [*quoteflag*]{} stored at address $a_{quoteflag}$ on tape as $z(a_{quoteflag})$. The semantics are: code in between two [*qot*]{}’s is quoted, not executed. More precisely, instructions appearing between the $m$-th ($m$ odd) and the $m+1$st [*qot*]{} are not executed; instead their instruction numbers are sequentially pushed onto data stack [*ds*]{}. Instruction [*nop()*]{} does nothing and may be used to structure programs.
In the context of instructions such as [*getq*]{} and [*bsf*]{}, let us quote Koopman [@Koopman:89] (reprinted with friendly permission by Philip J. Koopman Jr., 2002):
> [ ]{}
Some of the instructions introduced above are almost exactly doing what has been suggested by [@Tsukamoto:77]. Remarkably, they turn out to be quite useful in the experiments (Section \[experiments\]).
### Bias-Shifting Instructions to Modify Suffix Probabilities {#self}
The concept of online-generated probabilistic programs with [*“self-referential”*]{} instructions that modify the probabilities of instructions to be executed later was already implemented earlier by [@Schmidhuber:97bias]. Here we use the following primitives:
1. [*incQ(i)*]{} increases the current probability of $Q_i$ by incrementing $p[curp][i]$ and $sum[curp]$. Analogously for [*decQ(i)*]{} (decrement). It is illegal to set all $Q$ probabilities (or all but one) to zero; to keep at least two search options. [*incQ(i)*]{} and [*decQ(i)*]{} do not delete argument $i$ from [*ds*]{}, by not decreasing [*dp*]{}.
2. [*boostq(n)*]{} sequentially goes through all instructions of the $n$-th self-discovered frozen program; each time an instruction is recognized as some $Q_i$, it gets [**boosted**]{}: its numerator $p[curp][i]$ and the denominator $sum[curp]$ are increased by $n_Q$. (This is less specific than [*incQ(i)*]{}, but can be useful, as seen in the experiments, Section \[experiments\].)
3. [*pushpat()*]{} stores the current search pattern $pat[curp]$ by incrementing $patp$ and copying the sequence $pat[patp] := pat[curp]$; [*poppat()*]{} decrements $patp$, returning its old value. [*setpat(x)*]{} sets $curp := x$, thus defining the distribution for the next search, given the current task.
The idea is to give the system the opportunity to define several fairly arbitrary distributions on the possible search options, and switch on useful ones when needed in a given computational context, to implement conditional probabilities of tokens, given a computational history.
Of course, we could also [*explicitly*]{} implement tape-represented conditional probabilities of tokens, given previous tokens or token sequences, using a tape-encoded, modifiable [*probabilistic syntax diagram*]{} for defining modifiable [*n-grams*]{}. This may facilitate the act of ignoring certain meaningless program prefixes during search. In the present implementation, however, the system has to create / represent such conditional dependencies by invoking appropriate subprograms including sequences of instructions such as [*incQ()*]{}, [*pushpat()*]{} and [*setpat()*]{}.
Initial User-Defined Programs: Examples {#user}
---------------------------------------
The user can declare initial, possibly recursive programs by composing the tokens described above. Programs are sequentially written into $q$, starting with $q_1$ at address 1. To declare a new token (program) we write [*decl(m, n, [name]{}, body)*]{}, where [name]{} is the textual name of the code. Textual names are of interest only for the user, since the system immediately translates any new name into the smallest integer $> n_Q$ which gets associated with the topmost unused code address; then $n_Q$ is incremented. Argument $m$ denotes the code’s number of expected arguments on top of the data stack [*ds*]{}; $n$ denotes the number of return values; [*body*]{} is a string of names of previously defined instructions, and possibly one new name to allow for cross-recursion. Once the interpreter comes across a user-defined token, it simply calls the code in $q$ starting with that body’s first token; once the code is executed, the interpreter returns to the address of the next token, using the callstack $cs$. All of this is quite similar to the case of self-made functions defined by the system itself — compare instruction [*def*]{} in section \[control\].
Here are some samples of user-defined tokens or programs composed from the primitive instructions defined above. Declarations omit parantheses for argument lists of instructions.
1. [*decl(0, 1, [c999]{}, c5 c5 mul c5 c4 c2 mul mul mul dec ret)*]{} declares [c999()]{}, a program without arguments, computing constant 999 and returning it on top of data stack [*ds*]{}.
2. [ *decl(2, 1, [testexp]{}, by2 by2 dec c3 by2 mul mul up mul ret)* ]{} declares [testexp]{} [*(x,y)*]{}, which pops two values $x,y$ from [*ds*]{} and returns $[6x (4y - 1)]^2$.
3. [ *decl(1, 1, [fac]{}, up c1 ex rt0 del up dec [fac]{} mul ret)* ]{} declares a recursive function [fac]{}[*(n)*]{} which returns 1 if $n=0$, otherwise returns $n \times$ [fac]{}[*(n-1)*]{}.
4. [ *decl(1, 1, [fac2]{}, c1 c1 def up c1 ex rt0 del up dec topf dof mul ret)* ]{} declares [fac2]{}[*(n)*]{}, which defines self-made recursive code functionally equivalent to [fac]{}[*(n)*]{}, which calls itself by calling the most recent self-made function even before it is completely defined. That is, [fac2]{}[*(n)*]{} not only computes [fac]{}[*(n)*]{} but also makes a new [fac]{}-like function.
5. The following declarations are useful for defining and executing recursive procedures (without return values) that expect as many inputs as currently found on stack [*ds*]{}, and call themselves on decreasing problem sizes. [*defnp*]{} first pushes onto auxiliary stack [*Ds*]{} the number of return values (namely, zero), then measures the number $m$ of inputs on [*ds*]{} and pushes it onto [*Ds*]{}, then quotes (that is, pushes onto [*ds*]{}) the begin of the definition of a procedure that returns if its topmost input $n$ is 0 and otherwise decrements $n$. [*callp*]{} quotes a call of the most recently defined function / procedure. Both [*defnp*]{} and [*callp*]{} also quote code for making a fresh copy of the inputs of the most recently defined code, expected on top of [*ds*]{}. [*endnp*]{} quotes the code for returning, grabs from [*Ds*]{} the numbers of in and out values, and uses [*bsf*]{} to call the code generated so far on stack [*ds*]{} above the input parameters, applying this code (possibly located deep in $ds$) to a copy of the inputs pushed onto the top of $ds$.
*decl(-1,-1,defnp, c0 toD pushdp dec toD qot def up rt0 dec intpf cpn qot ret)*
decl(-1,-1,calltp, qot topf dof intpf cpn qot ret)
decl(-1,-1,endnp,qot ret qot fromD cpnb fromD up delD fromD ex bsf ret)
6. Since our entire language is based on integer sequences, there is no obvious distinction between data and programs. The following illustrative example demonstrates that this makes functional programming very easy:
[ *decl(-1, -1, [tailrec]{}, qot c1 c1 def up qot c2 outb qot ex rt0 del up dec topf dof qot c3 outb qot ret qot c1 outb c3 bsjmp)* ]{} declares a tail recursion scheme [tailrec]{} with a functional argument. Suppose the data stack [*ds*]{} holds three values $n$, [*val*]{}, and [*codenum*]{} above the current base pointer. Then [tailrec]{} will create a function that returns [*val*]{} if $n=0$, else applies the 2-argument function represented by [*codenum*]{}, where the arguments are $n$ and the result of calling the 2-argument function itself on the value $n-1$.
For example, the following code fragment uses [tailrec]{} to implement yet another version of [fac]{}[*(n)*]{}: [*(qot c1 mul qot [tailrec]{} ret)*]{}. Assuming $n$ on [*ds*]{}, it first quotes the constant [*c1*]{} (the return value for the terminal case $n=0$) and the function [*mul*]{}, then applies [tailrec]{}.
The primitives of Section \[language\] collectively embody a universal programming language, computationally as powerful as the one of [@Goedel:31] or [Forth]{} or [Ada]{} or C. In fact, a small fraction of the primitives would already be sufficient to achive this universality. Higher-level programming languages can be incrementally built based on the initial low-level [Forth]{}-like language.
To fully understand a given program, one may need to know which instruction has got which number. For the sake of completeness, and to permit precise re-implementation, we include the full list here:
[ *1: 1toD, 2: 2toD, 3: mvdsk, 4: xAD, 5: xSA, 6: bsf, 7: boostq, 8: add, 9: mul, 10: powr, 11: sub, 12: div, 13: inc, 14: dec, 15: by2, 16: getq, 17: insq, 18: findb, 19: incQ, 20: decQ, 21: pupat, 22: setpat, 23: insn, 24: mvn, 25: deln, 26: intpf, 27: def, 28: topf, 29: dof, 30: oldf, 31: bsjmp, 32: ret, 33: rt0, 34: neg, 35: eq, 36: grt, 37: clear, 38: del, 39: up, 40: ex, 41: jmp1, 42: outn, 43: inn, 44: cpn, 45: xmn, 46: outb, 47: inb, 48: cpnb, 49: xmnb, 50: ip2ds, 51: pip, 52: pushdp, 53: dp2ds, 54: toD, 55: fromD, 56: delD, 57: tsk, 58: c0, 59: c1, 60: c2, 61: c3, 62: c4, 63: c5, 64: exec, 65: qot, 66: nop, 67: fak, 68: fak2, 69: c999, 70: testexp, 71: defnp, 72: calltp, 73: endnp.* ]{}
|
---
abstract: 'The discrete modulation can make up for the shortage of transmission distance in measurement-device-independent continuous-variable quantum key distribution (MDI-CVQKD) that has an unique advantage against all side-channel attacks but also challenging for the further performance improvement. Here we suggest a quantum catalysis (QC) approach for enhancing the performance of the discrete-modulated (DM) MDI-CVQKD in terms of the achievable secret key rate and lengthening the maximal transmission distance. The numerical simulation results show that the QC-based MDI-CVQKD with discrete modulation that involves a zero-photon catalysis (ZPC) operation can not only obtain a higher secret key rate than the original DM protocol, but also contributes to the reasonable increase of the corresponding optimal variance. As for the extreme asymmetric and symmetric cases, the secret key rate and maximal transmission distance of the ZPC-involved DM MDI-CVQKD system can be further improved under the same parameters. This approach enables the system to tolerate lower reconciliation efficiency, which will promote the practical implementations with state-of-art technology.'
author:
- 'Wei Ye$^{1}$, Ying Guo$^{1}$, Yun Mao$^{2*}$, Hai Zhong$^{1\dag}$ and Liyun Hu$^{3\ddag}$'
title: '** Enhancing discrete-modulated continuous-variable measurement-device-independent quantum key distribution via quantum catalysis**'
---
Introduction
============
Quantum key distribution (QKD) [@1; @2; @3; @4], as one of the most mature applications in cryptography, aims to share secret keys between two distant legitimate users (Alice and Bob) even in the presence of an eavesdropper (Eve) [@5; @6; @7]. Thanks to the usage of efficient detection schemes, including the homodyne and heterodyne detections, the continuous variable (CV) QKD systems [@8; @9; @10; @11; @57] not only promise high key rates, but also can be easily compatible with the current optical communication technologies. In particular, since the Gaussian-modulated (GM) CVQKD protocol [@9] has been theoretically proved to be secure against collective attacks [@12] and coherent attacks [@13], this protocol paves the way for the development of commercial applications. For example, the field tests of the CVQKD system over 50 km commercial fiber have been achieved in recent years [@14]. Whereas, in practical implementations, the existence of imperfect detectors may lead to the potential security loopholes that could be successfully exploited by Eve to take some attack strategies, e.g., the local oscillator calibration attack [@15], the wavelength attack [@16], and the detector saturation attack [@17], which are still problems for realizing the practical CVQKD system.
In response to these problems, the measurement-device-independent (MDI) CVQKD protocols were proposed [@20; @21; @22; @23; @24; @25; @26; @27; @56], where the secret keys between Alice and Bob can be extracted by relying on the measurement results of an untrusted third party (Charlie). This protocol can be immune to all attacks on detectors, and its practical security proof has been analyzed in both the finite-size effect [@25; @26] and the composable security [@27]. However, in contrast to its discrete-variable (DV) counterpart [@18; @19], this MDI-CVQKD has limitations in the transmission distance, which is not sufficient for the requirements of the long-distance communication. Therefore, it is an urgent task to lengthen the maximal transmission distance with underlying technology.
Up till now, most of investigations have focused on performance improvement of the MDI-CVQKD system by means of quantum operations [@33; @34; @35], such as the phase-sensitive optical amplifier and the photon subtraction. To be more specific, using a phase-sensitive optical amplifier to compensate the imperfection of the Bell-state measurement (BSM) implemented by Charlie can improve the performance of the MDI-CVQKD system [@33]. In addition, the photon subtraction operations [@34; @35] have been used for lengthening the maximal transmission distance of the MDI-CVQKD system, and meanwhile these photon-subtracted operations can be emulated by the non-Gaussian postselection in order to circumvent the complexity of configurations. Despite existing the aforementioned advantages, the success probability of implementing photon-subtracted operations is less than $0.25$, which would lead to the limited performance improvements [@36; @37]. To eliminate this drawback, recently, the quantum catalysis [@36; @37; @38; @39; @40; @41] has been viewed as another useful method to extend the maximal transmission distance of the MDI-CVQKD system [@42], compared with the single-photon subtraction case. Unfortunately, a major problem, common to the aforementioned GM CVQKD, is that the reconciliation efficiency $\beta $ is low, especially on the long-distance transmission with a low signal-to-noise ratio.
To solve this problem, there are two approaches. One is to design effective reconciliation code, such as low-density parity-check code [@43] and multidimensional reconciliation code [@44], but this kind of design has a higher cost of hardwares and is hard to be realized in practice, and the alternative is to employ the discrete-modulated (DM) schemes [28,29,30,31,32,51,54,55]{}, including the four-state protocol and the eight-state protocol, which allows the distribution of secret keys over long distances in the regime of low signal-to-noise ratio with an efficient reconciliation efficiency. Recently, it has been shown [@32] that the DM-based MDI-CVQKD under the extreme asymmetric case rather than the symmetric case has an advantage over the GM-based one with respect to the maximal transmission distance. Motivated by the advantages of the DM schemes with an efficient reconciliation efficiency, in this paper, we put forward a feasible method using a zero-photon catalysis (ZPC) operation in order to further enhance the performance of MDI-CVQKD with discrete modulation. Our results under the same accessible parameters show that, as for the extreme asymmetric and symmetric cases, the usage of the ZPC operation on MDI-CVQKD with discrete modulation can not only give birth to the high secret key rates, but also contributes to the reasonable increase of the corresponding optimal variance. Furthermore, the maximal transmission distance can be lengthened when comparing with the traditional DM scheme. More interestingly, our protocol enables the MDI-CVQKD with discrete modulation to tolerate lower reconciliation efficiency.
The rest of this paper is structured as follows. In Sec. II, we first review the entanglement-based (EB) version of the original DM MDI-CVQKD protocol and then demonstrate the effects of the ZPC operation on MDI-CVQKD with discrete modulation in detail. In Sec. III, according to the optimality of Gaussian attack, we derive the asymptotic secret key rate of the ZPC-involved MDI-CVQKD with discrete modulation. In Sec. IV, the simulation results and discussions are provided. Finally, our conclusions are drawn in Sec. V.
\[Fig1\] {width="1.2\columnwidth"}
ZPC-involved MDI-CVQKD protocol with discrete modulation
========================================================
The EB version of MDI-CVQKD with discrete modulation is shown in Fig. 1(a), where Alice and Bob respectively generate a two-mode entangled state $\left
\vert \Phi _{4}\right \rangle _{AA_{1}}$ and $\left \vert \Phi
_{4}\right
\rangle _{BB_{1}}$, and then both of them respectively hold modes $A$ and $B$ while sending modes $A_{1}$ and $B_{1}$, along with the quantum channels of lengths $L_{AC}$ and $L_{BC}$, to an untrusted third party Charlie who proceeds to interfere with the incoming modes $A_{1}$ and $B_{1}$ at a symmetric beam splitter, and attains two output modes $C_{1}$ and $C_{2} $. After the $x$ ($p$) quadrature of mode $C_{1}$ ($C_{2}$) is measured by the BSM, Charlie proclaims the measured results $\left \{
X_{C_{1}},P_{C_{2}}\right \} $. Subsequently, Bob modifies mode $B$ to $\widetilde{B}$ by using a displacement operation $D\left( \gamma \right) $ according to the public results $\left \{ X_{C_{1}},P_{C_{2}}\right \} ,$ where $\gamma =g\left( X_{C_{1}}+iP_{C_{2}}\right) $ with the gain $g$ of a displacement operation. Through heterodyne detections, Alice and Bob respectively measure mode $A$ and $\widetilde{B}$ to attain the outcome results $\left \{ X_{A},P_{A}\right \} $ and $\left \{ X_{B},P_{B}\right \}
, $ and finally both of them have to implement the postprocessing to get a string of secret keys.
In what follows, we focus on the ZPC-involved MDI-CVQKD with discrete modulation. For the sake of discussions, here we take the EB version of the four-state protocol into account. As shown in Fig. 1(b), in the Alice’s station, Alice prepares for the two-mode entangled state $\left \vert \Phi
_{4}\right \rangle _{AA_{1}}$ on modes $A$ and $A_{1} $ with a modulated variance $V_{M}=2\alpha ^{2}=V_{A}-1$ given by$$\begin{aligned}
\left \vert \Phi _{4}\right \rangle _{AA_{1}}& =\underset{l=0}{\overset{3}{\sum }}\sqrt{\lambda _{l}}\left \vert \phi _{l}\right \rangle _{A}\left
\vert \phi _{l}\right \rangle _{A_{1}} \notag \\
& =\frac{1}{2}\underset{l=0}{\overset{3}{\sum }}\sqrt{\lambda _{l}}\left
\vert \varphi _{l}\right \rangle _{A}\left \vert \alpha _{l}\right \rangle
_{A_{1}}, \label{1}\end{aligned}$$where the non-Gaussian orthogonal state $\left \vert \varphi
_{l}\right
\rangle _{A}$ on mode $A$ is expressed as $$\left \vert \varphi _{l}\right \rangle _{A}=\frac{1}{2}\underset{j=0}{\overset{3}{\sum }}e^{i\left( 2l+1\right) j\pi /4}\left \vert \phi
_{j}\right \rangle _{A},\left( l=0,1,2,3\right) , \label{2}$$with $$\left \vert \phi _{l}\right \rangle =\frac{e^{-\alpha ^{2}/2}}{\sqrt{\lambda
_{l}}}\underset{m=0}{\overset{3}{\sum }}\frac{\left( -1\right) ^{m}\alpha
^{4m+l}}{\sqrt{\left( 4m+l\right) !}}\left \vert 4m+l\right \rangle ,
\label{3}$$$$\begin{aligned}
\lambda _{0,2}& =\frac{e^{-\alpha ^{2}}}{2}\left[ \cosh \alpha ^{2}\pm \cos
\alpha ^{2}\right] , \notag \\
\lambda _{1,3}& =\frac{e^{-\alpha ^{2}}}{2}\left[ \sinh \alpha ^{2}\pm \sin
\alpha ^{2}\right] , \label{4}\end{aligned}$$ and $\left \vert \alpha _{l}\right \rangle _{A_{1}}=$ $\left \vert \alpha
e^{i(2l+1)\pi /4}\right \rangle $ ($l=0,1,2,3$) on mode $A_{1}$ is a modulated coherent state with a positive number $\alpha $. Note that, one of the states $\left \vert \alpha _{l}\right \rangle _{A_{1}}$with the same probability $1/4$ is randomly sent to Charlie through an unsecure quantum channel. Thus, the corresponding covariance matrix $\Gamma _{AA_{1}}$ of the resending state $\left \vert \Phi _{4}\right \rangle _{AA_{1}}$ is$$\Gamma _{AA_{1}}=\left(
\begin{array}{cc}
XI_{2} & Z_{4}\sigma _{z} \\
Z_{4}\sigma _{z} & XI_{2}\end{array}\right) , \label{5}$$where $I_{2}=$diag$\left( 1,1\right) ,\sigma _{z}=$diag$\left( 1,-1\right) ,$ and $$X=1+2\alpha ^{2},~Z_{4}=2\alpha ^{2}\underset{k=0}{\overset{3}{\sum }}\lambda _{k-1}^{3/2}\lambda _{k}^{-1/2}. \label{6}$$
In order to lower requirements for device perfection of the quantum catalysis, we assume that, for our protocol, an untrusted third party, David, who is near Alice, controls the ZPC operation (magenta box) where a vacuum state $\left \vert 0\right \rangle _{D}$ in an auxiliary mode $D$ is injected into the beam splitter (BS) with a transmittance $T$, and simultaneously an ideal on/off detector placed at the output port of mode $D$ is used for registering the same state $\left \vert 0\right \rangle _{D}$. As shown in Refs. [@36; @37], the vacuum state $\left \vert
0\right
\rangle _{D} $ between the input and output ports seems to be unchanged, but promotes the transformation of quantum states between modes $A_{1}$ and $A_{2}$. Moreover, this ZPC operation can be described as an equivalent operator $$\widehat{O}_{0}=\text{Tr}\left[ \Pi ^{off}B\left( T\right) \right] =\sqrt{T}^{b^{\dag }b}, \label{7}$$where $B\left( T\right) =\exp \left[ \left( ad^{\dag }-a^{\dag }d\right)
\arccos \sqrt{T}\right] $ is a BS operator and $\Pi ^{off}=\left \vert
0\right \rangle _{D}\left \langle 0\right \vert $ is a projective operator on mode $D$. In addition, when such an operation is applied to an arbitrary coherent state $\left \vert \alpha \right \rangle $, one can obtain$$\begin{aligned}
\left \vert \psi \right \rangle _{out}& =\frac{\widehat{O}_{0}}{\sqrt{P_{d}}}\left \vert \alpha \right \rangle \notag \\
& =\frac{\exp [-\frac{\left \vert \alpha \right \vert ^{2}}{2}\left(
1-T\right) ]}{\sqrt{P_{d}}}\left \vert \sqrt{T}\alpha \right \rangle ,
\label{8}\end{aligned}$$where $P_{d}=\exp [-\left \vert \alpha \right \vert ^{2}\left( 1-T\right) ]$ presents the success probability of implementing such an operation. Evidently, with the assistant of the ZPC, the conversion of quantum states from $\left \vert \alpha \right \rangle $ to $\left \vert \sqrt{T}\alpha
\right \rangle $ can be achieved with the probability $P_{d}$. Thus, when one mode $A_{1}$ of the states $\left \vert \Phi _{4}\right \rangle
_{AA_{1}} $ is sent by Alice, the travelling state after succeeding in the ZPC operation can be expressed as $\left \vert \Phi _{4}\right \rangle
_{AA_{2}}=\sum_{l=0}^{3}\sqrt{\lambda _{l}}/2\left \vert \varphi
_{l}\right
\rangle _{A}\left \vert \widetilde{\alpha }_{l}\right \rangle
_{A_{2}}$ with a covariance matrix $$\Gamma _{AA_{2}}=\left(
\begin{array}{cc}
\widetilde{X}I_{2} & \widetilde{Z}_{4}\sigma _{z} \\
\widetilde{Z}_{4}\sigma _{z} & \widetilde{X}I_{2}\end{array}\right) , \label{9}$$where $\widetilde{X}$ and $\widetilde{Z}_{4}$ can be obtained by replacing $\alpha $ with $\widetilde{\alpha }$ in Eq. (\[5\]), and $\left \vert
\widetilde{\alpha }_{l}\right \rangle _{A_{2}}=\left \vert \widetilde{\alpha
}e^{i(2l+1)\pi /4}\right \rangle _{A_{2}}$ with $\widetilde{\alpha }=\sqrt{T}\alpha $. It should be noted that after performing the ZPC operation, the actual modulated variance of the travelling state $\left \vert \Phi
_{4}\right \rangle _{AA_{2}}$ turns out to be $\widetilde{V}_{M}=T\left(
V_{A}-1\right) .$ At the Bob’s station, for simplicity, we assume that the variance of the two-mode entangled state $\left \vert \Phi
_{4}\right
\rangle _{BB_{1}}$ is the same as that of the state $\left \vert
\Phi _{4}\right \rangle _{AA_{1}}$, i.e., $V_{A}=V_{B}=V$ throughout this paper. Besides, due to the fact that Alice and Bob carry out the same discrete modulation, the way of attaining the covariance matrix $\Gamma
_{BB_{1}}$ is the same as the Eq. (\[5\]).
The secret key rate of the ZPC-involved MDI-CVQKD system
========================================================
So far, we have established the schematic structure of the EB version ofthe ZPC-involved MDI-CVQKD system with discrete modulation. In this section, for the reverse reconciliation algorithm, we pay attention to the derivation of asymptotic secret key rates against one-mode collective Gaussian attacks [@12; @45] where two Markovian memoryless Gaussian quantum channels are not related [@46]. Here we consider the results that the both channels between Alice (or Bob) and Charlie are Gaussian under the condition of the modulated variance $V_{M}=V-1<0.5$ [@28; @29; @30; @31; @32], which arises from the indistinguishability of $Z_{4}$ and $Z_{G}=\sqrt{V^{2}-1}$. It means that the DM scheme can be viewed as the GM one when satisfying the condition of $V_{M}<1.5$. In Fig. 1(b), we assume that the losses of both channels are $\kappa =0.2$ dB/km, while $T_{A}=10^{-\kappa L_{AC}/10}$ ($T_{B}=10^{-\kappa L_{BC}/10}$) and $\xi _{A}$ ($\xi _{B}$) respectively represent the transmittance and excess noise of the channel between Alice (Bob) and Charlie. As referred to Refs. [20,21,22,23,24,25,26,27,33,34,35]{}, the EB version of the MDI-CVQKD system can be equivalent to a conventional one-way CVQKD system with the assumption that both the state $\left \vert \Phi _{4}\right \rangle _{BB_{1}}$ and the displacement operation $D\left( \gamma \right) $ in Bob’s station are untrusted except for the heterodyne detection, as shown in Fig. 1(c). For the equivalent system, $T_{C}$ and $\varepsilon _{th}$ are respectively the equivalent quantum channel transmittance and excess noise through the relations$$\begin{aligned}
T_{C}& =\frac{g^{2}T_{A}}{2}, \notag \\
\varepsilon _{th}& =1+\chi _{A}+\frac{T_{B}}{T_{A}}\left( \chi _{B}-1\right)
\notag \\
& +\frac{T_{B}}{T_{A}}\left( \sqrt{\frac{2\left( V_{B}-1\right) }{g^{2}T_{B}}}-\sqrt{V_{B}+1}\right) ^{2}, \label{10}\end{aligned}$$where $\chi _{j}=\left( 1-T_{j}\right) /T_{j}+\xi _{j}$ $(j=A,B)$. Further, by setting $g^{2}=2\left( V_{B}-1\right) /\left[ \left( V_{B}+1\right) T_{B}\right] $, one can derive the minimum $\varepsilon _{th}$, i.e., $$\varepsilon _{th}=\frac{T_{B}}{T_{A}}\left( \chi _{B}-1\right) +1+\chi _{A}.
\label{11}$$
Although the imperfection of the BSM in Charlie is an inescapable event, such an imperfection can be compensated by optical preamplifiers [35,50,53]{}. For simplicity, we consider the assumption of the perfection detection for performance analysis. Consequently, the total noise of the channel input in the shot-noise units (SNU) can be expressed as $\chi _{t}=$ $\left( 1-T_{C}\right) /T_{C}+\varepsilon _{th}$.
According to the aforementioned analysis, after performing the BSM and the displacement operation, the covariance matrix $\Gamma _{A\widetilde{B}}$ of the state $\left \vert \Phi _{4}\right \rangle _{A\widetilde{B}}$ is $$\begin{aligned}
\Gamma _{A\widetilde{B}}& =\left(
\begin{array}{cc}
aI_{2} & c\sigma _{z} \\
c\sigma _{z} & bI_{2}\end{array}\right) \notag \\
& =\left(
\begin{array}{cc}
\widetilde{X}I_{2} & \sqrt{T_{C}}\widetilde{Z}_{4}\sigma _{z} \\
\sqrt{T_{C}}\widetilde{Z}_{4}\sigma _{z} & T_{C}\left( \widetilde{X}+\chi
_{t}\right) I_{2}\end{array}\right) . \label{12}\end{aligned}$$We note that the ZPC operation belongs to a kind of Gaussian operation, which makes it possible to derive the secret key rates by using the results of the extremality of Gaussian quantum states [@47]. Thus, we have the secret key rate of the ZPC-involved MDI-CVQKD system with discrete modulation under reverse reconciliation against one-mode collective Gaussian attacks, i.e.,$$K=P_{d}\left \{ \beta I\left( A\text{:}B\right) -\chi \left( B\text{:}E\right) \right \} , \label{13}$$where the success probability $P_{d}$ has been given in Eq. (\[8\]), $\beta $ denotes a reverse-reconciliation efficiency, $I\left( A\text{:}B\right) $ denotes the Shannon mutual information between Alice and Bob, which can be derived as$$I\left( A\text{:}B\right) =\log _{2}\frac{a+1}{a+1-c^{2}/\left( Y+1\right) },
\label{14}$$and $\chi \left( B\text{:}E\right) $ denotes the Holevo bound between Bob and Eve. To obtain this Holevo bound, we assume that Eve is aware of the untrusted third party David, and can purify the whole system $\rho _{A\widetilde{B}ED}$, so that $$\begin{aligned}
\chi \left( B\text{:}E\right) & =S\left( E\right) -S\left( E|B\right) \notag
\\
& =S\left( A\widetilde{B}\right) -S\left( A|\widetilde{B}^{m_{B}}\right) ,
\notag \\
& =\underset{i=1}{\overset{2}{\sum }}G\left[ \frac{\lambda _{i}-1}{2}\right]
-G\left[ \frac{\lambda _{3}-1}{2}\right] , \label{15}\end{aligned}$$where $G\left[ x\right] =\left( x+1\right) \log _{2}\left( x+1\right) -x\log
_{2}x,S\left( A\widetilde{B}\right) $ is a function of the symplectic eigenvalues $\lambda _{1,2}$ of $\Gamma _{A\widetilde{B}}$ given by $$\lambda _{1,2}^{2}=\frac{1}{2}\left[ \vartheta \pm \sqrt{\vartheta
^{2}-4\zeta ^{2}}\right] , \label{16}$$with $\vartheta =a^{2}+b^{2}-2c^{2}$ and $\zeta =ab-c^{2}$, and Eve’s condition entropy $S(A|\widetilde{B}^{m_{B}})$ based on Bob’s measurement result $m_{B}$, is a function of the symplectic eigenvalues $\lambda _{3}$ of $\Gamma _{A}^{m_{B}}=aI_{2}-c\sigma _{z}\left( YI_{2}+I_{2}\right)
^{-1}c\sigma _{z}$, which is given by $$\lambda _{3}=\frac{a\left( b+1\right) -c^{2}}{b+1}. \label{17}$$
Numerical simulations and discussions
=====================================
In the traditional MDI-CVQKD system, most of investigations show that the extreme asymmetric case has the best performance when comparing with the symmetric case ($L_{AC}=L_{BC}$) [@20; @21; @22; @23; @24; @25; @26; @27; @33; @34; @35]. That is, if Charlie is close to Bob, then the extreme asymmetric case $\left( L_{BC}=0\right) $ is more suitable for the point-to-point communication in contrast to the symmetric case, enabling us to achieve the maximal transmission distance. Different from the former, the latter has an unique short-distance network application, e.g., the quantum repeater where the relay (Charlie) has to be placed in the middle of the legitimate communication parties. To extract the maximal secret key rates as many as possible, in the following, we demonstrate the optimal area and value of the variance $V$ that is an important factor of affecting the performance of MDI-CVQKD systems with discrete modulation, and then proceed with the security analysis, including the extreme asymmetric and symmetric cases.
Parameter optimization
----------------------
Now, let us examine the optimal area and value of $V$ in the ZPC-involved DM MDI-CVQKD system, which determines the transmitting power of the quantum signal, thereby impacting the performance of the CVQKD systems. We note that the actual modulated variance $\widetilde{V}_{M}$ of the proposed system is expressed as $\widetilde{V}_{M}=\widetilde{V}-1=T\left( V-1\right) $according to the relation of $\widetilde{\alpha }=\sqrt{T}\alpha $, which means that the value of $V$ can be expanded appropriately by using the ZPC operation on DM MDI-CVQKD. To understand this point, for the given $\beta
=0.95$, Figs. 2 (a) and 2(b) show the secret key rates as a function of the variance $V$ with the extreme asymmetric and symmetric cases, when optimizing over the transmittance $T$ shown in Figs. 2(c) and 2(d), respectively. As for the original DM MDI-CVQKD protocols (solid line), including the extreme asymmetric and symmetric cases, the optimal areas of $V $ are gradually compressed with the decrease of the transmission distance, whereas for the ZPC-involved DM MDI-CVQKD protocols (dashed line), the secret key rate decreases much more slowly with the increase of $V$. In other words, the ZPC-involved protocols have much larger optimal areas of $V$ than the original protocols, which implies that employing the ZPC operation on the DM MDI-CVQKD systems would bring about more flexible and stable performances. We also notice that, when $V$ exceeds a certain threshold, the secret key rate of the ZPC-involved DM MDI-CVQKD system can be higher than that of the original one, which reveals that the usage of the ZPC operation can offer a possible way to increase the secret key rate well. To be specific, for the original protocols, the optimal value of $V$ under the extreme asymmetric case (the symmetric case) is about $1.4$ ($1.5$) with several different transmission distances, e.g., $25$km, $30$km and $35$km ($0.1$km, $0.2$km and $0.3$km), as described in Ref. [@32]. While for the proposed protocols, we find that the correspondingly optimal $V$ under the extreme asymmetric case (the symmetric case) is about $2.5$ ($2.6$). As shown in Figs. 2(c) and 2(d), the corresponding transmittances are, respectively, given by $T\in \{0.275,0.266,0.258\}$ and $T\in
\{0.312,0.311,0.310\}$ so that the modulated variance $\widetilde{V}_{M}$ of the proposed protocols can also satisfy the constraint $\widetilde{V}_{M}<0.5 $. Thus, in the following, we shall show the best performance of the DM MDI-CVQKD system when taking into account these optimal variances.
Security analysis
-----------------
The security of the traditional QKD systems can be reflected from three aspects, i.e., the secret key rate, the tolerable excess noise and the transmission distance. In what follows, we focus on the security analysis of the ZPC-involved DM MDI-CVQKD system, including the extreme asymmetric and symmetric cases, compared with the original DM MDI-CVQKD.
In Figs. 3(a) and 3(b), we illustrate the secret key rate of both protocols versus the transmission distance with the special two cases, i.e., the extreme asymmetric case and the symmetric case, optimized over the transmittance $T$ depicted in Figs. 3(c) and 3(d), respectively. The solid lines denote that the original protocols, which can be surpassed by the protocols using the ZPC operation in terms of the secret key rate and the transmission distance. Compared with the original protocol, the transmission distance of the proposed protocol is closer to the PLOB bound [@52] that represents the ultimate limit of repeaterless communications. One reason is that, by regulating the transmittance $T$, the optimal variance $V$ for the latter can be obtained in the long-distance communication, and can be bigger than the former case ($V=1.4$ under the extreme asymmetric case and $V=1.5$ under the symmetric case). The other is that the ZPC operation can be indeed viewed as a noiseless linear attenuation, which has been used for improving the transmission distance [@48]. In addition, when comparing with the extreme asymmetric case, the performance of both protocols in the symmetric case is very poor under the same parameters, because the gap of the excess noise between the extreme asymmetric and symmetric cases becomes large with the increase of the transmission distance [@32]. We find that the performances of the above DM MDI-CVQKD systems, even with the ZPC operation, are affected by the increase of $\xi =0.002,0.0025$ and $0.003$, especially in terms of the maximal transmission distance. It reveals that the above-mentioned protocols are sensitive to the excess noise. To further understand this point, Figs. 4(a) and 4(b) show the tolerable excess noise as a function of the transmission distance with the extreme asymmetric and symmetric cases, when taking into account each possible transmittance $T$. We find that, with the help of the ZPC operation, the tolerable excess noise of the proposed protocol can be further improved, which implies that under the same acceptable excess noise, the ZPC-involved DM MDI-CVQKD protocol has obvious advantages over the original protocol in terms of the maximal transmission distance. In particular, when $\xi \approx 0.001$, the proposed protocol is able to support a robust DM MDI-CVQKD system over long distance about $56$km. However, there is a serious problem that, as for the extreme asymmetric and symmetric cases, once the excess noise of both protocols is greater than a certain threshold (about $0.015$ for the proposed protocol and $0.011$ for the original protocol), we can not achieve the positive secret key rate. To solve this problem, fortunately, a new two-way CVQKD to improve robustness against excess noise has been proposed [@49]. We also notice that, with the increase of transmission distance, the excess noise of both protocols in the symmetric case falls faster than that in the extreme asymmetric one, thereby making these protocols perform worse than the extreme asymmetric case with respect to long-distance communication.
The reconciliation efficiency $\beta $, on the other hand, is an important indicator of extracting secret key information. As shown in Figs 5(a) and 5(b), we give the secret key rate versus the reconciliation efficiency $\beta $ with the extreme asymmetric and symmetric cases, respectively. We find that, in the extreme asymmetric case, the available range of $\beta $ for both the proposed protocol (dashed lines) and the original protocol (solid lines) narrows as the transmission distance increases. Moreover, for the given transmission distance, the performance of the ZPC-involved DM MDI-CVQKD system is always better than that of the original one with respect to the secret key rate and the tolerant of $\beta $. This phenomenon may be caused by two aspects that the optimal $V$ can be legitimately expanded by using the ZPC operation on the DM MDI-CVQKD system (shown in Figs. 2(a) and 2(b)) and the system with discrete modulation works well at the extremely low signal-to-noise ratio with a high-efficiency error correction code in the reconciliation process. These results are also true for the symmetric case, as illustrated in Fig. 5(b). In this sense, it reveals that compared with the original protocol, the ZPC-involved DM MDI-CVQKD protocol in the symmetric case is more beneficial for short-distance communication, especially under the low reconciliation efficiency.
Conclusion
==========
We have suggested the performance improvement of the MDI-CVQKD with discrete modulation by performing the ZPC operation. We focus on the conventional four-state scheme as the representation of the DM-based CVQKD system. Due to the benefits of a high-efficiency error correction code in the reconciliation process, this four-state-based scheme may offers a possible way to extend the maximal transmission distance. In the context of an asymptotic-regime security analysis including the extreme asymmetric and symmetric cases, our results under the same accessible parameters show that the secret key rate of the ZPC-involved DM MDI-CVQKD protocol can be increased, compared with the original protocol. In addition, the performance of the ZPC-involved protocol is always superior to that of the original one in terms of the maximal transmission distance. Furthermore, we find that our protocol enables the DM MDI-CVQKD system to tolerate more lower reconciliation efficiency. However, the tolerable excess noise of the both protocols, especially for the symmetric case, decreases with the increase of the transmission distance, which restrains the effects of extending the security transmission distance. Fortunately, the two-way CVQKD protocol to tolerate more excess noise than the one-way protocol was first proposed by S. Pirandola [@49], which paves the way for our future investigations in the development of lengthening the security transmission distance.
This work was supported by the National Natural Science Foundation of China (Grant Nos. 61572529, 61821407, 11964013, 11664017), the Outstanding Young Talent Program of Jiangxi Province (Grant No. 20171BCB23034), the Training Program for Academic and Technical Leaders of Major Disciplines in Jiangxi Province, the Postgraduate Scientific Research Innovation Project of Hunan Province (Grant No. CX20190126) and the Postgraduate Independent Exploration and Innovation Project of Central South University (Grant No. 2019zzts070).
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|
---
author:
- Li Kai
- 'Xia, Qi-Qi'
- 'Liu, Jin-Zhong'
- 'Zhang, Yu'
- 'Gao, Xing'
- 'Hu, Shao-Ming'
- 'Guo, Di-Fu'
- 'Chen, Xu'
- 'Liu, Yuan'
title: 'Photometric investigations on two totally eclipsing contact binaries: V342 UMa and V509 Cam '
---
Introduction {#sect:intro}
============
W UMa contact binaries are comprised of two late type stars with spectral types from F to K. The two component stars are sharing a common convective envelope and have nearly equal effective temperatures although their masses are very different. The analysis of W UMa contact binaries is very necessary for modern astrophysics as they are probes for understanding tidal interactions, energy exchange, mass transfer, and angular momentum loss. The formation, evolution, ultimate fate, and magnetic activities of the W UMa contact binaries are still debatable issues (e.g., @Guinan+Bradstreet+1988 [@Bradstreet+Guinan+1994; @Eggleton+Kisseleva-Eggleton+2006; @Fabrycky+Tremaine+2007; @Qian+etal+2006; @Qian+etal+2007a; @Qian+etal+2014; @Qian+etal+2017; @Qian+etal+2018]). In order to solve these problems, the determination of physical parameters of a great deal of such type binaries is required.
The physical parameters, such as mass ratio, are poorly estimated for partially eclipsing contact binaries (e.g., @Pribulla+etal+2003 [@Terrell+Wilson+2005]). In addition, the spectroscopic mass ratio sometimes can not be reliably derived according to the broadened and blended spectral lines (e.g., @Dall+Schmidtobreick+2005 [@Rucinski+2010]). By the study of contact binaries that have been obtained both spectroscopic and photometric mass ratios, [@Pribulla+etal+2003] discovered that the photometric mass ratios of the totally eclipsing systems correspond to their spectroscopic ones. [@Terrell+Wilson+2005] determined similar result by discussing the relations between photometric and spectroscopic mass ratios. These results suggest that we can derive very precise and reliable physical parameters for totally eclipsing contact binaries only by the photometric light curves. Thanks to the Gaia mission (@Gaia [@Collaboration+etal+2018]), the parallaxes of more than one billion stars have been obtained, which allows researchers to estimate the absolute parameters of contact binaries even if there is no radial velocity observations (e.g., @Kjurkchieva+etal+2019a [@Kjurkchieva+etal+2019b]). Therefore, we chose two totally eclipsing binaries, V342 UMa and V509 Cam, to analyze their light curves and period variations and estimate their absolute parameters.
V342 UMa was firstly discovered as a W UMa type binary by [@Nelson+etal+2004] during the observations of a nearby star BH UMa. The period of 0.343854 days, the color index of $B-V=0.64$, and spectral type of G3 were obtained. A photometric study by them revealed that V342 UMa is a low mass ratio ($q=0.331$) W-subtype contact binary (the hotter component is the less massive one). It has been fifteen years after the discovery and the first photometric investigation of V342 UMa, we decided to investigate the light curves and orbital period changes of this target.
V509 Cam was firstly identified as an EW type eclipsing binary by [@Khruslov+2006] during an eclipsing binaries search in Camelopardalis. The variability amplitude of 0.6 mag and the orbital period of 0.35034 days were determined by him. At present, neither the light curve synthesis nor period variation analysis has been carried out for this star, we will analyze the light curves and orbital period variations of this target in this paper.
CCD observations of V342 UMa and V509 Cam {#sect:Obs}
=========================================
Charge-coupled device (CCD) photometry of V342 UMa and V509 Cam were carried out from 2018 to 2019 using the Weihai Observatory 1.0-m telescope of Shandong University (WHOT, @Hu+etal+2014), the Nanshan One-meter Widefield Telescope (NOWT, @Liu+etal+2014) at the Nanshan station of the Xinjiang Astronomical Observatory, the 60cm Ningbo Bureau of Education and Xinjiang Observatory Telescope (NEXT), and the 85 cm telescope at the Xinglong Station of National Astronomical Observatories (NAOs85cm) in China. The observational information is listed in Table \[Tab:obsevation\]. In order to record the observed images, $2K\times2K$ CCD cameras were used for WHOT, NEXT, and NAOs85cm, and a $4K\times4K$ CCD camera was applied to NOWT. The field of views are $12^{'}\times12^{'}$ for WHOT, $1.3^{\circ}\times1.3^{\circ}$ for NOWT, $22^{'}\times22^{'}$ for NEXT, and $32^{'}\times32^{'}$ for NAOs85cm. The effective subframe of NOWT is $30^{'}\times30^{'}$ during the observations. The filters we used are standard Johnson-Cousin-Bessel $BVR_cI_c$ systems. The standard IRAF routine was applied to process the observed data including zero and flat calibrations, and aperture photometry, then different magnitudes between the target and the comparison star and those between the comparison and check stars were obtained. The complete light curves of V342 UMa observed by NEXT and WHOT and those of V509 Cam observed by NOWT and NEXT are illustrated in Figure \[Fig1\] and Figure \[Fig2\], respectively. As seen in the two figures, the two targets show EW type light curves, and very clearly flat primary minima can be discovered. Based on our observations, six eclipsing minima were derived for V342 UMa, while ten were obtained for V509 Cam, all the minima were calculated by the K-W method (@Kwee+van [@Woerden+1956]) and are listed in Table \[Tab:ecl-times\].
Star Date Filters and Typical exposure time Type Uncertainties (mag)$^*$ Telescope
---------- -------------- ----------------------------------- --------------- ----------------------------------------- -----------
V342 UMa Mar 29, 2018 $R_c$40s minimum light $R_c$0.011 WHOT
May 08, 2018 $B$80s $V$60s $R_c$40s $I_c$40s light curve $B$0.007 $V$0.006 $R_c$0.007 $I_c$0.009 NEXT
May 21, 2018 $B$80s $V$60s $R_c$40s $I_c$40s light curve $B$0.010 $V$0.007 $R_c$0.007 $I_c$0.010 NEXT
May 27, 2018 $B$80s $V$60s $R_c$40s $I_c$40s light curve $B$0.008 $V$0.006 $R_c$0.007 $I_c$0.008 NEXT
Dec 28, 2018 $R_c$30s minimum light $R_c$0.005 NAOs85cm
Jan 20, 2019 $B$120s $V$60s $R_c$35s $I_c$25s light curve $B$0.005 $V$0.005 $R_c$0.005 $I_c$0.005 WHOT
V509 Cam Feb 08, 2018 $B$36s $V$22s $R_c$13s $I_c$12s light curve $B$0.006 $V$0.006 $R_c$0.006 $I_c$0.005 NOWT
Mar 05, 2018 $B$25s $V$25s $R_c$25s $I_c$25s minimum light $B$0.010 $V$0.009 $R_c$0.008 $I_c$0.008 NOWT
Mar 06, 2018 $B$14s $V$10s $R_c$10s $I_c$10s light curve $B$0.006 $V$0.005 $R_c$0.005 $I_c$0.005 NOWT
Apr 15, 2018 $B$70s $V$50s $R_c$40s $I_c$30s light curve $B$0.007 $V$0.006 $R_c$0.005 $I_c$0.007 NEXT
Apr 23, 2018 $B$70s $V$50s $R_c$40s $I_c$30s light curve $B$0.006 $V$0.005 $R_c$0.005 $I_c$0.007 NEXT
Jan 21, 2019 $R_c$40s minimum light $R_c$0.003 WHOT
: The Observational Log for V342 UMa and V509 Cam.[]{data-label="Tab:obsevation"}
$^*$ The uncertainties are the standard deviation of the differences between the comparison and check stars.
![The left figure displays the light curves of V342 UMa observed by NEXT on May 08, 21, and 27, 2018 (black, red and blue symbols respectively represent 20180508 observations, 20180521 observations, and 20180527 observations), while the right figure displays the light curves of V342 UMa observed by WHOT on Jan 20, 2019. Crosses refer to the $B$ band light curves, while open circles, triangles, and squares respectively represent the $V$, $R_c$, and $I_c$ bands light curves. []{data-label="Fig1"}](f1a.eps "fig:"){width="50.00000%"} ![The left figure displays the light curves of V342 UMa observed by NEXT on May 08, 21, and 27, 2018 (black, red and blue symbols respectively represent 20180508 observations, 20180521 observations, and 20180527 observations), while the right figure displays the light curves of V342 UMa observed by WHOT on Jan 20, 2019. Crosses refer to the $B$ band light curves, while open circles, triangles, and squares respectively represent the $V$, $R_c$, and $I_c$ bands light curves. []{data-label="Fig1"}](f1b.eps "fig:"){width="50.00000%"}
![The left panel displays the light curves of V509 Cam observed by NOWT on February 08, 2018, the middle panel shows the light curves of V509 Cam observed by NOWT on Mar 06, 2018, while the right panel plots the light curves of V509 Cam observed by NEXT on April 15 and 23, 2018 (black and red symbols respectively represent 20180415 observations and 20180423 observations). Crosses refer to the $B$ band light curves, while open circles, triangles, and squares respectively represent the $V$, $R_c$, and $I_c$ bands light curves. []{data-label="Fig2"}](f2a.eps "fig:"){width="33.00000%"} ![The left panel displays the light curves of V509 Cam observed by NOWT on February 08, 2018, the middle panel shows the light curves of V509 Cam observed by NOWT on Mar 06, 2018, while the right panel plots the light curves of V509 Cam observed by NEXT on April 15 and 23, 2018 (black and red symbols respectively represent 20180415 observations and 20180423 observations). Crosses refer to the $B$ band light curves, while open circles, triangles, and squares respectively represent the $V$, $R_c$, and $I_c$ bands light curves. []{data-label="Fig2"}](f2b.eps "fig:"){width="33.00000%"} ![The left panel displays the light curves of V509 Cam observed by NOWT on February 08, 2018, the middle panel shows the light curves of V509 Cam observed by NOWT on Mar 06, 2018, while the right panel plots the light curves of V509 Cam observed by NEXT on April 15 and 23, 2018 (black and red symbols respectively represent 20180415 observations and 20180423 observations). Crosses refer to the $B$ band light curves, while open circles, triangles, and squares respectively represent the $V$, $R_c$, and $I_c$ bands light curves. []{data-label="Fig2"}](f2c.eps "fig:"){width="33.00000%"}
Orbital period variations {#sect:orbtial}
=========================
Both of V342 UMa and V509 Cam have been identified more than ten years, no one has analyzed the orbital period variations at present. Then, we collected all published eclipsing times for V342 UMa and V509 Cam from literatures, and listed them in Table \[Tab:ecl-times\]. Moreover, WASP (Wide Angle Search for Planets, @Butters+etal+2010) has observed V342 UMa, we calculated two minima using the archive data and also listed in Table \[Tab:ecl-times\]. Combining our new observed ones, we obtained a total of 43 photoelectric or CCD eclipsing times for V342 UMa, and a total of 17 photoelectric or CCD eclipsing times for V509 Cam. Using the least-squares method, the linear ephemeris of V342 UMa taken from [@Nelson+etal+2004] was corrected to be: $$\textrm{Min.I}=2453054.83723(\pm0.00090)+0.34385184(\pm0.00000010)\textrm{E},$$ and the linear ephemeris of V509 Cam originated from O-C Gateway[^1] was amended to be: $$\textrm{Min.I}=2451492.27572(\pm0.00071)+0.35034717(\pm0.00000004)\textrm{E}.$$ All the O-C values calculated by the two equations are listed in Table \[Tab:ecl-times\], the corresponding curves are displayed in Figure \[Fig3\]. We can find that both of V342 UMa and V509 Cam show a parabolic trend. Then, quadratic ephemerides were applied to fit the O-C curves of the two targets, $$\textrm{Min.I}=-0.00043(\pm0.00095)+0.00000069(\pm0.00000038)\textrm{E}-4.81(\pm2.56)\times10^{-11}\textrm{E}^2 ,$$ $$\textrm{Min.I}=-0.00127(\pm0.00056)-0.00000041(\pm0.00000010)\textrm{E}+1.90(\pm0.43)\times10^{-11}\textrm{E}^2 .$$ When removing Equations (3) and (4), the residuals are listed in Table \[Tab:ecl-times\] and shown in the bottom panels of Figure \[Fig3\]. No cyclic variations can be detected in the residuals. According to the coefficients of the quadratic terms of Equations (3) and (4), we determined that the period of V342 UMa is secular decrease at a rate of $-1.02(\pm0.54)\times10^{-7}$ days/year and the period of V509 Cam is continuously increase at a rate of $3.96(\pm0.90)\times10^{-8}$ days/year.
![The left panel displays the O-C curve of V342 UMa, while the right panel refer to the O-C curve of V509 Cam. Open circles refer to eclipsing times from literatures, solid circles represent our data, while solid squares show the WASP eclipsing times. The errors of some points were not given in the literature and are fixed at 0.0010 when plotting this figure.[]{data-label="Fig3"}](f3a.eps "fig:"){width="50.00000%"} ![The left panel displays the O-C curve of V342 UMa, while the right panel refer to the O-C curve of V509 Cam. Open circles refer to eclipsing times from literatures, solid circles represent our data, while solid squares show the WASP eclipsing times. The errors of some points were not given in the literature and are fixed at 0.0010 when plotting this figure.[]{data-label="Fig3"}](f3b.eps "fig:"){width="50.50000%"}
------------ -------- --------- --------- ----------- ------------ ------------ -------- --------- --------- ----------- ------------
HJD Errors E O-C Residuals References HJD Errors E O-C Residuals References
2400000+ 2400000+
53053.8040 0.0100 -3 -0.0014 -0.0010 (1) 51492.2770 $-$ 0 -0.0013 0.0000 (15)
53053.9740 0.0100 -2.5 -0.0033 -0.0029 (1) 55937.8301 0.0003 12689 -0.0034 0.0001 (6)
53054.8371 0.0001 0 0.0001 0.0006 (1) 55996.3387 0.0005 12856 -0.0028 0.0007 (7)
53055.3529 0.0003 1.5 0.0002 0.0006 (1) 55996.5138 0.0013 12856.5 -0.0029 0.0006 (7)
53055.5253 0.0002 2 0.0006 0.0011 (1) 56017.7080 0.0004 12917 -0.0047 -0.0012 (6)
53057.7602 0.0002 8.5 0.0005 0.0009 (1) 57692.3700 0.0002 17697 -0.0021 0.0005 (16)
53057.9317 0.0002 9 0.0001 0.0005 (1) 57692.5435 0.0003 17697.5 -0.0038 -0.0012 (16)
53058.9635 0.0002 12 0.0003 0.0007 (1) 58158.3319 0.0002 19027 -0.0020 0.0003 (14)
53059.6509 0.0010 14 0.0000 0.0004 (1) 58158.5069 0.0001 19027.5 -0.0022 0.0001 (14)
53066.8716 0.0006 35 -0.0002 0.0002 (1) 58183.3815 0.0002 19098.5 -0.0022 0.0000 (14)
53067.0450 0.0006 35.5 0.0013 0.0017 (1) 58184.0819 0.0002 19100.5 -0.0025 -0.0002 (14)
53074.7807 0.0003 58 0.0003 0.0007 (1) 58184.2575 0.0002 19101 -0.0021 0.0002 (14)
53074.9528 0.0003 58.5 0.0005 0.0009 (1) 58184.4323 0.0002 19101.5 -0.0024 -0.0002 (14)
53077.7032 0.0003 66.5 0.0001 0.0005 (1) 58224.1969 0.0001 19215 -0.0022 0.0000 (14)
53077.8758 0.0002 67 0.0007 0.0011 (1) 58224.3725 0.0002 19215.5 -0.0018 0.0004 (14)
53404.8790 0.0002 1018 0.0009 0.0006 (2) 58232.2550 0.0002 19238 -0.0022 0.0001 (14)
53837.4389 0.0015 2276 -0.0049 -0.0015 (3) 58505.3504 0.0001 20017.5 -0.0023 -0.0004 (14)
54435.9180 0.0001 4016.5 0.0001 -0.0025 (4)
54438.6690 0.0009 4024.5 0.0003 0.0015 (3)
54499.8744 0.0002 4202.5 0.0001 0.0017 (5)
54815.8735 0.0002 5121.5 -0.0007 0.0005 (5)
55958.8411 0.0005 8445.5 0.0034 -0.0007 (6)
56009.3875 0.0046 8592.5 0.0036 0.0046 (7)
56009.5583 0.0039 8593 0.0025 0.0005 (7)
56013.6833 0.0002 8605 0.0012 -0.0010 (6)
56042.7440 0.0006 8689.5 0.0065 -0.0010 (6)
56387.4512 0.0045 9692 0.0022 -0.0018 (8)
56390.3725 0.0016 9700.5 0.0008 -0.0011 (8)
56761.3883 0.0011 10779.5 0.0004 0.0201 (9)
56783.3940 0.0012 10843.5 -0.0004 -0.0014 (9)
57091.4856 0.0019 11739.5 0.0000 -0.0004 (10)
57390.2931 $-$ 12608.5 0.0002 -0.0012 (11)
57450.2943 $-$ 12783 -0.0007 0.0025 (11)
57797.0720 $-$ 13791.5 0.0024 -0.0076 (12)
57797.2339 $-$ 13792 -0.0076 -0.0014 (12)
57806.8894 0.0003 13820 0.0200 -0.0024 (13)
57817.1834 $-$ 13850 -0.0016 -0.0019 (12)
58207.1096 0.0003 14984 -0.0033 -0.0002 (14)
58247.3406 0.0003 15101 -0.0030 -0.0015 (14)
58260.2368 0.0004 15138.5 -0.0013 -0.0017 (14)
58481.3317 0.0001 15781.5 -0.0030 -0.0016 (14)
58504.1976 0.0002 15848 -0.0033 -0.0058 (14)
58504.3696 0.0002 15848.5 -0.0032 -0.0012 (14)
------------ -------- --------- --------- ----------- ------------ ------------ -------- --------- --------- ----------- ------------
: Eclipsing Times of V342 UMa and V509 Cam[]{data-label="Tab:ecl-times"}
\(1) @Nelson+etal+2004; (2) @Krajci+2006; (3) This paper (WASP); (4) @Nelson+2008; (5) @Nelson+2009; (6) @Diethelm+2012; (7) @Hubscher+etal+2013; (8) @Hubscher+2014; (9) @Hubscher+Lehmann+2015; (10) @Hubscher+2016; (11) VSOLJ 63; (12) VSOLJ 64; (13) @Nelson+2018; (14) This paper (WHOT); (15) @Khruslov+2006; (16) OEJV 0179.\
Photometric solutions of V342 UMa and V509 Cam {#sect:photo}
==============================================
Based on our observations, two sets of complete light curves of V342 UMa and three sets of light curves of V509 Cam were obtained. The Wilson-Devinney (W-D) program (@Wilson+Devinney+1971 [@Wilson+1979; @Wilson+1990]) was used to model these light curves. Gaia DR 2 (@Gaia [@Collaboration+etal+2016; @Gaia; @Collaboration+etal+2018]) has observed these two targets and determined the mean temperatures of them, $T_m=5741$ K for V342 UMa and $T_m=6462$ K for V509 Cam. At first, the mean temperature was set as the temperature of the primary, $T_1$. The bolometric and bandpass limb-darkening coefficients were taken from [@van; @Hamme+1993]’s table, and the gravity-darkening coefficients and the bolometric albedos were set as $g_{1,2}=0.32$ and $A_{1,2} = 0.5$ for their convective envelopes (@Lucy+1967 [@Rucinski+1969]). Due to the lack of radial velocity curves, the $q$-search method was applied to determine the mass ratios of the two systems. When we obtained the final solutions, the temperatures of the two components were calculated using the following method (@Coughlin+etal+2011 [@Dimitrov+Kjurkchieva+2015]), $$\begin{aligned}
T_1&=&T_m+{c\Delta T\over c+1}, \\\nonumber
T_2&=&T_1-\Delta T,\end{aligned}$$ where $\Delta T=T_1-T_2$ and $c=L_2/L_1$ are derived by the W-D modeling.
Because both V342 UMa and V509 Cam were obtained two or more sets of complete light curves, and the light curves observed at different times are different. The physical parameters determined by different light curves may be different. Therefore, we chose one set of the complete light curves to determine the physical parameters, and the derived physical parameters were set as reference values to model the other light curves. For V342 UMa, the complete light curves observed in 2019 have higher quality comparing to those observed in 2018, so the 2019 light-curve was the chosen one. For V509 Cam, the complete light curves observed on March 06, 2018 are symmetric and have the highest precision among the three sets of light curves, so the 201803 light-curve was the chosen one. During the modeling, we used Mode 2 (detached configuration) for both targets at first and found that the solutions were quickly convergent at Mode 3 (contact configuration). The adjustable parameters are as follows: the orbital inclination, $i$, the temperature of the secondary, $T_2$, the dimensionless potential of the primary $\Omega_1$, and the monochromatic luminosity of the primary, $L_1$. Then, a series of of solutions with fixed values of mass ratio $q$ were carried out for them. The weighted sum of squared residuals, $\sum W_i(O-C)_i^2$, versus mass ratio $q$ of the two systems are respectively displayed in the left and right panels of Figure \[Fig4\]. As seen in Figure \[Fig4\], a very sharp minimum was determined for V342 UMa at $q=2.8$, while that was derived for V509 Cam at $q=2.5$. These two values were set as initial values and adjustable parameters, new solutions were performed. When the solutions were convergent, the physical parameters were obtained. The light curves of V342 UMa is asymmetric, adding a cool spot on the less massive primary component can reproduce the asymmetric light curves. The derived physical parameters are listed in Table \[Tab:photo-results\], and the synthetic light curves are respectively shown in Figure \[Fig5\] and Figure \[Fig6\].
To model the other light curves, the physical parameters derived above were set as reference values and the mass ratio $q$ was fixed. Due to the asymmetric light curves, spots model was applied. The best fitting results are also listed in Table \[Tab:photo-results\], and the corresponding fitting curves are respectively displayed in Figure \[Fig5\] and Figure \[Fig6\]. According to the previous discussion, the physical parameters determined by the 2019 light-curve of V342 UMa and the 201803 light-curve of V509 Cam should be more reliable. Therefore, the physical parameters determined by the 2019 light-curve of V342 UMa and the 201803 light-curve of V509 Cam were adopted as the final results. The geometric configurations of the two systems are respectively plotted in Figure \[Fig7\] and Figure \[Fig8\], the changes of the spot distributions can be clearly clarified.
Star
----------------------- ----------------- ----------------- ----------------- ----------------- -----------------
Parameters 201901 201805 201803 201802 201804
$T_1$(K) $5902\pm6$ $5899\pm8$ $6523\pm4$ $6498\pm5$ $6560\pm5$
$T_2$(K) $5662\pm11$ $5663\pm16$ $6433\pm9$ $6446\pm12$ $6415\pm11$
$q$ $2.748\pm0.015$ 2.748(fixed) $2.549\pm0.016$ 2.549(fixed) 2.549(fixed)
$i(^\circ)$ $84.2\pm0.2$ $82.9\pm0.3$ $82.1\pm0.1$ $82.3\pm0.1$ $82.8\pm0.2$
$\overline {L_2/L_1}$ $2.040\pm0.113$ $2.024\pm0.110$ $2.145\pm0.037$ $2.202\pm0.023$ $2.073\pm0.058$
$\Omega_1=\Omega_2$ $6.219\pm0.019$ $6.164\pm0.006$ $5.816\pm0.021$ $5.821\pm0.005$ $5.828\pm0.005$
$r_1$ $0.299\pm0.001$ $0.305\pm0.001$ $0.321\pm0.001$ $0.319\pm0.001$ $0.320\pm0.001$
$r_2$ $0.476\pm0.002$ $0.480\pm0.001$ $0.481\pm0.003$ $0.481\pm0.001$ $0.481\pm0.001$
$f$ $10.0\pm3.1\%$ $19.0\pm1.0\%$ $32.1\pm3.4\%$ $31.3\pm0.9\%$ $30.1\pm0.9\%$
Spot on star 1 on star 1 $-$ on star 2 on star 1
$\theta$(radian) $1.273\pm0.092$ $1.759\pm0.212$ $-$ $0.467\pm0.107$ $0.583\pm0.132$
$\phi$(radian) $5.896\pm0.064$ $2.134\pm0.056$ $-$ $1.225\pm0.057$ $1.658\pm0.039$
$r$(radian) $0.258\pm0.069$ $0.358\pm0.085$ $-$ $0.298\pm0.064$ $0.423\pm0.086$
$T_f(T_d/T_0)$ $0.832\pm0.099$ $0.789\pm0.124$ $-$ $0.811\pm0.089$ $0.824\pm0.078$
: Photometric results of V342 UMa and V509 Cam[]{data-label="Tab:photo-results"}
![ This figure displays the $\sum W_i(O-C)_i^2$ versus mass ratio $q$ of V342 UMa and V509 Cam. []{data-label="Fig4"}](f4a.eps "fig:"){width="50.00000%"} ![ This figure displays the $\sum W_i(O-C)_i^2$ versus mass ratio $q$ of V342 UMa and V509 Cam. []{data-label="Fig4"}](f4b.eps "fig:"){width="50.00000%"}
![The comparison between the synthetic and observed light curves for V342 UMa.[]{data-label="Fig5"}](f5a.eps "fig:"){width="50.00000%"} ![The comparison between the synthetic and observed light curves for V342 UMa.[]{data-label="Fig5"}](f5b.eps "fig:"){width="50.00000%"}
![The comparison between the synthetic and observed light curves for V509 Cam.[]{data-label="Fig6"}](f6a.eps "fig:"){width="33.00000%"} ![The comparison between the synthetic and observed light curves for V509 Cam.[]{data-label="Fig6"}](f6b.eps "fig:"){width="33.00000%"} ![The comparison between the synthetic and observed light curves for V509 Cam.[]{data-label="Fig6"}](f6c.eps "fig:"){width="33.00000%"}
![Geometrical configurations of V342 UMa. []{data-label="Fig7"}](f7.eps)
![Geometrical configurations of V509 Cam. []{data-label="Fig8"}](f8.eps)
Discussion and conclusions {#sect:discussion}
==========================
Two sets of complete $BVR_cI_c$ light curves of V789 Her and three sets of complete $BVR_cI_c$ light curves of V509 Cam were obtained and analyzed. We discovered that both of the two systems are W-subtype contact binaries, $q=2.748\pm0.015$ for V342 UMa and $q=2.549\pm0.016$ for V509 Cam. V342 UMa is a shallow contact binary with a fill-out factor of $f=10.0\pm3.1\%$, and V509 Cam is a medium contact binary with a contact degree of $f=32.1\pm3.4\%$. The two systems show totally eclipsing primary minima, the inclinations of them are higher than $82^\circ$, and the $q$-search curves exhibit very clear sharpness around the bottom. All these indicate that the photometric solutions derived only by the photometric light curves are reliable (e.g., @Pribulla+etal+2003 [@Terrell+Wilson+2005; @Zhang+etal+2017]). Our photometric results of V342 UMa, such as the less massive component is the hotter one, and the reciprocal of our mass ratio is $1/q\sim0.364$, are similar with those determined by [@Nelson+etal+2004]. By collecting all available eclipsing times of V342 UMa and V509 Cam, we studied the period changes and obtained that the period of V342 UMa is secular decrease at a rate of $-1.02(\pm0.54)\times10^{-7}$ days/year and the period of V509 Cam is continuously increase at a rate of $3.96(\pm0.90)\times10^{-8}$ days/year.
Absolute Parameters Estimation
------------------------------
Because of the lack of radial velocity curves of V342 UMa and V509 Cam, we can not directly determine the absolute parameters. However, due to the Gaia mission (@Gaia [@Collaboration+etal+2018]), we can estimate the absolute parameters based on the distance. Firstly, the absolute magnitude of the two systems can be derived according to the parallax determined by Gaia mission and the relation $M_{V}=m_{V}-5\log D+5-A_V$, where $m_{V}$ is the $V$ band visual magnitude which can be derived from [@Gettel+etal+2006], $D$ represent the distance which can be computed by the Gaia parallax, and $A_V$ is extinction value which can be derived from [@Chen+etal+2018]. Secondly, using the equations $M_{bol}=-2.5logL/L_\odot+4.74$ and $M_{bol}=M_V+BC_V$ ($M_{bol}$ is the absolute bolometric magnitude and $BC_V$ is the bolometric correction which can be interpolate from Table 5 of [@Pecaut+Mamajek+2013], the total luminosity of the binary can be calculated. Thirdly, the luminosity of each component ($L_1$ and $L_2$) can be determined by the luminosity ratio $L_2/L_1$ listed in Table \[Tab:photo-results\]. Fourthly, assuming black-body emission($L=4\pi\sigma T^4R^2$), the radius of each component can be estimated, hence the semi-major axis $a$ can be obtained based the absolute and relative radius of each component (an average value of $a$ was adopted). Finally, the mass of each component can be calculated by using Kepler’s third law $M_1+M_2=0.0134a^3/P^2$ and the mass ratio $q$. Following these steps, we obtained the absolute parameters of V342 UMa and V509 Cam. The absolute parameters, along with the parameters needed in the calculation process, are listed in Table \[Tab:abs-parameters\]. This method provide an opportunity to estimate absolute parameters of contact binaries without radial velocity curve observations and can be applied to other contact binaries with reliable photometric solutions.
Parameters $D$ $V_{max}$ $M_V$ $BC_V$ $M_{bol}$ $L_1$ $L_2$ $R_1$ $R_2$ $a$ $M_1$ $M_2$
------------ ----------- ----------- ------------ ---------- ------------ ------------- ------------- ------------- ------------- ------------- ------------- -------------
(pc) (mag) (mag) (mag) (mag) ($L_\odot$) ($L_\odot$) ($R_\odot$) ($R_\odot$) ($R_\odot$) ($M_\odot$) ($M_\odot$)
V342 UMa $685.6$ $13.418$ $4.134$ $-0.115$ $4.019$ $0.640$ $1.303$ $0.766$ $1.188$ $2.530$ $0.490$ $1.293$
$\pm14.0$ $-$ $\pm0.044$ $-$ $\pm0.044$ $\pm0.090$ $\pm0.157$ $\pm0.054$ $\pm0.072$ $\pm0.175$ $\pm0.103$ $\pm0.291$
V509 Cam $674.0$ $12.933$ $3.647$ $-0.041$ $3.606$ $0.904$ $1.938$ $0.746$ $1.122$ $2.328$ $0.388$ $0.997$
$\pm8.6$ $-$ $\pm0.028$ $-$ $\pm0.028$ $\pm0.068$ $\pm0.134$ $\pm0.028$ $\pm0.039$ $\pm0.093$ $\pm0.048$ $\pm0.129$
: Absolute parameters of V342 UMa and V509 Cam[]{data-label="Tab:abs-parameters"}
The Secular Period Changes
--------------------------
The period of V342 UMa is secular decrease at a rate of $-1.02(\pm0.54)\times10^{-7}$ days/year. Usually, the long-term period decrease is produced by conservative mass transfer or angular momentum loss (AML). If it is caused by conservative mass transfer, the mass transfer rate can be determined to be $dM_1/dt=3.01(\pm1.59)\times10^{-8}\,M_\odot$ yr$^{-1}$ by using the following equation, $$\begin{aligned}
{\dot{P}\over P}=-3\dot{M_1}({1\over M_1}-{1\over M_2}) .\end{aligned}$$ The positive sign indicates that the less massive primary component is receiving mass. Assuming the angular momentum and the total mass are constant and the more massive component transfers its mass on a thermal timescale, $\tau_{th}=3.39\times10^7$ years can be calculated using this relation $\tau_{th}={GM_2^2\over R_2L_2}$. Then, the mass transfer rate can be roughly derived to be $M_2/\tau_{th}=3.81\times10^{-8}\,M_\odot$ yr$^{-1}$, which is coincide with the result determined by Equation (6). Another possibility is the AML due to magnetic stellar winds. An approximate for calculating the period decrease rate was given by [@Guinan+Bradstreet+1988] as below, $$\begin{aligned}
{dP\over dt}\approx -1.1\times10^{-8}q^{-1}(1+q)^2(M_1+M_2)^{-5/3}k^2\times(M_1R_1^4+M_2R_2^4)P^{-7/3},\end{aligned}$$ where $k^2$ is the gyration constant. Taking $k^2=0.1$ from [@Webbink+1976] for low-mass main-sequence stars, we derived that the period decrease rate caused by AML is $-0.71\times10^{-7}$ days/year, which is similar to the observed value. Therefore, both the conservative mass transfer and AML can explain the long-term period decrease of V342 UMa. At present, we can not know which one is dominant only based on the observed period variation.
The period of V509 Cam is continuously increase at a rate of $3.96(\pm0.90)\times10^{-8}$ days/year. The long-term period increase is generally attributed to conservative mass transfer. Using Equation (6), we derived that the mass transfer rate is $dM_1/dt=-1.07(\pm0.24)\times10^{-8}\,M_\odot$ yr$^{-1}$. The negative sign exhibits that the less massive primary component is transferring mass to the more massive secondary one. V509 Cam is a late type contact binary, the AML due to magnetic stellar winds should also can happen in V509 Cam. Therefore, the determined mass transfer rate should be considered as a minimal value.
The secular period decrease rate, $-1.02(\pm0.54)\times10^{-7}$ days/year, of V342 UMa is very common in W-subtype contact binaries, such as $-1.69\times10^{-7}$ days/year for V502 Oph (@Zhou+etal+2016), $-0.62\times10^{-7}$ days/year for GU Ori (@Zhou+etal+2018), and $-1.78\times10^{-7}$ days/year for V1007 Cas (@Li+etal+2018). With decreasing period, V342 UMa will evolve from the present shallow contact configuration to high fill-out contact state. The long-term period increase rate, $3.96(\pm0.90)\times10^{-8}$ days/year, of V509 Cam is also very common in W-subtype contact binaries, such as $5.09\times10^{-8}$ days/year for EP And (@Lee+etal+2013), $7.7\times10^{-8}$ days/year for UX Eri (@Qian+etal+2007b), and $6.5\times10^{-8}$ days/year for LR Cam (@Yang+Dai+2010). With increasing period, V509 Cam may evolve into a broken-contact binary. There is another possibility that the long-term period variations of the two systems are only part of very long period periodic variations as a result of distant third body (@Liao+Qian+2010), continuous observations of the two targets are required to confirm this in the future.
Light Curve Variations of the Two Targets
-----------------------------------------
As seen in Figure \[Fig1\] and Figure \[Fig2\], we can discover that the light curves of V342 UMa and V509 Cam display very clear variations. For V342 UMa, the light curves observed in 2018 show a positive O’Connell effect (means the first light maximum, Max. I, is brighter than the second one, Max. II), while the light curves observed in 2019 reversed to exhibit a negative O’Connell effect (means Max. I is fainter than the Max. II). For V509 Cam, the light curves observed in February, 2018 show a negative O’Connell effect, then light curves observed in March, 2018 are almost symmetric. However, the light curves reverse to exhibit a positive O’Connell effect in April, 2018. The differences between Min.I and Min.II, Max.II and Max.I, Min.I and Max.I, and Min.II and Max.I were calculated and are listed in Table \[Tab:differences\]. Such changes in the light curves are very common in contact binaries, such as BX Peg (@Lee+etal+2004 [@Lee+etal+2009]), HH UMa (@Wang+etal+2015), RT LMi (@Qian+etal+2008). They are generally caused by magnetic activities and can be interpreted by the presence of spots. The changes of the light curves of V342 UMa and V509 Cam are both explained by spot variation.
Epoch Filter Min.I - Min.II Max.II - Max.I Min.I - Max.I Min.II - Max.I
---------- -------- ---------------- ---------------- --------------- ----------------
V342 UMa
2018 May $B$ 0.041 0.034 0.673 0.632
$V$ 0.027 0.031 0.631 0.604
$R_c$ 0.020 0.027 0.607 0.587
$I_c$ 0.038 0.026 0.609 0.571
2019 Jan $B$ 0.055 -0.012 0.655 0.600
$V$ 0.043 -0.009 0.614 0.572
$R_c$ 0.036 -0.008 0.591 0.555
$I_c$ 0.045 -0.007 0.584 0.539
V509 Cam
2018 Feb $B$ -0.008 -0.021 0.620 0.628
$V$ -0.007 -0.018 0.611 0.618
$R_c$ 0.000 -0.016 0.605 0.605
$I_c$ 0.004 -0.015 0.593 0.588
2018 Mar $B$ -0.004 0.001 0.642 0.646
$V$ -0.009 -0.001 0.610 0.619
$R_c$ 0.003 0.000 0.608 0.605
$I_c$ 0.003 0.000 0.591 0.588
2018 Apr $B$ 0.007 0.026 0.660 0.653
$V$ 0.013 0.021 0.641 0.628
$R_c$ 0.012 0.018 0.621 0.609
$I_c$ 0.021 0.016 0.613 0.592
: Light diffferences for V342 UMa and V509 Cam at different epochs[]{data-label="Tab:differences"}
Note. Min.I, Min.II, Max.I, and Max.II respectively denote the primary light minimum, the secondary light minimum, the light maximum after Min. I, and the light maximum after Min. II.
The Evolutionary Status
-----------------------
To study the evolutionary status of the two stars, the Hertzsprung-Russell (H-R) diagram and the color-density (C-D) diagram were constructed and are shown in the left and right panels of Figure \[Fig9\], respectively. The zero age main sequence (ZAMS) and the terminal age main sequence (TAMS) in the H-R diagram were taken from [@Girardi+etal+2000], while the ZAMS and TAMS in the C-D diagram were come from [@Mochnacki+1981]. In order to compare with other W-subtype contact binaries, the W-subtype low mass contact binaries (LMCBs) listed in [@Yakut+Eggleton+2005] are also displayed in Figure \[Fig9\]. The horizontal ordinate of the right panel of Figure \[Fig9\] is color-index, we converted the temperatures of the components of all systems including V342 UMa and V509 Cam to color-index based on Table 5 of [@Pecaut+Mamajek+2013]. The mean densities of the components were calculated using the relation provided by [@Mochnacki+1981], $$\begin{aligned}
\overline{\rho_1}={0.079\over V_1(1+q)P^2}\,g\,cm^{-3}, \qquad \overline{\rho_2}={0.079q\over V_2(1+q)P^2}\,g\,cm^{-3}.\end{aligned}$$ In Figure \[Fig9\], the ZAMS and TAMS are labeled as solid and dotted lines, respectively, and the circles refer to the more massive components (p), while the triangles represent the less massive ones (s). Both the H-R diagram and the C-D diagram reveal that the components of V342 UMa and V509 Cam are consistent with those of other W-subtype contact systems. The less massive components are close to the ZAMS, meaning that they are main-sequence or little evolved stars, while the more massive ones are located near the TAMS, indicating that they are at an advanced evolutionary stage. The evolutionary status of the components of W-subtype contact systems is similar with that of the A-subtype ones. Therefore, W-subtype phenomenon is still an open question. More observations and investigations on the two subtype contact binaries are needed.
![The H-R diagram and C-D diagram. The solid and dotted lines display the ZAMS and TAMS, respectively. The circles refer to the more massive components (p), while the triangles represent the less massive ones (s).[]{data-label="Fig9"}](f9a.eps "fig:") ![The H-R diagram and C-D diagram. The solid and dotted lines display the ZAMS and TAMS, respectively. The circles refer to the more massive components (p), while the triangles represent the less massive ones (s).[]{data-label="Fig9"}](f9b.eps "fig:")
In this paper, the investigations of the light curves and period variations of V342 UMa and V509 Cam are presented. We found that both of the two systems are W-subtype contact binaries and show very strong light curve changes. The period changes analysis reveal that V342 UMa shows long-term period decrease which can be caused by conservative mass transfer or AML due to magnetic stellar winds and that V509 Cam exhibits long-term period increase which can be attributed to conservative mass transfer. The absolute parameters of the two binaries were obtained based on the Gaia distances. In order to identify cyclic period changes of them, further observations are required.
This work is supported by Chinese Natural Science Foundation (No. 11703016), and by the Joint Research Fund in Astronomy (No. U1431105) under cooperative agreement between the National Natural Science Foundation of China (NSFC) and Chinese Academy of Sciences (CAS), and by the program of the Light in China¡¯s Western Region, No. 2015-XBQN-A-02, and by the Natural Science Foundation of Shandong Province (Nos. ZR2014AQ019, JQ201702), and by Young Scholars Program of Shandong University, Weihai (Nos. 20820162003, 20820171006), and by the program of Tianshan Youth (No. 2017Q091), and by the Open Research Program of Key Laboratory for the Structure and Evolution of Celestial Objects (No. OP201704). Thanks the referee very much for the very helpful comments and suggestions to improve our manuscript.
We acknowledge the support of the staff of the Xinglong 85cm telescope, NOWT, WHOT and NEXT. This work was partially supported by the Open Project Program of the Key Laboratory of Optical Astronomy, National Astronomical Observatories, Chinese Academy of Sciences.
This paper makes use of data from the DR1 of the WASP data (@Butters+etal+2010) as provided by the WASP consortium, and the computing and storage facilities at the CERIT Scientific Cloud, reg. no. CZ.1.05/3.2.00/08.0144 which is operated by Masaryk University, Czech Republic.
This work has made use of data from the European Space Agency (ESA) mission [*Gaia*]{} (<https://www.cosmos.esa.int/gaia>), processed by the [*Gaia*]{} Data Processing and Analysis Consortium (DPAC, <https://www.cosmos.esa.int/web/gaia/dpac/consortium>). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the [*Gaia*]{} Multilateral Agreement.
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[^1]: The website of O-C Gateway is http://var.astro.cz/ocgate/
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---
abstract: 'This article presents a novel framework for performing visual inspection around 3D infrastructures, by establishing a team of fully autonomous Micro Aerial Vehicles (MAVs) with robust localization, planning and perception capabilities. The proposed aerial inspection system reaches high level of autonomy on a large scale, while pushing to the boundaries the real life deployment of aerial robotics. In the presented approach, the MAVs deployed for the inspection of the structure rely only on their onboard computer and sensory systems. The developed framework envisions a modular system, combining open research challenges in the fields of localization, path planning and mapping, with an overall capability for a fast on site deployment and a reduced execution time that can repeatably perform the inspection mission according to the operator needs. The architecture of the established system includes: 1) a geometry-based path planner for coverage of complex structures by multiple MAVs, 2) an accurate yet flexible localization component, which provides an accurate pose estimation for the MAVs by utilizing an Ultra Wideband fused inertial estimation scheme, and 3) visual data post-processing scheme for the 3D model building. The performance of the proposed framework has been experimentally demonstrated in multiple realistic outdoor field trials, all focusing on the challenging structure of a wind turbine as the main test case. The successful experimental results, depict the merits of the proposed autonomous navigation system as the enabling technology towards aerial robotic inspectors.'
author:
- 'Christoforos Kanellakis\*'
- 'Emil Fresk\*'
- 'Sina Sharif Mansouri\*'
- Dariusz Kominiak
- George Nikolakopoulos
bibliography:
- 'mybib.bib'
title: 'Autonomous visual inspection of large-scale infrastructures using aerial robots'
---
Background & Related Works
==========================
Nowadays, Micro Aerial Vehicles (MAVs) are gaining more and more attention from the scientific community, constituting a fast-paced emerging technology that constantly pushes their limits for accomplishing complex tasks [@Kanellakis2017]. These platforms are characterized by their mechanical simplicity, agility, stability and outstanding autonomy to reach remote and distant places. Endowing MAVs with proper sensor suites, while navigating in indoors/outdoors, cluttered and complex environments, could establish them as a powerful aerial tool for a wide span of applications. Some characteristic examples of application scenarios for such a novel deployment of the aerial technology include infrastructure inspection [@mansouri2018cooperative; @TORRES2016441], public safety-surveillance [@michael2012collaborative], and search and rescue missions [@tomic2012toward].
One of the most common application areas that MAVs are employed, is in the filming industry, but there are efforts from other industries such as Mining, Oil, and Energy Providers, to invest in the commercialization of MAVs to perform remote inspection applications. Towards this vision, MAVs are powerful tools that have the profund potential to decrease the risks of human life, decrease the execution time and increase the efficiency of the overall inspection task, especially when compared to conventional methods [@sesar2016european]. Despite the fact that the research in the aerial robotics has reached significant milestones regarding localization [@perez2018architecture], planning [@achtelik2014motion] and perception [@scaramuzza2014vision; @GARCIAPULIDO2017152], successful real-life demonstrations of autonomous inspection systems have been rarely reported in the literature, with the majority of the applications focusing on impressive laboratory trials under full control environments and in most of the cases under the utilization of expensive motion capturing systems [@lupashin2014platform] or small scale and well defined outdoor environments [@teixeira2017real; @forster2015continuous].
Thus, one of the most important contributions of this article is the establishment of an aerial system capable to visually pre-inspect fully autonomously an outdoors large scale infrastructure, through the coordination of multiple aerial vehicles. Towards this contribution, the article will further contribute with the implementation of a novel and accurate localization enabled scheme for collaborative aerial inspection of infrastructure, a scheme that is based on Ultra WideBand (UWB) distance measurements and Inertial Measurement Unit (IMU) sensor fusion. In this approach, the aerial platforms navigate autonomously based on the UWB-Inertial fused state estimation, using a local UWB network, placed around the structure of inspection. A second contribution of this article is the experimental evaluation of a Collaborative Coverage Path Planner (C-CPP) algorithm that has the ability to guarantee the full coverage of the infrastructure by considering camera, geometry, collision, and other application posed constraints. The coverage path is generated for every MAV, based on the structure geometric characteristics, while identifying and assigning parts of the structure to different agents, leading to faster inspection times. As an outcome, the covered path guarantees an overlapping Field of View (FoV) to enable the generation of an off-line 3D model of the structure. The final contribution stems from the real life successful demonstration of a fully functional on-board visual sensor scheme that it is able to have the dual role of providing: a) low resolution compressed data for the visual assessment of the structure, and b) high resolution for post processing e.g. build 3D models and area image stitching. The overall concept of the proposed collaborative aerial inspection scheme is presented in Figure \[inspection\_concept\], where two collaborative MAVs are performing an aerial inspection of a wind turbine with a corresponding video [^1].
The rest of the article is structured as it follows. The overall system is described in Section \[Aerial Inspection System\]. More specifically, Section \[Cooperative Coverage Path Planner\] presents the novel geometric approach for the C-CPP problem for infrastructure inspection. Section \[Inertial Odometry\] provides an analysis on UWB fused inertial based localization for aerial platforms, while Section \[Visual inspection\] establishes the 3D reconstruction problem from multiple images and multiple MAVs. Section \[Experimental\] demonstrates the experimental setup and presents the experimental trials for the presented inspection system. Finally, the concluding remarks are presented in Section \[Conclusions\].
Aerial Inspection System {#Aerial Inspection System}
========================
This article, inspired by the increasing capabilities of MAVs, establishes an autonomous aerial inspection system, which is specialized in large scale industrial facilities. The system is realized by either a single agent or a team of agents and is characterized by advanced localization and structure coverage capabilities, all demonstrated in real life by inspecting a wind turbine power plant, where the aim of the system is to provide visual data to infrastructure owners for further analysis and asset management. The overall scheme of the proposed system is depicted in Figure \[fig:overallscheme\].
Field Trials and Open Challenges {#Challenges}
--------------------------------
During the development of the proposed aerial inspection framework, the wind turbine site located in Bure, Sweden have been visited multiple times. In these sites, the wind speed was measured up to 13 $\unit{m/s}$, while the wind turbine structure depicted in Figure \[inspection\_concept\], had a base diameter of approximately 4.5 $\unit{m}$, with a top diameter of approximately 1.5 $\unit{m}$ and with the height of the tower being 64 $\unit{m}$. Moreover, the length of each blade was 22 $\unit{m}$ with a corresponding cord length, at the root of the blade, of approximately 2 $\unit{m}$ and at the top of the blade of 0.2 $\unit{m}$, while the length of the hub and nacelle was approximately 4 $\unit{m}$.
Operating MAVs outside the lab, and especially around large scale infrastructures such as wind turbines, raises significant multidisciplinary research issues where one of the most important is to provide an accurate localization system that at the same time would be easily deployable. At the wind turbines the GPS solution fails at low height due to the multipath errors, which happen when the GPS receiver cannot distinguish a direct signal from a reflection, a fact that causes significant errors in the measurements. Usually, the GPS works well in positions where the interference from the building is small enough, however this is only at significant heights in this case. Moreover, the trending technology of visual inertial odometry, opposite to GPS, cannot provide reliable localization feedback in high altitudes. These algorithms base part of their processing in visual measurements by detecting areas of high contrast and texture. More specifically, in high altitudes this processing becomes unreliable, since they cannot detect and extract distinctive features from the environment due to lack of feature-rich local surfaces/areas, e.g. in the case of wind turbines which are simply described by a flat white color. This makes it difficult for the visual inertial odometry software to converge its state of movement to the actual state. As depicted in Figure \[fig:VIfeature\], the detected features are far-away, while there are no features on the wind turbine tower itself except for unstable boundary features, and egomotion causes very little feature movement to the background.
![A sample of features detected on the wind turbine tower and the surrounding environment.[]{data-label="fig:VIfeature"}](./Pictures/tower_features.jpg){width="0.5\linewidth"}
Furthermore, the challenges of the visual sensors, identified for localization, extends also to other visual processing tasks, such as 3D reconstruction, where during the performed experimental trials it was found that the depth and stereo sensors failed to provide a solid 3D model of the wind turbine. Additionally, MAVs provide a limited flight time, which can be affected by external disturbances, such as wind gusts, payloads and temperature of the environment. This limits the feasibility of the inspection mission with one MAV, especially in large scale structures, such as wind turbines. Moreover, strong wind gusts cause significant drift of the MAV from the predefined trajectory and it should be compensated by the MAV’s position controller. Thus the limited flight time and the deviation from the trajectory should taken under consideration or the overall system can fail to perform the inspection, or even worse result in a collision with the infrastructure that might cause damages to both the infrastructure and the aerial platform.
System Hardware
---------------
### MAV
For the envisioned aerial inspection system for large scale infrastructures, the Ascending Technologies NEO hexacopter was utilized as the MAV platform, where in Figure \[fig:NEOcombined2\] the overall specifications and the selected sensors are presented. This platform is capable of providing a flight time of up to 26$\unit{min}$ without payload and in ideal conditions, with a maximum payload capacity up to 2$\unit{kg}$. For onboard processing, the belly of the MAV contains an Intel NUC computer with a Core i7-5557U and 8 GB of RAM that runs Ubuntu Server 16.04 with the Robotic Operatic System (ROS) as its core. The platform has been equipped with a large set of different sensors, as depicted in Figure \[fig:NEOcombined2\], where each component will be explained in the sequel.
at (0,0) \[\] [![AscTec NEO platform equipped with the utilized full sensory system for the aerial inspection.[]{data-label="fig:NEOcombined2"}](./Pictures/NEO_sensors_clean.png "fig:"){width="0.7\columnwidth"}]{};
at (-1.4,-4.3) \[right\] [UWB node]{}; (-1.5,-4.3) – (-3.1,-3.8);
at (0,-1.3) \[\] [VI sensor]{}; (0,-1) – (0,0.7);
at (3,1.2) \[right\] [GoPro]{}; (2.9,1.2) – (0.3,1.5);
at (3,4.5) \[\] [PlayStation camera]{}; (3,4.3) – (0.3,2.4);
at (-3,4.5) \[\] [LIDAR]{}; (-3,4.3) – (-0.3,3.3);
(-3.8,-0.6) – (3.9,-0.6); at (2.5,-0.85) \[\] [0.87m]{};
### Localization system
Due to the feature-less surface of the wind turbines for visual odometry and the existence of multipath errors in the GPS measurements, as was discussed in the prequel, the localization algorithms based on cameras and GPS failed during the field trials, and thus the proposed localization system was based on UWB and IMU fusion. This component is extensively explained in Section \[Inertial Odometry\].
### Sensor suite
The proposed sensory suite for the aerial inspectors included 3 different cameras: a) the Visual-Inertial (VI) sensor, b) the GoPro Hero4, c) the PlayStation Eye, and an additional laser range finder RPLIDAR, as depicted in Figure \[fig:NEOcombined2\]. The VI sensor developed by Skybotix AG with a weight of 0.117$\unit{kg}$ was attached below the hexacopter with a 45$\unit{^\circ}$ tilt from the horizontal plane, which is a monochrome, high dynamic range, global shutter stereo camera with 120$\unit{^\circ}$ DFOV and with a resolution of 752x480 pixels, moreover it is housing an Analog Devices ADIS16445 tactical grade IMU. Both cameras and IMU were tightly temporally aligned with hardware synchronization, while the cameras were operated at 20$\unit{fps}$. The GoPro Hero4 camera was attached on top of the hexacopter facing forward with a weight of 0.2$\unit{kg}$, while it was capable of recording high-definition video at various resolutions, ranging from 720p to 4000p and at a rate of 15-120$\unit{fps}$, while during the experimental trials the camera was operated with a 2K resolution at 30$\unit{fps}$. The Playstation Eye camera was attached in the middle of the hexacopters housing, facing forward with a weight of 0.150$\unit{kg}$, this camera was operated at 20$\unit{fps}$ and with a resolution of 640x480 pixels. The variety in the specifications of the camera suite was motivated by the need to test their performance under challenging conditions, regarding the dataset collection. Thus, the main aim was to use the captured frames for direct visual inspection by experts in the structure maintenance, while the data from the VI sensor and the GoPro camera were also used to provide 3D models of the inspected parts. Finally, RPLDAR was a low cost laser sensor, which provides a $360^\circ$ scan field at a 5.5$\unit{Hz}$/10$\unit{Hz}$ rotating frequency with guaranteed 8 meter range. This laser scanner has also been tested during the experimental trials for enabling the online obstacle avoidance schemes.
Cooperative Coverage Path Planner {#Cooperative Coverage Path Planner}
---------------------------------
Towards the vision of the inspector MAV, the theoretical framework established in [@mansouri2018cooperative] is integrated in the autonomous inspector framework and experimentally tested in the complex case of a windturbine structure. Briefly, the coverage scheme is capable of providing a path for accomplishing a full coverage of the infrastructure, without any shape simplification, by slicing it by horizontal planes to identify branches of the infrastructure and assign specific areas to each agent. Complicated structures have multiple branches e.g. in wind turbine the base and each blade are considered as branches, where the proposed method identifies these branches and assign paths to $n$ agents. If the structure has one branch all $n$ agents are assigned to the same branch, otherwise the $n$ agents are equally distributed to different branches. Furthermore, to guarantee a full coverage to facilitate visual processing, the introduced path planning creates for each agent an overlapping visual inspection area. The novel established C-CPP scheme, in addition to the position references, provides also yaw references for each agent to assure a field of view, directed towards the structure surface.
at (0,0) \[\] ; at (-3.1,-4) \[\] [$\lambda_1$]{}; at (-3.1,-3.1) \[\] [$\lambda_2$]{}; at (-3.1,-0.1) \[\] [$\lambda_{i-k}$]{}; at (-3.1,3.7) \[\] [$\lambda_i$]{};
at (-3.1,1.5) \[\] [$\mathcal{C}_1$]{}; (-3.1,1.3) – (-1.6,0.1); at (-1,1.5) \[\] [$\mathcal{C}_2$]{}; (-1,1.3) – (-0.35,0.1); at (1,1.5) \[\] [$\mathcal{C}_3$]{}; (1,1.3) – (0.6,0.1);
at (1.4,-4.45) \[\] [$\Omega$]{}; (-0.2,-4.2) – (2.8,-4.2);
at (2.1,-3.4) \[\] [$\alpha$]{}; (2,-3.6) – (2,-4);
at (-2.3,-3.6) \[\] [$\Delta \lambda$]{}; (-2,-3.2) – (-2,-4);
For the use of the C-CPP, initially the general case of an aerial inspector equipped with a limited Field of View (FOV) sensor was considered, determined by an aperture angle $\alpha$ and a maximum range $r_{max}$. Furthermore, $\Omega \in \mathbb{R}^+$ is the user-defined offset distance ($\Omega < r_{max}$), from the infrastructure’s target surface and $\Delta \lambda$ is the distance between each inspected plane. $\Delta \lambda$ is equal to $\frac{\Omega}{\beta} \tan{\alpha/2}$, where the parameter $\beta \in [1,+\infty)$ represents the ratio of overlapping. The horizontal planes are defined as $\lambda_i$, with $i\in \mathbb{N}$. The 3D map of the infrastructure is provided as a set $\boldsymbol{S}$ with a finite collection of points, denoted as $\boldsymbol{S}=\{p_i\}$, and $p_i=[x_i,y_i,z_i]^\top \in \mathbb{R}^3$. Furthermore, $\mathcal{C}_j(x,y,z)$ with $j \in [1,m]$ are the points in each branch and $m$ is the overall number of branches in the structure. A graphical overview of this C-CCP scheme is presented in Figure \[fig:ccp\]. = \[diamond, draw, text width=8em, text badly centered, node distance=3cm, inner sep=0pt, minimum height=2em\] = \[rectangle, draw, text width=10em, text centered, rounded corners, minimum height=4em\] = \[rectangle, draw, text width=8em, text centered, rounded corners, minimum height=2em\] = \[rectangle, draw, text width=8em, text centered, rounded corners, minimum height=4em\] = \[draw, -latex’\] = \[draw, ellipse, node distance=3cm, minimum height=2em\]
UWB Inertial Odometry Framework {#Inertial Odometry}
-------------------------------
UWB Radio Frequency (RF) communication is based on using a wide band of the RF spectrum, rather than a single frequency as a carrier wave radio does, which has the temporal representation of a pulse and as a result is sometimes referred to as a pulse radio. Due to the high center frequency (3.1 to 4.8 GHz and 6.0 to 10.6 GHz) and the spectral width of the pulse (499.2 to 1331.2 MHz) the pulses have good spatial resolution, which makes them ideal for time stamping RF packets, referred to as messages, with high accuracy. This property of accurate timestamps, together with good reference clocks, give the ability to estimate the distance between two transceivers by exchanging 2 or more packets and thus it could be considered that the distance estimation is a byproduct of communication.
Furthermore, one major drawback of a carrier wave based radio is the problem of multipathing, where the carrier wave forms destructive interference with itself, effectively reducing the received signal strength, or introducing an unknown phase shift. This is a problem that is severely mitigated in the UWB radio, where the spatial length of each pulse is small enough for each pulse to be detected uniquely and this allows the receiver to reconstruct the pulse from multiple reflections. In a sense, the more the reflections are available, the stronger the received signal is, in contrary to GPS, which can give highly misleading measurements when close to tall structures.
In Figure \[fig:NEOcombined2\], the UWB node developed by LTU is depicted when mounted on the MAV. This hardware contains all the embedded electronics including the microprocessor, 3-axis accelerometer, 3-axis gyroscope, the UWB RF transceiver and the antenna to enable the UWB communication and localization, while this system is fully self contained and can directly be deployed for enabling full localization of the MAV state.
For a proper operation of the estimation framework, it is needed to have the UWB transceivers with known and fixed positions, called anchors (while the transceiver on the MAV is called a tag), spread out in the working area to act as known positions to measure distances for the later trilateration and fusion with the IMU, as described in [@fresk2017uwb]. This is directly analogous to GPS, while here the “satellites” (anchors) are placed as needed within the operating volume, conceptually presented in Figure \[fig:uwbsys\].
at (0,0) \[\] [![An overview of the UWB localization system, where $A_1$ - $A_5$ are the stationary anchors and $N_1$ is the tracked node mounted on the MAV, while the dashed lines highlight the measured distances.[]{data-label="fig:uwbsys"}](./Pictures/Panorama1.jpg "fig:"){width="0.7\columnwidth"}]{};
at (1.3,-0.8) ;
at (-1.3,-4.8) ; (-1.5,-4.3) – (0.9,-0.85);
at (1.3,-4.8) ; (1.0,-4.3) – (1,-1.0);
at (1.9,-3.0) ; (1.55,-2.5) – (1.1,-0.95);
at (-0.3,-2.5) ; (-0.4,-2.1) – (0.9,-0.7);
at (-1.5,-2.9) ; (-1.6,-2.5) – (0.85,-0.65);
Surface reconstruction {#Visual inspection}
----------------------
As stated in the prequel, this work targets the application scenario of autonomous inspection by single or multiple MAVs, where the objective of the inspection missions is the collection of high resolution visual data of regions of interest and the generation of 3D surface models. All available data will be used afterwards by inspection experts to analyze and detect possible defects on their assets. To this end, each aerial platform is equipped, but not limited, with a camera to record the required data from the infrastructure.
During the navigation of the MAVs around the structure, the raw visual stream is directly available for defect assessment. Regarding the surface reconstruction, the main approach to process the data considers a monocular camera Structure from Motion (SfM) [@schonberger2016structure], where the MAVs fly around covering specific parts, with the aim to collaboratively process all the captured data into a global representation. The selection of monocular mapping is driven by the application scale and the object characteristics. Generally, the perception of depth using stereo cameras is bounded to the stereo baseline, essentially reducing the configuration to monocular at far ranges and to this end, stereo algorithms cannot perform in cases with large structures and high altitudes. The employed SfM approach is an offline process that provides a sparse 3D reconstruction and accurate MAV poses, by using different camera viewpoints and consists of a massive optimization process. Finally, the data collected during the navigation mission is down sampled, since they contain redundant information from all the camera frames and there is a need to keep the resulting outcome within a reasonable time, while the sparse pointcloud is inserted into Multi View Stereo algorithms to perform dense reconstruction based on multi view stereo pairs [@seitz2006comparison].
System Software
---------------
The navigation system of the aerial inspector is integrated within the ROS framework, where two main components provide autonomous flight, namely an UWB inertial odometry estimator, based on the Multi Sensor Fusion Extended Kalman Filter (MSF-EKF) [@msf] and a linear Model Predictive Control (MPC) based position controller [@mav_linear_mpc_]. The sensor fusion node consists of an EKF filter that does tight inertial fusion from the hexacopter’s IMU during the state propagation and the UWB range measurements are utilized during the filter correction step. The outcome of the UWB inertial odometry are the position, orientation (pose), the linear/angular velocity (twist) of the aerial robot and the IMU biases. This consists of an error state Kalman filter performing sensor fusion as a generic software package that has the unique feature to handle delayed measurements, while staying within the desired computational bounds. The linear MPC position controller [@mav_linear_mpc_] generates attitude and thrust references for the NEO’s predefined low level attitude controller, with the aim to have separation of concerns, as the high level control and planning algorithms should have minimal knowledge of the low level controllers. The overall functional schematic of the experimental setup is presented in Figure \[fig:experimental\_setup\].
The C-CPP method, described in Section \[Cooperative Coverage Path Planner\], has been entirely implemented in MATLAB. The inputs for the method are a 3D model of the infrastructure of interest and specific parameters, which are the number of agents ($n$), the offset distance from the object ($\Omega$), the FOV of the camera ($\alpha$), the desired velocity of the aerial robot ($V_d$) and the position controller sampling time ($T_s$). The generated paths are sent to the NEO platforms through the utilization of the ROS framework.
Experimental results {#Experimental}
====================
Mission Preliminaries
---------------------
The presented aerial platform with the sensor systems and combined with the developed algorithmic components, described in previous section, constitutes the autonomous aerial inspection system named autonomous aerial inspector. The capabilities of the aerial inspectors have been publicly demonstrated for the case of wind turbine inspection in Sweden, where the mission scenario was two-fold by targeting the inspection of two separate parts of the structure, namely the wind turbine tower and the wind turbine blades. The requirements for the system were to provide a complete coverage of the inspected parts autonomously, while storing all necessary visual data for further analysis. Although, two agents were used for the specific case presented in this work, the presented inspection system can operate either in a single agent or multi-agent mode, depending on the application needs and the flying limitations of the MAVs.
The initial step for the deployment of the inspection system was to setup the ground station for monitoring the operations and fix 5 UWB anchors around the structure, with specific coordinates presented in Table \[tab:UWB anchor coordinates\], which constitute the infrastructure needed for the localization system of each aerial platform. The number of anchors as well as their position has been selected in a manner to guarantee UWB coverage around all parts of the wind turbine. From a theoretical point of view [@fresk2017uwb], only 3 anchors are needed, however it is common that one anchor will be behind the wind turbine for the MAV’s point of view, which gives rise to a minimum of 4 anchors to compensate, while a fifth anchor was added as redundancy. The resulting fixed anchor positions provide a local coordinate frame that guarantees repeatability of the system, and with the significant ability to revisit the same point multiple times, in case the data analysis shows issues that require further inspection. An important note for all the inspection cases on the wind turbine and for the system in operation is that the blades are locked in a star position, as shown in Figure \[fig:uwbsys\], which simplifies the 3D approximate modeling of the structure.
[ > p[0.15]{} < > p[0.1]{} < > p[0.1]{} < > p[0.1]{} < > p[0.1]{} < > p[0.1]{} < ]{} **Coordinate** & $\boldsymbol{A}_1$ & $\boldsymbol{A}_2$ & $\boldsymbol{A}_3$ & $\boldsymbol{A}_4$ & $\boldsymbol{A}_5$\
$x$ & & & & &\
$y$ & & & & &\
$z$ & & & & &\
\[tab:UWB anchor coordinates\]
In the proposed architecture, all the processing necessary for the navigation of the MAVs is performed onboard, while the overview of the mission and the commands from the mission operators (inspectors) is performed over a WiFi link, while the selection of WiFi is not a requirement and can be replaced with the communication link of choice e.g. 4G cellular communication. The UWB based inertial state estimation runs at the rate of the IMU, which in this case was 100 Hz, and the generated coverage trajectory has been uploaded to the MAV before take-off, which is followed as soon as the mission started by the command of the operator. The paths have been followed autonomously, without any intervention from the operators on the site, and the collected data have been saved onboard, while after downloading the mission data post processing is performed in the ground station or in the cloud. The data provided by the system can be used for position aware visual analysis, examining high resolution frames or they can be post-processed to generate 3D reconstructed models. The key feature to be highlighted from the inspection is that any detected fault can be fully linked with specific coordinates, which can be utilized by another round of inspections or for guiding the repair technician. The final, is a major contribution of the presented aerial inspection system, since this need is the fundamental information that is needed for enabling a safe and autonomous aerial inspection that has the potential to performed the human based ones.
Wind turbine inspection
-----------------------
For the specific case of wind turbines the C-CCP generated inspection paths have been obtained with two autonomous agents in order to reduce the needed flight time, and still be within the battery constrained flight time of the utilized MAV. However, due to the limited flight time of the MAVs in the field trials, the inspection problem has been split into the tower inspection and the blade inspection, where the specifics of each is presented in the sequel Table \[tab:info\_mission\], while both can be performed at the same time with more MAVs to reduce inspection time even further. A common characteristic for both of the cases is that the generated path for each MAV keeps a constant safety distance from the structure, while at the same time is keeping it in view of the visual sensors, and maximizing the safety distance between agents, which gives rise to the agents being on opposite side of the wind turbine at all times. The area in which the inspection is performed is generally of high wind and while the inspection of the tower is protected from wind, owing to the forest, the blade inspection is above the tree line. Thus, the aerial inspections have been specifically tuned to compensate strong wind gusts that were measured up to 13 m/s, where the tunning was targeting the MAV’s controller’s weight on angular rate that has been increased to significantly reduce the excessive angular movement.
[ > p[0.45]{} < > p[0.15]{} < > p[0.15]{} < ]{} **Mission Configuration** & **Tower** & **Blade**\
Number of agents & 2 & 1\
Inspection Time & &\
Safety Distance & &\
Velocity & &\
Starting height & &\
Finishing height & &\
\[tab:info\_mission\]
### Tower inspection
In the specific case of the wind turbine base and tower inspection, the generated paths are of a circular shape, as depicted in Figure \[fig:path\_reconst\_tower\], which is the result of the constant safety distance from the structure based on the C-CCP algorithm. As can be seen from the tracked trajectories the controllers perform well with an RMSE of 0.5464 m, while at the top of the trajectory a more significant error can be seen that is induced from the specific MAV transitioning above the tree-line, where a wind gust caused the deviation from the desired trajectory where the MAV compensates and finishes it’s inspection trajectory.
From the depicted reconstruction in Figure \[fig:path\_reconst\_tower\], it is possible to understand that the base of the wind turbine, which is feature rich, provides a good reconstruction result, while as the MAV continues to higher altitudes, the turbine tower loses texture due to its flat white color, causing the reconstruction algorithms to not provide a successful reconstruction. However, the visual camera streams do have position and orientation for every frame, as depicted in Figure \[fig:path\_reconst\_tower\] for some instances, which allows for a trained inspector to review the footage and be able to determine if there are spots which need extra inspection or repairs. For the reconstruction in Figure \[fig:path\_reconst\_tower\], the [@pizer1987adaptive] and [@schonberger2016structure] algorithms have been used, the former for pre-processing the images for enhance their contrast, while the latter was the SfM approach for providing the 3D model of the structure. The reconstruction took place on a PC with the configuration i7-7700 CPU and 32 GB of RAM, where the processing lasted approximately 4 hours.
at (0,0) \[align=center\] [\
\
]{};
(-3,-4.9) – (-2.5, -0.3); (0,-4.9) – (-2.4,5.3); (3,-4.9) – (1.3,4.25);
### Blade inspection
Compared to the base and tower inspection, for which the C-CCP algorithm generated circular trajectories, a similar approach was followed for the the blade inspections. This comes from the fact that the blade inspection is performed on the blade with a direction towards the ground and with the trailing edge of the blade towards the tower, which would cause the C-CCP algorithm to generate half-circle trajectories. However, in this case the same agent can inspect the final part of the tower by merging both tower and blade trajectories, as can be seen in Figure \[fig:wind3D2\], while minimizing the needed flight time and demonstrating at a full extend the concept of aerial cooperative autonomous inspection. With the available flight time of the MAV, it is possible to inspect the blade with only one operating MAV, allowing for the safety distance between agents to be adhered to, by the separation of the inspected parts. However, during the blade inspection, the tracking performance of the MAV was reduced to an RMSE of 1.368 m, due to the constant exposure to wind gusts and the turbulences generated by the structure, and as these effects were not measurable, until the effects are observed on the MAV, it has reduced the overall observed tracking capabilities of the aerial inspectors. The second effect of the turbulence was the excessive rolling and pitching of the MAV, which introduced a significant motion blur in the captured video streams, due to the fixed mounting of the camera sensor, introducing the need for adding a gimbal for stabilizing the inspection camera and reducing the motion blur. Finally, as can be seen in the camera frames in Figure \[fig:wind3D2\], there are no areas of high texture on the wind turbine tower or blades which caused 3D reconstruction to fail. However, the visual data captured is of high quality and suitable for review by an inspector.
at (0,0) \[align=center\] [\
\
]{};
(-3,-1) – (-2.5,-0.2); (0,-1) – (0.2,2.35); (3,-1) – (2.4,1.6);
Lessons Learned {#lessons}
===============
Throughout the experimental trials for this inspection scenario, many different experiences were gained that assisted in the development and tuning of the algorithms utilized. Based on this experience, an overview of the lessons learned is provided in the sequel with connections to the different utilized field algorithms.
MAV Control
-----------
When performing trajectory tracking and position control experiments indoors a dedicated laboratory many disturbances, which are significant in the field trials, can be neglected and this is especially true for strong wind gusts and turbulences caused by the structure. In the case of indoor experimental trials, the MAV can be tuned aggressively to minimize the position tracking error, while in the full scale outdoor experiments this kind of tuning would provide excessive rolling and pitching due to the controllers trying to fully compensate for the disturbances. However, this has the side effect of making the movements jerky and oscillatory, and overall reduce the operator’s trust in the system as it seems to be close to unstable. Furthermore, in the case that the controllers were tuned for a smooth trajectory following, larger tracking errors would have to be accepted in the trajectory following. During the field trials, some wind gust can even be above the operational limits of the MAV, causing excessive errors in the trajectory tracking. To reduce the effect in the outdoor experiments, the controller’s weight on angular rate was increased to significantly reduce the excessive movement, while in general the tuning of the high level control scheme, for the trajectory tracking, is a tedious task and it was found to be extremely sensitive to the existing weather conditions.
Planning
--------
The path planner provides a path to guarantee for a full coverage of the structure, however in the field trials, due to high wind gusts, there are variations between the performed trajectory and the reference. Thus, there is a need for an online path planner for considering these drifts and re-plan the path or to have a system that it is able to detect if a specific part of the structure has been neglected and provides extra trajectories to compensate. Additionally, due to the payload, the wind gusts and the low ambient temperature, the flight time was significantly less than the expected value from the MAV manufacturer. In certain worst cases, this time was down to 5 minutes, which is a severe limitation that should be considered in the path planning and task assignment to correctly select the correct number of agents for achieving a full coverage of the infrastructure.
System setup
------------
One of the most challenging issues when performing large scale infrastructure inspection is to keep a communication link with the agents performing the inspection, which is commonly used for monitoring the overall performance of the system. In this specific case, WiFi was the communication link of choice, mainly due to its simplicity of directly performing as expected, however it was quickly realized that the communication link was unstable due to height or occlusion of the MAV behind the wind turbine tower. To mitigate this issue, a different communication link should be used, e.g. the 4G cellular networks, and while WiFi can be used to upload mission trajectories it is not a reliable communication link at this scale.
Moreover, if it is desirable that the same mission can be executed again, the positions of the UWB anchors need to be kept. One possible way to achieve this is to consider the UWB anchors as supporting part of the infrastructure and have them permanently installed around the wind turbines, or to re-calibrate and consider the wind turbine as the origin, while only compensating for the rotation of the wind turbine depending on the mission setup.
3D reconstruction
-----------------
Various visual sensors have been tested in the challenging case of wind turbine. The most beneficial sensor proved to be the monocular camera system. More specifically, the fixed baseline for stereo cameras can limit the depth perception and eventually degenerate the stereo to monocular perception. The reconstruction performance can also vary slightly, depending on the flying environment due to visual feature differences, therefore a robust and reliable, invariant to rotations feature tracker should be used. Another important factor for the reconstruction is the camera resolution, since it poses the trade off between higher accuracy and higher computational costs. Additionally, the path followed around the structure affects the resulting 3D model, which in combination with the camera resolution can vary the reconstruction results. Generally, the cameras should be calibrated and it is preferred to have set manual focus and exposure to maintain the camera parameters for the whole dataset. For SfM techniques it is required a large motion in rotation and depth among sequential frames to provide reliable motion estimation and reconstruction.
Moreover, a low cost LIDAR solution, that was tested during the field trials, failed to operate due to sunlight interfering with the range measurements. This sensor technology, should be further examined with more tests since they could be useful in obstacle avoidance and cross-section analysis algorithms.
Localization
------------
While UWB positioning was the main localization system in the presented approach, it should be noted that this should not operate stand-alone. In the case of infrastructure inspection, one reference system should not act as a single point of failure, and it should be the aim to fuse as many sensors as possible. In the case of a wind turbine, the GPS does not provide a reliable position until the MAV is at significant height and the UWB localization system works best at lower height, hence it should be the aim to fuse both and utilize the sensor that is performing optimally depending on the current height. Moreover, neither UWB localization nor GPS provides a robust heading estimate, and the wind turbine causes magnetic disturbances that causing the magnetometers to fail and thus in this case visual inertial odometry is a robust solution to provide heading corrections since the landscape can be used as a stable attitude reference.
Conclusions {#Conclusions}
===========
This work presents a framework for autonomous visual inspection of a 3D infrastructure by utilizing multiple MAVs. To address this problem, the developed framework combined the fundamental tasks of path planning, localization and visual perception. Initially, a geometry-based path planner was employed for the collaborative coverage of complex structures, while the navigation of the platform has been performed through a localization component which provided accurate pose estimates of the MAVs by using a UWB-Inertial estimation scheme. Moreover, the inspection task considered compressed visual data streaming and visual data post processing for 3D model building. The performance of the proposed framework has the significant merit of being experimentally evaluated in realistic outdoor large scale infrastructure inspection experiments.
Acknowledgements {#Acknowledgements .unnumbered}
================
This work has received funding from the EU Horizon 2020 Research and Innovation Programme under the Grant Agreement No.644128, AEROWORKS. The authors would also like to thank Skellefte[å]{} Kraft for providing access to the windturbine site to perform the experimental trials.
[^1]: <https://youtu.be/z_Lu8HvJNoc>
|
---
abstract: |
We develop the *dichotomy spectrum* for random dynamical system and demonstrate its use in the characterization of pitchfork bifurcations for random dynamical systems with additive noise.\
Crauel and Flandoli [@Crauel_98_1] had shown earlier that adding noise to a system with a deterministic pitchfork bifurcation yields a unique attracting random fixed point with negative Lyapunov exponent throughout, thus “destroying” this bifurcation. Indeed, we show that in this example the dynamics before and after the underlying deterministic bifurcation point are topologically equivalent.\
However, in apparent paradox to [@Crauel_98_1], we show that there is after all a qualitative change in the random dynamics at the underlying deterministic bifurcation point, characterized by the transition from a hyperbolic to a non-hyperbolic dichotomy spectrum. This breakdown manifests itself also in the loss of uniform attractivity, a loss of experimental observability of the Lyapunov exponent, and a loss of equivalence under uniformly continuous topological conjugacies.
author:
- Mark Callaway
- Doan Thai Son
- 'Jeroen S.W. Lamb'
- |
Martin Rasmussen\
Department of Mathematics\
Imperial College London\
180 Queen’s Gate\
London SW7 2AZ\
United Kingdom
title: |
The dichotomy spectrum for random dynamical systems\
and pitchfork bifurcations with additive noise
---
[^1]
Introduction
============
Despite its importance for applications, relatively little progress has been made towards the development of a bifurcation theory for random dynamical systems. Main contributions have been made by Ludwig Arnold and co-workers [@Arnold_98_1], distinguishing between *phenomenological* (P-) and a *dynamical* (D-) bifurcations. P-bifurcations refer to qualitative changes in the profile of stationary probability densities [@Sri_90_1]. This concept carries substantial drawbacks such as providing reference only to static properties, and not being independent of the choice of coordinates. D-bifurcations refer to the bifurcation of a new invariant measure from a given invariant reference measure, in the sense of weak convergence, and are associated with a qualitative change in the Lyapunov spectrum. They have been studied mainly in the case of multiplicative noise [@Baxendale_94_1; @Crauel_99_1; @Wang_Unpub_1], and numerically [@Arnold_99_2; @Keller_99_1].
In this paper, we contribute to the bifurcation theory of random dynamical systems by shedding new light on the influential paper *Additive noise destroys a pitchfork bifurcation* by Crauel and Flandoli [@Crauel_98_1], in which the stochastic differential equation $$\label{sdeintro}
{\mathrm{d}}x= {\big(\alpha x-x^3\big)}{\mathrm{d}}t+ \sigma {\mathrm{d}}W_t\,,$$ with two-sided Wiener process $(W_t)_{t\in{\mathbb{R}}}$ on a probability space $(\Omega,{\mathcal{F}},{\mathbb{P}})$, was studied. In the deterministic (noise-free) case, $\sigma = 0$, this system has a pitchfork bifurcation of equilibria: if $\alpha<0$ there is one equilibrium ($x=0$) which is globally attractive, and if $\alpha>0$, the trivial equilibrium is repulsive and there are two additional attractive equilibria $\pm \sqrt{\alpha}$. [@Crauel_98_1] establish the following facts in the presence of noise, i.e. when $\sigma>0$:
- For all $\alpha\in{\mathbb{R}}$, there is a unique globally attracting random fixed point ${\{a_\alpha(\omega)\}}_{\omega\in\Omega}$.
- The Lyapunov exponent associated to ${\{a_\alpha(\omega)\}}_{\omega\in\Omega}$ is negative for all $\alpha\in{\mathbb{R}}$.
As a result, [@Crauel_98_1] concludes that the pitchfork bifurcation is destroyed by the additive noise. (This refers to the absence of D-bifurcation, as admits a qualitative change P-bifurcation, see [@Arnold_98_1 p. 473].) However, we are inclined to argue that the pitchfork bifurcation is not destroyed by additive noise, on the basis of the following additional facts concerning the dynamics near the bifurcation point, that we obtain in this paper:
- The attracting random fixed point ${\{a_\alpha(\omega)\}}_{\omega\in\Omega}$ is uniformly attractive only if $\alpha<0$ ([Theorem \[Theorem\_1\]]{}).
- At the bifurcation point there is a change in the practical observability of the Lyapunov exponent: when $\alpha<0$ all finite-time Lyapunov exponents are negative, but when $\alpha>0$ there is a positive probability to observe positive finite-time Lyapunov exponents, irrespectively of the length of time interval under consideration ([Theorem \[Finite-timeBifurcation\]]{}).
- The bifurcation point $\alpha=0$ is characterized by a qualitative change in the dichotomy spectrum associated to ${\{a_\alpha(\omega)\}}_{\omega\in\Omega}$ ([Theorem \[DichotomySpetrumc-Bifurcation\]]{}). In addition, we show that the dichotomy spectrum is directly related to the observability range of the finite-time Lyapunov spectrum ([Theorem \[theo\_1\]]{}).
In light of these findings, we thus argue for the recognition of qualitative properties of the dichotomy spectrum as an additional indicator for bifurcations of random dynamical systems. Spectral studies of random dynamical systems have focused mainly on Lyapunov exponents [@Arnold_98_1; @Cong_97_1], but here we develop an alternative spectral theory based on exponential dichotomies that is related to the Sacker–Sell (or dichotomy) spectrum for nonautonomous differential equations. The original construction due to R.J. Sacker and G.R. Sell [@Sacker_78_2] requires a compact base set (which can be obtained, for instance, from an almost periodic differential equation). Alternative approaches to the dichotomy spectrum [@Aulbach_01_2; @BenArtzi_93_1; @Rasmussen_09_1; @Rasmussen_10_2; @Siegmund_02_4] hold in the general non-compact case, and we use similar techniques for the construction of the dichotomy spectrum by combining them with ergodic properties of the base flow. We note that the relationship between the dichotomy spectrum and Lyapunov spectrum has also been explored in [@Johnson_87_1] in the special case that the base space of a random dynamical system is a compact metric space, but our setup does not require a topological structure of the base.
In analogy to the corresponding bifurcation theory for one-dimensional deterministic dynamical systems, we finally study whether the pitchfork bifurcation with additive noise can be characterized in terms of a breakdown of topologically equivalence. We recall that two random dynamical systems $(\theta,{{\varphi}}_1)$ and $(\theta,{{\varphi}}_2)$ are said to be topologically equivalent if there are families ${\{h_\omega\}}_{\omega\in\Omega}$ of homeomorphisms of the state space such that ${{\varphi}}_2(t, \omega, h_\omega(x)) = h_{\theta_t\omega}({{\varphi}}_1(t, \omega, x))$, almost surely. We establish the following results for the stochastic differential equation :
- Throughout the bifurcation, i.e. for $|\alpha|$ sufficiently small, the resulting dynamics are topologically equivalent ([Theorem \[Theorem2\]]{}).
- There does not exist a uniformly continuous topological conjugacy between the dynamics of cases with positive and negative parameter $\alpha$ ([Theorem \[theo1\]]{}).
These results lead us to propose the association of bifurcations of random dynamical systems with a breakdown of *uniform* topological equivalence, rather than the weaker form of general topological equivalence with no requirement on uniform continuity of the involved conjugacy. Note that uniformity of equivalence transformations plays an important role in the notion of equivalence for nonautonomous linear systems (i.e. in contrast to random systems, the base set of nonautonomous systems is not a probability but a topological space), see [@Palmer_79_1].
This paper is organised as follows. In [Section \[secinvpred\]]{}, invariant projectors and exponential dichotomies are introduced for random dynamical systems. [Section \[lin.sec1\]]{} is devoted to the development of the dichotomy spectrum. In [Section \[sec\_bif\]]{}, we discuss the pitchfork bifurcation with additive noise, reviewing the results of [@Crauel_98_1] and develop our main results in relationship to the dichotomy spectrum. Finally, in [Section \[sec\_1\]]{}, we discuss the existence (and absence) of (uniform) topological equivalence of the dynamics in the neighbourhood of the bifurcation point. Important preliminaries on random dynamical systems are provided in the appendix.
Exponential dichotomies for random dynamical systems {#secinvpred}
====================================================
In this section, we define invariant projectors and exponential dichotomies as tools to describe hyperbolicity and (un)-stable manifolds of linear random dynamical systems.
Let $(\Omega,{\mathcal{F}},{\mathbb{P}})$ be a probability space and $(X,d)$ be a metric space. A *random dynamical system* $(\theta,{{\varphi}})$ (RDS for short) consists of a metric dynamical system $\theta:{\mathbb{T}}\times \Omega \to \Omega$ (which models the noise, see Appendix) and a $({\mathcal{B}}({\mathbb{T}})\otimes{\mathcal{F}}\otimes{\mathcal{B}}(X),{\mathcal{B}}(X))$-measurable mapping ${{\varphi}}:{\mathbb{T}}\times \Omega\times X \to X$ (which models the dynamics of the system) fulfilling
- ${{\varphi}}(0,\omega,x)= x$ for all $\omega\in\Omega$ and $x\in X$,
- ${{\varphi}}(t+s,\omega,x) = {{\varphi}}(t,\theta_s\omega,{{\varphi}}(s,\omega,x))$ for all $t,s\in {\mathbb{T}}$, $\omega\in\Omega$ and $x\in X$.
Note that we frequently use the abbreviation ${{\varphi}}(t,\omega)x$ for ${{\varphi}}(t,\omega,x)$ (even if the random dynamical systems under consideration is nonlinear). We also say that a random dynamical system $(\theta,{{\varphi}})$ is ergodic if $\theta$ is ergodic.
For the spectral theory part of this paper, suppose that the phase space $X$ is given by the Euclidean space ${\mathbb{R}}^d$. A random dynamical system $(\theta, {{\varphi}})$ is called *linear* if for given $\alpha,\beta\in {\mathbb{R}}$, we have $${{\varphi}}(t,\omega)(\alpha x +\beta y)
=
\alpha{{\varphi}}(t,\omega)x+\beta{{\varphi}}(t,\omega)y$$ for all $t\in {\mathbb{T}}$, $\omega\in\Omega$ and $x,y\in{\mathbb{R}}^d$. Given a linear random dynamical system $(\theta,{{\varphi}})$, there exists a corresponding matrix-valued function $\Phi: {\mathbb{T}}\times \Omega\to {\mathbb{R}}^{d\times d}$ with $\Phi(t,\omega)x = {{\varphi}}(t,\omega)x$ for all $t\in{\mathbb{T}}$, $\omega\in\Omega$ and $x\in {\mathbb{R}}^d$.
Given a linear random dynamical system $(\theta, \Phi)$, an invariant random set $M$ (see Appendix) is called *linear random set* if for each $\omega\in {\mathbb{R}}$, the set $M(\omega)$ is a linear subspace of ${\mathbb{R}}^d$. Given linear random sets $M_1, M_2$, $$\omega \mapsto M_1(\omega) \cap M_2(\omega)
{\quad\text{and}\quad}\omega \mapsto M_1(\omega) + M_2(\omega)$$ are also linear random sets, denoted by $M_1\cap M_2$ and $M_1+M_2$, respectively. A finite sum $M_1 + \dots + M_n$ of linear random sets is called *Whitney sum* $M_1 \oplus \dots \oplus M_n$ if $M_1(\omega) \oplus \dots\oplus M_n(\omega) = {\mathbb{R}}^d$ holds for almost all $\omega\in\Omega$.
An *invariant projector* of $(\theta,{{\varphi}})$ is a measurable function $P: \Omega \to {\mathbb{R}}^{d \times d}$ with $$P(\omega)
=
P(\omega)^2
{\quad\text{and}\quad}P(\theta_t\omega)\Phi(t,\omega)
=
\Phi(t,\omega)P(\omega) {\quad \text{for all }\,}t \in{\mathbb{R}}{\text{ and }}\omega \in\Omega\,.$$ The *range* $${\mathcal{R}}(P) := {\big\{(\omega, x) \in \Omega \times {\mathbb{R}}^d: x \in {\mathcal{R}}P(\omega)\big\}}$$ and the *null space* $${\mathcal{N}}(P) := {\big\{(\omega, x ) \in \Omega \times {\mathbb{R}}^d: x \in {\mathcal{N}}P(\omega)\big\}}$$ of an invariant projector $P$ are linear random sets of $(\theta,{{\varphi}})$ such that ${\mathcal{R}}(P) \oplus {\mathcal{N}}(P) = \Omega \times {\mathbb{R}}^d$.
The following proposition says that, provided ergodicity, the dimensions of the range and the null space of an invariant projector are almost surely constant.
\[Lemma1\] Let $P:\Omega\to {\mathbb{R}}^{d\times d}$ be an invariant projector of an ergodic linear random dynamical system $(\theta,{{\varphi}})$. Then
- the mapping $\omega\mapsto {\operatorname{rk}}P(\omega)$ is measurable, and
- ${\operatorname{rk}}P(\omega)$ is almost surely constant.
\(i) We first show that the mapping $A\mapsto \dim A$ on ${\mathbb{R}}^{d\times d}$ is lower semi-continuous. For this purpose, let ${\{A_k\}}_{k\in{\mathbb{N}}}$ be a sequence of matrices in ${\mathbb{R}}^{d\times d}$ which converges to $A\in{\mathbb{R}}^{d\times d}$, and define $r:=\dim A$. Then there exist non-zero vectors $x_1,\dots,x_r$ such that $Ax_1,\dots,Ax_r$ are linearly independent, which implies that $\det [Ax_1,\dots,Ax_r,x_{r+1},\dots,x_d]\not=0$ for some vectors $x_{r+1},\dots, x_d\in{\mathbb{R}}^d$. Since $\lim_{k\to\infty} A_k=A$, one gets $$\lim_{k\to\infty} \det [A_k x_1,\dots,A_k x_r,x_{r+1},\dots,x_d]=\det [Ax_1,\dots,Ax_r,x_{r+1},\dots,x_d]\,.$$ Hence, there exists a $k_0\in{\mathbb{N}}$ such that vectors $A_kx_1,\dots,A_kx_r$ are linearly independent for $k\ge k_0$, and thus, $\dim A_k\geq r$ for all $k\ge k_0$. Consequently, the lower semi-continuity of the mapping $A \mapsto \dim A$ is proved. Therefore, the map ${\mathbb{R}}^{d\times d}\to {\mathbb{N}}, A\mapsto \dim A$ is the limit of a monotonically increasing sequence of continuous functions [@Tong_52_1] and thus is measurable. The proof of this part is complete. (ii) By invariance of $P$, we get that $$P(\theta_t\omega)=\Phi(t,\omega)P(\omega)\Phi(t,\omega)^{-1},$$ which implies that $\dim P(\theta_t\omega)=\dim P(\omega)$. This together with ergodicity of $\theta$ and measurability of the map $\omega\mapsto \dim P(\omega)$ as shown in (i) gives that $\dim P(\omega)$ is almost constant.
According to [Proposition \[Lemma1\]]{}, the rank of an invariant projector $P$ can be defined via $${\operatorname{rk}}P := \dim {\mathcal{R}}(P) := \dim {\mathcal{R}}P(\omega){\quad \text{for almost all }\,}\omega\in \Omega\,,$$ and one sets $$\dim {\mathcal{N}}(P) := \dim {\mathcal{N}}P(\omega) {\quad \text{for almost all }\,}\omega\in \Omega.$$
The following notion of an exponential dichotomy describes uniform exponential splitting of linear random dynamical systems.
\[ED\] Let $(\theta, \Phi)$ be a linear random dynamical system, and let $\gamma\in{\mathbb{R}}$ and $P_\gamma: \Omega \to {\mathbb{R}}^{d \times d}$ be an invariant projector of $(\theta,{{\varphi}})$. Then $(\theta,{{\varphi}})$ is said to admit an *exponential dichotomy* with growth rate $\gamma\in{\mathbb{R}}$, constants $\alpha>0$, $K\ge 1$ and projector $P_\gamma$ if for almost all $\omega\in\Omega$, one has $$\begin{aligned}
{\|\Phi(t, \omega)P_\gamma(\omega)\|}
&\le
K e^{(\gamma-\alpha)t} {\quad \text{for all }\,}t \ge 0\,,\\
{\|\Phi(t,\omega)({\mathds{1}}-P_\gamma(\omega))\|}
&\le
K e^{(\gamma+\alpha)t} {\quad \text{for all }\,}t\le 0\,.
\end{aligned}$$
The following proposition shows that the ranges and null spaces of invariant projectors are given by sums of Oseledets subspaces.
Let $(\theta,\Phi)$ be an ergodic linear random dynamical system which satisfies the integrability condition of Oseledets Multiplicative Ergodic Theorem (see Appendix). Let $\lambda_1>\dots>\lambda_p$ and $O_1(\omega),\dots, O_p(\omega)$ denote the Lyapunov exponents and the associated Oseledets subspaces of $(\theta,\Phi)$, respectively, and suppose that $\Phi$ admits an exponential dichotomy with growth rate $\gamma\in{\mathbb{R}}$ and projector $P$. Then the following statements hold:
- $\gamma\not\in\{\lambda_1,\dots,\lambda_p\}$.
- Define $k:=\max{\big\{i\in\{0,\dots,p\}: \lambda_i>\gamma\big\}}$ with the convention that $\lambda_0=\infty$. Then for almost all $\omega\in\Omega$, one has $${\mathcal{N}}P(\omega)=\bigoplus_{i=1}^{k}O_i(\omega) {\quad\text{and}\quad}{\mathcal{R}}P(\omega)=\bigoplus_{i=k+1}^{p}O_i(\omega)\,.$$
\(i) Suppose to the contrary that $\gamma=\lambda_k$ for some $k\in\{1,\dots,p\}$. Because of the Multiplicative Ergodic Theorem, we have $$\label{eqn1}
\lim_{t\to\infty}\frac{1}{t}\ln\|\Phi(t,\omega)v\|=\lambda_k=\gamma\quad {\quad \text{for all }\,}v\in O_k(\omega)\setminus\{0\}\,.$$ On the other hand, for all $v\in {\mathcal{R}}P_\gamma(\omega)$ we get $\|\Phi(t,\omega)v\|\leq Ke^{(\gamma-\alpha)t}\|v\|$ for all $t\geq 0$. Thus, $$\limsup_{t\to\infty}\frac{1}{t}\ln\|\Phi(t,\omega)v\|\leq \gamma-\alpha {\quad \text{for all }\,}v\in {\mathcal{R}}P(\omega)\,,$$ which together with implies that $O_k(\omega)\cap {\mathcal{R}}P(\omega)=\{0\}$. Similarly, using the fact that $$\lim_{t\to-\infty}\frac{1}{t}\ln\|\Phi(t,\omega)v\|=\lambda_k=\gamma {\quad \text{for all }\,}v\in O_k(\omega)\setminus\{0\}$$ and [Definition \[ED\]]{}, we obtain that $O_k(\omega)\cap {\mathcal{N}}P(\omega)=\{0\}$. Consequently, $O_k(\omega)=\{0\}$ and it leads to a contradiction.
\(ii) Let $v\in {\mathcal{R}}P(\omega)\setminus\{0\}$ be arbitrary. Then, according to Definition \[ED\] and the definition of $k$ we obtain that $$\label{Eq9}
\lim_{t\to\infty}\frac{1}{t}\ln\|\Phi(t,\omega)v\|\leq \gamma-\alpha<\lambda_k.$$ Now we write $v$ in the form $v=v_i+v_{i+1}+\dots+v_p$, where $i \in{\{1,\dots,p\}}$ with $v_i\not=0$ and $v_j\in O_j(\omega)$ for all $j=i,\dots,p$. Using the fact that for $j\in{\{i,\dots,p\}}$ with $v_j\not=0$ $$\lim_{t\to\infty}\frac{1}{t}\ln\|\Phi(t,\omega)v_j\|=\lambda_j\leq \lambda_i,$$ we obtain that $$\lim_{t\to\infty}\frac{1}{t}\ln\|\Phi(t,\omega)v\|=\lambda_i,$$ which together with implies that $i\geq k+1$ and therefore ${\mathcal{R}}P(\omega)\subset\bigoplus_{i=k+1}^{p}O_i(\omega)$. Similarly, we also get that ${\mathcal{N}}P(\omega)\subset \bigoplus_{i=1}^{k}O_i(\omega)$. On the other hand, $${\mathbb{R}}^d={\mathcal{N}}P(\omega)\oplus {\mathcal{R}}P(\omega)=\bigoplus_{i=1}^{k}O_i(\omega)\oplus \bigoplus_{i=k+1}^{p}O_i(\omega).$$ Consequently, we have ${\mathcal{R}}P(\omega)=\bigoplus_{i=k+1}^{p}O_i(\omega)$ and ${\mathcal{N}}P(\omega)\subset \bigoplus_{i=1}^{k}O_i(\omega)$. The proof is complete.
The monotonicity of the exponential function implies the following basic criteria for the existence of exponential dichotomies.
\[lin.lemma4\] Suppose that the linear random system $(\theta,\Phi)$ admits an exponential dichotomy with growth rate $\gamma$ and projector $P_\gamma$. Then the following statements are fulfilled:
- If $P_\gamma \equiv {\mathds{1}}$ almost surely, then $(\theta,\Phi)$ admits an exponential dichotomy with growth rate $\zeta$ and invariant projector $P_\zeta \equiv {\mathds{1}}$ for all $\zeta>\gamma$.
- If $P_\gamma \equiv 0$ almost surely, then $(\theta,\Phi)$ admits an exponential dichotomy with growth rate $\zeta$ and invariant projector $P_\zeta \equiv 0$ for all $\zeta<\gamma$.
Given $\gamma\in {\mathbb{R}}$, a function $g: {\mathbb{R}}\to {\mathbb{R}}^d$ is called *$\gamma^+$-exponentially bounded* if $\sup_{t \in {\mathbb{R}}\cap[0,\infty)} {\|g(t)\|}e^{-\gamma t} < \infty$. Accordingly, one says that a function $g: {\mathbb{R}}\to {\mathbb{R}}^d$ is *$\gamma^-$-exponentially bounded* if $\sup_{t \in {\mathbb{R}}\cap(-\infty,0]}\,{\|g(t)\|}e^{-\gamma t} < \infty$.
We define for all $\gamma \in{\mathbb{R}}$ $${\mathcal{S}}^\gamma := {\big\{(\omega, x) \in \Omega \times {\mathbb{R}}^d: \Phi(\cdot, \omega)x \text{ is } \gamma^+\text{-exponentially bounded}\big\}} \,,$$ and $${\mathcal{U}}^\gamma := {\big\{(\omega, x) \in \Omega \times {\mathbb{R}}^d: \Phi(\cdot, \omega)x \text{ is } \gamma^-\text{-exponentially bounded}\big\}} \,.$$ It is obvious that ${\mathcal{S}}^\gamma$ and ${\mathcal{U}}^\gamma$ are linear invariant random sets of $(\theta,{{\varphi}})$, and given $\gamma \le \zeta$, the relations ${\mathcal{S}}^\gamma \subset {\mathcal{S}}^\zeta$ and ${\mathcal{U}}^\gamma \supset {\mathcal{U}}^\zeta$ are fulfilled.
The relationship between the projectors of exponential dichotomies with growth rate $\gamma$ and the sets ${\mathcal{S}}^\gamma$ and ${\mathcal{U}}^\gamma$ will now be discussed.
\[lin.prop2\] If the linear random dynamical system $(\theta,\Phi)$ admits an exponential dichotomy with growth rate $\gamma$ and projector $P_\gamma$, then ${\mathcal{N}}(P_\gamma) = {\mathcal{U}}^\gamma$ and ${\mathcal{R}}(P_\gamma) = {\mathcal{S}}^\gamma$ almost surely.
Suppose that $(\theta,\Phi)$ admits an exponential dichotomy with growth rate $\gamma$, constants $\alpha$, $K$ and projector $P_\gamma$. This means that for almost all $\omega\in\Omega$, one has $$\begin{aligned}
{\|\Phi(t,\omega)P_\gamma(\omega)\|}
{\!\!\!&\le&\!\!\!}
Ke^{(\gamma-\alpha) t} {\quad \text{for all }\,}t\ge 0\,,\\
{\|\Phi(t,\omega)({\mathds{1}}-P_\gamma(\omega))\|}
{\!\!\!&\le&\!\!\!}
K e^{(\gamma+\alpha)t} {\quad \text{for all }\,}t\le 0\,.
\end{aligned}$$ We now prove the relation ${\mathcal{N}}(P_\gamma) = {\mathcal{U}}^\gamma$.\
($\supseteq$) Choose $(\omega, x) \in {\mathcal{U}}^\gamma$ arbitrarily. This implies ${\|\Phi(t,\omega)x\|} \le Ce^{\gamma t}$ for all $t \le 0$ with some real constant $C>0$. Write $x = x_1+x_2$ with $x_1 \in {\mathcal{R}}P_\gamma(\omega)$ and $x_2 \in {\mathcal{N}}P_\gamma(\omega)$. Hence, for all $t\le 0$, $$\begin{aligned}
{\|x_1\|}
&=
{\|\Phi(-t, \theta_t\omega)\Phi(t,\omega)P_\gamma(\omega)x\|}
=
{\|\Phi(-t, \theta_t\omega)P_\gamma(\theta_t\omega)\Phi(t,\omega)x\|}\\
&\le
K e^{-(\gamma-\alpha)t}{\|\Phi(t,\omega)x\|}
\le
C K e^{-(\gamma-\alpha)t}e^{\gamma t} =C K e^{\alpha t}\,.
\end{aligned}$$ The right hand side of this inequality converges to zero in the limit $t \to -\infty$. This implies $x_1=0$, and thus, $(\omega,x)\in{\mathcal{N}}(P_\gamma)$.\
($\subseteq$) Choose $(\omega, x) \in {\mathcal{N}}(P_\gamma)$. Thus, for all $t \le 0$, the relation ${\|\Phi(t,\omega)x\|} \le K
e^{(\gamma+\alpha)t}{\|x\|}$ is fulfilled. This means that $\Phi(\cdot,\omega)x$ is $\gamma^-$-exponentially bounded.\
The proof of statement concerning the range of the projector is treated analogously.
The dichotomy spectrum for random dynamical systems {#lin.sec1}
===================================================
We introduce the dichotomy spectrum for random dynamical systems in this section. For the definition of the dichotomy spectra, it is crucial for which growth rates, a linear random dynamical system $(\theta,\Phi)$ admits an exponential dichotomy. The growth rates $\gamma = \pm \infty$ are not excluded from our considerations; in particular, one says that $(\theta,\Phi)$ admits an exponential dichotomy with growth rate $\infty$ if there exists a $\gamma \in {\mathbb{R}}$ such that $(\theta,\Phi)$ admits an exponential dichotomy with growth rate $\gamma$ and projector $P_\gamma \equiv {\mathds{1}}$. Accordingly, one says that $(\theta,\Phi)$ admits an exponential dichotomy with growth rate $-\infty$ if there exists a $\gamma \in {\mathbb{R}}$ such that $(\theta,\Phi)$ admits an exponential dichotomy with growth rate $\gamma$ and projector $P_\gamma \equiv 0$.
\[def\_lin\_3\] Consider the linear random dynamical system $(\theta,\Phi)$. Then the *dichotomy spectrum* $(\theta,\Phi)$ is defined by $$\Sigma
:=
{\big\{\gamma \in {\overline{{\mathbb{R}}}}: (\theta,\Phi)\text{ does not admit an exponential dichotomy with growth rate }\gamma\big\}}\,.$$ The corresponding *resolvent sets* is defined by $\rho := {\overline{{\mathbb{R}}}}\setminus \Sigma$.
The aim of the following lemma is to analyze the topological structure of the resolvent sets.
\[lin.lemma2\] Consider the resolvent set $\rho$ of a linear random dynamical system $(\theta,\Phi)$. Then $\rho \cap {\mathbb{R}}$ is open. More precisely, for all $\gamma \in \rho\cap {\mathbb{R}}$, there exists an ${\varepsilon}>0$ such that $B_{\varepsilon}(\gamma) \subset \rho$. Furthermore, the relation ${\operatorname{rk}}P_\zeta = {\operatorname{rk}}P_\gamma$ is (almost surely) fulfilled for all $\zeta \in B_{\varepsilon}(\gamma)$ and every invariant projector $P_\gamma$ and $P_\zeta$ of the exponential dichotomies of $(\theta,\Phi)$ with growth rates $\gamma$ and $\zeta$, respectively.
Choose $\gamma \in \rho$ arbitrarily. Since $(\theta,\Phi)$ admits an exponential dichotomy with growth rate $\gamma$, there exist an invariant projector $P_\gamma$ and constants $\alpha>0$, $K\ge 1$ such that for almost all $\omega\in\Omega$, one has $$\begin{aligned}
{2}
{\|\Phi(t,\omega)P_\gamma(\omega)\|} &\le Ke^{(\gamma-\alpha) t} &{\quad \text{for all }\,}t\ge 0\,,\\
{\|\Phi(t,\omega)({\mathds{1}}-P_\gamma(\omega))\|} &\le K e^{(\gamma+\alpha)t} &{\quad \text{for all }\,}t\le 0 \,.
\end{aligned}$$ Set ${\varepsilon}:= \frac{1}{2}\alpha$, and choose $\zeta \in B_{\varepsilon}(\gamma)$. It follows that for almost all $\omega\in\Omega$, $$\begin{aligned}
{2}
{\|\Phi(t,\omega)P_\gamma(\omega)\|} &\le Ke^{(\zeta-\frac{\alpha}{2})t}&{\quad \text{for all }\,}& t\ge 0 \,,\\
{\|\Phi(t,\omega)({\mathds{1}}-P_\gamma(\omega))\|} &\le K e^{(\zeta+\frac{\alpha}{2})t} &{\quad \text{for all }\,}& t\le 0 \,.
\end{aligned}$$ This yields $\zeta \in \rho$, and it follows that ${\operatorname{rk}}P_\zeta = {\operatorname{rk}}P_\gamma$ for any projector $P_\zeta$ of the exponential dichotomy with growth rate $\zeta$. This finishes the proof of this lemma.
\[lin.lemma3\] Consider the resolvent set $\rho$ of a linear random dynamical system $(\theta,\Phi)$, and let $\gamma_1,\gamma_2 \in \rho \cap {\mathbb{R}}$ such that $\gamma_1<\gamma_2$. Moreover, choose invariant projectors $P_{\gamma_1}$ and $P_{\gamma_2}$ for the corresponding exponential dichotomies with growth rates $\gamma_1$ and $\gamma_2$. Then the relation ${\operatorname{rk}}P_{\gamma_1} \le {\operatorname{rk}}P_{\gamma_2}$ holds. In addition, $[\gamma_1,\gamma_2] \subset \rho$ is fulfilled if and only if ${\operatorname{rk}}P_{\gamma_1} = {\operatorname{rk}}P_{\gamma_2}$.
The relation ${\operatorname{rk}}P_{\gamma_1} \le {\operatorname{rk}}P_{\gamma_2}$ is a direct consequence of [Proposition \[lin.prop2\]]{}, since ${\mathcal{S}}^{\gamma_1} \subset {\mathcal{S}}^{\gamma_2}$ and ${\mathcal{U}}^{\gamma_1} \supset {\mathcal{U}}^{\gamma_2}$. Assume now that $[\gamma_1,\gamma_2] \subset \rho$. Arguing contrapositively, suppose that ${\operatorname{rk}}P_{\gamma_1} \not= {\operatorname{rk}}P_{\gamma_2}$, and choose invariant projectors $P_\gamma$, $\gamma \in (\gamma_1,\gamma_2)$, for the exponential dichotomies of $(\theta,\Phi)$ with growth rate $\gamma$. Define $$\zeta_0
:=
\sup\,{\big\{\zeta \in [\gamma_1,\gamma_2]: {\operatorname{rk}}P_\zeta \not= {\operatorname{rk}}P_{\gamma_2}\big\}}\,.$$ Due to [Lemma \[lin.lemma2\]]{}, there exists an ${\varepsilon}>0$ such that ${\operatorname{rk}}P_{\zeta_0} = {\operatorname{rk}}P_\zeta$ for all $\zeta \in B_{\varepsilon}(\zeta_0)$. This is a contradiction to the definition of $\zeta_0$. Conversely, let ${\operatorname{rk}}P_{\gamma_1} = {\operatorname{rk}}P_{\gamma_2}$. Because of ${\operatorname{rk}}P_{\gamma_1} = {\operatorname{rk}}P_{\gamma_2}$, [Proposition \[lin.prop2\]]{} yields ${\mathcal{N}}(P_{\gamma_1}) = {\mathcal{N}}(P_{\gamma_2})$ almost surely, and $P_{\gamma_2}$ is an invariant projector of the exponential dichotomy with growth rate $\gamma_1$. Thus, one obtains for almost all $\omega\in\Omega$, $${\|\Phi(t,\omega) P_{\gamma_2}(\omega)\|} \le K_1 e^{(\gamma_1 - \alpha_1)t} {\quad \text{for all }\,}t\ge 0$$ for some $K_1\ge 1$ and $\alpha_1 >0$. $P_{\gamma_2}$ is also projector of the exponential dichotomy on ${\mathbb{R}}_0^-$ with growth rate $\gamma_2$. Hence, for almost all $\omega\in\Omega$, one gets $${\big\|\Phi(t,\omega) ({\mathds{1}}-P_{\gamma_2}(\omega))\big\|} \le K_2 e^{(\gamma_2 + \alpha_2)t} {\quad \text{for all }\,}t\le0$$ with some $K_2\ge 1$ and $\alpha_2 >0$. For all $\gamma \in [\gamma_1, \gamma_2]$, these two inequalities imply by setting $K:= \max\,{\{K_1, K_2\}}$ and $\alpha:= \min \,{\{\alpha_1, \alpha_2\}}$ that for almost all $\omega\in\Omega$, $$\begin{aligned}
{2}
{\|\Phi(t,\omega) P_{\gamma_2}(\omega)\|} &\le K e^{(\gamma - \alpha)t} &{\quad \text{for all }\,}& t\ge0\,,\\
{\|\Phi(t,\omega)({\mathds{1}}-P_{\gamma_2}(\omega))\|} &\le K e^{(\gamma + \alpha)t} &{\quad \text{for all }\,}& t\le0\,.
\end{aligned}$$ This means that $\gamma \in \rho$, and thus, $[\gamma_1,\gamma_2] \subset \rho$.
For an arbitrarily chosen $a \in {\mathbb{R}}$, define $$[-\infty, a]:=(-\infty,a] \cup {\{-\infty\}}\,,\quad\quad\quad [a, \infty] := [a, \infty) \cup {\{\infty\}}$$ and $$[-\infty,-\infty]:= {\{-\infty\}}, \quad\quad\ [\infty,\infty]:= {\{\infty\}}, \quad\quad \quad [-\infty,\infty] := {\overline{{\mathbb{R}}}}\,.$$
The following *Spectral Theorem*, describes that the dichotomy spectrum consists of at least one and at most $d$ closed intervals.
\[lin.theo1\] Let $(\theta, \Phi)$ be a linear random dynamical system with dichotomy spectrum $\Sigma$. Then there exists an $n \in {\{1,\dots,d\}}$ such that $$\Sigma = [a_1, b_1] \cup \dots \cup [a_n,b_n]$$ with $-\infty \le a_1 \le b_1< a_2 \le b_2 <\dots<a_n\le b_n\le \infty$.
Due to [Lemma \[lin.lemma2\]]{}, the resolvent set $\rho \cap {\mathbb{R}}$ is open. Thus, $\Sigma \cap {\mathbb{R}}$ is the disjoint union of closed intervals. The relation $(-\infty, b_1] \subset \Sigma$ implies $[-\infty, b_1] \subset \Sigma$, because the assumption of the existence of a $\gamma \in {\mathbb{R}}$ such that $(\theta,{{\varphi}})$ admits an exponential dichotomy with growth rate $\gamma$ and projector $P_\gamma \equiv 0$ leads to $(-\infty,\gamma] \subset \rho$ using [Lemma \[lin.lemma4\]]{}, and this is a contradiction. Analogously, it follows from $[a_n, \infty) \subset \Sigma$ that $[a_n, \infty] \subset \Sigma$. To show the relation $n \le d$, assume to the contrary that $n \ge d+1$. Thus, there exist $$\zeta_1<\zeta_2<\dots<\zeta_d \in \rho$$ such that the $d+1$ intervals $(-\infty, \zeta_1)\,,\, (\zeta_1, \zeta_2)\,,\, \dots,\, (\zeta_d, \infty)$ have nonempty intersection with the spectrum $\Sigma$. It follows from Lemma \[lin.lemma3\] that $$0 \le {\operatorname{rk}}P_{\zeta_1} < {\operatorname{rk}}P_{\zeta_2} < \dots < {\operatorname{rk}}P_{\zeta_d} \le d$$ is fulfilled for invariant projectors $P_{\zeta_i}$ of the exponential dichotomy with growth rate $\zeta_i$, $i\in {\{1,\dots,n\}}$. This implies either ${\operatorname{rk}}P_{\zeta_1} =0$ or ${\operatorname{rk}}P_{\zeta_d} =d$. Thus, either $$[-\infty, \zeta_1] \cap \Sigma = \emptyset {\quad\text{or}\quad}[\zeta_d, \infty] \cap \Sigma = \emptyset$$ is fulfilled, and this is a contradiction. To show $n\ge 1$, assume that $\Sigma= \emptyset$. This implies ${\{-\infty, \infty\}} \subset \rho$. Thus, there exist $\zeta_1, \zeta_2 \in {\mathbb{R}}$ such that $(\theta,{{\varphi}})$ admits an exponential dichotomy with growth rate $\zeta_1$ and projector $P_{\zeta_1} \equiv 0$ and an exponential dichotomy with growth rate $\zeta_2$ and projector $P_{\zeta_2} \equiv {\mathds{1}}$. Applying [Lemma \[lin.lemma3\]]{}, one gets $(\zeta_1, \zeta_2) \cap \Sigma \not= \emptyset$. This contradiction yields $n\ge 1$ and finishes the proof of the theorem.
Each spectral interval is associated to a so-called spectral manifold, which generalises the stable and unstable manifolds obtained by the ranges and null spaces of invariant projectors of exponential dichotomies.
\[lin.theo6\] Consider the dichotomy spectrum $$\Sigma = [a_1, b_1] \cup\dots\cup [a_n,b_n]$$ of the linear random dynamical system $(\theta, \Phi)$ and define the invariant projectors $P_{\gamma_0} := 0$, $P_{\gamma_n}:= {\mathds{1}}$, and for $i\in{\{1,\dots,n-1\}}$, choose $\gamma_i \in (b_i,a_{i+1})$ and projectors $P_{\gamma_i}$ of the nonhyperbolic exponential dichotomy of $(\theta,{{\varphi}})$ with growth rate $\gamma_i$. Then the sets $${\mathcal{W}}_i := {\mathcal{R}}(P_{\gamma_i}) \cap {\mathcal{N}}(P_{\gamma_{i-1}}) {\quad \text{for all }\,}i \in {\{1,\dots,n\}}$$ are fiber-wise linear subset of ${\mathbb{R}}^d$, the so-called *spectral manifolds*, such that $${\mathcal{W}}_1 \oplus \dots \oplus {\mathcal{W}}_n = \Omega \times {\mathbb{R}}^d$$ and ${\mathcal{W}}_i \not= \Omega \times {\{0\}}$ for $i \in {\{1,\dots,n\}}$.
The sets ${\mathcal{W}}_1,\dots,{\mathcal{W}}_n$ obviously have linear fibers. Suppose that there exists an $i \in {\{1,\dots,n\}}$ with ${\mathcal{W}}_i = {\mathbb{R}}\times {\{0\}}$. In case $i=1$ or $i=n$, [Lemma \[lin.lemma4\]]{} implies $[-\infty, \gamma_1] \cap \Sigma = \emptyset$ or $[\gamma_{n-1},\infty] \cap \Sigma = \emptyset$, and this is a contradiction. In case $1<i<n$, due to [Lemma \[lin.lemma3\]]{}, one obtains $$\dim {\mathcal{W}}_i
=
\dim {\big({\mathcal{R}}(P_{\gamma_i}) \cap {\mathcal{N}}(P_{\gamma_{i-1}})\big)}
=
{\operatorname{rk}}P_{\gamma_i} + d - {\operatorname{rk}}P_{\gamma_{i-1}} - \dim {\big({\mathcal{R}}(P_{\gamma_i}) +{\mathcal{N}}(P_{\gamma_{i-1}})\big)} \ge 1\,,$$ and this is also a contradiction. Now the relation ${\mathcal{W}}_1 \oplus \dots \oplus {\mathcal{W}}_n = \Omega \times {\mathbb{R}}^d$ will be proved. For $1 \le i < j \le n$, due to \[lin.prop2\], the relations ${\mathcal{W}}_i \subset {\mathcal{R}}(P_{\gamma_i})$ and ${\mathcal{W}}_j \subset {\mathcal{N}}(P_{\gamma_{j-1}}) \subset {\mathcal{N}}(P_{\gamma_i})$ are fulfilled. This yields $${\mathcal{W}}_i \cap {\mathcal{W}}_j \subset {\mathcal{R}}(P_{\gamma_i}) \cap {\mathcal{N}}(P_{\gamma_i}) = {\mathbb{R}}\times {\{0\}}\,,$$ and one obtains $$\begin{aligned}
\Omega\times {\mathbb{R}}^d
{\!\!\!&=&\!\!\!}
{\mathcal{W}}_1 + {\mathcal{N}}(P_{\gamma_1}) = {\mathcal{W}}_1 + {\mathcal{N}}(P_{\gamma_1}) \cap {\big({\mathcal{R}}(P_{\gamma_2}) + {\mathcal{N}}(P_{\gamma_2})\big)}\\
{\!\!\!&=&\!\!\!}
{\mathcal{W}}_1 + {\mathcal{N}}(P_{\gamma_1}) \cap {\mathcal{R}}(P_{\gamma_2}) + {\mathcal{N}}(P_{\gamma_2}) = {\mathcal{W}}_1 + {\mathcal{W}}_2 + {\mathcal{N}}(P_{\gamma_2})\,.
\end{aligned}$$ Here, the fact that linear subspaces $E,F,G\subset {\mathbb{R}}^d$ with $E\supset G$ fulfill $E\cap(F+G)= (E\cap F) + G$ was used. It follows inductively that $$\Omega\times{\mathbb{R}}^d = {\mathcal{W}}_1 + \dots+ {\mathcal{W}}_n + {\mathcal{N}}(P_{\gamma_n}) = {\mathcal{W}}_1 + \dots+ {\mathcal{W}}_n\,.$$ This finishes the proof of this theorem.
If the linear random dynamical system $(\theta,\Phi)$ under consideration fulfills the conditions of the Multiplicative Ergodic Theorem, then [Proposition \[Lemma1\]]{} implies that the spectral manifolds ${\mathcal{W}}_i$ of the above theorem are given by Whitney sums of Oseledets subspaces.
The remaining part of this section on the dichotomy spectrum will be devoted to study boundedness properties of the spectrum. Firstly, a criterion for boundedness from above and below is provided by the following proposition.
Consider a linear random dynamical system $(\theta, \Phi)$, let $\Sigma$ denote the dichotomy spectrum of $(\theta, \Phi)$, and define $$\alpha^{\pm}(\omega)
:=
\left\{
\begin{array}{c@{\;:\;}l}
\ln^+{\big({\|\Phi(1,\omega)^{\pm 1}\|}\big)} & {\mathbb{T}}={\mathbb{Z}}\,,\\
\ln^+{\big(\sup_{t\in[0, 1]}{\|\Phi(t,\omega)^{\pm 1}\|}\big)} & {\mathbb{T}}={\mathbb{R}}\,.
\end{array}
\right.$$ Then $\Sigma$ is bounded from above if and only if $$\operatorname*{ess\,sup}_{\omega\in\Omega}\alpha^+(\omega)<\infty \,,$$ and $\Sigma$ is bounded from below if and only if $$\operatorname*{ess\,sup}_{\omega\in\Omega}\alpha^-(\omega)<\infty\,.$$ Consequently, if the dichotomy spectrum $\Sigma$ is bounded, then $\Phi$ satisfies the integrability condition of the Multiplicative Ergodic Theorem.
Suppose that $\Sigma$ is bounded from above. Then there exist $K>0$ and $\alpha\in{\mathbb{R}}$ such that $$\|\Phi(t,\omega)\|\leq K e^{\alpha t} {\quad \text{for almost all }\,}\omega\in\Omega\,,$$ which implies that $\operatorname*{ess\,sup}_{\omega\in\Omega}\alpha^+(\omega)\leq K e^{|\alpha|}$. On the other hand, suppose that $\operatorname*{ess\,sup}_{\omega\in\Omega}\alpha(\omega)<\infty$. Then there exists a measurable set $U$ of probability $1$ such that for all $\omega\in\Omega$ we have $\alpha^+(\omega)\leq e^{\alpha}$ for some positive number $\alpha$. Define $$\widetilde \Omega:=\bigcap_{n\in{\mathbb{Z}}}\theta^n U\,.$$ Due to the measure preserving property of $\theta$, we get that ${\mathbb{P}}{\big(\widetilde \Omega\big)}=1$. Let $\gamma>\alpha$ be arbitrary. Then for all $\omega\in\widetilde \Omega$, we have $$\|\Phi(t,\omega)\|\leq e^{\alpha t+\alpha} {\quad \text{for all }\,}t > 0 \,,$$ which implies that $\gamma\in {\mathbb{R}}\setminus \Sigma$. Hence, $\Sigma\subset (-\infty,\alpha]$. Similarly, we get that $\Sigma$ is bounded from below if and only if $\operatorname*{ess\,sup}_{\omega\in\Omega}\alpha^-(\omega)<\infty$. This finishes the proof of this proposition.
The following example shows that there exist linear random dynamical systems which satisfy the integrability condition of the Multiplicative Ergodic Theorem, but which have no bounded dichotomy spectrum.
\[Example1\] Let $(\Omega,\mathcal F,\mathbb P)$ be a probability space and $\theta:{\mathbb{R}}\times\Omega\to \Omega$ be a metric dynamical system which is ergodic and non-atomic. Then there exists, by using [@Halmos_60_1 Lemma 2, p. 71], a measurable set $U$ of the form $$\label{Eq10}
U=\bigcup_{k=1}^\infty\bigcup_{j=0}^{k}\theta_j U_k,$$ where $U_i$, $i\in{\mathbb{N}}$, are measurable sets such that
- for all $k,\ell\in{\mathbb{N}}$, $i\in{\{0,\dots, k\}}$ and $j\in{\{0,\dots,\ell\}}$, we have $$\theta_j U_k\cap \theta_i U_{\ell}=\emptyset \quad \mbox{whenever }\, k\not=\ell {\text{ or }}i\not=j\,,$$
- $0<\mathbb P(U_k)\leq \frac{1}{k^3}$ for all $k\in{\mathbb{N}}$.
We now define a random variable $a:\Omega\to {\mathbb{R}}$ by $$a(\omega):=
\left\{
\begin{array}{c@{\;:\;}l}
1 & \omega\in \Omega\setminus U\,, \\
k & k \mbox{ is even and } \omega\in\theta_j U_k\,, \\
\frac{1}{k} & k \mbox{ is odd and } \omega\in\theta_j U_k\,.
\end{array}
\right.$$ Using the random variable $a$, we define a discrete-time scalar linear random dynamical system $\Phi:{\mathbb{Z}}\times \Omega \to {\mathbb{R}}$ by $$\Phi(t,\omega)=\left\{
\begin{array}{c@{\;:\;}l}
a(\theta_{t-1}\omega) \cdots a(\omega) & t\geq 1\,, \\
1 & t=0\,, \\
a(\theta_{-1}\omega)^{-1}\cdots a(\theta_{t}\omega)^{-1} & t\leq -1\,.
\end{array}
\right.$$ A direct computation yields that $$\mathbb E \ln^+(\|\Phi(1,\omega)\|)
=
\sum_{k=1}^\infty (2k+1){\mathbb{P}}(U_{2k})\ln (2k)
\leq
\sum_{k=1}^\infty (2k+1)\frac{\ln (2k)}{8k^3}<\infty\,,$$ and $$\begin{aligned}
\mathbb E \ln^+(\|\Phi(1,\omega)^{-1}\|)
&=
\sum_{k=0}^\infty (2k+2){\mathbb{P}}(U_{2k+1})\ln (2k+1)\\
&\leq
\sum_{k=1}^\infty (2k+2)\frac{\ln (2k+1)}{(2k+1)^3}<\infty\,.
\end{aligned}$$ Then the linear system $\Phi$ satisfies the integrability condition of the Multiplicative Ergodic Theorem. The fact that the dichotomy spectrum of $\Phi$ is unbounded from above follows from $$\|\Phi(n,\omega)\|\geq k^{n}{\quad \text{for all }\,}\omega\in U_k {\text{ with }}k \mbox{ even and } 0\le n\le k\,.$$ Similarly, one can prove that the spectrum is unbounded from below.
Random pitchfork bifurcation {#sec_bif}
============================
We first review in [Subsection \[subsec1\]]{} the main results of [@Crauel_98_1], which concern the one-dimensional stochastic differential equation $$\label{Sde}
{\mathrm{d}}x= {\big(\alpha x-x^3\big)}{\mathrm{d}}t+ \sigma {\mathrm{d}}W_t\,,$$ depending on real parameters $\alpha$ and $\sigma$ and driven by a two-sided Wiener process $(W_t)_{t\in{\mathbb{R}}}$. This stochastic differential equation has a unique random fixed point ${\{a_\alpha(\omega)\}}_{\omega\in\Omega}$ for all $\alpha\in{\mathbb{R}}$. We then show in [Subsection \[subsec2\]]{} that there is a qualitative change in the random dynamics at the bifurcation point $\alpha=0$ in the sense that after the bifurcation, the attracting random fixed points ${\{a_\alpha(\omega)\}}_{\omega\in\Omega}$ have qualitatively different properties for $\alpha<0$ and $\alpha\ge 0$ with respect to uniform attraction, which is lost at the bifurcation point. We also associate this bifurcation in [Subsection \[subsec3\]]{} with non-hyperbolicity of the spectrum of the linearization at the bifurcation point.
Existence of a unique random attracting fixed point {#subsec1}
---------------------------------------------------
Consider the stochastic differential equation . We first look at the deterministic case $\sigma = 0$. Then for $\alpha<0$, the ordinary differential equation has one equilibrium ($x=0$) which is globally attractive. For positive $\alpha$, the trivial equilibrium becomes repulsive, and there are two additional equilibria, given by $\pm \sqrt{\alpha}$, which are attractive. This also means that the global attractor $K_\alpha$ of the deterministic equation undergoes a bifurcation from a trivial to a nontrivial object. It is given by $$K_\alpha:=
\left\{
\begin{array}{c@{\;:\;}l}
{\{0\}} & \alpha \le 0\,,\\ {\big[-\sqrt\alpha,\sqrt\alpha\big]} & \alpha> 0\,.
\end{array}
\right.$$
It was shown in [@Crauel_98_1] that such an attractor bifurcation does not persist for random attractors of the randomly perturbed system where $\sigma >0$, and we will explain the details now.
Firstly, the stochastic differential equation generates a random dynamical system $(\theta:{\mathbb{R}}\times \Omega\to\Omega,{{\varphi}}:{\mathbb{R}}\times \Omega \times {\mathbb{R}}\to {\mathbb{R}})$ which induces a Markov semigroup with transition probabilities $T(x, B)$ for $x\in {\mathbb{R}}$ and $B\in{\mathcal{B}}({\mathbb{R}})$. A probability measure $\rho$ on ${\mathcal{B}}(X)$ is called a *stationary measure* for the Markov semigroup if $$\rho(B) = \int_{\mathbb{R}}T(x,B) \,{\mathrm{d}}\rho(x) {\quad \text{for all }\,}B\in{\mathcal{B}}({\mathbb{R}})\,.$$ It can be shown [@Arnold_98_1 p. 474] that for any $\alpha,\sigma\in{\mathbb{R}}$, the Markov semigroup associated with admits a unique stationary measure $\rho_{\alpha,\sigma}$ with density $$\label{Eq3}
p_{\alpha,\sigma}(x)=N_{\alpha,\sigma}\exp{\big(\tfrac{1}{\sigma^2}(\alpha x^2-\tfrac{1}{2}{x^4})\big)}\,,$$ where $N_{\alpha,\sigma}$ is a normalization constant. This stationary measure corresponds to an invariant measure $\mu$ of the random dynamical system $(\theta,{{\varphi}})$ generated by . $\mu$ has the disintegration given by $$\mu_\omega=\lim_{t\to\infty}{{\varphi}}(t,\theta_{-t}\omega)\rho {\quad \text{for almost all }\,}\omega\in\Omega\,.$$ It was shown in [@Crauel_98_1] that $\mu_\omega$ is a Dirac measure concentrated on $a_\alpha(\omega)$, and linearizing along this invariant measure $\mu$ yields a negative Lyapunov exponent, given by $$\lambda_\alpha=-\frac{2}{\sigma^2}\int_{\mathbb{R}}(\alpha x-x^3)^2p_{\alpha,\sigma}(x)\,{\mathrm{d}}x\,.$$ Moreover, the family ${\{a_\alpha(\omega)\}}_{\omega\in\Omega}$ is the global random attractor (see Appendix), which implies that the attractor bifurcation associated with a deterministic pitchfork bifurcation (that is, $K_\alpha$ bifurcates from a non-trivial object to a singleton) is destroyed by noise.
Qualitative changes in uniform attractivity {#subsec2}
-------------------------------------------
In order to establish qualitative changes in the attractivity of the unique random attracting fixed point ${\{a_\alpha(\omega)\}}_{\omega\in\Omega}$, a detailed understanding about the location of this attractor is needed. For this purpose, we use similar techniques as developed in [@Tearne_05_1; @Tearne_08_1].
\[Prp1\] Consider for $\alpha\in{\mathbb{R}}$, and let ${\{a_\alpha(\omega)\}}_{\omega\in\Omega}$ be its unique random fixed point. Then for any ${\varepsilon}>0$ and $T\geq 0$, there exists a measurable set ${\mathcal{A}}\in {\mathcal{F}}_{-\infty}^{T}$ (see Appendix) of positive measure such that $$a_\alpha(\theta_s\omega)\in (-{\varepsilon},{\varepsilon}){\quad \text{for all }\,}s\in [0,T] {\text{ and }}\omega\in{\mathcal{A}}\,.$$
We first consider the case $\alpha\le 0$. According to [@Tearne_05_1 Theorem 12], there exists ${\mathcal{A}}\in {\mathcal{F}}_{-\infty}^{0}$ of positive measure such that $a(\omega)\in (-{\varepsilon}/3,{\varepsilon}/3)$ for all $\omega\in {\mathcal{A}}$. Define $${\mathcal{A}}^+:= {\Big\{\omega\in\Omega: \sup_{t\in[0,T]} |\omega(t)|\le \delta:= \frac{{\varepsilon}}{2 e^{-\alpha T}}\Big\}}\,.$$ Then $$|{{\varphi}}(t,\omega,a(\omega)) - \phi(t,a(\omega))|\le \delta -\alpha \int_0^t |{{\varphi}}(s,\omega,a(\omega)) - \phi(s,a(\omega))|\,{\mathrm{d}}s\,,$$ where $\phi(\cdot,x_0)$ denotes the solution of the initial value problem $$\dot x_t=\alpha x_t-x_t^3, \quad x(0)=x_0\,.$$ Thus, $$|{{\varphi}}(t,\omega,a(\omega))| \le |a(\omega)| + \delta e^{-\alpha t} < {\varepsilon}{\quad \text{for all }\,}t\in [0,T] {\text{ and }}\omega \in {\mathcal{A}}\cap {\mathcal{A}}^+\,.$$
This implies the assertion for $\alpha\le 0$. It remains to show the proposition for $\alpha>0$; the proof of this fact is divided in the following four steps.
*Step 1.* We will construct an absorbing set for the random dynamical system $(\theta, {{\varphi}})$. For this purpose, let $B_{\rho}(0)$ for some $\rho>0$ be a ball in ${\mathbb{R}}$, for which we will consider the pullback limit ${{\varphi}}(t,\theta_{-t}\omega)B_{\rho}(0)$. Consider the Langevin equation $$\label{Langevin}
{\mathrm{d}}z =-\alpha z \,{\mathrm{d}}t+\sigma {\mathrm{d}}W(t)\,,$$ and let $\psi:{\mathbb{R}}\times \Omega \times {\mathbb{R}}\to{\mathbb{R}}$ denote the associated random dynamical system, given by $$\psi(t,\omega)z_0=e^{-\alpha t} z_0+\sigma\int_0^t e^{-\alpha(t-s)}\,{\mathrm{d}}W(s)\,.$$ It follows that $$\psi(t,\theta_{-t}\omega)z_0=e^{-\alpha t} z_0+\sigma\int_{-t}^0 e^{\alpha s}\,{\mathrm{d}}W(s)\,,$$ which implies that $$z(\omega):=\lim_{t\to\infty} \psi(t,\theta_{-t}\omega)z_0= \sigma\int_{-\infty}^0 e^{\alpha s}\,{\mathrm{d}}W(s)$$ is the unique random fixed point of . Using the exponential martingale inequality, for almost all $\omega\in\Omega$, there are positive constants $A(\omega),B(\omega)$ such that $$\label{IteratedLogaLaw}
(\psi(t,\omega)z(\omega))^2\leq A(\omega)+ B(\omega)\ln (1+|t|) {\quad \text{for all }\,}t\in{\mathbb{R}}\,.$$ Fix $\tau\ge0$ and $\omega_0\in\Omega$, and define $v(t):={{\varphi}}(t,\theta_{-\tau}\omega_0)x_0- \psi(t,\theta_{-\tau}\omega_0) z(\theta_{-\tau}\omega_0)$ for all $t\in{\mathbb{R}}$, where $x_0\in B_\rho(0)$. Using the integral form of , we have $$\begin{aligned}
v(t)&={{\varphi}}(t,\theta_{-\tau}\omega_0)x_0-\psi(t,\theta_{-\tau}\omega_0)z(\theta_{-\tau}\omega_0)\\
&=\int_0^t {\big(\alpha {{\varphi}}(s,\theta_{-\tau}\omega_0)x_0-({{\varphi}}(s,\theta_{-\tau}\omega_0)x_0)^3\big)}\,{\mathrm{d}}s
+
\int_0^t \alpha \psi(s,\theta_{-\tau}\omega_0)z(\theta_{-\tau}\omega_0)\,{\mathrm{d}}s\,,
\end{aligned}$$ which yields that $$\label{Eq13}
\dot v(t)+\alpha v(t)= 2\alpha(v(t)+\psi(t,\theta_{-\tau}\omega_0)z(\theta_{-\tau}\omega_0))-(v(t)+\psi(t,\theta_{-\tau}\omega_0)z(\theta_{-\tau}\omega_0))^3\,.$$ Note that using Cauchy’s Inequality, we obtain that for all $v,z\in{\mathbb{R}}$ $$\begin{aligned}
3v^2z^2+\frac{\alpha^2}{3}
&\geq
2\alpha vz\,,\\
\frac{v^4}{2}+2\alpha^2
&\geq
2\alpha v^2\,,\\
\frac{v^4}{12}+ \frac{v^4}{12}+ \frac{v^4}{12}+\frac{3^{7}}{4}z^4
&\geq
3v^3z\,,\\
\frac{v^4}{4}+\frac{3\sqrt[3] 3}{4}z^4+\frac{3\sqrt[3] 3}{4}z^4+\frac{3\sqrt[3] 3}{4}z^4
&\geq
3vz^3\,.
\end{aligned}$$ Therefore, $$\left( 2\alpha(v+z)-(v+z)^3\right)v
\leq
C (1+z^2+z^4) {\quad \text{for all }\,}v,z\in{\mathbb{R}}\,,$$ where $C:=\max{\Big\{\frac{7}{3}\alpha^2,\frac{3^7}{4}+\frac{3\sqrt[3] 3}{4}\Big\}}$. Thus, from we derive that $$\begin{aligned}
v(t) \dot v(t) +\alpha v(t)^2
&\leq C (1+(\psi(t,\theta_{-\tau}\omega_0)z(\theta_{-\tau}\omega_0))^2+(\psi(t,\theta_{-\tau}\omega_0)z(\theta_{-\tau}\omega_0))^4)\,,\\[1ex]
&= C(1+z(\theta_{t-\tau}\omega_0)^2+z(\theta_{t-\tau}\omega_0)^4)
\end{aligned}$$ which implies that $$\begin{aligned}
v(\tau)^2
&\leq
e^{-2\alpha \tau} v(0)^2+2C\int_0^\tau e^{2\alpha (s-\tau)} \left(1+ z(\theta_{s-\tau}\omega_0)^2+z(\theta_{s-\tau}\omega_0)^4\right)\,{\mathrm{d}}s\\[1ex]
&\leq
e^{-2\alpha \tau} v(0)^2+2C\int_{-\infty}^0 e^{2\alpha s} \left(1+z(\theta_{s}\omega_0)^2+z(\theta_{s}\omega_0)^4\right)\,{\mathrm{d}}s\,,
\end{aligned}$$ where the existence of infinity integral follows from . Consequently, $$\begin{aligned}
({{\varphi}}(\tau,\theta_{-\tau}\omega_0)x_0)^2
&\leq
2 v(\tau)^2+2z(\omega_0)^2\\[1ex]
&\leq
2e^{-2\alpha \tau}v(0)^2+R(\omega_0)+2z(\omega_0)^2\\[1ex]
&\leq
4e^{-2\alpha \tau}(x_0^2+z(\theta_{-\tau}\omega_0)^2)+R(\omega_0)+2z(\omega_0)^2 ,
\end{aligned}$$ where $$R(\omega_0):=4C\int_{-\infty}^0 e^{2\alpha s}(1+z(\theta_s\omega_0)^2+z(\theta_s\omega_0)^4)\,{\mathrm{d}}s\,.$$ Since $|x_0|<\rho$ and $\limsup_{\tau\to\infty} e^{-2\alpha \tau} |z(\theta_{-\tau}\omega_0)|=0$ it follows that $B_{R(\omega_0)+2 z(\omega_0)+1}(0)$ is an absorbing set of . Thus, $a_\alpha(\omega)\in B_{R(\omega)+2z(\omega)+1}(0)$ for almost all $\omega\in\Omega$.
*Step 2.* In this step, we construct a measurable set $A_1\subset {\mathcal{F}}_{-\infty}^0$ of a positive probability such that $$a_\alpha(\omega)\in B_{1}(K) {\quad \text{for all }\,}\omega\in A_1\,.$$ Define $$A^-:=\{\omega: R(\omega)+2z(\omega)\leq {\mathbb{E}}[R+2z]\}.$$ Clearly, ${\mathbb{P}}(A^-)>0$ and we refer to [@Mao_97_1] for the existence of ${\mathbb{E}}[R+2z]$. Recall that $K$ denotes the global attractor for the deterministic case $\sigma = 0$. Then there exists $T_1>0$ such that $$\label{Eq11}
\phi{\big(t,B_{{\mathbb{E}}[R+2z] +1}(0)\big)}\subset B_{1/3}(K) {\quad \text{for all }\,}t\geq T_1\,,$$ where $\phi(\cdot,x_0)$ denotes the solution of the initial value problem $$\dot x_t=\alpha x_t-x_t^3, \quad x(0)=x_0\,.$$ Let $\delta_1>0$ be a positive constant satisfying that $$\label{Eq12}
\delta_1\leq \frac{1}{9\sigma e^{\alpha T_1}}.$$ Now, define $A^+:=\big\{\omega: \sup_{t\in [0,T_1]}|\omega(t)|\leq \delta_1\big\}$. From [@Ikeda_81_1 Section 6.8], the set $A^+$ has positive measure. Clearly, $A^-$ and $A^+$ are independent and therefore the set $A^-\cap A^+\in {\mathcal{F}}_{-\infty}^{T_1}$ is also of positive probability measure. Choose and fix an arbitrary $\omega\in A^-\cap A^+$. By the definition of $A^-$, we get that $a_\alpha(\omega)\in B_{{\mathbb{E}}[R+2z]+1}(0)$. Since $a_\alpha(\omega)$ is a random fixed point of ${{\varphi}}$, it follows that $$a_\alpha(\theta_t\omega)=a_\alpha(\omega)+\int_0^t {\big(\alpha a_\alpha(\theta_s\omega)-a_\alpha(\theta_s\omega)^3\big)}\,{\mathrm{d}}s+\sigma \omega(t)\,.$$ Define $u(t):=a_\alpha(\theta_t\omega)-\phi(t,a_\alpha(\omega))$. According to the definition of $\phi(t,\cdot)$, we obtain that $$u(t)=\int_0^t \alpha u(s)-u(s)\left(a_\alpha(\theta_s\omega)^2+a_\alpha(\theta_s\omega)\phi(s,a_\alpha(\omega))+\phi(s,a_\alpha(\omega))^2\right)\,{\mathrm{d}}s+\sigma w(t),$$ which together with the fact that $|w(t)|\leq \delta_1$ for all $t\in [0,T_1]$ implies that $$|u(t)|\leq \sigma\delta_1+\int_0^t\alpha |u(s)|\,{\mathrm{d}}s{\quad \text{for all }\,}t\in [0,T_1]\,.$$ Using Gronwall’s inequality, we get that $$|u(t)|\leq \sigma\delta_1 e^{\alpha t}\leq \frac{1}{3}{\quad \text{for all }\,}t\in [0,T_1]\,.$$ Therefore, by we get that $a_\alpha(\theta_{T_1}\omega)\in B_{1}(K)$. Consequently, the set $A_1:=\theta_{T_1}(A^-\cap A^+)$ satisfies the desired assertion in this step.
*Step 3.* In this step, we construct a measurable set $A_2\subset {\mathcal{F}}_{-\infty}^0$ of a positive probability such that $$a_\alpha(\omega)\in (-\delta_2,\delta_2){\quad \text{for all }\,}\omega\in A_2\,,$$ where $\delta_2:=\frac{{\varepsilon}e^{-\alpha T}}{2(1+|\sigma|)}$. For this purpose, let ${\varepsilon}_1\in {\mathbb{R}}_{>0}$ be arbitrary. According to the construction of the set $A$ in Step 2, we obtain that $a_\alpha(\omega)\in B_1(K)$ for all $\omega\in A$. This together with the fact that $$B_1(K)= B_1(K)\cup \bigcup_{n\in{\mathbb{Z}}} \big[n {\varepsilon}_1,(n+1){\varepsilon}_1\big]$$ implies that there exists $n\in{\mathbb{Z}}$ such that $$\label{Eq14}
|n|{\varepsilon}_1\leq \sqrt\alpha+1 {\quad\text{and}\quad}{\mathbb{P}}\left(\Big\{\omega\in A: a_\alpha(\omega)\in \big[n {\varepsilon}_1,(n+1){\varepsilon}_1\big]\Big\}\right)>0\,.$$ We will now only deal with the case $n\geq 0$ and refer a similar treatment for the case $n<0$. Let $\widetilde \phi$ denote the solution of the following integral equation $$\label{Eq15}
x(t)=n{\varepsilon}_1+\int_0^t {\big(\alpha x(s)-x(s)^3\big)} {\mathrm{d}}s -2(\alpha^{\frac{3}{2}}+1)t\,.$$ Define $T_{\min}:=\min{\big\{t\geq 0: \widetilde \phi(t)=0\big\}}$. We will show that $T_{\min}<1$. Suppose the contrary, i.e. $\widetilde \phi(t)>0$ for all $t\in [0,1]$. Using the inequality that $\alpha x-x^3\leq \alpha x$ for all $x\geq 0$ and , we get that $$\widetilde \phi(t)\in (0,n{\varepsilon}_1]{\quad \text{for all }\,}t\in [0,1]\,.$$ Therefore, $$\begin{aligned}
\widetilde \phi(1)
&=
n{\varepsilon}_1+\int_0^1 {\big(\alpha \widetilde\phi(s)-\widetilde\phi(s)^3\big)}{\mathrm{d}}s-2(\alpha^{\frac{3}{2}}+1)\\
&\leq
n{\varepsilon}_1+\alpha n{\varepsilon}_1-2(\alpha^{\frac{3}{2}}+1)<0\,,
\end{aligned}$$ which leads to a contradiction. Now we define $$\widetilde A_1:={\big\{\omega\in\Omega: \textstyle \sup_{t\in [0,T_{\min}]} {\big|\omega(t)+2{\big(\alpha^{\frac{3}{2}}+1\big)}t\big|}<{\varepsilon}_1\big\}}.$$ Note that for any $\omega\in A_1\cap \widetilde A_1$, we have $$\begin{aligned}
a_\alpha(\theta_{T_{\min}}\omega)
&=
a_\alpha(\omega)+\int_0^t {\big(\alpha a_\alpha(\theta_s\omega)-a_\alpha(\theta_s\omega)^3\big)}{\mathrm{d}}s+\omega(t)\\
&\leq
2{\varepsilon}_1+n{\varepsilon}_1+\int_0^t {\big(\alpha a_\alpha(\theta_s\omega)-a_\alpha(\theta_s\omega)^3\big)}{\mathrm{d}}s-2(\alpha^{\frac{3}{2}}+1)t\,.
\end{aligned}$$ Consequently, by choosing ${\varepsilon}_1$ sufficiently small we get that $|a_\alpha(\theta_{T_{\min}}\omega)|<\delta_2$ for all $\omega\in \omega\in A_1\cap \widetilde A_1$. Thus, the set $A_2:=\theta_{T_{\min}}(A_1\cap \widetilde A_1)$ will satisfy the desired assertion in this step.
*Step 4.* Define $$\widetilde A_2:={\big\{\omega\in\Omega: \textstyle\sup_{t\in [0,T]}|\omega(t)|\leq \delta_2\big\}}\,,$$ where $\delta_2$ is defined as in Step 3. Clearly, $A_2$ and $\widetilde A_2$ are independent and therefore the set $\mathcal A:=A_2\cap \widetilde A_2$ is also of positive probability measure. Choose and fix an arbitrary $\omega\in \mathcal A$. By the construction of $A_2$ as in Step 3, we get that $|a_\alpha(\omega)|<\delta_2$. Since $a_\alpha(\omega)$ is a random fixed point of ${{\varphi}}$ it follows that $$a_\alpha(\theta_t\omega)=a_\alpha(\omega)+\int_0^t {\big(\alpha a_\alpha(\theta_s\omega)-a_\alpha(\theta_s\omega)^3\big)}\,{\mathrm{d}}s+\sigma \omega(t)\,.$$ which implies that $$|a_\alpha(\theta_t\omega)|\leq (1+|\sigma|)\delta_2+\int_0^t\alpha |a_\alpha(\theta_s\omega)|\,{\mathrm{d}}s{\quad \text{for all }\,}t\in [0,T]\,.$$ Using Gronwall’s inequality, we get that $$|a_\alpha(\theta_t\omega)|\leq (1+|\sigma|)\delta_2 e^{\alpha t}< {\varepsilon}{\quad \text{for all }\,}t\in [0,T]\,.$$ Thus, we get that $a_\alpha(\theta_t\omega)\in (-{\varepsilon},{\varepsilon})$ for all $t\in[0,T]$, which completes the proof.
We now give a detailed description of the random bifurcation scenario for the stochastic differential equation by means of both *asymptotic* and *finite-time* dynamical behaviour. The asymptotic description implies that there is a qualitative change in the uniformity of attraction of the unique random attractor ${\{a_\alpha(\omega)\}}_{\omega\in\Omega}$. On the other hand, the finite-time description shows that after the bifurcation, even if the time interval is very large, the (asymptotic) Lyapunov exponent cannot be observed with non-vanishing probability (by a finite-time Lyapunov exponent); however, before the bifurcation, the (asymptotic) Lyapunov exponent can be approximated by the finite-time Lyapunov exponent. Finite-time Lyapunov exponents for random dynamical systems have not been considered in the literature so far, but play an important role in the description of Lagrangian Coherent Structures in fluid dynamics [@Haller_01_1].
Let ${\{a_\alpha(\omega)\}}_{\omega\in\Omega}$ denote the unique random attracting fixed point of a stochastic differential equation . Then ${\{a_\alpha(\omega)\}}_{\omega\in\Omega}$ is called *locally uniformly attractive* if there exists $\delta>0$ such that $$\lim_{t\to\infty}\sup_{x\in (-\delta,\delta)}\operatorname*{ess\,sup}_{\omega\in\Omega}|{{\varphi}}(t,\omega)(a_\alpha(\omega)+x)-a_\alpha(\theta_t\omega)|=0.$$
\[Theorem\_1\] Consider the stochastic differential equation with the unique random attracting fixed point ${\{a_\alpha(\omega)\}}_{\omega\in\Omega}$. Then the following statements hold:
- For $\alpha<0$, the random attractor ${\{a_\alpha(\omega)\}}_{\omega\in\Omega}$ is locally uniformly attractive; in fact, it is even globally uniformly exponential attractive, i.e. $$\label{Eq1}
|{{\varphi}}(t,\omega,x)-{{\varphi}}(t,\omega,a_\alpha(\omega))|\leq e^{\alpha t} |x-a_\alpha(\omega)|{\quad \text{for all }\,}x\in {\mathbb{R}}\,.$$
- For $\alpha>0$, the random attractor ${\{a_\alpha(\omega)\}}_{\omega\in\Omega}$ is not locally uniformly attractive.
\(i) Let $x\in{\mathbb{R}}$ be arbitrary such that $x\not=a_\alpha(\omega)$. Using the monotonicity of solutions, we may assume that ${{\varphi}}(t,\omega,x)>{{\varphi}}(t,\omega,a_\alpha(\omega))$ for all $t\ge0$. The integral form of , $${{\varphi}}(t,\omega)x=x+\int_0^t {\big(\alpha {{\varphi}}(s,\omega)x-({{\varphi}}(s,\omega)x)^3\big)}\,{\mathrm{d}}s+ \sigma \omega(t)$$ yields that $${{\varphi}}(t,\omega)x-{{\varphi}}(t,\omega)a_\alpha(\omega)\le x-a_\alpha(\omega)+\alpha \int_0^t {\big({{\varphi}}(s,\omega)x-{{\varphi}}(s,\omega)a_\alpha(\omega)\big)}\,{\mathrm{d}}s\,.$$ Using Gronwall’s inequality implies , which finished this part of the proof.
\(ii) Suppose to the contrary that there exists $\delta>0$ such that $$\lim_{t\to\infty}\sup_{x\in (-\delta,\delta)} \operatorname*{ess\,sup}_{\omega\in\Omega}|{{\varphi}}(t,\omega,a_\alpha(\omega)+x)-a_\alpha(\theta_t\omega)|=0\,,$$ which implies that there exists $N\in{\mathbb{N}}$ such that $$\label{Eq2}
\sup_{x\in (-\delta,\delta)} \operatorname*{ess\,sup}_{\omega\in\Omega}|{{\varphi}}(t,\omega,a_\alpha(\omega)+x)-a_\alpha(\theta_t\omega)|
< \frac{\sqrt \alpha}{4}{\quad \text{for all }\,}t\geq N\,.$$ According to Proposition \[Prp1\], there exists $\mathcal A\in {\mathcal{F}}_{-\infty}^0$ of positive probability such that $a_\alpha(\omega)\in (-\frac{\delta}{4},\frac{\delta}{4})$. Note that $-\sqrt\alpha$ and $\sqrt\alpha$ are two attractive fixed point for the deterministic differential equation $$\dot x=\alpha x-x^3\,.$$ Let $\phi(\cdot,x_0)$ denote the solution of the above deterministic equation which satisfies that $x(0)=x_0$. Then there exists $T>N$ such that $$\label{Eq16}
\phi(T,\delta/4)>\frac{\sqrt\alpha}{2} {\quad\text{and}\quad}\phi(T,-\delta/4)<-\frac{\sqrt\alpha}{2}\,.$$ For any ${\varepsilon}>0$, we define $$\mathcal A^+_{\varepsilon}:={\big\{\omega\in\Omega: \textstyle \sup_{t\in [0,T]}|\omega(t)|<{\varepsilon}\big\}}\,.$$ Clearly, $\mathcal A^+_{\varepsilon}\in {\mathcal{F}}_{0}^T$ has positive probability, and thus, ${\mathbb{P}}(\mathcal A\cap \mathcal A_{{\varepsilon}}^+)={\mathbb{P}}(\mathcal A){\mathbb{P}}(\mathcal A_{{\varepsilon}}^+)$ is positive. Due to compactness of $[0,T]$, there exists ${\varepsilon}>0$ such that for all $\omega\in {\mathcal{A}}^+_{\varepsilon}$, we have $$|{{\varphi}}(T,\omega,\delta/4)- \phi(T,\delta/4)|\leq \frac{\sqrt\alpha}{4} {\quad\text{and}\quad}|{{\varphi}}(T,\omega,-\delta/4)- \phi(T,-\delta/4)|< \frac{\sqrt\alpha}{4}\,,$$ which implies together with that $${{\varphi}}(T,\omega,\delta/4)> \frac{\sqrt\alpha}{4} {\quad\text{and}\quad}{{\varphi}}(T,\omega,-\delta/4)< -\frac{\sqrt\alpha}{4}\,.$$ Due to the fact that $|a_\alpha(\omega)|\leq \frac{\delta}{2}$ for all $\omega\in \mathcal A\cap \mathcal A_{{\varepsilon}}^+$, we get that for all $\omega\in \mathcal A\cap \mathcal A_{{\varepsilon}}^+$ $$\begin{aligned}
&\sup_{x\in (-\delta,\delta)}|{{\varphi}}(T,\omega,a_\alpha(\omega)+x)-a_\alpha(\theta_T\omega)|\\
\geq&
\max{\big\{{{\varphi}}(T,\omega,\delta/4)-a_\alpha(\theta_T\omega)|,|{{\varphi}}(T,\omega,-\delta/4)-a_\alpha(\theta_T\omega)|\big\}}\,.
\end{aligned}$$ Consequently, $$\sup_{x\in (-\delta,\delta)} \operatorname*{ess\,sup}_{\omega\in\Omega}|{{\varphi}}(t,\omega,a_\alpha(\omega)+x)-a_\alpha(\theta_t\omega)|> \frac{\sqrt\alpha}{4} \,,$$ which contradicts to and the proof is complete.
For the description of the bifurcation via finite-time properties, consider a compact time interval $I=[0,T]$ and define the corresponding *finite-time Lyapunov exponent* associated with the invariant measure $a_\alpha(\omega)$ by $$\lambda^{T,\omega}_\alpha:=\frac{1}{T}\ln{\left|\frac{\partial {{\varphi}}_\alpha}{\partial x}(T,\omega,a_{\alpha}(\omega))\right|}\,.$$ Clearly, the (classical) Lyapunov exponent $\lambda_\alpha^\infty$ associated with the random fixed point $a_\alpha(\omega)$ is given by $$\lambda_\alpha^\infty=\lim_{T\to\infty}\lambda^{T,\omega}_\alpha\,.$$ In contrast to classical Lyapunov exponent, the finite-time Lyapunov exponent is, in general, a non-constant random variable.
\[Finite-timeBifurcation\] Consider the stochastic differential equation with the unique random attracting fixed point ${\{a_\alpha(\omega)\}}_{\omega\in\Omega}$. For any finite time interval $[0,T]$, let $\lambda_\alpha^{T,\omega}$ denote the finite-time Lyapunov exponent associated with ${\{a_\alpha(\omega)\}}_{\omega\in\Omega}$. Then the following statements hold:
- For $\alpha<0$, the random attractor ${\{a_\alpha(\omega)\}}_{\omega\in\Omega}$ is finite-time attractive, i.e. $$\lambda_\alpha^{T,\omega}\leq \alpha<0 {\quad \text{for all }\,}\omega\in\Omega\,.$$
- For $\alpha>0$, the random attractor ${\{a_\alpha(\omega)\}}_{\omega\in\Omega}$ is not finite-time attractive, i.e. $${\mathbb{P}}\big\{\omega\in\Omega: \lambda_\alpha^{T,\omega}>0\big\}>0.$$
\(i) follows directly from [Theorem \[Theorem\_1\]]{} (i).
\(ii) Choose ${\varepsilon}:=\frac{\sqrt\alpha}{2}>0$. According to Proposition \[Prp1\], there exists a measurable set ${\mathcal{A}}\in {\mathcal{F}}_{-\infty}^T$ of positive probability such that $$a_\alpha(\theta_s\omega)\in (-{\varepsilon},{\varepsilon}){\quad \text{for all }\,}s\in [0,T]\,.$$ Let $\omega\in{\mathcal{A}}$ be arbitrary and we will estimate $\lambda_\alpha^{T,\omega}$. Note that the linearized equation along the random fixed point $a_\alpha(\omega)$ is given by $$\dot \xi(t)=(\alpha-3a_\alpha(\theta_t\omega)^2)\xi(t)\,.$$ We thus get $$\lambda_{\alpha}^{T,\omega}
=
\alpha-\frac{1}{T}\int_0^T 3 a_\alpha(\theta_t\omega)^2{\mathrm{d}}t
\geq
\frac{\alpha}{4}\,,$$ which completes the proof.
This theorem implies that the change in the signature of finite-time Lyapunov exponents indicates a qualitative change in the dynamics. This means that the bifurcation is observable in practice, since finite-time Lyapunov exponents are numerically computable quantities. Note that the numerical approximation of classical Lyapunov exponents is difficult in general. In the special case of random matrix products with positive matrices, however, [@Pollicott_10] established explicit bounds for the numerical approximation of (classical) Lyapunov exponents recently.
The dichotomy spectrum at the bifurcation point {#subsec3}
-----------------------------------------------
We will compute the dichotomy spectrum of the linearization around the unique random attracting fixed point ${\{a_\alpha(\omega)\}}$ of the system . As a direct consequence, we observe that hyperbolicity is lost at the bifurcated point $\alpha=0$.
\[DichotomySpetrumc-Bifurcation\] Let $\Phi_\alpha(t,\omega):=\frac{\partial\phi_\alpha}{\partial x}(t,\omega,a_\alpha(\omega))$ denote the linearized random dynamical system along the random fixed point $a_\alpha(\omega)$. Then the dichotomy spectrum $\Sigma_\alpha$ of $\Phi_\alpha$ is given by $$\Sigma_\alpha=[-\infty,\alpha]{\quad \text{for all }\,}\alpha\in {\mathbb{R}}\,.$$
From the linearized equation along $a_\alpha(\omega)$ $$\dot\xi(t)=(\alpha-3a_\alpha(\theta_t\omega)^2)\xi(t)\,,$$ we derive that $$\label{Eq17}
\Phi_\alpha(t,\omega)=\exp\left(\int_0^t \big(\alpha-3a_\alpha(\theta_s\omega)^2\big){\mathrm{d}}s\right)\,.$$ Consequently, $$|\Phi_\alpha(t,\omega)|\leq e^{\alpha |t|}{\quad \text{for all }\,}t\in{\mathbb{R}}\,,$$ which implies that $\Sigma_\alpha\subset (-\infty,\alpha]$. Thus, it is sufficient to show that $(-\infty,\alpha]\subset \Sigma_\alpha$. For this purpose, let $\gamma\in (-\infty,\alpha]$ be arbitrary. Suppose the opposite that $\Phi_\alpha$ admits an exponential dichotomy with growth rate $\gamma$ with an invariant projection $P_\gamma$ and positive constants $K,{\varepsilon}$. We now consider two cases: (i) $P_\gamma={\hbox{id}}$ and (ii) $P_\gamma=0$:
*Case (i).* $P_\gamma={\hbox{id}}$, i.e. we have $$\label{Eq18}
\Phi_\alpha(t,\omega)\leq K e^{(\gamma-{\varepsilon}) t}{\quad \text{for all }\,}t\geq 0\,.$$ Choose and fix $T>0$ such that $e^{\frac{{\varepsilon}}{4}T}>K$. According to Proposition \[Prp1\], there exists a measurable set $\mathcal A\subset {\mathcal{F}}_{-\infty}^T$ of positive measure such that $$a_\alpha(\theta_s\omega)\in {\big(-\sqrt{\varepsilon}/2, \sqrt{\varepsilon}/ 2\big)} {\quad \text{for all }\,}\omega\in\mathcal A {\text{ and }}s\in [0,T]\,.$$ From we derive that $$|\Phi_\alpha(T,\omega)| \geq e^{T\left(\alpha-\frac{3{\varepsilon}}{4}\right)}> K e^{(\gamma-{\varepsilon})T}\,,$$ which leads to a contradiction to .
*Case (ii)*: $P_\gamma=0$, i.e. we have $$\Phi_\alpha(t,\omega)\geq \frac{1}{K} e^{(\gamma-{\varepsilon}) t}{\quad \text{for all }\,}t\geq 0\,,$$ which together with implies that $$\label{Eq18b}
\frac{\ln K+(\alpha-\gamma)t}{3}\geq \int_0^t a_\alpha(\theta_s\omega)^2 \,{\mathrm{d}}s\,.$$ Choose and fix $T>0$ such that $$\frac{(T-1)^3}{3}>\frac{\ln K+(\alpha-\gamma)T}{3}\,.$$ Consider the following integral equation $$x(t)=\int_0^t {\big(\alpha x(s)-x(s)^3\big)} {\mathrm{d}}s +\frac{t^4}{4}-\alpha\frac{t^2}{2}+t\,.$$ Clearly, the explicit solution of the above equation is $x(t)=t$. Due to the compactness, there exists ${\varepsilon}>0$ such that for any $x(0)\in(-{\varepsilon},{\varepsilon})$ and $\omega(t)$ with $\sup_{t\in [0,T]}|\omega(t)-\frac{t^4}{4}+\alpha\frac{t^3}{3}-t|\leq {\varepsilon}$ then the solution $x(t)$ of the following equation $$x(t)=x(0)+\int_0^t (\alpha x(s)-x(s)^3)\,{\mathrm{d}}s+\omega(t)$$ satisfies that $\sup_{t\in [0,T]}|x(t)-t|\leq 1$. According to Proposition \[Prp1\], there exists a measurable set $\mathcal A^{-}_{\varepsilon}\subset {\mathcal{F}}_{-\infty}^0$ of positive measure such that $a_\alpha(\omega)\in (-{\varepsilon},{\varepsilon})$ for all $\omega\in\mathcal A$. Define $\mathcal A_+\subset {\mathcal{F}}_{0}^T$ by $$\mathcal A^+_{\varepsilon}:={\big\{\omega\in {\mathcal{F}}_{0}^T: \textstyle \sup_{t\in [0,T]}|\omega(t)-t^4/4+\alpha t^2/2-t|\leq {\varepsilon}\big\}}\,.$$ Therefore, for all $\omega\in \mathcal A^{-}_{\varepsilon}\cap\mathcal A^+_{\varepsilon}$, we get $$\sup_{t\in [0,T]}|a_\alpha(\theta_t\omega)-t|\leq 1\,,$$ which implies that $$\int_0^T a_\alpha(\theta_s\omega)^2\,{\mathrm{d}}s \ge \frac{(T-1)^3}{3} > \frac{\ln K+(\alpha-\gamma)T}{3}\,,$$ which leads to a contradiction to . The proof is complete.
We have seen in [Theorem \[Finite-timeBifurcation\]]{} that the bifurcation of manifests itself also via finite-time Lyapunov exponents: before the bifurcation, all finite-time Lyapunov exponents are negative, and after the bifurcation, one observes positive finite-time Lyapunov exponents with positive probability. This implies in particular that the set of all Lyapunov exponents observed almost surely within a *finite time* does not converge to the (asymptotic) Lyapunov exponents when time tends to infinity. The following theorem makes it precise that in contrast to asymptotic Lyapunov exponents, the dichotomy spectrum includes limits of the set of finite-time Lyapunov exponents.
\[theo\_1\] Let $(\theta, \Phi)$ be a linear random dynamical system on ${\mathbb{R}}^d$ with dichotomy spectrum $\Sigma$. Define the finite-time Lyapunov exponent $$\lambda(T,\omega,x) := \frac{1}{T} \ln \frac{{\|\Phi(t,\omega)x\|}}{{\|x\|}} {\quad \text{for all }\,}T>0,\,\omega\in\Omega{\text{ and }}x\in{\mathbb{R}}^d\setminus{\{0\}}\,.$$ Then $$\lim_{T\to\infty} \operatorname*{ess\,sup}_{\omega\in\Omega} \!\!\!\!\sup_{x\in{\mathbb{R}}^d\setminus{\{0\}}} \!\!\!\!\!\!\lambda(T,\omega, x) = \sup \Sigma$$ provided that $\sup \Sigma<\infty$ and $${\quad\text{and}\quad}\lim_{T\to\infty} \operatorname*{ess\,inf}_{\omega\in\Omega} \!\!\!\!\inf_{x\in{\mathbb{R}}^d\setminus{\{0\}}} \!\!\!\!\!\!\lambda(T,\omega, x) = \inf \Sigma$$ provided that $\inf \Sigma>-\infty$.
By definition of $\lambda(T,\omega,x)$, we get that for all $T,S\geq 0$ $$(T+S)\operatorname*{ess\,sup}_{\omega\in\Omega} \!\!\!\!\sup_{x\in{\mathbb{R}}^d\setminus{\{0\}}} \!\!\!\!\!\!\lambda(T+S,\omega, x)\leq T\operatorname*{ess\,sup}_{\omega\in\Omega} \!\!\!\!\sup_{x\in{\mathbb{R}}^d\setminus{\{0\}}} \!\!\!\!\!\!\lambda(T,\omega, x)+S\operatorname*{ess\,sup}_{\omega\in\Omega} \!\!\!\!\sup_{x\in{\mathbb{R}}^d\setminus{\{0\}}} \!\!\!\!\!\!\lambda(S,\omega, x).$$ This implies that the sequence $(T\operatorname*{ess\,sup}_{\omega\in\Omega} \sup_{x\in{\mathbb{R}}^d\setminus{\{0\}}} \lambda(T,\omega, x))_{T\geq 0}$ is subadditive. We thus obtain $$\lim_{T\to\infty} \operatorname*{ess\,sup}_{\omega\in\Omega} \!\!\!\!\sup_{x\in{\mathbb{R}}^d\setminus{\{0\}}} \!\!\!\!\!\!\lambda(T,\omega, x) = \limsup_{T\to\infty} \operatorname*{ess\,sup}_{\omega\in\Omega} \!\!\!\!\sup_{x\in{\mathbb{R}}^d\setminus{\{0\}}} \!\!\!\!\!\!\lambda(T,\omega, x).$$ We first prove that provided $\sup \Sigma<\infty$, we have $$\gamma:=\limsup_{T\to\infty} \operatorname*{ess\,sup}_{\omega\in\Omega} \sup_{x\in{\mathbb{R}}^d\setminus{\{0\}}} \lambda(T,\omega, x)=\sup \Sigma.$$ Since $\sup \Sigma<\infty$ it follows that there exists $K>0$ such that $$\label{Eq19}
{\|\Phi(t,\omega)x\|}\le K e^{t\sup\Sigma} {\quad \text{for all }\,}t\ge0\,.$$ Assume first that $\gamma < \sup \Sigma$. This means that there exists a $t_0>0$ such that for all $t\ge t_0$ and for almost all $\omega\in\Omega$, we have ${\|\Phi(t,\omega)x\|}\le e^{t/2(\gamma+\sup\Sigma)}$. Thus, together with , we obtain for all $t\ge 0$ and for almost all $\omega\in\Omega$ that $${\|\Phi(t,\omega)x\|}\le \widehat K e^{t/2(\gamma+\sup\Sigma)},\qquad \widehat K:=\max\{1,Ke^{t_0/2(\sup\Sigma-\gamma)}\}.$$ Hence, $\sup\Sigma\leq \gamma+\sup\Sigma$, which is a contradiction. Assume now that $\gamma > \sup \Sigma$. This means in particular that $\sup\Sigma < \infty$. Hence, there exists a $K>0$ such that for almost all $\omega\in\Omega$, we have $${\|\Phi(t,\omega)x\|} \le K e^{t/2(\gamma+\sup\Sigma)}{\|x\|}{\quad \text{for all }\,}x\in{\mathbb{R}}^d\,.$$ This leads to $\lambda(t,\omega,x)\le (\gamma+\sup\Sigma)/2$ for all $x\in{\mathbb{R}}^d\setminus\{0\}$, and thus, $$\gamma=\limsup_{T\to\infty} \operatorname*{ess\,sup}_{\omega\in\Omega} \sup_{x\in{\mathbb{R}}^d\setminus{\{0\}}} \lambda(T,\omega, x)\le (\gamma+\sup\Sigma)/2,$$ which proves the first equality. Similarly, one can show that $$\lim_{T\to\infty} \operatorname*{ess\,inf}_{\omega\in\Omega} \!\!\!\!\inf_{x\in{\mathbb{R}}^d\setminus{\{0\}}} \!\!\!\!\!\!\lambda(T,\omega, x) = \inf \Sigma$$ provided that $\inf \Sigma>-\infty$, which finishes the proof of this theorem.
In the following example, we construct explicitly a linear random dynamical system with $\sup \Sigma=\infty$ but $$\lim_{T\to\infty} \operatorname*{ess\,sup}_{\omega\in\Omega} \!\!\!\!\sup_{x\in{\mathbb{R}}^d\setminus{\{0\}}} \!\!\!\!\!\!\lambda(T,\omega, x) <\infty.$$ An example of a linear random dynamical system with $\inf \Sigma=-\infty$ but $$\lim_{T\to\infty} \operatorname*{ess\,inf}_{\omega\in\Omega} \!\!\!\!\inf_{x\in{\mathbb{R}}^d\setminus{\{0\}}} \!\!\!\!\!\!\lambda(T,\omega, x) >-\infty.$$ can be constructed analogously. This example shows the importance of the assumption $\sup \Sigma<\infty$ or $\inf \Sigma>-\infty$ in the above theorem.
Similarly to [Example \[Example1\]]{}, there exist infinitely many measurable sets ${\{U_n\}}_{n\in{\mathbb{N}}}$ of positive measure such that $U_n,\theta U_n,\theta^2U_n$ for $n\in {\mathbb{N}}$ are pairwise disjoint. We define a random mapping $A:\Omega\rightarrow {\mathbb{R}}$ as follows: $$A(\omega)=\left\{
\begin{array}{c@{\;:\;}l}
\frac{1}{n} & \omega\in U_n\cup \theta^2 U_n\,, n\in{\mathbb{N}}\,,\\[1.5ex]
n & \omega\in \theta U_n \,, n\in{\mathbb{N}}\,,\\[1.5ex]
1 & \hbox{otherwise}.
\end{array}
\right.$$ Let $\Phi$ denote the discrete-time random dynamical system generated by $A$. Since $\log\|A(\cdot)\|$ is neither bounded from above nor from below, we get that $\Sigma(\Phi)=(-\infty,\infty)$. On the other hand, it is easy to see that for all $T\geq 2$ we get that $$\operatorname*{ess\,sup}_{\omega\in\Omega}\log|\Phi(T,\omega)|=1,$$ which implies that $$\lim_{T\to\infty}\operatorname*{ess\,sup}_{\omega\in\Omega}\frac{1}{T}\log|\Phi(T,\omega)|=0.$$
Topological equivalence of random dynamical systems {#sec_1}
====================================================
This section deals with topological equivalence of random dynamical system [@Imkeller_01_2; @Imkeller_02_1; @Li_05_1; @Arnold_98_1]. This concept has not been used so far to study bifurcations of random dynamical systems, and the main aim of this section is to discuss topological equivalence for the stochastic differential equation from [Section \[sec\_bif\]]{}, given by $${\mathrm{d}}x= {\big(\alpha x-x^3\big)}{\mathrm{d}}t+ \sigma {\mathrm{d}}W_t\,.$$
The concept of topological equivalence for random dynamical systems [@Arnold_98_1 Definition 9.2.1] differs from the corresponding deterministic notion of topological equivalence in the sense that instead of one homeomorphism (mapping orbits to orbits), the random version is given by a family of homeomorphisms ${\{h_\omega\}}_{\omega\in\Omega}$. The precise definition is given as follows.
\[deftopeq\] Let $(\Omega,{\mathcal{F}},{\mathbb{P}})$ be a probability space, $\theta:{\mathbb{T}}\times \Omega\to\Omega$ a metric dynamical system and $(X_1,d_1)$, $(X_2,d_2)$ be metric spaces. Then two random dynamical systems $({{\varphi}}_1:{\mathbb{T}}\times\Omega\times X_1\to X_1,\theta)$ and $({{\varphi}}_2:{\mathbb{T}}\times\Omega\times X_1\to X_1,\theta)$ are called *topologically equivalent* if there exists a conjugacy $h:\Omega\times X_1\to X_2$ fulfilling the following properties:
- For almost all $\omega\in\Omega$, the function $x \mapsto h(\omega,x)$ is a homeomorphism from $X_1$ to $X_2$.
- The mappings $\omega \mapsto h(\omega, x_1)$ and $\omega \mapsto h^{-1}(\omega, x_2)$ are measurable for all $x_1 \in X_1$ and $x_2 \in X_2$.
- The random dynamical systems ${{\varphi}}_1$ and ${{\varphi}}_2$ are *cohomologous*, i.e. $${{\varphi}}_2(t, \omega, h(\omega, x)) = h(\theta_t\omega, {{\varphi}}_1(t, \omega, x)) {\quad \text{for all }\,}x\in X_1 \mbox{ and almost all } \omega\in\Omega\,.$$
A bifurcation is then described by means of a lack of topological equivalence at the bifurcation point. The following theorem says that near the bifurcation point $\alpha = 0$, all systems of are equivalent.
\[Theorem2\] Let $(\theta:{\mathbb{R}}\times \Omega\to\Omega, {{\varphi}}_\alpha:{\mathbb{R}}\times\Omega\times{\mathbb{R}}\to {\mathbb{R}})$ denote the random dynamical system generated by system . Then there exists an ${\varepsilon}>0$ such that for all $\alpha\in(-{\varepsilon},{\varepsilon})$, the random dynamical systems ${{\varphi}}_\alpha$ are topologically equivalent to the dynamical system $(e^{-t}x)_{t,x\in{\mathbb{R}}}$, i.e. there exists a conjugacy $h:\Omega\times {\mathbb{R}}\to{\mathbb{R}}$ such that for almost all $\omega\in\Omega$, we have $${{\varphi}}_\alpha(t,\omega,h(\omega,x))=h(\theta_t\omega,e^{-t}x){\quad \text{for all }\,}t,x\in{\mathbb{R}}\,.$$
Let $a_\alpha(\omega)$ denote the unique random fixed point of . According to the results in [@Crauel_98_1], we obtain that $${\mathbb{E}}a_\alpha(\omega)^2= \frac{\int_{-\infty}^\infty u^2 \exp\left(\frac{1}{\sigma^2}{\big(\alpha u^2-\frac{1}{2}u^4\big)}\right)\,{\mathrm{d}}u}{\int_{-\infty}^\infty\exp\left(\frac{1}{\sigma^2}{\big(\alpha u^2-\frac{1}{2}u^4\big)}\right)\,{\mathrm{d}}u}\,.$$ Therefore, $$\lim_{\alpha\to 0} {\mathbb{E}}a_\alpha(\omega)^2= \frac{\int_{-\infty}^\infty u^2 \exp{\big({-}\frac{u^4}{2\sigma^2}\big)}\,{\mathrm{d}}u}{\int_{-\infty}^\infty \exp{\big({-}\frac{u^4}{2\sigma^2}\big)}\,{\mathrm{d}}u}\,.$$ Then there exists an ${\varepsilon}>0$ such that for all $\alpha\in(-{\varepsilon},{\varepsilon})$, we have $$\delta:=\frac{{\mathbb{E}}a_\alpha(\omega)^2}{4}-\alpha>0\,.$$ For any $x\in{\mathbb{R}}$ and $(t,\omega)\in{\mathbb{R}}\times\Omega$, we define $$\label{Eq4}
\psi(t,\omega,x):={{\varphi}}_\alpha(t,\omega,x+a_\alpha(\omega))-a_\alpha(\theta_t\omega)\,.$$ By using the transformation function $g(\omega,x):=x-a_\alpha(\omega)$, the random dynamical systems ${{\varphi}}_\alpha$ and $\psi$ are topologically equivalent. Hence, it is sufficient to show that $\psi$ is topologically equivalent to the dynamical system $(e^{-t}x)_{t,x\in{\mathbb{R}}}$. We first summarise some properties of $\psi$:
- Since $a_\alpha(\omega)$ is a random fixed point of ${{\varphi}}_\alpha$, it follows that $$\psi(t,\omega,0)=0 {\quad \text{for all }\,}t\in{\mathbb{R}}{\text{ and }}\omega\in\Omega\,.$$
- Due to the monotonicity of ${{\varphi}}_\alpha$, for $x_1>x_2$, we have $$\psi(t,\omega,x_1)>\psi(t,\omega,x_2) {\quad \text{for all }\,}t\in{\mathbb{R}}{\text{ and }}\omega\in\Omega\,.$$
- From , we derive that $$\begin{aligned}
\psi(t,\omega,x)
&=
x+\int_0^t \psi(s,\omega,x) {\big(\alpha-a_\alpha(\theta_s\omega)^2-a_\alpha(\theta_s\omega){{\varphi}}_\alpha(s,\omega,a_\alpha(\omega)+x)-\\
&\qquad\qquad\qquad\qquad\qquad{{\varphi}}_\alpha(s,\omega,a_\alpha(\omega)+x)^2\big)}\,{\mathrm{d}}s\,.
\end{aligned}$$ Consequently, $$\begin{aligned}
\psi(t,\omega,x)
&=
x\exp{\Bigg(\int_0^t\alpha-a_\alpha(\theta_s\omega)^2-a_\alpha(\theta_s\omega){{\varphi}}_\alpha(s,\omega,a_\alpha(\omega)+x)\\
&
\qquad\qquad\qquad\qquad\qquad-{{\varphi}}_\alpha(s,\omega,a_\alpha(\omega)+x)^2\,{\mathrm{d}}s\Bigg)}\,.
\end{aligned}$$
According to Birkhoff’s ergodic theorem, there exists an invariant set $\widetilde \Omega$ of full measure such that $$\label{Eq3}
\lim_{t\to\pm\infty}\frac{1}{t}\int_0^t a_\alpha(\theta_s\omega)^2\,{\mathrm{d}}s={\mathbb{E}}a_\alpha(\omega)^2\,.$$ Choose and fix $\omega\in\widetilde\Omega$. From , there exists $T>0$ such that for all $|t|>T$ we have $$\left|\frac{1}{t}\int_0^t a_\alpha(\theta_s\omega)^2\,{\mathrm{d}}s-{\mathbb{E}}a_\alpha(\omega)^2\right|\leq \frac{\delta}{2}\,.$$ In what follows, we will show the following estimates on $\psi(t,\omega,x)$ for $x>0$:
- For $t\geq T$, we get $$\psi(t,\omega,x)
\le
x\exp{\left(\int_0^t \alpha-\frac{a_\alpha(\theta_s\omega)^2}{4}\,{\mathrm{d}}s\right)}\leq e^{-\delta t/2}x.$$
- For $t\leq -T$, we get $$\psi(t,\omega,x)
\geq
x\exp{\left(\int_0^t \alpha-\frac{a_\alpha(\theta_s\omega)^2}{4}\,{\mathrm{d}}s\right)}
\geq e^{-\delta t/2}x\,.$$
Consequently, we get that $$\lim_{r\to\infty}\int_r^\infty \psi(s,\omega,x)\,{\mathrm{d}}s=0
{\quad\text{and}\quad}\lim_{r\to-\infty}\int_r^\infty \psi(s,\omega,x)\,{\mathrm{d}}s=\infty.$$ Hence, there exists a unique $r(\omega,x)$ such that $$\label{Eq5}
\int_{r(\omega,x)}^\infty \psi(s,\omega,x)\,{\mathrm{d}}s=1.$$ Similarly, $r(\omega,x)$ for $x<0$ is defined to satisfy $$\label{Eq6}
\int_{r(\omega,x)}^\infty \psi(s,\omega,x)\,{\mathrm{d}}s=-1.$$ Using the cocycle property of $\psi$, we obtain that $$\label{Eq7}
r(\omega,x)=r(\theta_s\omega,\psi(s,\omega,x))+s.$$ Define a function $$g(\omega,x)
=
\left\{
\begin{array}{c@{\;:\;}l}
e^{r(\omega,x)} & x>0\,, \\
0 & x=0\,, \\
-e^{r(\omega,x)} & x<0\,.
\end{array}
\right.$$ We will now show that $g$ transforms the random dynamical system $\psi$ to the dynamical system $(e^{-t}x)_{t,x\in{\mathbb{R}}}$:
- For any $x>0$, we have $\psi(s,\omega,x)>0$ and thus from the definition of the function $g$ it follows that $$g(\theta_s\omega,\psi(s,\omega,x))
=
e^{r(\theta_s\omega,\psi(s,\omega,x))},$$ which implies together with that $$g(\theta_s\omega,\psi(s,\omega,x))=e^{r(\omega,x)-s}=e^{-s}\psi(s,\omega,x)\,.$$ Similarly, for $x<0$ we also have $g(\theta_s\omega,\psi(s,\omega,x))=e^{-s}\psi(s,\omega,x)$ for all $s\in{\mathbb{R}},\omega\in\Omega$.
- Choose and fix $\omega\in\widetilde \Omega$. We will show that $g_\omega:{\mathbb{R}}\rightarrow {\mathbb{R}}, x\mapsto g(\omega,x)$ is a homeomorphism.\
*Injectivity*: From the definition of $g$, it is easily seen that for $x_1>0>x_2$ we have $$g_\omega(x_1)>0>g_\omega(x_2).$$ On the other hand, based on strict monotonicity of $\psi$ we get that for $x_1>x_2>0$ $$\int_{r(\omega,x_2)}^\infty \psi(s,\omega,x_1)\,{\mathrm{d}}s>\int_{r(\omega,x_2)}^\infty \psi(s,\omega,x_2)\,{\mathrm{d}}s=1.$$ Consequently, $r(\omega,x_1)
>r(\omega,x_2)$ and thus $g_\omega(x_1)>g_\omega(x_2)$. Similarly, for $0>x_1>x_2$ we also have $g_\omega(x_1)>g_\omega(x_2)$. Therefore, $g_\omega$ is strictly increasing and thus injective.\
*Continuity*: We first show that $\lim_{x\to 0+}g_\omega(x)=0$. Let ${\varepsilon}>0$ be arbitrary. Choose $\tilde{T}>T$ such that $\frac{2}{\delta}e^{-\frac{\delta \tilde{T}}{2}}<\frac{1}{3}, e^{-\tilde{T}}<{\varepsilon},$ and for all $t\geq \tilde{T}$ we get $$\psi(t,\omega,x)\leq e^{-\frac{\delta t}{2}}x\,.$$ As a consequence, for all $x\in (0,1)$ we get $$\label{Eq8}
\int_{\tilde{T}}^\infty \psi(s,\omega,x)\,{\mathrm{d}}s\leq \int_{\tilde{T}}^\infty e^{-\frac{\delta s}{2}}\,{\mathrm{d}}s<\frac{{\varepsilon}}{3}\,.$$ Since $\lim_{x\to 0} \psi(s,\omega,x)=0$ and ${\big[{-}\tilde{T},\tilde{T}\big]}$ is a compact interval, there exists $\delta^*$ such that $$\int_{-\tilde{T}}^{\tilde{T}} \psi(s,\omega,\delta^*)\,{\mathrm{d}}s<\frac{{\varepsilon}}{3},$$ which together with implies that $$\int_{-\tilde{T}}^\infty \psi(s,\omega,x)\,{\mathrm{d}}s<\frac{2}{3}{\quad \text{for all }\,}x\in \big(0,\min(1,\delta^*)\big)\,.$$ Therefore, $r(\omega,x)<-\tilde{T}$ and thus $g_\omega(x)<{\varepsilon}$ for all $x\in \big(0,\min(1,\delta^*)\big)$. Hence, $\lim_{x\to 0+}g_\omega(x)=0$ and similarly we also have $\lim_{x\to 0-}g_\omega(x)=0$ and thus $g_\omega$ is continuous at $0$. The continuity of $g$ on the whole real line can be proved in a similar way.\
*Surjectivity*: It is easy to prove surjectivity from $$\lim_{x\to\infty} g_\omega(x)=\infty {\quad\text{and}\quad}\lim_{x\to-\infty} g_\omega(x)=-\infty\,.$$
This finishes the proof of this theorem.
This theorem implies that the stochastic differential equation does not admit a bifurcation at $\alpha=0$ which is induced by the above concept of topological equivalence. In addition, because of the observations in [Theorem \[DichotomySpetrumc-Bifurcation\]]{}, this concept of equivalence is not in correspondence with the dichotomy spectrum (linear systems which are hyperbolic and non-hyperbolic can be equivalent).
We will show now that the concept of a *uniform topological equivalence* is the right tool to obtain the bifurcations studied in this paper.
Let $(\Omega,{\mathcal{F}},{\mathbb{P}})$ be a probability space, $\theta:{\mathbb{T}}\times \Omega\to\Omega$ a metric dynamical system and $(X_1,d_1)$, $(X_2,d_2)$ be metric spaces. Then two random dynamical systems $({{\varphi}}_1:{\mathbb{T}}\times\Omega\times X_1\to X_1,\theta)$ and $({{\varphi}}_2:{\mathbb{T}}\times\Omega\times X_1\to X_1,\theta)$ are called *uniformly topologically equivalent* with respect to a random fixed point ${\{a_\alpha(\omega)\}}_{\omega\in\Omega}$ of ${{\varphi}}_1$ if there exists a conjugacy $h:\Omega\times X_1\to X_2$ fulfilling the following properties:
- For almost all $\omega\in\Omega$, the function $x \mapsto h(\omega,x)$ is a homeomorphism from $X_1$ to $X_2$.
- The mappings $\omega \mapsto h(\omega, x_1)$ and $\omega \mapsto h^{-1}(\omega, x_2)$ are measurable for all $x_1 \in X_1$ and $x_2 \in X_2$.
- The random dynamical systems ${{\varphi}}_1$ and ${{\varphi}}_2$ are *cohomologous*, i.e. $${{\varphi}}_2(t, \omega, h(\omega, x)) = h(\theta_t\omega, {{\varphi}}_1(t, \omega, x)) {\quad \text{for all }\,}x\in X_1 \mbox{ and almost all } \omega\in\Omega\,.$$
- We have $$\lim_{\delta\to0} \operatorname*{ess\,sup}_{\omega\in\Omega} \sup_{x\in B_\delta(a_\alpha(\omega))} d_2(h(\omega, x), h(\omega, a_\alpha(\omega))) = 0$$ and $$\lim_{\delta\to0} \operatorname*{ess\,sup}_{\omega\in\Omega} \sup_{x\in B_\delta(h(\omega, a_\alpha(\omega)))} d_1(h^{-1}(\omega, x), a_\alpha(\omega)) = 0\,.$$
Note that, in comparison to the concept of topological equivalence ([Definition \[deftopeq\]]{}), we added (iv) to take uniformity into account.
We show now that uniform topological equivalence preserves local uniform attractivity.
\[prop\_1\] Let $(\Omega,{\mathcal{F}},{\mathbb{P}})$ be a probability space, $\theta:{\mathbb{T}}\times \Omega\to\Omega$ a metric dynamical system and $(X_1,d_1)$, $(X_2,d_2)$ be metric spaces, and let $({{\varphi}}_1:{\mathbb{T}}\times\Omega\times X_1\to X_1,\theta)$ and $({{\varphi}}_2:{\mathbb{T}}\times\Omega\times X_2\to X_2,\theta)$ be two random dynamical systems which are *uniformly topologically equivalent* with respect to a random fixed point ${\{a_\alpha(\omega)\}}_{\omega\in\Omega}$ of ${{\varphi}}_1$. Let $h:\Omega\times X_1\to X_2$ denote the conjugacy. Then ${\{a_\alpha(\omega)\}}_{\omega\in\Omega}$ is locally uniformly attractive for ${{\varphi}}_1$ if and only if ${\{h(\omega, a_\alpha(\omega))\}}_{\omega\in\Omega}$ is locally uniformly attractive for ${{\varphi}}_2$.
Suppose that ${\{a_\alpha(\omega)\}}_{\omega\in\Omega}$ is locally uniformly attractive for ${{\varphi}}_1$ and let $\eta>0$. Then there exists a $\gamma>0$ such that $$\operatorname*{ess\,sup}_{\omega\in\Omega} \sup_{x\in B_\gamma(a_\alpha(\omega))} d_2(h(\omega, x), h(\omega,a_\alpha(\omega))) \le \eta\,.$$ Since ${\{a_\alpha(\omega)\}}_{\omega\in\Omega}$ is locally uniformly attractive for ${{\varphi}}_1$, there exists a $\delta>0$ and a $T>0$ such that $$\operatorname*{ess\,sup}_{\omega\in\Omega} \sup_{x\in B_\delta(a_\alpha(\omega))} d_1({{\varphi}}_1(t,\omega, x), a_\alpha(\theta_t\omega)) \le \frac{\gamma}{2} {\quad \text{for all }\,}t\ge T\,.$$ Hence, for all $t\ge T$, we have $$\operatorname*{ess\,sup}_{\omega\in\Omega} \sup_{x\in B_\delta(a_\alpha(\omega))} d_2(h(\theta_t\omega,{{\varphi}}_1(t,\omega,x)), h(\theta_t\omega,a_\alpha(\theta_t\omega))) \le \eta\,.$$ This means that for all $t\ge T$, we have $$\operatorname*{ess\,sup}_{\omega\in\Omega} \sup_{x\in B_\delta(a_\alpha(\omega))} d_2({{\varphi}}_2(t,\omega, h(\omega, x)), h(\theta_t\omega,a_\alpha(\omega))) \le \eta\,,$$ and there exists a $\beta>0$ such that $$\operatorname*{ess\,sup}_{\omega\in\Omega} \sup_{x\in B_\beta(h(\omega, a_\alpha(\omega)))} d_1(h^{-1}(\omega, x), a_\alpha(\omega)) \le \frac{\delta}{2}\,.$$ Finally, this means that for all $t\ge T$, we have $$\operatorname*{ess\,sup}_{\omega\in\Omega} \sup_{x\in B_\beta(h(\omega,a_\alpha(\omega)))} d_2({{\varphi}}_2(t,\omega, x), h(\theta_t\omega,a_\alpha(\omega))) \le \eta\,,$$ which finishes the proof that ${\{h(\omega, a_\alpha(\omega))\}}_{\omega\in\Omega}$ is locally uniformly attractive for ${{\varphi}}_2$.
As a corollary to this proposition, it follows that admits a bifurcation.
\[theo1\] The stochastic differential equation admits a random bifurcation at $\alpha=0$ which is induced by the concept of uniform topological equivalence.
This is a direct consequence of [Theorem \[Theorem\_1\]]{} and [Proposition \[prop\_1\]]{}.
****
*Metric dynamical systems.* Let ${\mathcal{B}}(Y)$ denote the Borel $\sigma$-algebra of a metric space $Y$. Consider a time set ${\mathbb{T}}={\mathbb{R}}$ or ${\mathbb{T}}={\mathbb{Z}}$, and let $(\Omega,{\mathcal{F}},{\mathbb{P}})$ be a probability space. A $({\mathcal{B}}({\mathbb{T}})\otimes {\mathcal{F}}, {\mathcal{F}})$-measurable function $\theta:{\mathbb{T}}\times \Omega \to \Omega$ is called a *measurable dynamical system* if $\theta(0,\omega) =\omega$ and $\theta(t+s,\omega) = \theta(t, \theta(s, \omega))$ for all $t,s\in{\mathbb{T}}$ and $\omega\in\Omega$. We use the abbreviation $\theta_t\omega$ for $\theta(t,\omega)$. A measurable dynamical system is said to be *measure preserving* or *metric* if ${\mathbb{P}}\theta(t,A) = {\mathbb{P}}A$ for all $t\in{\mathbb{T}}$ and $A\in{\mathcal{F}}$, and such a dynamical system is called *ergodic* if for any $A\in{\mathcal{F}}$ satisfying $\theta_t A = A$ for all $t\in{\mathbb{T}}$, one has ${\mathbb{P}}A\in{\{0,1\}}$. A particular metric dynamical system, which naturally is used when dealing with (one-dimensional) stochastic differential equations, is generated by the Brownian motion. More precisely, $\Omega:= C_0({\mathbb{R}},{\mathbb{R}}):={\{\omega\in C({\mathbb{R}},{\mathbb{R}}): \omega(0)=0\}}$. Let $\Omega$ be equipped with the compact-open topology and the Borel $\sigma$-algebra ${\mathcal{F}}:={\mathcal{B}}(C_0({\mathbb{R}}, {\mathbb{R}}))$. Let ${\mathbb{P}}$ denote the Wiener probability measure on $(\Omega, {\mathcal{F}})$. The metric dynamical system is then given by the Wiener shift $\theta : {\mathbb{R}}\times \Omega\to\Omega$, defined by $\theta(t, \omega(\cdot)):= \omega(\cdot+t)-\omega(t)$, and it is well-known that $\theta$ is ergodic [@Arnold_98_1]. On $(\Omega,{\mathcal{F}})$, we have the natural filtration $${\mathcal{F}}_s^t:= \sigma{\big(\omega(u)-\omega(v): s\le u,v\le t\big)} {\quad \text{for all }\,}s\le t\,,$$ with $\theta_u^{-1} {\mathcal{F}}_s^t= {\mathcal{F}}_{s+u}^{t+u}$.
*Invariant measures.* For a given random dynamical system ($\theta,{{\varphi}}$), let $\Theta:{\mathbb{T}}\times \Omega\times X \to \Omega\times X$ denote the corresponding *skew product flow*, given by $\Theta(t,\omega,x):= (\theta_t \omega, {{\varphi}}(t,\omega)x)$. This is a measurable dynamical system on the extended phase space $\Omega\times X$. A probability measure $\mu$ on $(\Omega\times X,{\mathcal{F}}\otimes {\mathcal{B}})$ is said to be an *invariant measure* if
- $\mu(\Theta_t A)=\mu(A)$ for all $t\in{\mathbb{T}}$ and $A\in {\mathcal{F}}\otimes{\mathcal{B}}$,
- $\pi_\Omega \mu = {\mathbb{P}}$,
where $\pi_\Omega \mu$ denotes the marginal of $\mu$ on $(\Omega,{\mathcal{F}})$. If the metric space $X$ is a Polish space, i.e., it is separable and complete, then an invariant measure $\mu$ admits a ${\mathbb{P}}$-almost surely unique *disintegration* [@Arnold_98_1 Proposition 1.4.3], that is a family of probability measures $(\mu_\omega)_{\omega\in\Omega}$ with $$\mu(A) = \int_\Omega\int_X {\mathds{1}}_A(\omega,x) \,{\mathrm{d}}\mu_\omega(x) \,{\mathrm{d}}{\mathbb{P}}(\omega)\,.$$
*Random sets.* A function $\omega \mapsto M(\omega)$ taking values in the subsets of the phase space $X$ of a random dynamical system is called a *random set* if $\omega \mapsto d(x,M(\omega))$ is measurable for each $x \in X$, and we use the term $\omega$-fiber of $M$ for the set $M(\omega)$. We call $M$ *closed* or *compact* if all $\omega$-fibers are closed or compact, respectively. A random set $M$ is called *invariant* with respect to the random dynamical system $(\theta,{{\varphi}})$ if ${{\varphi}}(t,\omega)M(\omega) = M(\theta_t\omega)$ for all $t\in {\mathbb{R}}$ and $\omega\in\Omega$.
*Random attractors.* A nonempty, compact and invariant random set $\omega\mapsto A(\omega)$ is called *global random attractor* for a random dynamical system $(\theta,{{\varphi}})$ with metric state space $(X,d)$, if it attracts all bounded sets in the sense of pullback attraction, i.e., for all bounded sets $B\subset X$, one has $$\lim_{t\to\infty} {\operatorname{dist}}({{\varphi}}(t,\theta_{-t}\omega)B, A(\omega)) = 0 {\quad \text{for almost all }\,}\omega\in \Omega\,,$$ where ${\operatorname{dist}}(C,D):= \sup_{c\in C} d(c,D)$ is the *Hausdorff semi-distance* of $C$ and $D$. A global random attractor (given it exists) is always unique [@Crauel_94_1]. The existence of random attractors is proved via so-called absorbing sets [@Flandoli_96_1]. A bounded set $B\subset X$ is called *absorbing set* if for almost all $\omega\in\Omega$ and any bounded set $D\subset X$, there exists a time $T>0$ such that $${{\varphi}}(t,\theta_{-t}\omega)D \subset B {\quad \text{for all }\,}t\ge T\,.$$ Given an absorbing set $B$, it follows that there exists a global random attractor ${\{A(\omega)\}}_{\omega\in\Omega}$, given by $$A(\omega):=\bigcap_{\tau\ge 0} \overline{\bigcup_{t\ge \tau} {{\varphi}}(t,\theta_{-t} \omega)B} {\quad \text{for almost all }\,}\omega\in\Omega\,.$$
*Lyapunov exponents and Multiplicative Ergodic Theory.* Given a linear random dynamical system $(\theta,\Phi)$ in ${\mathbb{R}}^d$, a *Lyapunov exponent* is given by $$\lambda=\lim_{t\to\pm\infty}\frac{1}{|t|}\ln{\|\Phi(t,\omega)x\|} {\quad \text{for some }\,}\omega\in\Omega{\text{ and }}x\in{\mathbb{R}}^d\,.$$ The Multiplicative Ergodic Theorem [@Oseledets_68_1] shows that there are only finitely many Lyapunov exponents provided the random dynamical system is ergodic and fulfills an integrability condition. More precisely, consider a linear random dynamical system $(\theta:{\mathbb{T}}\times \Omega\to\Omega, \Phi:{\mathbb{T}}\times \Omega \to {\mathbb{R}}^{d\times d})$, suppose that $\theta$ is ergodic and $\Phi$ satisfies the integrability condition $$\sup_{t\in[0,1]} \ln^+{\big({\|\Phi(t,\cdot)^{\pm 1}\|}\big)}\in L^1({\mathbb{P}})\,,$$ here $\ln^+(x):= \max{\{0,\ln(x)\}}$. Then the Multiplicative Ergodic Theorem states that almost surely, there exist at most $d$ Lyapunov exponents $\lambda_1<\lambda_2<\dots<\lambda_p$ and fiber-wise decomposition $${\mathbb{R}}^d=O_1(\omega)\oplus O_2(\omega)\oplus\dots\oplus O_p(\omega) {\quad \text{for almost all }\,}\omega\in\Omega$$ into Oseledets subspaces $O_i\subset {\mathbb{R}}^d$ such that for all $i \in{\{1,\dots,p\}}$ and almost all $\omega\in\Omega$, one has $$\lim_{t\to\pm\infty}\frac{1}{|t|}\ln{\|\Phi(t,\omega)x\|}=\lambda_i {\quad \text{for all }\,}0\not= x\in O_i(\omega)\,.$$
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[^1]: The fourth author was supported by an EPSRC Career Acceleration Fellowship (2010–2015)
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abstract: 'In this paper, we establish the higher order convergence rates in periodic homogenization of viscous Hamilton-Jacobi equations, which is convex and grows quadratically in the gradient variable. We observe that although the nonlinear structure governs the first order approximation, the nonlinear effect is absorbed as an external source term of a linear equation in the second and higher order approximation. Moreover, we find that the geometric shape of the initial data has to be chosen carefully according to the effective Hamiltonian, in order to achieve the higher order convergence rates.'
address:
- 'Department of Mathematical Sciences, Seoul National University, Seoul 08826, Korea'
- 'Department of Mathematical Sciences, Seoul National University, Seoul 08826, Korea & Center for Mathematical Challenges, Korea Institute for Advanced Study, Seoul 02455, Korea'
author:
- Sunghan Kim
- 'Ki-Ahm Lee'
title: 'Higher Order Convergence Rates in Theory of Homogenization III: viscous Hamilton-Jacobi Equations'
---
Introduction {#section:intro}
============
This paper concerns the higher order convergence rates of the homogenization of viscous Hamilton-Jacobi equations. The model problem is of the form, $$\label{eq:HJ}
\begin{dcases}
u_t^{\varepsilon}- {\varepsilon}{\operatorname{tr}}\left( A\left(\frac{x}{{\varepsilon}}\right) D^2 u^{\varepsilon}\right) + H \left( Du^{\varepsilon}, \frac{x}{{\varepsilon}} \right) = 0 & \text{in }{{\mathbb R}}^n\times(0,\infty),\\
u^{\varepsilon}= g & \text{on }{{\mathbb R}}^n\times\{t=0\}.
\end{dcases}$$ Here the diffusion matrix $A$ is periodic and uniformly elliptic, and the Hamiltonian $H$ is periodic in the spatial variable while it is convex and grows quadratically in the gradient variable. The initial data $g$ will be chosen to have smooth solutions for the effective Hamilton-Jacobi equation. At the end of this paper, we shall extend the result to the fully nonlinear, viscous Hamilton-Jacobi equation in the form of $$\label{eq:HJ-nl}
\begin{dcases}
u_t^{\varepsilon}+ H \left( {\varepsilon}D^2 u^{\varepsilon}, Du^{\varepsilon},\frac{x}{{\varepsilon}}\right) = 0 & \text{in }{{\mathbb R}}^n\times(0,\infty),\\
u^{\varepsilon}= g & \text{on }{{\mathbb R}}^n\times\{t=0\}.
\end{dcases}$$
This paper is in the sequel of the authors’ previous works [@KL1] and [@KL2], where the higher order convergence rates were achieved in the periodic homogenization of fully nonlinear, uniformly elliptic and parabolic, second order PDEs. We found it interesting in the previous works that even if we begin with a nonlinear PDE at the first order approximation, we no longer encounter such a nonlinear structure in the second and the higher order approximations. Instead, we always obtain a linear PDE with an external source term, which can be interpreted as the nonlinear effect coming from the error that is left undetected in the previous step of the approximation.
The previous papers were concerned of uniformly elliptic (or parabolic) PDEs that are nonlinear in the second order derivatives, where the nonlinear perturbation is still made in the same order of the linear structure. A key difference in the current paper is that we impose a nonlinear structure (in the gradient term) that has quadratic growth at the infinity, so that this nonlinearity cannot be attained by order 1 perturbations of a linear structure. We believe that the quadratic growth condition can be generalized to superlinear growth condition, only if the solution of the corresponding effective problem is smooth enough.
Another interesting fact we found in studying Hamilton-Jacobi equations is that the geometric shape of the initial data turns out to play an important role in achieving higher order convergence rates. In particular, what we observe in this paper is that the geometric shape of the initial data has to be selected according to the nonlinear structure of the effective Hamiltonian, which to the best of our knowledge has not yet been addressed in any existing literature. The main reason for this requirement is to ensure the solution of the effective problem to be sufficiently smooth such that one can proceed with the approximation as much as one desires.
In this paper, we establish higher order convergence rates when the initial data is convex, while the Hamiltonian is convex. However, a natural question is if one can generalize one of these structure conditions, which seems to be an interesting yet challenging problem. We shall come back to this in the forthcoming paper.
The periodic homogenization of (viscous) Hamilton-Jacobi equations is by now considered to be standard, and one may consult the classical materials [@LPV] and [@E] for a rigorous justification. For the notion of viscosity solutions and the standard theory in this framework we refer to [@CIL] and [@CC].
For the recent development in the rate of convergence in periodic homogenization of (viscous) Hamilton-Jacobi equations, we refer to [@CDI], [@CCDG], [@M], [@MT], and the references therein. Nevertheless, this is the first work on the higher order convergence rates in the regime of (viscous) Hamilton-Jacobi equations. For the higher order convergence rates for other type of equations, we refer to [@KL1], [@KL2] and the references therein.
The paper is organized as follows. In Section \[section:assump\], we introduce basic notation used throughout this paper, and list up the standing assumptions regarding the main problem . From Section \[section:prelim\] to Section \[section:cor\], we are concerned with the homogenization problem of . In Section \[section:prelim\], we summarize some standard results on the cell problem and the effective Hamiltonian. In Section \[section:reg\], we establish the regularity theory of interior correctors in the slow variable. Based on this regularity theory, we construct the higher order interior correctors in Section \[section:cor\] and prove Theorem \[theorem:cr\], which is the first main result. Finally in Section \[section:nl\], we generalize this result to the homogenization of , and prove Theorem \[theorem:cr-nl\], which is the second main result.
Basic Notation and Standing Assumption {#section:assump}
======================================
Throughout the paper, we set $n\geq 1$ to be the spatial dimension. The parameters $\lambda$, $\Lambda$, $\alpha$, $\alpha'$, $\beta$, $\beta'$, $K$, $L$, and $\bar\mu$ will be fixed positive constants, unless stated otherwise. By ${{\mathbb Z}}^n$ we denote the space of $n$-tuple of integers. By ${{\mathbb S}}^n$ we denote the space of all symmetric $n\times n$ matrices. By $C^\infty(X; C^{k,\mu}(Y))$, we denote the space of functions $f=f(x,y)$ on $X\times Y$ such that $f(\cdot,y)\in C^\infty(X)$ for all $y\in Y$ and $\{D_x^k f(x,\cdot)\}_{x\in X}$ is uniformly bounded in $C^{k,\mu}(Y)$.
From Section \[section:prelim\] to Section \[section:cor\], we study the higher order convergence rates in homogenization of . Throughout these sections, we assume that the diffusion matrix $A$ satisfies the following, for any $y\in{{\mathbb R}}^n$.
(i) $A$ is periodic: $$\label{eq:A-peri}
A(y+k) = A(y).$$
(ii) $A$ is uniformly elliptic: $$\label{eq:A-ellip}
\lambda I \leq A(y) \leq \Lambda I.$$
(iii) $A\in C^{0,1}({{\mathbb R}}^n)$ and $$\label{eq:A-C01}
{\left\Arrowvert {A}\right\Arrowvert}_{C^{0,1}({{\mathbb R}}^n)} \leq K.$$
On the other hand, we shall assume that the Hamiltonian $H$ verifies the following, for any $(p,y)\in{{\mathbb R}}^n\times{{\mathbb R}}^n$.
(i) $H$ is periodic in $y$: $$\label{eq:H-peri}
H(p,y+k) = H(p,y),$$ for any $k\in{{\mathbb Z}}^n$.
(ii) $H$ has quadratic growth in $p$: $$\label{eq:H-quad}
\alpha |p|^2 - \alpha' \leq H(p,y) \leq \beta|p|^2 + \beta'.$$
(iii) $H$ is convex in $p$: $$\label{eq:H-convex}
H(tp + (1-t)q,y) \leq t H(p,y) + (1-t) H(q,y),$$ for any $0\leq t\leq 1$ and any $q\in{{\mathbb R}}^n$.
(iv) $H\in C^\infty({{\mathbb R}}^n;C^{0,1}({{\mathbb R}}^n))$ and $$\label{eq:H-Ck-C01}
{\left\Arrowvert {D_p^k H(p,\cdot)}\right\Arrowvert}_{C^{0,1}({{\mathbb R}}^n)} \leq K \left(1+|p|^{(2-k)_+}\right),$$ for any nonnegative integer $k$.
The assumptions on the initial data $g$ will be given in the beginning of Section \[section:cor\], since we need to derive the effective Hamiltonian beforehand. On the other hand, the structure conditions for will be given in the beginning of Section \[section:nl\].
Preliminaries {#section:prelim}
=============
Let us begin with the well-known cell problem for our model equation , stated as below. This lemma is by now considered to be standard (for instance, see [@E1] and [@E2]), since the diffusion coefficient $A$ is uniformly elliptic and the Hamiltonian $H$ is convex. Nevertheless, we shall present a proof for the reader’s convenience.
\[lemma:cell\] For each $p\in{{\mathbb R}}^n$, there exists a unique real number, $\gamma$, for which the following PDE, $$\label{eq:cell}
-{\operatorname{tr}}(A(y) D^2w) + H (D w + p,y) = \gamma \quad\text{in }{{\mathbb R}}^n,$$ has a periodic viscosity solution $w\in C^{2,\mu}({{\mathbb R}}^n)$ for any $0<\mu<1$. Moreover, we have $$\label{eq:cell-Linf}
\alpha |p|^2 - \alpha' \leq \gamma \leq \beta |p|^2 + \beta'.$$ Furthermore, a periodic solution $w$ of is unique up to an additive constant, and satisfies $$\label{eq:cell-C2a}
(1+ |p|) \left({\left\Arrowvert { w - w(0) }\right\Arrowvert}_{L^\infty({{\mathbb R}}^n)} + {\left\Arrowvert { Dw}\right\Arrowvert}_{L^\infty({{\mathbb R}}^n)} \right) + {\left\Arrowvert {D^2 w}\right\Arrowvert}_{C^\mu({{\mathbb R}}^n)} \leq C( 1 + |p|^2),$$ where $C>0$ depends only on $n$, $\lambda$, $\Lambda$, $\alpha$, $\alpha'$, $\beta$, $\beta'$, $K$ and $\mu$.
Throughout the proof, $C$ will denote a positive, generic constant that depends at most on $n$, $\lambda$, $\Lambda$, $\alpha$, $\alpha'$, $\beta$, $\beta'$, $K$ and $\mu$, unless stated otherwise. Moreover, we shall fix $0<\mu<1$.
Let $p\in{{\mathbb R}}^n$ be given. We know [*a priori*]{} that periodic viscosity solutions of , if any, are unique up to an additive constant. Suppose that $w'$ is another periodic viscosity solution of . Then $v = w-w'$ satisfies the following linearized equation, $$- {\operatorname{tr}}(A(y) D^2 v) + B(y) \cdot Dv = 0 \quad \text{in }{{\mathbb R}}^n,$$ where $B(y) = \int_0^1 D_p H( t D_y w + (1-t) D_y w' + p,y) dt$. Now that $v$ is bounded, we deduce from the Liouville theorem that $v$ is a constant function on ${{\mathbb R}}^n$.
Henceforth, we prove the existence of a unique real number, $\gamma$, such that the cell problem admits a periodic viscosity solution. The existence is proved by considering the following approximation problem, $$\label{eq:wd-pde}
- {\operatorname{tr}}(A(y) D^2 w^\delta) + H(D w^\delta + p, y) + \delta w^\delta = 0 \quad\text{in }{{\mathbb R}}^n,$$ for each $\delta>0$. Due to , we know that $-\delta (\alpha |p|^2 - \alpha')$ and $-\delta (\beta|p|^2+\beta')$ are a supersolution and, respectively, a subsolution of . Thus, the comparison principle yields a unique viscosity solution, $w^\delta$, of , satisfying $$\label{eq:dwd-Linf}
-\beta |p|^2 - \beta' \leq \delta w^\delta \leq -\alpha |p|^2 + \alpha',$$ on ${{\mathbb R}}^n$. The uniqueness of $w^\delta$ implies its periodicity, that is, $w^\delta (y+k) = w^\delta(y)$ for all $y\in{{\mathbb R}}^n$ and all $k\in{{\mathbb Z}}^n$.
Let us remark here that $w^\delta\in C^{0,1}({{\mathbb R}}^n)$ and $$\label{eq:wd-osc}
{\left\Arrowvert {D w^\delta}\right\Arrowvert}_{L^\infty({{\mathbb R}}^n)} \leq C(1 + |p|),$$ where $C>0$ depends only on $n$, $\alpha$, $\alpha'$, $\beta$ and $\beta'$. Note that the uniform Lipschitz estimate has nothing to do with the periodicity of $w^\delta$. In fact, one may use the weak Bernstein method [@B] to verify this uniform regularity, due to the structure conditions , , and .
Now that $w^\delta$ is periodic, we may deduce from the interior gradient estimate for viscous Hamilton-Jacobi equations that Hence, by periodicity, ${\operatornamewithlimits{osc}}_{{{\mathbb R}}^n} w^\delta \leq C(1+|p|)$, which yields that $w^\delta - w^\delta (0) \in C^{0,1}({{\mathbb R}}^n)$ and $$\label{eq:wtd-C01}
{\left\Arrowvert {w^\delta - w^\delta(0)}\right\Arrowvert}_{L^\infty({{\mathbb R}}^n)} + {\left\Arrowvert {Dw^\delta}\right\Arrowvert}_{L^\infty({{\mathbb R}}^n)} \leq C(1+|p|),$$ where $C>0$ depend only on $n$, $\alpha$, $\alpha'$, $\beta$, $\beta'$ and $K$. Here $K$ is the constant appearing in the regularity assumption .
Due to , and , we know that $${\left\Arrowvert { H(D w^\delta + p,\cdot) + \delta w^\delta }\right\Arrowvert}_{L^\infty({{\mathbb R}}^n)} \leq C(1+ |p|^2),$$ where $C>0$ depends only on $n$, $\alpha$, $\alpha'$, $\beta$, $\beta'$ and $K$. Considering the second and the third terms on the left hand side of as an external force, we may apply the interior $C^{2,\mu}$ estimates and use the periodicity of $w^\delta$ to derive that $w^\delta - w^\delta(0) \in C^{1,\mu}({{\mathbb R}}^n)$ and $$\label{eq:wtd-C1a}
(1 + |p|) \left( {\left\Arrowvert {w^\delta - w^\delta(0)}\right\Arrowvert}_{L^\infty({{\mathbb R}}^n)} + {\left\Arrowvert {Dw^\delta}\right\Arrowvert}_{L^\infty({{\mathbb R}}^n)} \right)+ \left[D w^\delta\right]_{C^\mu({{\mathbb R}}^n)} \leq C(1+|p|^2).$$
Now the $C^{1,\mu}$ regularity of $w^\delta$ yields that $${\left\Arrowvert { H(D w^\delta + p,\cdot) + \delta w^\delta }\right\Arrowvert}_{C^\mu ({{\mathbb R}}^n)} \leq C(1+ |p|^2).$$ Hence, it follows from the interior $C^{2,\mu}$ estimates and the periodicity of $w^\delta$ that $w^\delta - w^\delta(0) \in C^{2,\mu}({{\mathbb R}}^n)$ and $$\label{eq:wtd-C1a}
(1 + |p|) \left( {\left\Arrowvert {w^\delta - w^\delta(0)}\right\Arrowvert}_{L^\infty({{\mathbb R}}^n)} + {\left\Arrowvert {Dw^\delta}\right\Arrowvert}_{L^\infty({{\mathbb R}}^n)} \right)+ {\left\Arrowvert {D^2 w^\delta}\right\Arrowvert}_{C^\mu({{\mathbb R}}^n)} \leq C(1+|p|^2).$$
Due to the compactness of both of the sequences $\{w^\delta - w^\delta(0)\}_{\delta>0}$ and $\{-\delta w^\delta \}_{\delta>0}$ in $C^{2,\mu}({{\mathbb R}}^n)$, we know that $w^\delta - w^\delta(0) {\rightarrow}w$ and $-\delta w^\delta {\rightarrow}\gamma$ in $C^{2,\mu'} ({{\mathbb R}}^n)$, for any $0<\mu'<\mu$, for some $w\in C^{2,\mu}({{\mathbb R}}^n)$ and some $\gamma\in{{\mathbb R}}$, along a subsequence. Now that viscosity solutions are stable under the uniform convergence, we know that $w$ is a viscosity solution of with the limit $\gamma$ on the right hand side. This proves the existence part of Lemma \[lemma:cell\].
To investigate the uniqueness of $\gamma$, we suppose towards a contradiction that there is another real number $\gamma'$, corresponding to the same $p$, such that has a periodic viscosity solution, say $w'$. Without losing any generality, let us assume $\gamma>\gamma'$. Then it is easy to see that $w'$ is a strict subsolution of . However, due to the periodicity of $w'-w$, $w'-w$ attains a local maximum at some point, whence we arrive at a contradiction. Thus, $\gamma$ must be unique.
The inequality follows immediately from the inequality and the fact that $-\delta w^\delta {\rightarrow}\gamma$ uniformly in ${{\mathbb R}}^n$. To see that the estimate holds, we first observe from the convergence of $w^\delta - w^\delta(0) {\rightarrow}w$ in $C^{2,\mu'}({{\mathbb R}}^n)$, for any $0<\mu'<\mu$, and the estimate that $w\in C^{2,\mu}({{\mathbb R}}^n)$ and satisfies . Note that we used $w(0) = 0$, which follows from the construction of $w$. Now if $w'$ is another periodic viscosity solution of , then due to the uniqueness that we have shown in the beginning of this proof, we have $w' - w'(0) = w$. Therefore, $w'$ satisfies , which completes the proof of this lemma.
Due to the uniqueness of $\gamma$ in Lemma \[lemma:cell\], we may define a functional $\bar{H}:{{\mathbb R}}^n{\rightarrow}{{\mathbb R}}$ in such a way that for each $p\in{{\mathbb R}}^n$, $\bar{H}(p)$ is the unique real number for which the following PDE, $$\label{eq:w-pde}
- {\operatorname{tr}}(A(y) D^2 w) + H( Dw + p, y ) = \bar{H}(p)\quad\text{in }{{\mathbb R}}^n,$$ has a periodic solution in $C^{2,\mu}({{\mathbb R}}^n)$ (for any $0<\mu<1$). Moreover, the second part of Lemma \[lemma:cell\] yields a functional $w:{{\mathbb R}}^n\times{{\mathbb R}}^n{\rightarrow}{{\mathbb R}}$ such that for each $p\in{{\mathbb R}}^n$, $w(p,\cdot)\in C^{2,\mu}({{\mathbb R}}^n)$ (for any $0<\mu<1$) is the unique periodic viscosity solution of that is normalized so as to satisfy $$\label{eq:w-0}
w(p,0) = 0.$$
Let us list up some basic properties of $\bar{H}$ that were already found in [@E]. We provide the proof for the sake of completeness.
\[lemma:Hb\] $\bar{H}$ satisfies the following properties.
(i) $\bar{H}$ has the same quadratic growth as that of $H$: $$\label{eq:Hb-quad}
\alpha|p|^2 - \alpha' \leq \bar{H}(p) \leq \beta|p|^2 + \beta',$$ for any $p\in{{\mathbb R}}^n$.
(ii) $\bar{H}$ is also convex: $$\label{eq:Hb-convex}
\bar{H} (tp + (1-t)q) \leq t \bar{H}(p) + (1-t) \bar{H}(q),$$ for any $0\leq t\leq 1$, and any $p,q\in{{\mathbb R}}^n$.
(iii) $\bar{H} \in C_{loc}^{0,1}({{\mathbb R}}^n)$ and $$\label{eq:Hb-C01}
| \bar{H}(p) - \bar{H}(q) | \leq C(1+|p| + |q|)|p-q|,$$ where $C>0$ depends only on $n$, $\lambda$, $\Lambda$, $\alpha$, $\alpha'$, $\beta$, $\beta'$ and $K$.
Notice that follows immediately from . Thus, we only prove ($\bar{H2}$) and .
For the notational convenience, let us write $w_p(y)= w(p,y)$. To prove , we assume to the contrary that there are some $p,q\in{{\mathbb R}}^n$ and $0<t<1$ such that $$\label{eq:Hb-false}
t\bar{H}(p) + (1-t )\bar{H}(q) < \bar{H}(tp + (1-t)q).$$ For the notational convenience, let us write $r = tp + (1-t)q$ and $\tilde{w}_r = tw_p + (1-t)w_q$. Then due to and , one can easily deduce that $\tilde{w}_r$ is a periodic viscosity solution of $$\label{eq:wtr-pde}
- {\operatorname{tr}}(A(y) D^2 \tilde{w}_r) + H (D \tilde{w}_r + r,y) < \bar{H}(r)\quad\text{in }{{\mathbb R}}^n.$$ In other words, $\tilde{w}_r$ is a strict viscosity subsolution of the PDE that $w_r$, which is precisely with $p$ replaced by $r$. Therefore, it follows from the comparison principle that $\tilde{w}_r - w_r$ cannot attain any local maximum. However, as $\tilde{w}_r - w_r$ being a non-constant continuous periodic function, it surely attains local maximum at some point, whence we arrive at a contradiction. Therefore, we must have for any $0\leq t\leq 1$ and any $p,q\in{{\mathbb R}}^n$.
Finally let us prove . To do so, we go back to the penalized problem . Analogous with the notation $w_p$, let us denote by $w_p^\delta$ the unique viscosity solution of corresponding to $p$. Due to the uniform gradient estimate and the regularity assumption , we have $$| H( Dw_p^\delta +p,y) - H(D w_q^\delta + q,y) | \leq C( 1+ |p| + |q| ) |p-q|,$$ where $C>0$ depends only on $n$, $\lambda$, $\Lambda$, $\alpha$, $\alpha'$, $\beta$, $\beta'$ and $K$. Therefore, we have $$\begin{split}
- {\operatorname{tr}}(A(y) D^2 w_p^\delta) + H(D w_p^\delta + q, y) + \delta w_p^\delta \leq C(1+|p| + |q|) |p-q|\quad\text{in }{{\mathbb R}}^n,
\end{split}$$ in the viscosity sense. In other words, $w_p^\delta - \delta^{-1} C ( 1+ |p| + |q|)|p-q|$ is a viscosity subsolution of with $p$ replaced by $q$. Hence, it follows from the comparison principle that $$\delta w_p^\delta - \delta w_q^\delta \leq C (1 + |p| + |q|)|p-q|,$$ on ${{\mathbb R}}^n$. Passing to the limit $\delta {\rightarrow}0$ in the last inequality, we arrive at $$\bar{H}(p) - \bar{H}(q) \leq C(1+ |p| + |q|) | p - q|$$ Similarly, one may also obtain that $$\bar{H}(q) - \bar{H}(p) \leq C(1+|p| + |q|) | p - q|,$$ proving . This completes the proof of Lemma \[lemma:Hb\].
Regularity in the Slow Variable {#section:reg}
===============================
In this section, we shall investigate the regularity of $\bar{H}$ and $w$ in the slow variable $p$. Such a regularity has been established in the authors’ previous works [@KL1] and [@KL2], for fully nonlinear elliptic and, respectively, parabolic PDEs. Let us first observe the continuity of $w$ in $p$ variable.
\[lemma:w-Linf-C2a\] $w\in C({{\mathbb R}}^n;C^{2,\mu}({{\mathbb R}}^n))$, for any $0<\mu<1$, and $$\label{eq:w-Linf-C2a}
(1+|p|) \left( {\left\Arrowvert {w (p,\cdot)}\right\Arrowvert}_{L^\infty({{\mathbb R}}^n)} {\left\Arrowvert { D_y w(p,\cdot)}\right\Arrowvert}_{L^\infty({{\mathbb R}}^n)} \right) + {\left\Arrowvert { D_y^2 w(p,\cdot) }\right\Arrowvert}_{C^\mu({{\mathbb R}}^n)} \leq C(1+|p|^2),$$ where $C>0$ depends only on $n$, $\lambda$, $\Lambda$, $\alpha$, $\alpha'$, $\beta$, $\beta'$, $K$ and $\mu$.
Let us fix $0<\mu<1$. The estimate follows immediately from and the choice of $w$ that $w(p,0) = 0$. Thus, we prove that $w$ is continuous in $p$ variable with respect to the $C^{2,\mu}$ norm in $y$ variable.
Let $\{p_k\}_{k=1}^\infty$ be a sequence of vectors in ${{\mathbb R}}^n$ converging to some $p_0\in{{\mathbb R}}^n$ as $k{\rightarrow}\infty$. Let us write, for the notational convenience, $w_k (y) = w(p_k,y)$ and $\gamma_k = \bar{H}(p_k)$ for $k=0,1,2,\cdots$. We already know from that $\gamma_k{\rightarrow}\gamma_0$ as $k{\rightarrow}\infty$. Hence, it suffices to prove that $w_k {\rightarrow}w_0$ in $C^{2,\mu'}({{\mathbb R}}^n)$ as $k{\rightarrow}\infty$, for any $0<\mu'<\mu$.
Due to , we know that $\{w_k\}_{k=1}^\infty$ is uniformly bounded in $C^{2,\mu}({{\mathbb R}}^n)$, for any $0<\mu<1$. Now that $w_k$ is periodic for all $k=1,2,\cdot,s$, the Arzela-Ascoli theorem yields that for any subsequence $\{v_k\}_{k=1}^\infty \subset \{w_k\}_{k=1}^\infty$ there are a further subsequence $\{v_{k_i}\}_{i=1}^\infty$ and a periodic function $v\in C^{2,\mu}({{\mathbb R}}^n)$ such that $v_{k_i}{\rightarrow}v$ in $C^{2,\mu}({{\mathbb R}}^n)$, for any $0<\mu<1$, as $i{\rightarrow}\infty$. Now that $p_{k_i}{\rightarrow}p_0$ and $\gamma_{k_i}{\rightarrow}\gamma_0$ as $i{\rightarrow}\infty$, we deduce from the stability of viscosity solutions that $w'$ and $\gamma'$ satisfies $$- {\operatorname{tr}}(A(y) D^2 v) + H(Dv + p_0,y) = \gamma_0 \quad \text{in }{{\mathbb R}}^n.$$ Since $v(0) = 0$, the second part of Lemma \[lemma:cell\] implies that $v = w_0$. This shows that any subsequence of $\{w_k\}_{k=1}^\infty$ contains a further subsequence that converges to $w_0$ in $C^{2,\mu'}({{\mathbb R}}^n)$, for any $0<\mu'<\mu$. Therefore, $w_k {\rightarrow}w_0$ in $C^{2,\mu'}({{\mathbb R}}^n)$, for any $0<\mu'<\mu$ as $k{\rightarrow}\infty$, which completes the proof.
Next we prove that $\bar{H}$ and $w$ are continuously differentiable in $p$.
\[lemma:w-C1-C2a\] $\bar{H} \in C^1({{\mathbb R}}^n)$ and $$\label{eq:Hb-C1}
| D_p \bar{H}(p)| \leq C(1+|p|),$$ where $C>0$ depends only on $n$, $\lambda$, $\Lambda$, $\alpha$, $\alpha'$, $\beta$, $\beta'$ and $K$. Moreover, $w\in C^1({{\mathbb R}}^n; C^{2,\mu}({{\mathbb R}}^n))$, for any $0<\mu<1$, such that for any $L>0$ and any $p\in B_L$, $$\label{eq:w-C1-C2a}
{\left\Arrowvert { D_p w(p,\cdot)}\right\Arrowvert}_{C^{2,\mu}({{\mathbb R}}^n)} \leq C_L,$$ where $C_L>0$ depends only on $n$, $\lambda$, $\Lambda$, $\alpha$, $\alpha'$, $\beta$, $\beta'$, $K$, $\mu$ and $L$.
Let us fix $0<\mu<1$. Throughout this proof, we shall write by $C_{*,\cdots,*}$ a positive constant depending at most on the parameters on the subscripts, if any, as well as $n$, $\lambda$, $\Lambda$, $\alpha$, $\alpha'$, $\beta$, $\beta'$, $K$ and $\mu$. We will also let it differ from one line to another, unless stated otherwise.
Fix $L>0$, $p\in B_L$, $0<\mu<1$ and $1\leq k\leq n$. Write $w_h (y) = w(p+h e_k,y)$ and $\gamma_h = \bar{H}(p+he_k)$ for any $h\in{{\mathbb R}}$ with $|h|\leq 1$. Also write $W_h (y) = h^{-1} (w_h(y) - w_0(y))$, and $\Gamma_h = h^{-1}(\gamma_h - \gamma_0)$. Then $W_h$ turns out to be a periodic viscosity solution to $$\label{eq:Wh-pde}
- {\operatorname{tr}}(A(y) D^2 W_h) + B_h(y)\cdot (D W_h + e_k) = \Gamma_h\quad\text{in }{{\mathbb R}}^n,$$ where $$B_h(y) = \int_0^1 D_p H( t D_y w_h + (1-t) D_y w_0 + p + the, y)dt.$$
It follows from and that $B_h \in C^\mu({{\mathbb R}}^n)$ and $$\label{eq:Bh-Ca}
{\left\Arrowvert { B_h }\right\Arrowvert}_{C^\mu({{\mathbb R}}^n)} \leq C( 1+ |p|),$$ for any $h\in{{\mathbb R}}$ with $|h|\leq 1$. Moreover, we know from that $$| \Gamma_h | \leq C(1 + |p|),$$ for any $h\in{{\mathbb R}}$ with $0<|h|\leq 1$. Let us point out that the constant $C$ in the last inequality for $\Gamma_h$ depends only on $n$, $\alpha$, $\alpha'$, $\beta$, $\beta'$ and $K$.
One may notice that belongs to the same class of , whence it follows from Lemma \[lemma:cell-lin\] below that $W_h\in C^{2,\mu}({{\mathbb R}}^n)$ and $$\label{eq:Wh-C2a}
{\left\Arrowvert { W_h }\right\Arrowvert}_{C^{2,\mu}({{\mathbb R}}^n)} \leq C_L,$$ for any $h\in{{\mathbb R}}$ with $0<|h|\leq 1$. On the other hand, from the fact that Lemma \[lemma:w-Linf-C2a\] implies $D w_h {\rightarrow}D w_0$ in $C^{1,\mu}({{\mathbb R}}^n)$, we know that $B_h {\rightarrow}B_0$ in $C^\mu({{\mathbb R}}^n)$, where $B_0$ is defined by $$B_0(y) = D_p H (D_y w_0 + p, y).$$ As with the estimate , we also know that $$\label{eq:B0-Ca}
{\left\Arrowvert { B_0 }\right\Arrowvert}_{C^\mu({{\mathbb R}}^n)} \leq C(1+|p|).$$
According to the Arzela-Ascoli theorem, there is some $W_0\in C^{2,\mu}({{\mathbb R}}^n)$ such that $W_h {\rightarrow}W_0$ in $C^{1,\mu'}({{\mathbb R}}^n)$ for any $0<\mu'<\mu$, along a subsequence. Moreover, we may choose $\Gamma_0\in{{\mathbb R}}$ such that $\Gamma_h {\rightarrow}\Gamma_0$ along a further subsequence. Then by the stability of viscosity solutions, $W_0$ becomes a periodic solution to $$\label{eq:W0-pde}
- {\operatorname{tr}}(A(y) D^2 W_0) + B_0 (y) \cdot (D_y W_0 + e_k) = \Gamma_0\quad\text{in }{{\mathbb R}}^n.$$
Now that belongs to the same class of , it follows from Lemma \[lemma:cell-lin\] below that $\Gamma_0$ is unique, and satisfies $$\label{eq:Gamma-Linf}
|\Gamma_0| \leq C(1+|p|),$$ due to . From the uniqueness of the limit $\Gamma_0$, we infer that $\Gamma_h {\rightarrow}\Gamma_0$ without extracting any subsequence. By definition, $\Gamma_0 = D_{p_k} \bar{H}(p)$.
Moreover, since any limit $W_0$ of $\{W_h\}_{0<|h|\leq 1}$ satisfies $W_0(0) = 0$, we also have from the last part of Lemma \[lemma:cell-lin\] below that $W_0$ is unique, and belongs to $C^{2,\mu}({{\mathbb R}}^n)$, with the estimate $$\label{eq:W0-C2a}
{\left\Arrowvert {W_0}\right\Arrowvert}_{C^{2,\mu}({{\mathbb R}}^n)} \leq C_L.$$ Owing to the uniqueness of the limit $W_0$, again we conclude that $W_h{\rightarrow}W_0$ in $C^{2,\mu}({{\mathbb R}}^n)$ along the full sequence, which implies that $W_0 = D_{p_k} w(p,\cdot)$.
The continuity of $D_{p_k} \bar{H}$ and $D_{p_k} w$ in variable $p$ can be proved similarly as in the proof of Lemma \[lemma:w-Linf-C2a\]. To avoid repeating arguments, we omit the details and leave this part to the reader.
\[lemma:cell-lin\] Let $B\in C^\mu({{\mathbb R}}^n)$ be a periodic, vector-valued mapping. Then for each $p\in{{\mathbb R}}^n$, there exists a unique real number, $\gamma$, for which the following PDE, $$\label{eq:cell-lin}
-{\operatorname{tr}}(A(y) D^2 v) + B(y)\cdot (D v + p) = \gamma\quad\text{in }{{\mathbb R}}^n,$$ admits a periodic viscosity solution $v\in C^{2,\mu}({{\mathbb R}}^n)$. Moreover, $\gamma$ satisfies $$\label{eq:cell-lin-Linf}
|\gamma| \leq |p| {\left\Arrowvert {B}\right\Arrowvert}_{L^\infty({{\mathbb R}}^n)}.$$ Furthermore, a periodic viscosity solution $v$ of is unique up to an additive constant, and satisfies $$\label{eq:cell-lin-C2a}
{\left\Arrowvert { v - v(0) }\right\Arrowvert}_{C^{2,\mu}({{\mathbb R}}^n)} \leq C |p|,$$ where $C>0$ depends only on $n$, $\lambda$, $\Lambda$, $\mu$ and ${\left\Arrowvert {B}\right\Arrowvert}_{C^\mu({{\mathbb R}}^n)}$.
The proof is essentially the same with that of Lemma \[lemma:cell\], and hence it is omitted.
In what follows, let us write $\bar{B}(p) = D_p \bar{H}(p)$, $v(p,y) = D_pw(p,y)$ and $$\label{eq:b}
B(p,y) = D_p H ( D_y w(p,y) + p, y).$$ In view of the proof of Lemma \[lemma:w-C1-C2a\], we may understand $\bar{B}(p)$ as the unique real vector in ${{\mathbb R}}^n$ for which the following (decoupled) system, $$\label{eq:v-pde}
-{\operatorname{tr}}(A(y) D^2 v) + B(p,y)\cdot ( D_y v + I ) = \bar{B}(p),$$ has a periodic viscosity solution, where $I$ is the identity matrix in ${{\mathbb S}}^n$. Moreover, $v(p,\cdot)$ can be considered as the unique periodic viscosity solution of such that $$\label{eq:v-0}
v(p,0) = 0.$$
It is remarkable that after linearization in , we end up with a cell problem whose gradient part has a linear growth, as shown in . Moreover, one may expect that the linear structure of the “new” cell problem will be preserved throughout the linearization we do in the future to obtain higher regularity of $\bar{H}$ and $w$ in $p$. This is the brief idea behind the proof for the following proposition. One may find a similar proposition for uniformly elliptic, fully nonlinear PDEs in the authors’ previous work [@KL1] and [@KL2].
\[lemma:w-Ck-C2a\] $\bar{H}\in C^\infty({{\mathbb R}}^n)$ and $w\in C^\infty({{\mathbb R}}^n;C^{2,\mu}({{\mathbb R}}^n))$, for any $0<\mu<1$, such that for any $k=0,1,2,\cdots$, any $L>0$ and any $p\in B_L$, $$\label{eq:w-Ck-C2a}
\left| D_p^k \bar{H}(p) \right| + {\left\Arrowvert { D_p^k w(p,\cdot) }\right\Arrowvert}_{C^{2,\mu}({{\mathbb R}}^n)} \leq C_{k,L},$$ where $C_{k,L}>0$ depends only on $n$, $\alpha$, $\alpha'$, $\beta$, $\beta'$, $K$, $\mu$, $k$ and $L$.
We follow the proof of Lemma \[lemma:w-C1-C2a\]. Due to Lemma \[lemma:w-C1-C2a\] and the regularity assumption , we already know that $B\in C^1({{\mathbb R}}^n;C^{2,\mu}({{\mathbb R}}^n))$, for any $0<\mu<1$, with $B$ defined in . Thus, in order to run the same argument in the proof of Lemma \[lemma:w-C1-C2a\], we need the Lipschitz regularity of $\bar{B} = D_p\bar{H}$ in $p$. However, this can be shown as in the proof for of Lemma \[lemma:Hb\]. This is because we can also understand the constant vector $\bar{B}(p)$ as the limit of $\{-\delta v^\delta\}_{\delta>0}$, with $v^\delta$ being the unique periodic viscosity solution of $$\label{eq:vd-pde}
-{\operatorname{tr}}(A(y) D^2 v^\delta) + B(p,y) \cdot (D_y v^\delta + I) + \delta v^\delta = 0\quad\text{in }{{\mathbb R}}^n.$$
Once we know that $\bar{B}$ is Lipschitz in $p$, then it follows from Lemma \[lemma:w-C1-C2a\] and the elliptic regularity theory that the difference quotient $V_h = h^{-1}( v_h - v_0)$, being a periodic viscosity solution of $$\label{eq:Vh-pde}
-{\operatorname{tr}}(A(y) D^2 V_h) + B_h(y) \cdot D_y V_h + B_h(y) \cdot (D_y v_0(y) + I) = \bar{B}_h\quad\text{in }{{\mathbb R}}^n,$$ with $v_h = v(p+he_k,\cdot)$, $B_h = B(p+he_k,\cdot)$, $B_h = h^{-1} (B_h - B_0)$ and $\bar{B}_h = h^{-1}(\bar{B}_h - \bar{B}_0)$, is uniformly bounded in $C^{2,\mu}({{\mathbb R}}^n)$. Hence, we deduce from the stability of viscosity solutions that any pair $(V_0,\bar{B}_0)$ of $\{V_h\}_{0<|h|\leq 1}$ and, respectively, $\{\bar{B}_h\}_{0<|h|\leq 1}$ must satisfy $$\label{eq:V0-pde}
-{\operatorname{tr}}(A(y) D^2 V_0) + B_0(y) \cdot D_y V_0 + B_0(y) \cdot (D_y v_0(y) + I) = \bar{B}_0\quad\text{in }{{\mathbb R}}^n.$$
Since belongs to the same class of , we know from Lemma \[lemma:cell-lin\] that $V_0$ and $\bar{B}_0$ are unique. Thus, we derive the differentiability of $\bar{B}$ and $v$ in $p$. Arguing as in the proof of Lemma \[lemma:w-Linf-C2a\], we may also observe that $D_p \bar{B}$ and $D_p v$ are continuous in $p$.
One may now iterate this argument to obtain higher regularity of $\bar{B}$ and $v$ in $p$, which automatically implies that of $\bar{H}$ and $w$. We leave out the details to the reader.
Interior Corrector and Higher Order Convergence Rate {#section:cor}
====================================================
In this section, we construct the higher order interior correctors for the homogenization problem , based on the regularity result achieved in Section \[section:reg\].
We begin with the effective Hamilton-Jacobi equation for , which is given by $$\label{eq:eff-HJ}
\begin{dcases}
{\partial}_t \bar{u}_0 + \bar{H} ( D\bar{u}_0 ) = 0 & \text{in }{{\mathbb R}}^n\times(0,\infty),\\
\bar{u}_0 = g & \text{on }{{\mathbb R}}^n\times\{t=0\}.
\end{dcases}$$ The characteristic curve, which starts from $x\in{{\mathbb R}}^n$, is given by $$\label{eq:char}
\xi(t;x) = x + D_p \bar{H} ( D_x g(x)) t.$$ Note that this is indeed a line with direction $D_p \bar{H} ( D_x g(x))$. Moreover, the gradient of $\bar{u}$ is constant along this curve. To be specific, we have $$\label{eq:char-Dub}
D_x\bar{u}(\xi(t;x),t) = D_x g(x).$$
It is noteworthy that the initial data, $g$, does not play any role when deriving the effective Hamiltonian $\bar{H}$, as shown in Section \[section:reg\]. This allows us to choose the initial data $g$ [*a posteriori*]{} so as to make sure that $$\label{eq:char-inj}
\{\xi(t;x) : t>0\} \bigcap \{ \xi(t;x') : t>0\} = \emptyset,$$ if and only if $x\neq x'$, as well as that $$\label{eq:char-surj}
\bigcup_{x\in{{\mathbb R}}^n} \{\xi(t;x) : t>0\} = {{\mathbb R}}^n.$$
One may easily observe that there are infinitely many initial data $g$ that satisfy the conditions and , once $\bar{H}$ is determined. A trivial example is any affine function whose gradient is a non-vanishing point of $D_p \bar{H}$. Note that the non-vanishing set of $D_p \bar{H}$ is always open and non-empty, since $\bar{H}$ is convex and grows quadratically at the infinity. A rather non-trivial example is any smooth, convex and globally Lipschitz function whose gradients are contained in the non-vanishing set of $D_p \bar{H}$.
Once we have the initial data $g$, we know from the characteristic equations for that $\bar{u}_0\in C^\infty({{\mathbb R}}^n\times[0,\infty))$. Setting $$\label{eq:Bb}
\bar{B}(x,t) = D_p \bar{H}(D_x \bar{u}_0(x,t)),$$ we obtain $\bar{B} \in C^\infty({{\mathbb R}}^n\times[0,\infty))$, according to Lemma \[lemma:w-Ck-C2a\].
In order to have a regular solution for the first order linear PDE whose drift term is associated with $\bar{B}$, we require that $$\label{eq:Bb-0}
\bar{B} (x,t) \neq 0,$$ for any $(x,t)\in{{\mathbb R}}^n\times(0,\infty)$. Since the image of $\bar{B}$ on ${{\mathbb R}}^n\times(0,\infty)$ coincides with that of $D_p \bar{H} ( D_x g)$ on ${{\mathbb R}}^n$, we ask $D_x g$ not to be the critical points of $D_p \bar{H}$.
Let us list up the conditions for $g$ to be imposed in the rest of this paper:
(i) $g$ is convex: $$\label{eq:g-convex}
g(tx+ (1-t)x') \leq t g(x) + (1-t) g(x'),$$ for any $0\leq t\leq 1$ and any $x,x'\in{{\mathbb R}}^n$.
(ii) $g \in C^\infty({{\mathbb R}}^n)\cap {\operatorname{Lip}}({{\mathbb R}}^n)$, and there is $L>0$ such that $$\label{eq:g-Ck}
{\left\Arrowvert {D_x^k g}\right\Arrowvert}_{L^\infty({{\mathbb R}}^n)} \leq L,$$ for any $k=1,2,\cdots$. Moroever, $g$ is normalized such that $$\label{eq:g-0}
g(0) = 0.$$
(iii) $D_x g$ is not the critical points of $D_p \bar{H}$: $$\label{eq:DHDg-0}
D_p \bar{H} (D_x g(x)) \neq 0,$$ for any $x\in{{\mathbb R}}^n$.
Under the assumptions – on $g$, altogether with the properties – and of $\bar{H}$, we know from the standard regularity theory for Hamilton-Jacobi equations that $\bar{u}_0 \in C^\infty({{\mathbb R}}^n\times[0,\infty))$, and in particular, we have, for each $i,j=0,1,2,\cdots$, and any $T>0$, $$\label{eq:ub0-Ck}
\left| D_x^i{\partial}_t^j \bar{u}_0(x,t) \right| \leq C_{i,j,T},$$ uniformly for all $(x,t)\in{{\mathbb R}}^n\times[0,T]$, where $C_{i,j,T}$ is a positive constant depending at most on $n$, $\alpha$, $\alpha'$, $\beta$, $\beta'$, $K$, $L$, $i$, $j$ and $T$.
Moreover, due to and , we know that $\bar{B} \in C^\infty({{\mathbb R}}^n\times[0,\infty))$ and, for each $i,j=0,1,2,\cdots$, and any $T>0$, $$\label{eq:Bb-Ck}
\left| D_x^i {\partial}_t^j \bar{B} (x,t) \right| \leq C_{i,j,T},$$ uniformly for all $(x,t)\in {{\mathbb R}}^n \times[0,T]$, where $C_{i,j,T}$ is another positive constant determined by the same parameters listed above.
In what follows, we shall seek a sequence of the interior correctors for the homogenization problem . The first order interior corrector $w_1$ will be in the form of $$\label{eq:w1}
w_1 (x,t,y) = \phi_1(x,t,y) + \bar{u}_1(x,t),$$ where $\phi_1$ denotes $$\label{eq:phi1}
\phi_1(x,t,y) = w(D_x \bar{u}_0(x,t),y),$$ with $w = w(p,y)$ being the periodic (viscosity) solution of normalized so as to satisfy . Here $\bar{u}_1$ is an effective data that is not determined yet. Let us remark that one may choose $\bar{u}_1$ by any regular data, if one stops seeking interior correctors at this step. However, if one would like to go further and construct the second order corrector $w_2$, one needs to select $\bar{u}_1$ specifically by the solution of an effective limit equation, which arises from the solvability condition of $w_2$.
We will continuously observe such a relationship between the consecutive correctors. In fact, the $k$-th order interior corrector $w_k$, for $k\geq 2$, will be in the form of $$\label{eq:wk}
w_k (x,t,y) = \phi_k(x,t,y) + \chi(x,t,y)\cdot D_x\bar{u}_{k-1}(x,t) + \bar{u}_k(x,t),$$ where $\phi_k(x,t,\cdot)$ will be the periodic viscosity solution of a certain cell problem normalized so as to satisfy $\phi_k(x,t,0) = 0$, and $\chi:{{\mathbb R}}^n\times[0,\infty)\times{{\mathbb R}}^n{\rightarrow}{{\mathbb R}}^n$ will be defined by $$\label{eq:chi}
\chi(x,t,y) = v( D_x\bar{u}_0(x,t),y),$$ with $v=v(p,y)$ being the periodic solution of normalized so as to satisfy . Here $\bar{u}_{k-1}$ will be determined specifically such that the cell problem for $\phi_k$ is solvable, while $\bar{u}_k$ will be “free” to choose before one tries to construct the $(k+1)$-th corrector $w_{k+1}$.
It is noteworthy that, owing to Lemma \[lemma:w-Ck-C2a\], we have $\chi \in C^\infty({{\mathbb R}}^n\times[0,\infty);C^{2,\mu}({{\mathbb R}}^n))$ and, for any $i,j=0,1,2,\cdots$ and any $T>0$, $$\label{eq:chi-Ck-C2a}
{\left\Arrowvert { D_x^i {\partial}_t^j \chi(x,t,\cdot)}\right\Arrowvert}_{C^{2,\mu}({{\mathbb R}}^n)} \leq C_{i,j,R,T},$$ uniformly for all $(x,t)\in {{\mathbb R}}^n \times[0,T]$. In addition, we know from and that for each $(x,t)\in{{\mathbb R}}^n\times(0,\infty)$, $\chi(x,t,\cdot)$ is the unique periodic viscosity solution of $$\label{eq:chi-pde}
-{\operatorname{tr}}(A(y) D_y^2 \chi) + B(x,t,y)\cdot( D_y\chi + I) = \bar{B}(x,t)\quad\text{in }{{\mathbb R}}^n,$$ which also satisfies $$\label{eq:chi-0}
\chi(x,t,0) = 0.$$
For the rest of this section, we will justify the existence of the higher order interior correctors in a rigorous way. The corresponding work has been done by the authors in [@KL1] and [@KL2] in the framework of fully nonlinear, uniformly elliptic, second order PDEs in non-divergence form.
To simplify the notation, let us write $$\label{eq:w0}
w_0(x,t,y) = \bar{u}_0(x,t),$$ and by $W_k$, for $k=0,1,2,\cdots$, the vector-valued mapping, $$\label{eq:Wk}
W_k (x,t,y) = D_y w_{k+1} (x,t,y) + D_x w_k(x,t,y).$$ Note from , and that $$\label{eq:W0}
W_0 (x,t,y) = D_y \phi_1 (x,t,y) + D_x \bar{u}_0(x,t).$$
We shall also write by $B_k$, for $k=1,2,\cdots$, the mapping, $$\label{eq:Bk}
B_k(x,t,y) = D_p^k H(W_0(x,t,y)),$$ where $D_p^k H$ is understood in the sense of Fréchet derivatives, and to make the notation coherent to the notation of $\bar{B}$, we will write $$\label{eq:B}
B (x,t,y) = B_1(x,t,y).$$ Let us also remark that, due to , and , we have $B_k\in C^\infty({{\mathbb R}}^n\times[0,\infty);C^\mu({{\mathbb R}}^n))$, for any $0<\mu<1$. In particular, we obtain, for any $i,j=0,1,2,\cdots$, any $k=1,2,\cdots$ and any $T>0$, $$\label{eq:Bk-Ck-Ca}
{\left\Arrowvert { D_x^i {\partial}_t^j B_k(x,t,\cdot)}\right\Arrowvert}_{C^\mu({{\mathbb R}}^n)} \leq C_{i,j,k,T},$$ uniformly for all $(x,t)\in {{\mathbb R}}^n \times[0,T]$, where $C_{i,j,k,T}>0$ depends only on $n$, $\lambda$, $\Lambda$, $\alpha$, $\alpha'$, $\beta$, $\beta'$, $K$, $L$, $\mu$, $i$, $j$, $k$ and $T$.
\[lemma:wk\] Suppose that $A$, $H$ and $g$ satisfy – , – and, respectively, – . Then there exists a sequence $\{w_k\}_{k=1}^\infty$ satisfying the following.
(i) $w_k \in C^\infty({{\mathbb R}}^n\times[0,\infty); C^{2,\mu}({{\mathbb R}}^n))$, for any $0<\mu<1$, and $$\label{eq:wk-Ck-C2a}
{\left\Arrowvert { D_x^i {\partial}_t^j w_k(x,t,\cdot)}\right\Arrowvert}_{C^{2,\mu}({{\mathbb R}}^n)} \leq C_{i,j,k,T},$$ for each $i,j=0,1,2,\cdots$, any $T>0$, and uniformly for all $(x,t)\in{{\mathbb R}}^n\times[0,T]$, where $C_{i,j,k,T}>0$ depends only on $n$, $\lambda$, $\Lambda$, $\alpha$, $\alpha'$, $\beta$, $\beta'$, $K$, $L$, $\mu$, $i$, $j$, $k$ and $T$.
(ii) $w_k$ satisfies $$\label{eq:wk-0}
w_k(x,0,0) = 0.$$
(iii) For each $(x,t)\in{{\mathbb R}}^n\times(0,\infty)$, $w_k(x,t,\cdot)$ is a periodic solution of $$\label{eq:w1-pde}
{\partial}_t w_0(x,t,y) - {\operatorname{tr}}(A(y) D_y^2 w_1) + H( D_y w_1 + D_x w_0(x,t,y),y ) = 0 \quad\text{in }{{\mathbb R}}^n,$$ for $k=1$, and $$\label{eq:wk-pde}
\begin{split}
&{\partial}_t w_{k-1}(x,t,y) - {\operatorname{tr}}(A(y) D_y^2 w_k) \\
&+ B(x,t,y) \cdot (D_y w_k + D_x w_{k-1}(x,t,y)) + \Phi_{k-1}(x,t,y) = 0 \quad\text{in }{{\mathbb R}}^n,
\end{split}$$ for $k\geq 2$, where $$\label{eq:Phik}
\begin{split}
\Phi_{k-1} (x,t,y) &= -2{\operatorname{tr}}(A(y) (D_xD_y w_{k-1} (x,t,y) + D_x^2 w_{k-2} (x,t,y))) \\
&\quad + \sum_{l=2}^{k-1} \frac{1}{l!} \sum_{\substack{ i_1 + \cdots + i_l = k-1 \\ i_1,\cdots,i_1\geq 1}} B_l (x,t,y) ( W_{i_1}(x,t,y),\cdots, W_{i_l} (x,t,y)),
\end{split}$$ with the last summation term understood as zero when $k=2$.
\[remark:wk\] The summation term in the definition of $\Phi_k$ amounts to the nonlinear effect of the Hamiltonian $H$ in $p$. In view of , one may easily observe that the whole summation term becomes zero when $H$ is linear in $p$. The choice of $\Phi_k$ is specifically designed to achieve , which will eventually leads us to the higher order convergence rate for the homogenization problem . We will also see later in and that the choice of $\Phi_k$ changes according to the type of nonlinearity that needs to be taken care of.
Throughout this proof, we shall fix $0<\mu<1$, and denote by $C_{*,\cdots,*}$ a positive constant depending only on the subscripts as well as the parameters $n$, $\lambda$, $\Lambda$, $\alpha$, $\alpha'$, $\beta$, $\beta'$, $K$, $L$ and $\mu$. We will also allow it to vary from one line to another, for notational convenience.
Define $\phi_1$ by . Since $\bar{u}_0\in C^\infty({{\mathbb R}}^n\times[0,\infty))$, we know from that $\phi_1 \in C^\infty({{\mathbb R}}^n\times[0,\infty);C^{2,\mu}({{\mathbb R}}^n))$. Moreover, it follows from that for each $i,j=0,1,2,\cdots$, and any $T>0$, $$\label{eq:phi1-Ck-C2a}
{\left\Arrowvert { D_x^i{\partial}_t^j \phi_1(x,t,\cdot) }\right\Arrowvert}_{C^{2,\mu}({{\mathbb R}}^n)} \leq C_{i,j,T},$$ uniformly for all $(x,t)\in{{\mathbb R}}^n\times[0,T]$. In view of the definition of $w_0$ in , $\phi_1(x,t,\cdot)$ is a periodic viscosity solution of , for each $(x,t)\in{{\mathbb R}}^n\times(0,\infty)$, as $\bar{u}_0$ being the solution of .
Let us now fix $k\geq 1$ and suppose that we have already found $\{w_l\}_{l=0}^{k-1}$ that satisfies the assertions (i) and (ii) of this lemma. Moreover, assume that we have already obtained $\bar{u}_{k-1} \in C^\infty({{\mathbb R}}^n\times[0,\infty))$ such that $$\label{eq:ubk-1-Ck}
\left| D_x^i {\partial}_t^j \bar{u}_{k-1} (x,t) \right| \leq C_{i,j,k,T},$$ for any $i,j=0,1,2,\cdots$, any $T>0$ and any $(x,t)\in {{\mathbb R}}^n\times[0,T]$. Additionally, suppose that we have also found $\phi_k\in C^\infty({{\mathbb R}}^n\times[0,\infty);C^{2,\mu}({{\mathbb R}}^n))$ such that for each $(x,t)\in{{\mathbb R}}^n\times[0,\infty)$, $\phi_k (x,t,\cdot)$ is a periodic function normalized by $$\label{eq:phik-0}
\phi_k(x,t,0) = 0,$$ and that we have, for any $i,j=0,1,2,\cdots$ and any $T>0$, $$\label{eq:phik-Ck-C2a}
{\left\Arrowvert { D_x^i {\partial}_t^j \phi_k(x,t,\cdot)}\right\Arrowvert}_{C^{2,\mu}({{\mathbb R}}^n)} \leq C_{i,j,k,T},$$ uniformly for all $(x,t)\in {{\mathbb R}}^n \times[0,T]$.
Define $\tilde{w}_k$ by $$\label{eq:wt1}
\tilde{w}_1 (x,t,y) = \phi_1(x,t,y),$$ if $k=1$, and by $$\label{eq:wtk}
\tilde{w}_k (x,t,y) = \phi_k(x,t,y) + \chi(x,t,y) \cdot D_x \bar{u}_{k-1}(x,t),$$ if $k\geq 2$. We deduce from , and that $\tilde{w}_k \in C^\infty({{\mathbb R}}^n\times[0,\infty);C^{2,\mu}({{\mathbb R}}^n))$ and satisfies $$\label{eq:wtk-Ck-C2a}
{\left\Arrowvert { D_x^i {\partial}_t^j \tilde{w}_k(x,t,\cdot)}\right\Arrowvert}_{C^{2,\mu}({{\mathbb R}}^n)} \leq C_{i,j,k,T},$$ for any $i,j=0,1,2,\cdots$, any $T>0$ and any $(x,t)\in {{\mathbb R}}^n \times[0,T]$.
In view of the estimate , we observe that $\tilde{w}_k$ satisfies the assertion (i) of Lemma \[lemma:wk\]. Moreover, it follows from the hypothesis , and the fact that $\tilde{w}_k$ verifies the assertion (ii) of this lemma as well. Henceforth, we shall assume, as the last hypothesis for this induction argument, that $\tilde{w}_k$ satisfies the assertion (iii) of this lemma.
In order to find $\bar{u}_k$, we first define $$\label{eq:fk}
\begin{split}
f_k (x,t,y) &= {\partial}_t \tilde{w}_k (x,t,y) + B(x,t,y)\cdot D_x \tilde{w}_k(x,t,y) \\
&\quad -2{\operatorname{tr}}( A(y) (D_xD_y \tilde{w}_k (x,t,y) + D_x^2 w_{k-1} (x,t,y))) \\
&\quad + \sum_{l=2}^k \frac{1}{l!} \sum_{\substack{ i_1 + \cdots + i_l = k \\ i_1,\cdots,i_1\geq 1}} B_l (x,t,y) ( W_{i_1}(x,t,y),\cdots, W_{i_l} (x,t,y)).
\end{split}$$ Using , , , , and together with the induction hypothesis , we deduce that $f_k \in C^\infty({{\mathbb R}}^n\times[0,\infty);C^\mu({{\mathbb R}}^n))$ and $$\label{eq:fk-Ck-Ca}
{\left\Arrowvert { D_x^i {\partial}_t^j f_k(x,t,\cdot)}\right\Arrowvert}_{C^\mu({{\mathbb R}}^n)} \leq C_{i,j,k,T},$$ for any $i,j=0,1,2,\cdots$, any $T>0$ and any $(x,t)\in {{\mathbb R}}^n \times[0,T]$.
Now that $f_k$ is periodic in $y$, we may consider the following cell problem: there exists a unique function $\bar{f}_k:{{\mathbb R}}^n\times(0,\infty){\rightarrow}{{\mathbb R}}$ such that for each $(x,t)\in{{\mathbb R}}^n\times[0,\infty)$, the PDE, $$\label{eq:phik-pde}
-{\operatorname{tr}}(A(y) D_y^2 \phi_{k+1}) + B(x,t,y) \cdot D_y \phi_{k+1} + f_k(x,t,y) = \bar{f}_k (x,t) \quad\text{in }{{\mathbb R}}^n,$$ has a periodic viscosity solution. Following the argument in the proof of Lemma \[lemma:cell-lin\], we see that the cell problem is solvable. Moreover, if we normalize $\phi_{k+1}$ so as to satisfy $$\label{eq:phik+1-0}
\phi_{k+1}(x,t,0) = 0,$$ such a periodic viscosity solution $\phi_{k+1}$ is unique. Furthermore, applying the regularity theory in the slow variable established in Lemma \[lemma:w-Ck-C2a\], we deduce from and that $\bar{f}_k \in C^\infty({{\mathbb R}}^n\times(0,\infty))$ and $\phi_{k+1} \in C^\infty({{\mathbb R}}^n\times[0,\infty);C^{2,\mu}({{\mathbb R}}^n))$. In particular, we have, for any $i,j=0,1,2,\cdots$ and any $T>0$, $$\label{eq:fbk-phik-Ck-C2a}
\left| D_x^i {\partial}_t^j \bar{f}_k (x,t) \right| + {\left\Arrowvert { D_x^i {\partial}_t^j \phi_{k+1}(x,t,\cdot)}\right\Arrowvert}_{C^{2,\mu}({{\mathbb R}}^n)} \leq C_{i,j,k,T},$$ uniformly for all $(x,t)\in {{\mathbb R}}^n \times[0,T]$.
With $\bar{f}_k$ at hand, we consider the first order linear PDE, $$\label{eq:ubk-pde}
\begin{cases}
{\partial}_t \bar{u}_k + \bar{B}(x,t)\cdot D_x \bar{u}_k + \bar{f}_k(x,t) =0 & \text{in }{{\mathbb R}}^n\times(0,\infty),\\
\bar{u}_k = 0 & \text{on }{{\mathbb R}}^n\times\{t=0\},
\end{cases}$$ where $\bar{B}$ is defined by . Recall from that $\bar{B}$ vanishes nowhere in ${{\mathbb R}}^n\times(0,\infty)$. Thus, it follows from the classical existence theory for the first order linear PDE that there exists a unique solution $\bar{u}_k \in C^\infty({{\mathbb R}}^n\times[0,\infty))$ of such that $$\label{eq:ubk-Ck}
\left| D_x^i {\partial}_t^j \bar{u}_k (x,t) \right| \leq C_{i,j,k,T},$$ for any $i,j=0,1,2,\cdots$, any $T>0$ and any $(x,t)\in {{\mathbb R}}^n\times[0,T]$.
Define $w_k$ by $$\label{eq:wk-re}
w_k(x,t,y) = \tilde{w}_k(x,t,y) + \bar{u}_k(x,t),$$ which coincides with the expression and for any $k\geq 1$. Using and , we see that $w_k$, defined by , verifies the assertions (i) and (ii) of Lemma \[lemma:wk\]. Besides, let us notice that $$\label{eq:fk-re}
f_k(x,t,y) = {\partial}_t \tilde{w}_k(x,t,y) + B(x,t,y)\cdot D_x \tilde{w}_k(x,t,y) + \Phi_k(x,t,y),$$ where $\Phi_k$ is defined by , since we have $D_y \tilde{w}_k (x,t,y) = D_y w_k(x,t,y)$.
To this end, let us set $\tilde{w}_{k+1}$ by $$\label{eq:wtk+1}
\tilde{w}_{k+1} (x,t,y) = \phi_{k+1}(x,t,y) + \chi(x,t,y) \cdot D_x \bar{u}_k(x,t).$$ Then we observe from , , and that $$\begin{split}
& {\partial}_t w_k(x,t,y) - {\operatorname{tr}}( A(y) D_y^2 \tilde{w}_{k+1}(x,t,y)) \\
& + B(x,t,y)\cdot( D_y \tilde{w}_{k+1}(x,t,y) + D_x w_k(x,t,y) ) + \Phi_k(x,t,y) \\
& = {\partial}_t\bar{u}_k(x,t) - {\operatorname{tr}}(A(y) D_y^2 \phi_{k+1}(x,t,y)) + B(x,t,y)\cdot \phi_{k+1} + f_k(x,t,y) \\
&\quad + (- {\operatorname{tr}}( A(y) D_y^2 \chi(x,t,y)) + B(x,t,y) \cdot (D_y\chi(x,t,y) + I))\cdot D_x\bar{u}_k(x,t) \\
& = {\partial}_t\bar{u}_k(x,t) + \bar{f}_k(x,t) + \bar{B}(x,t)\cdot D_x\bar{u}_k(x,t) \\
& = 0.
\end{split}$$ Hence, we have proved that $\tilde{w}_{k+1}$ satisfies the assertion (iii) of Lemma \[lemma:wk\].
Recall that we have started with $\{w_l\}_{k=0}^{k-1}$, $\bar{u}_{k-1}$, $\phi_k$ and $\tilde{w}_k$, and obtained $w_k$, $\bar{u}_k$, $\phi_{k+1}$ and $\tilde{w}_{k+1}$ that satisfy all the induction hypotheses. Moreover, we have established the initial case for the induction hypotheses in the beginning of this proof. Thus, the proof is completed by the induction principle.
We shall call $w_k$, chosen from Lemma \[lemma:wk\], the $k$-th order interior corrector for the homogenization problem , due to the following lemma. Although the computation involved in the proof below is similar to what can be found in [@KL1 Section 3.3] and [@KL2 Section 4.1], we present it in detail for the sake of completeness.
\[lemma:wk-cor\] Let $\{w_k\}_{k=1}^\infty$ be chosen as in Lemma \[lemma:wk\]. Then for each integer $m\geq 1$ and each $0<{\varepsilon}\leq\frac{1}{2}$, the function $\eta_m^{\varepsilon}$, defined by $$\label{eq:etame}
\eta_m^{\varepsilon}(x,t) = \bar{u}_0(x,t) + \sum_{k=1}^m {\varepsilon}^k w_k\left(x,t,\frac{x}{{\varepsilon}}\right),$$ is a viscosity solution of $$\label{eq:etame-pde}
\begin{dcases}
{\partial}_t \eta_m^{\varepsilon}-{\varepsilon}{\operatorname{tr}}\left( A\left(\frac{x}{{\varepsilon}}\right) D^2 \eta_m^{\varepsilon}\right) + H\left( D \eta_m^{\varepsilon},\frac{x}{{\varepsilon}}\right) = \psi_m^{\varepsilon}\left(x,t,\frac{x}{{\varepsilon}}\right) & \text{in }{{\mathbb R}}^n\times(0,\infty),\\
\eta_m^{\varepsilon}= g & \text{on }{{\mathbb R}}^n\times\{t=0\},
\end{dcases}$$ where $\psi_m^{\varepsilon}\in C({{\mathbb R}}^n\times[0,\infty);L^\infty({{\mathbb R}}^n))$ satisfies, for any $T>0$, $$\label{eq:psime-Linf}
{\left\Arrowvert {\psi_m^{\varepsilon}(x,t,\cdot)}\right\Arrowvert}_{L^\infty({{\mathbb R}}^n)} \leq C_{m,T}{\varepsilon}^m,$$ uniformly for all $(x,t)\in {{\mathbb R}}^n\times[0,T]$, where $C_{m,T}>0$ is a constant depending only on $n$, $\lambda$, $\Lambda$, $\alpha$, $\alpha'$, $\beta$, $\beta'$, $K$, $L$, $\mu$, $m$ and $T$.
Aligned with the notation of $W_k$, let us denote by $X_k$, the matrix-valued mapping, $$\label{eq:Xk}
X_k(x,t,y) = D_y^2 w_{k+1}(x,t,y) + (D_xD_y + D_yD_x) w_k(x,t,y) + D_x^2 w_{k-1}(x,t,y),$$ for $k=1,2,\cdots$, with $w_{-1}$ being understood as the identically zero function. One may notice from , and that $$\label{eq:X0}
X_0(x,t,y) = D_y^2 \phi_1(x,t,y).$$
Fix $m\geq 1$ and $0<{\varepsilon}\leq\frac{1}{2}$. For the moment, we shall replace $w_{m+1}$ and $w_{m+2}$ by the identically zero functions, only to simplify the exposition. With this replacement, we have $W_m = D_x w_m$, $X_m = (D_xD_y + D_yD_x) w_m + D_x^2 w_{m-1}$ and $X_{m+1} = D_x^2 w_m$.
In view of and , we have $$D \eta_m^{\varepsilon}(x,t) = \sum_{k=0}^m {\varepsilon}^k W_k\left(x,t,\frac{x}{{\varepsilon}}\right),$$ and $${\varepsilon}D^2 \eta_m^{\varepsilon}(x,t) = \sum_{k=0}^{m+1} {\varepsilon}^k X_k\left(x,t,\frac{x}{{\varepsilon}}\right).$$ Let us define $\Psi_k$ by $$\label{eq:Psi0}
\Psi_0(x,t,y) = - {\operatorname{tr}}(A(y) X_0(x,t,y)) + H(W_0(x,t,y),y),$$ if $k=0$, and by $$\label{eq:Psik}
\begin{split}
\Psi_k(x,t,y) &= - {\operatorname{tr}}(A(y) X_k(x,t,y)) \\
&\quad + \sum_{l=1}^k \frac{1}{l!} \sum_{\substack{i_1+\cdots+i_l=k \\ i_1,\cdots,i_l\geq 1}} B_l(x,t,y)(W_{i_1}(x,t,y),\cdots,W_{i_l}(x,t,y)),
\end{split}$$ if $1\leq k\leq m-1$. Using $\Psi_k$, one may rephrase the PDEs and $$\label{eq:wk-pde-re}
{\partial}_t w_k (x,t,y) + \Psi_k (x,t,y) = 0 \quad\text{in }{{\mathbb R}}^n,$$ for $0\leq k\leq m-1$.
Denoting by $T_{m-1}(p_0,p)$ the $(m-1)$-th order Taylor polynomial of $H$ in $p$ at $p_0$, namely, $$T_{m-1}(p_0,p) (y) = \sum_{k=0}^{m-1} \frac{1}{k!} D_p^k H(p_0,y) (p,\cdots,p),$$ we have $$\label{eq:Tm}
\begin{split}
& T_{m-1} \left( W_0(x,t,y), \sum_{k=1}^m {\varepsilon}^k W_k(x,t,y) \right)(y) - \sum_{k=0}^{m-1} {\varepsilon}^k {\operatorname{tr}}(A(y) X_k (x,t,y)) \\
& = \sum_{k=0}^{m-1} {\varepsilon}^k \Psi_k(x,t,y) + \sum_{k=2}^m \sum_{\substack{m\leq i_1+\cdots+i_k\leq km \\ 1\leq i_1,\cdots,i_k\leq m}} \frac{{\varepsilon}^{i_1+\cdots+i_k}}{k!} B_k(x,t,y)(W_{i_1}(x,t,y),\cdots,W_{i_k}(x,t,y)).
\end{split}$$ Hence, we apply the Taylor expansion of $H$ in $p$ at $W_0$ up to $(m-1)$-th order and derive that $$\label{eq:HDetame}
-{\varepsilon}{\operatorname{tr}}\left( A\left(\frac{x}{{\varepsilon}}\right) D^2 \eta_m^{\varepsilon}(x,t)\right) + H\left( D\eta_m^{\varepsilon}(x,t) ,\frac{x}{{\varepsilon}}\right) = \sum_{k=0}^{m-1} {\varepsilon}^k\Psi_k\left(x,t,\frac{x}{{\varepsilon}}\right) + E_m^{\varepsilon}\left(x,t,\frac{x}{{\varepsilon}}\right),$$ where $E_m^{\varepsilon}$ is defined so as to satisfy $$\label{eq:Eme}
\begin{split}
&E_m^{\varepsilon}(x,t,y) - R_{m-1}\left(W_0(x,t,y),\sum_{k=1}^m{\varepsilon}^k W_k(x,t,y)\right)(y) + \sum_{k=m}^{m+1} {\varepsilon}^k {\operatorname{tr}}(A(y) X_k(x,t,y)) \\
&= \sum_{k=2}^m \sum_{\substack{m\leq i_1+\cdots+i_k\leq km \\ 1\leq i_1,\cdots,i_k\leq m}} \frac{{\varepsilon}^{i_1+\cdots+i_k}}{k!} B_k(x,t,y)(W_{i_1}(x,t,y),\cdots,W_{i_k}(x,t,y)),
\end{split}$$ with $R_{m-1}(p_0,p)$ being the $(m-1)$-th order remainder term of $H$ in $p$ at $p_0$.
Now using , we observe that $\eta_m^{\varepsilon}$ solves with$$\label{eq:psime}
\psi_m^{\varepsilon}(x,t,y) = {\varepsilon}^m {\partial}_t w_m (x,t,y) + E_m^{\varepsilon}(x,t,y).$$ Note that the initial condition of is satisfied, due to that of and the assertion (ii) of Lemma \[lemma:wk\]. Hence, we are only left with proving the estimate for $\psi_m^{\varepsilon}$.
It is clear that implies $$\label{eq:ptwm-Linf}
{\left\Arrowvert { {\partial}_t w_m(x,t,\cdot) }\right\Arrowvert}_{L^\infty({{\mathbb R}}^n)} \leq C_{m,T},$$ for any $T>0$ and any $(x,t)\in {{\mathbb R}}\times[0,T]$, where $C_{m,T}>0$ is a constant depending only on $n$, $\lambda$, $\Lambda$, $\alpha$, $\alpha'$, $\beta$, $\beta'$, $K$, $L$, $\mu$, $m$ and $T$. On the other hand, using , and , and noting that ${\varepsilon}^{i_1+\cdots+i_k} \leq {\varepsilon}^m$ for any $1\leq i_1,\cdots,i_k\leq m$ satisfying $m\leq i_1+\cdots +i_k\leq km$, we deduce from that $$\label{eq:Eme-Linf}
{\left\Arrowvert { E_m^{\varepsilon}(x,t,\cdot) }\right\Arrowvert}_{L^\infty({{\mathbb R}}^n)} \leq C_{m,T}{\varepsilon}^m,$$ for any $T>0$ and any $(x,t)\in \bar{B}_R\times[0,T]$, with $C_{m,T}>0$ being yet another constant depending only on the same parameters listed above. This finishes the proof.
With the aid of Lemma \[lemma:wk-cor\], we prove the first main result of this paper.
\[theorem:cr\] Suppose that the diffusion coefficient $A$, the Hamiltonian $H$ and the initial data $g$ satisfy – , – , and respectively – . Under these circumstances, let $\{u^{\varepsilon}\}_{{\varepsilon}>0}$ be the sequence of the viscosity solutions of . Then with the viscosity solution $\bar{u}_0$ of and the sequence $\{w_k\}_{k=1}^\infty$ of $k$-th order interior correctors chosen in Lemma \[lemma:wk\], we have, for each integer $m\geq 1$, any $0<{\varepsilon}\leq\frac{1}{2}$ and any $T>0$, $$\label{eq:cr}
\left| u^{\varepsilon}(x,t) - \bar{u}_0(x,t) - \sum_{k=1}^m {\varepsilon}^k w_k \left(x,t,\frac{x}{{\varepsilon}}\right) \right| \leq C_{m,T}{\varepsilon}^m,$$ uniformly for all $(x,t)\in{{\mathbb R}}^n\times [0,T]$, where $C_{m,T}>0$ depends only on $n$, $\lambda$, $\Lambda$, $\alpha$, $\alpha'$, $\beta$, $\beta'$, $K$, $L$, $\mu$, $m$ and $T$.
The proof follows from Lemma \[lemma:wk-cor\] and the comparison principle for viscosity solutions. Let $\eta_m^{\varepsilon}$ be as in . Due to , we see that $\eta_m^{\varepsilon}+ C_{m,T}{\varepsilon}^m t$ and $\eta_m^{\varepsilon}- C_{m,T}{\varepsilon}^m t$ are a viscosity supersolution and, respectively, a viscosity subsolution of . Thus, the comparison principle yields that $$|u^{\varepsilon}(x,t) - \eta_m^{\varepsilon}(x,t)| \leq T C_{m,T}{\varepsilon}^m,$$ uniformly for all $(x,t)\in{{\mathbb R}}^n\times[0,T]$, which finishes the proof.
Generalization to Fully Nonlinear Hamiltonian {#section:nl}
=============================================
In this section, we generalize Theorem \[theorem:cr\] to the fully nonlinear, viscous Hamilton-Jacobi equation, , whose gradient term is convex and grows quadratically at the infinity. Henceforth, we shall assume that the nonlinear functional $H$ satisfies the following conditions, for any $(M,p,y)\in{{\mathbb S}}^n\times{{\mathbb R}}^n\times{{\mathbb R}}^n$.
(i) $H$ is periodic in $y$: $$\label{eq:H-peri-nl}
H(M,p,y+k) = H(M,p,y),$$ for any $k\in{{\mathbb Z}}^n$.
(ii) $H$ is uniformly elliptic in $M$: $$\label{eq:H-ellip-nl}
\lambda{\left\Arrowvert {N}\right\Arrowvert} \leq H(M,p,y) - H(M + N,p,y) \leq \Lambda{\left\Arrowvert {N}\right\Arrowvert},$$ for any $N\in{{\mathbb S}}^n$ with $N\geq 0$.
(iii) $H$ has interior $C^{2,\bar\mu}$ estimates: for any $r>0$, any $(M_0,p_0,y_0)\in{{\mathbb S}}^n\times{{\mathbb R}}^n\times{{\mathbb R}}^n$ with $a= H(M,p,y_0)$, and any $v_0 \in C({\partial}B_r(y_0))$, there exists a viscosity solution $v\in C(\bar{B}_r(y_0)) \cap C^2( B_r(y_0)) \cap C^{2,\bar\mu}(\bar{B}_{r/2}(y_0))$ of $$\begin{cases}
H( D^2 v + M,p,y_0) = a &\text{in }B_r(y_0),\\
v = v_0 & \text{on }{\partial}B_r(y_0),
\end{cases}$$ such that $${\left\Arrowvert {v}\right\Arrowvert}_{C^{2,\bar\mu}(\bar{B}_{r/2}(y_0))} \leq K{\left\Arrowvert {v_0}\right\Arrowvert}_{L^\infty({\partial}B_r(y_0))}.$$
(iv) $H$ is convex in $p$: $$\label{eq:H-convex-nl}
H(M,tp+(1-t)q,y) \leq tH(M,p,y) + (1-t)H(M,q,y),$$ for any $0\leq t\leq 1$ and any $q\in{{\mathbb R}}^n$.
(v) $H$ has quadratic growth in $p$: $$\label{eq:H-quad-nl}
\alpha|p|^2 - \alpha' \leq H(0,p,y) \leq \beta|p|^2 + \beta'.$$
(vi) $H\in C^\infty({{\mathbb S}}^n\times{{\mathbb R}}^n;C^{0,1}({{\mathbb R}}^n))$ and $$\label{eq:H-Ck-C01-nl}
{\left\Arrowvert {D_M^k D_p^l H(M,p,\cdot)}\right\Arrowvert}_{C^{0,1}({{\mathbb R}}^n)} \leq K\left( 1 + {\left\Arrowvert {M}\right\Arrowvert}^{(1-k)_+} + |p|^{(2-l)_+} \right),$$ for any pair $(k,l)$ of nonnegative integers.
We shall impose the conditions – to the initial data $g$, as we did in the preceding section, once the effective Hamiltonian $\bar{H}$ is determined. The effective Hamiltonian $\bar{H}$ is derived by solving the cell problem . Since the proof is analogous to that of Lemma \[lemma:cell\], we shall omit the details. Let us remark that although the Hamiltonian $H$ is now fully nonlinear in the Hessian variable $M$, we still have the $C^{2,\mu}$ regularity (for any $0<\mu<\bar\mu$) of periodic viscosity solutions to the cell problem , since $H$ has interior $C^{2,\bar\mu}$ estimates for fixed coefficients (assumption (iii)), and it satisfies Lipschitz regularity in $y$ (assumption (vi)). We refer to [@CC Theorem 8.1], for details on this regularity theory. Besides, the reader who is unfamiliar with hypothesis (iii) can simply replace it by convexity in $M$.
\[lemma:cell-nl\] For each $p\in{{\mathbb R}}^n$, there exists a unique real number $\gamma$, for which the following PDE, $$\label{eq:cell-nl}
H(D^2 w, Dw + p, y ) = \gamma \quad\text{in }{{\mathbb R}}^n,$$ has a periodic solution $w\in C^{2,\mu}({{\mathbb R}}^n)$, for some $0<\mu<1$ depending only on $n$, $\lambda$ and $\Lambda$. Moreover, $\gamma$ satisfies and, furthermore, a periodic solution $w$ of is unique up to an additive constant and satisfies .
As in Section \[section:reg\], we shall denote by $\bar{H}$ the effective Hamiltonian of $H$. That is, $\bar{H}:{{\mathbb R}}^n{\rightarrow}{{\mathbb R}}$ is a function defined in such a way that for each $p\in{{\mathbb R}}^n$, $\bar{H}(p)$ is the unique real number for which the following PDE, $$\label{eq:w-pde-nl}
H(D^2 w, Dw + p, y) =\bar{H}(p) \quad\text{in }{{\mathbb R}}^n,$$ has a periodic viscosity solution in $C^{2,\mu}({{\mathbb R}}^n)$. Moreover, we shall also denote by $w:{{\mathbb R}}^n\times{{\mathbb R}}^n{\rightarrow}{{\mathbb R}}$ by the functional such that for each $p\in{{\mathbb R}}^n$, $w(p,\cdot) \in C^{2,\mu}({{\mathbb R}}^n)$ is the unique periodic solution of that is normalized so as to satisfy .
Following the same arguments in their proofs, one may prove that $\bar{H}$ and $w$ satisfy Lemma \[lemma:Hb\] and Lemma \[lemma:w-Linf-C2a\], except for that $w\in C({{\mathbb R}}^n;C^{2,\mu}({{\mathbb R}}^n))$ for some fixed $0<\mu<1$, rather than any $0<\mu<1$. This is because the proofs of those lemmas do not rely on the linear structure of the diffusion coefficient, but more on its uniform ellipticity. A more important observation is the generalization of Lemma \[lemma:w-Ck-C2a\], which amounts to the regularity of $\bar{H}$ and $w$ in the slow variables.
\[lemma:w-Ck-C2a-nl\] $\bar{H}\in C^\infty({{\mathbb R}}^n)$ and $w\in C^\infty({{\mathbb R}}^n;C^{2,\mu}({{\mathbb R}}^n))$, for any $0<\mu<\bar\mu$, such that holds, for any $k=0,1,2,\cdots$, any $L>0$ and any $p\in B_L$.
Let us fix $0<\mu<\bar\mu$. It suffices to prove that $\bar{H}$ and $w$ verify Lemma \[lemma:w-C1-C2a\]. Moreover, to see this fact, it is enough to show that the linearization argument in the proof of Lemma \[lemma:w-C1-C2a\] also works out when the Hamiltonian $H$ depends nonlinearly on the Hessian variable $M$.
Let $w_h$, $\gamma_h$, $W_h$ and $\Gamma_h$ be as in the proof of Lemma \[lemma:w-C1-C2a\]. Then by linearizing the cell problem (in both of the Hessian and the gradient variables), we observe that $W_h$ solves $$\label{eq:Wh-pde-nl}
-{\operatorname{tr}}(A_h(y) D^2 W_h) + B_h(y) \cdot (DW_h + e_k) = \Gamma_h\quad\text{in }{{\mathbb R}}^n,$$ where $$A_h(y) = \int_0^1 - D_M H ( t D_y^2 w_h + (1-t) D_y w_0, D_y w_h + p, y) dt,$$ and $$B_h(y) = \int_0^1 D_p H ( D_y w_0, t D_y w_h + (1-t) D_y w_0 +p + the,y) dt.$$ In comparison of with , one may see that the only major difference here is that the diffusion coefficient, $A_h$, here is not fixed but depends on the parameter $h$.
Nevertheless, $A_h$ is uniformly elliptic not only in $y$ but also in $h$, due to the assumption . This implies that Lemma \[lemma:cell-lin\] is still applicable, and thus $W_h\in C^{2,\mu}({{\mathbb R}}^n)$ and satisfies uniformly for $h$.
Moreover, since $w$ satisfies , it follows from the regularity assumption of $H$ that $A_h\in C^\mu({{\mathbb R}}^n)$ and $$\label{eq:Ah-Ca}
{\left\Arrowvert {A_h}\right\Arrowvert}_{C^\mu({{\mathbb R}}^n)} \leq C,$$ where $C>0$ depends only on $n$, $\lambda$ and $\Lambda$. For the same reason, we deduce that $B_h \in C^\mu({{\mathbb R}}^n)$ and satisfies . Furthermore, since $w \in C({{\mathbb R}}^n; C^{2,\mu}({{\mathbb R}}^n))$, we have $A_h{\rightarrow}A_0$ and $B_h{\rightarrow}B_0$ in $C^{\mu'}({{\mathbb R}}^n)$, for any $0<\mu'<\mu$, with $$A_0(y) = - D_M H ( D_y w_0, D_y w_0 + p, y),$$ and $$B_0(y) = D_p H ( D_y w_0, D_y w_0 + p ,y).$$
The rest of the proof follows similarly with that of of Lemma \[lemma:w-C1-C2a\]. In particular, we obtain unique $W_0\in C^{2,\mu}({{\mathbb R}}^n)$ and $\Gamma_0\in {{\mathbb R}}$ such that $W_0$ is the periodic solution $$\label{eq:W0-pde-nl}
-{\operatorname{tr}}(A_0(y) D^2 W_0) + B_0(y) \cdot (DW_0 + e_k) = \Gamma_0\quad\text{in }{{\mathbb R}}^n,$$ satisfying $W_0(0) = 0$. We leave out the details to the reader.
Now we are in position to construct the higher order interior correctors of the homogenization problem . We shall now let $g$ satisfy the structure conditions – , with $\bar{H}$ being the effective Hamiltonian chosen to satisfy the cell problem . Next we shall denote by $\bar{u}_0$ the solution of , with the updated data $\bar{H}$ and $g$, and write by $\bar{B}$ the function defined by . Once again, we have $\bar{u}_0\in C^\infty({{\mathbb R}}^n\times[0,\infty))$ and $\bar{B}\in C^\infty({{\mathbb R}}^n\times[0,\infty))$ with the estimates and .
Let $w_0$, $\{W_k\}_{k=0}^\infty$ and $\{X_k\}_{k=0}^\infty$ denote those defined in , and, respectively, , where the sequence $\{w_k\}_{k=1}^\infty$ of higher order interior correctors will be given as below.
Now that the Hamiltonian $H$ is nonlinear in $M$, we need to apply the Taylor expansion not only in the variable $p$ but also in the variable $M$, in order to obtain the PDEs (or, more precisely, the cell problems) for the higher order interior correctors. In this direction, we consider the coefficient $B_{k,l}$ defined by $$\label{eq:Bkl-nl}
B_{k,l} (x,t,y) = D_M^kD_p^l H(X_0(x,t,y),W_0(x,t,y),y),$$ for $k,l=0,1,2,\cdots$. In particular, we shall write $$\label{eq:A-nl}
A(x,t,y) = -B_{1,0}(x,t,y) = - D_M H(X_0(x,t,y),W_0(x,t,y),y),$$ and $$\label{eq:B-nl}
B(x,t,y) = B_{0,1}(x,t,y) = D_p H(X_0(x,t,y),W_0(x,t,y),y).$$ Note that $A$ is uniformly elliptic with the same ellipticity bounds as those of $H$.
\[lemma:wk-nl\] Suppose that $H$ and $g$ satisfy – and, respectively, – . Then there exists a sequence $\{w_k\}_{k=1}^\infty$ satisfying the following.
(i) $w_k \in C^\infty({{\mathbb R}}^n\times[0,\infty); C^{2,\mu}({{\mathbb R}}^n))$, for any $0<\mu<\bar\mu$, and satisfies the estimate , for any $i,j=0,1,2,\cdots$, any $T>0$ and any $(x,t)\in{{\mathbb R}}^n\times[0,T]$.
(ii) $w_k$ is normalized so as to satisfy .
(iii) For each $(x,t)\in{{\mathbb R}}^n\times(0,\infty)$, $w_k(x,t,\cdot)$ is a periodic solution of $$\label{eq:w1-pde-nl}
{\partial}_t w_0(x,t,y) + H( D_y^2 w_1, D_y w_1 + D_x w_0(x,t,y), y ) = 0 \quad\text{in }{{\mathbb R}}^n,$$ for $k=1$, and $$\label{eq:wk-pde-nl}
\begin{split}
&{\partial}_t w_{k-1}(x,t,y) - {\operatorname{tr}}(A(x,t,y) D_y^2 w_k) \\
&+ B(x,t,y) \cdot (D_y w_k + D_x w_{k-1}(x,t,y)) + \Phi_{k-1}(x,t,y) = 0 \quad\text{in }{{\mathbb R}}^n,
\end{split}$$ for $k\geq 2$, where $$\label{eq:Phik-nl}
\begin{split}
\Phi_{k-1} (x,t,y) &= -2{\operatorname{tr}}(A(x,t,y) (D_xD_y w_{k-1} (x,t,y) + D_x^2 w_{k-2} (x,t,y))) \\
&\quad + \sum_{l=2}^{k-1} \frac{1}{l!} \sum_{\substack{ i_1 + \cdots + i_l = k-1 \\ i_1,\cdots,i_1\geq 1}} \sum_{r=0}^l B_{r,l-r} (x,t,y) ( X_{i_1}(x,t,y),\cdots, X_{i_r} (x,t,y), \\
&\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad W_{i_{r+1}}(x,t,y),\cdots, W_{i_l}(x,t,y))
\end{split}$$ with the last summation term understood as zero when $k=2$.
\[remark:wk-nl\] As mentioned in Remark \[remark:wk\], $\Phi_k$ now takes care of the nonlinear effect produced by $H$ in both $M$ and $p$ variables. Moreover, the summation term in the definition of $\Phi_k$ is specifically constructed to have , by which we will eventually derive the higher order convergence rates for the homogenization problem .
The proof follows essentially the same induction argument presented in that of Lemma \[lemma:wk\]. To avoid any repeating argument, we shall only point out the major difference from the proof of Lemma \[lemma:wk\], and ask the reader to fill in the details.
Here we define $\phi_1$ by with $w$ being the (normalized) periodic solution of (instead of ), and accordingly set $w_1$ by with some $\bar{u}_1$ to be determined. Then we observe that $W_0$ and $X_0$ verify the expressions and, respectively, . Moreover, we verify that $B_{l,k-l}$ satisfy the estimate , for any $l=0,1,\cdots,k$ and any $k=1,2,\cdots$.
The function $f_k$, which takes cares of all the nonlinear effect caused in the $k$-th step of approximation, is now replaced by $$\label{eq:fk-nl}
\begin{split}
f_k (x,t,y) &= {\partial}_t \tilde{w}_k (x,t,y) + B(x,t,y)\cdot D_x \tilde{w}_k(x,t,y) \\
&\quad -2{\operatorname{tr}}( A(x,t,y) (D_xD_y \tilde{w}_k (x,t,y) + D_x^2 w_{k-1} (x,t,y))) \\
&\quad + \sum_{l=2}^{k-1} \frac{1}{l!} \sum_{\substack{ i_1 + \cdots + i_l = k-1 \\ i_1,\cdots,i_1\geq 1}} \sum_{r=0}^l B_{r,l-r} (x,t,y) ( X_{i_1}(x,t,y),\cdots, X_{i_r} (x,t,y), \\
&\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad W_{i_{r+1}}(x,t,y),\cdots, W_{i_l}(x,t,y)),
\end{split}$$ Due to the periodicity of $f_k$ in $y$, we consider the following cell problem: there exists a unique $\bar{f}_k:{{\mathbb R}}^n\times(0,\infty){\rightarrow}{{\mathbb R}}^n$ such that for each $(x,t)\in{{\mathbb R}}^n\times(0,\infty)$, the PDE, $$\label{eq:phik-pde-nl}
-{\operatorname{tr}}(A(x,t,y) D_y^2 \phi_{k+1}) + B(x,t,y) \cdot D_y \phi_{k+1} + f_k(x,t,y) = \bar{f}_k(x,t) \quad\text{in }{{\mathbb R}}^n,$$ has a periodic viscosity solution. The rest of the proof can be derived by following that of Lemma \[lemma:wk\], whence we omit the details.
The next lemma is the corresponding version of Lemma \[lemma:wk-cor\] for fully nonlinear Hamiltonian $H$.
\[lemma:wk-cor-nl\] Let $\{w_k\}_{k=1}^\infty$ be chosen as in Lemma \[lemma:wk-nl\]. Then for each integer $m\geq 1$ and each $0<{\varepsilon}\leq\frac{1}{2}$, the function $\eta_m^{\varepsilon}$, defined by , is a viscosity solution of $$\label{eq:etame-pde-nl}
\begin{dcases}
{\partial}_t \eta_m^{\varepsilon}+ H\left( {\varepsilon}D^2 \eta_m^{\varepsilon}, D \eta_m^{\varepsilon}, \frac{x}{{\varepsilon}}\right) = \psi_m^{\varepsilon}\left(x,t,\frac{x}{{\varepsilon}}\right) & \text{in }{{\mathbb R}}^n\times(0,\infty),\\
\eta_m^{\varepsilon}= g & \text{on }{{\mathbb R}}^n\times\{t=0\},
\end{dcases}$$ where $\psi_m^{\varepsilon}\in C({{\mathbb R}}^n\times[0,\infty);L^\infty({{\mathbb R}}^n))$ satisfies , for any $T>0$ and all $(x,t)\in {{\mathbb R}}^n\times[0,T]$.
As in the proof of Lemma \[lemma:wk-nl\], we shall mention the key points that need to be modified from the proof of Lemma \[lemma:wk-cor\], in order to take care of the nonlinear effect in the Hessian variable of $H$. Let us begin by fixing $m\geq 1$ and $0<{\varepsilon}\leq\frac{1}{2}$, and replacing $w_{m+1}$ and $w_{m+2}$ by the identically zero functions, again for the notational convenience.
We shall define $\Psi_k$ by $$\label{eq:Psi0-nl}
\Psi_0(x,t,y) = H( X_0(x,t,y), W_0(x,t,y), y),$$ if $k=0$, and by $$\label{eq:Psik-nl}
\begin{split}
\Psi_k(x,t,y) &= \sum_{l=1}^{k-1} \frac{1}{l!} \sum_{\substack{ i_1 + \cdots + i_l = k-1 \\ i_1,\cdots,i_1\geq 1}} \sum_{r=0}^l B_{r,l-r} (x,t,y) ( X_{i_1}(x,t,y),\cdots, X_{i_r} (x,t,y), \\
&\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad W_{i_{r+1}}(x,t,y),\cdots, W_{i_l}(x,t,y))
\end{split}$$ if $1\leq k\leq m-1$. Then it follows from the PDEs and that holds for $0\leq k\leq m-1$.
Applying the Taylor expansion of $H$ in $(M,p)$ at $(X_0,W_0)$ up to $(m-1)$-th order, and after some calculations similar to those in , we obtain that $$\label{eq:HDetame-nl}
H \left( {\varepsilon}D^2 \eta_m^{\varepsilon}(x,t), D\eta_m^{\varepsilon}(x,t),\frac{x}{{\varepsilon}}\right) = \sum_{k=0}^{m-1} {\varepsilon}^k \Psi_k\left(x,t,\frac{x}{{\varepsilon}}\right) + E_m^{\varepsilon}\left(x,t,\frac{x}{{\varepsilon}}\right),$$ where $E_m^{\varepsilon}$ is defined so as to satisfy $$\label{eq:Eme-nl}
\begin{split}
&E_m^{\varepsilon}(x,t,y) - R_{m-1}\left((X_0(x,t,y),W_0(x,t,y)),\left(\sum_{k=1}^{m+1} {\varepsilon}^k X_k(x,t,y), \sum_{k=1}^m{\varepsilon}^k W_k(x,t,y)\right)\right)(y)\\
&= \sum_{k=2}^m \sum_{\substack{m\leq i_1+\cdots+i_k\leq km \\ 1\leq i_1,\cdots,i_k\leq m}} \frac{{\varepsilon}^{i_1+\cdots+i_k}}{k!} \sum_{l=0}^k B_{l,k-l}(x,t,y)(X_{i_1}(x,t,y)\cdots,X_{i_l}(x,t,y), \\
&\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad W_{i_{l+1}}(x,t,y),\cdots,W_{i_k}(x,t,y)),
\end{split}$$ where $R_{m-1}((M_0,p_0),(M,p))$ denotes the $(m-1)$-th order remainder term of $H$ in $(M,p)$ at $(M_0,p_0)$.
We deduce from that $\eta_m^{\varepsilon}$ solves with $\psi_m^{\varepsilon}$ defined by . The rest of the proof follows similarly to that of Lemma \[lemma:wk-cor\]. In particular, we have , since $B_{l,k-l}$ and $w_k$ satisfy the estimate and, respectively, . We leave out the details to the reader.
Finally, we generalize Theorem \[theorem:cr\] to the regime of fully nonlinear, viscous Hamilton-Jacobi equation, as stated below.
\[theorem:cr-nl\] Suppose that the Hamiltonian $H$ and the initial data $g$ satisfy – and, respectively, – . Under these circumstances, let $\{u^{\varepsilon}\}_{{\varepsilon}>0}$ be the sequence of the viscosity solutions of . Then with the viscosity solution $\bar{u}_0$ of and the sequence $\{w_k\}_{k=1}^\infty$ of $k$-th order interior correctors chosen in Lemma \[lemma:wk-nl\], we have, for each integer $m\geq 1$, any $0<{\varepsilon}\leq\frac{1}{2}$ and any $T>0$, $$\label{eq:cr-nl}
\left| u^{\varepsilon}(x,t) - \bar{u}_0(x,t) - \sum_{k=1}^m {\varepsilon}^k w_k \left(x,t,\frac{x}{{\varepsilon}}\right) \right| \leq C_{m,T}{\varepsilon}^m,$$ uniformly for all $(x,t)\in{{\mathbb R}}^n\times [0,T]$, where $C_{m,T}>0$ depends only on $n$, $\lambda$, $\Lambda$, $\alpha$, $\alpha'$, $\beta$, $\beta'$, $K$, $L$, $\mu$, $\bar\mu$, $m$ and $T$.
The proof follows the same comparison argument as that in the proof of Theorem \[theorem:cr\]. Let $\eta_m^{\varepsilon}$ be as in Lemma \[lemma:wk-cor-nl\]. According to Lemma \[lemma:wk-cor-nl\], $\eta_m^{\varepsilon}+ C_{m,T}{\varepsilon}^m t$ and $\eta_m^{\varepsilon}- C_{m,T}{\varepsilon}^m t$ are a viscosity supersolution and, respectively, a viscosity subsolution of , for some constant $C_{m,T}>0$ depending only on $n$, $\lambda$, $\Lambda$, $\alpha$, $\alpha'$, $\beta$, $\beta'$, $K$, $L$, $\mu$, $\bar\mu$, $m$ and $T$. Therefore, the comparison principle yields that $$|u^{\varepsilon}(x,t) - \eta_m^{\varepsilon}(x,t)| \leq T C_{m,T}{\varepsilon}^m,$$ uniformly for all $(x,t)\in{{\mathbb R}}^n\times[0,T]$. This completes the proof.
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|
---
abstract: |
This survey paper reports on the properties of the fourth-order Bessel-type linear ordinary differential equation, on the generated self-adjoint differential operators in two associated Hilbert function spaces, and on the generalisation of the classical Hankel integral transform.
These results are based upon the properties of the classical Bessel and Laguerre second-order differential equations, and on the fourth-order Laguerre-type differential equation. From these differential equations and their solutions, limit processes yield the fourth-order Bessel-type functions and the associated differential equation.
address: 'W.N. Everitt, School of Mathematics and Statistics, University of Birmingham, Edgbaston, Birmingham B15 2TT, England, UK'
author:
- 'W.N. Everitt'
date: '25 August 2005 (File: C:$\backslash$Swp50$\backslash$Bessel$\backslash$munich9.tex)'
title: |
Fourth-order Bessel-type special functions:\
a survey
---
Introduction\[sec1\]
====================
This survey paper is based on joint work with the following named colleagues:
Jyoti Das, University of Calcutta, India
D.B. Hinton, University of Tennessee, USA
H. Kalf, University of Munich, Germany
L.L. Littlejohn, Utah State University, USA
C. Markett, Technical University of Aachen, Germany
M. Plum, University of Karlsruhe, Germany
M. van Hoeij, Florida State University, USA
History\[sec2\]
===============
We see below that the structured definition of the general-even order Bessel-type special functions is dependent upon the Jacobi and Laguerre classical orthogonal polynomials, and the Jacobi-type and Laguerre-type orthogonal polynomials.
These latter orthogonal polynomials were first defined by H.L. Krall in 1940, see [@HK] and [@HK1], and later studied in detail by A.M. Krall in 1981, see [@AK], and by Koornwinder in 1984, see [@THK]. In this respect see the two survey papers, [@EL1] of 1990 and [@PAT] of 1999.
The Bessel-type special functions of general even-order were introduced by Everitt and Markett in 1994, see [@EM].
The properties of the fourth-order Bessel-type functions have been studied by the present author and the seven colleagues named in Section \[sec1\] above, in the papers [@DEHLM], [@EKLM] and [@EKLM1].
The fourth-order differential equation\[sec3\]
==============================================
The fourth-order Bessel-type differential equation takes the form$$(xy^{\prime\prime}(x))^{\prime\prime}-((9x^{-1}+8M^{-1}x)y^{\prime
}(x))^{\prime}=\Lambda xy(x)\ \text{for all}\ x\in(0,\infty) \label{eq3.1}$$ where $M\in(0,\infty)$ is a positive parameter and $\Lambda\in\mathbb{C},$ the complex field, is a spectral parameter. The differential equation (\[eq3.1\]) is derived in the paper [@EM Section 1, (1.10a)], by Everitt and Markett.
This linear, ordinary differential equation on the interval $(0,\infty
)\subset\mathbb{R},$ the real field, is written in Lagrange symmetric (formally self-adjoint) form, or equivalently Naimark form, see [@MAN Chapter V].
The structured Bessel-type functions of all even orders, and their associated linear differential equations, were introduced in the paper [@EM Section 1] through linear combinations of, and limit processes applied to, the Laguerre and Laguerre-type orthogonal polynomials, and to the classical Bessel functions. This process is best illustrated through the following diagram, see [@EM Section 1, Page 328] (for the first two lines of this table see the earlier work of Koornwinder [@THK] and Markett [@CM]):$$\left\{
\begin{array}
[c]{ccc}\begin{array}
[c]{c}\text{Jacobi polynomials}\\
k(\alpha,\beta)(1-x)^{\alpha}(1+x)^{\beta}\end{array}
& \longrightarrow &
\begin{array}
[c]{c}\text{Jacobi-type polynomials}\\
k(\alpha,\beta)(1-x)^{\alpha}(1+x)^{\beta}\\
+M\delta(x+1)+N\delta(x-1)
\end{array}
\\
\downarrow & & \downarrow\\\begin{array}
[c]{c}\text{Laguerre polynomials}\\
k(\alpha)x^{\alpha}\exp(-x)
\end{array}
& \longrightarrow &
\begin{array}
[c]{c}\text{Laguerre-type polynomials}\\
k(\alpha)x^{\alpha}\exp(-x)+N\delta(x)
\end{array}
\\
\downarrow & & \downarrow\\\begin{array}
[c]{c}\text{Bessel functions}\\
\kappa(\alpha)x^{2\alpha+1}\end{array}
& \longrightarrow &
\begin{array}
[c]{c}\text{Bessel-type functions}\\
\kappa(\alpha)x^{2\alpha+1}+M\delta(x)
\end{array}
\end{array}
\right. \label{eq3.2}$$ The symbol entry (here $k$ and $\kappa$ are positive numbers depending only on the parameters $\alpha$ and $\beta$) under each special function indicates a non-negative (generalised) weight, on the interval $(-1,1)$ or $(0,\infty),$ involved in:
- the orthogonality property of the special functions
- the weight coefficient in the associated differential equations.
It is important to note in this diagram that:
- a horizontal arrow $\longrightarrow$ indicates a definition process either by a linear combination of special functions of the same type but of different orders, or by a linear-differential combination of special functions of the same type and order (alternatively by an application of the Darboux transform, see [@GH])
- a vertical arrow $\downarrow$ indicates a confluent limit process of one special function to give another special function
- the use of the symbol $M\delta(\cdot)$ is a notational device to indicate that the monotonic function on the real line $\mathbb{R}$ defining the weight has a jump at an end-point of the interval concerned, of magnitude $M>0$
- the combination of any vertical arrow $\downarrow$ with a horizontal arrow $\longrightarrow$ must give a consistent single entry.
Information about the Jacobi-type and Laguerre-type orthogonal polynomials, and their associated differential equations, is given in the Everitt and Littlejohn survey paper [@EL1]; see in particular the references in this paper to the introduction of the fourth-order Laguerre-type differential equation by H.L. and A.M. Krall, Koornwinder and by Littlejohn. The general Laguerre-type differential equation is introduced in the paper [@KK] by Koekoek and Koekoek; the order of this linear differential equation is determined by $4+2\alpha$ with $\alpha\in\mathbb{N}_{0}=\{0,1,2,\cdots\}.$
It is significant that the general order Bessel-type functions also satisfy a linear differential equation of order $4+2\alpha$ (with $\alpha\in
\mathbb{N}_{0}$), being an inheritance from the order of the general Laguerre-type equation.
The purpose of this survey paper is to discuss the properties of the Bessel-type linear differential equation in the special case when $\alpha=0,$ as given in the bottom right-hand corner of the diagram; this is the fourth-order differential equation (\[eq3.1\]) and involves the weight coefficient $\kappa(0)x$; its solutions should, in some sense, have orthogonality properties with respect to the generalised weight function $\kappa(0)x+M\delta(0),$ where $M>0$ is the parameter appearing in the differential equation (\[eq3.1\]); see [@EM Section 4].
Our knowledge of the special function solutions of the Bessel-type differential equation (\[eq3.1\]) is now more complete than at the time the paper [@EM] was written. However, the results in [@EM Section 1, (1.8a)], with $\alpha=0,$ show that the function defined by$$J_{\lambda}^{0,M}(x):=[1+M(\lambda/2)^{2}]J_{0}(\lambda x)-2M(\lambda
/2)^{2}(\lambda x)^{-1}J_{1}(\lambda x)\ \text{for all}\ x\in(0,\infty),
\label{eq3.3}$$ is a solution of the differential equation (\[eq3.1\]), for all $\lambda
\in\mathbb{C},$ and hence for all $\Lambda\in\mathbb{C},$ and all $M>0.$ Here:
- the parameter $M>0$
- the parameter $\lambda\in\mathbb{C}$
- the spectral parameter $\Lambda$ and the parameter $M,$ in the equation (\[eq3.1\]), and the parameters $M$ and $\lambda,$ in the definition (\[eq3.3\]), are connected by the relationship$$\Lambda\equiv\Lambda(\lambda,M)=\lambda^{2}(\lambda^{2}+8M^{-1})\ \text{for
all}\ \lambda\in\mathbb{C}\ \text{and all}\ M>0 \label{eq3.4}$$
- $J_{0}$ and $J_{1}$ are the classical Bessel functions (of the first kind), see [@GNW Chapter III].
Similar arguments to the methods given in [@EM] show that the function defined by$$Y_{\lambda}^{0,M}(x):=[1+M(\lambda/2)^{2}]Y_{0}(\lambda x)-2M(\lambda
/2)^{2}(\lambda x)^{-1}Y_{1}(\lambda x)\ \text{for all}\ x\in(0,\infty),
\label{eq3.5}$$ is also a solution of the differential equation (\[eq3.1\]), for all $\lambda\in\mathbb{C},$ and hence for all $\Lambda\in\mathbb{C}$ and all $M>0;$ here, again, $Y_{0}$ and $Y_{1}$ are classical Bessel functions (of the second kind), see [@GNW Chapter III].
The earlier studies of the fourth-order differential equation (\[eq3.1\]) failed to find any explicit form of two linearly independent solutions, additional to the solutions $J_{\lambda}^{0,M}$ and $Y_{\lambda}^{0,M}.$ However, results of van Hoeij, see [@MvH1] and [@MvH], using the computer algebra program Maple have yielded the required two additional solutions, here given the notation $I_{\lambda}^{0,M}$ and $K_{\lambda}^{0.M},$ with explicit representation in terms of the classical modified Bessel functions $I_{0},K_{0}$ and $I_{1},K_{1}.$ These two additional solutions are defined as follows, where as far as possible we have followed the notation used for the solutions $J_{\lambda}^{0,M}$ and $Y_{\lambda}^{0,M},$
- given $\lambda\in\mathbb{C}$, with $\arg(\lambda)\in\lbrack
0,2\pi),$ $M\in(0,\infty)$ and using the principal value of $\sqrt{\cdot},$ define$$c\equiv c(\lambda,M):=\sqrt{\lambda^{2}+8M^{-1}}\ \text{and}\ d\equiv
d(\lambda,M):=1+M(\lambda/2)^{2} \label{eq3.5a}$$
- define the solution $I_{\lambda}^{0,M}$, for all $x\in
(0,\infty),$$$\begin{array}
[c]{lll}I_{\lambda}^{0,M}(x) & := & -dI_{0}(cx)+\tfrac{1}{2}cMx^{-1}I_{1}(cx)\\
& := & -[1+M(\lambda/2)^{2}]I_{0}\left( x\sqrt{\lambda^{2}+8M^{-1}}\right) \\
& & +\sqrt{M\left( 2+M(\lambda/2)^{2}\right) }x^{-1}I_{1}\left(
x\sqrt{\lambda^{2}+8M^{-1}}\right)
\end{array}
\label{eq3.5b}$$
- define the solution $K_{\lambda}^{0,M}$, for all $x\in
(0,\infty),$$$\begin{array}
[c]{lll}I_{\lambda}^{0,M}(x) & := & dK_{0}(cx)+\tfrac{1}{2}cMx^{-1}K_{1}(cx)\\
& := & [1+M(\lambda/2)^{2}]K_{0}\left( x\sqrt{\lambda^{2}+8M^{-1}}\right) \\
& & +\sqrt{M\left( 2+M(\lambda/2)^{2}\right) }x^{-1}K_{1}\left(
x\sqrt{\lambda^{2}+8M^{-1}}\right)
\end{array}
\label{eq3.5c}$$
\[rem3.1\]We have
1. The four linearly independent solutions $J_{\lambda}^{0.M},Y_{\lambda
}^{0.M},I_{\lambda}^{0.M},K_{\lambda}^{0.M}$ provide a basis for all solutions of the original differential equation (\[eq3.1\]), subject to the $(\Lambda,\lambda)$ connection given in (\[eq3.4\]).
2. These four solutions are real-valued on their domain $(0,\infty)$ for all $\lambda\in\mathbb{R}.$
3. The domain $(0,\infty)$ of the solutions $J_{\lambda}^{0,M}$ and $I_{\lambda}^{0,M}$ can be extended to the closed half-line $[0,\infty)$ with the properties$$J_{\lambda}^{0,M}(0)=I_{\lambda}^{0,M}(0)=1\ \text{for all}\ \lambda
\in\mathbb{R}\ \text{and all}\ M\in(0,\infty).$$
The classical Bessel differential equation, with order $\alpha=0,$ written in a form comparable to the fourth-order equation (\[eq3.1\]), is best taken from the left-hand bottom corner of the diagram (\[eq3.2\]); from [@EM Section 1, (1.2)] with $\alpha=0$ we obtain$$-(xy^{\prime}(x))^{\prime}=\lambda^{2}xy(x)\ \text{for all}\ x\in(0,\infty);
\label{eq3.6}$$ here $\lambda\in\mathbb{C}$ is the spectral parameter. It is to be observed that, formally, if the fourth-order Bessel-type equation (\[eq3.1\]) is multiplied by the parameter $M>0$ and then $M$ tends to zero, we obtain essentially the classical Bessel equation of order zero (\[eq3.6\]), on using the spectral relationship (\[eq3.4\]) between the parameters $\lambda$ and $\Lambda.$ This Bessel differential equation (\[eq3.6\]) has solutions $J_{1}(\lambda x)$ and $Y_{1}(\lambda x)$ for all $x\in(0,\infty)$ and all $\lambda\in\mathbb{C}.$
For the need to apply the Frobenius series method of solution we also consider the differential equation (\[eq3.1\]) on the complex plane $\mathbb{C}$:$$\begin{array}
[c]{r}w^{(4)}(z)+2z^{-1}w^{(3)}(z)-(9z^{-2}+8M^{-1})w^{\prime\prime}(z)\\
+(9z^{-3}-8M^{-1}z^{-1})w^{\prime}(z)-\Lambda w(z)=0
\end{array}
\label{eq3.7}$$ for all $z\in\mathbb{C}.$ In this form the equation has a *regular singularity* at the origin $0,$ and an *irregular singularity* at the point at infinity $\infty$ of the complex plane $\mathbb{C};$ all other points of the plane are *regular* or *ordinary* points for the differential equation. It should be noted that the classical Bessel differential equation (\[eq3.6\]) has the same classification when considered in the complex plane $\mathbb{C}.$
A calculation shows that the Frobenius indicial roots for the regular singularity of the differential equation (\[eq3.7\]) at the origin $0,$ are $\{4,2,0,-2\}.$ The application of the Frobenius series method, using the computer programs [@BELM] and Maple (see [@MvH]), yield four linearly independent series solutions of (\[eq3.7\]), each with infinite radius of convergence in the complex plane $\mathbb{C}.$ If these solutions are labelled to hold for the Bessel-type differential equation (\[eq3.1\]) then we have four solutions $\{y_{r}(\cdot,\Lambda,M):r=4,2,0,-2\},$ to accord with the indicial roots, to give the theorem, see [@DEHLM Section 3]:
\[th3.1\]For all $\Lambda\in\mathbb{C}$ and all $M>0,$ the differential equation $(\ref{eq3.1})$ has four linearly independent solutions $\{y_{r}(\cdot,\Lambda,M):r=4,2,0,-2\}$, defined on $(0,\infty)\times
\mathbb{C},$ with the following series properties as $x\rightarrow0^{+},$ where the $O$-terms depend upon the complex spectral parameter $\Lambda$ and the parameter $M,$$$\left\{
\begin{array}
[c]{lll}y_{4}(x,\Lambda,M) & = & x^{4}+\tfrac{1}{3}M^{-1}x^{6}+O(x^{8})\\
y_{2}(x,\Lambda,M) & = & kx^{2}+O(x^{4}\left\vert \ln(x)\right\vert )\\
y_{0}(x,\Lambda,M) & = & l+O(x^{4}\left\vert \ln(x)\right\vert )\\
y_{-2}(x,\Lambda,M) & = & mx^{-2}+O(\left\vert \ln(x)\right\vert ).
\end{array}
\right. \label{eq3.8}$$ Here the fixed numbers $k,l,m\in\mathbb{R}$ and are independent of the parameters $\Lambda$ and $M;$ these numbers are produced by the Frobenius computer program [@BELM] and have the explicit values:$$k=-(27720)^{-1},\ l=(174636000)^{-1},\ m=-(9779616000)^{-1}.$$
Higher-order differential equations\[sec3a\]
============================================
As mentioned in Section \[sec3\] above there exist Bessel-type linear differential equations of all even-orders $4+2\alpha$, where $\alpha
\in\mathbb{N}_{0}$ is any non-negative integer. The definition and some properties of these differential equations, and the associated Bessel-type functions, are considered in detail in [@EM Sections 2 and 3].
Here we give the form of the sixth-order and eighth-order differential equations, as given in [@EM Section 1, (1.10b) and (1.10c).]. (Note that there is a printing error in the display (1.10b); the numerical factor $255$ is to be replaced by 225. Also printing errors in the display (1.10c) which are now to be corrected using the form of the differential equation (\[eq3.8b\]) below.)
- The sixth-order equation derived from the corrected differential expression for [@EM Section 1, (1.9) and (1.10b)] is$$\begin{array}
[c]{r}-(x^{3}y^{(3)}(x))^{(3)}+(33xy^{\prime\prime}(x))^{\prime\prime}-((225x^{-1}-96M^{-1}x^{3})y^{\prime}(x))^{\prime}\\
=(\lambda^{6}+M^{-1}2^{4}\left( 3!\right) \lambda^{2})x^{3}y(x)\ \text{for
all}\ x\in(0,\infty),
\end{array}
\label{eq3.8a}$$ where, as before, the parameters $M\in(0,\infty)$ and $\lambda\in\mathbb{C}.$
When this equation is considered in the complex plane $\mathbb{C}$ the Frobenius indicial roots for the regular singularity at the origin $0$ are $\{6,4,2,0,-2,-4\},$ using the methods provided by [@MvH].
- The eighth-order equation derived from the corrected differential expression for [@EM Section 1, (1.9) and (1.10c)] is $$\begin{array}
[c]{r}(x^{5}y^{(4)}(x))^{(4)}-(78x^{3}y^{(3)}(x))^{(3)}+(1809xy^{\prime\prime
}(x))^{\prime\prime}\\
-((11025x^{-1}-2^{6}(4!)M^{-1}x^{5})y^{\prime}(x))^{\prime}\\
=(\lambda^{8}+M^{-1}2^{6}(4!)\lambda^{2})x^{5}y(x)\ \text{for all}\ x\in(0,\infty),
\end{array}
\label{eq3.8b}$$ where, as before, the parameters $M\in(0,\infty)$ and $\lambda\in\mathbb{C}.$
When this equation is considered in the complex plane $\mathbb{C}$ the Frobenius indicial roots for the regular singularity at the origin $0$ are $\{8,6,4,2,0,-2,-4,-6\},$ using the methods provided by [@MvH].
We note that, formally, if the equations in (\[eq3.8a\]) and (\[eq3.8b\]) are multiplied by $M>0,$ and then letting $M$ tend to zero we obtain, respectively, the two Sturm-Liouville differential equations, see [@EM Section 1, (1.2)],$$-(x^{3}y^{\prime}(x))^{\prime}=\lambda^{2}x^{3}y(x)\ \text{and}\ -(x^{5}y^{\prime}(x))^{\prime}=\lambda^{2}x^{5}y(x)\ \text{for all}\ x\in(0,\infty).
\label{eq3.8c}$$ For the solutions of these equations in classical Bessel functions see [@EM Section 1, (1.4)].
The fourth-order differential expression $L_{M}$\[sec4\]
========================================================
We define the differential expression $L_{M}$ with domain $D(L_{M})$ as follows:$$D(L_{M}):=\{f:(0,\infty)\rightarrow\mathbb{C}:f^{(r)}\in AC_{\text{loc}}(0,\infty)\;\text{for}\;r=0,1,2,3\}, \label{eq4.1}$$ and for all $f\in D(L_{M})$$$L_{M}[f](x):=(xf^{\prime\prime}(x))^{\prime\prime}-\left( (9x^{-1}+8M^{-1}x)f^{\prime}(x)\right) ^{\prime}\;(x\in(0,\infty)); \label{eq4.2}$$ it follows that$$L_{M}:D(L_{M})\rightarrow L_{\text{loc}}^{1}(0,\infty). \label{eq4.3}$$
The Green’s formula for $L_{M}$ on any compact interval $[\alpha,\beta
]\subset(0,+\infty)$ is given by$$\int_{\alpha}^{\beta}\left\{ \overline{g}(x)L_{M}[f](x)-f(x)\overline
{L_{M}[g]}(x)\right\} dx=\left. [f,g](x)\right\vert _{\alpha}^{\beta},
\label{eq4.5}$$ where the symplectic form $[\cdot,\cdot](\cdot):D(L_{M})\times D(L_{M})\times(0,+\infty)\rightarrow\mathbb{C}$ is defined by$$\begin{aligned}
\lbrack f,g](x) & :=\overline{g}(x)(xf^{\prime\prime}(x))^{\prime
}-(x\overline{g}^{\prime\prime}(x))^{\prime}f(x)\nonumber\\
& \quad-x\left( \overline{g}^{\prime}(x)f^{\prime\prime}(x)-\overline
{g}^{\prime\prime}(x)f^{\prime}(x)\right) \nonumber\\
& \quad-\left( 9x^{-1}+8M^{-1}x\right) \left( \overline{g}(x)f^{\prime
}(x)-\overline{g}^{\prime}(x)f(x)\right) . \label{eq4.6}$$
The Dirichlet formula for $L_{M}$ on any compact interval $[\alpha
,\beta]\subset(0,+\infty)$ is given by$$\begin{array}
[c]{c}{\displaystyle\int_{\alpha}^{\beta}}
\left\{ xf^{\prime\prime}(x)\overline{g}^{\prime\prime}(x)+\left(
9x^{-1}+8M^{-1}x\right) f^{\prime}(x)\overline{g}^{\prime}(x)\right\} dx\\
=\left. [f,g]_{D}(x)\right\vert _{\alpha}^{\beta}+{\displaystyle\int_{\alpha}^{\beta}}
L_{M}[f](x)\overline{g}(x)\,dx,
\end{array}
\label{eq4.7}$$ where the Dirichlet form $[\cdot,\cdot]_{D}:D(L_{M})\times D_{0}(L_{M})\times(0,+\infty)\rightarrow\mathbb{C}$ is defined by, for $f\in D(L_{M})$ and $g\in D_{0}(L_{M}),$ with$$D_{0}(L_{M}):=\{g:(0,+\infty)\rightarrow\mathbb{C}:g^{(r)}\in AC_{\text{loc}}(0,+\infty)\;\text{for}\;r=0,1\} \label{eq4.8}$$ and$$\lbrack f,g]_{D}(x):=-\overline{g}(x)\left( xf^{\prime\prime}(x)\right)
^{\prime}+\overline{g}^{\prime}(x)xf^{\prime\prime}(x)+\overline{g}(x)\left(
9x^{-1}+8M^{-1}x\right) f^{\prime}(x). \label{eq4.9}$$
Hilbert function spaces\[sec5\]
===============================
The spectral properties of the fourth-order Bessel differential equation$$(xy^{\prime\prime}(x))^{\prime\prime}-((9x^{-1}+8M^{-1}x)y^{\prime
}(x))^{\prime}=\Lambda xy(x)\ \text{for all}\ x\in(0,\infty), \label{eq5.1}$$ with $\Lambda\in\mathbb{C}$ as the spectral parameter, are considered in two Hilbert function spaces:
1. The Lebesgue weighted space$$L^{2}((0,\infty);x):=\left\{ f:(0,+\infty)\rightarrow\mathbb{C}:\int
_{0}^{\infty}x\left\vert f(x)\right\vert ^{2}\,dx<+\infty\right\}
\label{eq5.2}$$ with inner-product and norm defined by, for all $f,g\in L^{2}((0,\infty);x),$$$(f,g):=\int_{0}^{\infty}xf(x)\overline{g}(x)~\,dx\quad\text{and}\quad\left\Vert f\right\Vert :=(f,f)^{1/2}. \label{eq5.3}$$ This space takes into account the weight function $x$ on the right-hand side of (\[eq5.1\]).
2. The Lebesgue-Stieltjes jump space $L^{2}([0,\infty);m_{k})$, as suggested by the results in [@EM Section 4].
Let the monotonic non-decreasing function $\hat{m}_{k}:[0,\infty
)\rightarrow\lbrack0,\infty)$ be defined by, where $k>0$ is a real parameter,$$\begin{aligned}
\hat{m}_{k}(x) & =-k\ \text{for }x=0\\
& =x^{2}/2\ \text{for all}\ x\in(0,+\infty).\end{aligned}$$ Then $\hat{m}_{k}$ generates a Baire measure $m_{k}$ on the $\sigma$-algebra $\mathcal{B}$ of Borel sets on the interval $[0,\infty)$; in turn this measure generates a Lebesgue-Stieltjes integral for Borel measurable functions.
The Hilbert function space $L^{2}([0,\infty);m_{k})$ is defined on all functions with the properties:$$\begin{array}
[c]{ll}(i) & f:[0,\infty)\rightarrow\mathbb{C\ }\text{and is Borel measurable
on}\ [0,\infty)\\
(ii) &
{\displaystyle\int_{0}^{\infty}}
x\left\vert f(x)\right\vert ^{2}dx<+\infty.
\end{array}$$ The norm and inner-product in $L^{2}([0,\infty);m_{k})$ are defined by$$\left\Vert f\right\Vert _{k}^{2}:=\int_{[0,\infty)}\left\vert f(x)\right\vert
^{2}dm_{k}(x)=k\left\vert f(0)\right\vert ^{2}+\int_{0}^{\infty}x\left\vert
f(x)\right\vert ^{2}dx \label{eq5.4}$$ and$$(f,g)_{k}:=\int_{[0,\infty)}f(x)\overline{g}(x)~dm_{k}(x)=kf(0)\overline
{g}(0)+\int_{0}^{\infty}xf(x)\overline{g}(x)~dx. \label{eq5.5}$$ Note that the first integrals in both these definitions are Lebesgue-Stieltjes integrals taken over the set $[0,\infty),$ whilst the second integrals can be taken as Lebesgue integrals.
Differential operators generated by $L_{M}$\[sec5a\]
====================================================
The Lagrange symmetric differential expression $L_{M}$ generates self-adjoint operators in both the Hilbert function spaces $L^{2}((0,\infty);x)$, and in $L^{2}([0,\infty);m_{k})$ for all $k\in(0,\infty).$
In the space $L^{2}((0,\infty);x)$ the expression $L_{M}$ generates a continuum $\{T\}$ of self-adjoint operators, including the significant Friedrichs operator $F$; these properties are developed and considered in Sections \[sec6\] to \[sec11\] below.
For each $k\in(0,\infty)$ the expression $L_{M}$ generates a unique self-adjoint operator $S_{k}$ in the space $L^{2}([0,\infty);m_{k});$ the properties of this operator are considered in Sections \[sec12\] and \[sec13\].
Differential operators in $L^{2}((0,\infty);x)$\[sec6\]
=======================================================
The maximal and the minimal differential operators, denoted respectively $T_{1}$ and $T_{0},$ as generated by the differential expression $L_{M}$ in the Hilbert function space $L^{2}((0,\infty);x),$ are defined as follows, see [@MAN Chapter V, Section 17]:
- $T_{1}:D(T_{1})\subset L^{2}((0,\infty);x)\rightarrow
L^{2}((0,\infty);x)$ by$$D(T_{1}):=\{f\in D(L_{M}):f,x^{-1}L_{M}(f)\in L^{2}((0,\infty);x)\}
\label{eq6.1}$$ and$$T_{1}f:=x^{-1}L_{M}(f)\ \text{for all}\ f\in D(T_{1}). \label{eq6.2}$$ From the Green’s formula (\[eq4.5\]) if follows that the limits$$\lbrack f,g](0^{+}):=\lim_{x\rightarrow0}[f,g](x)\ \text{and}\ [f,g](\infty
):=\lim_{x\rightarrow\infty}[f,g](x) \label{eq6.3}$$ both exist and are finite in $\mathbb{C}$ for all $f,g\in D(T_{1}).$
- $T_{0}:D(T_{0})\subset L^{2}((0,\infty);x)\rightarrow
L^{2}((0,\infty);x)$ by$$\begin{array}
[c]{r}D(T_{0}):=\{f\in D(T_{1}):\lim_{x\rightarrow0}[f,g](x)=0\ \text{and\qquad
\qquad}\\
\lim_{x\rightarrow\infty}[f,g](x)=0\text{ for all}\ f,g\in D(T_{1})\},
\end{array}
\label{eq6.4}$$
and$$T_{0}f:=x^{-1}L_{M}(f)\ \text{for all}\ f\in D(T_{0}). \label{eq6.5}$$
From standard results we have the operator properties, see [@MAN Chapter V],$$T_{0}\subseteq T_{1},T_{0}^{\ast}=T_{1}\ \text{and}\ T_{1}^{\ast}=T_{0},
\label{eq6.6}$$ thereby noting that both $T_{0}$ and $T_{1}$ are closed linear operators in $L^{2}((0,\infty);x).$
Self-adjoint operators in $L^{2}((0,\infty);x)$\[sec7\]
=======================================================
In the weighted space $L^{2}((0,\infty);x)$ the Lagrange symmetric (formally self-adjoint) differential expression has the following endpoint classifications at the singular endpoints $0$ and $+\infty$ (for additional details see [@DEHLM Section 6]):
- At $0^{+}$ the singular endpoint is limit-3 in $L^{2}((0,\infty);x)$
- At $+\infty$ the singular endpoint is Dirichlet and strong limit-2 in $L^{2}((0,\infty);x).$
Based on this information the self-adjoint extensions of the closed symmetric operator $T_{0}$ are determined by the GKN-theorem on singular boundary conditions as given in [@MAN Chapter V] and [@EM1]. In particular, for the operators $T_{0}$ and $T_{1},$ any self-adjoint operator $T=T^{\ast}$ generated by $L_{M}$ in $L^{2}((0,\infty);x)$ is a one-dimensional extension of $T_{0}$ or, equivalently, a one-dimensonal restriction of $T_{1}.$ Let the domain $D(T)$ as a restriction of the domain $D(T_{1})$ be determined by$$D(T):=\{f\in D(T_{1}):[f,\varphi](0)=0\}, \label{eq7.1}$$ where the function $\varphi\in D(T_{1})$ is a non-null element of the quotient space $D(T_{1})\diagup D(T_{0})$ which satisfies the GKN symmetry condition$$\lbrack\varphi,\varphi](0)=0. \label{eq7.2}$$ Then the differential operator $T$ defined by$$Tf:=x^{-1}L_{M}[f]\ \text{for all}\ f\in D(T) \label{eq7.3}$$ satisfies $T^{\ast}=T,$ and is self-adjoint in the Hilbert space $L^{2}((0,\infty);x).$ All such self-adjoint operators are determined in this way on making an appropriate choice of the boundary condition function $\varphi.$
Boundary properties at $0^{+}$\[sec8\]
======================================
The results of the following theorem are essential to obtaining the explicit forms of the boundary conditions at $0^{+}$ to determine all self-adjoint extensions of $T_{0}.$
\[th8.1\]Let $f\in D(T_{1});$ then the values of $f,f^{\prime},f^{\prime\prime}$ can be defined at the point $0$ so that the following results hold:
- $f\in AC[0,1]$
- $f^{\prime}\in AC[0,1]$ and $f^{\prime}(0)=0$
- $f^{\prime\prime}\in AC_{\text{loc}}(0,1]$ and $f^{\prime
\prime}\in C[0,1]$
- $f^{(3)}\in AC_{\text{loc}}(0,1]$ and $\lim_{x\rightarrow0^{+}}(xf^{(3)}(x))=0.$
For the proof of this theorem see [@DEHLM Section 8].
We consider the functions $1,x,x^{2}$ on the interval $[0,1]$ but patched, see the Naimark patching lemma [@MAN Chapter V, Section 17.3, Lemma 2], to zero on $[2,\infty)$ in such a manner that the patched functions belong to the domain $D(T_{1})$; we continue to use the symbols $1,x,x^{2}$ for the patched functions.
A calculation shows that the results given in the next lemma are satisfied:
\[lem8.1\]The patched functions $1,x,x^{2}$ have the following limit properties in respect of the symplectic form $[\cdot,\cdot]$ and the maximal domain $D(T_{1})$:
- $1,x^{2}\in D(T_{1})$ but $x\notin D(T_{1})$
- $[1,1](0^{+})=[x,x](0^{+})=[x^{2},x^{2}](0^{+})=0$
- $[x,x^{2}](0^{+})=0$ and $[1,x^{2}](0^{+})=16$
- $[1,x](0^{+})$ does not exist.
The lemmas and corollaries now given below are taken from [@DEHLM Section 9], where proofs are given in detail.
The results of Theorem \[th8.1\] and Lemma \[lem8.1\] now provide a basis for the two-dimensional quotient space $D(T_{1})/D(T_{0})$;$$D(T_{1})/D(T_{0})=\mathrm{span}\{1,x^{2}\}=\{a+bx^{2}:a,b\in\mathbb{C}\}.
\label{eq8.1}$$ The linear independence of the functions $\{1,x^{2}\}$ within the the quotient space follows from the property $[1,x^{2}](0^{+})=16\neq0.$
A calculation now gives, recall Theorem \[th8.1\],
\[lem8.2\]Let $f\in D(T_{1});$ then the following identities hold:
- $[f,1](0^{+})=-8f^{\prime\prime}(0)$
- $[f,x^{2}](0^{+})=16f(0).$
Similarly we have
\[lem8.3\]Let $f,g\in D(T_{1});$ then
- $[f,g](0^{+})=8\left[ f(0)\overline{g}^{\prime\prime
}(0)-f^{\prime\prime}(0)\overline{g}(0)\right] $
- $\left[ f,g\right] _{D}(0^{+})=8f^{\prime\prime}(0)\overline{g}(0).$
We have the corollaries:
\[cor8.1\]The domain of the minimal operator $T_{0}$ is determined explicitly by$$D(T_{0})=\{f\in D(T_{1}):f(0)=0\ \text{and}\ f^{\prime\prime}(0)=0\}.
\label{eq8.2}$$
\[cor8.3\]For all $f\in D(T_{1})$ $$\int_{0}^{\infty}\left\{ x\left\vert f^{\prime\prime}(x)\right\vert
^{2}+\left( 9x^{-1}+8M^{-1}x\right) \left\vert f^{\prime}(x)\right\vert
^{2}\right\} dx<\infty. \label{eq8.4}$$
\[cor8.4\]For all $f,g\in D(T_{1})$ the Dirichlet formula takes the form$$(T_{1}f,g)=8f^{\prime\prime}(0)\overline{g}(0)+\int_{0}^{\infty}\left\{
xf^{\prime\prime}(x)\overline{g}^{\prime\prime}(x)+\left( 9x^{-1}+8M^{-1}x\right) f^{\prime}(x)\overline{g}^{\prime}(x)\right\} dx.
\label{eq8.5}$$
Explicit boundary condition functions at $0^{+}$\[sec9\]
========================================================
We can now determine all forms of the boundary condition function $\varphi$ satisfying the symmetry condition (\[eq7.2\]) to determine the domain of all self-adjoint extensions $T$ of the minimal operator $T_{0}.$
\[lem9.1\]All self-adjoint extensions $T$ of $T_{0}$ generated by the differential expression $L_{M}$ in $L^{2}((0,\infty;x)$ are determined by, using the patched functions $1,x^{2},$$$D(T):=\{f\in D(T_{1}):[f,\varphi](0^{+})=0\ \text{where}\
\begin{array}
[t]{ll}(i) & \varphi(x)=\alpha+\beta x^{2}\\
(ii) & \alpha,\beta\in\mathbb{R}\ \text{and}\ \alpha^{2}+\beta^{2}\neq0\}.
\end{array}
\ \label{eq9.1}$$ and$$\left( Tf\right) (x):=x^{-1}L_{M}(f)(x)\ \text{for all}\ x\in(0,\infty
)\ \text{and all}\ f\in D(T). \label{eq9.2}$$
There is an equivalent form of this last result, using the results of Lemma \[lem8.2\]:
\[lem9.2\]All self-adjoint extensions $T$ of $T_{0}$ generated by the differential expression $L_{M}$ in $L^{2}((0,\infty;x)$ are determined by$$D(T):=\{f\in D(T_{1}):\begin{array}
[t]{ll}(i) & -\alpha f^{\prime\prime}(0)+2\beta f(0)=0\\
(ii) & \alpha,\beta\in\mathbb{R}\ \text{and}\ \alpha^{2}+\beta^{2}\neq0\}.
\end{array}
\label{eq9.3}$$ and$$\left( Tf\right) (x):=x^{-1}L_{M}(f)(x)\ \text{for all}\ x\in(0,\infty
)\ \text{and all}\ f\in D(T). \label{eq9.4}$$
\[rem9.1\]We note the two special cases:
- When $\alpha=0$ the boundary condition is $f(0)=0;$ this boundary condition plays a special role, and gives an explicit form of the domain of the Friedrichs extension $F$ of $T_{0};$ see Section \[sec11\] below.
- When $\beta=0$ the boundary condition is $f^{\prime\prime
}(0)=0.$
Spectral properties of the fourth-order Bessel-type operators\[sec10\]
======================================================================
\[th10.1\]The minimal operator $T_{0},$ defined in $(\ref{eq6.4})$ and $(\ref{eq6.5}),$ is bounded below in the space $L^{2}((0,\infty);x)$ by the null operator $O,$ *i.e.*$$(T_{0}f,f)\geq0\ \text{for all}\ f\in D(T_{0}). \label{eq10.1}$$
Since $T_{0}$ is a restriction of the maximal operator $T_{1}$ the result of Corollary \[cor8.4\] can be applied to give, using also Corollary \[cor8.1\],$$\begin{array}
[c]{c}(T_{0}f,f)=8f^{\prime\prime}(0)\overline{f}(0)+{\displaystyle\int_{0}^{\infty}}
\left\{ x\left\vert f^{\prime\prime}(x)\right\vert ^{2}+\left(
9x^{-1}+8M^{-1}x\right) \left\vert f^{\prime}(x)\right\vert ^{2}\right\}
dx\\
={\displaystyle\int_{0}^{\infty}}
\left\{ x\left\vert f^{\prime\prime}(x)\right\vert ^{2}+\left(
9x^{-1}+8M^{-1}x\right) \left\vert f^{\prime}(x)\right\vert ^{2}\right\}
dx\geq0\
\end{array}$$ for all $f\in D(T_{0}).$
\[th10.2\]
1. Let $T$ be a self-adjoint extension of $T_{0};$ then:
- The essential spectrum $\sigma_{\text{ess}}(T)$ is given by$$\sigma_{\text{ess}}(T)=\sigma_{\text{cont}}(T)=[0,\infty). \label{eq10.2}$$
- There are no embedded eigenvalues of $T$ in the essential spectrum.
- $T$ has at most one eigenvalue; if this eigenvalue is present then it is simple and lies in the interval $(-\infty,0).$
2. Every point $\mu\in(-\infty,0)$ is the eigenvalue of some unique self-adjoint extension $T$ of $T_{0}.$
The proof of this theorem is given in detail in [@DEHLM Section 13].
The Friedrichs extension $F$\[sec11\]
=====================================
The closed symmetric operator $T_{0}$ is bounded below in $L^{2}((0,\infty);x),$ see Theorem \[th10.1\], and the general theory of such operators implies the existence of a distinguished self-adjoint extension $F,$ called the Friedrichs extension of $T_{0}.$
This Friedrichs operator has the properties:
- $T_{0}\subset F=F^{\ast}\subset T_{1}$
- $D(F)=\{f\in D(T_{1}):f(0)=0\}$
- The essential spectrum $\sigma_{\text{ess}}(F)$ is given by$$\sigma_{\text{ess}}(F)=\sigma_{\text{cont}}(F)=[0,\infty) \label{eq11.1}$$
- $F$ has no eigenvalues.
For a discussion of the definition and properties of this Friedrichs extension see [@DEHLM Section 15].
Self-adjoint operator $S_{k}$ in $L^{2}([0,\infty);m_{k})$\[sec12\]
===================================================================
In this section, given any $k\in(0,\infty),$ we define the operator $S_{k}$ generated by the differential expression $L_{M}$ in the Hilbert function space $L^{2}([0,\infty);m_{k}),$ where this space is defined in Section \[sec5\] above.
\[def12.1\]Let $k\in(0,\infty)$ be given$;$ then the operator $S_{k}$$$S_{k}:D(S_{k})\subset L^{2}([0,\infty);m_{k})\rightarrow L^{2}([0,\infty
);m_{k}) \label{eq12.1}$$ is defined by $($see $(\ref{eq6.1})$ and $(\ref{eq6.2}),$ and Theorem $\ref{th8.1}$ for the definition and properties of the domain $D(T_{1})\subset
L^{2}((0,\infty);x))$
- $D(S_{k}):=D(T_{1})$
- for all $f\in D(S_{k})$$$\left\{
\begin{array}
[c]{lll}\left( S_{k}f\right) (x) & := & -8k^{-1}f^{\prime\prime}(0)\ \text{for}\ x=0\\
& := & x^{-1}L_{M}[f](x)\ \text{for all}\ x\in(0,\infty).
\end{array}
\right. \label{eq12.2}$$
\[th12.1\]For all $k\in(0,\infty)$:
- The linear manifold $D(S_{k})$ is dense in $L^{2}([0,\infty
);m_{k}).$
- The operator $S_{k}$ is hermitian in $L^{2}([0,\infty);m_{k}).$
- The operator $S_{k}$ is symmetric in $L^{2}([0,\infty);m_{k}).$
- The operator $S_{k}$ is bounded below in $L^{2}([0,\infty
);m_{k})$$$(S_{k}f,f)_{k}\geq0\ \text{for all}\ f\in D(S_{k}). \label{eq12.3}$$
For the proof of this theorem see [@EKLM Theorem 5.2].
\[th12.2\]Let $k\in(0,\infty)$ be given; then the symmetric operator $S_{k}$ on the domain $D(S_{k})$ is self-adjoint in the Hilbert function space $L^{2}([0,\infty);m_{k}).$
For the proof of this theorem see [@EKLM Theorem 5.4].
\[th12.3\]Let $k\in(0,\infty)$ be given; then the operator $S_{k}$ on the domain $D(S_{k})$ is the unique self-adjoint operator generated by the differential expression $L_{M}$ in the Hilbert function space $L^{2}([0,\infty);m_{k}).$
For the proof of this theorem see [@EKLM Theorem 5.5].
Spectral properties of the self-adjoint operator $S_{k}$\[sec13\]
=================================================================
The spectral properties of the self-adjoint operator $S_{k}$ in $L^{2}([0,\infty);m_{k})$ are given by
\[th13.1\]For any $k\in(0,\infty)$ let the self-adjoint operator $S_{k}$ in $L^{2}([0,\infty);m_{k})$ be defined as in Definition $\ref{def12.1}$ above; then the spectrum $\sigma(S_{k})$ of $S_{k}$ has the following properties$:$
- $S_{k}$ has no eigenvalues
- the essential spectrum of $S_{k}$ is given by$$\sigma_{\text{ess}}(S_{k})=\sigma_{\text{cont}}(S_{k})=[0,\infty).
\label{eq13.1}$$
For the proof of this theorem see [@EKLM Theorem 6.1].
Distributional orthogonality relationships\[sec14\]
===================================================
Recall that from the properties of the classical Bessel function $J_{0}$ we have the result that $J_{0}(\cdot)$ $\notin L^{2}((0,\infty);x)$. However from [@EM Section 1, (1.7)] we have the following distributional (Schwartzian) orthogonal relationship for the classical Bessel function $J_{0},$ in the space $\mathcal{D}^{\prime}$ of distributions,$$\lambda\int_{0}^{\infty}xJ_{0}(\lambda x)J_{0}(\mu x)~dx=\delta(\lambda
-\mu)\ \text{for all}\ \lambda,\mu\in(0,\infty); \label{eq14.1}$$ here $\delta\in\mathcal{D}^{\prime}$ is the Dirac delta distribution. This is the generalised orthogonality property for the solutions $J_{0}$ of the classical Bessel differential equation, of order $0,$ given by (\[eq3.6\]); this result mirrors the spectral properties of this equation, when considered on the half-line $(0,\infty),$ in the space $L^{2}((0,\infty);x);$ in particular the result that every self-adjoint extension $T$ of the corresponding minimal operator $T_{0}$ has the property $\sigma_{\text{ess}}(T)=[0,\infty).$
The distributional proof of (\[eq14.1\]) is discussed in the forthcoming paper [@EKLM1], where the result is also related to the properties of infinite integrals of Bessel functions as originated by Hankel, see [@GNW Chapter XIII].
As above for the Bessel function $J_{0}$ we have, from the explicit representation (\[eq3.3\]), the fourth-order Bessel-type function $J_{\lambda}^{0,M}\notin L^{2}([0,\infty);m_{k})$ for all $k,M\in(0,\infty).$
To obtain a distributional orthogonality for $J_{\lambda}^{0,M}$, given any $M>0$, it is necessary to choose a special value of the parameter $k,$ *i.e.* $k=M/2.$ Then it is shown in [@EM Section 4, Corollary 4.3] that we have the following distributional (Schwartzian) orthogonal relationship for the fourth-order Bessel-type function $J_{\lambda}^{0,M},$ in the space $\mathcal{D}^{\prime}$ of distributions,$$\begin{array}
[c]{r}\lambda\left[ 1+M(\lambda/2)^{2}\right] ^{-2}\left\{
{\displaystyle\int_{0}^{\infty}}
xJ_{\lambda}^{0,M}(x)J_{\mu}^{0,M}(x)~dx+\tfrac{1}{2}MJ_{\lambda}^{0,M}(0)J_{\mu}^{0,M}(0)\right\} \\
=\delta(\lambda-\mu)\ \text{for all}\ \lambda,\mu\in(0,\infty).
\end{array}
\label{eq14.2}$$ The distributional proof of (\[eq14.2\]) is discussed in the forthcoming paper [@EKLM1].
As a formal representation it follows that (\[eq14.2\]) may be written as, using the inner-product for the space $L^{2}\left( [0,\infty);m_{M/2}\right)
,$$$\lambda\left[ 1+M(\lambda/2)^{2}\right] ^{-2}\left( J_{\lambda}^{0,M}(\cdot),J_{\mu}^{0,M}(\cdot)\right) _{M/2}=\delta(\lambda-\mu)~\text{for
all}\ \lambda,\mu\in(0,\infty). \label{eq14.3}$$
As another connection between the classical Bessel (\[eq3.6\]) and the fourth-order Bessel-type (\[eq3.1\]) differential equations it is to be noted that, formally, the orthogonality result (\[eq14.2\]) tends to the orthogonality result (\[eq14.1\]), as the parameter $M$ tends to zero.
The generalised Hankel transform\[sec15\]
=========================================
From the general theory of symmetric integrable-square transforms given in [@ECT Chapter VIII] one form of the classical Hankel transform, for the Bessel function $J_{0}$ and working in the Hilbert function space $L^{2}((0,\infty;x),$ is:
- Let $f\in L^{2}((0,\infty);x)$ then the Hankel transform $g\in
L^{2}((0,\infty);s)$ is given by, for $s\in(0,\infty),$$$g(s)=\int_{0}^{\infty}\xi J_{0}(s\xi)f(\xi)~d\xi\label{eq15.1}$$ with convergence of the integral in $L^{2}((0,\infty);s)$
- With $g\in L^{2}((0,\infty);s)$ the inverse transform, to recover $f,$ is given by, for $x\in(0,\infty),$$$f(x)=\int_{0}^{\infty}sJ_{0}(xs)g(s)~ds \label{eq15.2}$$
with convergence of the integral in $L^{2}((0,\infty);x)$
- The Parseval relation holds between $g$ and $f$$$\int_{0}^{\infty}x\left\vert f(x)\right\vert ^{2}dx=\int_{0}^{\infty
}s\left\vert g(s)\right\vert ^{2}ds. \label{eq15.3}$$
There is also a direct convergence form of the Hankel transform which is best written as, starting with $f\in L^{1}((0,\infty);x),$$$f(x)=\int_{0}^{\infty}sJ_{0}(xs)~ds\int_{0}^{\infty}\xi J_{0}(s\xi)f(\xi
)~d\xi\label{eq15.4}$$ with $x\in(0,\infty).$ Here the integrals are Lebesgue or limits of Lebesgue integrals as discussed in [@ECT Chapter VIII].
There is an equivalent generalised Hankel transform involving the fourth-order Bessel-type function $J_{\lambda}^{0,M}(\cdot)$ and working now in the Hilbert function space $L^{2}\left( [0,\infty);m_{M/2}\right) ;$ note again these results require the unique choice of $k=M/2.$
The complete discussion of the following results for the generalised Hankel transform are to be found in the forthcoming paper [@EKLM1].
To state these results the Lebesgue-Stieltjes Hilbert function space $L^{2}((0,\infty);n)$ is required. Let the function $\hat{n}:[0,\infty
)\rightarrow\lbrack0,\infty)$ be defined by$$\hat{n}(\lambda):=\tfrac{1}{2}\lambda^{2}\left[ 1+M(\lambda/2)^{2}\right]
^{-1}\ \text{for all}\ \lambda\in\lbrack0,\infty); \label{eq15.5}$$ then$$\hat{n}^{\prime}(\lambda)=\lambda\left[ 1+M(\lambda/2)^{2}\right] ^{-2}\geq0\ \text{for all}\ \lambda\in\lbrack0,\infty)$$ so that $\hat{n}$ is monotonic increasing on $[0,\infty)$ and generates a Baire measure on the $\sigma$-algebra $\mathcal{B}$ of Borel sets on the interval $[0,\infty).$ The Hilbert space $L^{2}((0,\infty);n)$ is then defined as the set of all Borel measurable complex-valued functions $f$ on $[0,\infty)$ such that$$\int_{\lbrack0,\infty)}\left\vert f(\lambda)\right\vert ^{2}dn(\lambda
)<+\infty,$$ with norm and inner-product defined by$$\left\Vert f\right\Vert _{n}^{2}:=\int_{0}^{\infty}\left\vert f(\lambda
)\right\vert ^{2}dn(\lambda)=\int_{0}^{\infty}\left\vert f(\lambda)\right\vert
^{2}\lambda\left[ 1+M(\lambda/2)^{2}\right] ^{-2}d\lambda\label{eq15.6}$$$$(f,g)_{n}=\int_{[0,\infty)}f(\lambda)\overline{g}(\lambda)dn(\lambda)=\int
_{0}^{\infty}f(\lambda)\overline{g}(\lambda)\lambda\left[ 1+M(\lambda
/2)^{2}\right] ^{-2}d\lambda.$$
\[rem15.0\]This norm $\left\Vert \cdot\right\Vert _{n}$ and inner-product $(\cdot,\cdot)_{n}$ for the space $L^{2}((0,\infty);n)$ are not to be confused with the norm $\left\Vert \cdot\right\Vert _{k}$ and inner-product $(\cdot,\cdot)_{k},$ introduced in Section \[sec5\], for the space $L^{2}([0,\infty);m_{k}).$
We note that the weight function $\lambda\longmapsto\lambda\left[
1+M(\lambda/2)^{2}\right] ^{-2}$ in the integral in (\[eq15.6\]) is the factor in the distributional orthogonal relationships (\[eq14.2\]) and (\[eq14.3\]).
1. The $L^{2}$-theory of the generalised Hankel transform is given by the following results:
\[th15.1\]Let $f\in L^{2}\left( [0,\infty);m_{M/2}\right) .$ Then there exists exactly one function $g\in L^{2}((0,\infty);n)$ with the property that$$\int_{0}^{\infty}\left\vert g(\lambda)\right\vert ^{2}dn(\lambda
)=\int_{[0,\infty)}\left\vert f(x)\right\vert ^{2}dm_{M/2}(x); \label{eq15.7}$$ here $g$ is defined by, for almost all $\lambda\in(0,\infty),$$$\left( \mathcal{F}_{M}f\right) (\lambda):=g(\lambda)=\int_{[0,\infty
)}J_{\lambda}^{0,M}(x)f(x)dm_{M/2}(x), \label{eq15.8}$$ thereby defining also the generalised Hankel operator $$\mathcal{F}_{M}:L^{2}\left( [0,\infty);m_{M/2}\right) \rightarrow
L^{2}((0,\infty);n). \label{eq15.9}$$
In addition $g$ satisfies$$\int_{0}^{\infty}g(\lambda)dn(\lambda)=f(0). \label{eq15.10}$$
\[rem15.1\]Note that the result (\[eq15.8\]) has to be interpreted as follows, in $(i)$ and $(ii)$:
- ${\displaystyle\int_{[0,X]}}
J_{\lambda}^{0,M}(x)f(x)dm_{M/2}(x)\in L^{2}((0,\infty);n)\ $for all$\ X\in\lbrack0,\infty)$
- $$\lim_{X\rightarrow\infty}\int_{0}^{\infty}\left\vert g(\lambda)-\int
_{[0,X]}J_{\lambda}^{0,M}(x)f(x)dm_{M/2}(x)\right\vert ^{2}dn(\lambda)=0.$$
- Note that the Cauchy-Schwarz inequality shows that if $g\in
L^{2}((0,\infty);n)$ then $g\in L^{1}((0,\infty);n).$
\[th15.2\]Let $g\in L^{2}((0,\infty);n).$ Then there is exactly one function $f\in L^{2}\left( [0,\infty);m_{M/2}\right) $ with the property that $(\ref{eq15.7})$ is satisfied; here $f$ is defined by$$\left( \mathcal{G}_{M}g\right) (x):=f(x):=\left\{
\begin{array}
[c]{ll}{\displaystyle\int_{0}^{\infty}}
g(\lambda)dn(\lambda) & \text{if}\ x=0\\
& \\{\displaystyle\int_{0}^{\infty}}
J_{\lambda}^{0,M}(x)g(\lambda)dn(\lambda) & \text{for}\ x\in(0,\infty),
\end{array}
\right. \label{eq15.11}$$ thereby defining also the inverse generalised Hankel operator$$\mathcal{G}_{M}:L^{2}((0,\infty);n)\rightarrow L^{2}\left( [0,\infty
);m_{M/2}\right) . \label{eq15.12}$$
\[rem15.2\]Note that the result (\[eq15.11\]) has to be interpreted as follows:
- ${\displaystyle\int_{0}^{\Lambda}}
J_{\lambda}^{0,M}(x)g(\lambda)dn(\lambda)\in L^{2}\left( [0,\infty
);m_{M/2}\right) \ $for all$\ \Lambda\in(0,\infty)$
- $$\lim_{\Lambda\rightarrow\infty}\int_{0}^{\infty}\left\vert f(x)-{\displaystyle\int_{0}^{\Lambda}}
J_{\lambda}^{0,M}(x)g(\lambda)dn(\lambda)\right\vert ^{2}xdx=0.$$
2. The direct convergence of the generalised Hankel transform is given by the following results:
\[th15.3\]Let $\gamma\in(0,\infty).$ If $f:(0,\infty)\rightarrow
\mathbb{R}$ has the property$$x\longmapsto\sqrt{x}f(x)\in L^{1}(0,\infty) \label{eq15.13}$$ and is of bounded variation in a neighbourhood of $\gamma,$ then$$\tfrac{1}{2}[f(\gamma+0)+f(\gamma-0)]=\int_{0}^{\infty}J_{\lambda}^{0,M}(\gamma)\left( \int_{0}^{\infty}J_{\lambda}^{0,M}(x)f(x)xdx\right)
dn(\lambda). \label{eq15.14}$$
Let $\mu\in(0,\infty).$ If $g:(0,\infty)\rightarrow\mathbb{R}$ has the property$$\lambda\longmapsto\frac{\sqrt{\lambda}g(\lambda)}{1+M(\lambda/2)^{2}}\in
L^{1}(0,\infty) \label{eq15.15}$$
and is of bounded variation in a neighbourhood of $\mu,$ then$$\begin{array}
[c]{l}\tfrac{1}{2}[g(\mu+0)+g(\mu-0)]\\
\qquad\qquad={\displaystyle\int_{0}^{\infty}}
J_{\mu}^{0,M}(x)\left(
{\displaystyle\int_{0}^{\infty}}
J_{\lambda}^{0,M}(x)g(\lambda)dn(\lambda)\right) dm_{M/2}(x).
\end{array}
\label{eq15.16}$$
\[rem15.3\]The integrals in Theorem \[th15.3\] are either Lebesgue integrals or limits of such integrals over compact intervals of $(0,\infty).$
\[cor15.1\]
- If $\gamma\in(0,\infty)$ is a point of continuity of the function $f$ then $(\ref{eq15.8}),~(\ref{eq15.11})$ and $(\ref{eq15.14})$ imply$$\begin{aligned}
(\mathcal{G}_{M}(\mathcal{F}_{M}f))(\gamma) & =\int_{0}^{\infty}J_{\lambda
}^{0,M}(\gamma)(\mathcal{F}_{M}f)(\lambda)dn(\lambda)\nonumber\\
& =(M/2)f(0)\int_{0}^{\infty}J_{\lambda}^{0,M}(\gamma)dn(\lambda)+\nonumber\\
& \;\;\;\int_{0}^{\infty}J_{\lambda}^{0,M}(\gamma)\left( \int_{0}^{\infty
}J_{\lambda}^{0,M}(x)f(x)xdx\right) dn(\lambda)\nonumber\\
& =f(\gamma), \label{eq15.17}$$ since$$\int_{0}^{\infty}J_{\lambda}^{0,M}(\eta)dn(\lambda)=0\ \text{for all}\ \eta
\in(0,\infty). \label{eq15.18}$$
- If $\gamma=0$ then$$(\mathcal{G}_{M}(\mathcal{F}_{M}f))(0)=(M/2)f(0)\int_{0}^{\infty}dn(\lambda)=f(0)$$ since$$\int_{0}^{\infty}J_{\lambda}^{0,M}(0)\left( \int_{0}^{\infty}J_{\lambda
}^{0,M}(x)f(x)xdx\right) dn(\lambda)=0,$$ from $(\ref{eq15.18}),$ the use of the Fubini integral theorem and noting that $J_{\lambda}^{0,M}(0)=1.$
- If $\mu\in(0,\infty)$ is a point of continuity of $g$ then $(\ref{eq15.8}),~(\ref{eq15.11})$ and $(\ref{eq15.16})$ imply$$(\mathcal{F}_{M}(\mathcal{G}_{M}g))(\mu)=g(\mu).$$
The Plum partial differential equation\[sec16\]
===============================================
The Plum equation is a fourth-order linear partial differential in the Euclidean space $\mathbb{R}^{2}$ of two dimensions, derived from a linear partial differential expression which is connected with the fourth-order Bessel-type ordinary differential equation.
If the Laplacian $\nabla^{2}$ partial differential expression is written in polar co-ordinates$$\nabla^{2}=\frac{\partial^{2}}{\partial r^{2}}+\frac{1}{r}\frac{\partial
}{\partial r}+\frac{1}{r^{2}}\frac{\partial^{2}}{\partial\theta^{2}}
\label{eq16.1}$$ then the Plum equation has the form, with $u=u(r,\theta),$$$\nabla^{4}u-\gamma\nabla^{2}u-\frac{4\gamma}{r^{2}}u=\Lambda u. \label{eq16.2}$$ Here $\gamma>0$ is determined by $\gamma=8M^{-1}$ where $M>0$ is the parameter in the fourth-order Bessel equation (\[eq3.1\]), and $\Lambda\in\mathbb{C}$ is a spectral parameter.
Written out the equation (\[eq16.2\]) becomes, see [L1]{},$$\begin{aligned}
& \dfrac{\partial^{4}u}{\partial r^{4}}+\dfrac{2}{r}\dfrac{\partial^{3}u}{\partial r^{3}}+\left( -\dfrac{1}{r^{2}}-\gamma\right) \dfrac
{\partial^{2}u}{\partial r^{2}}+\left( \dfrac{1}{r^{3}}-\dfrac{\gamma}{r}\right) \dfrac{\partial u}{\partial r}+\dfrac{1}{r^{4}}\dfrac{\partial
^{4}u}{\partial\theta^{4}}\nonumber\\
& +\dfrac{2}{r^{2}}\dfrac{\partial^{4}u}{\partial\theta^{2}\partial r^{2}}-\dfrac{2}{r^{3}}\dfrac{\partial^{3}u}{\partial\theta^{2}\partial r}+\left(
\dfrac{4}{r^{4}}-\dfrac{\gamma}{r^{2}}\right) \dfrac{\partial^{2}u}{\partial\theta^{2}}-\dfrac{4\gamma}{r^{2}}u\nonumber\\
& =\Lambda u. \label{eq16.3}$$
From the results given in [@MP] and [@L1] assume that a solution for (\[eq16.2\]) is of the separated form$$u(r,\theta)=v(r)w(\theta) \label{eq16.4}$$ where $w$ is required to be a solution of the second-order Sturm-Liouville differential equation$$-w^{\prime\prime}(\theta)=4w(\theta). \label{eq16.5}$$ Note that $w$ is then of the general form$$w(\theta)=A\cos(2\theta)+B\sin(2\theta) \label{eq16.6}$$ for, say, $\theta\in\lbrack0,\pi]$ and scalars $A,B.$
Also note that the factor $4$ in the equation (\[eq16.5\]) is critical, and has to be fixed, for the separation method to be effective.
Substitution of (\[eq16.4\]) into (\[eq16.3\]) yields, see [@L1 Section 1] and [@MP],$$\begin{aligned}
& \left( v^{(4)}(r)+\dfrac{2}{r}v^{(3)}(r)+\left( -\dfrac{1}{r^{2}}-\gamma\right) v^{\prime\prime}(r)+\left( \dfrac{1}{r^{3}}-\dfrac{\gamma}{r}\right) v^{\prime}(r)+\dfrac{16}{r^{4}}v(r)\right) w(\theta)\nonumber\\
& +\left( -\dfrac{8}{r^{2}}v^{\prime\prime}(r)+\dfrac{8}{r^{3}}v^{\prime
}(r)+\left( -\dfrac{16}{r^{4}}+\dfrac{4\gamma}{r^{2}}\right) v(r)-\dfrac
{4\gamma}{r^{2}}v(r)\right) w(\theta)\nonumber\\
& =\Lambda v(r)w(\theta)\ \text{for all}\ r\in(0,\infty)\ \text{and
all}\ \theta\in\lbrack0,\pi]. \label{eq16.7}$$
For (\[eq16.7\]) to hold requires that the function $v(\cdot),$ on gathering up terms, has to satisfy the ordinary differential equation, see [@L1 Section 1, (4)], for all $r\in(0,\infty),$$$v^{(4)}(r)+\dfrac{2}{r}v^{(3)}(r)+\left( -\dfrac{9}{r^{2}}-\gamma\right)
v^{\prime\prime}(r)+\left( \dfrac{9}{r^{3}}-\dfrac{\gamma}{r}\right)
v^{\prime}(r)-\Lambda v(r)=0. \label{eq16.8}$$ This last equation may be written in the Lagrange symmetric form$$(rv^{\prime\prime}(r))^{\prime\prime}-((9r^{-1}+\gamma r)v^{\prime
}(r))^{\prime}=\Lambda rv(r)\ \text{for all}\ r\in(0,\infty), \label{eq16.9}$$ which is the Bessel fourth-order differential equation (\[eq3.1\]) when $\gamma=8M^{-1}.$
Thus separated solutions of the partial differential equation (\[eq16.2\]) can be written in the form$$u(r,\theta)=v(r)w(\theta)\ \text{for all}\ r\in(0,\infty)\ \text{and}\ \theta\in\lbrack0,2\pi], \label{eq16.10}$$ where $w(\cdot)$ is any trigonometrical solution (\[eq16.6\]) of (\[eq16.5\]), and $v(\cdot)$ is any solution of the fourth-order Bessel equation (\[eq16.9\]) for any choice of the spectral parameter $\Lambda.$
Defining the partial differential expression $P_{\gamma}[\cdot],$ for $\gamma\in(0,\infty),$ by$$P_{\gamma}[u]:=\left( \nabla^{4}u-\gamma\nabla^{2}u-\frac{4\gamma}{r^{2}}u\right) \label{eq16.11}$$ it is shown in [@L1] that $P_{\gamma}$ is a formally symmetric linear partial differential expression in $L^{2}(\mathbb{E}^{2})$, using polar co-ordinates $(r,\theta).$
Some early studies indicate that there may be problems in applied mathematics, for which the partial differential equation $P_{\gamma}[u]=\Lambda u$ is involved in one or more of the associated mathematical models.
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|
---
abstract: 'The reliability of quiet Sun magnetic field diagnostics based on the lines at 6302 Å has been questioned by recent work. We present here the results of a thorough study of high-resolution multi-line observations taken with the new spectro-polarimeter SPINOR, comprising the 5250 and 6302 Å spectral domains. The observations were analyzed using several inversion algorithms, including Milne-Eddington, LTE with 1 and 2 components, and MISMA codes. We find that the line-ratio technique applied to the 5250 Å lines is not sufficiently reliable to provide a direct magnetic diagnostic in the presence of thermal fluctuations and variable line broadening. In general, one needs to resort to inversion algorithms, ideally with realistic magneto-hydrodynamical constrains. When this is done, the 5250 Å lines do not seem to provide any significant advantage over those at 6302 Å. In fact, our results point towards a better performance with the latter (in the presence of turbulent line broadening). In any case, for very weak flux concentrations, neither spectral region alone provides sufficient constraints to fully disentangle the intrinsic field strengths. Instead, we advocate for a combined analysis of both spectral ranges, which yields a better determination of the quiet Sun magnetic properties. Finally, we propose the use of two other lines (at 4122 and 9000 Å) with identical line opacities that seem to work much better than the others.'
author:
- 'H. Socas-Navarro, J. M. Borrero'
- 'A. Asensio Ramos, M. Collados, I. Domínguez Cerde\~ na, E. V. Khomenko, M. J. Martínez Gonz'' alez, V. Martínez Pillet, B. Ruiz Cobo, J. S'' anchez Almeida'
title: 'Multi-Line Spectro-Polarimetry of the Quiet Sun at 5250 and 6302 Å'
---
Introduction {#sec:intro}
============
The empirical investigation of quiet Sun[^1] magnetism is a very important but extremely challenging problem. A large (probably dominant) fraction of the solar magnetic flux resides in magnetic accumulations outside active regions, forming network and inter-network patches (e.g., [@SNSA02 [-@SNSA02]]{}). It is difficult to obtain conclusive observations of these structures, mainly because of two reasons. First, the size of the magnetic concentrations is much smaller than the spatial resolution capability of modern spectro-polarimetric instrumentation. Estimates obtained with inversion codes yield typical values for the filling factor of the resolution element between $\sim$1% and 30% . The interpretation of the polarization signal becomes non-trivial in these conditions and one needs to make use of detailed inversion codes to infer the magnetic field in the atmosphere. Second, the observed signals are extremely weak (typically below $\sim$1% of the average continuum intensity), demanding both high sensitivity and high resolution. Linear polarization is rarely observed in visible lines, so one is usually left with Stokes $I$ and $V$ alone.
[@S73 ([-@S73])]{} proposed to use the pair of lines at 5247 and 5250 Å which have very similar excitation potentials and oscillator strengths (and, therefore, very similar opacities) but different Landé factors, to determine the intrinsic field strength directly from the Stokes $V$ line ratio. That work led to the subsequent popularization of this spectral region for further studies of unresolved solar magnetic structures. Later, the pair of lines at 6302 Å became the primary target of the Advanced Stokes Polarimeter (ASP, [@ELT+92 [-@ELT+92]]{}), mainly due to their lower sensitivity to temperature fluctuations. The success of the ASP has contributed largely to the currently widespread use of the 6302 Å lines by the solar community.
Recent advances in infrared spectro-polarimetric instrumentation now permit the routine observation of another very interesting pair of lines, namely those at 15648 and 15653 Å (hereafter, the 1.56 $\mu$m lines). Examples are the works of [@LR99 ([-@LR99])]{}; [@KCS+03 ([-@KCS+03])]{}. The large Land' e factors of these lines, combined with their very long wavelengths, result in an extraordinary Zeeman sensitivity. Their Stokes $V$ profiles exhibit patterns where the $\sigma$-components are completely split for fields stronger than $\sim$400 G at typical photospheric conditions. They also produce stronger linear polarization. On the downside, this spectral range is accesible to very few polarimeters. Furthermore, the 1.56 $\mu$m lines are rather weak in comparison with the above-mentioned visible lines.
Unfortunately, the picture revealed by the new infrared data often differs drastically from what was being inferred from the 6302 Å observations (e.g., [@LR99 [-@LR99]]{}; [@KCS+03 [-@KCS+03]]{}; [@SNSA02 [-@SNSA02]]{}; [@SNL04 [-@SNL04]]{}; [@DCSAK03 [-@DCSAK03]]{}), particularly in the inter-network. [@SNSA03 ([-@SNSA03])]{} proposed that the discrepancy in the field strengths inferred from the visible and infrared lines may be explained by magnetic inhomogeneities within the resolution element (typically 1). If multiple field strengths coexist in the observed pixel, then the infrared lines will be more sensitive to the weaker fields of the distribution whereas the visible lines will provide information on the stronger fields (see also the discussion about polarimetric signal increase in the 1.56 $\mu$m lines with weakening fields in [@SAL00 [-@SAL00]]{}). This conjecture has been tested recently by [@DCSAK06 ([-@DCSAK06])]{} who modeled simultaenous observations of visible and infrared lines using unresolved magnetic inhomogeneities.
A recent paper describing numerical simulations by [@MGCRC06 ([-@MGCRC06])]{} casts some doubts on the results obtained using the 6302 Å lines. Our motivation for the present work is to resolve this issue by observing simultaneously the quiet Sun at 5250 and 6302 Å. We know that unresolved magnetic structure might result in different field determinations in the visible and the infrared, but the lines analyzed in this work are close enough in wavelengths and Zeeman sensitivities that one would expect to obtain the same results for both spectral regions.
Methodology {#sec:method}
===========
Initially, our goal was to observe simultaneously at 5250 and 6302 Å because we expected the 5250 Å lines to be a very robust indicator of intrinsic field strength, which we could then use to test under what conditions the 6302 Å lines are also robust. Unfortunately, as we show below, it turns out that in most practical situations the 5250 Å lines are not more robust than the 6302 Å pair. This left us without a generally valid reference frame against which to test the 6302 Å lines. We then decided to employ a different approach for our study, namely to analyze the uniqueness of the solution obtained when we invert the lines and how the solutions derived for both pairs of lines compare to each other. In doing so, there are some subtleties that need to be taken into consideration.
An inversion technique necessarily resorts on a number of physical assumptions on the solar atmosphere in which the lines are formed and the (in general polarized) radiative transfer. This implies that the conclusions obtained from applying a particular inversion code to our data may be biased by the modeling implicit in the inversion. Therefore, a rigorous study requires the analysis of solutions from a wide variety of inversion procedures. Ideally, one would like to cover at least the most frequently employed algorithms. For our purposes here we have chosen four of the most popular codes, spanning a wide range of model complexity. They are described in some detail in section \[sec:obs\] below.
When dealing with Stokes inversion codes, there are two very distinct problems that the user needs to be aware of. First, it may happen that multiple different solutions provide satisfactory fits to our observations. This problem is of a very fundamental nature. It is not a problem with the inversion algorithm but with the observables themselves. Simply put, they do not carry sufficient information to discriminate among those particular solutions. The only way around this problem is to either supply additional observables (e.g., more spectral lines) or to restrict the allowed range of solutions by incorporating sensible constraints in the physical model employed by the inversion. A second problem arises when the solution obtained does not fit the observed profiles satisfactorily. This can happen because the physical constraints in the inversion code are too stringent (and thus no good solution exists within the allowed range of parameters), or simply because the algorithm happened to stop at a secondary minimum. This latter problem is not essential because one can always discard “bad” solutions (i.e., those that result in a bad fit to the data) and simply try again with a different initialization. In this work we are interested in exploring the robustness of the observables in as much as they relate to the former problem, i.e. the underlying uniqueness of the solution. We do this by performing a large number of inversions with random initializations and analyzing the dependence of the solutions with the merit function $\chi^2$ (defined below).
Ideally, one would like to see that for small values of $\chi^2$, all the solutions are clustered around a central value with a small spread (the behavior for large $\chi^2$ is not very relevant for our purposes here). This should happen regardless of the inversion code employed and the pair of lines analyzed. If that were the case we could conclude that our observables are truly robust. Otherwise, one needs to be careful when analyzing data corresponding to that particular scenario.
Observations and Data Reduction {#sec:obs}
===============================
The observations used in this work were obtained during an observing run in March 2006 with the Spectro-Polarimeter for Infrared and Optical Regions (SPINOR, see [@SNEP+06 [-@SNEP+06]]{}), attached to the Dunn Solar Telescope (DST) at the Sacramento Peak Observatory (Sunspot, NM USA, operated by the National Solar Observatory). SPINOR allows for the simultaneous observation of multiple spectral domains with nearly optimal polarimetric efficiency over a broad range of wavelengths.
The high-order adaptive optics system of the DST ([@RHR+03 [-@RHR+03]]{}) was employed for image stabilization and to correct for atmospheric turbulence. This allowed us to attain sub-arcsecond spatial resolution during some periods of good seeing.
The observing campaign was originally devised to obtain as much information as possible on the unresolved properties of the quiet Sun magnetic fields. In addition to the 5250 and 6302 Å domains discussed here, we also observed the line sensitive to hyperfine structure effects ([@LATC02 [-@LATC02]]{}) at 5537 Å and the line pair at 1.56 $\mu$m.
SPINOR was operated in a configuration with four different detectors which were available at the time of observations (see Table \[tab:detectors\]): The Rockwell TCM 8600 infrared camera, with a format of 1024$\times$1024 pixels, was observing the 1.56 $\mu$m region. The SARNOFF CAM1M100 of 1024$\times$512 pixels was used at 5250 Å. Finally, the two dual TI TC245 cameras of 256$\times$256 pixels (the original detectors of the Advanced Stokes Polarimeter) were set to observe at 5537 and 6302 Å. Unfortunately, we encountered some issues during the reduction of the 5537 Å and 1.56 $\mu$m data and it is unclear at this point whether or very not they are usable. Therefore, in the remainder of this paper we shall focus on the analysis of the data taken at 5250 and 6302 Å. The spectral resolutions quoted in the table are estimated as the quadratic sum of the spectrograph slit size, diffraction limit and pixel sampling.
In order to have good spectrograph efficiency at all four wavelengths simultaneously we employed the 308.57 line mm$^{-1}$ grating (blaze angle 52$^{\rm o}$), at the expense of obtaining a relatively low dispersion and spectral resolution (see Table \[tab:detectors\] for details).
[lccccc]{} ROCKWELL & 15650 & 190 & 150 & 150 & 187\
SARNOFF & 5250 & 53 & 31 & 15 & 145\
TI TC245 & 5537 & 40 & 29 & 5 & 95\
TI TC245 & 6302 & 47 & 24 & 6 & 95\
Standard flatfield and bias correction proceduress were applied to the images. Subsequent processing included the removal of spectrum curvature and the alignment of both polarized beams, using for this purpose a pair of hairlines inserted across the slit. Calibration operations were performed to determine the SPINOR polarimetric response matrix by means of calibration optics located at the telescope beam exit port. In this manner we can decontaminate the datasets from instrumental polarization introduced by the polarimeter. Finally, it is also important to consider the contamination introduced by the telescope. To this aim we obtained telescope calibration data with an array of linear polarizers situated over the DST entrance window. By rotating these polarizers to different angles, it is possible to feed light in known polarization states into the system. A cross-dispersing prism was placed in front of one of the detectors, allowing us to obtain calibration data simultaneously across the entire visible spectrum. Details on the procedure may be found in [@SNEP+06 ([-@SNEP+06])]{}.
In this paper we focus on two scan operations near disk center, one over a relatively large pore (at solar heliocentric coordinates longitude -25.40, latitude -3.68) and the other of a quiet region (coordinates longitude 0.01, latitude -7.14). The pore map was taken with rather low spatial resolution ($\sim$1.5) but exhibits a large range of polarization signal amplitudes. The quiet map, on the other hand, has very good spatial resolution ($\sim$0.6) but the polarization signals are much weaker. The noise level, measured as the standard deviation of the polarization signal in continuum regions, is approximately 7$\times$10$^{-4}$ and 5$\times$10$^{-4}$ times the average quiet Sun continuum intensity at 5250 and 6302 Å, respectively. We used several different inversion codes for the various tests presented here, namely: SIR (Stokes Inversion based on Response functions, [@RCdTI92 [-@RCdTI92]]{}); MELANIE (Milne-Eddington Line Analysis using a Numerical Inversion Engine) and LILIA (LTE Inversion based on the Lorien Iterative Algorithm, [@SN01a [-@SN01a]]{}); and MISMA (MIcro-Structured Magnetic Atmosphere, [@SA97 [-@SA97]]{}). The simplest of these algorithms is MELANIE, which implements a Milne-Eddington type of inversion similar to that of [@SL87 ([-@SL87])]{}. The free parameters considered include a constant along the line of sight magnetic field vector, magnetic filling factor, line-of-sight velocity and several spectral line parameters that represent the thermal properties of the atmosphere (Doppler width $\Delta \lambda_D$, line-to-continuum opacity ratio $\eta_0$, source function $S$ and damping $a$). The lines at 6302 Å belong to the same multiplet and their $\eta_0$ are related by a constant factor. Assuming that the formation height is similar enough for both lines we can also consider that they have the same $\Delta \lambda_D$, $S$ and $a$. In this manner the same set of free parameters can be used to fit both lines. In the case of the 5250 Å lines we only invert the pair and assume that both lines have identical opacities $\eta_0$.
SIR considers a model atmosphere defined by the depth stratification of variables such as temperature, pressure, magnetic field vector, line-of-sight velocity and microturbulence. Atomic populations are computed assuming LTE for the various lines involved, making it possible to fit observations of lines from multiple chemical elements with a single model atmosphere that is common to all of them. Unlike MELANIE, one can produce line asymmetries by incorporating gradients with height of the velocity and other parameters. LILIA is a different implementation of the SIR algorithm. It works very similarly with some practical differences that are not necessary to discuss here.
MISMA is another LTE code but has the capability to consider three atmospheric components (two magnetic and one non-magnetic) that are interlaced on spatial scales smaller than the photon mean free path. Perhaps the most interesting feature of this code for our purposes here is that it implements a number of magneto-hydrodynamic (MHD) constrains, such as momentum, as well as mass and magnetic flux conservation. In this manner it is possible to derive the full vertical stratification of the model atmosphere from a limited number of free parameters (e.g., the magnetic field and the velocity at the base of the atmosphere).
In all of the inversions presented here we employed the same set of atomic line parameters, which are listed in Table \[tab:atomic\].
[cccccc]{} & 4121.8020 & 2.832 & -1.300 & $^3$P$_2$ & $^3$F$_3$\
& 5246.7680 & 3.714 & -2.630 & $^4$P$_{1/2}$ & $^4$P$_{3/2}$\
& 5247.0504 & 0.087 & -4.946 & $^5$D$_2$ & $^7$D$_3$\
& 5247.2870 & 2.103 & -0.927 & $^5$F$_3$ & $^5$F$_2$\
& 5247.5660 & 0.961 & -1.640 & $^5$D$_0$ & $^5$P$_1$\
& 5250.2089 & 0.121 & -4.938 & $^5$D$_0$ & $^7$D$_1$\
& 5250.6460 & 2.198 & -2.181 & $^5$P$_2$ & $^5$P$_3$\
& 6301.5012 & 3.654 & -0.718 & $^5$P$_2$ & $^5$D$_2$\
& 6302.4916 & 3.686 & -1.235 & $^5$P$_1$ & $^5$D$_0$\
& 8999.5600 & 2.832 & -1.300 & $^3$P$_2$ & $^3$P$_2$\
Results {#sec:results}
=======
Figures \[fig:map1\] and \[fig:map2\] show continuum maps and reconstructed magnetograms of the pore and quiet Sun scans in the 5250 Å region. Notice the much higher spatial resolution in the quiet Sun observation (Fig \[fig:map1\]). Similar maps, but with a somewhat smaller field of views, can be produced at 6302 Å.
The first natural step in the analysis of these observations, before even considering any inversions, is to calculate the Stokes $V$ amplitude ratio of the line 5250.2 to 5247.0 Å (hereafter, the line ratio). One would expect to obtain a rough idea of the intrinsic magnetic field strength from this value alone. A simple calibration was derived by taking the thermal stratification of the Harvard-Smithsonian Reference Atmosphere (HSRA, see [@GNK+71 [-@GNK+71]]{}) and adding random (depth-independent) Gaussian temperature perturbations with an amplitude (standard deviation) of $\pm$300 K, different magnetic field strengths and a fixed macroturbulence of 3 km s$^{-1}$ (this value corresponds roughly to our spectral resolution). Line-of-sight gradients of temperature, field strength and velocity are also included. Figure \[fig:calib\] shows the line ratio obtained from the synthetic profiles as a function of the magnetic field employed to synthesize them. The line ratio is computed simply as the peak-to-peak amplitude ratio of the Stokes $V$ profiles. Notice that when the same experiment is carried out with a variable macroturbulent velocity the scatter increases considerably, even for relatively small values of up to 1 km s$^{-1}$ (right panel). The syntheses of Fig \[fig:calib\] consider the partial blends of all 6 lines in the 5250 Å spectral range.
Figure \[fig:mapratios\] shows the observed line ratios for both scans. According to our calibration (see above), ratios close to 1 indicate strong fields of nearly (at least) $\sim$2 kG, whereas larger values would suggest the presence of weaker fields, down to the weak-field saturation regime at (at most) $\sim$500 G corresponding to a ratio of 1.5. Figure \[fig:mapratios\] is somewhat disconcerting at first sight. The pore exhibits the expected behavior with strong $\sim$2 kG fields at the center that decrease gradually towards the outer boundaries until it becomes weak. The network and plage patches, on the other hand, contain relatively large areas with high ratios of 1.4 and even 1.6 at some locations. This is in sharp contrast with the strong fields ($\sim$1.5 kG) that one would expect in network and plage regions (e.g., [@SSH+84 [-@SSH+84]]{}; [@SAL00 [-@SAL00]]{}; [@BRRCC00 [-@BRRCC00]]{}).
In view of these results we carried out inversions of the Stokes $I$ and $V$ profiles of the spectral lines in the 5250 Å region emergent from the pore. We used the code LILIA considering a single magnetic component with a constant magnetic field embedded in a (fixed) non-magnetic background, taken to be the average quiet Sun. The resulting field-ratio scatter plot is shown in Figure \[fig:calibpore\]. Note that the scatter in this case is much larger than that obtained with the HSRA calibration above. It is important to point out that the line ratio depicted in the figure is that measured on the synthetic profiles. Therefore, the scatter cannot be ascribed to inaccuracies of the inversion. As a verification test we picked one of the models with kG fields that produced a line ratio of $\simeq$1.4 and synthesized the emergent profiles with a different code (SIR), obtaining the same ratio. We therefore conclude that, when the field is strong, many different atmospheres are able to produce similar line ratios if realistic thermodynamic fluctuations and turbulence are allowed in the model. For weak fields, the ratio tends to a value of $\simeq$1.5 without much fluctuations.
Accepting then that we could no longer rely on the line ratio of 5247.0 and 5250.2 Å as an independent reference to verify the magnetic fields obtained with the 6302 Å lines, we considered the result of inverting each spectral region separately. Figure \[fig:porefields\] depicts the scatter plot obtained. At the center of the pore, where we have the strongest fields (right-hand side of the figure), there is a very good correlation between the results of both measurements. However, those points lay systematically below the diagonal of the plot. This may be explained by the different “formation heights” of the lines. The lines at 5250 Å generally form somewhat higher than those at 6302 Å. If the field strength decreases with height, one would expect to retrieve a slightly lower field strength when using 5250 Å. Unfortunately, the correlation breaks down for the weaker fields. In Figure \[fig:poreerrors\] we can see that both sets of lines yield approximately the same field strengths (with a slightly lower values for the 5250 inversions, as discussed above) for longitudinal fluxes above some $\sim$500 G. Below this limit our diagnosis is probably not sufficiently robust.
If instead of the intrinsic field we consider the longitudinal magnetic flux, we obtain a fairly good agreement between both spectral regions. The Milne-Eddington inversions with MELANIE yield a Pearsons correlation coefficient of 0.89. In principle, the agreement is somewhat worse for the LILIA inversions, with a correlation coefficient of 0.60. However, we have found that this is due to a few outlayer points. Removing them results in a correlation coefficient of 0.82.
The situation becomes more complicated in the network. The results of inverting a network patch with MELANIE and LILIA can be seen in Figure \[fig:fluxes\]. The inversions with MELANIE (upper panels) do not agree very well with each other. The correlation coefficient is only 0.23. The 5250 Å map (upper right panel in the figure) looks considerably more noisy and rather homogeneous, compared to the 6302 Å map (upper left). The LILIA inversions (lower panels) exhibit somewhat better agreement (correlation is 0.65), but again look very noisy at 5250 Å.
Figure \[fig:NWa\] depicts average profiles over a network patch. The 5250 line ratio for this profile is 1.21. We started by exploring the uniqueness of the magnetic field strength inferred with the simplest algorithm, MELANIE. Each one of these average profiles was inverted 100 times with random initializations. The results are presented in Figures \[fig:uniqME5250\] and \[fig:uniqME6302\], which show the values obtained versus the goodness of the fit, defined as the merit function $\chi^2$, which in this work is defined as: $$\label{eq:chisq}
\chi^2={1 \over N_p} \sum_{i=1}^{N_p}{ (I_i^{obs}-I_i^{syn})^2 \over
\sigma_i^2} \, ,$$ where $N_p$ is the number of wavelengths and $\sigma_i$ have been taken to be 10$^{-3}$, so that a value of $\chi^2$=1 would represent on average a good fit at the 10$^{-3}$ level. The average profiles inverted here have a much lower noise (near 10$^{-4}$) and thus it is some times possible to obtain $\chi^2$ smaller than 1.
The $\chi^2$ represented in the plots is the one corresponding to the Stokes $V$ profile only (although the inversion codes consider both $I$ and $V$, but $I$ is consistently well reproduced and does not help to discriminate among the different solutions). Most of the fits correspond to kG field, indicating that inversions of network profiles are very likely to yield high field strengths. However, there exists a very large spread of field strength values that provide reasonably good fits to the observed data. This is especially true for the 5250 Å lines, for which it is possible to fit the observations virtually equally well with fields either weaker than 500 G or stronger than 1 kG. In the case of 6302 Å the best solutions are packed around $\simeq$1.5 kG, although other solutions of a few hecto-Gauss (hG) are only slightly worse than the best fit.
It could be argued that Milne-Eddington inversions are too simplistic to deal with network profiles, since they are known to exhibit fairly strong asymmetries (both in area and in amplitude) that cannot be reproduced by a Milne-Eddington model. With this consideration in mind, we made a similar experiment using the LILIA and SIR codes. Figures \[fig:uniqLI5250\] and \[fig:uniqLI6302\] show the results obtained with LILIA. The magnetic and non-magnetic atmospheres have been forced to have the same thermodynamics in order to reduce the possible degrees of freedom (on the downside, this introduces an implicit assumption on the solar atmosphere). Again, the 6302 Å lines seem to yield somewhat more robust inversions. The best fits correspond to field strengths of approximately 1.5 kG, with weaker fields delivering somewhat lower fit quality. The 5250 Å lines give nearly random results (although they tend to be clustered between 500 and 800 G there is a tail of good fits with up to almost 1400 G). Similar results are obtained using SIR. In order to give an idea of what the different $\chi^2$ values mean, we present some of the fits in Fig \[fig:fits\].
The thermal stratifications obtained in the 6302 Å inversions are relatively similar, although the weaker fields require a hotter upper photosphere than the stronger fields (see Fig \[fig:tlil6302\]). On the other hand, the 5250 Å inversions do not exhibit a clear correlation between the magnetic fields and temperature inferred (Fig \[fig:tlil5250\]).
The MISMA code was also able to find good solutions with either weak or strong fields. The smallest $\chi^2$ values correspond systematically to kG fields, but there are also some reasonably good fits ($\chi^2 \simeq 1$) obtained with weak fields of $\sim$500 G (Figs \[fig:misma6302\] and \[fig:misma5250\]). However, we found that all the weak-field solutions for 6302 Å have a temperature that increases outwards in the upper photosphere (Fig \[fig:misma6302\]). This might be useful to discriminate between the various solutions. The 5250 Å lines, on the other hand, do not exhibit this behavior (Fig \[fig:misma5250\]).
In principle it would seem plausible to discard the models with outward increasing temperature using physical arguments. This would make us conclude from the 6302 Å lines that the fields are actually very strong, between $\sim$1.5 and 2.5 kG (it would not be possible to draw similar conclusions from the 5250 Å lines). In any case, it would be desirable to have a less model-dependent measurement that could be trusted regardless of what the thermal stratification is.
Interestingly, when we invert all four lines simultaneously, both at 5250 and 6302 Å, the weak-field solutions disappear from the low-$\chi^2$ region of the plot and the best solutions gather between approximately 1 and 1.4 kG (see Fig \[fig:mismaboth\]).
[@SNARMS06 ([-@SNARMS06])]{} present a list of spectral lines with identical excitation potentials and oscillator strengths. We decided to test one of the most promising pairs, namely the lines at 4122 and 9000 Å. The choice was made based on their equivalent widths, Land' e factors and also being reasonably free of blends in the quiet solar spectrum. Simulations similar to those in Figure \[fig:calib\] showed that the line-ratio technique is still incapable of retrieving an unambiguous field strength due to the presence of line broadening. Also, even if the [*line*]{} opacities are the same, the continuum opacities are significantly different at such disparate wavelengths, but in any case the line opacity is much stronger than the background continuum where the Stokes $V$ lobes are formed.
When we tested the robustness of inversion codes applied to this pair of lines, we obtained extremely reliable results as described below. Unfortunately we do not have observations of these two lines and therefore resorted on synthetic profiles to perform the tests. We used a 2-component reference model, where both components have the thermal structure of HSRA. The magnetic comonent has an arbitrary magnetic field and line-of-sight velocity. The field has a linear gradient and goes from $\simeq$1.6 kG at the base of the photosphere to 1 kG at continuum optical depth $\log(\tau_{5000})$=-4. The velocity field goes from 2.5 km s$^{-1}$ to 0.4 km s$^{-1}$ at $\log(\tau_{5000})$=-4. The magnetic component has a filling factor of 0.2. The reference macroturbulence is 3 km s$^{-1}$, which is roughly the value inferred from our observations. The spectra produced by this model at 5250 and 6302 Å are very similar to typical network profiles. We considered the reference 4122 and 9000 Å profiles as simulated observations and inverted them with two different methods.
In order to make the test as realistic as possible, we gave the inversion code a somewhat erroneous non-magnetic profile. Specifically, we multiplied the profile from the reference non-magnetic component by a factor of 0.9 and shifted it 1 pixel towards the red. The wavelength shift corresponds to roughly 0.36 km s$^{-1}$ in both regions. This distorted non-magnetic profile was given as input to the inversion codes. We inverted the reference profiles using LILIA with 100 different initializations. Only $\sim$30% of the inversions converged to a reasonably low value of $\chi^2$, with the results plotted in Figure \[fig:lilia9000\]. We can see that the inversions are extremely consistent over a range of $\chi^2$ much larger than in the previous cases. In fact, none of the solutions are compatible with weak fields, suggesting that these lines are much better at discriminating intrinsic field strengths.
A much more demanding verification for the diagnostic potential of these new lines is to use a simpler scenario in the inversion than in the synthesis of the reference profiles, incorporating typical uncertainties in the calculation. After all, the real Sun will always be more complex than our simplified physical models. Thus, inverting the reference profiles with the Milne-Eddington code is an appropriate test. For this experiment we not only supplied the same “distorted” non-magnetic profile as above, but we also introduced a systematic error in the $\log(gf)$ of the lines. We forced the opacity of the 9000 Å line to be 20% lower than that of 4122 (instead of taking them to be identical, as their tabulated values would indicate). Again, we performed 100 different inversions of the reference set of profiles with random initializations. In this case the inversion results are astonishingly stable, with 98 out of the 100 inversions converging to a $\chi^2$ within 15% of the best fit. The single-valued magnetic field obtained for those 98 inversions has a median of 1780 G with a standard deviation of only 3 G. The small scatter does not reflect the systematic errors introduced by several factors, including: a)the inability of the Milne-Eddington model to reproduce the comparatively more complex referece profiles; b)the artificial error introduced in the atomic parameters of the 9000 Å line; c)the distortion (scale and shift) of the non-magnetic profile provided to the inversion code.
A final caveat with this new pair is that, even though the synthetic atlas of [@SNARMS06 ([-@SNARMS06])]{} indicates that the 9000 Å line Stokes $V$ profile is relatively free of blends, this still needs to be confirmed by observations (there is a very prominent line nearby that may complicate the analysis otherwise).
Conclusions {#sec:conc}
===========
The ratio of Stokes $V$ amplitudes at 5250 and 5247 Å is a very good indicator of the intrinsic field strength in the absence of line broadening, e.g. due to turbulence. However, line broadening tends to smear out spectral features and reduce the Stokes $V$ amplitudes. This reduction is not the same for both lines, depending on the profile shape. If the broadening could somehow be held constant, one would obtain a line-ratio calibration with very low scatter. However, if the broadening is allowed to fluctuate, even with amplitudes as small as 1 km s$^{-1}$, the scatter becomes very large. Fluctuations in the thermal conditions of the atmosphere further complicate the analysis. This paper is not intended to question the historical merits of the line-ratio technique, which led researchers to learn that fields seen in the quiet Sun at low spatial resolution are mostly of kG strength with small filling factors. However, it is important to know its limitations. Otherwise, the interpretation of data such as those in Figure \[fig:mapratios\] could be misleading. Before this work, most of the authors were under the impression that measuring the line ratio of the 5250 Å lines would always provide an accurate determination of the intrinsic field strength.
With very high-resolution observations, such as those expected from the Advanced Technology Solar Telescope (ATST, [@KRK+03 [-@KRK+03]]{}) or the Hinode satellite, there is some hope that most of the turbulent velocity fields may be resolved. In that case, the turbulent broadening would be negligible and the line-ratio technique would be more robust. However, even with the highest possible spatial resolution, velocity and temperature fluctuations along the line of sight will still produce turbulent broadening.
From the study presented here we conclude that, away from active region flux concentrations, it is not straightforward to measure intrinsic field strengths from either 5250 or 6302 Å observations taken separately. Weak-flux internetwork observations would be even more challenging, as demonstrated recently by [@MG07 ([-@MG07])]{}. Surprisingly enough, the 6302 Å pair of lines is more robust than the 5250 Å lines in the sense that it is indeed possible to discriminate between weak and strong field solutions if one is able to rule out a thermal stratification with temperatures that increase outwards. Even so, this is only possible when one employs an inversion code that has sufficient MHD constrains (an example is the MISMA implementation used here) to reduce the space of possible solutions.
The longitudinal flux density obtained from inversions of the 6302 Å lines is better determined than those obtained with 5250 Å. This happens regardless of the inversion method employed, although using a code like LILIA provides better results than a simpler one such as MELANIE. The best fits to average network profiles correspond to strong kG fields, as one would expect.
An interesting conclusion of this study is that it is possible to obtain reliable results by inverting simultaneous observations at both 5250 and 6302 Å. Obviously this would be possible with relatively sophisticated algorithms (e.g., LTE inversions) but not with simple Milne-Eddington inversions.
The combination of two other lines, namely those at 4122 and 9000 Å, seems to provide a much more robust determination of the quiet Sun magnetic fields. Unfortunately, these lines are very distant in wavelength and few spectro-polarimeters are capable of observing them simultaneously. Examples of instrument with this capability are the currently operational SPINOR and THEMIS, as well as the planned ATST and GREGOR. Depending on the evolution time scales of the structures analyzed it may be possible for some other instruments to observe the blue and red lines alternatively.
This work has been partially funded by the Spanish Ministerio de Educación y Ciencia through project AYA2004-05792
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[^1]: In this work, we use the term “quiet Sun” to refer to the solar surface away from sunspots and active regions.
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---
abstract: 'A comprehensive set of experiments on the effect of high-frequency surface acoustic waves, SAWs, in the spin relaxation in Mn$_{12}$-acetate is presented. We have studied the quantum magnetic deflagration induced by SAWs under various experimental conditions extending the data shown in a very recent paper \[A. Hernández-Mínguez *et. al.*, Phys. Rev. Lett. **95**, 217205 (2005)\]. We have focused our study on the dependence of both the ignition time and the propagation speed of the magnetic avalanches on the frequency, amplitude, and duration of the SAW pulses in experiments performed under different temperatures and external magnetic fields.'
author:
- 'A. Hernández-Mínguez'
- 'F. Macià'
- 'J. M. Hernandez'
- 'J. Tejada'
- 'P. V. Santos'
title: 'Phonon-Induced Quantum Magnetic Deflagration in Mn$_{12}$'
---
Introduction
============
Molecular nanomagnets are well-defined discrete molecules consisting of several transition metal ions interacting through organic and/or inorganic ligands. The so called single molecule magnets, SMMs, are characterized by two energy scales: i) the strong exchange interactions between metal ions within a molecule, and ii) an anisotropic spin-orbit coupling that results in pronounced magnetic anisotropy. Some of these SMMs have large spin ($S$) values as well as a high energy anisotropy barrier to prevent spontaneous magnetization reversal at low temperatures.[@SESSOLI] These well characterized objects have an intermediate size between microscale and macroscale and thus allow for the study of phenomena at the border between macroscopic quantum tunnelling and the conventional quantum mechanics of spin. The first unambiguous evidence for resonant magnetization tunnelling in molecular magnets came from the stepwise magnetic hysteresis experiments at different temperatures performed on a Mn$_{12}$-acetate sample, which has $S=10$ and an anisotropy barrier height .[@TEJADA] Since then, numerous experiments have been performed on several SMMs demonstrating the existence of very interesting phenomena, which are well accounted for by theoretical calculations. For a review see Refs. .
To understand the physics of the SMMs, let us assume the simplest Hamiltonian with the magnetic field applied parallel to the easy anisotropy axis: $$\mathcal{H} = -DS_{z}^{2} - H_{z}S_{z}$$
This Hamiltonian yields pairs of degenerate levels for fields $H_z=nD$ with $n=0,1,\ldots, S$. When all molecules occupy the spin states with a negative magnetic quantum number $m$, the sample is said to be magnetized in the negative direction. By applying a magnetic field in the positive $z$ direction, the molecules eventually relax to the spin states with positive $m$. This process may occur via thermal transitions over the energy barrier or through quantum tunnelling between the states on different sides of the energy barrier. The tunnelling adds to the thermal activation when the levels at the two sides of the barrier are on resonance, that is at $H_{z}=nD$. To account for tunnelling, one must include additional terms in the spin Hamiltonian, which do not commute with $S_z$. These terms may be associated with transverse anisotropy, like quadratic and higher-order terms in $S_x$ and $S_y$ or the transversal Zeeman energy term $-H_xS_x$.
From the experimental point of view, there are two approaches to study spin tunnelling: by 1) measuring the magnetic relaxation at fixed temperature and magnetic field or by 2) sweeping the magnetic field through the resonant field and detecting the fraction of molecules, $\eta$, that change their magnetic moment. $\eta$ depends on the overlap of the wave functions of the resonant spin states at the two sides of the barrier, which defines the tunnelling splitting, $\Delta$, as well as on the sweep rate of the magnetic field energy, $v$, according to the expression
$$\eta=1-\exp(-\frac{\pi\Delta^{2}}{2\hbar v})$$
Since the 1990s, [@art:Fominaya; @art:delBarco] it has been known that for fast sweeping rates, the magnetic reversal in large crystals takes place in a short time scale (typically less than ) through so-called magnetic avalanches. In this process, the initial relaxation of the magnetization towards the direction of the magnetic field results in the release of heat that further accelerates the magnetic relaxation. Recent local magnetic measurements on Mn$_{12}$-acetate crystals have demonstrated that during an avalanche the magnetization reversal occurs inside a narrow interface that propagates through the crystal at a constant speed of a few meters per second.[@YOKO] This phenomenon has been named “magnetic deflagration” because of its parallelism to classical chemical deflagration. Deflagration is a subsonic combustion that propagates through thermal conductivity (a steady flame heats the next layer of cold material and ignites it). Classically, combustion describes the exothermic chemical reaction between a substance and a gas (usually O$_2$). The gas oxidize the substance (the fuel), leading to heat release. During magnetic deflagration, the magnetic field “oxidizes” spins originally in a meta-stable state and makes them flip. The released heat then feedbacks the magnetization reversal process.
Recently, a novel method for the controlled ignition of avalanches at constant magnetic field by means of surface acoustic waves, SAWs, has been reported.[@AHM] These acoustomagnetic experiments are performed by using hybrid piezoelectric interdigital transducers deposited on a LiNbO$_3$ substrate, with the Mn$_{12}$ single crystals directly glued onto the piezoelectric.[@AHM; @JMH] The investigations have clearly shown that the propagation speed of the avalanches exhibits maxima for magnetic fields corresponding to the tunnelling resonances of Mn$_{12}$-acetate. The results suggest, therefore, a novel physical phenomenon: deflagration assisted by quantum tunnelling. In this paper, we study these well-controlled ignited magnetic avalanches assisted by spin tunnelling in Mn$_{12}$-acetate under various experimental conditions.
Experimental set up
===================
The acoustomagnetic experiments were performed by using hybrid piezoelectric interdigital transducers (IDT) deposited on 128 YX-cut LiNbO$_{3}$ substrates[@PIEZO] with dimensions $4\times12\times1$ mm$^{3}$. Several single crystals of Mn$_{12}$-acetate with average dimensions of were studied. Each crystal was measured independently. The sample was glued directly onto the IDT using a commercial silicon grease. The microwaves for the SAW generation were transported to the transducers by coaxial cables (cf. Fig. \[fig:setup\]). Experiments have been carried out using a commercial rf-SQUID Quantum Design magnetometer at temperatures between 2 and 2.7 K.
![Experimental setup. 1-coaxial cable; 2-conducting stripes; 3-interdigital transducers (IDT); 4-LiNbO$_3$ substrate; 5-Mn$_{12}$ crystal. The $c$-axis of the Mn$_{12}$ crystal is oriented parallel to the applied magnetic field $H_z$. \[fig:setup\]](Fig1){width="\columnwidth"}
The frequency response of the IDTs was measured using an Agilent network analyzer. The frequency dependence of the reflection coefficient $S_{11}$, displayed in Fig. \[fig:S11&DM\](a), shows that the coaxial cables introduce an attenuation smaller than 10 dB. The IDT generates SAWs at multiple harmonics of the fundamental frequency of up to a maximum frequency of approximately . Next to the $S_{11}$ determination, we also measured the frequency dependence of the magnetization changes induced by the SAWs in the superparamagnetic regime of Mn$_{12}$ (i.e., for temperatures $>3$ K). In this case, the microwave pulses used to excite the IDT were generated by a commercial Agilent signal generator, which allows the selection of the shape, duration, and energy of the pulses in the frequency range between to . Most of the experiments were performed using rectangular microwave pulses of duration . Fast magnetization measurements (time resolution of ) were carried out at constant temperature and magnetic field by continuously recording the voltage variation detected by the rf-SQUID. The negative magnetization changes ($-\Delta M$, see Fig. \[fig:S11&DM\]b) show peaks at the resonance frequencies of $S_{11}$, thus clearly demonstrating that they are induced by the acoustic field. The magnitude of the magnetization variation for each peak depends on how much acoustic energy is absorbed by the Mn$_{12}$ single crystal. During the experiments, the temperature of the IDT attached to the sample and the temperature of the helium gas that provided heat exchange were independently monitored.[@JMH]
![(a) Reflection coefficient, $S_{11}$, of the LiNbO$_3$/Mn$_{12}$ hybrids (including the effects of the coaxial assembly used to feed the microwave power) and (b) magnetization variation measured by the magnetometer as a function of the frequency of the microwave pulses sent to the interdigital transducer (IDT). The temperature and applied magnetic field are $T=6$ K and $H=1$ T, respectively.\[fig:S11&DM\]](Fig2.eps){width="\columnwidth"}
To study magnetic avalanches, we first saturated the sample magnetization at -2 T (i.e., below the blocking temperature) and then swept the magnetic field at a constant rate of 300 Oe/s up to a predefined value $H$. A few seconds later, we applied a rectangular microwave pulse with well-controlled frequency, energy, and duration to excite the SAWs and trigger the magnetic avalanche. These experiments have established a new method for igniting magnetization avalanches in molecular magnets with total control of the magnetic field and initial temperature. Furthermore, by varying the frequency, duration, and nominal power of the pulse we have been able to study the dynamics of the avalanche ignition process.
Discussion
==========
Time evolution of the avalanches
--------------------------------
Figure \[fig:tev\] shows the time evolution of the magnetization during avalanches ignited at $T=2$ K with SAWs pulses of $t_{p}=5$ ms at different magnetic fields. As the whole experiment is performed in a time on the order of a few seconds (i.e., much shorter than the relaxation time in the absence of acoustic excitation), it can be carried out at any temperature below the blocking temperature $T_B=3$ K. From the magnetic point of view, the important experimental fact is that since the magnetization is measured with a resolution time of 1 $\mu$s, we have been able to detect three different stages during magnetization reversal:
![Time evolution of the magnetization, $M$, for different magnetic fields at $T=2$ K. The magnetization is normalized to the total magnetization change during the avalanche, $M_\mathrm{aval}$. The duration of the SAWs pulse is $t_p=5$ ms and the frequency is $\nu=449$ MHz. $t=0$ ms corresponds to the instant at which the microwave pulse is applied. The results show that the rate of change of the magnetization peaks at the resonant fields of $H=4600$ Oe and $H=9200$ Oe.\[fig:tev\]](Fig3){width="\columnwidth"}
1. The first stage corresponds to the time elapsed between the application of the microwave pulse and the first detection of magnetization changes. During this time, the SAWs thermalize and the magnetization reversal process nucleates within a small region of the Mn$_{12}$ crystal. To define this nucleation time, we have adopted the criterium that it corresponds to the time interval between the application of the SAWs pulse and the detection of the first magnetization change.
2. The second stage corresponds to the time interval between nucleation and the formation of a stable deflagration front propagating with constant velocity through the crystal.[@YOKO] During this time, the nucleation bubble containing reversed spins becomes larger and larger until a steady interface is formed and it propagates along the sample (the “flame” in magnetic deflagration). The heat energy liberated during spin reversal corresponds to the Zeeman energy $\Delta E=g\mu_BH_z\Delta S$ per spin. The temperature increase induced by the spin reversal leads to the expansion of the nucleation bubble across the width of the Mn$_{12}$ crystal (cf. Fig. \[fig:setup\]). In contrast to the subsequent steady-state propagation along the $c$-axis, this propagation of the front flame depends on the geometry of the crystal and on its magnetic history. For instance, the time to reach that flame will depend on the width of the sample. We call the [*ignition time*]{}, $t_\mathrm{ign}$, the total duration of these two first steps.
3. The third stage corresponds to the so-called magnetic deflagration, where the flame propagates at constant velocity along the $c$-axis of the Mn$_{12}$ single crystal. During this time, the rate of magnetization variation is constant and the flame front has a finite width $\delta$. Contrary to the previous stage, the temperature remains constant at value $T_f$ considerably greater than those achieved during the flame formation.
Figure \[fig:vel&tign\](a) shows the velocity $v=l/\Delta t$ of the deflagration front deduced from the magnetization data displayed in Fig. \[fig:tev\]. Here, $l$ is the length of the crystal and the avalanche time, $\Delta t$, is defined as the time needed by the magnetization to change between the 20% and the 80% of the total variation, $M_{aval}$. The velocity $v$ increases with the applied magnetic field and presents peaks at the resonant field values. The dependence on magnetic field is well-fitted (see solid line in Fig. \[fig:vel&tign\](a)) by the law: [@YOKO]
$$v=\sqrt{\frac{\kappa}{\tau_0}}\exp{\left[-\frac{U(H)}{2k_BT_f}\right]}
\label{Eq3}$$
where $\kappa$ is the thermal diffusivity. $U(H)$, $\tau_0$ and $T_f$ are, respectively, the energy barrier, the attempt frequency, and the temperature of the “flame”, which are related to the “chemical reaction time” $\tau$ by the expression $\tau=\tau_0\exp\left[U(H)/k_BT_f\right]$. In the case of Mn$_{12}$, $\kappa\sim10^{-5}$ m$^2/s$, $\tau_0\sim10^{-7}$ s, and the field dependence of the energy barrier, $U(H)$, is well known.[@art:delBarco; @art:Sales] The temperature of the flame increases linearly with magnetic field. According to the fitting, the temperature of the flame front is about 6.8 K for $H=$4600 Oe, and increases to 10.9 K for $H=$9200 Oe.
Figure. \[fig:vel&tign\]b shows the dependence of the ignition time (i.e., the time delay between the application of the microwave pulse and the observation of a steady flame front) on magnetic field. The ignition time decreases with increasing magnetic field, and shows minima at the resonant fields. The later are a consequence of the enhanced spin reversal probability at the resonant fields. In other words, the ignition time reproduces the dependence of the effective barrier height for magnetic transitions on the magnetic field.
![(a) Velocity and (b) ignition time ($t_\mathrm{ign}$) of the deflagration front as a function of magnetic field recorded at $T=2$ K by using $t_p=5$ ms SAWs pulses with a frequency $\nu=449$ MHz. The vertical dotted lines mark the resonant field values. The solid line displays a fit to Eq. \[Eq3\].\[fig:vel&tign\]](Fig4){width="\columnwidth"}
The applied acoustic energy {#subsec:energy}
---------------------------
To analyze the dependence of the avalanches on the acoustic energy supplied by the SAWs, we applied SAWs pulses of different durations, $t_{p}$, and power, $P$, and the time evolution of the magnetization for magnetic fields at and out of resonance was recorded.
A very interesting point is that, at a given field and temperature, there is a minimum value for the acoustic energy $P
t_p$ required to trigger the avalanche. As illustrated in Fig. \[fig:dMvstp&P\](a) and Fig. \[fig:dMvstp&P\](b), this threshold energy can be surpassed by varying either the amplitude or the duration of the acoustic pulses. The acoustic energy supplied by the piezoelectric in excess of the threshold energy for ignition has no further influence on the avalanche dynamics, which becomes determined only by temperature and by the amplitude of the magnetic field.
These results can be easily understood if we consider that the threshold energy is the minimal energy required to create the initial bubble of reversed spins. Once the nucleation process is overcome, the energy released by spin reversal is much greater than the acoustic energy delivered by the SAWs. This process is analogous to the ignition of a combustion process: the material to be burned has to be externally heated only until the combustion begins. After that, the amount of energy released by the exothermic reaction is large enough to maintain the combustion front propagating through the material.
![Dependence of magnetization changes during the avalanche on the (a) power ($P$, recorded for $t_{p}=1$ ms) and (b) duration ($t_p$, recorded for for $P=20$ dBm) of the SAWs pulses. The measurements were carried out at $T=2$ K under magnetic fields of $H=8800$ Oe (solid squares) and $H=11000$ Oe (open squares).\[fig:dMvstp&P\]](Fig5){width="\columnwidth"}
Temperature dependence {#subs:Tdep}
----------------------
![Ignition time versus applied magnetic field, $H$, for different temperatures. The duration of the pulse and frequency are $t_p=10$ ms, $\nu=224$ MHz. The inset shows the magnitude of the avalanche for different temperatures and magnetic fields: $H=4800$ Oe (solid squares), $H=6400$ Oe (open squares), $H=9200$ Oe (solid circles), $H=11000$ Oe (open circles), $H=14000$ Oe (solid triangles).\[fig:tignvsT\]](Fig6){width="\columnwidth"}
![Time evolution of magnetization for different temperatures under an applied field $H=8000$ Oe. The values of the initial temperature are: $T=2.0$ K (solid squares), $T=2.1$ K (open squares), $T=2.2$ K (solid circles), $T=2.3$ K (open circles), $T=2.4$ K (solid triangles), $T=2.5$ K (open triangles), $T=2.7$ K (solid rhombus).[]{data-label="fig:tevvsT"}](Fig7 "fig:"){width="\columnwidth"}\
In order to study the dependence of the deflagration on the initial amount of reversed spins, we have produced magnetic avalanches at different temperatures and magnetic fields by applying acoustic pulses of fixed duration and amplitude. Figure \[fig:tignvsT\] illustrates the typical behavior found at low temperatures (i.e., below 2.3 K). Under these conditions, the ignition time is temperature independent and decreases exponentially with applied magnetic field, with minima at the resonant field values.
Nevertheless, as the blocking temperature is approached, the ignition time becomes sensitive not only to the magnetic field but also to temperature. In particular, the features associated with the second and third resonances (at $H=9.5$ kOe and $H=13.5$ kOe, respectively) change from minima in the $t_\mathrm{ign}$ traces recorded at low temperatures ($T<2.3$ K) to maxima in traces measured above 2.3 K. At first sight, one should attribute this behavior to the enhanced relaxation that must occur at the resonant fields before the SAWs pulse is applied: since the concentration of metastable spins reduces, it takes longer to create the nucleation bubble required to launch the front flame. In fact, the total magnetization change during the avalanche, $M_{aval}$, decreases slightly with temperature at low magnetic fields, as indicated in the inset of Fig. \[fig:tignvsT\]. However, at high fields and temperatures above $2.3$ K, $M_{aval}$ presents an anomalous behavior, which has been consistently observed in several crystals. This behavior is presently not understood and makes unclear the relationship between $t_{ign}$ and the initial magnetic state of the sample in this temperature range.
Figure. \[fig:tevvsT\] shows the time evolution of the magnetization for avalanches ignited at different temperatures under a fixed magnetic field . Like the ignition time, the velocity of the avalanche depends weakly on the temperature for low magnetic fields, but slowers down at high fields and temperatures.
Dependence on SAWs frequency
----------------------------
![Velocity of the flame front, $v$, versus applied magnetic field, $H$, for different SAWs frequencies. The temperature and SAWs pulse duration are $T=2$ K and $t_p=5$ ms, respectively. The SAWs frequencies and powers are: $\nu=224$ MHz and $P=13.7$ dBm (solid squares); $\nu=449$ MHz and $P=15.2$ dBm (open circles); $\nu=895$ MHz and $P=20$ dBm (solid triangles).\[fig:velvsfrec\]](Fig8){width="\columnwidth"}
![Ignition time ($t_\mathrm{ign}$) versus applied magnetic field, $H$, and different frequencies of the SAWs. The temperature is $T=2.1$ K and $t_p=10$ ms. The values of frequency are: $\nu=114$ MHz (solid squares), $\nu=225$ MHz (open squares), $\nu=336$ MHz (solid circles), $\nu=449$ MHz (open circles), $\nu=671$ MHz (solid triangles) and $\nu=783$ MHz (open triangles). The inset shows the ignition time for different frequencies at $H=10200$ Oe.\[fig:tignvsfrec\]](Fig9){width="\columnwidth"}
To study the effect of the SAWs frequency on the magnetic avalanches, it is necessary to ensure that, for each frequency, we couple the same amount of acoustic energy to the Mn$_{12}$ crystals. For that purpose, we fixed the duration $t_{p}$ of the SAWs pulses and selected their amplitude in order to produce the same magnetization variation when the sample is in the superparamagnetic regime. The results are shown in Figs. \[fig:velvsfrec\] and \[fig:tignvsfrec\]. As expected, the propagation velocity (Fig. \[fig:velvsfrec\]) is essentially independent of the frequency of the SAWs, since it is primarily determined by the internal energy liberated during spin reversal. In contrast, the ignition time (Fig. \[fig:tignvsfrec\]), which depends on the external energy supplied by the SAWs, shows a strong frequency dependence. Furthermore, for each magnetic field, the ignition time shows a non-monotonic behavior with frequency, with a minimum at 225 MHz.
Conclusion
==========
We have investigated the dynamics of magnetic avalanches induced by acoustic waves in Mn$_{12}$ crystals. The magnetic avalanche becomes a fully deterministic process under acoustic excitation. When the acoustic power exceeds the threshold for the nucleation of the avalanche process, the avalanche dynamics can, therefore, be investigated for different values of the magnetic field and temperature. Well below the blocking temperature, the velocity of the deflagration front of reversing magnetization and the ignition time do only depend on the applied magnetic field. The ignition time depends also on the frequency of the SAWs.
We are indebted to W. Seidel and S. Krauss for the fabrication of the transducers for SAWs generation. A. H-M. thanks the Spanish Ministerio de Educación y Ciencia for a research grant. J. M. H. thanks the Ministerio de Educación y Ciencia and the University of Barcelona for a Ramón y Cajal research contract. F. M. and J. T. thank SAMCA Enterprise for financial support.
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---
abstract: 'We study the nonequilibrium phase transition in the one-dimensional contact process with quenched spatial disorder by means of large-scale Monte-Carlo simulations for times up to $10^9$ and system sizes up to $10^7$ sites. In agreement with recent predictions of an infinite-randomness fixed point, our simulations demonstrate activated (exponential) dynamical scaling at the critical point. The critical behavior turns out to be universal, even for weak disorder. However, the approach to this asymptotic behavior is extremely slow, with crossover times of the order of $10^4$ or larger. In the Griffiths region between the clean and the dirty critical points, we find power-law dynamical behavior with continuously varying exponents. We discuss the generality of our findings and relate them to a broader theory of rare region effects at phase transitions with quenched disorder.'
author:
- Thomas Vojta
- Mark Dickison
title: Critical behavior and Griffiths effects in the disordered contact process
---
Introduction {#sec:intro}
============
The nonequilibrium behavior of many-particle systems has attracted considerable attention in recent years. Of particular interest are continuous phase transitions between different nonequilibrium states. These transitions are characterized by large scale fluctuations and collective behavior over large distances and times very similar to the behavior at equilibrium critical points. Examples of such nonequilibrium transitions can be found in population dynamics and epidemics, chemical reactions, growing surfaces, and in granular flow and traffic jams (for recent reviews see, e.g., Refs.[@SchmittmannZia95; @Marro_book; @Dickman; @chopard_book; @Hinrichsen00; @Odor; @tauber_rev])
A prominent class of nonequilibrium phase transitions separates active fluctuating states from inactive, absorbing states where fluctuations cease entirely. Recently, much effort has been devoted to classifying possible universality classes of these absorbing state phase transitions [@Hinrichsen00; @Odor]. The generic universality class is directed percolation (DP) [@dp]. According to a conjecture by Janssen and Grassberger [@conjecture], all absorbing state transitions with a scalar order parameter, short-range interactions, and no extra symmetries or conservation laws belong to this class. Examples include the transitions in the contact process [@contact], catalytic reactions [@ziff], interface growth [@tang], or turbulence [@turb]. In the presence of conservation laws or additional symmetries, other universality classes can occur, e.g., the parity conserving class [@Takayasu92; @Zhong95; @Tauber96] or the $Z_2$-symmetric directed percolation (DP2) class [@KimPark94; @Menyhard94; @Hinrichsen97].
In realistic systems, one can expect impurities and defects, i.e., quenched spatial disorder, to play an important role. Indeed, it has been suggested that disorder may be one of the reasons for the surprising rarity of experimental realizations of the ubiquitous directed percolation universality class [@hinrichsen_exp]. The only verification so far seems to be found in the spatio-temporal intermittency in ferrofluidic spikes [@spikes].
The investigation of disorder effects on the DP transition has a long history, but a coherent picture has been slow to emerge. According to the Harris criterion [@harris; @Noest86], a clean critical point is stable against weak disorder if the spatial correlation length critical exponent $\nu_\perp$ fulfills the inequality $$d\nu_\perp > 2,$$ where $d$ is the spatial dimensionality. The DP universality class violates the Harris criterion in all dimensions $d<4$, because the exponent values are $\nu_\perp \approx
1.097$ (1D), 0.73 (2D), and 0.58 (3D) [@Hinrichsen00]. A field-theoretic renormalization group study [@janssen97] confirmed the instability of the DP critical fixed point. Moreover, no new critical fixed point was found. Instead the renormalization group displays runaway flow towards large disorder, indicating unconventional behavior. Early Monte-Carlo simulations [@Noest86] showed significant changes in the critical exponents while later studies [@moreira] of the two-dimensional contact process with dilution found logarithmically slow dynamics in violation of power-law scaling. In addition, rare region effects similar to Griffiths singularities [@Griffiths] were found to lead to slow dynamics in a whole parameter region in the vicinity of the phase transition [@moreira; @Noest88; @bramson; @webman; @cafiero].
Recently, an important step towards understanding spatial disorder effects on the DP transition has been made by Hooyberghs et al. [@hooyberghs]. These authors used the Hamiltonian formalism [@alcaraz] to map the one-dimensional disordered contact process onto a random quantum spin chain. Applying a version of the Ma-Dasgupta-Hu strong-disorder renormalization group [@SDRG], they showed that the transition is controlled by an infinite-randomness critical point, at least for sufficiently strong disorder. This type of fixed point leads to activated (exponential) rather than power-law dynamical scaling. For weaker disorder, Hooyberghs et al. [@hooyberghs] used computer simulations and predicted non-universal continuously varying exponents, with either power-law or exponential dynamical correlations.
In this paper, we present the results of large-scale Monte-Carlo simulations of the one-dimensional contact process with quenched spatial disorder. Using large systems of up to $10^7$ sites and very long times (up to $10^9$) we show that the critical behavior at the nonequilibrium phase transition is indeed described by an infinite-randomness fixed point with activated scaling. Moreover, we provide evidence that this behavior is universal, i.e., it occurs even in the weak-disorder case. However, the approach to this universal asymptotic behavior is extremely slow, with crossover times of the order of $10^4$ or larger which may explain why nonuniversal (effective) exponents have been seen in previous work. We also study the Griffiths region between the clean and the dirty critical points. Here, we find power-law dynamical behavior with continuously varying exponents, in agreement with theoretical predictions.
This paper is organized as follows. In section \[sec:theory\], we introduce the model. We then contrast power-law scaling as found at conventional critical points with activated scaling arising from infinite-randomness critical points. We also summarize the predictions for the Griffiths region. In section \[sec:mc\], we present our simulation method and the numerical results together with a comparison to theory. We conclude in section \[sec:conclusions\] by discussing the generality of our findings and their relation to a broader theory of rare region effects at phase transitions with quenched disorder.
Theory {#sec:theory}
======
Contact process with quenched spatial disorder
----------------------------------------------
We start from the clean contact process [@contact], a prototypical system in the DP universality class. It can be interpreted, e.g., as a model for the spreading of a disease. The contact process is defined on a $d$-dimensional hypercubic lattice. Each lattice site $\mathbf r$ can be active (occupied by a particle) or inactive (empty). In the course of the time evolution, active sites can infect their neighbors, or they can spontaneously become inactive. Specifically, the dynamics is given by a continuous-time Markov process during which particles are created at empty sites at a rate $\lambda n/
(2d)$ where $n$ is the number of active nearest neighbor sites. Particles are annihilated at rate $\mu$ (which is often set to unity without loss of generality). The ratio of the two rates controls the behavior of the system.
For small birth rate $\lambda$, annihilation dominates, and the absorbing state without any particles is the only steady state (inactive phase). For large birth rate $\lambda$, there is a steady state with finite particle density (active phase). The two phases are separated by a nonequilibrium phase transition in the DP universality class at $\lambda=\lambda_c^0$. The central quantity in the contact process is the average density of active sites at time $t$ $$\rho(t) = \frac 1 {L^d} \sum_{\mathbf{r}} \langle n_\mathbf{r}(t) \rangle$$ where $n_\mathbf{r}(t)$ is the particle number at site $\mathbf{r}$ and time $t$, $L$ is the linear system size, and $\langle \ldots \rangle$ denotes the average over all realizations of the Markov process. The longtime limit of this density (i.e., the steady state density) $$\rho_{\rm stat} = \lim_{t\to\infty} \rho(t)$$ is the order parameter of the nonequilibrium phase transition.
Quenched spatial disorder can be introduced by making the birth rate $\lambda$ a random function of the lattice site $\mathbf{r}$. We assume the disorder to be spatially uncorrelated; and we use a binary probability distribution $$P[\lambda({\mathbf r})] = (1-p)\, \delta[\lambda({\mathbf r})-\lambda] + p\,
\delta[\lambda({\mathbf r}) - c\lambda] \label{eq:impdist}$$ where $p$ and $c$ are constants between 0 and 1. This distribution allows us to independently vary spatial density $p$ of the impurities and their relative strength $c$. The impurities locally *reduce* the birth rate, therefore, the nonequilibrium transition will occur at a value $\lambda_c$ that is larger than the clean critical birth rate $\lambda_c^0$.
Conventional power-law scaling
------------------------------
In this subsection we summarize the phenomenological scaling theory of an absorbing state phase transition that is controlled by a conventional fixed point with power-law scaling (see, e.g., Ref. [@Hinrichsen00]). The clean contact process falls into this class.
In the active phase and close to critical point $\lambda_c$, the order parameter $\rho_{\rm stat}$ varies according to the power law $$\rho_{\rm stat} \sim (\lambda-\lambda_c)^\beta \sim \Delta^\beta$$ where $\Delta=(\lambda-\lambda_c)/\lambda_c$ is the dimensionless distance from the critical point, and $\beta$ is the critical exponent of the particle density. In addition to the average density, we also need to characterize the length and time scales of the density fluctuations. Close to the transition, the correlation length $\xi_\perp$ diverges as $$\xi_\perp \sim |\Delta|^{-\nu_{\perp}}~.$$ The correlation time $\xi_\parallel$ behaves like a power of the correlation length, $$\xi_\parallel \sim \xi_\perp^z, \label{eq:powerlawscaling}$$ i.e., the dynamical scaling is of power-law form. Consequently the scaling form of the density as a function of $\Delta$, the time $t$ and the linear system size $L$ reads $$\rho(\Delta,t,L) = b^{\beta/\nu_\perp} \rho(\Delta b^{-1/\nu_\perp},t b^z, L b)~.
\label{eq:rho}$$ Here, $b$ is an arbitrary dimensionless scaling factor.
Two important quantities arise from initial conditions consisting of a single active site in an otherwise empty lattice. The survival probability $P_s$ describes the probability that an active cluster survives when starting from such a single-site seed. For directed percolation, the survival probability scales exactly like the density [@betaprime], $$P_s(\Delta,t,L) = b^{\beta/\nu_\perp} P_s(\Delta b^{-1/\nu_\perp},t b^z, L b)~.
\label{eq:Ps}$$ Thus, for directed percolation, the three critical exponents $\beta$, $\nu_\perp$ and $z$ completely characterize the critical point.
The pair connectedness function $C(\mathbf{r'},t',\mathbf{r},t)=\langle
n_{\mathbf{r'}}(t') \, n_{\mathbf{r}}(t) \rangle$ describes the probability that site $\mathbf{r}'$ is active at time $t'$ when starting from an initial condition with a single active site at $\mathbf{r}$ and time $t$. For a clean system, the pair connectedness is translationally invariant in space and time. Thus, it only depends on two arguments $C(\mathbf{r},t',\mathbf{r},t)=C(\mathbf{r'}-\mathbf{r},t-t')$. Because $C$ involves a product of two densities, its scale dimension is $2\beta/\nu_\perp$, and the full scaling form reads [@hyperscaling] $$C(\Delta,\mathbf{r},t,L) = b^{2\beta/\nu_\perp} C(\Delta b^{-1/\nu_\perp}, \mathbf{r}b, t
b^z, L b)~.$$ The total number of particles $N$ when starting from a single seed site can be obtained by integrating the pair connectedness $C$ over all space. This leads to the scaling form $$N(\Delta,t,L) = b^{2\beta/\nu_\perp - d} N(\Delta b^{-1/\nu_\perp},t b^z, L b)~.
\label{eq:N}$$
At the critical point, $\Delta=0$, and in the thermodynamic limit, $L\to\infty$, the above scaling relations lead to the following predictions for the time dependencies of observables: The density and the survival probability asymptotically decay like $$\rho(t) \sim t^{-\delta}, \qquad P_s(t) \sim t^{-\delta}$$ with $\delta=\beta/(\nu_\perp z)$. In contrast, the number of particles in a cluster starting from a single seed site increases like $$N(t) \sim t^\Theta$$ where $\Theta=d/z - 2\beta/(\nu_\perp z)$ is the so-called critical initial slip exponent.
Highly precise estimates of the critical exponents for clean one-dimensional directed percolation have been obtained by series expansions [@Jensen99]: $\beta=0.276486$, $\nu_\perp=1.096854$, $z=1.580745$, $\delta=0.159464$, and $\Theta=0.313686$.
Activated scaling {#subsec:activated}
-----------------
In this subsection we summarize the scaling theory for an infinite-randomness fixed point with activated scaling, as has been predicted to occur in absorbing state transitions with quenched disorder [@hooyberghs]. It is similar to the scaling theory for the quantum phase transition in the random transverse field Ising model [@dsf9295].
At an infinite-randomness fixed point, the dynamics is extremely slow. The power-law scaling (\[eq:powerlawscaling\]) gets replaced by activated dynamical scaling $$\ln(\xi_\parallel) \sim \xi_\perp^\psi, \label{eq:activatedscaling}$$ characterized by a new exponent $\psi$. This exponential relation between time and length scales implies that the dynamical exponent $z$ is formally infinite. In contrast, the static scaling behavior remains of power law type.
Moreover, at an infinite-randomness fixed point the probability distributions of observables become extremely broad, so that averages are dominated by rare events such as rare spatial regions with large infection rate. In such a situation, averages and typical values of a quantity do not necessarily agree. Nonetheless, the scaling form of the *average* density at an infinite-randomness critical point is obtained by simply replacing the power-law scaling combination $t b^z$ by the activated combination $\ln(t)
b^\psi$ in the argument of the scaling function: $$\rho(\Delta,\ln(t),L) = b^{\beta/\nu_\perp} \rho(\Delta b^{-1/\nu_\perp},\ln(t) b^\psi, L
b)~. \label{eq:rho_activated}$$ Analogously, the scaling forms of the average survival probability and the average number of sites in a cluster starting from a single site are $$\begin{aligned}
P_s(\Delta,\ln(t),L) &=& b^{\beta/\nu_\perp} P_s(\Delta b^{-1/\nu_\perp},\ln(t) b^\psi,L
b) \label{eq:Ps_activated}\\ N(\Delta,\ln(t),L) &=& b^{2\beta/\nu_\perp -d} N(\Delta
b^{-1/\nu_\perp},\ln(t) b^\psi,L b) ~~. \label{eq:N_activated}\end{aligned}$$
These activated scaling forms lead to logarithmic time dependencies at the critical point (in the thermodynamic limit). The average density and the survival probability asymptotically decay like $$\rho(t) \sim [\ln(t)]^{-\bar\delta}, \qquad P_s(t) \sim [\ln(t)]^{-\bar\delta}
\label{eq:logdecay}$$ with $\bar\delta=\beta/(\nu_\perp \psi)$ while the average number of particles in a cluster starting from a single seed site increases like $$N(t) \sim [\ln(t)]^{\bar\Theta} \label{eq:clustersize}$$ with $\bar\Theta=d/\psi-2\beta/(\nu_\perp \psi)$. These are the relations we are going to test in this paper.
Within the strong disorder renormalization group approach of Ref. [@hooyberghs], the critical exponents of the disordered one-dimensional contact process can be calculated exactly. Their numerical values are $\beta=0.38197$, $\nu_\perp=2$, $\psi=0.5$, $\bar\delta=0.38197$, and $\bar\Theta=1.2360$.
Griffiths region {#subsec:Griffiths}
----------------
The inactive phase of our disordered contact process with the impurity distribution (\[eq:impdist\]) can be divided into two regions. For birth rates below the clean critical point, $\lambda<\lambda_c^0$, the behavior is conventional. The system approaches the absorbing state exponentially fast in time. The decay time increases with $\lambda$ and diverges as $|\lambda-\lambda_c^0|^{-z\nu_\perp}$ where $z$ and $\nu_\perp$ are the exponents of the clean critical point [@Bray88; @Dickison05].
The more interesting region is the so-called Griffiths region [@Noest86; @Griffiths] which occurs for birth rates between the clean and the dirty critical points, $\lambda_c^0 < \lambda < \lambda_c$. The system is globally still in the inactive phase, i.e., the system eventually decays into the absorbing state. However, in the thermodynamic limit, one can find arbitrarily large spatial regions devoid of impurities. For $\lambda_c^0 < \lambda < \lambda_c$, these so-called rare regions are locally in the active phase. Because they are of finite size, they cannot support a non-zero steady state density but their decay is very slow because it requires a rare, exceptionally large density fluctuation.
The contribution of the rare regions to the time evolution of the density can be estimated as follows [@Noest86; @Noest88]. The probability $w$ for finding a rare region of linear size $L_r$ devoid of impurities is (up to pre-exponential factors) given by $$w(L_r) \sim \exp( -\tilde p L_r^d)$$ where $\tilde p$ is a nonuniversal constant which for our binary disorder distribution is given by $\tilde p = - \ln(1-p)$. The long-time decay of the density is dominated by these rare regions. To exponential accuracy, the rare region contribution to the density can be written as $$\rho(t) \sim \int dL_r ~L_r^d ~w(L_r) \exp\left[-t/\tau(L_r)\right] \label{eq:rrevo}$$ where $\tau(L_r)$ is the decay time of a rare region of size $L_r$. Let us first discuss the behavior at the clean critical point, $\lambda_c^0$, i.e., at the boundary between the conventional inactive phase and the Griffiths region. At this point, the decay time of a single, impurity-free rare region of size $L_r$ scales as $\tau(L_r) \sim L_r^z$ as follows from finite size scaling [@barber]. Here $z$ is the clean critical exponent. Using the saddle point method to evaluate the integral (\[eq:rrevo\]), we find the leading long-time decay of the density to be given by a stretched exponential, $$\ln \rho(t) \sim - \tilde{p}^{z/(d+z)}~ t^{d/(d+z)}~, \label{eq:stretched}$$ rather than a simple exponential decay as for $\lambda<\lambda_c^0$.
Inside the Griffiths region, i.e., for $\lambda_c^0<\lambda<\lambda_c$, the decay time of a single rare region depends exponentially on its volume, $$\tau(L_r) \sim \exp(a L_r^d)$$ because a coordinated fluctuation of the entire rare region is required to take it to the absorbing state [@Noest86; @Noest88; @Schonmann85]. The nonuniversal prefactor $a$ vanishes at the clean critical point $\lambda_c^0$ and increases with $\lambda$. Close to $\lambda_c^0$, it behaves as $a \sim \xi_\perp^{-d} \sim
(\lambda-\lambda_c^0)^{d\nu_\perp}$ with $\nu_\perp$ the clean critical exponent. Repeating the saddle point analysis of the integral (\[eq:rrevo\]) for this case, we obtain a power-law decay of the density $$\rho(t) \sim t^{-\tilde p/a} = t^{-d/z'} \label{eq:griffithspower}$$ where $z'=da/\tilde p$ is a customarily used nonuniversal dynamical exponent in the Griffiths region. Its behavior close to the *dirty* critical point $\lambda_c$ can be obtained within the strong disorder renormalization group method [@hooyberghs; @dsf9295]. When approaching the phase transition, $z'$ diverges as $z'
\sim |\lambda-\lambda_c|^{-\psi\nu_\perp}$ where $\psi$ and $\nu_\perp$ are the exponents of the dirty critical point.
Monte-Carlo simulations {#sec:mc}
=======================
Method and overview
-------------------
We now turn to the main part of the paper, extensive Monte-Carlo simulations of the one-dimensional contact process with quenched spatial disorder. There is a number of different ways to actually implement the contact process on the computer (all equivalent with respect to the universal behavior). We follow the widely used algorithm described, e.g., by Dickman [@dickman99]. Runs start at time $t=0$ from some configuration of occupied and empty sites. Each event consists of randomly selecting an occupied site $\mathbf{r}$ from a list of all $N_p$ occupied sites, selecting a process: creation with probability $\lambda(\mathbf{r})/[1+ \lambda(\mathbf{r})]$ or annihilation with probability $1/[1+ \lambda(\mathbf{r})]$ and, for creation, selecting one of the neighboring sites of $\mathbf{r}$. The creation succeeds, if this neighbor is empty. The time increment associated with this event is $1/N_p$. Note that in this implementation of the disordered contact process both the creation rate and the annihilation rate vary from site to site in such a way that their sum is constant (and equal to one).
Using this algorithm, we have performed simulations for system sizes between $L=1000$ and $L=10^7$. We have studied impurity concentrations $p=0.2, 0.3, 0.4, 0.5, 0.6$ and 0.7 as well as relative impurity strengths of $c=0.2, 0.4, 0.6$ and 0.8. To explore the extremely slow dynamics associated with the predicted infinite-randomness critical point, we have simulated very long times up to $t=10^9$ which is, to the best of our knowledge, at least three orders of magnitude in $t$ longer than previous simulations of the disordered contact process. In all cases we have averaged over a large number of different disorder realizations, details will be mentioned below for each specific set of calculations.
Figure \[fig:pd\] gives an overview over the phase diagram resulting from our simulations.
![Phase diagrams of the disordered contact process. Left: Birth rate $\lambda$ vs. impurity concentration $p$ for fixed impurity strength $c=0.2$. Right: $\lambda$ vs. $c$ for fixed $p=0.3$.[]{data-label="fig:pd"}](phaseboundary.eps){width="\columnwidth"}
As expected, the critical birthrate $\lambda_c$ increases with increasing impurity concentration $p$. It also increases with decreasing birth rate on the impurities, i.e., a decreasing relative strength $c$. For $p=0$ or $c=1$, we reproduce the well-known clean critical birth rate $\lambda_c^0 \approx 3.298$ [@Jensen93]. In the following subsections we discuss the behavior in the vicinity of the phase transition in more detail.
Time evolution starting from full lattice
-----------------------------------------
In this subsection we discuss simulations which follow the time evolution of the average density starting from a full lattice. This means, at time $t=0$, all sites are active and $\rho(0)=1$.
Figure \[fig:overviewp03c02\] gives an overview of the time evolution of the density for a system of $10^6$ sites with $p=0.3,c=0.2$, covering the $\lambda$ range from the conventional inactive phase, $\lambda < \lambda_c^0$ all the way to the active phase, $\lambda>\lambda_c$.
![Overview of the time evolution of the density for a system of $10^6$ sites with $p=0.3$ and $c=0.2$. The clean critical point $\lambda_c^0\approx 3.298$ and the dirty critical point $\lambda_c \approx 5.24$ are specially marked.[]{data-label="fig:overviewp03c02"}](overviewp03c02.eps){width="\columnwidth"}
The data are averages over 480 runs, each with a different disorder realization. For birth rates below and at the clean critical point $\lambda_c^0\approx 3.298$, the density decay is very fast, clearly faster then a power law. Above $\lambda_c^0$, the decay becomes slower and asymptotically seems to follow a power-law. For even larger birth rates the decay seems to be slower than a power law while the largest birth rates give rise to a nonzero steady state density, i.e., the system is in the active phase.
### Griffiths region {#griffiths-region}
Let us investigate the different parameter regions in more detail, beginning with the behavior at the clean critical point $\lambda_c^0$, i.e., at the boundary between the Griffiths region and the conventional absorbing phase. According to eq.(\[eq:stretched\]), the density should asymptotically decay like a stretched exponential. To test this behavior, we plot the logarithm of the density as a function of $t^{d/(d+z)}$ where $d=1$ and $z\approx 1.581$ is the dynamical exponent of the clean one-dimensional contact process. Figure \[fig:evo\_lc0\] shows the resulting graphs for system size $L=10^7$, $c=0.2$, and several impurity concentrations $p=0.2 \ldots 0.7$.
![Time evolution of the density at the clean critical point $\lambda_c^0=3.298$ for systems of $10^7$ sites with $c=0.2$ and several $p$. The straight lines are fits to the stretched exponential $\ln \rho(t) \sim -E t^{d/(d+z)}$ predicted in eq.(\[eq:stretched\]) with $d=1$ and the clean $z=1.580$. Inset: Decay constant $E$ vs. $\tilde{p}^{z/(d+z)}$. []{data-label="fig:evo_lc0"}](evo_lc0.eps){width="\columnwidth"}
The data are averages over 960 runs, each with a different disorder realization. The figure shows that the data follow a stretched exponential behavior $\ln \rho = -E
t^{0.3875}$ over more than four orders of magnitude in $\rho$, in good agreement with eq. (\[eq:stretched\]). The decay constant $E$, i.e., the slope of these curves, increases with increasing impurity concentration $p$. The inset of figure \[fig:evo\_lc0\] shows the relation between $E$ and $\tilde p =-\ln(1-p)$. In good approximation, the values follow the power law $E \sim \tilde p^{z/(d+z)}=\tilde
p^{0.6125}$ predicted in (\[eq:stretched\]).
We now turn to the behavior inside the Griffiths region, $\lambda_c^0 < \lambda <
\lambda_c$. Figure \[fig:griffiths\] shows a double-logarithmic plot of the density time evolution for birth rates $\lambda = 3.5 \ldots 5.1$ and $p=0.3, c=0.2$. The system sizes are between $10^6$ and $10^7$ lattice sites, and we have averaged over 480 disorder realizations.
![Log-log plot of the density time evolution in the Griffiths region for systems with $p=0.3, c=0.2$ and several birth rates $\lambda$. The system sizes are $10^7$ sites for $\lambda=3.5, 3.7$ and $10^6$ sites for the other $\lambda$ values. The straight lines are fits to the power law $\rho(t) \sim t^{-1/z'}$ predicted in eq.(\[eq:griffithspower\]). Inset: Dynamical exponent $z'$ vs. birth rate $\lambda$.[]{data-label="fig:griffiths"}](griffiths.eps){width="\columnwidth"}
For all birth rates $\lambda$ shown, the long-time decay of the density asymptotically follows a power-law, as predicted in eq. (\[eq:griffithspower\]), over several orders of magnitude in $\rho$ (except for the largest $\lambda$ where we could observe the power law only over a smaller range in $\rho$ because the decay is too slow).
The nonuniversal dynamical exponent $z'$ can be obtained by fitting the long-time asymptotics of the curves in figure \[fig:griffiths\] to eq.(\[eq:griffithspower\]). The inset of figure \[fig:griffiths\] shows $z'$ as a function of the birth rate $\lambda$. As discussed in section \[subsec:Griffiths\], $z'$ increases with increasing $\lambda$ throughout the Griffiths region with an apparent divergence around $\lambda \approx 5.2$. Unfortunately, our data did not allow us to make quantitative comparisons with the predictions for the $\lambda$-dependence of $z'$ because we could not reliably determine $z'$ sufficiently close to either the clean critical point or the dirty critical point. Close to the clean critical point $\lambda_c^0$, the crossover to the asymptotic power law occurs at very low densities, thus the system size limits how close one can get to $\lambda_c^0$. Conversely, for larger $\lambda$ close to the dirty critical point $\lambda_c$, the crossover to the asymptotic power law occurs at very long times. Thus, the maximum simulation time limits how close to the dirty critical point one can still extract $z'$.
### Dirty critical point
After having discussed the Griffiths region, we now turn to the most interesting parameter region, the vicinity of the dirty critical point. In contrast to the clean critical birth rate $\lambda_c^0$ which is well known from the literature [@Jensen93], the dirty critical birth rate $\lambda_c$ is not known a priori. In order to find $\lambda_c$ and at the same time test the predictions of the activated scaling picture of section \[subsec:activated\], we employ the logarithmic time dependence of the density, eq. (\[eq:logdecay\]). In figure \[fig:criticalp03c02\] we plot $\rho^{-1/\bar\delta}$ with the predicted $\bar\delta=0.38197$ against $\ln(t)$ for a system of $10^4$ sites with $p=0.3$ and $c=0.2$.
![$\rho^{-1/\bar\delta}$ vs. $\ln(t)$ for a system of $10^4$ sites with $p=0.3$ and $c=0.2$. The filled circles mark the critical birth rate $\lambda_c=5.24$, and the straight line is a fit of the long-time behavior to eq. (\[eq:logdecay\])[]{data-label="fig:criticalp03c02"}](criticalp03c02.eps){width="\columnwidth"}
Because the dynamics at the dirty critical point is expected to be extremely slow, we have simulated up to $t=10^8$ ($10^9$ for the critical curve). As before, the data are averages over 480 runs, each with a different disorder realization.
In this type of plot, the logarithmic time dependence (\[eq:logdecay\]) is represented by a straight line. Subcritical data should curve upward from the critical straight line, while supercritical data should curve downward and eventually settle to a constant long-time limit. From the data in figure \[fig:criticalp03c02\] we conclude that the dirty critical point indeed follows the activated scaling scenario associated with an infinite-randomness critical point. The critical birthrate is $\lambda_c=5.24\pm 0.01$. At this $\lambda$, the density follows eq. (\[eq:logdecay\]) over almost four orders of magnitude in $t$.
The statistical error of the plotted average densities can be estimated from the standard deviation of $\rho(t)$ between the 480 separate runs. For the critical curve, $\lambda=5.24$, in figure \[fig:criticalp03c02\], the error of the average density remains below $0.002$ which corresponds to about a symbol size in the figure (at the long-time end of the plot). We have also checked for possible finite-size effects by repeating the calculation for a smaller system size of $L=10^3$. Within the statistical error the results for $L=10^3$ and $L=10^4$ are identical, from which we conclude that our data are not influenced by finite size effects.
### Universality
We now turn to the questions of universality: Is the activated scaling scenario valid for all impurity concentrations $p$ and strengths $c$, and is the value of the critical exponent $\bar\delta$ the same for all cases? To answer these questions we have repeated the above critical point analysis for different sets of the disorder parameters $p$ and $c$. In this subsection, we first show that all these data are in agreement with the activated scaling scenario with an universal exponent $\bar\delta$. We then discuss whether they could interpreted in a conventional power-law scaling scenario as well.
In the first set of calculations we have kept the impurity concentration at $p=0.3$, but we have varied their relative strength from $c=0.2$ to 0.8. In all cases, the density decay at the respective dirty critical birth rate $\lambda_c$ follows the logarithmic law (\[eq:logdecay\]) with the predicted $\bar\delta=0.38197$ over several orders of magnitude in $t$. Figure \[fig:allcritical\] shows these critical curves for systems with $c=0.2, 0.4, 0.6$ and 0.8.
![Time evolution of the density for systems of $10^4$ sites with $p=0.3$ and $c=0.2, 0.4, 0.6$ and 0.8 at their respective critical points plotted as $\rho^{-1/\bar\delta}$ vs. $\ln(t)$ as in figure \[fig:criticalp03c02\].[]{data-label="fig:allcritical"}](allcritical.eps){width="\columnwidth"}
The seemingly larger fluctuations for the curves with higher $c$ are caused by the way the data are plotted: The large negative exponent $-1/\bar\delta$ strongly stretches the low-density part of the ordinate.
We have obtained analogous results from the second set of runs where we kept $c=0.2$ constant but varied the impurity concentration from $p=0.2$ to 0.5. From these simulation results we conclude that the data are in agreement with the activated scaling scenario for all studied parameter values including the case of weak disorder. (Note that for $p=0.3, c=0.8$, the disorder-induced shift of the critical birthrate is small, ($\lambda_c -\lambda_c^0)/\lambda_c^0 \approx 0.07$.) Moreover, our results are compatible with a universal value of 0.38197 for the exponent $\bar\delta$.
However, we would like to emphasize that while our data do *not* show any indication of non-universality, we cannot exclude some variation of $\bar\delta$ with the disorder strength. This is caused by the fact that even though we observe the logarithmic time dependence (\[eq:logdecay\]) over almost four orders of magnitude in $t$, this corresponds only to about a factor of 2 to 3 in $\ln(t)$. This is a very small range for extracting the exponent of the power-law relation between $\rho$ and $\ln(t)$. More specifically, the asymptotic critical time dependence of the density for $p=0.3, c=0.2$ can be fitted by $\rho = (A*\ln(t) + B)^{-0.38197}$ with $A\approx 5.12$ and $B\approx
-20.1$ (this is the straight line in figure \[fig:criticalp03c02\]). Comparison with eq. (\[eq:logdecay\]) shows that $B$ represents a correction to scaling. For a reliable extraction of the exponent one would want the $\ln(t)$ term to be at least one order of magnitude larger than $B$ at the very minimum. This corresponds to $\ln(t)
\approx 40$ or $t \approx 10^{17}$ which is clearly unreachable in a Monte Carlo simulation for the foreseeable future.
We now turn to the question of whether the numerical data could also be interpreted in terms of conventional power-law scaling. To this end, we first replot the critical density decay curves as identified above [@CPIDENT] in standard log-log form in Fig.\[fig:allcritical\_power\].
![Replot of the data of figure \[fig:allcritical\] in log-log form. The dashed line shows a power-law decay with the [*clean*]{} critical exponent $\delta=0.159$.[]{data-label="fig:allcritical_power"}](allcritical_power.eps){width="\columnwidth"}
This figure shows that the density decay at early times follows a power law with the *clean* decay exponent $\delta=0.159$. However, after some disorder dependent crossover time $t_x$ the curves show pronounced upward curvature. This curvature persists to the largest times and signifies an asymptotic decay that is slower than any power law. The crossover time $t_x$ which increases with decreasing disorder $c\to 1$ nicely agrees with the onset of the asymptotic logarithmic behavior in Fig. \[fig:allcritical\] for all disorder strengths. Thus, the time dependence of the density $\rho(t)$ directly crosses over from the clean critical (power law) behavior at short times to the activated (logarithmic) behavior (\[eq:logdecay\]) at large times. We find no indication of any power laws other than the clean one, not even at transient times. We thus conclude that our data, at least for times up to $t=10^8$ are not compatible with power-law scaling at the dirty critical point.
It would be highly desirable to discriminate between power-law and activated scaling by performing a full scaling analysis of the data using relations (\[eq:rho\],\[eq:Ps\],\[eq:N\]) or (\[eq:rho\_activated\],\[eq:Ps\_activated\],\[eq:N\_activated\]), respectively. Unfortunately, such an analysis is impossible because the extremely slow dynamics leads to the abovementioned strong corrections to scaling in the accessible range of times up to $t=10^9$.
Time evolution starting from a single particle
----------------------------------------------
In addition to the simulations staring from a full lattice, we have also performed simulations of the time evolution starting from a single active site in an otherwise empty lattice. In these calculations, we have monitored the survival probability $P_s$ and the number of sites $N$ in the active cluster as functions of time. According to eqs. (\[eq:logdecay\]) and (\[eq:clustersize\]), these quantities are expected to behave as $P_s(t) \sim [\ln(t)]^{-1/\bar\delta}$ and $N(t) \sim [\ln(t)]^{\bar\Theta}$ at the dirty critical point.
Figure \[fig:p03c02local\] shows plots of $P_s^{-\bar\delta}$ and $N^{1/\bar\Theta}$ vs. $\ln(t)$ for a system with $p=0.3$ and $c=0.2$.
![Survival probability $P_s$ and particle number $N$ in the active cluster for a simulation starting from single active site. The disorder parameters are $p=0.3$ and $c=0.2$.[]{data-label="fig:p03c02local"}](p03c02local.eps){width="\columnwidth"}
The data are averages over 480 different disorder realizations. For each realization we have performed 200 runs starting from a single active site at a random position. The system size $L=10^6$ was several orders of magnitude larger than the largest active cluster, thus we have effectively simulated an infinite-size system. Figure \[fig:p03c02local\] shows that $P_s$ and $N$ indeed follow the predicted logarithmic laws with $\bar\delta=0.38197$ and $\bar\Theta=1.236$ at the critical birthrate $\lambda_c=5.24$. Subcritical and supercritical data curve away from the critical straight lines as expected. Thus, the simulations starting from a single site also confirm the activated scaling scenario resulting from an infinite-randomness critical point.
Steady state
------------
Lastly, we have studied the behavior of the steady state density in the active phase. Close to the critical point it is expected to vary as $\rho_{\rm stat} \sim
(\lambda-\lambda_c)^\beta$ with $\beta=0.38197$. These calculations require a particularly high numerical effort because the approach to the steady is logarithmically slow close to the critical point. Therefore, we have simulated up to $t=10^9$ for birthrates close to $\lambda_c$. Figure \[fig:p03c02steady\] shows the time evolution of the density in the active phase for a system with $p=0.3$ and $c=0.2$, averaged over 480 disorder realizations (one run per realization).
![Time evolution of the density in the active phase for a system with $p=0.3$ and $c=0.2$. The system sizes are $10^4$ for $\lambda\ge 5.8$ and $10^3$ otherwise. Inset: Steady state density vs. distance from critical point. The solid line is a fit of the data for $\lambda-\lambda_c<0.3$ to the expected dirty power law $\rho_{\rm stat} \sim
(\lambda-\lambda_c)^\beta$ with $\beta=0.38197$. The dashed line is a fit of the data for $\lambda-\lambda_c>0.3$ to a power law with the clean $\beta=0.2765$.[]{data-label="fig:p03c02steady"}](p03c02steady.eps){width="\columnwidth"}
The steady state densities can be obtained by averaging the constant part of each of the curves. The data clearly demonstrate the extremely slow approach to the steady state. For $\lambda \le 5.4$ the steady state is not reached (within the statistical error) even after $t = 10^9$. Therefore, our steady state density values for these $\lambda$ are only rough estimates.
The inset of figure \[fig:p03c02steady\] shows the resulting steady state densities plotted vs. the distance from the critical point. This graph nicely demonstrates the crossover from clean to dirty critical behavior. The data away from the critical point ($\lambda-\lambda_c>0.3$) can be well fitted by a power law with the clean value of the critical exponent $\beta=0.2765$. When approaching the critical point the curve becomes steeper, and for $\lambda-\lambda_c < 0.3$ the data can be reasonably well fitted by a power law with the expected dirty critical exponent $\beta=0.38197$. The position of the crossover can be compared to an estimate from the time dependence at $\lambda_c$ in figure \[fig:criticalp03c02\]. The critical curve reaches its asymptotic logarithmic form at about $t_x \approx 10^4$ which corresponds to a crossover density of $\rho_x
\approx 0.3$ in rough agreement with the crossover density in the inset of figure \[fig:p03c02steady\].
Conclusions {#sec:conclusions}
===========
To summarize, we have presented the results of large-scale Monte-Carlo simulations of a one-dimensional contact process with quenched spatial disorder for large systems of up to $L=10^7$ sites and very long times up to $t=10^9$. These simulations show that the critical behavior at the nonequilibrium phase transition is controlled by an infinite-randomness fixed point with activated scaling and ultraslow dynamics, as predicted in Ref. [@hooyberghs]. Moreover, the simulations provide evidence that this behavior is universal. The logarithmically slow time dependencies (\[eq:logdecay\],\[eq:clustersize\]) are valid (with the same exponent values) for all parameter sets investigated including the weak-disorder case. However, the approach to this universal asymptotic behavior is extremely slow, with crossover times of the order of $10^4$ or larger. Since most of the earlier Monte-Carlo simulations did not exceed $t=10^5 \ldots 10^6$ this may explain why nonuniversal (effective) exponents were seen in previous work. We have also presented results for the Griffiths region between the clean and the dirty critical points. Here, we have found power-law dynamical behavior with continuously varying exponents, in agreement with theoretical predictions [@Noest86; @Noest88].
We now discuss the relation of our results to a more general theory of rare region effects at phase transitions with quenched disorder. In Ref. [@VojtaSchmalian05] a general classification of phase transitions in quenched disordered systems with short-range interactions [@RKKY] has been suggested, based on the effective dimensionality $d_{\rm eff}$ of the rare regions. Three cases can be distinguished.
\(i) If $d_{\rm eff}$ is below the lower critical dimension $d_c^-$ of the problem, the rare region effects are exponentially small because the probability of a rare region decreases exponentially with its volume but the contribution of each region to observables increases only as a power law. In this case, the critical point is of conventional power-law type. Examples in this class include, e.g., the classical equilibrium Ising transition with point defects where $d_{\rm eff}=0$ and $d_c^-=1$.
\(ii) In the second class, with $d_{\rm eff}=d_c^-$, the Griffiths effects are of power-law type because the exponentially rarity of the rare regions in $L_r$ is overcome by an exponential increase of each region’s contribution. In this class, the critical point is controlled by an infinite-randomness fixed point with activated scaling. Examples include the quantum phase transition in the random transverse field Ising model ($d_{\rm eff}=d_c^-=1)$ [@dsf9295; @taufootnote] as well as the transition discussed in this paper (where $d_{\rm eff}=d_c^-=0$).
\(iii) Finally, for $d_{\rm eff}>d_c^-$, the rare regions can undergo the phase transition independently from the bulk system. This leads to a destruction of the sharp phase transition by smearing. This behavior occurs at the equilibrium Ising transition with plane defects where $d_{\rm eff}=2,d_c^-=1$ [@us_planar], the quantum phase transition in itinerant magnets [@us_rounding], and for the contact process with extended (line or plane) defects [@Dickison05; @contact_pre].
Thus, the results of this paper do fit into the general rare-region based classification scheme of phase transitions with quenched disorder and short-range interactions [@VojtaSchmalian05]. These arguments also suggest that the behavior of the phase transition in a higher-dimensional disordered contact process should be controlled by an infinite-randomness fixed point as well.
We conclude by pointing out that the unconventional behavior found in this paper may explain the striking absence of directed percolation scaling [@hinrichsen_exp] in at least some of the experiments. However, the extremely slow dynamics will prove to be a challenge for the verification of the activated scaling scenario not just in simulations but also in experiments.
Acknowledgements {#acknowledgements .unnumbered}
================
This work has been supported in part by the NSF under grant nos. DMR-0339147 and PHY99-07949 as well as by the University of Missouri Research Board. Thomas Vojta is a Cottrell Scholar of Research Corporation. We are also grateful for the hospitality of the Aspen Center for Physics and the Kavli Institute for Theoretical Physics, Santa Barbara during the early stages of this work.
[99]{} B. Schmittmann and R.K.P. Zia, in [*Phase transitions and critical phenomena*]{}, edited by C. Domb and J.L. Lebowitz, Vol. 17, (Academic, New York 1995)
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The Modular number, Congruence number, and Multiplicity One The Modular number and congruence number Amod Agashe [^1] Agashe
[*To Ken Ribet,\
on the occasion of his sixtieth birthday*]{}
Let $N$ be a positive integer and let $f$ be a newform of weight $2$ on $\Gamma_0(N)$. In earlier joint work with K. Ribet and W. Stein, we introduced the notions of the modular number and the congruence number of the quotient abelian variety $A_f$ of $J_0(N)$ associated to the newform $f$. These invariants are analogs of the notions of the modular degree and congruence primes respectively associated to elliptic curves. We show that if $p$ is a prime such that every maximal ideal of the Hecke algebra of characteristic $p$ that contains the annihilator ideal of $f$ satisfies multiplicity one, then the modular number and the congruence number have the same $p$-adic valuation.
Amod Agashe Insert Current Address Kenneth A. Ribet Insert Current Address William A. Stein Department of Mathematics Harvard University Cambridge, MA 02138 [was@math.harvard.edu]{}
Introduction and results {#sec:intro}
========================
Let $N$ be a positive integer and let $X_0(N)$ denote the modular curve over ${{\bf{Q}}}$ associated to the classification of isomorphism classes of elliptic curves with a cyclic subgroup of order $N$. The Hecke algebra ${{\bf{T}}}$ of level $N$ is the subring of the ring of endomorphisms of $J_0(N)=\operatorname{Jac}(X_0(N))$ generated by the Hecke operators $T_n$ for all $n\geq 1$. Let $f$ be a newform of weight $2$ for $\Gamma_0(N)$ and let $I_f$ denote ${\rm Ann}_{{\bf{T}}}(f)$. Then the quotient abelian variety $A_f = J_0(N)/I_f J_0(N)$ is called the newform quotient associated to $f$. If $f$ has integer Fourier coefficients, then $A_f$ is an elliptic curve and in fact by [@breuil-conrad-diamond-taylor] any elliptic curve over ${{\bf{Q}}}$ is isogenous to such an elliptic curve for some $f$. The dual abelian variety ${{A^{\vee}_f}}$ of $A_f$ may be viewed as an abelian subvariety of $J_0(N)$. Recall that the [*exponent*]{} of a finite group $G$ is the smallest positive integer $n$ such that multiplication by $n$ annihilates every element of $G$. The exponent of the group ${{A^{\vee}_f}}\cap I_f J$ is called the [*modular exponent*]{} of $A_f$ and its order is called the [*modular number*]{} (see [@ars:moddeg §3]). When $f$ has integer Fourier coefficients, so that $A_f$ is an elliptic curve, we will sometimes denote $A_f$ by $E$ for emphasis. In that case, composing the embedding $X_0(N){\hookrightarrow}J_0(N)$ that sends $\infty$ to $0$ with the quotient map $J_0(N) {\rightarrow}E$, we obtain a surjective morphism of curves ${\phi_{\scriptscriptstyle E}}: X_0(N) {\rightarrow}E$, whose degree is called the [*modular degree*]{} of $E$. The modular exponent ${{\tilde{n}_{\scriptscriptstyle{E}}}}$ of $E$ is equal to the modular degree, and the modular number ${{n_{\scriptscriptstyle{E}}}}$ is the square of the modular degree (see [@ars:moddeg §3]). In general, for any newform $f$, the modular number ${n_{\scriptscriptstyle{A_f}}}$ is a perfect square (e.g., see [@agst:bsd Lemma 3.14]). Let $S_2({{\bf{Z}}})$ denote the group of cuspforms of weight $2$ on $\Gamma_0(N)$ with integral Fourier coefficients, and if $G$ is a subgroup of $S_2({{\bf{Z}}})$, let $G^\perp$ denote the subgroup of $S_2({{\bf{Z}}})$ consisting of cuspforms that are orthogonal to every $g$ in $G$ with respect to the Petersson inner product. The exponent of the quotient group $$\frac{S_2({{\bf{Z}}})}{S_2({{\bf{Z}}})[I_f] + S_2({{\bf{Z}}})[I_f]^\perp}$$ is called the [*congruence exponent*]{} of $A_f$ (really, that of $f$) and its order is called the [*congruence number*]{} (see [@ars:moddeg §3]). If $f$ has integer Fourier coefficients, so that $A_f$ is an elliptic curve, then ${{r_{\scriptscriptstyle{A_f}}}}$ is the largest integer $r$ such that there exists a cuspform $g \in S_2({{\bf{Z}}})$ that is orthogonal to $f$ under the Petersson inner product and whose $n$-th Fourier coefficient is congruent modulo $r$ to the $n$-th Fourier coefficient of $f$ for all positive integers $n$. We say that a prime is a [*congruence prime for $A_f$*]{} if it divides the congruence number ${{r_{\scriptscriptstyle{A_f}}}}$.
Congruence primes have been studied by Doi, Hida, Ribet, Mazur and others (see, e.g., [@ribet:modp §1]), and played an important role in Wiles’s work [@wiles:fermat] on Fermat’s last theorem. Frey and Mai-Murty have observed that an appropriate asymptotic bound on the modular degree is equivalent to the $abc$-conjecture (see [@frey:boston p.544] and [@murty:congruence p.180]). Thus congruence primes and the modular degree are quantities of significant interest. Theorem 3.6 of [@ars:moddeg] says that the modular exonent ${{\tilde{n}_{\scriptscriptstyle{A_f}}}}$ divides the congruence exponent ${{\tilde{r}_{\scriptscriptstyle{A_f}}}}$ and if $p$ is a prime such that $p^2 {{\not|\ }}N$, then $\operatorname{ord}_p({{\tilde{n}_{\scriptscriptstyle{A_f}}}}) = \operatorname{ord}_p({{\tilde{r}_{\scriptscriptstyle{A_f}}}})$.
One might wonder if similar relations hold for the modular number ${{r_{\scriptscriptstyle{A_f}}}}$ and congruence number ${n_{\scriptscriptstyle{A_f}}}$ (as opposed to modular/congruence exponents). As mentioned earlier, if $A_f$ is an elliptic curve, then ${n_{\scriptscriptstyle{A_f}}}= {{\tilde{n}_{\scriptscriptstyle{A_f}}}}^2$ and so, considering that ${{\tilde{n}_{\scriptscriptstyle{A_f}}}}{{\ |\ }}{{\tilde{r}_{\scriptscriptstyle{A_f}}}}$ and ${{\tilde{r}_{\scriptscriptstyle{A_f}}}}{{\ |\ }}{{r_{\scriptscriptstyle{A_f}}}}$, one sees that ${n_{\scriptscriptstyle{A_f}}}\mid {{r_{\scriptscriptstyle{A_f}}^2}}$. One might wonder if ${n_{\scriptscriptstyle{A_f}}}$ divides ${{r_{\scriptscriptstyle{A_f}}^2}}$ even if $A_f$ is not an elliptic curve (i.e., has dimension more than one). It turns out that the answer is no: as mentioned in [@ars:moddeg Remark 3.7] we have
\[eg\] There is a newform of degree $24$ in $S_2(\Gamma_0(431))$ such that $${n_{\scriptscriptstyle{A_f}}}= (2^{11}\cdot 6947)^2 \,\,{{\not|\ }}\,\, r^2_{A_f} = (2^{10}\cdot
6947)^2.$$
We say that a maximal ideal ${{\mathfrak{m}}}$ of ${{\bf{T}}}$ satisfies [*multiplicity one*]{} if $J_0(N)[{{\mathfrak{m}}}]$ is of dimension two over ${{\bf{T}}}/{{\mathfrak{m}}}$. The reason one calls this “multiplicitly one” is that if the canonical two dimensional representation $\rho_{{\mathfrak{m}}}$ over ${{\bf{T}}}/{{\mathfrak{m}}}$ attached to ${{\mathfrak{m}}}$ (e.g., see [@ribet:modreps Prop. 5.1]) is irreducible, then $J_0(N)[{{\mathfrak{m}}}]$ is a direct sum of copies of $\rho_{{\mathfrak{m}}}$ (e.g., see [@ribet:modreps Thm. 5.2]), and a maximal ideal ${{\mathfrak{m}}}$ of ${{\bf{T}}}$ satisfies [*multiplicity one*]{} precisely if the multiplicity of $\rho_{{\mathfrak{m}}}$ in this decomposition is one. Even if $\rho_{{\mathfrak{m}}}$ is reducible, the definition of multiplicity one given above is relevant (e.g., see [@mazur:eisenstein Cor. 16.3]). It was remarked in [@ars:moddeg] that concerning Example \[eg\] above where ${n_{\scriptscriptstyle{A_f}}}{{\not|\ }}{{r_{\scriptscriptstyle{A_f}}^2}}$, the level $431$ is prime and by [@kilford:non-gorenstein], mod $2$ multiplicity one fails for $J_0(431)$. In this article, we show that multiplicity one is the only obstruction for the divisibility ${n_{\scriptscriptstyle{A_f}}}\mid {{r_{\scriptscriptstyle{A_f}}^2}}$ to fail. In fact, we show something stronger:
\[thm:main2\] Let $p$ be a prime such that every maximal ideal ${{\mathfrak{m}}}$ with residue characteristic $p$ that contains $I_f$ satisfies multiplicity one. Then $\operatorname{ord}_p({n_{\scriptscriptstyle{A_f}}}) = \operatorname{ord}_p({{r_{\scriptscriptstyle{A_f}}^2}})$.
The theorem above follows from the more general Theorem \[thm:main\] below. Example \[eg\] above shows that the multiplicity one hypothesis cannot be completely removed from the theorem. Also, in the context of Example \[eg\], our theorem gives a new proof that mod $2$ multiplicity fails for $J_0(431)$ (the original proof being the one in [@kilford:non-gorenstein]). Note that in [@ars:moddeg], the authors found examples of failure of multiplicity one using Propostion 5.9 of loc. cit., which implies that if the modular exponent does not equal the congruence exponent for some newform $f$, then there is a maximal ideal of ${{\bf{T}}}$ that not satisfy multiplicity one. However, we could not have detected the failure of multiplicity one in Example \[eg\] by checking if the modular [*exponent*]{} equals the congruence [*exponent*]{}, since the equality holds in the example for any newform $f$ by [@ars:moddeg Thm. 3.6(b)], considering that the level is prime in the example. At the same time, consideration of the modular [*number*]{} and the congruence [*number*]{} did dectect the failure of multiplicity one. It would be interesting to do more calculations to see when ${n_{\scriptscriptstyle{A_f}}}{{\not|\ }}{{r_{\scriptscriptstyle{A_f}}^2}}$, as this may give new instances of failure of multiplicity one.
We remark that our theorem gives information about the [*order*]{} of a certain intersection of abelian subvarieties of $J_0(N)$ in terms of congruences between modular forms (in fact, we give information in a more general setting in Section \[sec:quotients\]). We expect that the relation between a particular such intersection and certain congruences will be useful in understanding the “visible factor” in [@agmer] (in loc. cit., we were able to say something about the primes that divide this factor, as opposed to saying something about the entire factor), and hope that such relations will be useful in other contexts as well.
It is known that multiplicity one holds in several situations. We content ourselves by pointing out that by the main theorem in Section 1.2 of [@mazur-ribet], a maximal ideal ${{\mathfrak{m}}}$ with residue characteristic $p$ satisfies multiplicity one if either $p {{\not|\ }}N$ or $p || N$ and $\rho_{{\mathfrak{m}}}$ is not modular of level $N/p$. We also have:
\[prop:ribet\] Let $p$ be an odd prime and ${{\mathfrak{m}}}$ be a maximal ideal of ${{\bf{T}}}$ with residue characteristic $p$ such that $\rho_{{\mathfrak{m}}}$ is irreducible. Assume that either\
(i) $p {{\not|\ }}N$ or\
(ii) $p || N$ and $I_f \subseteq {{\mathfrak{m}}}$ for some newform $f$.\
Then ${{\mathfrak{m}}}$ satisfies multiplicity one.
If $p \nmid N$, then the claim follows from Theorem 5.2(b) of [@ribet:modreps], so let us assume that $p || N$. Let $X_0(N)_{{{\bf{Z}}}_p}$ denote the minimal regular resolution of the compactified coarse moduli scheme over ${{\bf{Z}}}_p$ associated to $\Gamma_0(N)$ as in [@deligne-rapoport § IV.3] and let $\Omega_{X_0(N)_{{{\bf{Z}}}_p}/{{\bf{Z}}}_p}$ denote the relative dualizing sheaf of $X_0(N)_{{{\bf{Z}}}_p}$ over ${{\bf{Z}}}_p$ (it is the sheaf of regular differentials as in [@mazur-ribet §7]). We denote by $X_0(N)_{{{{{{\bf{F}}}}_p}}}$ the special fiber of $X_0(N)_{{{\mathbf{Z}_{p}}}}$ at the prime $p$ and by $\Omega_{X_0(N)/{{{{{\bf{F}}}}_p}}}$ the relative dualizing sheaf of $X_0(N)_{{{{{{\bf{F}}}}_p}}}$ over ${{{{{\bf{F}}}}_p}}$. It is shown in [@ars:moddeg §5.2.2] that under the hypotheses above, ${\rm dim}_{{{\bf{T}}}/{{\mathfrak{m}}}} H^0(X_0(N)_{{{{{{\bf{F}}}}_p}}}, \Omega_{X_0(N)_{{{{{{\bf{F}}}}_p}}}/{{{{{\bf{F}}}}_p}}}) [{{\mathfrak{m}}}] \leq 1$. Let $J_{{{\bf{Z}}}_p}$ denote the Néron model of $J_0(N)$ over ${{\bf{Z}}}_p$ and let $J_{{{\bf{Z}}}_p}^0$ denote its identity component. Then the natural morphism ${\rm Pic}^0_{X_0(N)/{{{\mathbf{Z}_{p}}}}} {\rightarrow}J_{{{\mathbf{Z}_{p}}}}$ identifies ${\rm Pic}^0_{X_0(N)/{{\mathbf{Z}_{p}}}}$ with $J_{{{\mathbf{Z}_{p}}}}^0$ (see, e.g., [@neronmodels §9.4–9.5]). Passing to tangent spaces along the identity section over ${{\mathbf{Z}_{p}}}$, we obtain an isomorphism $H^1(X_0(N)_{{{\mathbf{Z}_{p}}}}, {{\mathcal{O}}}_{X_0(N)_{{{\mathbf{Z}_{p}}}}}) {\cong}{\rm Tan}(J_{{{\mathbf{Z}_{p}}}})$. Reducing both sides modulo $p$ and applying Grothendieck duality, we get ${\rm Tan}(J_{{{\bf{F}}}_p}) {\cong}{\rm Hom}(H^0(X_0(N)_{{{\bf{F}}}_p}, \Omega_{X_0(N)/{{{{{\bf{F}}}}_p}}}),
{{{{{\bf{F}}}}_p}})$. Thus from the above discussion, we see that ${\rm Tan}(J_{{{\bf{F}}}_p})/ {{\mathfrak{m}}}{\rm Tan}(J_{{{\bf{F}}}_p})$ has dimension at most one over ${{\bf{T}}}/{{\mathfrak{m}}}$. Since ${\rm Tan}(J_{{{\mathbf{Z}_{p}}}})$ is a faithful ${{\bf{T}}}{\otimes}{{\bf{Z}}}_p$-module, we see that ${\rm Tan}(J_{{{\bf{F}}}_p})/ {{\mathfrak{m}}}{\rm Tan}(J_{{{\bf{F}}}_p})$ is non-trivial, hence it is one dimensional over ${{\bf{T}}}/{{\mathfrak{m}}}$. With this input, the proof of multiplicity one in Theorem 2.1 of [@wiles:fermat], which is in the $\Gamma_1(N)$ context, but is a formal argument involving abelian varieties (apart from the input above), carries over in the $\Gamma_0(N)$ context with the obvious modifications (in particular, replacing $X_1(N/p,p)_{{{\bf{Z}}}_p}$ in loc. cit. by $X_0(N)_{{{\bf{Z}}}_p}$) to prove our claim (see p. 487-488 of loc. cit., as well as [@tilouine:gorenstein], where the input above is the equation (\*\*) on p. 339).
We remark that the condition that $p^2 {{\not|\ }}N$ in condition (ii) of the proposition above cannot be removed, as follows from the counterexamples in [@ars:moddeg §2.2]. From Theorem \[thm:main2\] and Proposition \[prop:ribet\], we obtain:
Let $p$ be an odd prime. Suppose that either\
(i) $p {{\not|\ }}N$ or\
(ii) $p || N$ and ${{A^{\vee}_f}}[{{\mathfrak{m}}}]$ is irrreducible for every maximal ideal ${{\mathfrak{m}}}$ of ${{\bf{T}}}$ with residue characteristic $p$.\
Then $\operatorname{ord}_p({n_{\scriptscriptstyle{A_f}}}) = \operatorname{ord}_p({{r_{\scriptscriptstyle{A_f}}^2}})$.
The corollary is clear from Theorem \[thm:main2\] and Proposition \[prop:ribet\] in the case where $p {{\not|\ }}N$, so let us assume that $p || N$. By Theorem \[thm:main2\] and Proposition \[prop:ribet\], it suffices to show that $\rho_{{\mathfrak{m}}}$ is irreducible for every maximal ideal ${{\mathfrak{m}}}$ of ${{\bf{T}}}$ with residue characteristic $p$ such that $I_f \subseteq {{\mathfrak{m}}}$.
Let ${{\mathfrak{m}}}$ be such a maximal ideal. Then note that ${{A^{\vee}_f}}[{{\mathfrak{m}}}]$ is non-trivial since ${{\bf{T}}}/I_f$ acts faithfully on ${{A^{\vee}_f}}$. Let $D$ denote the direct sum of ${{A^{\vee}_f}}[{{\mathfrak{m}}}]$ and its Cartier dual. Let $\ell$ be a prime that does not divide $Np$ and let ${{{\rm Frob}_\ell}}$ denote the Frobenius element of ${\rm Gal}({{\overline{{{\bf{Q}}}}}}/{{\bf{Q}}})$ at $\ell$. As discussed in [@mazur:eisenstein p. 115], by the Eichler-Shimura relation, the characteristic polynomial of ${{{\rm Frob}_\ell}}$ acting on $D$ is $(X^2 -a_\ell X + \ell)^d =0$, where $a_\ell$ is the image of $T_\ell$ in ${{\bf{T}}}/{{\mathfrak{m}}}$ and $d$ is the ${{\bf{T}}}/{{\mathfrak{m}}}$-dimension of ${{A^{\vee}_f}}[{{\mathfrak{m}}}]$. But this is also the characteristic polynomial of ${{{\rm Frob}_\ell}}$ acting on the direct sum of $d$ copies of $\rho_{{\mathfrak{m}}}$. By the Chebotarev density theorem and the Brauer-Nesbitt theorem, the semisimplification of $D$ is $\rho_{{\mathfrak{m}}}^d$. Thus the semisimplification of ${{A^{\vee}_f}}[{{\mathfrak{m}}}]$ is a direct sum of certain number of copies of $\rho_{{\mathfrak{m}}}$. But ${{A^{\vee}_f}}[{{\mathfrak{m}}}]$ is irreducible by hypothesis, so $\rho_{{\mathfrak{m}}}= {{A^{\vee}_f}}[{{\mathfrak{m}}}]$. Thus $\rho_{{\mathfrak{m}}}$ is also irreducible, as was to be shown.
The corollary above is the analog of Theorem 3.6(b) of [@ars:moddeg], which says that $\operatorname{ord}_p({{\tilde{n}_{\scriptscriptstyle{A_f}}}}) = \operatorname{ord}_p({{\tilde{r}_{\scriptscriptstyle{A_f}}}})$ provided $p^2 {{\not|\ }}N$, in the setting of modular/congruence [*numbers*]{} as opposed to modular/congruence [*exponents*]{} (although, note that we have an extra irreducibility hypothesis in our corollary). We remark that the proofs of both results use “multiplicity one for differentials” (as defined in [@ars:moddeg §5.2]).
If the level $N$ is prime, then more can be said. By Prop. II.14.2 and Corollary II.16.3 of [@mazur:eisenstein], every maximal ideal ${{\mathfrak{m}}}$ such that $\rho_{{\mathfrak{m}}}$ is reducible also satisfies multiplicity one. Thus in view of Theorem \[thm:main2\] and Proposition \[prop:ribet\], we obtain the following:
Suppose the level $N$ is prime and let $p$ be an odd prime. Then $\operatorname{ord}_p({n_{\scriptscriptstyle{A_f}}}) = \operatorname{ord}_p({{r_{\scriptscriptstyle{A_f}}^2}})$.
Also, much is known in this situation if $\rho_{{\mathfrak{m}}}$ is irreducible and ${{\mathfrak{m}}}$ has residue characteristic is $2$ – we refer to [@kilford:non-gorenstein] and the references therein for details. But note that by the examples in [@kilford:non-gorenstein] or by Example \[eg\] and Theorem \[thm:main2\], multiplicity one need not hold for a maximal ideal ${{\mathfrak{m}}}$ of residue characteristic $2$ with $\rho_{{\mathfrak{m}}}$ irreducible even if the level $N$ is prime.
In Section \[sec:quotients\], we describe a more general setup, which includes newform quotients of $J_1(N)$, and state a more general version of Theorem \[thm:main2\] (Theorem \[thm:main\] below). In Section \[sec:proof\], we give the proof of Theorem \[thm:main\].\
[*Acknowldegements*]{}: We are grateful to K. Ribet for indicating the proof of Lemma \[lemma:ribet\] below, and in appreciation of his help in other situations as well over the years, it is a pleasure to dedicate this paper to him. We would also like to thank J. Tilouine for some discussion regarding the proof of Proposition \[prop:ribet\] above.
A more general setup {#sec:quotients}
====================
For the benefit of the reader, we repeat below some of the discussion in [@ars:moddeg Section 3].
For $N\geq 4$, let $\Gamma$ be either $\Gamma_0(N)$ or $\Gamma_1(N)$. Let $X$ denote the modular curve over ${{\bf{Q}}}$ associated to $\Gamma$, and let $J$ be the Jacobian of $X$. Let $J_f$ denote the standard abelian subvariety of $J$ attached to $f$ by Shimura [@shimura:intro Thm. 7.14]. Up to isogeny, $J$ is the product of factors $J_f^{e(f)}$ where $f$ runs over the set of newforms of level dividing $N$, taken up to Galois conjugation, and $e(f)$ is the number of divisors of $N/N(f)$, where $N(f)$ is the level of $f$. Let $A$ be the sum of $J_f^{e(f)}$ for some set of $f$’s (taken up to Galois conjugation), and let $B$ be the sum of all the other $J_f^{e(f)}$’s. Clearly $A + B = J$. The $J_f$’s are simple (over ${{\bf{Q}}}$), hence $A \cap B$ is finite. By [@ars:moddeg Lemma 3.1], ${{\rm End}}(J)$ preserves $A$ and $B$, where if $C$ is an abelian variety over ${{\bf{Q}}}$, by ${{\rm End}}(C)$ we mean the ring of endomorphisms of $C$ defined over ${{\bf{Q}}}$. If $f$ is a newform of weight $2$ on $\Gamma$ and $A_f$ is its associated newform quotient, then ${{A^{\vee}_f}}$ and $I_f J$ provide an example of $A$ and $B$ respectively as above, as shown in the discussion following Lemma 3.1 in [@ars:moddeg].
The [*modular exponent*]{} ${{\tilde{n}_{\scriptscriptstyle{A}}}}$ of $A$ is defined as the exponent of $A \cap B$ and the [*modular number*]{} ${{n_{\scriptscriptstyle{A}}}}$ of $A$ is its order (see [@ars:moddeg §3]). Note that the definition is symmetric with respect to $A$ and $B$. In fact, the definition depends on both $A$ and $B$, unlike what the notation may suggest—we have suppressed the dependence on $B$ for ease of notation, with the understanding that there is a natural choice of $B$. If $f$ is a newform, then by the modular exponent/number of $A_f$, we mean that of $A = {{A^{\vee}_f}}$, with $B = I_f J$, which agrees with our earlier definition. If $R$ is a subring of ${{\bf{C}}}$, let $S_2(R)=S_2(\Gamma;R)$ denote the subgroup of $S_2(\Gamma; {{\bf{C}}})$ consisting of cups forms whose Fourier expansions at the cusp $\infty$ have coefficients in $R$. Let ${{\bf{T}}}$ denote the Hecke algebra corresponding to the group $\Gamma$. There is a ${{\bf{T}}}$-equivariant bilinear pairing $$\begin{aligned}
\label{eqn:pairing}
{{\bf{T}}}\times
S_2({{\bf{Z}}})\to{{\bf{Z}}}\end{aligned}$$ given by $(t,g)\mapsto a_1(t(g))$, which is perfect (e.g., see [@abbull Lemma 2.1] or [@ribet:modp Theorem 2.2]). Let ${{\bf{T}}}_A$ denote the image of ${{\bf{T}}}$ in ${{\rm End}}(A)$, and let ${{\bf{T}}}_B$ be the image of ${{\bf{T}}}$ in ${{\rm End}}(B)$ (since ${{\bf{T}}}\subset {{\rm End}}(J)$, ${{\bf{T}}}$ preserves $A$ and $B$). Since $A + B = J$, the natural map ${{\bf{T}}}{\rightarrow}{{\bf{T}}}_A \oplus {{\bf{T}}}_B$ is injective, and moreover, its cokernel is finite (since $A \cap B$ is finite).
Let $S_A = {{\rm Hom}}({{\bf{T}}}_A,{{\bf{Z}}})$ and $S_B = {{\rm Hom}}({{\bf{T}}}_B,{{\bf{Z}}})$ be the subgroups of $S_2({{\bf{Z}}})$ obtained via the pairing in (\[eqn:pairing\]). By [@ars:moddeg Lemma 3.3], we have an isomorphism $$\label{eqn:congexp}
\frac{S_2({{\bf{Z}}})} { S_A + S_B} {\cong}\frac{{{\bf{T}}}_A \oplus {{\bf{T}}}_B}{{{\bf{T}}}}\ .$$ By definition [@ars:moddeg], the exponent of either of the isomorphic groups in (\[eqn:congexp\]) is the [*congruence exponent*]{} ${{\tilde{r}_{\scriptscriptstyle{A}}}}$ of $A$ and the order of either group is the [*congruence number*]{} ${{r_{\scriptscriptstyle{A}}}}$. Note that this definition is also symmetric with respect to $A$ and $B$, and again, the definition depends on both $A$ and $B$, unlike what the notation may suggest – we have suppressed the dependence on $B$ with the implicit understanding that $B$ has been chosen (given $A$). If $f$ is a newform, then by the congruence exponent/number of $A_f$, we mean that of $A = {{A^{\vee}_f}}$, with $B = I_f J$. In this situation, ${{\bf{T}}}_A = {{\bf{T}}}/I_f$ and $S_A = S_2({{\bf{Z}}})[I_f]$. Also, ${{\rm Hom}}({{\bf{T}}}_B,{{\bf{Z}}})$ is the unique saturated Hecke-stable complement of $S_2({{\bf{Z}}})[I_f]$ in $S_2({{\bf{Z}}})$, hence must equal $S_2({{\bf{Z}}})[I_f]^{\perp}$. This shows that the new definition of the congruence number/exponent generalizes our earlier definition for $A_f$.
Let $I_A = \operatorname{Ann}_{{\bf{T}}}(A)$ and $I_B = \operatorname{Ann}_{{\bf{T}}}(B)$. Theorem 3.6(a) of [@ars:moddeg] says that the modular exponent ${{\tilde{n}_{\scriptscriptstyle{A}}}}$ divides the congruence exponent ${{\tilde{r}_{\scriptscriptstyle{A}}}}$, and Propostion 5.9 of loc. cit. says that if $p$ is a prime such that all maximal ideals ${{\mathfrak{m}}}$ of ${{\bf{T}}}$ containing $I_A + I_B$ satisfy multiplicity one, then $\operatorname{ord}_p({{\tilde{r}_{\scriptscriptstyle{A}}}}) = \operatorname{ord}_p({{\tilde{n}_{\scriptscriptstyle{A}}}})$. Our main theorem deals with the case of modular/congruence numbers as opposed to modular/congruence exponents. In view of the case of newform quotients discussed in Section \[sec:intro\], one would like to understand the relation between the modular number ${{n_{\scriptscriptstyle{A}}}}$ and the [*square*]{} of the congruence number ${{r_{\scriptscriptstyle{A}}}}$. As mentioned earlier, it is not true that ${{n_{\scriptscriptstyle{A}}}}$ divides ${{r_{\scriptscriptstyle{A}}^2}}$ in general. At the same time, we have:
\[thm:main\] Let $p$ be an odd prime such that every maximal ideal ${{\mathfrak{m}}}$ with residue characteristic $p$ that contains $I_A + I_B$ satisfies multiplicity one. Then $\operatorname{ord}_p({{n_{\scriptscriptstyle{A}}}}) = \operatorname{ord}_p({{r_{\scriptscriptstyle{A}}^2}})$.
This theorem is proved in the next section. It is an analog of Propostion 5.9 of [@ars:moddeg] mentioned above in the context of modular/congruence numbers as opposed to modular/congruence exponents. For results on multiplicity one in the $\Gamma = \Gamma_1(N)$ context, see, e.g., [@tilouine:gorenstein] and the references therein.
Proof of Theorem \[thm:main\] {#sec:proof}
=============================
We continue to use the notation introduced in previous sections. The following lemma is easily extracted from [@emerton:optimal], and is the key input in our proof of Theorem \[thm:main\]:
\[lem:emerton\] Let $I$ be a saturated ideal of ${{\bf{T}}}$ and let $J[I]^0$ denote the abelian subvariety of $J$ that is the connected component of $J[I]$. Then the quotient $J[I]/J[I]^0$ is supported at maximal ideals of ${{\bf{T}}}$ that do not satisfy multiplicity one.
It is shown in the proof of Theorem A of [@emerton:optimal] that if ${{\mathfrak{m}}}$ satifies multiplicity one, then ${{\mathfrak{m}}}$ is good for $J$ (in the notation of loc. cit.). The lemma now follows from Corollary 2.3 of loc. cit. (note that [@emerton:optimal] is in the $\Gamma_0(N)$ context, but the ideas behind the argument above work in the $\Gamma_1(N)$ situation as well).
\[prop:coker\] The cokernel of the injection $A \cap B {\rightarrow}J[I_A + I_B]$ is supported at maximal ideals ${{\mathfrak{m}}}$ of ${{\bf{T}}}$ containing $I_A + I_B$ that do not satisfy multiplicity one.
Consider the natural map $B \cap J[I_A] {\rightarrow}J[I_A]/A$. It’s kernel is $B \cap J[I_A] \cap A = B \cap A$, and hence we have an injection: $$\begin{aligned}
\label{eqn1}
\frac{B \cap J[I_A]}{B \cap A} {\hookrightarrow}\frac{J[I_A]}{A}.\end{aligned}$$ Also, the natural map $J[I_A + I_B] = J[I_B][I_A] {\rightarrow}J[I_B]/B$ has kernel $B \cap J[I_B][I_A] = B \cap J[I_A]$, and hence we have an injection $$\begin{aligned}
\label{eqn2}
\frac{J[I_A+I_B]}{B \cap J[I_A]} {\hookrightarrow}\frac{J[I_B]}{B}\end{aligned}$$ Now $A$ is the connected component of $J[I_A]$ and similarly $B$ is the connected component of $J[I_B]$. Thus, by Lemma \[lem:emerton\], the quotient groups on the right side of (\[eqn1\]) and (\[eqn2\]) are supported at maximal ideals of ${{\bf{T}}}$ that do not satisfy multiplicity one. Then, by the injections (\[eqn1\]) and (\[eqn2\]), the cokernel of of the injection $A \cap B {\rightarrow}J[I_A + I_B]$ is supported at maximal ideals ${{\mathfrak{m}}}$ that do not satisfy multiplicity one. Also, any maximal ideal in the support of $J[I_A + I_B]$ contains $I_A + I_B$. Our lemma follows.
The following lemma is perhaps known to experts; its proof was indicated to us by K. Ribet.
\[lemma:ribet\] Let $I$ be an ideal of ${{\bf{T}}}$ of finite index. Suppose that every maximal ideal ${{\mathfrak{m}}}$ of ${{\bf{T}}}$ that contains $I$ satisfies multiplicity one (i.e., $J[{{\mathfrak{m}}}]$ has order $|{{\bf{T}}}/{{\mathfrak{m}}}|^2$). Then $J[I]$ has order $|{{\bf{T}}}/I|^2$.
If ${{\mathfrak{m}}}$ is a maximal ideal of ${{\bf{T}}}$, then let ${{J_{{{\mathfrak{m}}}}}}$ denote the ${{\mathfrak{m}}}$-divisible group attached to $J$. It suffices to show that ${{J_{{{\mathfrak{m}}}}}}[I]$ is of order $|{{{{\bf{T}}}_{{{\mathfrak{m}}}}}}/I {{{{\bf{T}}}_{{{\mathfrak{m}}}}}}|^2$ for each maximal ideal ${{\mathfrak{m}}}$ containing $I$. Let ${{\mathfrak{m}}}$ be such a maximal ideal and let ${{J^{\vee}_{{{\mathfrak{m}}}}}}$ denote the Pontryagin dual of ${{J_{{{\mathfrak{m}}}}}}$. Considering that $J[{{\mathfrak{m}}}]$ is free of rank $2$ over ${{\bf{T}}}/{{\mathfrak{m}}}$, by a standard argument due to Mazur that uses Nakayama’s lemma (e.g., see [@tilouine:gorenstein p. 333 and p. 341]), ${{J^{\vee}_{{{\mathfrak{m}}}}}}$ is free of rank two over ${{{{\bf{T}}}_{{{\mathfrak{m}}}}}}$. Then ${{J^{\vee}_{{{\mathfrak{m}}}}}}/I {{J^{\vee}_{{{\mathfrak{m}}}}}}$ is free of rank two over ${{{{\bf{T}}}_{{{\mathfrak{m}}}}}}/I {{{{\bf{T}}}_{{{\mathfrak{m}}}}}}$, and in particular has order $|{{{{\bf{T}}}_{{{\mathfrak{m}}}}}}/I {{{{\bf{T}}}_{{{\mathfrak{m}}}}}}|^2$. But ${{J_{{{\mathfrak{m}}}}}}[I]$ is Pontryagin dual to ${{J^{\vee}_{{{\mathfrak{m}}}}}}/I {{J^{\vee}_{{{\mathfrak{m}}}}}}$, and hence has the same order $|{{{{\bf{T}}}_{{{\mathfrak{m}}}}}}/I {{{{\bf{T}}}_{{{\mathfrak{m}}}}}}|^2$, as was to be shown.
Taking $I = I_A + I_B$ in Lemma \[lemma:ribet\], we see that $$\begin{aligned}
\label{eqn3}
\operatorname{ord}_p \Big(\big|J[I_A + I_B]\big|\Big) = \operatorname{ord}_p \bigg( \Big| \frac{{{\bf{T}}}}{I_A + I_B} \Big|^2 \bigg)
= \operatorname{ord}_p({{r_{\scriptscriptstyle{A}}^2}}), \end{aligned}$$ where the last equality follows since we have an isomorphism $$\frac{{{\bf{T}}}}{I_A + I_B} \stackrel{\simeq}{\longrightarrow}
\frac{{{\bf{T}}}_A \oplus {{\bf{T}}}_B}{{{\bf{T}}}}$$ obtained by sending $t \in {{\bf{T}}}$ to $(\pi_A(t),0) \in {{\bf{T}}}_A \oplus {{\bf{T}}}_B$, where $\pi_A$ is the projection map ${{\bf{T}}}{\rightarrow}{{\bf{T}}}_A$. Also by Proposition \[prop:coker\] and the hypothesis that every maximal ideal ${{\mathfrak{m}}}$ with residue characteristic $p$ that contains $I_A + I_B$ satisfies multiplicity one, we have $$\begin{aligned}
\label{eqn4}
\operatorname{ord}_p\Big(\big|J[I_A + I_B]\big|\Big) = \operatorname{ord}_p\big( |A \cap B|\big) = \operatorname{ord}_p({{n_{\scriptscriptstyle{A}}}}),\end{aligned}$$ where the last equality follows by the definition of ${{n_{\scriptscriptstyle{A}}}}$. The theorem now follows from (\[eqn3\]) and (\[eqn4\]).
[BCDT01]{}
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[^1]: This material is based upon work supported by the National Science Foundation under Grant No. 0603668.
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**Locally Supported Wavelets for the Separation of Spherical Vector Fields with Respect to their Sources**\
C. Gerhards
**Abstract** We provide a space domain oriented separation of magnetic fields into parts generated by sources in the exterior and sources in the interior of a given sphere. The separation itself is well-known in geomagnetic modeling, usually in terms of a spherical harmonic analysis or a wavelet analysis that is spherical harmonic based. In contrast to these frequency oriented methods, we use a more spatially oriented approach in this paper. We derive integral representations with explicitly known convolution kernels. Regularizing these singular kernels allows a multiscale representation of the internal and external contributions to the magnetic field with locally supported wavelets. This representation is applied to a set of CHAMP data for crustal field modeling.\
**Key Words** Green’s function, single layer kernel, locally supported wavelets, magnetic field, spherical decomposition\
**Mathematics Subject Classification (2000)** 41A30, 42C40, 86A99
Introduction
============
The Earth’s magnetic field is a complex structure consisting of various contributions, such as the dominating core field, the crustal field, and effects from iono- and magnetospheric processes. A major task in understanding the geomagnetic field is the separation of these contributions. An overview on different approaches to this is given, e.g., in [@olsen10]. A first step is the mathematical separation of magnetic field measurements taken at satellite altitude into contributions from sources in the exterior of the orbit and contributions from sources in the interior. Generally, we assume the magnetic field $b$ to be governed by the pre-Maxwell equations $$\begin{aligned}
\nabla\wedge b&=&\mu_0j,
\\\nabla\cdot b&=&0,\end{aligned}$$ with $j$ describing the source current density and $\mu_0$ the vacuum permeability ($\wedge$ denotes the vector product). If no source currents $j$ are present, one has $b=\nabla U$, for some harmonic potential $U$, and the typical approach to modeling the magnetic field is the so-called Gauss representation of the corresponding potential in terms of scalar spherical harmonics $Y_{n,k}$ (see, e.g., [@back96] and [@sabaka10]). Generally, however, satellite data is collected in a source region of the magnetic field. Then the Mie decomposition allows a decomposition of the magnetic field into a poloidal part $p_b$ and a toroidal part $q_b$. The toroidal part describes the magnetic field due to poloidal current densities $p_j$, while the poloidal part can be split into a part $p_b^{ext}$ that is due toroidal sources in the exterior of the satellite’s orbit, and a part $p_b^{int}$ that is due to toroidal sources in the interior. A more detailed description can be found, e.g., in [@back86] and [@back96]. In this setting, the quantities $p_b^{int}$, $p_b^{ext}$ and $q_b$ can be expanded in a system of vector spherical harmonics $\tilde{y}_{n,k}^{(1)}$, $\tilde{y}_{n,k}^{(2)}$ and $\tilde{y}_{n,k}^{(3)}$, respectively (a system that actually originates in quantum mechanics; see, e.g., [@edmonds57]).
However, due to the global nature of scalar and vector spherical harmonics, they are not the best choice for modeling strongly localized structures, such as the Earth’s crustal field, or modeling from only locally available data. Several multiscale approaches with spatially better localizing kernels have been developed to improve this drawback, e.g., in [@cham05], [@hol03] for potential fields, in [@may06], [@maymai] for the above described separation with respect to the sources, and in [@bay01], [@mai05] for a representation of ionospheric magnetic fields and current densities. A comprehensive introduction of kernel functions for such methods can also be found in [@free98].
It is the aim of this paper to transfer the multiscale approach described in [@may06], [@maymai], which is based on a construction of scaling and wavelet kernels in frequency domain (i.e., based on an adequate superposition of the vector spherical harmonics $\tilde{y}_{n,k}^{(i)}$), to a setting where the scaling and wavelet kernels are constructed entirely in space domain. For that purpose, the vector spherical harmonics $\tilde{y}_{n,k}^{(i)}$, $i=1,2,3$, are described by operators $\tilde{o}^{(i)}$, $i=1,2,3$. A decomposition of the magnetic field in terms of these operators, in combination with the spherical Helmholtz decomposition, allows an integral expression of the quantities $p_b^{int}$, $p_b^{ext}$, $q_b$. Motivated by [@free06], a regularization of the convolution kernels appearing in this integral expression provides a multiscale representation with wavelets that are locally supported in space. The multiscale representation is described in detail in Section \[sec:multiscale\]. There, we also apply the derived algorithm to a set of real CHAMP satellite data. The preparatory construction of the regularized kernels and a decomposition with respect to the operators $\tilde{o}^{(i)}$ is described in Sections \[sec:kernels\] and \[sec:decomps\]. Section \[sec:basics\] provides fundamental aspects on Legendre polynomials and scalar and vector spherical harmonics.
Preliminaries {#sec:basics}
=============
By $P_n:[-1,1]\to\mathbb{R}^3$, $n\in\mathbb{N}_0$, we denote the set of Legendre polynomials of degree $n$, by $Y_{n,k}:\Omega\to\mathbb{R}$, $n\in\mathbb{N}_0$, $k=1,\ldots,2n+1$, an orthonormal set of spherical harmonics of degree $n$ and order $k$ ($\Omega_R=\{x\in\mathbb{R}^3|\,|x|=R\}$ denotes the sphere of radius $R>0$ and $\Omega=\Omega_1$ the unit sphere). The fundamental connection between these two function systems is the so-called addition theorem, $$\begin{aligned}
\sum_{k=1}^{2n+1}Y_{n,k}(\xi)Y_{n,k}(\eta)=\frac{2n+1}{4\pi}P_n(\xi\cdot\eta),\quad\xi,\eta\in\Omega.\end{aligned}$$ This allows us to expand zonal kernels (i.e., functions $F:\Omega\times\Omega\to\mathbb{R}$ that satisfy $F(\xi,\eta)=G(\xi\cdot\eta)$, $\xi,\eta\in\Omega$, for an adequate function $G:[-1,1]\to\mathbb{R}$) in terms of Legendre polynomials. Known closed representations for certain series of Legendre polynomials can then be used to derive closed representations for some zonal kernels appearing in this paper. One of these series is the generating series for the Legendre polynomials, $$\begin{aligned}
\sum_{n=0}^\infty h^nP_n(t)=\frac{1}{\sqrt{1+h^2-2ht}},\quad t\in[-1,1],\,h\in(-1,1).\end{aligned}$$ From this, one can derive various further representations that are, e.g., listed in [@hansen]. Of importance to us are the following ones.
\[lem:genseriesgreenfunc\] For $t\in(-1,1)$, we have $$\begin{aligned}
\sum_{n=1}^\infty \frac{1}{n}P_n(t)&=&\ln\left(\frac{\sqrt{2}\sqrt{1-t}-1+t}{1-t^2}\right)+\ln\left(2\right),\label{eqn:genseriesgreenfunc1hto0}
\\\sum_{n=1}^\infty \frac{1}{n+1}P_n(t)&=&\ln\left(1+\frac{\sqrt{2}}{\sqrt{1-t}}\right)-1.\label{eqn:genseriesgreenfunc2hto0}\end{aligned}$$
Furthermore, the generating series for the Legendre polynomials yields an expansion of the single layer kernel. This is of interest since it allows an integral definition of the single layer operator and a definition in terms of pseudodifferential operators.
\[lem:seriespotfunc\] Let $x,y\in\mathbb{R}^3$ with $|x|<|y|$. Then $$\frac{1}{|x-y|}=\frac{1}{|y|}\sum_{n=0}^\infty\left(\frac{|x|}{|y|}\right)^nP_n\left(\frac{x}{|x|}\cdot\frac{y}{|y|}\right).$$
The set of the previously mentioned spherical harmonics yields a complete orthonormal system in $L^2(\Omega)=\{F:\Omega\to\mathbb{R}|\int_\Omega |F(\eta)|^2d\omega(\eta)<\infty\}$. The modeling of magnetic fields, however, is in first place a vectorial problem. For that purpose, we introduce two different complete sets of vector spherical harmonics. The first set requires the operators $$\begin{aligned}
o^{(1)}_\xi F(\xi)&=&\xi F(\xi),\quad\xi\in\Omega,\label{eqn:o1operator}
\\o^{(2)}_\xi F(\xi)&=&\nabla^*_\xi F(\xi),\quad\xi\in\Omega,\label{eqn:o2operator}
\\o^{(3)}_\xi F(\xi)&=&L^*_\xi F(\xi),\quad\xi\in\Omega,\label{eqn:o3operator}\end{aligned}$$ for $F:\Omega\to\mathbb{R}$ a sufficiently smooth scalar function, $\nabla^*$ the surface gradient (i.e., the tangential part of the gradient $\nabla$; more precisely, $\nabla_x=\xi\frac{\partial}{\partial r}+\frac{1}{r}\nabla_\xi^*$, for $x=r\xi\in\mathbb{R}^3$, with $r=|x|$, $\xi=\frac{x}{|x|}$), and $L^*$ the surface curl gradient (acting as $L_\xi^*=\xi\wedge\nabla_\xi^*$ in a point $\xi\in\Omega$). A complete orthonormal system in $l^2(\Omega)=\{f:\Omega\to\mathbb{R}^3|$ $\int_\Omega|f(\eta)|^2d\omega(\eta)<\infty\}$ is then given via $$\begin{aligned}
y_{n,k}^{(i)}=(\mu_n^{(i)})^{-\frac{1}{2}}o^{(i)}Y_{n,k},\quad i=1,2,3,\,n\in\mathbb{N}_{0_i},\,k=1,\ldots,2n+1,\end{aligned}$$ where $0_i$ is an abbreviation for $0_1=0$ and $0_i=1$, $i=2,3$, and $\mu_n^{(i)}$ denotes the normalization constants $\mu_n^{(1)}=1$ and $\mu_n^{(i)}=n(n+1)$, $i=2,3$. Concerning the notation, upper case letters, such as $F$, $Y_{n,k}$, generally denote scalar valued functions, lower case letters, such as $f$, $y_{n,k}^{(i)}$, denote vector valued functions, and bold face letters denote tensor valued functions. The same notation holds for the function spaces $C^{(k)}(\Omega)$, $c^{(k)}(\Omega)$ of $k$-times continuously differentiable functions and the spaces $L^2(\Omega)$, $l^2(\Omega)$ of square integrable functions.
The second set of vector spherical harmonics requires the modified operators $$\begin{aligned}
\tilde{o}^{(1)}&=&o^{(1)}\left(D+\textnormal{\footnotesize $\frac{1}{2}$}\right)-o^{(2)},\label{eqn:to1operator}
\\\tilde{o}^{(2)}&=&o^{(1)}\left(D-\textnormal{\footnotesize $\frac{1}{2}$}\right)+o^{(2)},\label{eqn:to2operator}
\\\tilde{o}^{(3)}&=&o^{(3)},\label{eqn:to3operator}\end{aligned}$$ where $$\begin{aligned}
D=\left(-\Delta^*+\textnormal{\footnotesize $\frac{1}{4}$}\right)^{\frac{1}{2}}.\label{eqn:d}\end{aligned}$$ By $\Delta^*$ we denote the Beltrami operator $\nabla^*\cdot\nabla^*$. The operator $D$ is treated in more detail in Subsection \[sec:singlelayer\]. A second complete orthonormal system in $l^2(\Omega)$ is then given via $$\begin{aligned}
\tilde{y}_{n,k}^{(i)}=(\tilde{\mu}_n^{(i)})^{-\frac{1}{2}}\tilde{o}^{(i)}Y_{n,k},\quad i=1,2,3,\,n\in\mathbb{N}_{0_i},\,k=1,\ldots,2n+1,\label{eqn:ytilde}\end{aligned}$$ where $\tilde{\mu}_n^{(i)}$ denotes the normalization constants $\tilde{\mu}_n^{(1)}=(n+1)(2n+1)$, $\tilde{\mu}_n^{(2)}=n(2n+1)$ and $\tilde{\mu}_n^{(3)}=n(n+1)$. The advantage of this basis system is its connection to the inner and outer harmonics, i.e., the functions $H_{n,k}^{int}(x)=\frac{1}{R}\big(\frac{|x|}{R}\big)^nY_{n,k}\big(\frac{x}{|x|}\big)$, $x\in{\Omega_R^{int}}=\{x\in\mathbb{R}^3|\,|x|<R\}$, and $H_{n,k}^{ext}(x)=\frac{1}{R}\big(\frac{R}{|x|}\big)^{n+1}Y_{n,k}\big(\frac{x}{|x|}\big)$, $x\in{\Omega^{ext}_R}=\{x\in\mathbb{R}^3|\,|x|>R\}$, which yield solutions to the inner and outer Dirichlet boundary value problem, respectively (i.e., boundary values $H_{n,k}^{int}=H_{n,k}^{ext}=Y_{n,k}$ on $\Omega_R$ and $\Delta H_{n,k}^{int}=0$ in $\Omega_R^{int}$, $\Delta H_{n,k}^{ext}=0$ in $\Omega_R^{ext}$). We have $$\begin{aligned}
\nabla_xH^{int}_{n,k}(x)&=&\frac{1}{R^2}\left(\frac{r}{R}\right)^{n-1}(\tilde{\mu}_n^{(2)})^{\frac{1}{2}}\tilde{y}_{n,k}^{(2)}(\xi),\qquad r=|x|,\,x=r\xi\in\overline{\Omega_R^{int}},\label{eqn:hint}
\\-\nabla_xH^{ext}_{n,k}(x)&=&\frac{1}{R^2}\left(\frac{R}{r}\right)^{n+2}(\tilde{\mu}_n^{(1)})^{\frac{1}{2}}\tilde{y}_{n,k}^{(1)}(\xi),\qquad r=|x|,\, x=r\xi\in\overline{\Omega_R^{ext}}.\label{eqn:hext}\end{aligned}$$ For a more comprehensive introduction of the function systems mentioned in this section, the reader is referred to, e.g., [@free98] and the references therein. The special importance of the last set of vector spherical harmonics in geomagnetic modeling is well emphasized, e.g., in [@back96], [@may06] and [@maymai]. In this paper, however, they are only to be understood as a motivation for the Helmholtz decomposition and a modified decomposition with respect to $\tilde{o}^{(i)}$. Our main goal is to actually avoid spherical harmonic representations.
Regularized Kernels {#sec:kernels}
===================
Green’s function for the Beltrami operator and the single layer kernel are especially useful when working with differential equations involving the operators $\nabla^*$, $L^*$, $\Delta^*$ and $D$. We briefly recapitulate some of the properties of these functions and the corresponding operators before we introduce a regularization for both kernels separately and for their combination. To achieve integral representations for the scalars of the classical Helmholtz decomposition, it is actually sufficient to only have Green’s function. The single layer kernel becomes necessary when we introduce a decomposition that pays tribute to interior and exterior sources.
Green’s Function {#sec:greenfunc}
----------------
By *Green’s function with respect to the Beltrami operator* we denote the uniquely defined function $G(\Delta^*;\cdot):[-1,1)\to\mathbb{R}$ satisfying the properties
- $\eta\mapsto G(\Delta^*;\xi\cdot\eta)$ is twice continuously differentiable on the set $\{\eta\in\Omega|$ $1-\xi\cdot\eta>0\}$, and $$\Delta^*_\eta G(\Delta^*;\xi\cdot\eta)= -\frac{1}{4 \pi}, \quad 1-\xi\cdot\eta>0,$$ for any fixed $\xi \in \Omega$,
- for any fixed $\xi \in \Omega$, the function $$\eta\mapsto G(\Delta^*;\xi\cdot\eta)- \frac{1}{4 \pi}\ln(1-\xi \cdot \eta),$$ is continuously differentiable on $\Omega$,
- for any fixed $\xi \in \Omega$, $$\frac{1}{4\pi}\int_\Omega G(\Delta^*;\xi\cdot\eta)d\omega(\eta) = 0 .$$
One can verify the following explicit representation, $$\begin{aligned}
\label{eqn:beltramigreenfct}
G(\Delta^*;\xi\cdot\eta)&=&\frac{1}{4\pi}\ln(1-\xi\cdot\eta)+\frac{1}{4\pi}(1-\ln(2)),\quad1-\xi\cdot\eta>0.\label{eqn:repmodgreenfunc}\end{aligned}$$ The bilinear series expansion reads $$\begin{aligned}
G(\Delta^*; \xi\cdot\eta)&=&\sum_{n=1}^\infty\sum_{k=1}^{2n+1}\frac{1}{-n(n+1)}Y_{n,k}(\xi)Y_{n,k}(\eta),\quad 1-\xi\cdot\eta>0.\end{aligned}$$ Observing that $\eta\mapsto\Delta^*_\eta G(\Delta^*;\xi\cdot\eta)$ only varies by the constant $-\frac{1}{4 \pi}$ from the Dirac distribution motivates the following theorems which express a sufficiently smooth function by its integral mean value and a correction term involving Green’s function. For more details, the reader is again referred to [@free98] and the references therein.
\[thm:fundthmdelta\] Let $F$ be of class $C^{(2)}(\Omega)$. Then $$F(\xi)=\frac{1}{4\pi}\int_\Omega F(\eta)d\omega(\eta)+\int_\Omega G(\Delta^*;\xi\cdot\eta)\Delta^*_\eta F(\eta)\,d\omega(\eta),\quad\xi\in\Omega.$$
\[thm:fundthmgradcurl\] Let $F$ be of class $C^{(1)}(\Omega)$. Then $$\begin{aligned}
F(\xi)&=&\frac{1}{4\pi}\int_\Omega F(\eta)d\omega(\eta)-\int_\Omega \Lambda_\eta^*G(\Delta^*;\xi\cdot\eta)\cdot \Lambda_\eta^*F(\eta)\,d\omega(\eta),\quad\xi\in\Omega,\end{aligned}$$ where $\Lambda^*$ denotes one of the operators $\nabla^*$ or $L^*$.
These theorems directly yield simple integral representations for solutions to the spherical differential equations with respect to $\nabla^*$, $L^*$, and $\Delta^*$.
Next, we present a spatial regularization of $G(\Delta^*;\cdot)$ around its singularity. This is a crucial step for the later definition of the scaling and wavelet kernels of the multiscale representation.
\[def:reggreen\] Let $R^\rho$, $\rho>0$, be of class $C^{(n)}([-1,1])$, $n\in\mathbb{N}$ fixed, satisfying $$\lim_{\rho\to0+}\rho^{\frac{k}{2}}\int_{1-\rho}^1\left|\left(\frac{d}{dt}\right)^kR^\rho(t)\right|\,dt=0,\quad k=0,1,\label{eqn:limitrrho}$$ and $$\left[\left(\frac{d}{dt}\right)^k R^\rho(t)\right]_{t=1-\rho}=\left[\left(\frac{d}{dt}\right)^kG(\Delta^*;t)\right]_{t=1-\rho},\quad k=0,1,\ldots,n.$$ Then the function $$\begin{aligned}
G^\rho(\Delta^*;\xi\cdot\eta)=\left\{\begin{array}{ll}
G(\Delta^*;\xi\cdot\eta),&1-\xi\cdot\eta\geq\rho,
\\[1.25ex]R^\rho(\xi\cdot\eta),&1-\xi\cdot\eta<\rho,
\end{array}\right.\end{aligned}$$ is called *regularized Green’s function (of order $n$)*. $R^\rho$ is called the regularization function.
A typical choice for $R^\rho$ is the Taylor series of $G(\Delta^*;\cdot)$ centered at $1-\rho$ and truncated at some power $n$. An exemplary plot for different scaling parameters $\rho$ can be found in Figure \[fig:reggreenfunc\]. Similar regularizations, but only for Taylor polynomials up to degree $2$, have been used in other areas of geosciences, e.g., in [@fehl07], [@fehl08], and [@free06]. To be able to state a multiscale decomposition, it has to be guaranteed that convolutions with the regularized kernels converge to convolutions with the original kernels. The proofs are based on the fact that $\eta\mapsto G(\Delta^*;\xi\cdot\eta)$, $\eta\mapsto \nabla_\xi^*G(\Delta^*;\xi\cdot\eta)$ and $\eta\mapsto L_\xi^*G(\Delta^*;\xi\cdot\eta)$ are integrable on the sphere $\Omega$, uniformly with respect to $\xi\in\Omega$, and can be found in [@free09] and [@freeger10].
\[lem:convg\] Let $G^\rho(\Delta^*; \cdot)$ be of class $ C^{(1)}([-1, 1])$ and $F$ of class $C^{(0)}(\Omega)$. Then we have $$\begin{aligned}
\lim_{\rho\to0+}\sup_{\xi\in\Omega}\left|\int_\Omega G^\rho(\Delta^*;\xi\cdot\eta)F(\eta)d\omega(\eta)-\int_\Omega G(\Delta^*;\xi\cdot\eta)F(\eta)d\omega(\eta)\right|=0.\end{aligned}$$
\[lem:convdiffg\] Let $F$ be of class $C^{(0)}(\Omega)$ and $G^\rho(\Delta^*; \cdot)$ of class $ C^{(1)}([-1, 1])$. Then $$\begin{aligned}
\lim_{\rho\to0+}\sup_{\xi\in\Omega}\left|\int_\Omega \Lambda^*_\xi G^\rho(\Delta^*;\xi\cdot\eta)F(\eta)d\omega(\eta)- \Lambda^*_\xi\int_\Omega G(\Delta^*;\xi\cdot\eta)F(\eta)d\omega(\eta)\right|&=&0,\end{aligned}$$ where $\Lambda^*$ denotes one of the operators $\nabla^*$ or $L^*$.
Relations for higher order derivatives are simple consequences of the above lemmas by use of well-known surface versions of Green’s formulas that shift the differentiation from the convolution kernel to the convolved function $F$. Thus, they also require a higher smoothness of $F$.
\[cor:convbeltramig\] Let $G^\rho(\Delta^*; \cdot)$ be of class $ C^{(2)}([-1,1])$ and $F$ of class $C^{(1)}(\Omega)$. Then $$\begin{aligned}
\lim_{\rho\to0+}\sup_{\xi\in\Omega}\left|\int_\Omega \Delta^*_\xi G^\rho(\Delta^*;\xi\cdot\eta)F(\eta)d\omega(\eta)- \Delta^*_\xi\int_\Omega G(\Delta^*;\xi\cdot\eta)F(\eta)d\omega(\eta)\right|&=&0.\end{aligned}$$
\[cor:convtensbeltramig\] Let $G^\rho(\Delta^*; \cdot)$ be of class $ C^{(2)}([-1,1])$ and $f$ of class $c^{(1)}(\Omega)$. Then $$\begin{aligned}
\lim_{\rho\to0+}\sup_{\xi\in\Omega}&\!\!\!\!\bigg|&\!\!\!\!\int_\Omega \big(\big(\Lambda_1^{*}\big)_\xi\otimes \big(\Lambda_2^{*}\big)_\eta G^\rho(\Delta^*;\xi\cdot\eta)\big)f(\eta)d\omega(\eta)
\\&&-\big(\Lambda_1^{*}\big)_\xi\int_\Omega \big(\Lambda_2^{*}\big)_\eta G(\Delta^*;\xi\cdot\eta)\cdot f(\eta)d\omega(\eta)\bigg|=0,\end{aligned}$$ where $\Lambda_1^*$ and $\Lambda_2^*$ denote one of the operators $\nabla^*$ or $L^*$ ($\otimes$ denotes the tensor product $x\otimes y=xy^T$, for $x,y\in\mathbb{R}^3$).
An adequate choice of $R^\rho$ admits an explicit statement on the convergence rate. More precisely, if $\int_{1-\rho}^1\left|R^\rho(t)\right|\,dt=\mathcal{O}(\rho)$ and $\int_{1-\rho}^1\left|\frac{d}{dt}R^\rho(t)\right|\,dt=\mathcal{O}(1)$, one can find $$\begin{aligned}
\left|\int_\Omega G^\rho(\Delta^*;\xi\cdot\eta)F(\eta)d\omega(\eta)-\int_\Omega G(\Delta^*;\xi\cdot\eta)F(\eta)d\omega(\eta)\right|&=&\mathcal{O}(\rho\ln(\rho)),
\\\left|\int_\Omega \Lambda^*_\xi G^\rho(\Delta^*;\xi\cdot\eta)F(\eta)d\omega(\eta)- \Lambda^*_\xi\int_\Omega G(\Delta^*;\xi\cdot\eta)F(\eta)d\omega(\eta)\right|&=&\mathcal{O}(\rho^{\frac{1}{2}}),\end{aligned}$$ for $F$ of class $C^{(0)}(\Omega)$. If $F$ is of class $C^{(1)}(\Omega)$, it even holds $$\begin{aligned}
\left|\int_\Omega \Lambda^*_\xi G^\rho(\Delta^*;\xi\cdot\eta)F(\eta)d\omega(\eta)- \Lambda^*_\xi\int_\Omega G(\Delta^*;\xi\cdot\eta)F(\eta)d\omega(\eta)\right|&=&\mathcal{O}(\rho\ln(\rho)).\end{aligned}$$ The conditions on the regularization function are satisfied, e.g., by the choice of $R^\rho$ as the truncated Taylor series of $G(\Delta^*;\cdot)$.
Single Layer Kernel {#sec:singlelayer}
-------------------
By the singel layer kernel we denote the convolution kernel of the integral operator $D^{-1}$, with $D$ formally given as in (\[eqn:d\]). Observing that $\Delta^*Y_{n,k}=-n(n+1)Y_{n,k}$, the fractional pseudodifferential operator $D$, mapping the Sobolev space $H_s(\Omega)$ into $H_{s-1}(\Omega)$, can be defined via $$\begin{aligned}
DF=\left(-\Delta^*+\textnormal{\footnotesize $\frac{1}{4}$}\right)^{\frac{1}{2}}F=\sum_{n=0}^\infty\sum_{k=1}^{2n+1}\big(n+\textnormal{\footnotesize$\frac{1}{2}$}\big)\,(F,Y_{n,k})_{L^2(\Omega)}Y_{n,k},\end{aligned}$$ for $F$ of class $H_s(\Omega)$, where $(\cdot,\cdot)_{L^2(\Omega)}$ denotes the inner product $(F,G)_{L^2(\Omega)}=\int_\Omega F(\eta)G(\eta)d\omega(\eta)$. Its inverse $D^{-1}$, mapping $H_{s-1}(\Omega)$ into $H_{s}(\Omega)$, is correspondingly given by $$\begin{aligned}
D^{-1}F=\left(-\Delta^*+\textnormal{\footnotesize $\frac{1}{4}$}\right)^{-\frac{1}{2}}F=\sum_{n=0}^\infty\sum_{k=1}^{2n+1}\frac{1}{n+\textnormal{\footnotesize$\frac{1}{2}$}}\,(F,Y_{n,k})_{L^2(\Omega)}Y_{n,k}, \label{eqn:singlelayerpseudodiff}\end{aligned}$$ for $F$ of class $H_{s-1}(\Omega)$. From the addition theorem and the power series in Lemma \[lem:seriespotfunc\], it is easy to derive the integral representation $$\begin{aligned}
\sum_{n=0}^\infty\sum_{k=1}^{2n+1}\frac{1}{n+\frac{1}{2}}(F,Y_{n,k})_{L^2(\Omega)}Y_{n,k}(\xi)=\frac{1}{2\sqrt{2}\pi}\int_\Omega \frac{1}{\sqrt{1-\xi\cdot\eta}}F(\eta)d\omega(\eta),\quad \xi\in\Omega.\end{aligned}$$ The function $$\begin{aligned}
S(\xi\cdot\eta)=\frac{1}{\sqrt{2}}\frac{1}{\sqrt{1-\xi\cdot\eta}},\quad 1-\xi\cdot\eta>0,\end{aligned}$$ is from now on called the *single layer kernel*, and denotes the starting point for our further considerations. The integral operator $D^{-1}$ is called the single layer operator. For a more general and detailed overview on spherical pseudodifferential operators and the definition of Sobolev spaces, the reader is referred to, e.g., [@free98] and [@svensson]. Since we are dealing with continuously differentiable functions in the remainder of this paper, it should be remarked that $D^{-1}$ actually maps $C^{(k)}(\Omega)$ into $C^{(k)}(\Omega)$, $k\in\mathbb{N}_0$.
In analogy to Green’s function, one can define a spatial regularization of the single layer kernel.
\[def:regsingkernel\] Let $\rho>0$ and $R^\rho$ a non-negative function of class $C^{(n)}([-1,1])$, $n\in\mathbb{N}$ fixed, satisfying $$\begin{aligned}
\label{eqn:propregsing}
\lim_{\rho\to0+}\rho^k\int_{1-\rho}^1\left|\left(\frac{d}{dt}\right)^kR^\rho(t)\right|\,dt=0,\quad k=0,1,\end{aligned}$$ and $$\left[\left(\frac{d}{dt}\right)^k R^\rho(t)\right]_{t=1-\rho}=\left[\left(\frac{d}{dt}\right)^kS(t)\right]_{t=1-\rho},\quad k=0,1,\ldots,n.$$ Then the function $$\begin{aligned}
S^\rho(\xi\cdot\eta)=\left\{\begin{array}{ll}
\frac{1}{\sqrt{2}}\frac{1}{\sqrt{1-\xi\cdot\eta}},&1-\xi\cdot\eta\geq\rho,
\\[1.25ex]R^\rho(\xi\cdot\eta),&1-\xi\cdot\eta<\rho,
\end{array}\right.\end{aligned}$$ is called *regularized single layer kernel (of order $n$)*.
The regularizing function $R^\rho$ is generally chosen as the Taylor series of $S$ centered at $1-\rho$ and truncated at degree $n$. An exemplary plot for different scaling parameters $\rho$ can be found in Figure \[fig:reggreenfunc\]. The special cases of a linear or quadratic regularization have been applied to multiscale methods in physical geodesy, e.g., in [@freewolf09], [@free09]. In the Euclidean space $\mathbb{R}^3$, the kernel $S$ can be related to the fundamental solution of the Laplace operator $\Delta$. A different kind of regularization for that kernel is treated in [@akram10]. In our setting, we obtain the following limit relation in the same manner as for the Green function case in the previous subsection.
\[lem:convs\] Let $S^\rho$ be of class $ C^{(1)}([-1, 1])$ and $F$ of class $C^{(0)}(\Omega)$. Then we have $$\begin{aligned}
\lim_{\rho\to0+}\sup_{\xi\in\Omega}\left|\int_\Omega S^\rho(\xi\cdot\eta)F(\eta)d\omega(\eta)-\int_\Omega S(\xi\cdot\eta)F(\eta)d\omega(\eta)\right|=0.\end{aligned}$$
For the relations involving the surface gradient and the surface curl gradient, one has to observe that $\eta\mapsto \nabla_\xi^*S(\xi\cdot\eta)$ and $\eta\mapsto L_\xi^*S(\xi\cdot\eta)$ are not integrable on the sphere $\Omega$. However, if $F$ is of class $C^{(1)}(\Omega)$ and $\mathbf{t}_\xi\in\mathbb{R}^{3\times3}$ denotes the rotation matrix with $\mathbf{t}_\xi\xi={\varepsilon}^3=(0,0,1)^T$, we obtain $$\begin{aligned}
\nabla_\xi^*\int_\Omega S(\xi\cdot\eta) F(\eta)d\omega(\eta)&=&\nabla_\xi^*\int_\Omega S(\xi\cdot\mathbf{t}_\xi^T\eta)F(\mathbf{t}_\xi^T\eta)d\omega(\eta)\nonumber
\\&=&\nabla_\xi^*\int_\Omega S(\eta_3)F(\mathbf{t}_\xi^T\eta)d\omega(\eta)\\&=&\int_\Omega S(\eta_3)\nabla_\xi^*F(\mathbf{t}_\xi^T\eta)d\omega(\eta),\quad\xi\in\Omega,\nonumber\end{aligned}$$ where $\eta=(\eta_1,\eta_2,\eta_3)^T\in\Omega$. Furthermore, regularizing the single layer kernel yields $$\begin{aligned}
\int_\Omega \nabla_\xi^*S^\rho(\xi\cdot\eta) F(\eta)d\omega(\eta)&=&\nabla_\xi^*\int_\Omega S^\rho(\xi\cdot\eta) F(\eta)d\omega(\eta)
\\&=&\nabla_\xi^*\int_\Omega S^\rho(\eta_3)F(\mathbf{t}_\xi^T\eta)d\omega(\eta)
\\&=&\int_\Omega S^\rho(\eta_3)\nabla_\xi^*F(\mathbf{t}_\xi^T\eta)d\omega(\eta),\quad\xi\in\Omega,\end{aligned}$$ so that the previous lemma implies the desired relations.
\[lem:convsgradsing\] Let $F$ be of class $C^{(1)}(\Omega)$ and $S^\rho$ of class $C^{(1)}([-1,1])$. Then we have $$\begin{aligned}
\lim_{\rho\to0+} \sup_{\xi\in\Omega} \left|\int_\Omega \Lambda^*_\xi S^\rho(\xi\cdot\eta) F(\eta)d\omega(\eta)-\Lambda_\xi^*\int_\Omega S(\xi\cdot\eta) F(\eta)d\omega(\eta)\right|&=&0,\end{aligned}$$ where $\Lambda^*$ denotes one of the operators $\nabla^*$ or $L^*$.
Relations for higher order differential operators follow analogously. Of more interest to us are combinations of the single layer operator with Green’s function for the Beltrami operator. It holds, e.g., that $$\begin{aligned}
\label{eqn:singgreenrel}
\lim_{\rho\to0+}\sup_{\xi\in\Omega}\left|\int_\Omega \left(\Lambda_\xi^* D_\xi^{-1}G^\rho(\Delta^* ;\xi\cdot\eta)\right)\,F(\eta)d\omega(\eta)-\Lambda_\xi^* D_\xi^{-1}\int_\Omega G(\Delta^*;\xi\cdot\eta)F(\eta)d\omega(\eta)\right|=0.\,\,\,\,\end{aligned}$$ However, it is difficult to explicitly calculate $D_\xi^{-1}G^\rho(\Delta^* ;\xi\cdot\eta)$, as it would be required for our later applications. $D_\xi^{-1}G(\Delta^* ;\xi\cdot\eta)$, on the other hand, can be calculated, and a regularization afterwards yields a similar limit relation.
For $\xi,\eta\in\Omega$ we have, $$\begin{aligned}
D_\xi^{-1}G(\Delta^*;\xi\cdot\eta)=\frac{1}{2\pi}\ln\left((1+\xi\cdot\eta)\left(\frac{1}{2}-\frac{1}{1-2S(\xi\cdot\eta)}\right)\right)-\frac{1}{2\pi}.\end{aligned}$$
Lemma \[lem:genseriesgreenfunc\] and the pseudodifferential representation (\[eqn:singlelayerpseudodiff\]) imply $$\begin{aligned}
D_\xi^{-1}G(\Delta^*;\xi\cdot\eta)&=&\sum_{n=1}^\infty\frac{1}{n+\frac{1}{2}}\frac{2n+1}{4\pi}\frac{1}{-n(n+1)}P_n(\xi\cdot\eta)
\\&=&\frac{1}{2\pi}\sum_{n=1}^\infty\frac{1}{n+1}P_n(\xi\cdot\eta)-\frac{1}{2\pi}\sum_{n=1}^\infty\frac{1}{n}P_n(\xi\cdot\eta)
\\&=&\frac{1}{2\pi}\ln\left(1+\frac{\sqrt{2}}{\sqrt{1-\xi\cdot\eta}}\right)-\frac{1}{2\pi}\ln\left(\frac{\sqrt{2}\sqrt{1-\xi\cdot\eta}-1+\xi\cdot\eta}{1-(\xi\cdot\eta)^2}\right)
-\frac{1}{2\pi}(1+\ln\left(2\right))
\\&=&\frac{1}{2\pi}\ln\left((1+\xi\cdot\eta)\left(\frac{1}{2}-\frac{1}{1-2S(\xi\cdot\eta)}\right)\right)-\frac{1}{2\pi},\pagebreak[0]\end{aligned}$$which is well-defined for every $\xi,\eta\in\Omega$.
The above derived representation implies that $(\xi,\eta)\mapsto D_\xi^{-1}G(\Delta^*;\xi\cdot\eta)$ is zonal and of class $C^{(1)}(\Omega\times\Omega)$. Some lengthy but basic computations yield $$\begin{aligned}
\nabla_\xi^*D_\xi^{-1}G(\Delta^*;\xi\cdot\eta)=\frac{1}{2\pi}\left(\frac{1}{2}-S(\xi\cdot\eta)-\frac{1}{2+4S(\xi\cdot\eta)}\right)(\eta-(\xi\cdot\eta)\xi),\quad \xi,\eta\in\Omega.\end{aligned}$$ A further application of the surface gradient causes a singularity of type $\mathcal{O}((1-\xi\cdot\eta)^{-\frac{1}{2}})$. Therefore, we do the following regularization for $\rho>0$, $$\begin{aligned}
\label{eqn:regsinggreen}
s_{\nabla^*}^\rho(\xi,\eta)=\frac{1}{2\pi}\left(\frac{1}{2}-S^\rho(\xi\cdot\eta)-\frac{1}{2+4S^\rho(\xi\cdot\eta)}\right)(\eta-(\xi\cdot\eta)\xi),\quad \xi,\eta\in\Omega.\end{aligned}$$ For this kernel we can calculate $$\begin{aligned}
\nabla_\xi^*\otimes s_{\nabla^*}^\rho(\eta,\xi)&=&\frac{1}{2\pi}\left(\frac{1}{2}-S^\rho(\xi\cdot\eta)-\frac{1}{2+4S^\rho(\xi\cdot\eta)}\right)\nabla_\xi^*\otimes(\xi-(\xi\cdot\eta)\eta)
\\&&+\frac{1}{2\pi}\left(-\big(S^\rho\big)'(\xi\cdot\eta)+\frac{4\big(S^\rho\big)'(\xi\cdot\eta)}{(2+4S^\rho(\xi\cdot\eta))^2}\right)(\eta-(\xi\cdot\eta)\xi)\otimes (\xi-(\xi\cdot\eta)\eta),\end{aligned}$$ where $\big(S^\rho\big)'$ denotes the one-dimensional derivative of $S^\rho$. The analogous procedure works for the surface curl gradient, and we have for $\rho>0$ that $$\begin{aligned}
\label{eqn:regsinggreen2}
s_{L^*}^\rho(\xi,\eta)=\frac{1}{2\pi}\left(\frac{1}{2}-S^\rho(\xi\cdot\eta)-\frac{1}{2+4S^\rho(\xi\cdot\eta)}\right)(\xi\wedge\eta), \quad \xi,\eta\in\Omega,\end{aligned}$$ and $$\begin{aligned}
L_\xi^*\otimes s_{L^*}^\rho(\eta,\xi)&=&\frac{1}{2\pi}\left(\frac{1}{2}-S^\rho(\xi\cdot\eta)-\frac{1}{2+4S^\rho(\xi\cdot\eta)}\right)L_\xi^*\otimes(\eta\wedge\xi)
\\&&+\frac{1}{2\pi}\left(-\big(S^\rho\big)'(\xi\cdot\eta)+\frac{4\big(S^\rho\big)'(\xi\cdot\eta)}{(2+4S^\rho(\xi\cdot\eta))^2}\right)(\xi\wedge\eta)\otimes (\eta\wedge\xi).\end{aligned}$$
Thus, relation (\[eqn:singgreenrel\]) can be formulated in the following numerically more advantageous way.
\[lem:limsinggreen\] Let $F$ be of class $C^{(0)}(\Omega)$ and $S^\rho$ of class $C^{(1)}([-1,1])$. Then we get with $s_{\nabla^*}^\rho(\cdot,\cdot)$ and $s_{L^*}^\rho(\cdot,\cdot)$ as in (\[eqn:regsinggreen\]) and (\[eqn:regsinggreen2\]), respectively, that $$\begin{aligned}
\lim_{\rho\to0+}\sup_{\xi\in\Omega}\left|\int_\Omega s_{\Lambda^*}^\rho(\xi,\eta)F(\eta)\,d\omega(\eta)-\Lambda_\xi^*D_\xi^{-1}\int_\Omega G(\Delta^*;\xi\cdot\eta) F(\eta)\,d\omega(\eta)\right|&=&0,\end{aligned}$$ where $\Lambda^*$ denotes one of the operators $\nabla^*$ or $L^*$.
The relation we are actually aiming at, and which we require in later applications, is the following tensorial one.
\[lem:limdiffsinggreen \] Let $f$ be of class $c^{(1)}(\Omega)$ and $S^\rho$ of class $C^{(1)}([-1,1])$. Then we get with $s_{\nabla^*}^\rho(\cdot,\cdot)$ and $s_{L^*}^\rho(\cdot,\cdot)$ as in (\[eqn:regsinggreen\]) and (\[eqn:regsinggreen2\]), respectively, that $$\begin{aligned}
\lim_{\rho\to0+}\sup_{\xi\in\Omega}\left|\int_\Omega \left(\Lambda_\xi^*\otimes s_{\Lambda^*}^\rho(\eta,\xi)\right) f(\eta)\,d\omega(\eta)-\Lambda_\xi^*\int_\Omega \left(\Lambda_\eta^*D_\xi^{-1}G(\Delta^*;\xi\cdot\eta)\right)\cdot f(\eta)\,d\omega(\eta)\right|&=&0,\end{aligned}$$ where $\Lambda^*$ is one of the operators $\nabla^*$ or $L^*$.
Since $|\nabla_\xi^*\otimes (\xi-(\xi\cdot\eta)\eta)|$ and $|f(\eta)|$ are uniformly bounded with respect to $\xi,\eta\in\Omega$ by some constant $M>0$, we get the following estimate for $\xi\in\Omega$ and $\rho>0$, $$\begin{aligned}
&&\!\!\!\!\!\!\!\!\!\!\left|\int_\Omega \left(\nabla_\xi^*\otimes s_{\nabla^*}^\rho(\eta,\xi)\right) f(\eta)\,d\omega(\eta)-\int_\Omega \left(\nabla_\xi^*\otimes\nabla_\eta^*D_\xi^{-1}G(\Delta^*;\xi\cdot\eta)\right) f(\eta)\,d\omega(\eta)\right|\label{eqn:tenslimgreensing}
\\&\leq&\int_{\eta\in\Omega\atop 1-\xi\cdot\eta\leq \rho}\left|S(\xi\cdot\eta)+\frac{1}{2+4S(\xi\cdot\eta)}-S^\rho(\xi\cdot\eta)-\frac{1}{2+4S^\rho(\xi\cdot\eta)}\right|\nonumber
\\&&\qquad\qquad\times\left|\nabla_\xi^*\otimes (\xi-(\xi\cdot\eta)\eta)\right|\,|f(\eta)|\,d\omega(\eta)\nonumber
\\&&+\int_{\eta\in\Omega\atop 1-\xi\cdot\eta\leq \rho}\left|S(\xi\cdot\eta)^3-\frac{4S(\xi\cdot\eta)^3}{(2+4S(\xi\cdot\eta\nonumber))^2} -\big(S^\rho\big)'(\xi\cdot\eta)+\frac{4\big(S^\rho\big)'(\xi\cdot\eta)}{(2+4S^\rho(\xi\cdot\eta))^2}\right|
\\&&\qquad\qquad\quad\times\left|(\eta-(\xi\cdot\eta)\xi)\otimes (\xi-(\xi\cdot\eta)\eta)\right|\,|f(\eta)|\,d\omega(\eta)\nonumber
\\[1.25ex]&\leq& M^2 \int_{\eta\in\Omega\atop 1-\xi\cdot\eta\leq \rho}\left|S(\xi\cdot\eta)+\frac{1}{2+4S(\xi\cdot\eta)}-S^\rho(\xi\cdot\eta)-\frac{1}{2+4S^\rho(\xi\cdot\eta)}\right|\,d\omega(\eta)\nonumber
\\&&+\,M\int_{\eta\in\Omega\atop 1-\xi\cdot\eta\leq \rho}\left|S(\xi\cdot\eta)^3-\frac{4S(\xi\cdot\eta)^3}{(2+4S(\xi\cdot\eta))^2} -\big(S^\rho\big)'(\xi\cdot\eta)+\frac{4\big(S^\rho\big)'(\xi\cdot\eta)}{(2+4S^\rho(\xi\cdot\eta))^2}\right|\nonumber
\\&&\qquad\qquad\qquad\times|\eta-(\xi\cdot\eta)\xi|\,|\xi-(\xi\cdot\eta)\eta|d\omega(\eta).\nonumber\end{aligned}$$ Observing $$|\eta-(\xi\cdot\eta)\xi|\,|\xi-(\xi\cdot\eta)\eta|=\frac{1}{2S(\xi\cdot\eta)^2}(1+\xi\cdot\eta),$$ the integrability of $\eta\mapsto S(\xi\cdot\eta)$ on the sphere $\Omega$, and the properties for $S^\rho$ from Definition \[def:regsingkernel\], we see that the integrals above vanish as $\rho$ tends to zero. Due to the zonality of the kernels, this convergence is uniform with respect to $\xi\in\Omega$. Furthermore, the convergence of (\[eqn:tenslimgreensing\]) to zero additionally yields $$\begin{aligned}
\int_\Omega \left(\nabla_\xi^*\otimes\nabla_\eta^*D_\xi^{-1}G(\Delta^*;\xi\cdot\eta)\right) f(\eta)\,d\omega(\eta)=\nabla_\xi^*\int_\Omega \left(\nabla_\eta^*D_\xi^{-1}G(\Delta^*;\xi\cdot\eta)\right)\cdot f(\eta)\,d\omega(\eta),\end{aligned}$$ and therefore, the desired statement. The assertion for the surface curl gradient follows analogously.
In analogy to the Green’s function case, an adequate choice of $R^\rho$ admits an explicit statement on the convergence rate. More precisely, if $\int_{1-\rho}^1\left|R^\rho(t)\right|\,dt=\mathcal{O}(\rho)$ and $\int_{1-\rho}^1\left|\frac{d}{dt}R^\rho(t)\right|\,dt=\mathcal{O}(1)$, one obtains $$\begin{aligned}
\left|\int_\Omega \Lambda^*_\xi S^\rho(\xi\cdot\eta) F(\eta)d\omega(\eta)-\Lambda_\xi^*\int_\Omega S(\xi\cdot\eta) F(\eta)d\omega(\eta)\right|&=&\mathcal{O}(\rho^{\frac{1}{2}}).\end{aligned}$$ For the combination of Green’s function and the single layer kernel the same convergence rate holds true, $$\begin{aligned}
\left|\int_\Omega \left(\Lambda_\xi^*\otimes s_{\Lambda^*}^\rho(\eta,\xi)\right) f(\eta)\,d\omega(\eta)-\Lambda_\xi^*\int_\Omega \left(\Lambda_\eta^*D_\xi^{-1}G(\Delta^*;\xi\cdot\eta)\right)\cdot f(\eta)\,d\omega(\eta)\right|&=&\mathcal{O}(\rho^{\frac{1}{2}}),\end{aligned}$$ for $F$ of class $C^{(1)}(\Omega)$ and $f$ of class $c^{(1)}(\Omega)$. The conditions on the regularization function are satisfied, e.g., by the choice of $R^\rho$ as the truncated Taylor series of $S$.
Spherical Decompositions {#sec:decomps}
========================
We introduce two decompositions relating to the two sets of vector spherical harmonics from Section \[sec:basics\]. The first one, relating to $y^{(i)}_{n,k}$ and the operators $o^{(i)}$, respectively, is the well-known spherical Helmholtz decomposition. It decomposes a vector field into its radial part and two tangential parts. The second one, relating to $\tilde{y}_{n,k}^{(i)}$ and the operators $\tilde{o}^{(i)}$, respectively, is crucial for the separation with respect to the sources and is presented in some detail in this section. A more general overview on similar spherical decompositions can be found, e.g., in [@gerhards10].
\[thm:helmholtzglob\] Let $f$ be of class $c^{(1)}(\Omega)$. Then there exist uniquely defined scalar fields $F_1$ of class $C^{(1)}(\Omega)$ and $F_2,F_3$ of class $C^{(2)}(\Omega)$ satisfying $$\frac{1}{4\pi}\int_{\Omega}F_i(\eta)d\omega(\eta)=0,\quad i=2,3,$$ such that $$\begin{aligned}
f(\xi)&=&o^{(1)}_\xi F_1(\xi)+o^{(2)}_\xi F_2(\xi)+o^{(3)}_\xi F_3(\xi),\quad \xi\in\Omega.\end{aligned}$$
A proof of this decomposition can be found, e.g., in [@back96]. Using Green’s function for the Beltrami operator and Theorem \[thm:fundthmgradcurl\] yields the following representations for the Helmholtz scalars, $$\begin{aligned}
F_1(\xi)&=&\xi\cdot f(\xi),\quad\xi\in\Omega,\label{eqn:h1}
\\F_2(\xi)&=&-\int_\Omega\big(\nabla_\eta^*G(\Delta^*;\xi\cdot\eta)\big)\cdot f(\eta)\,d\omega(\eta),\quad\xi\in\Omega,\label{eqn:hemlf2rep}
\\F_3(\xi)&=&-\int_\Omega\big(L_\eta^*G(\Delta^*;\xi\cdot\eta)\big)\cdot f(\eta)\,d\omega(\eta),\quad\xi\in\Omega.\label{eqn:hemlf3rep}\end{aligned}$$
If $F_1$ additionally has vanishing integral mean value, i.e., $\int_\Omega F_1(\eta)d\omega(\eta)=0$ (as is the case for functions satisfying the pre-Maxwell equations), there exists a function $U$ of class $C^{(2)}(\Omega)$ with $\Delta^*U=F_1$, such that Theorem \[thm:fundthmdelta\] implies $$\begin{aligned}
F_1(\xi)&=&\Delta_\xi^*\int_\Omega G(\Delta^*;\xi\cdot\eta)\eta\cdot f(\eta)d\omega(\eta),\quad\xi\in\Omega.\label{eqn:h4}\end{aligned}$$
While the orthogonality is the main property of the Helmholtz decomposition, a representation with respect to the operators $\tilde{o}^{(i)}$ is of special interest in geomagnetic modeling. In order to obtain a representation of the corresponding scalars, we rewrite (\[eqn:to1operator\])–(\[eqn:to3operator\]) as $$\begin{aligned}
o^{(1)}&=&\textnormal{\footnotesize $\frac{1}{2}$}\tilde{o}^{(1)} D^{-1}+\textnormal{\footnotesize $\frac{1}{2}$}\tilde{o}^{(2)} D^{-1},\label{eqn:tto1operator}
\\o^{(2)}&=&\textnormal{\footnotesize $\frac{1}{2}$}\tilde{o}^{(1)} \left(\textnormal{\footnotesize $\frac{1}{2}$}D^{-1}-1\right)+\textnormal{\footnotesize $\frac{1}{2}$}\tilde{o}^{(2)} \left(\textnormal{\footnotesize $\frac{1}{2}$}D^{-1}+1\right),\label{eqn:tto2operator}
\\o^{(3)}&=&\tilde{o}^{(3)}.\label{eqn:tto3operator}\end{aligned}$$ This gives us the necessary representation to prove the following decomposition theorem.
\[thm:althelmholtz\] Let $f$ be of class $c^{(1)}(\Omega)$. Then there exist uniquely defined scalar fields $\tilde{F}_1,\tilde{F}_2$ of class $C^{(1)}(\Omega)$ and $\tilde{F}_3$ of class $C^{(2)}(\Omega)$ satisfying $$\begin{aligned}
\frac{1}{4\pi}\int_{\Omega}\tilde{F}_3(\eta)d\omega(\eta)&=&0,
\\\frac{1}{4\pi}\int_{\Omega}\tilde{F}_1(\eta)-\tilde{F}_2(\eta)d\omega(\eta)&=&0,\end{aligned}$$ such that $$\begin{aligned}
f(\xi)&=&\tilde{o}^{(1)}_\xi \tilde{F}_1(\xi)+\tilde{o}^{(2)}_\xi \tilde{F}_2(\xi)+\tilde{o}^{(3)}_\xi \tilde{F}_3(\xi),\quad \xi\in\Omega.\end{aligned}$$ The scalars $\tilde{F}_1,\tilde{F}_2$ and $\tilde{F}_3$ can be represented by $$\begin{aligned}
\tilde{F}_1&=&\textnormal{\footnotesize $\frac{1}{2}$}D^{-1}F_1+\textnormal{\footnotesize $\frac{1}{4}$}D^{-1}F_2-\textnormal{\footnotesize $\frac{1}{2}$}F_2,\label{eqn:tf1}
\\\tilde{F}_2&=&\textnormal{\footnotesize $\frac{1}{2}$}D^{-1}F_1+\textnormal{\footnotesize $\frac{1}{4}$}D^{-1}F_2+\textnormal{\footnotesize $\frac{1}{2}$}F_2,\label{eqn:tf2}
\\\tilde{F}_3&=&F_3,\label{eqn:tf3}\end{aligned}$$ with $F_1,F_2$ and $F_3$ being the uniquely determined functions of the Helmholtz decomposition in Theorem \[thm:helmholtzglob\].
Applying the Helmholtz decomposition to $f$ and using (\[eqn:tto1operator\])–(\[eqn:tto3operator\]), we get on $\Omega$, $$\begin{aligned}
f&=&o^{(1)}F_1+o^{(2)}F_2+o^{(3)}F_3
\\&=&\frac{1}{2}\tilde{o}^{(1)}D^{-1}F_1+\frac{1}{2}\tilde{o}^{(2)}D^{-1}F_1+\frac{1}{2}\tilde{o}^{(1)}\left(\frac{1}{2}D^{-1}-1\right)F_2+\frac{1}{2}\tilde{o}^{(2)}\left(\frac{1}{2}D^{-1}+1\right)F_2+\tilde{o}^{(3)}F_3
\\&=&\tilde{o}^{(1)}\left(\frac{1}{2}D^{-1}F_1+\frac{1}{4}D^{-1}F_2-\frac{1}{2}F_2\right)+\tilde{o}^{(2)}\left(\frac{1}{2}D^{-1}F_1+\frac{1}{4}D^{-1}F_2+\frac{1}{2}F_2\right)+\tilde{o}^{(3)}F_3.\end{aligned}$$ This implies a decomposition as stated in the theorem. Due to the uniqueness of the Helmholtz representation, it follows directly that $\tilde{F}_3$ is defined uniquely when having a vanishing integral mean value. For the uniqueness of $\tilde{F}_1$ and $\tilde{F}_2$ it is sufficient to show that $f(\xi)=0$, $\xi\in\Omega$, only has the trivial decomposition with respect to the operators $\tilde{o}^{(i)}$, $i=1,2,3$. If $$\tilde{o}^{(1)}_\xi\tilde{F}_1(\xi)+\tilde{o}^{(2)}_\xi\tilde{F}_2(\xi)+\tilde{o}^{(3)}_\xi\tilde{F}_3(\xi)=0, \quad\xi\in\Omega,$$ we get from (\[eqn:to1operator\])–(\[eqn:to3operator\]) that $$o^{(1)}_\xi\left(\left(D_\xi+\textnormal{\footnotesize $\frac{1}{2}$}\right)\tilde{F}_1(\xi)+\left(D_\xi-\textnormal{\footnotesize $\frac{1}{2}$}\right)\tilde{F}_2(\xi)\right)+o^{(2)}_\xi\left(\tilde{F}_2(\xi)-\tilde{F}_1(\xi)\right)+o^{(3)}_\xi\tilde{F}_3(\xi)=0,\quad\xi\in\Omega.$$ The uniqueness of the Helmholtz decomposition then implies $$\begin{aligned}
\tilde{F}_2(\xi)-\tilde{F}_1(\xi)&=&0,\quad\xi\in\Omega,
\\\left(D_\xi+\textnormal{\footnotesize $\frac{1}{2}$}\right)\tilde{F}_1(\xi)+\left(D_\xi-\textnormal{\footnotesize $\frac{1}{2}$}\right)\tilde{F}_2(\xi)&=&0,\quad \xi\in\Omega,\end{aligned}$$ if $\frac{1}{4\pi}\int_{\Omega}\tilde{F}_1(\eta)-\tilde{F}_2(\eta)d\omega(\eta)=0$, which gives us $$D_\xi\tilde{F}_1(\xi)=0,\quad\xi\in\Omega.$$ Thus, $\tilde{F}_1(\xi)=0$, $\xi\in\Omega$, since $D$ is injective, and it follows $\tilde{F}_2(\xi)=0$, $\xi\in\Omega$, so that uniqueness is given for this decomposition.
The theorem above yields, by use of (\[eqn:h1\])–(\[eqn:hemlf3rep\]), a representation of the scalars $\tilde{F}_i$. Of importance in the later application, however, are the vectorial quantities $\tilde{o}^{(i)}\tilde{F}_i$. Thus, we first calculate from (\[eqn:tf1\]) that $$\begin{aligned}
\tilde{o}^{(1)}\tilde{F}_1(\xi)&=&\frac{1}{2}\xi(\xi\cdot f(\xi))+\frac{1}{4}\xi D_\xi^{-1}(\xi\cdot f(\xi))
-\frac{1}{8}\xi D_\xi^{-1}\int_\Omega\nabla_\eta^*G(\Delta^*;\xi\cdot\eta)\cdot f(\eta)d\omega(\eta)\nonumber
\\&&+\frac{1}{2}\xi D_\xi\int_\Omega\nabla_\eta^*G(\Delta^*;\xi\cdot\eta)\cdot f(\eta)d\omega(\eta)\label{eqn:tf1d}
\\&&-\frac{1}{2}\nabla_\xi^*D_\xi^{-1}(\xi\cdot f(\xi)) +\frac{1}{4}\nabla_\xi^*D_\xi^{-1}\int_\Omega\nabla_\eta^*G(\Delta^*;\xi\cdot\eta)\cdot f(\eta)d\omega(\eta)\nonumber
\\&&-\frac{1}{2}\nabla_\xi^*\int_\Omega\nabla_\eta^*G(\Delta^*;\xi\cdot\eta)\cdot f(\eta)d\omega(\eta),\qquad\xi\in\Omega.\nonumber\end{aligned}$$ The expression in the second row, involving the operator $D$, is unfortunate since we have no explicit representation for the corresponding regularized convolution kernel. Observing $$D=D^{-1}\Big(-\Delta^*+\frac{1}{4}\Big),$$ this can be circumvented by rewriting $$\begin{aligned}
&&\!\!\!\!\!\!\!\!\!\!\!\!\!\!D_\xi\int_\Omega \nabla_\eta^*G(\Delta^*;\xi\cdot\eta) \cdot f(\eta)d\omega(\eta)
\\&=&\frac{1}{4}D_\xi^{-1}\int_\Omega\nabla_\eta^* G(\Delta^*;\xi\cdot\eta) \cdot f(\eta)d\omega(\eta)+D_\xi^{-1}\Delta_\xi^*\int_\Omega G(\Delta^*;\xi\cdot\eta) \nabla_\eta^*\cdot f_{tan}(\eta)d\omega(\eta)
\\&=&\frac{1}{4}D_\xi^{-1}\int_\Omega\nabla_\eta^* G(\Delta^*;\xi\cdot\eta) \cdot f(\eta)d\omega(\eta)+D_\xi^{-1}\nabla_\xi^*\cdot f_{tan}(\xi),\qquad\xi\in\Omega,\end{aligned}$$ where Theorem \[thm:fundthmdelta\] has been used in the last step, assuming $f$ to be of class $c^{(2)}(\Omega)$. Application of the above to (\[eqn:tf1d\]) provides an easier calculable expression $$\begin{aligned}
\tilde{o}^{(1)}\tilde{F}_1(\xi)&=&\frac{1}{2}\xi(\xi\cdot f(\xi))+\frac{1}{4}\xi D_\xi^{-1}(\xi\cdot f(\xi))+\frac{1}{2}\xi D_\xi^{-1}\nabla_\xi^*\cdot f_{tan}(\xi)-\frac{1}{2}\nabla_\xi^*D_\xi^{-1}(\xi\cdot f(\xi))
\\&&+\frac{1}{4}\nabla_\xi^*\int_\Omega\nabla_\eta^*D_\xi^{-1}G(\Delta^*;\xi\cdot\eta)\cdot f(\eta)d\omega(\eta)-\frac{1}{2}\nabla_\xi^*\int_\Omega\nabla_\eta^*G(\Delta^*;\xi\cdot\eta)\cdot f(\eta)d\omega(\eta).\end{aligned}$$ Since the occurring differential operators and the integration cannot be interchanged without restriction, we need to switch to the regularized versions of the single layer kernel and the Green function for the Beltrami operator. Then it is valid to set $$\begin{aligned}
\tilde{f}^{(1)}_\rho(\xi)&=&\frac{1}{2}\xi(\xi\cdot f(\xi))+\frac{1}{8\pi}\xi \int_\Omega S^\rho(\xi\cdot\eta) \,\eta\cdot f(\eta)d\omega(\eta)-\frac{1}{4\pi}\xi \int_\Omega \nabla_\eta^*S^\rho(\xi\cdot\eta)\cdot f(\eta)d\omega(\eta)\nonumber
\\&&-\frac{1}{4\pi}\int_\Omega \nabla_\xi^*S^\rho(\xi\cdot\eta)\,\eta\cdot f(\eta)d\omega(\eta)+\frac{1}{4}\int_\Omega\left(\nabla_\xi^*\otimes s_{\nabla^*}^\rho(\eta,\xi)\right) f(\eta)d\omega(\eta)\label{eqn:tf1approx}
\\&&-\frac{1}{2}\int_\Omega\left(\nabla_\xi^*\otimes\nabla_\eta^*G^\rho(\Delta^*;\xi\cdot\eta)\right) f(\eta)d\omega(\eta),\qquad\xi\in\Omega,\nonumber\end{aligned}$$ for $\rho>0$. If $G^\rho(\Delta^*;\cdot)$ is of class $C^{(2)}([-1,1])$ and $S^\rho$ of class $C^{(1)}([-1,1])$, the considerations in Section \[sec:kernels\] imply $$\begin{aligned}
\lim_{\rho\to0+}\sup_{\xi\in\Omega}\left|\tilde{f}^{(1)}_\rho(\xi)-\tilde{o}^{(1)}\tilde{F}_1(\xi)\right|=0.\label{eqn:limittf1rho}\end{aligned}$$
Analogous computations for $\tilde{o}^{(2)}\tilde{F}_2$ yield a regularization $$\begin{aligned}
\tilde{f}^{(2)}_\rho(\xi)&=&\frac{1}{2}\xi(\xi\cdot f(\xi))-\frac{1}{8\pi}\xi \int_\Omega S^\rho(\xi\cdot\eta) \,\eta\cdot f(\eta)d\omega(\eta)+\frac{1}{4\pi}\xi \int_\Omega \nabla_\eta^*S^\rho(\xi\cdot\eta)\cdot f(\eta)d\omega(\eta)\nonumber
\\&&+\frac{1}{4\pi}\int_\Omega \nabla_\xi^*S^\rho(\xi\cdot\eta)\,\eta\cdot f(\eta)d\omega(\eta)-\frac{1}{4}\int_\Omega\left(\nabla_\xi^*\otimes s_{\nabla^*}^\rho(\eta,\xi)\right) f(\eta)d\omega(\eta)\label{eqn:tf2approx}
\\&&-\frac{1}{2}\int_\Omega\left(\nabla_\xi^*\otimes\nabla_\eta^*G^\rho(\Delta^*;\xi\cdot\eta)\right) f(\eta)d\omega(\eta),\qquad\xi\in\Omega,\nonumber\end{aligned}$$ and $$\begin{aligned}
\lim_{\rho\to0+}\sup_{\xi\in\Omega}\left|\tilde{f}^{(2)}_\rho(\xi)-\tilde{o}^{(2)}\tilde{F}_2(\xi)\right|=0.\label{eqn:limittf2rho}\end{aligned}$$ Finally, the determination of $\tilde{o}^{(3)}\tilde{F}_3$ corresponds to the calculation of the toroidal part of $f$. From (\[eqn:tf3\]) and (\[eqn:hemlf3rep\]), we get $$\tilde{o}^{(3)}\tilde{F}_3(\xi)=-L_\xi^*\int_\Omega L_\eta^*G(\Delta^*;\xi\cdot\eta)\cdot f(\eta)d\omega(\eta),\quad\xi\in\Omega.$$ Corollary \[cor:convtensbeltramig\] then implies for the regularized version $$\begin{aligned}
\tilde{f}^{(3)}_\rho(\xi)=-\int_\Omega\left(L_\xi^*\otimes L_\eta^*G^\rho(\Delta^*;\xi\cdot\eta)\right)f(r\eta)\,d\omega(\eta),\quad\xi\in\Omega,\label{eqn:tf3approx}\end{aligned}$$ that for $G^\rho(\Delta^*;\cdot)$ of class $C^{(2)}([-1,1])$, $$\lim_{\rho\to0+}\sup_{\xi\in\Omega}\left|\tilde{f}^{(3)}_\rho(\xi)-\tilde{o}^{(3)}\tilde{F}_3(\xi)\right|=0.\label{eqn:limittf3rho}$$ Summarizing, we can state the following theorem.
\[lem:althelmapprox\] Let $f$ be of class $c^{(2)}(\Omega)$, with $\int_\Omega\eta\cdot f(\eta)d\omega(\eta)=0$. Furthermore, let the regularized Green function $G^\rho(\Delta^*;\cdot)$ be of class $C^{(2)}([-1,1])$ and the single layer kernel $S^\rho$ of class $C^{(1)}([-1,1])$. Then $$\begin{aligned}
f(\xi)=\tilde{o}^{(1)}\tilde{F}_1(\xi)+\tilde{o}^{(2)}\tilde{F}_2(\xi)+\tilde{o}^{(3)}\tilde{F}_3(\xi),\quad\xi\in\Omega,\end{aligned}$$ with $$\lim_{\rho\to0+}\sup_{\xi\in\Omega}\left|\int_\Omega\boldsymbol{\Phi}_\rho^{(i)}(\xi,\eta)f(\eta)d\omega(\eta)-\tilde{o}^{(i)}\tilde{F}_i(\xi)\right|=0,$$ for $i=1,2,3$. The convolution kernels are given by $$\begin{aligned}
\boldsymbol{\Phi}_\rho^{(1)}(\xi,\eta)&=&\xi\otimes\eta\left(\frac{1}{2}\Delta^*_\xi G^{\rho}(\Delta^*;\xi\cdot\eta)+\frac{1}{8\pi}S^{\rho}(\xi\cdot\eta)\right)-\frac{1}{4\pi}\xi\otimes\nabla_\eta^*S^{\rho}(\xi\cdot\eta)
\\&&+\frac{1}{4}\nabla_\xi^*\otimes s_{\nabla^*}^{\rho}(\eta,\xi)-\frac{1}{4\pi}\nabla_\xi^*S^{\rho}(\xi,\eta)\otimes\eta-\frac{1}{2}\nabla_\xi^*\otimes\nabla_\eta^*G^{\rho}(\Delta^*;\xi\cdot\eta),\quad \xi,\eta\in\Omega,
\\\boldsymbol{\Phi}_\rho^{(2)}(\xi,\eta)&=&\xi\otimes\eta\left(\frac{1}{2}\Delta^*_\xi G^{\rho}(\Delta^*;\xi\cdot\eta)-\frac{1}{8\pi}S^{\rho}(\xi\cdot\eta)\right)+\frac{1}{4\pi}\xi\otimes\nabla_\eta^*S^{\rho}(\xi\cdot\eta)
\\&&-\frac{1}{4}\nabla_\xi^*\otimes s_{\nabla^*}^{\rho}(\eta,\xi)+\frac{1}{4\pi}\nabla_\xi^*S^{\rho}(\xi,\eta)\otimes\eta-\frac{1}{2}\nabla_\xi^*\otimes\nabla_\eta^*G^{\rho}(\Delta^*;\xi\cdot\eta),\quad \xi,\eta\in\Omega,
\\[1.25ex]\boldsymbol{\Phi}_\rho^{(3)}(\xi,\eta)&=&-L_\xi^*\otimes L_\eta^*G^{\rho}(\Delta^*;\xi\cdot\eta),\quad \xi,\eta\in\Omega.\end{aligned}$$
Since $\int_\Omega\eta\cdot f(\eta)d\omega(\eta)=\int_\Omega F_1(\eta)d\omega(\eta)=0$, representation (\[eqn:h4\]) implies $$\begin{aligned}
\frac{1}{2}\xi(\xi\cdot f(\xi))&=&\frac{1}{2}\xi\Delta_\xi^*\int_\Omega G(\Delta^*;\xi\cdot\eta)\eta\cdot f(\eta),\quad\xi\in\Omega.\end{aligned}$$ Substituting the Green function by its regularized counterpart, we obtain $$\begin{aligned}
\frac{1}{2}\xi\Delta_\xi^*\int_\Omega G^\rho(\Delta^*;\xi\cdot\eta)\eta\cdot f(\eta)\,d\omega(\eta)&=&\frac{1}{2}\xi\int_\Omega \Delta_\xi^*G^\rho(\Delta^*;\xi\cdot\eta)\eta\cdot f(\eta)\,d\omega(\eta)
\\&=&\int_\Omega \left(\xi\otimes\eta\left(\frac{1}{2}\Delta_\xi^*G^\rho(\Delta^*;\xi\cdot\eta)\right)\right) f(\eta)\,d\omega(\eta),\quad\xi\in\Omega.\end{aligned}$$ Analogously, the remaining integral expressions in (\[eqn:tf1approx\]) can be written in terms of convolutions with tensorial kernels if this is not already the case. This yields the kernel $\boldsymbol{\Phi}_\rho^{(1)}(\cdot,\cdot)$ for $\tilde{f}_\rho^{(1)}$. Corollary \[cor:convbeltramig\] and (\[eqn:limittf1rho\]) provide the desired limit relation for $\tilde{o}^{(1)}\tilde{F}_1$. The same holds true for $\tilde{o}^{(2)}\tilde{F}_2$ and $\tilde{o}^{(3)}\tilde{F}_3$.
Multiscale Representation for the Separation of Sources {#sec:multiscale}
=======================================================
The kernels from Theorem \[lem:althelmapprox\] are the main ingredient to the upcoming multiscale representation. They actually denote the so-called scaling kernels, while the differences for different parameters $\rho$ denote the corresponding wavelet kernels. But before we go into detail, we briefly want to motivate why the decomposition with respect to the operators $\tilde{o}^{(i)}$ can be called a separation with respect to the sources.
From now on, we denote the magnetic field by $b$ of class $c^{(2)}(\mathbb{R}^3)$, the corresponding source current density by $j$ of class $c^{(1)}(\mathbb{R}^3)$, and by $\mu_0$ we mean the vacuum permeability. Furthermore, we assume the pre-Maxwell equations $$\begin{aligned}
\nabla_x\wedge b(x)&=&\mu_0j(x),\quad x\in\mathbb{R}^3,
\\\nabla_x\cdot b(x)&=&0,\quad x\in\mathbb{R}^3,\end{aligned}$$ to be satisfied. As mentioned in the introduction, the Mie decomposition (see, e.g., [@back86] and [@back96]) yields poloidal fields $p_b$, $p_j$ and toroidal fields $q_b$, $q_j$, such that $$\begin{aligned}
b(x)&=&p_b(x)+q_b(x),\quad x\in\mathbb{R}^3,
\\j(x)&=&p_j(x)+q_j(x),\quad x\in\mathbb{R}^3.\end{aligned}$$ Making use of the law of Biot-Savart (see, e.g., [@jackson75]) and the fact that the poloidal magnetic field $p_b$ is solely produced by tangential toroidal current densities $q_j$, the poloidal magnetic field can be split up as follows, $$\begin{aligned}
p_b(x)=p_b^{int}(R;x)+p_b^{ext}(R;x),\quad x\in\mathbb{R}^3\setminus\Omega_R,\end{aligned}$$ where $$\begin{aligned}
\nabla_x\wedge p_b^{int}(R;x)&=&\left\{\begin{array}{ll}\mu_0q_j(x),&x\in\Omega_R^{int},
\\0,&x\in\Omega_R^{ext},\end{array}\right.\label{eqn:curlpbint}
\\\nabla_x\wedge p_b^{ext}(R;x)&=&\left\{\begin{array}{ll}0,&x\in\Omega_R^{int},
\\\mu_0q_j(x),&x\in\Omega_R^{ext}.\end{array}\right.\label{eqn:curlpbext}\end{aligned}$$ In other words, $p_b^{int}(R;\cdot)$ denotes the part of the magnetic field that is due to source currents in the interior of the satellite’s orbit $\Omega_R$, and $p_b^{ext}(R;\cdot)$ the part due to source currents in the exterior. A more detailed description can be found, e.g., in [@back96] and [@may06]. Since $p_b^{int}(R;\cdot)$ is still divergence-free, equation (\[eqn:curlpbint\]) implies that a harmonic potential $U^{int}(R;\cdot):\Omega_R^{ext}\to\mathbb{R}$ exists, such that $p_b^{int}(R;x)=\nabla_xU^{int}(R;x)$, for $x\in\Omega_R^{ext}$. The potential $U^{int}$ can be expanded with respect to the outer harmonics $H_{n,k}^{ext}$, and the application of the gradient in combination with (\[eqn:hext\]) then implies that $p_b^{int}(R;\cdot)$ can be expanded in $\Omega_R^{ext}$ with respect to $\tilde{y}_{n,k}^{(1)}$. Analogously, $p_b^{ext}(R;\cdot)$ relates to an expansion in $\Omega_R^{int}$ with respect to $\tilde{y}_{n,k}^{(2)}$. The remaining toroidal part $q_b$ can be interpreted as the part induced by poloidal source currents $p_j$ crossing the sphere $\Omega_R$, and corresponds to the vector spherical harmonics $\tilde{y}_{n,k}^{(3)}$. To sum up, additionally observing that the poloidal and toroidal fields are continuous up to $\Omega_R$, we find $$\begin{aligned}
\label{eqn:intextpol2}
b(x)=p_b^{int}(R;x)+p_b^{ext}(R;x)+q_b(x),\quad x\in\Omega_R,\end{aligned}$$ with $\tilde{y}_{n,k}^{(i)}$, $i=1,2,3$, $n\in\mathbb{N}_{0_i}$, $k=1,\ldots,2n+1$, being the appropriate basis system for this split-up. Finally, the definition of the vector spherical harmonics in (\[eqn:ytilde\]) implies that Theorem \[thm:althelmholtz\] yields the exact same decomposition as (\[eqn:intextpol2\]). More precisely, the $\tilde{o}^{(1)}$-part denotes the contribution due to sources in $\Omega_R^{int}$, the $\tilde{o}^{(2)}$-part the contribution due to sources in $\Omega_R^{ext}$, and the $\tilde{o}^{(3)}$-part the contribution due to source currents crossing the sphere $\Omega_R$.
Multiscale Representation
-------------------------
Now, we turn to the actual multiscale representation. We discretize the regularized kernels from Theorem \[lem:althelmapprox\], by choosing parameters $\rho=2^{-J}$, for $J\in\mathbb{N}_{0}$. The *scaling kernels (of scale $J$)* are then defined by $$\begin{aligned}
\boldsymbol{\Phi}_J^{int}(\xi,\eta)&=&\boldsymbol{\Phi}_{2^{-J}}^{(1)}(\xi,\eta),\quad\xi,\eta\in\Omega,
\\\boldsymbol{\Phi}_J^{ext}(\xi,\eta)&=&\boldsymbol{\Phi}_{2^{-J}}^{(2)}(\xi,\eta),\quad\xi,\eta\in\Omega,
\\\boldsymbol{\Phi}_J^{q}(\xi,\eta)&=&\boldsymbol{\Phi}_{2^{-J}}^{(3)}(\xi,\eta),\quad\xi,\eta\in\Omega.\end{aligned}$$
These kernels still have global support. The announced locally supported wavelets are obtained by taking the difference of two such scaling kernels. A *wavelet kernel (of scale $J$)* denotes one of the following kernels $$\begin{aligned}
\boldsymbol{\Psi}_J^{int}(\xi,\eta)&=&\boldsymbol{\Phi}_{J+1}^{int}(\xi,\eta)-\boldsymbol{\Phi}_J^{int}(\xi,\eta),\quad\xi,\eta\in\Omega,
\\\boldsymbol{\Psi}_J^{ext}(\xi,\eta)&=&\boldsymbol{\Phi}_{J+1}^{ext}(\xi,\eta)-\boldsymbol{\Phi}_J^{ext}(\xi,\eta),\quad\xi,\eta\in\Omega,
\\\boldsymbol{\Psi}_J^{q}(\xi,\eta)&=&\boldsymbol{\Phi}_{J+1}^{q}(\xi,\eta)-\boldsymbol{\Phi}_J^{q}(\xi,\eta),\quad\xi,\eta\in\Omega.\end{aligned}$$ Due to the regularization of the Green function and the single layer kernel, these wavelets clearly have local support in a spherical cap of radius $2^{-J}$. More precisely, we find that $\textnormal{supp}\big( \boldsymbol{\Psi}_J^{i}(\xi,\cdot)\big)\subset\{\eta\in\Omega|1-\xi\cdot\eta<2^{-J}\}$, for $i\in\{int,ext,q\}$ and $\xi\in\Omega$. An illustration of the kernels is given in Figure \[fig:kernels1\].
Each scaling kernel generates a scaling transform. These are given by $$\begin{aligned}
P_J^{i}b(x)&=&\int_\Omega \boldsymbol{\Phi}^{i}_J(\xi,\eta) b(R\eta)d\omega(\eta),\quad x=R\xi\in\Omega_R.\label{eqn:scaltransformint}\end{aligned}$$ The corresponding wavelet transforms read $$\begin{aligned}
R_J^{i}b(x)&=&\int_\Omega \boldsymbol{\Psi}^{i}_J(\xi,\eta) b(R\eta)d\omega(\eta),\quad x=R\xi\in\Omega_R.\label{eqn:wavelettransformint}\end{aligned}$$ The idea of the multiscale approach is to resolve the modeled quantities at different spatial resolutions. Therefore, the scaling kernels are only used to provide a trend approximation of the coarse features at some small initial scale $J_0$. The spatially stronger localized features are subsequently added by the wavelet transforms. This is reflected in the following relations, $$\begin{aligned}
P_J^{i}b(x)&=&P_{J_0}^{i}b(x)+\sum_{j=J_0}^{J-1}R_j^{i}b(x),\quad x=R\xi\in\Omega_R,\end{aligned}$$ for $i\in\{int,ext,q\}$. Different from the multiresolution constructed, e.g., in [@may06] and [@maymai], the scale spaces $V_J^i=\{P_J^ib|b\in c^{(2)}(\Omega_R)\}$ in our approach are not necessarily nested in the sense $V_J^i\subset V_{J+1}^i$. The advantage here is the local support of the wavelet kernels, which implies that the evaluation of the wavelet transforms $R_j^{i}b(x)$, for $x=R\xi\in\Omega_R$, only requires data in a spherical cap around $\xi\in\Omega$ with scale-dependent spherical radius $2^{-j}$. Thus, regions with a higher data density can be resolved up to higher scales, i.e., up to a higher spatial resolution, without suffering errors from the lower data densities in surrounding areas. The general concept is illustrated by the following tree algorithm
0.9mm
(150,20) (0,0)[(0,0)\[c\][$P^i_{J_0}F$]{}]{} (20,0)[(0,0)\[c\][$+$]{}]{} (40,0)[(0,0)\[c\][$P^i_{J_0+1}F$]{}]{} (62,0)[(0,0)\[c\][$+$]{}]{} (82,0)[(0,0)\[c\][$P^i_{J_0+2}F$]{}]{} (104,0)[(0,0)\[c\][$+$]{}]{} (123,0)[(0,0)\[c\][$\ldots$]{}]{} (0,15)[(0,0)\[c\][$R^i_{J_0}F$]{}]{} (40,15)[(0,0)\[c\][$R^i_{J_0+1}F$]{}]{} (82,15)[(0,0)\[c\][$R^i_{J_0+2}F$]{}]{} (5,0)[(1,0)[12]{}]{} (47,0)[(1,0)[12]{}]{} (89,0)[(1,0)[12]{}]{} (23,0)[(1,0)[10]{}]{} (65,0)[(1,0)[10]{}]{} (107,0)[(1,0)[10]{}]{} (5,13)[(1,-1)[12]{}]{} (47,13)[(1,-1)[12]{}]{} (89,13)[(1,-1)[12]{}]{} (128,0)[(1,0)[10]{}]{} (146,0)[(0,0)\[c\][$P^i_{J_{max}}F$.]{}]{}
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The maximal scale $J=J_{max}$ at which $P_Jb(x)$ can be evaluated is determined by the amount of data points in the vicinity of $x=R\xi\in\Omega_R$. Sufficiently many data points in the support of $\boldsymbol{\Psi}^{i}_{J_{max}-1}(\xi,\cdot)$ are required to guarantee a numerical meaningful evaluation of the integral in the wavelet transform $R_{J_{max}-1}^{i}b(x)$.
To conclude this subsection, we summarize the results in the following theorem, which is mainly a reformulation of Theorem \[lem:althelmapprox\] in terms of the above described multiscale setting for the separation of the magnetic field with respect to its sources.
\[thm:intpart\] Let $b$ be of class $c^{(2)}(\mathbb{R}^3)$, and $j$ of class $c^{(1)}(\mathbb{R}^3)$, satisfying the pre-Maxwell equations $$\begin{aligned}
\nabla_x\wedge b(x)&=&\mu_0j(x),\quad x\in\mathbb{R}^3,
\\\nabla_x\cdot b(x)&=&0,\quad x\in\mathbb{R}^3.\end{aligned}$$ If $P_J^{int}$, $P_J^{ext}$, $P_J^{q}$, $R_J^{int}$, $R_J^{ext}$, $R_J^{q}$ are defined as in (\[eqn:scaltransformint\]), (\[eqn:wavelettransformint\]), then $$\begin{aligned}
b(x)&=&p_b^{int}(R;x)+p_b^{ext}(R;x)+q_b(x),\quad x\in\Omega_R,\end{aligned}$$ for a fixed $R>0$, with $$\begin{aligned}
p_b^{int}(R;x)&=&P_{J_0}^{int}b(x)+\sum_{j=J_0}^{\infty}R_j^{int}b(x),\quad x\in\Omega_R,
\\p_b^{ext}(R;x)&=&P_{J_0}^{ext}b(x)+\sum_{j=J_0}^{\infty}R_j^{ext}b(x),\quad x\in\Omega_R,
\\q_b(x)&=&P_{J_0}^{q}b(x)+\sum_{j=J_0}^{\infty}R_j^{q}b(x),\quad x\in\Omega_R.\end{aligned}$$
Crustal Field Modeling from CHAMP Data
--------------------------------------
In this subsection, we apply the above derived multiscale approach to a set of CHAMP satellite measurements. The used data set is similar to the one used in [@may06] and [@maymai] and has been collected between June 2001 and December 2001. It has been pre-processed at the GFZ Potsdam by Stefan Maus to fit the purpose of crustal field modeling (see, e.g., [@maus06] for a detailed description). Due to the almost spherical orbit of the CHAMP satellite, we can assume all data to be given on a sphere of radius $R_E+450$km, where $R_E=6371.2$km denotes the mean Earth radius. For the discretization of the integrals appearing in the multiscale representation, we use the integration rule described in [@drihea], which requires an equiangular data grid (and reflects the data situation of satellite measurements). In this example, we use a grid with $180$ grid points in latitudinal as well as longitudinal direction. Centered around each grid point, we select a spherical rectangle with a diameter of $2.5^\circ$ in latitude and longitude and average all measurements in the cell using a M-estimation with Huber’s weight function (see, e.g., [@hogg]). The resulting input data set is shown in Figure \[fig:binput\].
The results obtained from the multiscale representation are illustrated in Figures \[fig:brmultiscal\]–\[fig:bs9diff\]. For the sake of brevity, we only indicate the radial and the south-north component of the magnetic field, and in Figure \[fig:brmultiscal\] only the radial component. Figure \[fig:brmultiscal\] also illustrates best the different spatial resolutions of the multiscale representation.
The initial trend approximation at scale $J_0=2$ only resolves very coarse features, while the subsequent wavelet transforms resolve more and more localized features, such that the scales $J=5,6,7$ mainly focus on the strongest crustal field anomalies over Central Africa and Eastern Europe, as well as North America and Australia. In oceanic regions, there is hardly any contribution at these scales, indicating that there the crustal field is of a rather coarse nature (i.e., of large wavelength when arguing in frequency domain). Furthermore, one finds that the structure of the wavelet contributions hardly changes for scales higher than $J=5$. This might be an indicator of the general spatial extend of the anomalies of the crustal field signal at satellite altitude (when comparing the resolved features with the size of the support of the wavelet kernels). Figure \[fig:bs9\] shows the final approximation $P_{J_{max}}^{int}b$ of the internal magnetic field contributions at the highest scale $J_{max}=9$. The difference to the input data set is indicated in Figure \[fig:bs9diff\]. It actually illustrates the performance of the separation with respect to the sources. One can recognize strong polar fields that are clearly not due to the Earth’s crustal field and are probably induced by polar ionospheric current systems. Furthermore, one finds bands of positive and negative field strength oriented parallel to the dipole equator. This is a typical signature of magnetospheric ring currents.
Thus, the multiscale approach of this paper can be used to improve pre-processed crustal magnetic field data. The decomposition with respect to $\tilde{o}^{(i)}$ is actually able to filter out contributions originating outside the satellite’s orbit or at satellite altitude, while the wavelet transformations give a clearer impression of the local features of the crustal field.
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|
---
abstract: 'The order of letters is not always relevant in a communication task. This paper discusses the implications of order irrelevance on source coding, presenting results in several major branches of source coding theory: lossless coding, universal lossless coding, rate-distortion, high-rate quantization, and universal lossy coding. The main conclusions demonstrate that there is a significant rate savings when order is irrelevant. In particular, lossless coding of $n$ letters from a finite alphabet requires $\Theta(\log n)$ bits and universal lossless coding requires $n + o(n)$ bits for many countable alphabet sources. However, there are no universal schemes that can drive a strong redundancy measure to zero. Results for lossy coding include distribution-free expressions for the rate savings from order irrelevance in various high-rate quantization schemes. Rate-distortion bounds are given, and it is shown that the analogue of the Shannon lower bound is loose at all finite rates.'
author:
- 'Lav R. Varshney, and Vivek K Goyal, [^1][^2][^3]'
bibliography:
- 'abrv.bib'
- 'lrv\_lib.bib'
title: |
Benefiting from Disorder:\
Source Coding for Unordered Data
---
compression, lossless coding, multisets, order statistics, quantization, rate-distortion theory, universal coding, types
“ceiiinosssttuv” *— Robert Hooke in 1676, establishing priority for the statement*\
*“ut tensio sic vis” (as is the extension, so is the force) published in 1678 [@Moyer1977].*
Introduction
============
Are there situations where [claude shannon]{} is no different than [a sound channel]{}; where [maximum entropy]{} might be considered a reasonable reconstruction for [momentary mixup]{}?[^4] If one is interested in communicating a textual source, an anagram is not a sufficient representation. If however, one is simply interested in classifying the language, an anagram may be sufficient because it gives the same first-order language approximation.[^5] That is to say, the values of the letters (a multiset) may be important when the order of the letters (a permutation) is not. The order of source letters is irrelevant in a multitude of scenarios beyond such language representation and related texture representation applications [@EfrosL1999]. Examples include warehouse inventories [@Lempel1986]; records in scientific or financial databases [@BuchsbaumFG2003]; collections of multimedia files; arrival processes [@Rubin1974]; visual languages that have multiset grammars [@MarriottM1998]; data in a parallel computing paradigm [@BanatreM1993]; and the channel state in chemical channels [@PermuterCVW2006]. Moreover, it has been suggested that when humans use data for recognition or recall tasks [@JuddS1959], or for judgments of coincidences [@GriffithsT2001; @GriffithsT2007], the order of symbols is not relevant.
If a sample consists of independent observations from the same distribution, then associated minimum variance unbiased estimators are symmetric in the observations [@Rao1965]. Therefore when coding for estimation, the multiset of observations is all that need be represented, cf. [@HanA1998]. Moving beyond the point-to-point case, in distributed inference, often particle-based [@ArulampalamMGC2002; @IhlerFMW2005] and kernel-based [@Parzen1962; @IhlerFW2004] representations of densities must be communicated. As $$p(x) = \sum_{i} \phi\left(x - x_i\right)
= \sum_{i} \phi\left(x - x_{\pi(i)}\right)$$ for any permutation $\pi(\cdot)$, the multiset of representation coefficients $\{x_i\}$ may be communicated rather than the sequence of these values $(x_i)$. This extends to any destination that performs permutation-invariant computations.
The aim of this paper is to develop ramifications of order irrelevance on source coding problems. We consider lossless coding, universal lossless coding, high-rate and low-rate quantization, and rate distortion theory. In all of these, the rate requirement is obviously reduced by making order irrelevant. The reduction can be dramatic: for lossless coding of $n$ symbols, the required rate is changed from $O(n)$ to $O(\log n)$; for lossy coding, with large enough $n$ arbitrarily small mean squared error (MSE) can be achieved with zero rate. These examples are made precise in the sequel.
Notation and Formalism
----------------------
We consider the encoding of multisets and sequences of letters of size $n$ drawn from an alphabet $\mathcal{X}$. When $\mathcal{X}$ is discrete, we take it to be the (possibly-infinite) set $\{1,\,2,\,\ldots,|\mathcal{X}|\}$ without loss of generality. In addition to standard uses of parentheses, we will also use parentheses and braces to distinguish between sequences and multisets. For the ordered sequence $X_1,\,X_2,\,\ldots,\,X_n$ that is often denoted $X_1^n$ in the information theory literature, we write $(X_i)_{i=1}^n$. When these $n$ symbols are taken as an unordered multiset, we write $\{X_i\}_{i=1}^n$. The range limits are often omitted. We will refer to the distribution that describes $(X_i)_{i=1}^n$ as the parent distribution.
There are two perspectives that can be taken to relate (order-irrelevant) source coding of $\{X_i\}_{i=1}^n$ with (standard) source coding of $(X_i)_{i=1}^n$. In either case we take the sample space $\Omega$ to be the set of all sequences $\mathcal{X}^n$. Under the first perspective, which we take when discussing lossless coding, we define the event algebra, $\mathcal{F}$, based on permutation-invariant equivalence classes of sequences, rather than the sequences themselves. Since events are defined in terms of multisets, the source coding problem is formally no different than the standard one, though the results are interestingly different.
For lossy coding, we take an alternative perspective where the event algebra is based on sequences. Order irrelevance is incorporated by considering fidelity criteria with a permutation-invariance property. These fidelity criteria cannot be stated in single-letter terms, thus calling the mathematical tractability into question. However, like the non-single-letter fidelity criteria in [@BergerY1972; @Kieffer1978b], our fidelity criteria are tractable and also bear semantic significance on several applications.
All logarithms use base 2, and all rates are thus given in bits. In this paper, rates are generally not normalized by the number of symbols $n$. The reason for this departure from convention will soon become clear: the total number of bits required in some problems scales sublinearly with $n$. We use standard asymptotic notation such as $o(\cdot)$, $O(\cdot)$, $\Omega(\cdot)$ and $\Theta(\cdot)$ [@OrlitskySVZ2006].
Outline
-------
The remainder of the paper is organized as follows. In Section \[sec:ordervalue\], we propose the transformation of a sequence into an order and a multiset of values, and we show that the order and values are independent when letters are produced i.i.d. Sections \[sec:entropyrate\] and \[sec:universal\] address lossless coding. First we consider lossless coding with known distribution for both finite and countably-infinite alphabets in Section \[sec:entropyrate\]. Section \[sec:universal\] considers the universal setting and provides both a positive result (achievability of a rather low coding rate of 1 bit per letter) and a negative result (unachievability of negligible redundancy).
In Sections \[sec:lossy\] and \[sec:universalLossy\], we turn to lossy coding. In Section \[sec:RDzero\] it is shown that, for a large class of sources, a natural rate-distortion function is the trivial zero-zero point. This inspires restriction to finite-sized blocks in Sections \[sec:finitefinite\] and \[sec:finiteuncountable\]. These sections discuss the rate-distortion functions for the discrete- and uncountable-alphabet cases, respectively. Section \[sec:finiteuncountable\] also presents several high-rate quantization analyses. Universal lossy coding is considered in Section \[sec:universalLossy\]. Finally, Section \[sec:seq2set\] concludes the paper with a discussion of intermediates between full relevance and full irrelevance of order; additional connections to related work; and a summary of the main results.
Many results presented here appeared first in [@VarshneyG2006b; @VarshneyG2006; @Varshney2006; @VarshneyG2007].
Separating Order and Value {#sec:ordervalue}
==========================
Consider source variables $X_1,\,X_2,\,\ldots,\,X_n$ drawn from a common alphabet $\mathcal{X}$ according to any joint distribution. A realization $(x_i)_{i=1}^n$ can be decomposed into a multiset of values $\{x_i\}_{i=1}^n$ and an order $j$. This can be expressed as $$\label{eq:transform}
(x_1,\,x_2,\,\ldots,\,x_n)
=
(y_{i_1},\,y_{i_2},\,\ldots,\,y_{i_n})
\longrightarrow
\left( {\begin{array}{*{20}c}
{i_1 } & {i_2 } & \cdots & {i_n } \\
{x_1 } & {x_2 } & \cdots & {x_n } \\
\end{array}} \right) = \left( {\begin{array}{*{20}c}
{j} \\
{\left\{x_i\right\}_{i=1}^n} \\
\end{array}} \right)
\mbox{,}$$ where $(y_i)_{i=1}^n$ is $(x_i)_{i=1}^n$ put into a canonical order.[^6] The indices $i_1,\,i_2,\,\ldots,\,i_n$ are a permutation of the integers $1,\,2,\,\ldots,\,n$ and a deterministic function of $(x_i)_{i=1}^n$; when the $x_i$s are not distinct, we require any deterministic mechanism for choosing amongst the permutations such that (\[eq:transform\]) holds. The ordering is collapsed into a single variable $j$, which defines a chance variable $J$.
The decomposition into order and value can be interpreted as the generation of “transform coefficients.” Whether decomposing signals into low frequency and high frequency [@Dudley1939]; predictable and unpredictable [@Oliver1952]; style and content [@TenenbaumF2000]; object and texture [@YanS1977]; or dictionary and pattern [@OrlitskySZ2004], divide-and-conquer approaches have been used to good advantage in many source coding scenarios. Here, we are concerned with situations in which the order $J$ is irrelevant and hence allocated no bits. In contrast, allocating all the bits to $J$ yields permutation source codes [@BergerJW1972]. Other rate allocations are discussed in [@VarshneyG2006b].
If the joint distribution of $(X_i)_{i=1}^n$ is exchangeable, $J$ and $\left\{X_i\right\}_{i=1}^n$ are statistically independent chance variables [@JiTK1995]. Thus they could be coded separately without loss of optimality. This is expressed in information theoretic terms by the following theorem.
\[thm:Hdecomp\] For exchangeable sources, the order and the value are independent, and the sequence entropy $H((X_i)_{i=1}^n)$ can be decomposed into the value entropy $H(\{X_i\}_{i=1}^n)$ and the order entropy $H(J)$: $$\label{eq:Hdecomp}
H\left( (X_i)_{i=1}^n \right) = H\left( \{X_i\}_{i=1}^n \right) + H(J) \mbox{.}$$
Suppressing unnecessary subscripts, we can write $$\begin{aligned}
\label{eq:Hdecomp}
H((X)) & \stackrel{(a)}{=} H((X)) + H(\{X\}) - H((X)|J)
\ = \ H(\{X\}) + I((X) \; ; \; J) \notag \\
& = H(\{X\}) + H(J) - H(J|(X))
\ \stackrel{(b)}{=} \ H(\{X\}) + H(J) \mbox{.} \notag\end{aligned}$$ Step (a) follows from noting that $H(\{X\}) = H((X)|J)$ for exchangeable sources, since all orderings are equiprobable and uninformative about the value. Step (b) follows from the fact $H(J|(X)) = 0$, since the sequence determines the order. The other steps are simple informational manipulations.
When we disregard order, we are just left with a multiset. Type classes—also known variously as histograms or empirical distributions in statistics and as rearrangement classes or Abelian classes in combinatorics—are complete, minimal sufficient statistic for multisets. For discrete-alphabet sources, types are convenient mathematical representations for multisets; several results for these sources will depend on counting numbers of type classes.
Despite being sufficient statistics, types are not useful in the representation of continuous-alphabet sources. As Csiszár [@Csiszar1998] writes: “extensions of the type concept to continuous alphabets are not known.” In the case that $\mathcal{X} = \mathbb{R}$, we make extensive use of the basic distribution theory of order statistics. When the sequence of random variables $X_1,\ldots,X_n$ is arranged in ascending order as $X_{(1:n)} \le \cdots \le X_{(n:n)}$, $X_{(r:n)}$ is called the $r$th order statistic. It can be shown that the order statistics for exchangeable variates are complete, minimal sufficient statistics [@DavidN2003]. For alphabets of vectors of real numbers, there is no simple canonical form for expressing the minimal sufficient statistic since there is no natural ordering of vectors [@Barnett1976].
Lossless Coding {#sec:entropyrate}
===============
Consider the lossless coding of multisets of $n$ letters drawn from the discrete alphabet $\mathcal{X}$. Since there are $n!$ permutations of a sequence of length $n$, it would seem that a rate savings of $\log(n!)$ relative to sequence coding might be possible. Specifically, the upper bound $H(J) \leq \log(n!)$ combined with (\[eq:Hdecomp\]) gives the lower bound $$H(\{X_i\}_{i=1}^n) \ge H((X_i)_{i=1}^n) - \log n! \mbox{.}
\label{eq:orderlowerbound}$$ Since this lower bound can be negative, there must be more to the story.
The lower bound is not tight due to the positive chance of ties among members of a multiset drawn from a discrete parent. If the chance of ties is small (if $|\mathcal{X}|$ is sufficiently large and $n$ is sufficiently small), the lower bound is a good approximation.
For any given source distribution and $n$, a multiset of samples $\{x_i\}_{i=1}^n$ can be cast as a superletter drawn from an alphabet of multisets, itself having a known distribution. By the lossless block-to-variable source coding theorem [@Shannon1948], the entropy of the superletter is an asymptotically tight lower bound on the rate required for representation. Since the type specifies the multiset, as mentioned in Section \[sec:ordervalue\], we can write $$H(\{X_i\}_{i=1}^n) = H(K_1,K_2,\ldots,K_{|\mathcal{X}|}) \mbox{,}
\label{eq:multinomialEquiv}$$ where $K_i$ is the number of occurrences of $x_i$ in $n$ trials. While we will exhibit a few explicit calculations, our main interest is in relating the rate requirement to the sample size $n$. For this we first consider finite alphabets and then infinite alphabets.
Finite Alphabets
----------------
If the multiset is drawn i.i.d., the distribution of types is given by a multinomial distribution derived from the parent distribution [@deMoivre1756 Problem VI]. Suppose $x_i \in \mathcal{X}$ has probability $p_i$ in the parent. Then $$\Pr[K_i = k_i] = \binom{n}{k_1,k_2,\ldots,k_{|\mathcal{X}|}} \prod_{i=1}^{|\mathcal{X}|}p_i^{k_i}, \qquad \mbox{for $i = 1,\ldots, |\mathcal{X}|$,}$$ for any type $(k_1,k_2,\ldots,k_{\left|\mathcal{X}\right|})$ of non-negative integers with sum $n$.
The simplest case of a binary source ($|\mathcal{X}| = 2$) gives $K_1 \sim \binomial(n,p)$ and $K_2 = n - K_1$, where $p = \Pr[X_i = 1]$. Then since $K_2$ is a deterministic function of $K_1$, we have the simplification $H(K_1,K_2) = H(K_1)$. Now we have $$H(\{X_i\}_{i=1}^n)
= H(K_1)
= \Half \log(2\pi e p(1-p) n) + \sum_{k=1}^\infty a_k n^{-k}
\label{eq:binomialEntropy}$$ for some constants $a_1,\,a_2,\,\ldots$. The leading term can be obtained with the de Moivre approximation of a binomial random variable with a Gaussian random variable [@deMoivre1756 pp. 243–259]; the full expansion requires more sophisticated techniques [@JacquetS1999].
To emphasize the dependence on $n$, note that the rate in (\[eq:binomialEntropy\]) is $\Half \log n + c + o(1)$, where $c$ is some constant. We will now see that $O(\log n)$ lossless coding rate extends to all finite-alphabet multiset sources.
Let $|\mathcal{X}|$ be finite. Then $H(\{X_i\}_{i=1}^n) = O(\log n)$. \[thm:HRzero\]
Denote the alphabet of distinct types by $\mathcal{K}(\mathcal{X},n)$. By simple combinatorics [@Csiszar1998], $$|\mathcal{K}(\mathcal{X},n)| = \binom{n + |\mathcal{X}| - 1}{|\mathcal{X}| - 1} \le (n+1)^{|\mathcal{X}|} \mbox{.}
\label{eq:NofTypes}$$ Recalling the equality (\[eq:multinomialEquiv\]), the desired entropy $H(\{X_i\}_{i=1}^n)$ is upper-bounded by the logarithm of $|\mathcal{K}(\mathcal{X},n)|$. Thus, $$H(\{X_i\}_{i=1}^n) \leq |\mathcal{X}| \log(n+1)
= O(\log n)$$ since $|\mathcal{X}|$ is finite.
Note that the theorem holds for any source, not just for i.i.d. sources. For a non-trivial i.i.d. source we can use the calculation (\[eq:binomialEntropy\]) to show an $\Omega(\log n)$ lower bound, so in fact we have $H(\{X_i\}_{i=1}^n) = \Theta(\log n)$.
For an i.i.d. source, the upper bound in the proof is quite loose. To achieve the bound with equality, each of the types would have to be equiprobable; however by the strong asymptotic equipartition property [@Yeung2002], collectively, all non-strongly typical types will occur with arbitrarily small probability. The number of types in the strongly typical set is polynomial in $n$, so any upper bound would still be $\Theta(\log n)$.
Countable Alphabets {#sec:losslessCountable}
-------------------
Theorem \[thm:HRzero\] with its presented proof obviously does not extend to infinite alphabets. To get an interesting bound we must do more than enumerate types.
Define the entropy rate of a multiset as $$H(\mathfrak{X}) = \lim_{n \rightarrow \infty} \tfrac{1}{n}H(\{X_i\}_{i=1}^n) \mbox{.}$$ Theorem \[thm:HRzero\] shows that finite-alphabet sources yield multisets with zero entropy rate. Using a dictionary–pattern decomposition, we will show a related result for countable-alphabet sources.
A sequence may be decomposed into a *dictionary*, $\Delta$, and a *pattern*, $(\Psi_i)$, where the dictionary specifies which letters from the alphabet have appeared and the pattern specifies the order in which these letters have appeared [@OrlitskySZ2004]. For a sequence $(x_i)$, dictionary entry $\delta_k \in \mathcal{X}$ is the $k$th distinct letter in the sequence and pattern entry $\psi_i \in \mathbb{Z}^{+}$ is the dictionary index of the $i$th letter in the sequence. Note that the type of a pattern, denoted as $\{\Psi_i\}$, and the pattern of a multiset, denoted as $\Psi(\{X_i\})$, are the same. It can be seen that a multiset is determined by $\Delta$ and $\{\Psi_i\}$; the order of the pattern, $J(\Psi)$, is not needed.
Based on [@OrlitskySVZ2006], we show that the entropy rate of a multiset generated by a discrete finite-entropy stationary process and the entropy rate of its pattern are equal. First note that the dictionary of a sequence and the dictionary of its associated multiset are the same; we use $\Delta$ to signify either one. Since $\{X_i\}$ determines $\Psi(\{X_i\})$ and since given $\{\Psi_i\}$, there is a one-to-one correspondence between $\{X_i\}$ and $\Delta$, $$H(\{X_i\}) = H(\{\Psi_i\}) + H(\{X_i\} \mid \{\Psi_i\}) = H(\{\Psi_i\}) + H(\Delta \mid \{\Psi_i\}) \mbox{.}$$ If we can show that $H(\Delta | \{\Psi_i\})$ is $o(n)$, then it will follow that the entropy rate of the multiset, $H(\mathfrak{X})$, is equal to the entropy rate of the pattern of the multiset $$H(\mbox{\Neptune}) = \lim_{n\to\infty}\tfrac{1}{n} H(\{\Psi_i\}_{i=1}^n) \mbox{.}$$
Noting the fact that the dictionary $\Delta$ is independent of the order of the pattern, $J(\Psi)$, we establish that $$\begin{aligned}
\lim_{n\to\infty}\tfrac{1}{n} H(\Delta|(\Psi_i)_{i=1}^n) &= \lim_{n\to\infty}\tfrac{1}{n} H(\Delta | J((\Psi_i)_{i=1}^n),\{\Psi_i\}_{i=1}^n) \\
&= \lim_{n\to\infty}\tfrac{1}{n} H(\Delta \mid \{\Psi_i\}_{i=1}^n) \mbox{,}\end{aligned}$$ where the first step follows since the order and type of pattern determine the pattern, and the second step is due to independence. The result [@OrlitskySVZ2006 Theorem 9] shows that for all finite-entropy, discrete stationary processes, the asymptotic per-letter values of $H((\Psi_i)))$ and $H((X_i))$ are equal. Hence, $$\lim_{n\to\infty}\tfrac{1}{n} H(\Delta \mid (\Psi_i)_{i=1}^n) = 0 \mbox{,}$$ and so $$\lim_{n\to\infty}\tfrac{1}{n} H(\Delta \mid \{\Psi_i\}_{i=1}^n) = 0 \mbox{.}$$ This implies that $H(\Delta | \{\Psi_i\})$ is $o(n)$ and yields the following theorem.
\[thm:coincide\] The entropy rate of a multiset generated by a discrete finite-entropy stationary process and the entropy rate of its pattern coincide: $$H(\mathfrak{X}) = H(\mbox{\Neptune}) \mbox{.}$$
Computing the entropy rate of the multiset or equivalently of the pattern of the multiset can be difficult. See [@Shamir2007] and references therein for a discussion on computing the entropy of patterns; the entropy computation for patterns of multisets is closely related.
Universal Lossless Coding {#sec:universal}
=========================
The previous section considered source coding for multisets when the source distribution was known. Most prominently in estimation and inference but also in other applications, the underlying distribution is not known. In this section, we discuss universal source coding of multisets, first presenting an achievability result for countable alphabets and then showing that redundancy, defined in a stronger sense than usual, cannot be driven to zero.
Universal Achievability {#sec:achievability}
-----------------------
We propose a source coding scheme that achieves some degree of compression for all members of a source class at the same time. We will not compare to the entropy bound, holding off detailed discussion of redundancy until Section \[sec:Conv\].
Consider classes of countable-alphabet i.i.d. sources that meet Kieffer’s condition for universal encodability for the sequence representation problem [@Kieffer1978; @HeY2005]. For these source classes, the redundancy for encoding $(X_i)_{i=1}^n$ is $o(n)$. We formulate a universal scheme for the multiset representation problem and demonstrate an achievability result, making use of the dictionary–pattern decomposition.
As we saw in Section \[sec:losslessCountable\], a multiset can be represented as the concatenation of the pattern of the multiset and the dictionary. Consider the rate requirements of these two parts separately. First, let us bound the rate that is required to represent the pattern of the multiset (the type of the pattern). We can make use of the fact that there are $2^{n-1}$ types of patterns. This enumeration follows because the types are sequences of positive integers that sum to $n$. These can appear in any order, thus we are counting ordered partitions. It is well known that there are $2^{n-1}$ ordered partitions, which can be seen as determining arrangements of $n-1$ possible separations of $n$ places. Thus the rate requirement for an enumerative universal scheme representing the type of the pattern is $n-1$ bits.
Now to determine the rate requirement of the dictionary given the type of pattern. If the underlying distribution were known, we saw in Theorem \[thm:coincide\] that $H(\Delta | \{\Psi_i\}_{i=1}^n)$ is $o(n)$. It was shown in [@OrlitskySZ2004] that there is an $O(\sqrt{n})$ upper bound on the pattern redundancy, independent of $|\mathcal{X}|$. Since this is sublinear, the asymptotic per-letter redundancy in coding a class of sequences $(X_i)_{i=1}^{n}$ from a countable alphabet coincides with the asymptotic per-letter redundancy in coding the dictionary given the pattern. Since we are considering a class that meets Kieffer’s condition, we find the redundancy in coding the dictionary given the pattern is $o(n)$. This also carries over to coding the dictionary given the pattern of the multiset, due to the independence between dictionary and order of pattern that we had put forth in Section \[sec:losslessCountable\]. Since both $H(\Delta|\{\Psi_i\}_{i=1}^n)$ and the redundancy in coding the dictionary given the pattern of the multiset are $o(n)$, the total rate requirement for coding the dictionary given the pattern of the multiset is $o(n)$.
Adding together the rate requirements for the two parts yields the following achievability theorem. The rate requirement is universally reduced from $[0,\infty)$ bits per letter for the sequence problem to $1$ bit per letter for the multiset problem.
\[thm:universalAchievability\] Given any i.i.d. source class that is universally encodable as a sequence, the multiset $\{X_i\}_{i=1}^n$ can be encoded with $n + o(n)$ bits.
A representation consists of the concatenation of the type of pattern and the dictionary given the type of pattern. The first part requires $n-1$ bits. The second part requires $o(n)$ bits. The total rate is then $n + o(n)$.
Coding a multiset is equivalent to coding a type or histogram. An interpretation of Theorem \[thm:universalAchievability\] is thus that histograms, from a certain class, with total weight $n$ can be encoded with $n + o(n)$ bits. Figure \[fig:histogram\] shows a histogram. An encoding method that shows the plausibility of $n + o(n)$ total rate is to encode the histogram one letter at a time, starting from the left end and moving to the right. After each letter, use [0]{} to indicate that there is another occurrence of the same letter and use [1]{} to move on to the next letter. If every symbol in the alphabet appears at least once, the rate is $n$; as long as the right-most letter encountered does not grow too quickly with $n$, an $n + o(n)$ rate is achieved.
Unattainability of Negligible Redundancy {#sec:Conv}
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For finite alphabets, we showed that the multiset entropy rate is zero for any source; this is a crude consequence of Theorem \[thm:HRzero\]. We also saw, in the proof of Theorem \[thm:HRzero\], that simply enumerating the type classes requires zero rate per multiset letter asymptotically. Hence such a universal scheme requires zero rate for any finite-alphabet source. However, we cannot conclude from “zero equals zero” that the excess rate due to not knowing the source distribution is negligible.
In this section we take a finer look at universal lossless coding of multisets. We will see that for a finite-alphabet source, the redundancy cannot be made a negligible fraction of the coding rate. Since the coding rate with full distributional knowledge is $\Theta(\log n)$, we come to this conclusion using new information and redundancy measures that have normalization by $\log n$ rather than by $n$. We will find that zero-redundancy universal coding of multisets is *not* possible with respect to the class of memoryless multisets, using the more stringent redundancy definition. Zero redundancy is thus also not possible for more general classes of sources such as sources with memory or with infinite alphabets.
### Log-Blocklength Normalized Information Measures
We formulate several definitions and extend the source coding theorems to these definitions. Let $Z_1^1,\,Z_1^2,\,\ldots$ represent a sequence of random variables over a sequence of alphabets $\mathcal{Z}_1,\,\mathcal{Z}_2,\,\ldots$. (For sequence coding we would have $Z_1^n = (X_i)_{i=1}^n$ and $\mathcal{Z}_n = \mathcal{X}^n$ is an alphabet of sequences. For multiset coding we would have $Z_1^n = \{X_i\}_{i=1}^n$ and $\mathcal{Z}_n = \mathcal{K}(\mathcal{X},n)$ is an alphabet of types.) Define the log-blocklength normalized entropy rate as $$\mathfrak{H}(\mathfrak{Z}) = \lim_{n \to \infty} \frac{H(Z_1^n)}{\log n}$$ when the limit exists. With conditioning on another random variable $\Theta$, define the log-blocklength normalized conditional entropy rate as $$\mathfrak{H}(\mathfrak{Z} \mid \Theta) = \lim_{n \to \infty} \frac{H(Z_1^n \mid \Theta)}{\log n}$$ when the limit exists. Similarly, define the log-blocklength normalized information rate as $$\mathfrak{I}(\mathfrak{Z};\Theta) = \lim_{n \to \infty} \frac{I(Z_1^n;\Theta)}{\log n}$$ when the limit exists. While these definitions parallel the standard definitions, none of the limits would generally exist for sequence coding because the numerators grow linearly with $n$.
For each $n \in \mathbb{Z}^+$, let $\phi_n$ be a source code for random variable $Z_1^n$. For this sequence of source codes, the average codeword lengths are $$C_{\phi,n} = \sum_{\mathcal{Z}_n} p_{Z_1^n}(z_1^n) \ell(z_1^n) \mbox{,}$$ where $\ell(\cdot)$ is the length of the codeword assigned to the source realization $z_1^n$. Shannon’s fixed-to-variable source coding theorem [@Shannon1948] establishes that there exists a sequence of source codes that satisfy the following inequalities for all $n$: $$H(Z_1^n) \le C_{\phi,n} \le H(Z_1^n) + 1 \mbox{.}$$ Dividing through by $\log n$ yields $$\frac{H(Z_1^n)}{\log n} \le \frac{C_{\phi,n}}{\log n} \le \frac{H(Z_1^n) + 1}{\log n} \mbox{.}$$ Taking the limit of large blocklength ($n \rightarrow \infty$), we see that when the limits exist there is a sequence of source codes that achieves $\mathfrak{H}(\mathfrak{Z})$.
### Log-Blocklength Normalized Redundancy Measures
Define the redundancy of a source code, $r_{\phi,n}$, as the excess average codeword length that is required over the minimum $H(Z_1^n)$: $$r_{\phi,n} = C_{\phi,n} - H(Z_1^n) \mbox{.}$$ Finally, define the log-blocklength normalized redundancy of a sequence of source codes as $$\lim_{n\to\infty}\frac{r_{\phi,n}}{\log n} = \lim_{n\to\infty}\frac{C_{\phi,n} - H(Z_1^n)}{\log n}
= \lim_{n\to\infty}\frac{C_{\phi,n}}{\log n} - \mathfrak{H}(\mathfrak{Z}) \stackrel{\Delta}{=} \mathfrak{C}_{\phi} - \mathfrak{H}(\mathfrak{Z}) \mbox{.}$$ By the manipulations of the source coding theorem that we had made previously, we know that there is a sequence of codes with $\mathfrak{C} = 0$. The code used to develop the upper bound in the source coding theorem, however, requires that $p_{Z_1^n}(z_1^n)$ is known.
Now we define performance measures for source coding for a class of source distributions, rather than just a single source distribution. The definitions parallel those of [@Davisson1973]. Suppose that the source distribution is chosen from a class that is parameterized by $\Theta \in \mathcal{T}$. For each $\theta$, there is a conditional distribution $$p(z_1^n \mid \theta) = \Pr\left[Z_1^n = z_1^n \mid \Theta = \theta\right] \mbox{.}$$ The parameter $\theta$ is fixed but unknown, when generating the source realization. Moreover, there may be a distribution on this parameter, $p_{\Theta}(\theta)$. Let $\Phi_n$ be the set of all uniquely decipherable codes on $Z_1^n$. Then, the average log-blocklength normalized redundancy of a code $\phi \in \Phi_n$ for the class of sources described by $p_{\Theta}(\theta)$ is $$\mathcal{L}_{\phi,n}(p_{\Theta}) = \int_{\mathcal{T}} \frac{r_{\phi,n}}{\log n} p_{\Theta}(\theta) \, d\theta \mbox{.}$$
The minimum $n$th-order average log-blocklength normalized redundancy is $$\mathcal{L}_{n}^{*}(p_{\Theta}) = \inf_{\psi \in \Phi_n} \mathcal{L}_{\psi,n}(p_{\Theta}) \mbox{.}$$ Finally, the minimum average log-blocklength normalized redundancy is $$\mathcal{L}^{*}(p_{\Theta}) = \lim_{n\to\infty} \mathcal{L}_{n}^{*}(p_{\Theta}).$$ If $\mathcal{L}^{*}(p_{\Theta}) = 0$, then a sequence of codes that achieves the limit is called *weighted log-blocklength normalized universal*. Now let $T$ be the set of all probability distributions defined on the alphabet $\mathcal{T}$. Then the $n$th-order maximin log-blocklength normalized redundancy of $T$ is $$\mathcal{L}_{n}^{-} = \sup_{q_{\Theta} \in T} \mathcal{L}_{n}^{*}(q_{\Theta}) \mbox{.}$$ If it exists, then the maximin log-blocklength normalized redundancy is $$\mathcal{L}^{-} = \lim_{n\to\infty}\mathcal{L}_{n}^{-} \mbox{.}$$ If $\mathcal{L}^{-} = 0$, then a sequence of codes that achieves the limit is called *maximin log-blocklength normalized universal*. The $n$th-order minimax log-blocklength normalized redundancy of $\mathcal{T}$ is $$\mathcal{L}_{n}^{+} = \inf_{\phi\in\Phi_n}\sup_{\theta\in\mathcal{T}} \frac{r_{\phi,n}(\theta)}{\log n}$$ and the minimax log-blocklength normalized redundancy of $\mathcal{T}$ is $$\mathcal{L}^{+} = \lim_{n\to\infty}\mathcal{L}_{n}^{+} \mbox{.}$$ If $\mathcal{L}^{+} = 0$, then a sequence of codes that achieves the limit is called *minimax log-blocklength normalized universal*.
### Redundancy-Capacity Theorems
The senses of universality that we have defined obey an ordering relation.
\[thm:relation\] The log-normalized redundancy quantities satisfy $$\mathcal{L}_n^{+} \ge \mathcal{L}_n^{-} \ge \mathcal{L}_n^{*}(p_{\Theta})$$ and $$\mathcal{L}^{+} \ge \mathcal{L}^{-} \ge \mathcal{L}^{*}(p_{\Theta}) \mbox{.}$$
Minor modification of [@Davisson1973 Theorem 1].
Armed with definitions and relations among several notions of log-blocklength normalized universality, we now study when it is possible to achieve universality. We give a theorem that gives a necessary and sufficient condition on the existence of weighted log-blocklength normalized universal codes.
\[thm:MIuniversal\] The minimum $n$th-order average log-blocklength normalized redundancy is bounded as $$\frac{I(Z_1^n;\Theta)}{\log n} \le \mathcal{L}_{n}^{*}(p_{\Theta}) \le \frac{I(Z_1^n;\Theta)}{\log n} + \frac{1}{\log n} \mbox{.}$$ A necessary and sufficient condition for the existence of weighted log-blocklength normalized universal codes is that $$\mathcal{L}^{*}(p_{\Theta}) = \lim_{n\to\infty} \mathcal{L}_{n}^{*}(p_{\Theta}) = \lim_{n\to\infty} \frac{I(Z_1^n;\Theta)}{\log n} = \mathfrak{I}(Z;\Theta) = 0 \mbox{.}$$
Minor modification of [@Davisson1973 Theorem 2].
Theorem \[thm:MIuniversal\] can be extended to conditions for minimax and maximin log-blocklength normalized universality and can also be strengthened by suitable modification of theorems in [@Gallager1979; @MerhavF1995].
### Class of Memoryless Multisets {#sec:universalMemorylessMultisets}
Consider the class of memoryless, binary multisets. The parameter $\theta$ is the Bernoulli trial parameter. Now suppose that there is a distribution over the parameter space $q_{\Theta}(\theta) \in T$ that is uniform over $[0,1]$. This gives a mixed source where all type classes are equiprobable, as shown now.
Let the realizations of $\{X_i\}_{i=1}^n$ be expressed as $z \in \{0,\ldots,n\}$, the number of ones. $$\Pr[\{X_i\}_{i=1}^n = z] = \int_{0}^{1} q_{\Theta}(\theta) \Pr[\{X_i\}_{i=1}^n = z \mid \Theta = \theta] \, d\theta \mbox{,}$$ where $q_{\Theta}(\theta)$ simply equals one over the range of integration, and $\Pr[\{X_i\}_{i=1}^n = z \mid \Theta = \theta]$ is given by a binomial distribution: $$\Pr[\{X_i\}_{i=1}^n = z \mid \Theta = \theta] = \binom{n}{z}\theta^z (1-\theta)^{n-z} \mbox{.}$$ So, $$\Pr[\{X_i\}_{i=1}^n = z] = \int_{0}^{1} \binom{n}{z}\theta^z (1-\theta)^{n-z} \, d\theta = \binom{n}{z} B(z+1,n-z+1) = \frac{1}{1+n} \mbox{,}$$ where the beta function $B(\cdot,\cdot)$ has been used. Thus, the result is that the type classes are equiprobable.
Since the types are equiprobable, the entropy $H(\{X_i\}_{i=1}^n)$ is just the logarithm of the number of the types: $$H(\{X_i\}_{i=1}^n) = \log \binom{n+|\mathcal{X}|-1}{|\mathcal{X}|-1} = \log(n+1) \mbox{.}$$ The entropy conditioned on $\Theta = \theta$ is simply the entropy of a binomial random variable [@JacquetS1999] and so the conditional entropy is $$\begin{aligned}
H(\{X_i\}_{i=1}^n \mid \Theta)
&= \int H(\{X_i\}_{i=1}^n \mid \theta) q_{\Theta}(\theta) \, d\theta
= \int \left[\tfrac{1}{2}\log\left(2\pi e n \theta(1-\theta)\right) + {\ts\sum_{k\ge 1}a_k n^{-k}}\right]q_{\Theta}(\theta) \, d\theta \\
&= \tfrac{1}{2}\log n + \sum_{k\ge 1}n^{-k} \int a_k q_{\Theta}(\theta) \, d\theta + \int \tfrac{1}{2}\log\left(2\pi e \theta(1-\theta)\right) q_{\Theta}(\theta) \, d\theta \\
&= \tfrac{1}{2}\log n + o(\log n)\mbox{,}\end{aligned}$$ where the $a_k$s are known constants given in [@JacquetS1999].
Now the mutual information is $I(\{X_i\}_{i=1}^n;\Theta) = H(\{X_i\}_{i=1}^n) - H(\{X_i\}_{i=1}^n \mid \Theta)$, so the log-blocklength normalized information rate is given by $$\begin{aligned}
\mathfrak{I}
&= \lim_{n\to\infty} \frac{I(\{X_i\}_{i=1}^n;\Theta)}{\log n}
= \lim_{n\to\infty} \frac{\log(n+1) - \tfrac{1}{2}\log n - o(\log n)}{\log n} \\
&= \lim_{n\to\infty} \frac{\log(n+1) - \tfrac{1}{2}\log n}{\log n}
= \lim_{n\to\infty} -\frac{1}{2} + \frac{\log(n+1)}{\log n}
= \frac{1}{2}.\end{aligned}$$ Since this is greater than $0$, we have shown that the weakest form of universality is not possible, by Theorem \[thm:MIuniversal\]. Thus by Theorem \[thm:relation\], stronger forms of universality are not possible either. Since the class of binary memoryless sets is a subset of more general source classes such as memoryless; Markov; and stationary, ergodic, universal source coding over these source classes is not possible either.
We can calculate the weighted redundancy for classes of memoryless sources with larger alphabet sizes. Using the $p_{\Theta}(\theta)$ that yields equiprobable multisets and the conditional entropy given by the entropy of a multinomial random variable [@JacquetS1999], we are interested in $$\lim_{n\to\infty} \frac{\log\binom{n+|\mathcal{X}|-1}{|\mathcal{X}|-1} - \Half(|\mathcal{X}|-1)\log(K n) - o(1)}{\log n} = \frac{|\mathcal{X}|-1}{2} \mbox{,}$$ where $K$ is a known constant. As we can see, this redundancy grows without bound as the alphabet size increases. Perhaps unsurprisingly, this redundancy expression is reminiscent of the unnormalized redundancy expression for i.i.d. sequences [@DrmotaS2004]: $$\frac{|\mathcal{X}|-1}{2} \log\frac{n}{2\pi} + \log\frac{\Gamma^{|\mathcal{X}|}\left(1/2\right)}{\Gamma\left({|\mathcal{X}|}/{2}\right)} + o_{|\mathcal{X}|}(1) \mbox{.}$$ We see that the richness of a class of sources as sequences is the same as the richness of that class as multisets.
Let us also comment that in the deterministic case of individual multisets, rather than the probabilistic classes of sources that we have been considering, the same non-achievability result applies. This follows from the arguments summarized in [@MerhavF1998].
Lossy Coding {#sec:lossy}
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In the two previous sections, we have considered lossless representation of multiset sources with discrete alphabets. Now in this section and Section \[sec:universalLossy\], we look at lossy coding with both discrete and continuous alphabets.
Large-Size Multiset Asymptotics {#sec:RDzero}
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### Multiset Mean Squared Error
Assume that the source alphabet $\mathcal{X}$ is a subset of the real numbers. In cases of interest—for example, when every $(X_i)_{i=1}^n$ has a probability density—multisets drawn from these alphabets are almost surely sets. (Simply, ties have zero probability.) Thus, the type is a list of $n$ values that each occurred once. This list is conveniently represented with order statistics. Recall that $X_{(r:n)}$ denotes the $r$th order statistic from a block of $n$, which is the $r$th-largest of the set $\{X_i\}_{i=1}^n$.
Define a word distortion measure as $$\label{eq:rhodef}
\rho_n(x_1^n,y_1^n) = \frac{1}{n}\sum_{i = 1}^n (x_{(i:n)} - y_{(i:n)})^2 \mbox{,}$$ and an associated fidelity criterion as $$\label{eq:F1defA}
F_1 = \{\rho_n(x_1^n,y_1^n), n=1,2,\ldots\}\mbox{.}$$ Although not a single-letter fidelity criterion, it is single-letter mean square error on the block of order statistics.
If $\{X_i\}_{i=1}^n$ is reconstructed by $\{Y_i\}_{i=1}^n$, the incurred distortion is $$D_n = \frac{1}{n} \sum_{i=1}^n \E\left[\left(X_{(i:n)} - Y_{(i:n)}\right)^2\right] \mbox{.}$$ If we use no rate, then the best choice for the reconstruction is simply $y_{(i:n)} = \E[X_{(i:n)}]$, $i=1,\,2,\,\ldots,\,n$, and the average incurred distortion reduces to $$D_n(R=0) = \frac{1}{n}\sum_{i=1}^n \var\left(X_{(i:n)}\right) \mbox{.}$$ Before proceeding with a general proof that $D(R=0)=\lim_{n\to\infty}D_n(R=0) = 0$, we give some examples. For multisets with elements drawn i.i.d. from the uniform distribution with support $\left[-\sqrt{3},\sqrt{3}\right]$, from the Gaussian distribution with mean zero and variance one, and from the exponential distribution with mean one, Figure \[fig:AvgV\] shows the $n$th-order distortion-rate function. This is the average variance of the order statistics. It can be shown that all of these bounded, monotonically decreasing sequences of real numbers, $\{D_n(0)\}$, have limit $0$ [@Varshney2006]. In fact all of these sequences decay as $\Theta(1/n)$. Hence for zero rate, there is zero distortion incurred.
![Distortion at zero rate $D_n(0)$ as function of multiset size $n$ for several sources.[]{data-label="fig:AvgV"}](AvgVbw.eps){width="3.0in"}
The result that the rate-distortion function is the zero-zero point, along with the distortion decay as a function of block size being $\Theta(1/n)$, also holds for a large class of other sources. If we assume that the cumulative distribution function of the source is always differentiable (i.e. the density function $p_X(x)$ exists) and that $p_X(x) > 0$ for all $x \in \mathbb{R}$, then the same result holds. This follows from the asymptotic fixed variance normality of $\sqrt{n}$-normalized central order statistics [@Vaart1998 Corollary 21.5]. Notice that although this class of sources is very large, two of our examples were not members. Thus, we formulate an even more widely-applicable theorem on zero rate-zero distortion, though we no longer have a characterization of the decay rate.
The general theorem will be based on the quantile function of the i.i.d. parent process; this is the generalized inverse of the cumulative distribution function, $F_X(x)$, $$Q(w) = F^{-1}(w) = \inf\{x:F_X(x)\ge w \} \mbox{.}$$ The empirical quantile function, defined in terms of order statistics is $$Q_n(w) = X_{(\lfloor wn\rfloor + 1:n)} = F_n^{-1}(w) \mbox{,}$$ where $F_n(\cdot)$ is the empirical distribution function. The quantile function $Q(\cdot)$ is continuous if and only if the distribution function has no flat portions in the interior. The main step of the proof will be a Glivenko-Cantelli like theorem for empirical quantile functions [@Mason1982].
\[lemma:mason\] Let the letters to be coded, $X_1,\,X_2,\,\ldots,\,X_k$, be generated in an i.i.d. fashion according to $F_X(x)$ with associated quantile function $Q(w)$. Let $X_1$ satisfy $$\E\left[|\min(X_1,0)|^{1/\nu_1}\right] < \infty \quad \mbox{and} \quad \E\left[(\max(X_1,0))^{1/\nu_2}\right] < \infty
\label{eq:bndmom}$$ for some $\nu_1 > 0$ and $\nu_2 > 0$ and have continuous quantile function $Q(w)$. Then the sequence of distortion-rate values for the coding of size-$n$ sets drawn from the parent distribution satisfy $$\lim_{n \to \infty} D_n(R=0) = 0.$$
For any nonnegative function $\omega$ defined on $(0,1)$, define a weighted Kolmogorov-Smirnov like statistic $$S_n(\omega) = \sup_{0 < w < 1} \omega(w)\left|Q_n(w) - Q(w)\right| \mbox{.}$$ For each $\nu_1 > 0$, $\nu_2 > 0$, and $w \in (0,1)$, define the weight function $$\omega_{\nu_1,\nu_2}(w) = w^{\nu_1}(1-w)^{\nu_2} \mbox{.}$$ Assume that $Q$ is continuous, choose any $\nu_1 > 0$ and $\nu_2 > 0$, and define $$\gamma = \limsup_{n \to \infty} S_n(\omega_{\nu_1,\nu_2}) \mbox{.}$$ Then by a result of Mason [@Mason1982], $\gamma = 0$ with probability $1$ when (\[eq:bndmom\]) holds. Our assumptions on the parent process meet this condition, so $\gamma = 0$ with probability $1$. This implies that $$\label{eq:convergenceOfOS}
\limsup_{n \to \infty } |X_{(\lfloor wn \rfloor + 1:n)} - Q(w)| \le 0 \mbox{ for all } w \in (0,1) \mbox{ w.p.} 1 \mbox{,}$$ and since the absolute value is nonnegative, the inequality holds with equality. According to (\[eq:convergenceOfOS\]), for sufficiently large $n$, each order statistic takes a fixed value with probability 1. The bounded moment condition on the parent process, (\[eq:bndmom\]), implies a bounded moment condition on the order statistics. Almost sure convergence to a fixed quantity, together with the bounded moment condition on the events of probability zero imply convergence in second moment of all order statistics. This convergence in second moment to a deterministic distribution implies that the variance of each order statistic is zero, and thus the average variance is zero.
We have established that asymptotically in $n$, the point $(R=0,D=0)$ is achievable, which leads to the following theorem.
\[thm:zerozero\] Under fidelity criterion $F_1$, $R(D) = 0$ for an i.i.d. source that meets the bounded moment condition (\[eq:bndmom\]) and has continuous quantile function.
By the nonnegativity of the distortion function, $D(R) \ge 0$. By Lemma \[lemma:mason\], $D(0) \le 0$, so $D(0) = 0$. Since $D(R)$ is a non-increasing function, $D(R) = 0$, and so $R(D) = 0$.
Due to the generality of the Glivenko-Cantelli like theorem that we used, the result will stand for a very large class of distortion measures. One only needs to ensure that the set of outcomes of probability zero is not problematic.
### Arbitrary Single-Letter Distortions {#sec:lossyArbitrary}
While Theorem \[thm:zerozero\] applies to a large class of real-value parent distributions, it depends on the multiset squared error distortion measure (\[eq:rhodef\]) to make convergence of moments of the order statistics relevant. With arbitrary single-letter distortion measures we obtain a result analogous to Theorem \[thm:HRzero\] in that it shows that an $O(\log n)$ rate is sufficient for coding $n$ letters accurately.
Consider a source and single-letter distortion function $d: \mathcal{X} \times \hat{\mathcal{X}} \rightarrow \mathbb{R}^+$ such that the rate distortion function (for encoding as a *sequence*) is $R_X(D)$. For coding this source without regard to order, define a word distortion measure $$\label{eq:rhodefGeneral}
\rho_n(x_1^n,y_1^n) = \min_{\pi} \frac{1}{n}\sum_{i=1}^n d(x_i,y_{\pi(i)}),$$ where $\pi$ is a permutation on $\{1,\,2,\,\ldots,\,n\}$, and an associated fidelity criterion $$\label{eq:F1defB}
F_1 = \{\rho_n(x_1^n,y_1^n), n=1,2,\ldots\}\mbox{.}$$ (We have re-used the notation from (\[eq:rhodef\])–(\[eq:F1defA\]) since the per-letter MSE on order statistics defined there is a special case.) Denote the minimum (total) rate for encoding $\{X_i\}_{i=1}^n$ with $\E[\rho_n(X_1^n,\hat{X}_1^n)] \leq D$ by $R_{ \{X_i\}_{i=1}^n }(D)$. Then the following theorem bounds the growth of $R_{ \{X_i\}_{i=1}^n }(D)$ as a function of $n$.
\[thm:lossy\_logn\] If $R_X(D)$ is finite, then for any $\epsilon > 0$, $$R_{\{X_i\}_{i=1}^n}(D + \epsilon) = O(\log n) \mbox{.}$$
Let $D$ be such that $R = R_X(D)$ is finite and let $\epsilon > 0$. The achievability of $R_X(D)$ means there is sequence of dimension-$n$ quantizers with $2^{nR}$ codewords such that $\lim_{n \rightarrow \infty} \E[d(X_1^n,\hat{X}_1^n)] \leq D$. Thus, there exists finite $N$ such a dimension-$N$ quantizer with $2^{NR}$ codewords achieves distortion at most $D + \epsilon$. Applying this quantizer to blocks of length $N$ of the source creates a finite-alphabet source that can be communicated as a multiset with $O(\log n)$ bits (Theorem \[thm:HRzero\]). The distortion with respect to (\[eq:rhodefGeneral\]) does not exceed $D + \epsilon$. In fact, a somewhat more stringent distortion measure is held to at most $D + \epsilon$; this measure is of the form (\[eq:rhodefGeneral\]) with permutations $\pi$ limited to rearrangements that keep blocks of length $N$ intact.
Coding of Finite-Size Multisets for Discrete-Alphabet Sources {#sec:finitefinite}
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In Sections \[sec:entropyrate\], \[sec:universal\], and \[sec:RDzero\] we allowed the multiset size to go to infinity. Such source coding incurs infinite delay, and in the case of Section \[sec:RDzero\] the source coding problem is trivialized. If we are concerned with delay, we would want to code short blocks at a time. In this section and the subsequent section, we investigate bounds on coding when we restrict the multiset size to be fixed and finite. Then our asymptotic results are based on increasing the number of independent realizations of these finite-sized multisets.
If we are concerned about lossless representation of each fixed multiset, then the rate requirement is simply lower bounded by the entropy. As we had discussed in Section \[sec:entropyrate\], if the multiset elements are drawn i.i.d., then the entropy is the same as the entropy of a multinomial random variable. For example, Bernoulli$(p)$ multisets of size $K$ have $H(\{X_i\}_{i=1}^K) \approx \tfrac{1}{2} \log_2(2\pi e K p(1 - p))$. The entropy lower bound assumes that we require the fixed-size multisets to be uniquely decipherable. If we insist on a slightly weaker requirement, where these multisets might become permuted, we can require multiset decipherability of the multiset representations. Notwithstanding the falsity of [@Lempel1986 Conjecture 3] (shown in [@Restivo1989]), the gains below entropy are minimal, and so we do not pursue this weaker requirement further.
Rather than lossless coding, one might be interested in lossy coding of fixed-size multisets from discrete alphabets. We define a fidelity criterion for $K$-size multisets $$F_{2} = \left\{\frac{K}{n}\sum_{i=1}^{n/K} d_K(x_{iK-K+1}^{iK},y_{iK-K+1}^{iK}), n=K,2K,\ldots\right\}\mbox{.}$$ The word distortion measure, $d_K$, used to define the fidelity criterion takes value zero when $x$ and $y$ are in the same type class and one otherwise. One can also express the word distortion measure in group theoretic terms using permutation groups, if desired. This notion of fidelity casts the problem into a frequency of error framework on the types. Assuming that the multisets to be coded are independent and identically distributed, this is simply an i.i.d. discrete (finite or countable) source with error frequency distortion, so the reverse waterfilling solution of Erokhin [@Erokhin1958] applies. The rate-distortion function is given parametrically as $$\begin{aligned}
D_{\theta} &= 1 - S_{\theta} + \theta(N_{\theta} - 1) \notag \\
R_{\theta} &= -\sum_{\ell:p(\ell)>\theta} p(\ell)\log p(\ell) + (1 - D_{\theta})\log(1 - D_{\theta}) + (N_{\theta} - 1)\theta \log \theta \mbox{,} \notag\end{aligned}$$ where $N_{\theta}$ is the number of types whose probability is greater than $\theta$ and $S_{\theta}$ is the sum of the probabilities of these $N_{\theta}$ types. The parameter $\theta$ goes from $0$ to $p(\ell^\ddagger)$ as $D$ goes from $0$ to $D_{\rm max} = 1 - p(\ell^\dagger)$; the most probable type is denoted $\ell^\dagger$ and the second most probable type is denoted $\ell^\ddagger$. If the letters within the multisets are also i.i.d., the probability values needed for the reverse waterfilling characterization are computed using the multinomial distribution.
Only the most probable source types are used in the representation alphabet. It is known that the probability of type class $k$ drawn i.i.d. from the a finite-alphabet parent $p_X$ is bounded as follows [@Csiszar1998]: $$\tfrac{1}{|\mathcal{K}(\mathcal{X},n)|}2^{-nD(p_k\|p_X)} \le \Pr[\ell] \le 2^{-nD(p_k\|p_X)} \mbox{,}$$ where $p_k$ is a probability measure derived by normalizing the type $k$. The multiset types used in the representation alphabet are given by the type classes in the typical set $$T_{p_X}^{\epsilon(\theta)} = \left\{k: D(p_k\|p_X) \le \epsilon(\theta)\right\} \mbox{.}$$ Since multiset sources are successively refinable under error frequency distortion [@EquitzC1991], scalable coding would involve adding types into the representation alphabet.
In addition to $F_2$, we can define other fidelity criteria that reduce the multiset rate-distortion problem to well-known discrete memoryless source rate-distortion problems. As a simple example, consider multisets of length $K = 2$ and consisting of i.i.d. equiprobable binary elements. Then there are three letters in the alphabet of types: $\{0,0\}$, $\{0,1\}$, and $\{1,1\}$, which can be represented by their Hamming weights, $\{0,1,2\}$. The probabilities of these three letters are $\{1/4,1/2,1/4\}$. Define the word distortion function using the Hamming weight, $w_H(\cdot)$,: $$\delta(x_1^2,y_1^2) = \left| w_H(x_1^2) - w_H(y_1^2)\right| \mbox{.}$$ The fidelity criterion is $$F_{3} = \left\{\frac{2}{n}\sum_{i=1}^{n/2} \delta(x_{2i-1}^{2i},y_{2i-1}^{2i}), n=2,4,\ldots\right\}\mbox{.}$$ This is a single-letter fidelity criterion on the Hamming weights and is in fact the well-studied problem known as the Gerrish problem [@Berger1971 Problem 2.8]. One can easily generate equivalences to other known problems as well.
Coding of Finite-Size Multisets for Continuous-Alphabet Sources {#sec:finiteuncountable}
---------------------------------------------------------------
Now turning our attention to fixed-size multisets with continuous alphabets, we first see what simple quantization schemes can do, then develop some high-rate quantization theory results and finally compute some rate distortion theory bounds.
### Low-Rate Low-Dimension Quantization for Fixed-Size Multisets
Using the previously defined word distortion (\[eq:rhodef\]) on blocks of length $K$, a new fidelity criterion is $$F_{4} = \left\{\frac{K}{n}\sum_{i=1}^{n/K} \rho_K(x_{iK-K+1}^{iK},y_{iK-K+1}^{iK}), n=K,2K,\ldots\right\}\mbox{.}$$ This is average MSE on the block of order statistics. Notice that the fidelity criterion is defined only for words that have lengths that are multiples of the block size $K$.
For low rates and coding one set at a time, we can find optimal MSE quantizers through the Lloyd-Max optimization procedure [@GershoG1992]. The quantizers generated in this way are easy to implement for practical source coding, and they also provide an upper bound on the rate-distortion function. Designing the quantizers requires knowledge of the distributions of order statistics, which can be derived from the parent distribution [@DavidN2003]. For $X_1,\,X_2,\,\ldots,\,X_K$ that are drawn i.i.d. according to the cumulative distribution function $F_X(x)$, the marginal cumulative distribution function of $X_{(r:K)}$ is given in closed form by $$F_{(r:K)}(x) = \sum_{i=r}^{K}\binom{K}{i}F_X^i(x)\left[1-F_X(x)\right]^{K-i} = I_{F_X(x)}(r,K-r+1)\mbox{,}$$ where $I_p(a,b)$ is the incomplete beta function. Subject to the existence of the parent density $f_X(x)$, the marginal density of $X_{(r:K)}$ is $$\label{eq:margpdf}
f_{(r:K)}(x) = \frac{1}{B(r,K-r+1)}\left[1 - F_X(x)\right]^{K-r} F_X^{r - 1}(x)f_X(x) \mbox{,}$$ where $B(a,b)$ is the beta function. The joint density of all $K$ order statistics is $$\label{eq:jointpdf}
f_{(1:K),\ldots,(K:K)}(x_1,\ldots,x_K) =
\left\{ \begin{array}{ll}
K! \prod_{i=1}^Kf_X(x_i), & x_1^K \in \mathfrak{R}; \\
0, & \mbox{else}. \end{array} \right.$$ The region of support, $\mathfrak{R} = \{x_1^K: x_1 \le \cdots \le x_K\}$, is a convex cone that occupies $(1/K!)$th of $\mathbb{R}^K$. The order statistics also have the Markov property [@DavidN2003] with transition probability $$\label{eq:transpdf}
f_{X_{(r+1:K)}|X_{(r:K)}=x}(y) = (K-r)\left[\frac{1-F_X(y)}{1-F_X(x)}\right]^{K-r-1} \frac{f_X(y)}{1-F_X(x)}, \quad \mbox{for $y>x$.}$$
In a standard quantization setup, the sorting filter (\[eq:transform\]) would be applied first to generate the transform coefficients and then further source coding would be performed. Since sorting quantized numbers is easier than sorting real-valued numbers, we would prefer to be able to interchange the operations. Based on the form of the joint distribution of order statistics, (\[eq:jointpdf\]), we can formulate a statement about when sorting and quantization can be interchanged without loss of optimality. If the order statistics are to be quantized individually using scalar quantization, then interchange without loss can be made in all cases [@Gandhi1997]. Scalar quantization, however, does not take advantage of the Markovian dependence among elements to be coded. We consider coding the entire set together, referring to $K$ as the dimension of the order statistic vector quantizer.
If the representation points for an MSE-optimal $(R \mbox{ rate},\, K \mbox{ dimension})$ order statistic quantizer are the intersection of $\mathfrak{R}$ with the representation points for an MSE-optimal $(R + \log K!,\, K)$ quantizer for the unordered variates, then we can interchange sorting and quantization without loss of optimality. This condition can be interpreted as a requirement of permutation polyhedral symmetry on the quantizer of the unordered variates. This form of symmetry requires that there are corresponding representation points of the unordered variate quantizer in each of the $K!$ convex cones that partition $\mathbb{R}^K$ on the basis of permutation. The polyhedron with vertices that are corresponding points in each of the $K!$ convex cones is a permutation polyhedron. In fact, the distortion performance of the MSE-optimal $(R,\,K)$ order statistic quantizer is equal to the distortion performance of the best $(R + \log K!,\, K)$ unordered quantizer constrained to have the required permutation symmetry. An example where the symmetry condition is met is for the standard bivariate Gaussian distribution shown in Figure \[fig:gaussOSvq\].
![Quantization for bivariate standard Gaussian order statistics. Optimal one-bit quantizer (white) achieves $(R = 1,\, D=(2\pi-4)/\pi)$. Optimal two-bit quantizer (black) for unordered variates achieves $(R = 2,\, D=(2\pi-4)/\pi)$. Since representation points for order statistic quantizer are the intersection of the cone (shaded) and the representation points for the unordered quantizer, the distortion performance is the same.[]{data-label="fig:gaussOSvq"}](gaussOSvq.eps){width="2.5in"}
### High-Rate Quantization Theory for Fixed-Size Multisets {#sec:highRateQuant}
Based on the basic distributional properties of order statistics, (\[eq:margpdf\])–(\[eq:transpdf\]), the differential entropies of order statistics can be derived. The individual marginal differential entropies are $$h(X_{(r:K)}) = \int f_{(r:K)}(x) \log f_{(r:K)}(x) dx \mbox{,}$$ where no particular simplification is possible unless the parent distribution is specified. The average marginal differential entropy, however, can be expressed in terms of the differential entropy of the parent distribution and a constant that depends only on $K$ [@WongC1990]: $$\bar{h}(X_{(1:K)},\ldots,X_{(K:K)}) = \frac{1}{K}\sum_{i = 1}^K h(X_{(i:K)}) = h(X_1) - \log K - \frac{1}{K}\sum_{i = 1}^K \log \binom{K-1}{i-1} + \frac{K - 1}{2} \mbox{.}
\label{eq:hbar}$$ The subtractive constant is positive and increasing in $K$, and not dependent on the parent distribution. The individual conditional differential entropies, as derived in [@Varshney2006], are $$\begin{aligned}
&h(X_{(r+1:K)}|X_{(r:K)}) = -\log(K-r) - N_h(K) + N_h(K-r) +1 - \frac{1}{K-r} \\ \notag
&- \frac{K!}{\Gamma(K-r)\Gamma(r)} \int_{-\infty}^{\infty} \int_{x}^{\infty} f_X(y)\log(y) \left[1 - F_X(y)\right]^{K-r-1} dy F_X^{r-1}(x) f_X(x) dx \mbox{,}\end{aligned}$$ where $\Gamma(\cdot)$ is the gamma function and $N_h(k) = \sum_{m=1}^k 1/m$ is the harmonic number. As in the individual marginal case, further simplification of this expression requires the parent distribution to be specified. Again, as in the marginal case, the total conditional differential entropy can be expressed in terms of the parent differential entropy and a constant that depends only on $K$. Due to Markovianity, the sum of the individual conditional differential entropies is in fact the joint differential entropy: $$h\left(X_{(1:K)},\ldots,X_{(K:K)}\right) = h(X_{(1:K)}) + \sum_{i=1}^{K-1} h(X_{(i+1:K)}|X_{(i:K)}) = Kh(X_1) - \log K! \mbox{.}$$ Notice that an analogous statement (\[eq:orderlowerbound\]) was a lower bound in the discrete alphabet case; equality holds in the continuous case since there are no ties.
High-rate quantization results follow easily from the differential entropy calculations. To develop results, we introduce four quantization schemes in turn, measuring performance under fidelity criterion $F_{4}$. In particular, we sequentially introduce a *shape advantage*, a *memory advantage*, and a *space-filling advantage* as in [@LookabaughG1989].[^7] As a baseline, take the naïve scheme of direct uniform scalar quantization of the arbitrarily-ordered sequence with quantization step size $\epsilon$. The average rate and distortion per source symbol of the naïve scheme are $R_1 = h(X_1) - \log \epsilon$, and $D_1 = \epsilon^2/12$. Now instead uniformly scalar quantize the deterministically-ordered sequence (the order statistics). This changes the shape of the marginal distributions that we are quantizing, and thus we get a shape advantage. The average rate per source symbol for this scheme is $$R_2 = \bar{h}(X_{(1:K)},\ldots,X_{(K:K)}) - \log \epsilon = R_1 - \log K - \frac{1}{K}\sum_{i = 1}^K \log \binom{K-1}{i-1} + \frac{K - 1}{2}\mbox{.}$$ The distortion is the same as the naïve scheme, $D_2 = D_1$. As a third scheme, scalar quantize the order statistics sequentially, using the previous order statistic as a form of side information. Even though the encoding requires waiting for the entire block so as to sort, decoding can proceed symbol-by-symbol, reducing delay. We assume that the previous order statistics are known exactly to both the encoder and decoder. Since the order statistics form a Markov chain, this single-letter sequential transmission exploits all available memory advantage. The rate for this scheme is $$R_3 = \tfrac{1}{K}h(X_{(1:K)},\ldots,X_{(K:K)}) - \log \epsilon = R_1 - \tfrac{1}{K}\log K!\mbox{.}$$ Again, $D_3 = D_1$. Finally, the fourth scheme would vector quantize the entire sequence of order statistics collectively. Since we have exploited all shape and memory advantages, the only thing we can gain is space-filling gain. The rate is the same as the third scheme, $R_4 = R_3$, however the distortion is less. This distortion reduction is a function of $K$, is related to the best packing of polytopes, and is not known in closed form for most values of $K$; see [@LookabaughG1989 Table I] and more recent work on packings. We denote the distortion as $D_4 = D_1/G(K)$, where $G(K)$ is a function greater than unity. The performance improvements of these schemes are summarized in Table \[tab:advantage\]. Notice that all values in Table \[tab:advantage\] depend only on the multiset length $K$ and not on the parent distribution.
Rate Reduction $(-)$ Distortion Reduction $(\times)$
------------------ ------------------------------------------------------------------------------- ---------------------------------
Scheme 1 $0$ $1$
Scheme 2 (s) $\log K + \frac{1}{K}\sum_{i = 1}^K \log \binom{K-1}{i-1} - \frac{K - 1}{2}$ $1$
Scheme 3 (s,m) $(\log K!)/K$ $1$
Scheme 4 (s,m,f) $(\log K!)/K$ $1/G(K)$
: Comparison between the naïve scalar quantization (Scheme 1) and several other quantization schemes. The symbols (s), (m), and (f) denote shape, memory, and space-filling advantages.[]{data-label="tab:advantage"}
We have introduced several quantization schemes and calculated their performance in the high-rate limit. It was seen that taking the fidelity criterion into account when designing the source coder resulted in rate savings that did not depend on the parent distribution. These rate savings can be quite significant for large blocklengths $K$.
### Rate Distortion for Fixed-Size Multisets
It is quite difficult to obtain the full rate-distortion function for the $F_{4}$ fidelity criterion; however, upper and lower bounds may be quite close to each other for particular source distributions. As an example, consider the rate-distortion function for the independent bivariate standard Gaussian distribution that was considered in Figure \[fig:gaussOSvq\]. The rate-distortion function under $F_{4}$ is equivalent to the rate-distortion function for the order statistics under the MSE fidelity criterion, as shown. For clarity of expression, let $X = (X_i)_{i=1}^K$ and $\hat{X} = (Y_i)_{i=1}^K$ in the unsorted domain and $Z = \{X_i\}_{i=1}^K$ and $\hat{Z} = \{Y_i\}_{i=1}^K$ in the sorted domain. Clearly, the fidelity constraint is naturally expressed in the $Z$ domain. The affirmatively answered question is whether the mutual information in the rate distortion optimization can be switched from $I(X;\hat{Z})$ to $I(Z;\hat{Z})$: $$\begin{aligned}
I(X;\hat{Z}) &= h(X) + h(\hat{Z}) - h(X,\hat{Z}) \\ \notag
&\stackrel{(a)}{=} h(Z) + h(J) + h(\hat{Z}) - h(X,\hat{Z}) \\ \notag
&= h(Z) + h(J) + h(\hat{Z}) - \left[h(X,\hat{Z}|J) + I(X,\hat{Z};J)\right] \\ \notag
&= h(Z) + h(J) + h(\hat{Z}) - h(Z,\hat{Z}) - I(X,\hat{Z};J) \\ \notag
&= h(Z) + h(J) + h(\hat{Z}) - h(Z,\hat{Z}) - h(J) + h(J|X,\hat{Z}) \\ \notag
&\stackrel{(b)}{=} I(Z;\hat{Z}) + h(J) - h(J) = I(Z;\hat{Z}) \mbox{.}\end{aligned}$$ Step (a) is due to Theorem \[thm:Hdecomp\] and step (b) follows since $h(J|X,\hat{Z})$ is zero.
The Shannon lower bound is simply $$R_{\rm SLB}(D) = \log(1/D),$$ the Gaussian rate-distortion function under the MSE fidelity criterion, reduced by $\log K!$ bits (one bit). Note that since the order statistic source cannot be written as the sum of two independent processes, one of which has the properties of a Gaussian with variance $D$,[^8] the Shannon lower bound is loose everywhere [@GerrishS1964], though it becomes asymptotically tight in the high-rate limit.
The covariance matrix of the Gaussian order statistics can be computed in closed form as $$\Lambda = \left[ {\begin{array}{*{20}c}
{1 - 1/\pi} & {1/\pi} \\
{1/\pi} & {1 - 1/\pi} \\
\end{array}} \right] \mbox{,}$$ with eigenvalues $1$ and $1 - 2/\pi$. Reverse waterfilling yields the Shannon upper bound $$R_{\rm SUB}(D) = \left\{ {\begin{array}{*{20}c}
{\tfrac{1}{2}\log\left(\frac{2 - 4/\pi}{D}\right) + \tfrac{1}{2}\log\left(\frac{2}{D}\right),} & {0 \le D \le 2 - 4/\pi } \\
{\tfrac{1}{2}\log\left(\frac{1}{D - 1 + 2/\pi}\right),} & {2 - 4/\pi \le D \le 2 - 2/\pi} \\
{0,} & {D \ge 2 - 2/\pi.} \\
\end{array}} \right.$$ This bound is tight at the point achieved by zero rate. Since the Gaussian order statistics for $K=2$ have small non-Gaussianity, the Shannon lower bound and the Shannon upper bound are close to each other, as shown in Figure \[fig:rdorder2\]. For moderately small distortion values, we can estimate the rate-distortion function quite well.
![Shannon upper and lower bounds for the Gaussian order statistic rate-distortion function. The point achievable by single set code of Figure \[fig:gaussOSvq\] is also shown connected to the zero rate point, which is known to be tight. Note that rate is not normalized per source letter.[]{data-label="fig:rdorder2"}](rdorder.eps){width="3.0in"}
The fact that the Shannon lower bound is loose everywhere applies not only to the particular example we considered, but to any problem. That is to say, $\log(K!)$ bits cannot be saved below the rate distortion function for the usual squared error fidelity criterion.
\[thm:SLBloose\] The Shannon lower bound to the rate distortion function is loose everywhere for any source, under the fidelity criterion $F_4$.
The support of the joint distribution (\[eq:jointpdf\]) for any order statistic source is the convex cone $\mathfrak{R} = \{x_1^K: x_1 \le \cdots \le x_K\}$. The support of a Gaussian distribution is all of $\mathbb{R}^K$. For the Shannon lower bound to be tight, the source must be decomposable as the sum of two independent processes, one of which has the properties of a Gaussian [@GerrishS1964]. Since the Gaussian density has support over all space, it cannot be convolved with another density (non-negative) to yield a third density that has support over only part of space.
Universal Lossy Coding {#sec:universalLossy}
======================
The final setting in which we investigate the ramifications of order irrelevance is universal lossy coding. The general goal in universal source coding is to find encoding algorithms that perform well for all members of a class of sources [@NeuhoffGD1975]. Here we have the modest goal of demonstrating that $O(\log n)$ rate requirements extend quite generally to lossy coding. The results we present are not intended to be conclusive, but rather are included to wind up our tour of source coding.
Recall the main result of Section \[sec:RDzero\]: under a *per-letter* MSE fidelity criterion, zero distortion is achievable with zero *total* rate for a large class of sources. This result is obtained with the number of letters $n$ growing without bound and the source distribution known. An interpretation of this is that using the known distribution to pseudorandomly “simulate” the source at the destination is sufficient for achieving zero distortion. Now we consider a universal setting in which this approach will not work because the source distribution is not known at the destination.
Instead of giving a result for real-valued sources and multiset MSE, we jump directly to a more general result. Let $d$ be a single-letter distortion measure, and let $R^*(D)$ denote an operational rate distortion function that is achievable by fixed-rate codes uniformly over a class of sources (coding the sources as sequences). As in Section \[sec:lossyArbitrary\], for coding without regard to order, consider the single-letter distortion measure $\rho_n$ and associated fidelity criterion $F_1$ given in (\[eq:rhodefGeneral\])–(\[eq:F1defB\]). Denote by $R^*_{ \{X_i\}_{i=1}^n }(D)$ the minimum (total) rate for encoding $\{X_i\}_{i=1}^n$ with $\E[\rho_n(X_1^n,\hat{X}_1^n)] \leq D]$ for every source in the class. Then we obtain the following result analogous to Theorem \[thm:lossy\_logn\]:
\[thm:lossyUniversal\] If $R^*(D)$ is finite, then for any $\epsilon > 0$, $$R^*_{\{X_i\}_{i=1}^n}(D + \epsilon) = O(\log n) \mbox{.}$$
Let $D$ be such that $R = R^*(D)$ is finite and let $\epsilon > 0$. The achievability of $R^*(D)$ *uniformly over the class* means that there is a finite dimension $N$ at which distortion $D+\epsilon$ is achieved at rate $R$ for every source in the class. Thus only minor adjustments to the proof of Theorem \[thm:lossy\_logn\] are needed.
As a simple application consider a set of real-valued parent distributions that share a bounded support. An arbitrarily small multiset MSE can be obtained uniformly over all the sources with $O(\log n)$ rate. This follows from Theorem \[thm:lossyUniversal\] because the finiteness of $R^*(D)$ for any positive distortion $D$ can be demonstrated by uniform quantization of the support of the source class.
Concluding Comments: From Sequences to Multisets {#sec:seq2set}
================================================
We have completed a tour through the major areas of source coding while discussing how things are changed by irrelevance of the order of source letters. To conclude, we discuss three conceptual transitions between sequences and multisets and then summarize.
Types $\rightarrow$ Markov Types $\rightarrow$ Sequences {#sec:Markovtypes}
--------------------------------------------------------
In the Shannon-style language approximations that were mentioned in the opening, a first-order approximation corresponds to a multiset of letters of the original alphabet, whereas an approximation of the same order as the length of the sequence is the sequence itself. In between these extremes, there are many possibilities: A second-order approximation is a multiset of digrams (ordered pairs of source letters), a third-order approximation is a multiset of trigrams, etc. Thus, as the approximation order is increased, the lengths of segments within which the ordering of letters is relevant increases.
For a fixed alphabet, increasing the approximation order also causes the number of distinct outcomes to increase. For first-order approximations to $n$ source letters drawn from alphabet $\mathcal{X}$, the number of distinct outcomes is the number of types. For an $\ell$th-order approximation, the number of distinct outcomes is the number of *Markov type classes* [@DavissonLS1981; @Shields1990; @JacquetS2004]. Markov type classes are also known as *kinds* in cognitive science [@GriffithsT2001; @GriffithsT2007] and used to visualize motifs in computational genomics [@SchneiderS1990]. The enumeration of Markov types is not simply expressed [@JacquetS2004], but can be upper bounded by $(n + 1)^{|\mathcal{X}|^{\ell}}$. The number of Markov types gives an upper bound on the rate requirements for lossless coding and is computed exactly in Figure \[fig:Sets\_rate\]. There is no source that can achieve the enumeration upper bound for different values of $\ell$ simultaneously since it is impossible to have equiprobable sequences and multisets at the same time. For a real source, like the empirical source from the Zenith radio experiments in telepathy [@Goodfellow1938], the entropy is much lower than the bounds; see Table \[table:zenith\].
![Logarithm of the number of binary sequences, multisets, or $k$-gram multisets as a function of the number of binary letters.[]{data-label="fig:Sets_rate"}](Sets_rate_fig.eps){width="3.0in"}
-- -- --
-- -- --
: Entropy of Zenith Radio Telepathy Data[]{data-label="table:zenith"}
Partially Commutative Alphabets
-------------------------------
Rather than varying ordering requirements by varying the segments over which order is relevant, one can allow particular letters in the alphabet to commute in position with other letters. As discussed in [@Savari2004], a source with such a partially-commutative alphabet can be described by a noncommutation graph. As edges are removed from this graph, the importance of the order of letters decreases. In the case of the empty noncommutation graph, the order is irrelevant and the so-called lexicographic normal form associated with the noncommutation graph is simply the sequence sorted into order. The distinct outcomes associated with a noncommutation graph are called interchange classes and the moment-generating function for the number of interchange classes is equal to the inverse of the Möbius polynomial corresponding to a function of the noncommutation graph. Interchange entropies are discussed in detail by Savari [@Savari2004].
If noncommutation graphs are defined on sliding windows of source letters rather than on individual letters, the problem becomes one of source coding with a $0$-$1$ context-dependent fidelity criterion [@Shannon1959; @BergerY1972]; sliding windows that have distortion zero between them commute. Since the sliding windows overlap, however, the commutation relations must be constrained to remain consistent. Just as sources with partially-commutative alphabets lead to type classes when the noncommutation graph is empty, sources with empty noncommutation graphs on sliding windows lead to Markov type classes.
Quantum Physics {#sec:quantum}
---------------
In statistical physics, the Maxwell-Boltzmann statistics are used for non-interacting, identical bosons in the classical limit and correspond to sequences, whereas the Bose-Einstein statistics are used when quantum effects are manifested and correspond to multisets. In the classical regime, bosons of the same energy level, $x \in \mathcal{X}$, may be distinguished by their different positions in space. That is to say the order of particles is important. As the concentration of particles increases, some particles become so close that they can no longer be distinguished in position, and degeneracy results. Thus degeneracy measures the importance of order in representing bosons. When the concentration of particles exceeds the quantum concentration, i.e. when the interparticle distance is less than the thermal de Broglie wavelength, the bosons become indistinguishable and so representable by a multiset. The combinatorics of indistinguishability as a function of particle concentration is given in the so-called partition function for bosons. The partition function allows distributional characterizations of intermediate levels of particle concentration to be made.
Incidentally, at even greater concentrations than the quantum concentration, the probability mass of letters appearing in the multiset concentrates on a single letter, as determined by the average boson occupation number. This is known as Bose-Einstein condensation.
Summary
-------
Partial or full order irrelevance has significant qualitative impact. For lossless coding of $n$ letters from a finite-alphabet source, the rate requirement grows only logarithmically with $n$ rather than linearly with $n$ (Theorem \[thm:HRzero\]); the rate reduction as a ratio is thus arbitrarily large. In a universal setting, the rate reduction is again arbitrarily large: for a source satisfying Kieffer’s condition for sequence representation [@Kieffer1978], universal coding with $n + o(n)$ bits is achievable (Theorem \[thm:universalAchievability\]). This should be compared with $cn$ bits when order is relevant, where constant $c$ could be arbitrarily large. Despite this positive statement about universal coding, it is impossible for the redundancy to be a negligible fraction of the coding rate (Section \[sec:universalMemorylessMultisets\]).
For lossy coding subject to per-letter MSE distortion, irrelevance of order can trivialize the source coding problem for a large class of sources. Specifically, under rather weak moment conditions on the parent distribution, zero distortion is achieved even with zero rate as $n \rightarrow \infty$ (Theorem \[thm:zerozero\]). This is not of practical importance because a source coder will process only a finite amount of data at once. High-resolution analyses of various quantization schemes for a block of size $K$ are presented in Section \[sec:highRateQuant\]. Through the inclusion of shape and memory advantage—and under the assumption of high rate—a rate savings of $\log K!$ bits can be achieved relative to naïve scalar quantization. However, in a rate distortion setting the “full” savings of $\log K!$ bits can only be achieved as the rate approaches infinity, not at any finite rate (Theorem \[thm:SLBloose\]).
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors thank Alon Orlitsky for fruitful discussions; in particular, the results in Section \[sec:achievability\] were developed in collaboration with him. The authors also thank Sanjoy K. Mitter for several discussions.
[^1]: This work was supported in part by an NSF Graduate Research Fellowship, NSF Grant CCR-0325774, the Texas Instruments Leadership University Consortium Program, and the Centre Bernoulli at École Polytechnique Fédérale de Lausanne.
[^2]: The material in this paper was presented in part at the Information Theory and its Applications Inaugural Workshop, La Jolla, California, February 2006; the IEEE Data Compression Conference, Snowbird, Utah, March 2006; and the 2007 Information Theory and its Applications Workshop, La Jolla, California, January/February 2007.
[^3]: L. R. Varshney (email: lrv@mit.edu) and V. K Goyal (email: vgoyal@mit.edu) are with the Department of Electrical Engineering and Computer Science and the Research Laboratory of Electronics, Massachusetts Institute of Technology. L. R. Varshney is also with the Laboratory for Information and Decision Systems, Massachusetts Institute of Technology.
[^4]: Anagrams due to R. J. McEliece, 2004 Shannon Lecture.
[^5]: In Shannon’s sense of language approximation [@Shannon1948], first-order approximation requires that the distribution of letters matches the source, second-order approximation requires that the distribution of digrams matches the source, and so on; see also classical criticisms to this method of approximation [@Chomsky1956].
[^6]: We could say that $y$ is the sorted version of $x$, but we want to emphasize that there is no need at this point for a *meaningful* order for $\mathcal{X}$.
[^7]: Note that vector quantizer advantages are discussed in terms of distortion for fixed rate in [@LookabaughG1989], but we present some of these advantages in terms of rate for fixed distortion.
[^8]: Even though $X_{(1:2)} = \tfrac{1}{2}(X_1 + X_2) - \tfrac{1}{2}|X_1 - X_2|$ and $X_{(2:2)} = \tfrac{1}{2}(X_1 + X_2) + \tfrac{1}{2}|X_1 - X_2|$, and the first terms are Gaussian, the troublesome part is the independence.
|
---
author:
- 'P. P. Marvol$^1$'
title: A Tip of the TOE
---
physics/0309055
Institute for Applied Quantum Acoustics,\
Pyramid College Cacania, Hatetepe Boulevard 12,\
CC-12000 Cacania City, CACANIA
[*Abstract.*]{} Using standard methods from string theory, this paper presents a comprehensive survey about the most important aspects of the theory of sand. Special interest is put on the examination of the sand-wind duality and the interaction of ordinary (non-supersymmetric) sand with the heat field of the earth, as solution of the field inequalities of a stone. This will lead us, in a natural way, to completely new insights into the theory of sandstorms.
Keywords: [*many particle systems; string theory; field theory; quantum chaos; GRT; SUSY; thermal QFT; GRB; quantum cosmology; noncommutative QT;*]{}
Introduction
============
The matter of sand has ever since fascinated mankind. Already twenty thousand years ago attempts to classify the grainy zoo have been made, [@abc]. In the Cretacious Age many advances with respect to sand have been undertaken, especially it has been proved by [@unspellable] that chalk and sand are—at a fundamental level—two manifestations of the same thing. Unfortunately, since the end of the Cretatious Age, sand research activities suffered from a considerable lack of financial resources, a time period which is called the Big Sand Crisis (BSC). This led to the the well-known revolutionary movement “Every Scientist Needs a Camel”, [@marx]. After all, at least a few new concepts have been invented, among them the so-called [*wheel*]{}, which then, however, has been discarded due to its impracticability, [@tuareg]. (However, there are some unteachable fanatics who still try to demonstrate the usefulness of wheels on sand in the annual Paris-Dakar competition.)
Some centuries after the BSC a completely new impulse came from ancient Lei Li Ga, [@lei], who was the first to put special interest in the dynamics of sand.
The so far highly disordered efforts in sand have been fibre bundled by the famous philosoph Goe-The, who formulated the central question of today’s complex theory of sand, [@fist]:
[*Herauszufinden, was den Sand im Innersten zusammenband.$^{2}$* ]{}\
Starting from this point, many interesting theories about sand dynamics have been developed. One of the most ambitious (and ambiguous) models was the well-known sandstring theory developed by the old Egyptian priest Edua ’Wit X [@cheops]. With the postulate of the non-existence of infinitely small grains of sand, several severe problems of ordinary sand-theory (e.g. incurable sand-divergencies, causality problems etc.) seemed to be repaired. However, over the millennia, this model did not bear any fruits in sand. (After all, there have been some interesting side results for weavers, in particular new techniques for knotting.) This led to a complete restart of research in the field of sand. Fortunately, at least some interesting questions have been solved independently of sandstring theory, [@cameroon], [@unknown], [@acta].
Modern, post-string sand theories have also their origin in Egypt. In particular, a comprehensive quantum theory of the dynamics of sand dunes has been developed, [@annals], [@quant]. Piles of stones (so called “pyramides”) have been constructed for a macroscopic test of the dynamic theory, especially for the prominent dune-tunnel effect. However, the pyramid-models emerged as too roughly textured, so there could not be proven anything with these (although they are still very impressive). Indeed, experimental clues for the correctness of the Egyptian Theory of Quantum Sand Dynamics (ETQSD) has been found only recently by Swedish Inger Zeil, [@zei]. Of course, to people living in North-Africa, his results have always been obvious. However, as they aroused some interest outside the Big Desert, he eventually will receive the prize without any bells.
Presently, research in sand dynamics came to an interlocutory end with a stone’s$^3$ field inequalities, [@stone], which form the basis of our approach.
The paper is organized as follows. In section two, the main aspects of the standard sand-wind duality are recapitulated. In section three a stone’s field inequalities (specialized for the heat kernel of the earth) are presented. From these fundamental considerations, section four will lead us in a natural way to a completely new and attractive theory of sandstorms. In section five follows a brief conclusion and outlook with respect to some recent aspects of sand theory will be given.
Sand-Wind Duality
=================
This section is devoted to a brief recapitulation of the sand-wind duality of grain dynamics. Since sand is quantized (’grained’), one has to use the well-known cat-equation, which reads (in natural units) $$(W + V_H)|\mathrm{sand}\rangle = -i\frac\partial{\partial t}|\mathrm{sand}
\rangle.$$ $W$ is the wind operator, $V_H$ stands for the heat field operator. We use the idempotency of sand dunes (two sand dunes thrown upon each other give again a sand dune, since the superfluous grains merely drain in the sea of sand, as introduced by the great wizard Pamdirac) and $WV_H=V_HW=0$ (heat and wind never occur at the same time). Now, multiplying this equation by its complex conjugate and dressing the cat with its bra leads immediately to $$\langle \mathrm{sand}|W|\mathrm{sand}\rangle =
\langle \mathrm{sand}|(\frac\partial{\partial t})^2\mathrm{|sand}\rangle.$$ Due to the quadratic time derivative on the right hand side of this equation, by interpreting the operators as states and the states as operators we can immediately read off the duality between sand and wind. Most interesting consequences of this phenomenon follow from the fact that a (sand theoretical) distinction between the sand-field and the wind-field is impossible. This is the basis of our modern understanding of sandstorms.
In classical sand theory, it has been believed that the genesis of sandstorms lies merely in the wind field, interacting with the grains of sand, pushing them up and around. Modern understanding of sandstorms using the sand-wind duality of grain dynamics, however, has taught that the wind is a mere manifestation of the sand field dynamics. Thus, the usual theory of sandstorms is simply that the interaction of the sand-wind-field with itself is sufficient to create a sandstorm. Indeed, it has been shown by [@private] that the movement of the grains of sand functions as source term for the wind field, not the other way as suggested by classical theory: It’s the grains generating the wind.
So far, this is somwhat awkward but, after all, well understood. Unfortunately, combinig the principles of grain dynamics with the Relative Generality Theory of a stone, we will find that this simple, non-general model is not sufficient to a full understanding of sandstorms.
The Heatfield
=============
In the preceding section we have deliberately assumed that the reader is familiar with some of the basic notions describing the coupling of wind to sand. However, in order to fully exploit the origin of the breakdown of sand-wind-duality in sandstorms, we shall need a more thorough understanding of the generalistic physics of sand with respect to the heat field, i.e. of a stone’s theory and its grainization. An excellent recent introduction can be found in [@fein] – in chapter 42, of course.
Let’s begin with a brief historical survey. As it is well known, the experiments of Michael’s son (an ethiopian sprinter) and a cheetah [@mic] have given a first evidence that nothing moves faster than sand and that all grains (in a sandstorm) have the same speed, at least within the experimental accuracy of that time. This led to a relatively special view of matters. However, it still remained unclear how such a theory could be combined with the interaction of wind, sand and heat in the spirit of the old ton theory [@cern]. Only a few millennia later, this problem has been solved in an ingeniously simple manner by a stone.
Obviously, the sand in the desert is rather hot, in fact even hotter than the air surrounding it. Up to a stone’s era many scientists desperately tried to understand this phenomenon. A stone turned it into a new paradigm instead. In brief, his idea can be stated as follows: The heat moves the grains, but the grains are the source for the heatfield at the same time. (If we could switch off the dynamics and wait long enough then the air would be as hot as the sand.) More quantitatively, a stone’s inequalities read $$\begin{aligned}
T_g^{\mu\nu} &\geq& T_w^{\mu\nu} \quad , \\
G^{\mu\nu} &\approx& T^{\mu\nu}_{\mbox{?}} \end{aligned}$$ from which one immediately derives his famous saying “E=m ceh ceh”. (The ancient letter $=$ has no analogue nowadays.) Note that the sand is still treated as a classical (in the sense of continuous) field (of characteristic zero) here. The coupling of the spin-2-heatfield to wind, which is spin-1, i.e. a vector-field, is described by the force $ F^\mu = h^{\mu\nu} W_{\nu} $ acting on the wind quanta.
Due to its many nontrivial predictions, the success of a stone’s theory was overwhelming. Among these predictions were the delay of the time shown by a sandglass in an external heatfield and, most striking, the deflection of light in heatfields, often referred to as “Fata Morgana”. (By the way, this effect is also the origin of the common misbelief that dromedars have two hunches.)
Even more so, it turned out that the grainization of this theory was no great deal $^4$. A typical effect that can only be understood in the framework of the full theory of grain dynamics, as it is a typical grain effect of the sand-wind-interaction, is the movement of sandpiles [@quant]. We have already given a detailed account of this application in the preceding section. We have also shown therein that the resulting sand-wind duality leads to our present understanding of the genesis of sandstorms.
But we also mentioned, that this model has some drawbacks to which we shall turn our attention now.
First of all and honestly speaking, the above described mechanism for the genesis of sandstorms is only of limited theoretical value. In fact, no one has ever been able to prove that such extreme states of the sand-wind-field like sandstorms really do exist. (Actually, it is not even clear, whether this theory describes anything realistic at all.) A modern researcher can hardly take a “problem”of this kind serious, of course, but unfortunately it is not the end of the story:
Evidently, there is only a finite amount of energy in the heatfield of the earth. It came as a big surprise, when it has been discovered that certain kinds of sandstorm, so called grain ray bursts (GRB) had energies seemingly exceeding this upper bound, if a stone’s (grainized) theory was valid in this regime.
For a few hundred years this discovery aroused a lot of confusion and many scientists even started to doubt the spherical shape of the earth, until Re-Ez [@reez] remembered the forgotten singularity theorems of and $\;$ , [@Penhawk], for the heatfield . The former, , has also shown that such singularities will “radiate” large amounts of sand.
Assuming that the GRBs stem from the creation of sand due to the presence of a singularity of the heatfield he overcame all problems and eventually caused the (long overlooked) second revolution in the field of sand.
In fact, it soon became clear that there do exist many visible singularities of the heatfield. For instance, the sun is nothing but such a singularity, arising periodically and then moving at the sky. Due to the emittance of sand$^5$ , it disappears again after a certain time, depending on the state of the heatfield, i.e. the temperature. (In summer it will last longer than in winter.)
Quite recently, even the existence of a very massive heat-kernel in the center of earth has been established.
Most importantly and even more surprisingly, however, it turned out that many ordinary (non-GBR) sandstorms can be traced back to such singularities. This motivates our conjecture that [*all*]{} sandstorms are caused by such singularities.
Finally, the interpretation of the genesis of sandstorms as caused by singularities also resolved another puzzle:
Since sand is fermionic (it obviously respects the exclusion principle and, of course, the grains have spin $\frac{1}{2}$) and interacts only via heat-interactions, the system should be stable, seemingly in contradiction to the existence of sandstorms. However, sandstorms are only a local instability in the superfield, just like oases [@acta]. (For this reason, the energy stored in a sandstorm does not exceed the maximal available energy.) Globally, the system is in equilibrium! This fact might be of some importance.
Sandstorms Revisited
====================
So let’s assume that in the center of each sandstorm a singularity of the heatfield is sitting.$^6$ As it is well known, in the eye of a (sand)storm the wind vanishes, and thus it is commonly believed that $WV_H=V_HW=0$ holds even there. But that is wrong, since $V_H$ is infinite in the center of a sandstorm. Accordingly, the sand-wind-duality breaks down in this region, and “that is the poodle’s core ”[@fist].
Before we can explore this fact and its consequences in detail we should describe the mechanism of the emergence of these singularities, however. This mechanism is quite similar to the mechanism discovered by Msw the Elder [@msw], when he tried to understand the so-called solar-new-tiro puzzle. (The problem why the sun emits so few sand, which is solved by a resonance mechanism. At the same time the Msw-resonance explains why the sun emerges only periodically and why it is red around the times of its appearance, respectively disappearance.)
Thus, we immediately infer that small fluctuations in the heatfield will eventually lead, by a similar resonance, to the emergence of a singularity if we apply the following well-established result [@myself]:
[**Theorem 1.**]{} Under the above assumptions, there exists a point $x\in \mathbf{R}^3 $ in which the degree of singularity of the heatfield-distribution $\langle
\mathrm{sand} |h^{\mu\nu}
|\mathrm{sand} \rangle $ is strictly larger than $C_{storm}$.
Thus, the genesis of a sandstorm will be inevitable. Moreover, it is now evident, that the standard sand-wind duality is broken, viz. $$\lim_{r\to x }\langle \mathrm{sand} \label{result}
|V_H W |\mathrm{sand} \rangle (r) \neq 0.$$ Hence, [**the grains in a sandstorm are driven by the wind!**]{} This is the most nontrivial and surprising result of this paper. It can and should be verified empirically.
Conclusion and Outlook
======================
In this paper we presented in a highly suggestive and self-explaining way a new idea about the emergence of sandstorms. We have shown that the movement of the grainized sand is caused by the wind field. This result seems to be in consistency with the predictions of the original (but wrong) classical sand-theory. Note, however, that this “accordance” is not even qualitative: There are no similarities of the classical theory with our completely new and astonishing result (\[result\]).
Of course, this paper has only sketched some basic features of the new sand theory. A complete review is in preparation. There furthergoing questions about noncommutative sand, the super-connection between quicksand and worme holes and some quite interesting (although avantgardistic) new features in sand theory will be presented [@greek]. Moreover, we will overcome some of the flaws this paper is suffering from, e.g. a renormalization of the infantilities due to the self-citation [@myself] will be done. This works the following way: Assuming that the problem is already removed at the first order. Then it follows from an inductive proof that no infantilities will arise at any higher order due to recursive citations. Now, curing the problem at the first loop level is rather simple. This completes the proof of renormalizability of this work.
We should also admit that this work suffers from a considerable lack of computer assistance. Unfortunately, the current generation of the abacus (used here at the Pyramid College) seems unable to perform simulations of such complexity. Not to speak of resolving a heart-tree in the fog.
[**Acknowledgements**]{}\
I would like to thank my colleagues Mario Paschke and Volkmar Putz who made me aware of the reference [@myself]. Without their support and friendship I would not exist. Furthermore I have to thank my namesake P.P. Longstockings, from whom much inspiration has been drawn in making this paper: [*“No one really knows...”*]{}
[99]{}
see fig.1, facsimile taken from [**Sand in our times 15, 199985 .**]{}
$\ \ $
.
K. Murx, [*“Proletariat and Sand,”*]{} work in regress.
Abou Tsa Id, [*“The Wheel and its Possible Applications,”* ]{} [**Tuareg Journal of Modern Physics A 8000 ,**]{} stone tables 12-12.5.
Lei Li Ga, [*“But still it moves,”*]{} [**Crackpot Rev. Let. of Common Sand Vol. MCMDXXXIII.**]{}
Goe-The, [*“Fist I,”*]{} [**Reclam excavations,**]{} edited by the police department of Leipzig.
Edua ’Wit X, [*“M-, F-, Sand-Strings and all that*]{}”, [**Pyramide of Cheops, corridor 6B (periodically under water),**]{} table 1267.
Shrinkhead No. 7, [*“Some least interesting questions of sand”*]{}, [**Cameroon Jungle Drums 6000 .**]{}
reference unknown, but widely distributed.
He, who drank too much of the muddy water (if the hyroglyphes are translated correctly), [*“Do oases really exist? IV : The unreasonable effectiveness of washing clothes with sand.”*]{}, [**Acta Mesopotamica Vol. 4.8 %.**]{}
Pamdirac, [*“Spells on Grain mechanics”*]{}, [**Beduin Acad. Press.**]{}
A most recent review – though written in a somewhat old-fashioned language – can be found under:
[**www.marxist.com/science/dialecticalmaterialism.html**]{}
I. Zeil \[[*innumerable references*]{}\], [**Comm. in Priv. Mat. Vol. 12 - 400.**]{}
\[Author unknown\], [*“The Relative Generality Theory of the Heat Field,”*]{} [**found on a large stone near the Sahara.**]{}
Probably going back to Apu, who had told this to H. Simpson, who had told it to his wife, who had told it to her hair-cutter...
1237 anonymous authors, [*“On Temperature and Evaporation of apples stored in new and old tons,”*]{} [**Clear and Evident Research of Nubia,**]{} reprint of sheepskins 11-27.
Re-Ez, [*“I don’t get the problem ”*]{} contribution to the [**2. Masai-meeting of heat and wind**]{} 2000 .
& $\;$, [**Severe Lectures in Sand,**]{} Knight, 12 .
Msw the Elder [*“What the great god Fe y Nman told me,”*]{} [**Annals of Nil-Pollution, Thebia-Series,**]{} pp. 387-402.
P.P.Marvol [*“A Tip of the TOE,”*]{} physics/0309055.
X. Inachos & Y. Theseus, “[*A Brief Introduction to Brain Theory,*]{}” [**Artemis-Temple-Library of Athens**]{}, destroyed by the Spanish Inquisition.
Figure 1 (The so-called $\Omega^{- -}$-facsimile).
|
---
abstract: |
*Given a hypersurface $M$ of null scalar curvature in the unit sphere $\mathbb{S}^n$, $n\ge 4$, such that its second fundamental form has rank greater than 2, we construct a singular scalar-flat hypersurface in $\Rr^{n+1}$ as a normal graph over a truncated cone generated by $M$. Furthermore, this graph is 1-stable if the cone is strictly 1-stable.*
[**MSC 2000:**]{} 53C21, 53C42.
author:
- 'Jorge H. Lira'
- Marc Soret
title: 'Examples of scalar-flat hypersurfaces in $\mathbb{R}^{n+1}$'
---
Introduction
============
A consistent theme of research is the use of refined perturbation techniques in the study of constant mean curvature surfaces and metrics with positive constant scalar curvature. New and complex examples and deep results on structure of moduli space of solutions had been achieved with the aid of those techniques.
A kind of prototype of this type of construction may be found at the seminal paper [@CHS]. There, the authors prove the existence of minimal hypersurfaces with an isolated singularity in $\Rr^{n+1}$. These examples arise as perturbations of cones over minimal hypersurfaces of $\Ss^n$.
Our contribution here focuses on a similar construction but for scalar-flat singular hypersurfaces in Euclidean space $\Rr^{n+1}$. We consider a truncated cone $\bar M^*$ in $\mathbb{R}^{n+1}$ generated by a hypersurface $M$ of $\mathbb{S}^n$ that satisfies $S_2=0$ and then we take normal graphs over that cone. [*A priori*]{} estimates plus a fixed point theorem assure the existence of a graph with “small” boundary data which also satisfies the equation $S_2=0$.
We recall that $S_2$ is one of the elementary symmetric functions $S_r$, $1\le r \le n$, of the principal curvatures of a hypersurface in $\Rr^{n+1}$. An interesting feature of $S_2$ is that this curvature is [*intrinsic*]{} and coincides with the scalar curvature of the hypersurface.
Our aim here is to provide a test case that gives an evidence that the well succeeded perturbation methods alluded above may be also applicable to deal with some geometric problems involving fully nonlinear elliptic equations. The results we obtained are in some sense local. Global issues may be addressed only if we are able to overcome serious technical difficulties.
[**Theorem 1.**]{} *Let $M$ be a scalar-flat hypersurface in $\Ss^n$, $n\ge 4$. Suppose that the rank of the second fundamental form of $M$ is greater than or equal to $3$. Let $\psi$ be a function in $C^{2,\alpha}(M)$. There exists $\Lambda<1$ depending on $M$ such that for each $\lambda\in
[0,\Lambda)$ there exists a function $u_{\lambda}$ defined in $\bar
M^*$ whose graph $\bar M^*_\lambda$ has null scalar curvature and boundary given by $\Pi_J(u_\lambda)=\Pi_J(\lambda \psi)$, for some integer $J$.*
Here, $\Pi_J$ is the projection map defined in p. 10.
This paper has the following presentation. In Section 2, we deduce the null scalar curvature equation $\bar R(u)=0$ for the normal graph of a function $u$ defined over $\bar M^*$. The linearized equation involves the Jacobi operator $L$ in $\bar M^*$ which turns to be elliptic in view of the hypothesis concerning the rank of the second fundamental form of $M$. Section 3 is devoted to solve in $\bar M^*$ a Dirichlet problem for the Jacobi operator with boundary data $\psi$. Following closely [@CHS], the idea is that an adequate control of the data $f$ near the singular point in $\bar
M^*$ permits to solve $Lu=f$ in terms of separation of variables techniques. Second order estimates for the resulting Fourier series $u$ may be obtained in suitably weighted Hölder spaces. Applying these estimates to the problem $$\label{aux} Lu = Q(v),\quad u|_M = \psi,$$ where $v$ is a function in a weighted Hölder space and $Q$ collects all nonlinear terms in $\bar R(v)=0$, we reduce the nonlinear problem to that one of finding a fixed point for the map that associates $v$ to the solution of (\[aux\]). This is achieved by showing that for small boundary data $\psi$, this map is a contraction.
In the last section we relate the stability of the normal graphs with the stability of the hypersurface $M\subset \mathbb{S}^n$. There, stability refers to the functional $\mathcal{A}_1$ defined by the integral of the mean curvature.
[**Theorem 2.**]{} *If $\bar M^*$ is strictly $1$-stable, then the graph $\bar M^*_\lambda$ of the function $u_{\lambda}$ given in Theorem 1 is strictly $1$-stable for $\lambda$ sufficiently small.*
We point out that the results presented here may be easily adapted to the other higher order mean curvatures $S_r, \, r\ge 3$. It is interesting to produce examples with singular sets with small codimension as Nathan Smale did for minimal hypersurfaces in [@Smale]. This is the subject of current research by the authors.
The corrections and suggestions by the anonymous referee improved sensibly the reading of the paper. We express here our gratitude to him.
Scalar-flat cones
=================
The scalar curvature equation.
------------------------------
Let $M$ be a compact hypersurface of the unit sphere $\Ss^{n}$ in the Euclidean space $\Rr^{n+1}$. The [*cone*]{} over $M$ is the hypersurface $\bar M$ in $\Rr^{n+1}$ parametrized by $$X(t,\theta) = t\,\theta, \quad t\in \Rr^+,\, \theta\in M.$$ Let $N$ be an unit normal vector field to $M$. Parallel transporting $N$ along the rays $t\mapsto t\,\theta$ gives rise to a normal vector field to $\bar M$. One then defines the first and second fundamental forms of $\bar M$ respectively by $$I = \ae \dd X,\dd X\ad, \quad II =-\ae \dd N,\dd X\ad.$$ Let $x^1,\ldots, x^{n-1}$ be local coordinates in $M$ with corresponding coordinate vector fields denoted by $\partial_1,\ldots, \partial_{n-1}$. A local frame tangent to $\bar
M$ may be given by adding the vector field $\partial_t$ to that coordinate local frame. In terms of such a frame, the first quadratic form is represented by the matrix $$\big(\bar g_{\mu\nu}\big)=\left(\begin{array}{cc} t^2\,\theta_{ij} &
0
\\
0 & 1\end{array}\right)$$ and the second fundamental form has components $$\big(\bar b_{\mu\nu}\big)=\left(\begin{array}{cc} t\,b_{ij} & 0
\\
0 & 0\end{array}\right),$$ where $\theta_{ij}=\ae \partial_i,\partial_j\ad$ and $b_{ij}=-\ae
\partial_j N,\partial_j\ad$ are the components of the first and second fundamental forms of the immersion $M\subset\mathbb{S}^n$. Thus, the Weingarten map $\bar A$ of $\bar M$ has local components given by $\bar a^\mu_\nu = \bar g^{\mu\rho}\bar b_{\rho\nu}$. We then compute $$\big(\bar a^\mu_\nu\big)=\left(\begin{array}{cc}
\frac{1}{t}\,a^i_{j} & 0
\\
0 & 0\end{array}\right),$$ where $a^i_j = \theta^{ik}\, b_{jk}$ are the components of the Weingarten map $A$ of $M$ defined by $N_i = - a^j_i \partial_j$.
If we denote by $\lambda_1,\ldots, \lambda_{n-1}$ the eigenvalues of $A$, then the eigenvalues of $\bar A$ are $$0,\frac{1}{t}\,\lambda_1,\ldots, \frac{1}{t}\,\lambda_{n-1}.$$ The $r$-th mean curvature $\bar H_r$ of $\bar M$ is defined by $$\bar H_r = \frac{1}{{n \choose r}}\,\bar S_r,\quad 1\le r\le n,$$ where $\bar S_r$ are the elementary symmetric functions of the eigenvalues of $\bar A$ relative to $I$ given by $$\det\big(\textrm{Id}-s\,\bar A\big) =1 - s\, \bar S_1 + s^{2}\,\bar
S_2+\ldots + (-s)^{n-1} \bar S_{n-1}+ (-s)^n\, \bar S_n.$$ Denoting by $H_r$ and $S_r$ the corresponding functions on $M$, one easily proves that $$\bar S_r = \frac{1}{t^r}\, S_r, \quad 1\le r\le n-1$$ and $\bar S_n =0$. For a given multi-index $i_1<\ldots <i_r$ with $1\le i_k\le n$, we denote $$D_{i_1\ldots i_r} = \det\big(\theta_{1j}\ldots b_{i_1j}\ldots
b_{i_rj}\ldots \theta_{n-1j}\big),$$ that is, $D_{i_1\ldots i_r}$ is the determinant of the matrix obtained replacing in $(\theta_{ij})$ the columns numbered by $i_1,\ldots,i_r$ by the corresponding columns in $(b_{ij})$.
In terms of these determinants, one calculates $$\det(\theta_{ij})\, S_r =\sum_{i_1<\ldots<i_r}\, D_{i_1\ldots i_r}.$$ We suppose that $M$ satisfies $S_2=0$. Thus, the cone $\bar M$ is a scalar-flat manifold, that is, it holds that $\bar S_2=0$.
The scalar curvature equation for normal graphs over cones.
-----------------------------------------------------------
From now on, we will be mainly concerned with linearizing the equation $\bar S_2=0$ near $\bar M$. Given a function $u:\bar M\to \Rr$ with sufficiently small $C^2$ norm, its [*normal graph*]{} is defined as the hypersurface $$\bar M_u =\{ X(t,\theta) + u(t,\theta) \, N : t\in \Rr^+,\,\theta\in
M\}.$$ We denote by $\bar S_2(u)$ the scalar curvature of $\bar M_u$. We then proceed to linearize the equation $\bar S_2(u)=0$ and to describe the nonlinear part of this equation.
We begin by determining the quadratic fundamental forms in $\bar
M_u$. The tangent space to $\bar M_u$ is spanned by the vector fields $\theta + u_t \, N$ and $$\begin{aligned}
%t\,\big(\delta_i^j-\frac{u}{t}\, a^j_i\big)\, \partial_j + u_i\, N =
t\,\big(\delta_i^j-u\,\bar a^j_i\big)\,\partial_j + u_i\,N,\end{aligned}$$ where $u_t =\frac{\partial u}{\partial t}$ and $u_i =\frac{\partial
u}{\partial x^i}$. The induced metric in $\bar M_u$ has components $$\begin{aligned}
\bar g_{\mu\nu}(u)=\bar g_{\mu\nu} + \delta \bar
g_{\mu\nu},%\left(\begin{array}{cc} t^2\,\theta_{ij} -2tu\, b_{ij}+
%u^2 \,r_{ij}
%+ u_i u_j & u_t u_i\\
%u_i u_t & 1+ u_t^2\end{array}\right).\end{aligned}$$ where $$\begin{aligned}
\big(\delta \bar g_{\mu\nu}\big)=\left(\begin{array}{cc} -2 u \bar
b_{ij}+ u^2 \bar r_{ij}
+ u_i u_j & u_t u_i\\
u_i u_t & u_t^2\end{array}\right)\end{aligned}$$ and $\bar r_{ij} =t^2\theta_{kl} \bar a_i^k \bar a^l_j$ are the components of the third fundamental form $\ae \dd N,\dd N\ad$ of $\bar M$. More briefly, we may write $$\delta\bar g_{\mu\nu} =- 2u\bar b_{\mu\nu}+u^2\bar r_{\mu\nu} +
u_\mu u_\nu.$$ Let $\bar R_{\mu\nu}$ be the Ricci tensor of $\bar M$. If we denote $\bar R = S_2$ and $\bar R(u)= \bar S_2(u)$ then it follows that $$\begin{aligned}
\bar R (u) = \bar R + \delta \bar R,\end{aligned}$$ where $$\begin{aligned}
\delta \bar R = \bar g^{\mu\nu}\delta \bar R_{\mu\nu} +\delta \bar
g^{\mu\nu}\, \bar R_{\mu\nu}.\end{aligned}$$ A classical tensorial identity (see [@Bleecker], p. 398) states that $$\label{deltag} \bar g^{\mu\nu}\,\delta \bar R_{\mu\nu}
=\bar\nabla_\rho W^\rho$$ where $\bar\nabla$ denotes the Riemannian covariant derivative in $\bar M$ with respect to the metric $(\bar g_{\mu\nu})$ and $$W^\rho =\bar g^{\rho\sigma}\bar g^{\mu\nu}\bar\nabla_\nu \delta \bar
g_{\mu\sigma}- \bar g^{\rho\nu}\bar g^{\mu\sigma}\bar\nabla_\nu
\delta \bar g_{\mu\sigma}.$$ In what follows, we use the abbreviated notation $\bar\nabla^\rho
=\bar g^{\rho\mu}\bar\nabla_\mu$.
Since $\bar\nabla\bar g =0$ we may commute the covariant derivatives and the components $\bar g^{\mu\nu}$ in the formula above (\[deltag\]), obtaining $$\begin{aligned}
\bar g^{\mu\nu}\,\delta \bar R_{\mu\nu} & = & \bar\nabla_\rho\bar
g^{\rho\sigma}\bar g^{\mu\nu}\bar\nabla_\nu \delta \bar
g_{\mu\sigma}- \bar\nabla_\rho \bar g^{\rho\nu}\bar
g^{\mu\sigma}\bar\nabla_\nu \delta \bar
g_{\mu\sigma}\\
&=&\bar\nabla_\rho\bar\nabla^\mu \bar g^{\rho\sigma}\delta \bar
g_{\mu\sigma}-\bar\nabla^\nu
\bar\nabla_\nu \bar g^{\mu\sigma}\delta \bar g_{\mu\sigma}\\
& = & -2\bar\nabla_\rho\bar\nabla^\mu \bar g^{\rho\sigma} \bar
b_{\mu\sigma}u+2\bar\nabla^\nu \bar\nabla_\nu \bar g^{\mu\sigma}
\bar b_{\mu\sigma}u +Q_1\\
& = & -2\bar\nabla_\rho\bar\nabla^\mu \bar
a^{\rho}_{\mu}u+2\bar\nabla^\nu \bar\nabla_\nu \bar a^{\mu}_{\mu}u
+Q_1\\
& =&-2\bar\nabla_\rho\bar\nabla^\mu \bar
a^{\rho}_{\mu}u+2\bar\nabla^\nu \bar\nabla_\nu \bar S_1\,u
+Q_1 \\
& = & 2\bar\nabla_\rho\bar\nabla^\mu \big(\delta^{\rho}_{\mu}\bar
S_1-\bar a^{\rho}_{\mu}\big)u+Q_1 \\
&=& 2\bar\nabla^\rho \bar\nabla_\mu \bar T^\mu_\rho \, u+Q_1,\end{aligned}$$ where $\bar T^\rho_\mu$ are the components of the $(1,1)$ tensor field $$\bar T_1 = \bar S_1 \,\textrm{Id}-\bar A$$ and $$\begin{aligned}
Q_1 &=& \bar\nabla_\rho\bar\nabla^\mu \big(u^2 \bar r^\rho_\mu +
u^\rho u_\mu\big)-\bar\nabla^\rho\bar\nabla_\rho\big(u^2 \bar
r^\mu_\mu + u^\mu u_\mu\big).\end{aligned}$$ However, we have $$\begin{aligned}
& & \bar\nabla_\rho\bar\nabla^\mu \big(u^\rho u_\mu\big)
-\bar\nabla^\rho\bar\nabla_\rho\big(u^\mu u_\mu\big)=\bar
g^{\mu\nu}\bar g^{\rho\tau}\bar\nabla_{\rho}\bar\nabla_\nu(u_\tau
u_\mu)-\bar g^{\rho\tau}\bar g^{\mu\nu}
\bar\nabla_\tau\bar\nabla_\rho(u_\mu u_\nu)\\
& & \,\,\,\,=\bar g^{\mu\nu}\bar g^{\rho\tau}(u_{\tau;\nu\rho}
u_\mu+u_{\tau;\nu} u_{\mu;\rho}+u_{\tau;\rho} u_{\mu;\nu}+u_\tau
u_{\mu;\nu\rho})\\
& & \,\,\,\,\,\,\,\,\,-\bar g^{\rho\tau}\bar g^{\mu\nu}
(u_{\mu;\rho\tau} u_\nu+u_{\mu;\rho} u_{\nu;\tau}+u_{\mu;\tau}
u_{\nu;\rho}+u_\mu u_{\nu;\rho\tau})\\
& & \,\,\,\,=\bar g^{\mu\nu}\bar g^{\rho\tau}(u_{\tau;\nu\rho}
u_\mu+u_\tau u_{\mu;\nu\rho})-\bar g^{\rho\tau}\bar g^{\mu\nu}
(u_{\mu;\rho\tau} u_\nu+u_\mu u_{\nu;\rho\tau})\\
& & \,\,\,\,\,\,\,\,\,+\bar g^{\mu\nu}\bar g^{\rho\tau}(u_{\tau;\nu}
u_{\mu;\rho}+u_{\tau;\rho} u_{\mu;\nu})-\bar g^{\rho\tau}\bar
g^{\mu\nu} (u_{\mu;\rho} u_{\nu;\tau}+u_{\mu;\tau} u_{\nu;\rho}).\end{aligned}$$ Using Ricci identity $$u_{\nu;\tau\rho}- u_{\nu;\rho\tau}=\bar
R_{\tau\rho\sigma\nu}u^\sigma$$ where $\bar R_{\tau\rho\sigma\nu}$ is the Riemann curvature tensor in $\bar M$, we rewrite the terms with third order derivatives as follows $$\begin{aligned}
& & \bar g^{\mu\nu}\bar g^{\rho\tau}(u_{\tau;\nu\rho} u_\mu+u_\tau
u_{\mu;\nu\rho})-\bar g^{\rho\tau}\bar g^{\mu\nu} (u_{\mu;\rho\tau}
u_\nu+u_\mu u_{\nu;\rho\tau})\\
& & \,\,\,\, =\bar g^{\mu\nu}\bar g^{\rho\tau}(u_{\tau;\nu\rho}
u_\mu-u_\mu u_{\nu;\rho\tau})+\bar g^{\mu\nu}\bar g^{\rho\tau}u_\tau
u_{\mu;\nu\rho}-\bar g^{\rho\tau}\bar g^{\mu\nu} u_{\mu;\rho\tau}
u_\nu
\\
& & \,\,\,\, =\bar g^{\mu\nu}\bar g^{\rho\tau}(u_{\nu;\tau\rho}
u_\mu- u_{\nu;\rho\tau}u_\mu)+\bar g^{\mu\nu}\bar g^{\rho\tau}
u_{\mu;\nu\rho}u_\tau-\bar g^{\rho\tau}\bar g^{\mu\nu}
u_{\mu;\rho\tau} u_\nu\\
& & \,\,\,\,=\bar g^{\mu\nu}\bar g^{\rho\tau}\bar
R_{\tau\rho\sigma\nu}u^\sigma u_\mu+\bar g^{\mu\nu}\bar g^{\rho\tau}
u_{\mu;\rho\nu}u_\tau+\bar g^{\mu\nu}\bar g^{\rho\tau} \bar
R_{\nu\rho\sigma\mu}u^\sigma u_\tau-\bar g^{\rho\tau}\bar g^{\mu\nu}
u_{\mu;\rho\tau} u_\nu\\
& & \,\,\,\,=\bar g^{\rho\tau}\bar R_{\tau\rho\sigma\nu} u^\sigma
u^\nu+\bar g^{\mu\nu}\bar g^{\rho\tau} u_{\mu;\rho\nu}u_\tau+\bar
g^{\mu\nu} \bar R_{\nu\rho\sigma\mu}u^\rho u^\sigma -\bar
g^{\rho\tau}\bar g^{\mu\nu} u_{\mu;\rho\tau} u_\nu.\end{aligned}$$ The antisymmetry of the curvature tensor in the last two indices implies that $\bar g^{\rho\tau}\bar R_{\tau\rho\sigma\nu} u^\sigma
u^\nu=0$. Therefore, one has $$\begin{aligned}
& & \bar g^{\mu\nu}\bar g^{\rho\tau}(u_{\tau;\nu\rho} u_\mu+u_\tau
u_{\mu;\nu\rho})-\bar g^{\rho\tau}\bar g^{\mu\nu} (u_{\mu;\rho\tau}
u_\nu+u_\mu u_{\nu;\rho\tau})\\
& & \,\,\,\,=\bar g^{\mu\nu} u^{\rho} u_{\mu;\rho\nu}-\bar
g^{\rho\tau}u^{\mu} u_{\mu;\rho\tau} +\bar R_{\rho\sigma}u^\rho
u^\sigma\\
& & \,\,\,\,=\bar g^{\mu\nu} u^{\rho} u_{\mu;\rho\nu}-\bar
g^{\rho\tau}u^{\mu} u_{\rho;\mu\tau} +\bar R_{\rho\sigma}u^\rho
u^\sigma\\
& & \,\,\,\, =\bar R_{\rho\sigma}u^\rho u^\sigma.\end{aligned}$$ Thus, one concludes that $$\begin{aligned}
& & \bar\nabla_\rho\bar\nabla^\mu \big(u^\rho u_\mu\big)
-\bar\nabla^\rho\bar\nabla_\rho\big(u^\mu u_\mu\big) =\bar
R_{\rho\sigma}u^\rho u^\sigma+\bar g^{\mu\nu}\bar
g^{\rho\tau}(u_{\tau;\nu} u_{\mu;\rho}+u_{\tau;\rho}
u_{\mu;\nu})\\
& & \,\,\,\,\,\,\,\,-\bar g^{\rho\tau}\bar g^{\mu\nu} (u_{\mu;\rho}
u_{\nu;\tau}+u_{\mu;\tau} u_{\nu;\rho})\\
& & \,\,\,\,=\bar R_{\rho\sigma}u^\rho u^\sigma+u^\rho_{\,\,;\nu}
u^\nu_{\,\,;\rho}+u^\rho_{\,\,;\rho} u^\nu_{\,\,;\nu}-\bar
g^{\rho\tau} (u^\nu_{\,\,;\rho} u_{\tau;\nu}+u^\nu_{\,\,;\tau}
u_{\rho;\nu})\\
& & \,\,\,\, =\bar R_{\rho\sigma}u^\rho u^\sigma+u^\rho_{\,\,;\nu}
u^\nu_{\,\,;\rho}+u^\rho_{\,\,;\rho}
u^\nu_{\,\,;\nu}-u^\nu_{\,\,;\rho}
u^\rho_{\,\,;\nu}-u^\nu_{\,\,;\tau} u^\tau_{\,\,;\nu}\\
& & \,\,\,\,=\bar R_{\rho\sigma}u^\rho u^\sigma+u^\rho_{\,\,;\rho}
u^\nu_{\,\,;\nu}-u^\nu_{\,\,;\rho} u^\rho_{\,\,;\nu}.\end{aligned}$$ These calculations imply that $$\begin{aligned}
Q_1 & = & \bar\nabla_\rho\bar\nabla^\mu \big(u^2 \bar r^\rho_\mu
\big)-\bar\nabla^\rho\bar\nabla_\rho\big(u^2 \bar
r^\mu_\mu\big)+\bar R_{\rho\sigma}u^\rho u^\sigma+u^\rho_{\,\,;\rho}
u^\nu_{\,\,;\nu}-u^\nu_{\,\,;\rho} u^\rho_{\,\,;\nu}\\
& = & u^\rho_{\,\,;\rho} u^\nu_{\,\,;\nu}-u^\nu_{\,\,;\rho}
u^\rho_{\,\,;\nu} +2uu^\mu_{\,\,;\rho}(\bar
r^\rho_\mu-\delta^\rho_\mu \bar r)+u^\rho u^\mu (2\bar
r_{\rho\mu}-2\bar g_{\rho\mu}\bar r+\bar R_{\rho\mu})\\
& & \,\,\,\, +4uu^\rho(\bar r^\mu_{\rho;\mu}-\bar
r^\mu_{\mu;\rho})+u^2(\bar g^{\mu\nu}\bar r^\rho_{\mu;\nu\rho}-\bar
g^{\mu\nu}\bar r_{;\mu\nu}),\end{aligned}$$ where $\bar r = \bar r^\mu_\mu$.
It is a well-known fact that the tensor $\bar T_1$ is divergence-free. Indeed, one computes using Codazzi’s equation $$(\delta^\rho_\mu \bar S_1-\bar a^\rho_\mu)_{;\rho}= \delta^\rho_\mu
\bar a^\nu_{\nu;\rho}- \bar a^\rho_{\mu;\rho}= \bar a^\nu_{\nu;\mu}-
\bar a^\rho_{\rho;\mu}=0.$$ Using this, one gets $$\begin{aligned}
\bar g^{\mu\nu}\,\delta \bar R_{\mu\nu} & = & 2\bar\nabla^\rho\big(
(\bar\nabla_\mu \bar T^\mu_\rho \big) u+\bar T^\mu_\rho
(\bar\nabla_\mu u)\big)+Q_1= 2\bar\nabla^\rho \bar T^\mu_\rho
\bar\nabla_\mu u +Q_1\\
&=& 2\bar\nabla_\rho \big(\bar T_\mu^\rho \bar\nabla^\mu u \big)+Q_1
= 2\textrm{div}\,\bar T_1\bar\nabla u+Q_1.\end{aligned}$$ On the other hand, we infer from Gauss equation that $$\begin{aligned}
\bar R_{\mu\nu} = \bar g^{\rho\sigma}\,\bar
R_{\mu\rho\nu\sigma}=\bar g^{\rho\sigma}\big(\bar b_{\mu\nu}\bar
b_{\rho\sigma}-\bar b_{\mu\sigma}\bar b_{\nu\rho}\big)=\bar
b_{\mu\nu}\,\bar S_1-\bar r_{\mu\nu}\end{aligned}$$ and since $$\delta \bar g^{\mu\nu}\bar R_{\mu\nu}= \delta\bar
g^{\mu\nu}g_{\mu\rho}\bar R^\rho_\nu=- \delta g_{\mu\rho}\bar
g^{\mu\nu}\bar R^\rho_\nu= - \delta g_{\mu\rho}\bar R^{\mu\rho},$$ one obtains $$\begin{aligned}
\delta \bar g^{\mu\nu}\, \bar R_{\mu\nu} & = & 2u \bar S_1 \bar
b_{\mu\nu}\bar b^{\mu\nu} -2u\bar b_{\mu\nu}\bar
r^{\mu\nu}+Q_2=2u\bar S_1 \textrm{tr} \bar A^2 -2u\textrm{tr} \bar
A^3 +Q_2\\
&=& 2\textrm{tr}\big((\bar S_1 \textrm{Id}-\bar A)\bar A^2\big)+Q_2=
2\textrm{tr}\bar T_1\bar A^2+Q_2,\end{aligned}$$ where $$Q_2 = -\bar R^{\mu\rho}\big(u^2\bar r_{\mu\rho}+ u_\rho u_\mu\big).$$ Since we are assuming that $\bar S_2=0$ one easily verifies that $$\textrm{tr}\,\bar T_1\bar A^2 = -3\bar S_3.$$ We then conclude that the equation $\bar R(u)=0$ may be written as $$\label{nonl-eqtn} Lu + Q(u) =0,$$ where $$L u =\textrm{div} \,\bar T_1\bar\nabla u-3\bar S_3 u$$ is the Jacobi operator for the scalar curvature and $Q=Q_1+Q_2$.
The quadratic term $Q$ has the form $$\begin{aligned}
\label{Q} Q(u,\bar\nabla u,\bar\nabla^2 u) =|\Delta_{\bar M}
u|^2-|\bar\nabla^2 u|^2+\textrm{tr}\, P_0(u)\cdot\bar\nabla^2
u+P_1(u,\bar\nabla u)\end{aligned}$$ where $\Delta_{\bar M}$ is the Laplace-Beltrami operator in $\bar M$ and $$\begin{aligned}
(P_0)^\rho_\mu=2(\bar r^\rho_\mu-\delta^\rho_\mu \bar r)u\end{aligned}$$ and $$\begin{aligned}
& & P_1 = (2\bar r_{\rho\mu}-2\bar g_{\rho\mu}\bar r)u^\rho
u^\mu+4(\bar r^\mu_{\rho;\mu}-\bar r^\mu_{\mu;\rho})uu^\rho\\
& & \,\,\,\,+(\bar g^{\mu\nu}\bar r^\rho_{\mu;\nu\rho}-\bar
g^{\mu\nu}\bar r_{;\mu\nu}-\bar R^{\mu\nu}\bar r_{\mu\nu})u^2.\end{aligned}$$
The Dirichlet problem for the Jacobi operator.
==============================================
As we proved above, a normal graph $\bar M_u$ is scalar-flat if $u$ satisfies the fully nonlinear equation (\[nonl-eqtn\]). Our goal in this section is to solve the corresponding linearized equation for small boundary data by using Fourier analysis in some suitably weighted spaces.
Following the notation previously fixed, we denote $$\bar L_1 u = \textrm{div} \,\bar T_1\bar\nabla u.$$ The corresponding tensor and operator in $M$ are respectively $$T_1 = S_1 \textrm{Id}- A$$ and $$L_1 u = \textrm{div}\, T_1\nabla u,$$ where the divergence and gradient are taken this time on $M$. In [@BdC], it is proved that the operators $L$ and $\bar L_1$ decomposes as follows $$\bar L_1 u = \frac{1}{t}\, S_1 u_{tt} + \frac{n-2}{t^2} \, S_1 u_t
+ \frac{1}{t^3} \,L_1 u(t,\cdot)$$ and $$\label{Jtt} L u = \frac{1}{t}\, S_1 u_{tt} + \frac{n-2}{t^2} \, S_1
u_t + \frac{1}{t^3} \,L_1 u(t,\cdot)-3\frac{1}{t^3}S_3 u.$$ From now on, we assume that $S_3$ never vanishes along $M$ or equivalently that $\textrm{rk}A\ge 3$. In [@HL], it is proved that this assumption assures the ellipticity of the second-order differential operator $L$. This is a crucial ingredient in our analysis. We point out that there are examples of hypersurfaces fitting our assumptions in $\Ss^n$ like certain products of spheres.
As an example, if we fix the lowest dimension $n=4$, we may consider the product of spheres $M=\mathbb{S}^2(a_1)\times\mathbb{S}^1(a_2)$ immersed in $\mathbb{S}^4$, where $a_1 =\sqrt{1/3}$ and $a_2
=\sqrt{2/3}$. With these choices one has $S_2 =0$ and $$S_1 = 2\sqrt{2} -\sqrt{1/2}\quad \textrm{and}\quad S_3 =-\sqrt{2}.$$ For a detailed explanation on these products of spheres, we refer the reader to [@BdC].
We begin our analysis of the equation (\[nonl-eqtn\]) by solving first the non-homogeneous linear Dirichlet problem for the Jacobi operator $$\begin{aligned}
\label{jlinear} %\left\{ \begin{array}{ccc}
L u = f \,\, \textrm{in}\,\,\bar M^*,\quad u = \psi \,\,
\textrm{in}\,\, M
%\end{array}\right.\end{aligned}$$ where $\bar M^*$ is the truncated cone obtained restricting the variable $t$ to $(0,1]$. Using (\[Jtt\]), we reduce the linear equation $Lu = f$ to $$\label{radial} t^2 S_1 u_{tt} + (n-2) t S_1 u_t +L_1 u(t,\cdot) -
3S_3 u = t^3 f(t, \cdot).$$ The hypothesis on $S_3$ implies that $S_1$ also never vanishes. We then may choose an orientation for $M$ in such a way that $S_1 >
0$. Hence, the operator in $M$ defined by $$-S_1^{-1}(L_1 -3S_3)$$ has $L^2(M,S_1\dd \theta)$ discrete spectra given by a set of diverging eigenvalues $$\mu_1\le \mu_2\le\ldots\to +\infty$$ with corresponding eigenfunctions $\{\phi_1,\phi_2,\ldots\}$. These facts permit to separate variables in (\[radial\]) and reduce the problem to the determination of a Fourier series for $u$. We will see that a formal solution of (\[radial\]) in Fourier series gives rise to convergent solutions if we consider functions $f= f(t,\theta)$ such that $$\label{hyp1}
\vert f\vert _t ^2 : = \int_{M} f(t,\theta)^2
S_1(\theta)^{-1} \dd \theta <\infty,\quad t\in (0,1].$$ Let $m>2$ and $\epsilon>0$ be real constants to be chosen later. It is required too that the function $t\mapsto \vert f\vert_t$ satisfies $$\sup_{(0,1)} t^{2 -m -\epsilon} \vert f\vert_t <\infty.$$ This implies that $f(0,\cdot) =0$ and $$\label{hyp2} ||f||: =\left( \int^1_0 t^{4-2m} \vert f\vert_t^2 \dd t
\right)^{1/2}<\infty.$$ Under the assumptions above on $f$, it is possible to decompose it in its Fourier series $$\frac{f}{S_1} = \sum_{j=1}^\infty f_j(t) \phi_j(\theta)$$ with $f_j(t) = \int f\phi_j\,\dd \theta$. Let $u$ be a formal solution $$u(t,\theta) = \sum_j a_j(t) \phi_j(\theta)$$ of equation (\[radial\]). Thus, the coefficients $a_j$ are determined by the sequence of ODE’s $$\label{ode} t^2 a_j'' + (n-2) t a_j' - \mu_j a_j = t^3 f_j,\quad j=
1,2,\ldots$$ The homogeneous equations associated to (\[ode\]) have solutions of the form $t^{\gamma_j}$ where $\gamma_{j}$ is root of the characteristic equation $\gamma^2 + (n-3)\gamma - \mu_j = 0$. Its roots are the [*indicial roots*]{} $$\label{iroots} \gamma_j = -\frac{n-3}{2} \pm \sqrt{
\frac{(n-3)^2}{4}+\mu_j}.$$ We observe that $\gamma_j$ may be complex since $\mu_j$ may be negative. In these cases, one has $\Re \gamma_j = (3-n)/2$. Since the eigenvalues $\mu_j$ diverge to $+\infty$, there exists an index $J$ such that $\Re (\gamma_{J+1}) = \gamma_{J+1}
>0$. This index may be chosen so that for a given $m>2$ it holds that $$\frac{3-n}{2}\le\ldots\le \Re (\gamma_J) < m < \Re (\gamma_{J+1})\le
\Re (\gamma_{J+2})\le...$$ From now on, we consider these choices for $m$ and $J$.
In order to find a particular solution of the non-homogeneous equation (\[ode\]), we consider functions of the form $a_j(t) =
t^{\gamma_j}v_j(t)$. Plugging this expression of $a_j$ in (\[ode\]) we obtain $$t^{\gamma_j+2}v_j'' + (2\gamma_j + n - 2) t^{\gamma_j+1}v_j' = t^3
f_j$$ and after multiplying this equation by $t^{\gamma_j+ n-4}$ one has $$(t^{n-2 +2\gamma_j}v_j')' = t^{n-1+\gamma_j} f_j.$$ Integrating twice we get $$v_j = \alpha_j +\int^t_{\beta_j} s^{2-n-2\gamma_j}\int^s_0
\tau^{n-1+\gamma_j} f_j\,\dd\tau\,\dd s, \quad j=1,2,\ldots$$ where $\alpha_j$ and $\beta_j$ are constants of integration to be specified in the sequel. We conclude that the formal solution $u=\sum_j a_j \,\phi_j$ to equation (\[radial\]) has coefficients of the form $$\label{solution}
a_j(t) = \Re\bigg(\alpha_j t^{\gamma_j}+
t^{\gamma_j}\int^t_{\beta_j}s^{2-n-2\gamma_j}\,\int^s_0
\tau^{n-1+\gamma_j} f_j(\tau)\,\dd\tau\,\dd s\bigg).$$ We claim that the integrals in the definition of these coefficients converge in $(0,1]$ if we choose $\alpha_j = \beta_j =0$ for $j\le
J$ and $\beta_j=1$ for $j\ge J+1$. In fact, one has $$\begin{aligned}
f_j(t) = \int_M \frac{f}{\sqrt{S_1}}\phi_j \sqrt{S_1} \,\dd\theta
\le \sqrt{\int_M \frac{f^2}{S_1}\,\dd\theta} \,\,\sqrt{\int_M
\phi^2_j S_1\,\dd\theta}=|f|_t.\end{aligned}$$ Thus, using the hypothesis (\[hyp2\]) and Cauchy-Schwarz inequality we estimate, for a constant $c$ that does not depend on $f$, $$\begin{aligned}
\int^s_0 \tau^{n-1+\gamma_j}f_j(\tau)\,\dd\tau &\le & \sqrt{\int^s_0
\tau^{2(n-1+\gamma_j)}\tau^{2m-4}\, \dd\tau}
\,\,\sqrt{\int^s_0 \tau^{4-2m}|f|_\tau^2\,\dd\tau}\\
& = & c||f||s^{n-3+m+\gamma_j+\frac{1}{2}},\end{aligned}$$ where we used the fact that $m>\Re{\gamma_j}$ for $j\le J$ in order to assure convergence of the integral at $s=0$. This estimate implies that $$\begin{aligned}
\label{est-aj} t^{\gamma_j}\int^t_{\beta_j}
s^{2-n-2\gamma_j}\,\int^s_0 \tau^{n-1+\gamma_j}f_j(\tau)\,
\dd\tau\,\dd s \le c||f|| \,t^{\gamma_j}\int^t_{\beta_j}
s^{m-\gamma_j-\frac{1}{2}}\, \dd s.\end{aligned}$$ For $j\le J$, the right hand side converges at $t=0$ if one sets $\beta_j=0$. For $j\ge J+1$, it converges if we consider $\beta_j
=1$. This proves the claim.
The values of $\alpha_j$ for $j\ge J+1$ are determined by $$\begin{aligned}
\label{boundary} \alpha_j :=\int_{M}\lim_{t\to 1}
u(t,\cdot)\,\phi_j,\quad j\ge J+1.\end{aligned}$$ Let $\Pi_J$ be the projection of $L^2(M,S_1\dd\theta)$ in the linear subspace spanned by the eigenfunction $\phi_j,\,j\ge J+1$. Thus, $$\Pi_J (u) = \Pi_J(\psi)$$ if and only if $$\Pi_J(\psi) = \sum_{j=J+1}^\infty \alpha_j \phi_j.$$ Thus, since $\psi\in L^2(M,S_1\dd\theta)$, one has $$\sum_{j= J+1}^\infty \alpha_j^2 <\infty.$$ In this case, we then had verified that the problem (\[jlinear\]) has as solution the convergent Fourier series $u$ defined by the coefficients $a_j$ above.
In particular we have found a solution to the equation $Lu=0$ with boundary Dirichlet data $\psi$ referred to in what follows as the $L$-harmonic extension of $\psi$. In other terms we denote by $H_J
(\psi)$ the Fourier series solution of $$Lu=0 \,\,\textrm{ in }\,\, \bar M^*, \quad \Pi_J(u) = \Pi_J
(\psi)\,\,\textrm{ in }\,\, M.$$ Notice that our previous calculations imply that $$\label{HJ} H_J(\psi) = \sum_{j=J+1}^\infty \alpha_j
t^{\gamma_j}\phi_j.$$ and $H_J$ is a right inverse to $\Pi_J$.
In order to obtain integral estimates for $u$, we notice that since $$|uS_1|_t^2 =\int_M u^2(t,\theta)S_1(\theta)\,\dd\theta =
\sum_{j=1}^\infty a_j^2(t)$$ it follows that $$|u|_t^2\le c \sum_{j=1}^\infty a_j^2(t)$$ where $c= 1/(\inf_M S_1^2(\theta))$. On the other hand, using (\[solution\]) and (\[est-aj\]), one obtains from Cauchy-Schwarz inequality $$t^{-m}|u|_t \le t^{-m}\sqrt{\sum_{j=1}^\infty a_j^2(t)}\le c||f||
+\sqrt{\sum_{j=J+1}^\infty \alpha_j^2},$$ where $c>0$ is a positive constant which depends on $M, m$ and $J$. In a similar way, using (\[solution\]) and (\[HJ\]) one proves that $$t^{-m}|u-H_J(\psi)|_t \le c||f||.$$
We summarize the facts above in the following proposition.
\[C0\] Let $m>2$ be a constant and let $J$ be an integer such that $$0< \Re(\gamma_J) < m < \Re(\gamma_{J+1})$$ for $\gamma_j$ given by (\[iroots\]). Given a function $f$ defined in $\bar M^*$ satisfying $$\sup_{(0,1)}t^{2-m-\epsilon}|f|_t <\infty$$ and a function $\psi\in L^2(M,S_1\dd\theta)$, the series $$u= \sum_{j=1}^\infty a_j\, \phi_j$$ with $a_j$ defined by (\[solution\]) is the unique solution of $$\label{jpilinear} Lu= f \textrm{ in }
\bar M^*\quad \textrm{ and }\quad
\Pi_J(u) = \Pi_J(\psi) \textrm{ in } M$$ satisfying $$\label{ut} \sup_{(0,1)} t^{-m}|u|_t<\infty.$$ Moreover, we have the following estimates for $u$ $$\begin{aligned}
& & t^{-m}|u|_t \le c\,(||f|| + |\Pi_J(\psi)|),\nonumber\\
& & t^{-m}|u-H_J(\psi)|_t \le c\,||f||,\end{aligned}$$ where the constant $c$ does not depend on $f$.
[*Proof of the uniqueness.*]{} In view of the previous discussion, it remains to prove the uniqueness of the solution. If we consider two solutions $u_1$ and $u_2$ of the equation $Lu =f$, then their difference $v= u_1 - u_2$ is decomposed as $v = \sum_j
b_j \,\phi_j$ where the functions $b_j$ are solutions of the homogeneous ODE associated to (\[ode\]). Notice that (\[ut\]) implies that $u_1$ and $u_2$ vanish at the origin. Thus, $b_j(t)\to 0$ as $t\to 0$ for all $j$. Moreover, if $j\ge J+1$ then $\gamma_j$ is real and positive. So, $\mu_j$ is necessarily positive. Therefore the maximum principle guarantees that $b_j = 0$ for all $j\ge J+1$. For $j\le J$ we have that $b_j$ is of the form $b_j = c t^{\gamma_j} + \tilde c t^{\tilde \gamma_j}$ where $\gamma_j, \, \tilde \gamma_j$ are the roots of the characteristic equation. Thus $|t^{-m}b_j |\to \infty$ unless that $b_j = 0$ for $j\le J$. So, we have proved the proposition.
Following [@CHS] we now define some weighted Hölder spaces in terms of that it is possible to obtain second order estimates for the solution of the linear problem.
More precisely, we introduce as in [@CHS] and [@PR], the norms $$\begin{aligned}
\label{pnorms} & & |v|_{k,\alpha,t}= \sum_{l=0}^k t^l |\bar\nabla^l
v |_{0,A_t}+t^{k+\alpha}[\bar\nabla^k u]_{\alpha, A_t},\end{aligned}$$ for $t\in (0,1/2)$, $k$ a positive integer and $\alpha\in (0,1)$. Here, $A_t$ is the truncated cone corresponding to $t< |X| < 2t$ and $|\cdot|_{0,\alpha,A_t}$ denotes the usual Hölder norm in $A_t$.
\[C2\] Under the hypothesis of the Proposition \[C0\], the function $u$ satisfies $$\begin{aligned}
\label{C2est} & & t^{-m}|u|_{2,\alpha,t}\le c\,(||f||_{\alpha} +
|\Pi_J(\psi)|),\nonumber\\
& & t^{-m}|u - H_J(\psi)|_{2,\alpha,t} \le c\, ||f||_{\alpha},\end{aligned}$$ for $t\in (0,\frac{1}{2})$, $\psi\in C^{2,\alpha}(M)$ and $$||f||_{\alpha}\equiv \sup_{0<t<1/2}t^{2-m-\epsilon}|f|_{0,\alpha,t}$$ where $\epsilon$ is a fixed positive number. The constants do not depend on $f$.
[*Sketch of the proof.*]{} A similar estimate for the Laplacian could be found in [@P] and [@PR]. We may obtain the estimates for elliptic linear operators with constant coefficients and only second order terms. The general case could be handled by freezing coefficients in $L$. For usual Hölder norms, this method is nicely exposed in Chapters 4 and 6 of [@GT].
Solving the nonlinear problem
=============================
Using the weighted Hölder spaces we just defined above, we then introduce the subspace $B$ of $C^{2,\alpha}(\bar M^*)$ consisting of the functions $v$ for which $$\label{norm} ||v|| = \sup_{0<t<1/2} t^{-m}|v|_{2,\alpha,t}$$ is finite.
We define a map $U$ in the unit ball in $B$ in the following way: given a function $v\in B$ with $||v||<1$, $U(v)$ is the solution of the linear problem $$L U = Q(v) \,\, \textrm{ in }\,\, M^*, \quad \Pi_J(U) = \Pi_J
(\psi)\,\,\textrm{ in }\,\, M$$ as defined in Proposition 1. Our task now is to exhibit a convex subset $K$ of the unit ball in $B$ so that $U|_K$ is a contraction map.
With this purpose, we begin by estimating $Q(v)$ for $v$ with $||v||<1$. We have, using that $t<1$, $$\begin{aligned}
& & |Q(v)|_{0,\alpha,t}\le 2|\bar\nabla^2 v |^2_{0,\alpha,t} +
|P_0|_{0,\alpha,t}|\bar\nabla^2 v|_{0,\alpha,t} +
|P_1|_{0,\alpha,t} \\
%& &\,\,\,\,\le |M|_{0,\alpha,t}|\bar\nabla^2 v|^2_{0,\alpha,t}
%+t^{-1}
%|N|_{0,\alpha,t} |\bar\nabla^2 v|_{0,\alpha,t} + t^{-2}|P|_{0,\alpha,t} \\
& & \,\,\,\,\le 2 (t^{-2}|v|_{2,\alpha,t})^2 +
|P_0|_{0,\alpha,t}t^{-2}|v|_{2,\alpha,t} + |P_1|_{0,\alpha,t}\\
& & \,\,\,\,\le 2 t^{-4}|v|_{2,\alpha,t}^2 +
C_0t^{-2}|v|_{0,\alpha,t}|v|_{2,\alpha,t} + C_1 (|v|_{0,\alpha,t} + |\bar\nabla v|_{0,\alpha,t})^2\\
& & \,\,\,\,\le 2 t^{-4}|v|_{2,\alpha,t}^{2} + C_0 t^{-2}
|v|^2_{2,\alpha,t}+ C_1
(1+t^{-1})^2|v|^2_{2,\alpha,t} \\
& & \,\,\,\,\le \mu |v|_{2,\alpha,t}^2 \le \mu t^{2m}||v||^2,\end{aligned}$$ where $C_0, C_1$ and $\mu$ are positive constants depending only on $M$.
We choose $\epsilon$ such that $m+2\ge \epsilon$. Since $t<1$ we have $t^{2m}\le t^{m-2+\epsilon}$. Thus we obtain $$\label{Bnorm} |Q(v)|_{0,\alpha,t}\le \mu t^{m-2+\epsilon}||v||^2$$ and similarly one easily verifies that $$\label{BHnorm} |Q(v) - Q(w)|_{0,\alpha,t}\le \mu (||v||+ ||w||)(||v
- w||)t^{m-2+\epsilon}.$$ It follows from estimates stated in Proposition \[C2\] that $U(v)$ satisfies $$\begin{aligned}
||U(v) - H_J\psi|| &=& \sup_{0<t<1/2} t^{-m}|U(v) -
H_J\psi|_{2,\alpha,t}\le c
||f||_{\alpha}\\
& = & c \sup_{0<t<1/2}t^{2-m-\epsilon}|Q(v)|_{0,\alpha,t}\le c\mu
||v||^2.\end{aligned}$$ Moreover since $L(U(v)-U(w))= Q(v) -Q(w)$ and $\Pi_J (U(v))=
\Pi_J(U(w))$ then using the first estimate in Proposition 2 we obtain $$\begin{aligned}
||U(v) -U(w)|| & = & \sup_{t} t^{-m}|U(v) -U(w)|_{2,\alpha,t}\le
c||Q(v)-Q(w)||_{\alpha}\\ & = & c \sup_{t}
t^{2-m-\epsilon}|Q(v)-Q(w)|_{0,\alpha,t} \\
& \le & c\mu (||v|| +||w||)(||v -w||).\end{aligned}$$ In view of the last inequality, it is necessary to distinguish two cases. We suppose first that $c\mu <\lambda/2$ for some constant $\lambda<1$. Then, given $u,v$ with $||u||\le 1$ and $||v||\le 1$ we have $$||U(u) - U(v)|| \le \lambda
||u-v||.$$ Moreover, $$||U(v)|| \le c\mu ||v||^2 + ||H_J\psi||\le 1$$ if we assume that $$||H_J\psi||\le 1-c\mu ||v||^2.$$ Since $||v||\le 1$ the last inequality holds if we suppose $$\label{psi-1} ||H_J\psi||\le 1-c\mu,$$ which is true for suficiently small $\psi$. Hence, assuming this we conclude that $U|_K : K\to K$ is a contraction map where $K$ is the intersection of the unit open ball in $B$ with the affine subspace $\mathcal{P}=\{v\in B:\Pi_J v = \Pi_J \psi\}$. Notice that the smallness of $\psi$ also guarantees that $K$ is not empty.
Now, we suppose that $c\mu \ge 1/2$. In this case, we assume that $||v||\le a$ for some constant $a$ to determine. One gets $$||U(v)|| \le c\mu ||v||^2 +||H_J\psi|| \le c\mu a^2 +
||H_J\psi||.$$ Thus in order that $||U(v)|| \le a$ it is sufficient that $$c\mu a^2 -a + ||H_J\psi||\le 0.$$ Then $a$ must be choosen as $a\le \frac{1+\sqrt{1-4c\mu
||H_J\psi||}}{2c\mu}$. We must assume that $$||H_J\psi||\le \frac{1}{4c\mu}$$ in order to assure that the square root above is well-defined. Since $$\frac{1}{2c\mu}< \frac{1+\sqrt{1-4c\mu
||H_J\psi||}}{2c\mu},$$ we may choose $a=1/(2c\mu)$. So, we must suppose simultaneously that $||v||\le 1$ and that $||v||\le a$. However, the hypothesis $c\mu \ge 1/2$ implies that $a=1/(2c\mu) \le
1$. So, we prove that $U(K_1) \subset K_1$ and $U|_{K_1}$ is a contraction mapping, where $K_1$ is the intersection of the ball of radius $a$ in $B$ with the affine plane $\mathcal{P}$.
In both cases, we had just verified that $U$ defines a contraction map in properly chosen convex sets of the Banach space $B$. So, by Leray’s fixed point theorem (see, e.g., [@GT], Chapter 11), we assure the existence of a solution for the equation (\[nonl-eqtn\]).
Let $M$ be a scalar-flat hypersurface in $\Ss^n$, $n\ge 4$. Suppose that the rank of the second fundamental form of $M$ is greater than or equal to $3$. Let $\psi$ be a function in $C^{2,\alpha}(M)$. There exists $\Lambda<1$ depending on $M$ such that for each $\lambda\in
[0,\Lambda)$ there exists a function $u_{\lambda}$ defined in $\bar
M^*$ such that the graph $\bar M^*_\lambda$ of $u_{\lambda}$ has null scalar curvature and boundary given by $\Pi_J(u_\lambda)=\Pi_J(\lambda \psi)$, for some integer $J$.
Stability of scalar-flat cones
==============================
It is well-known that scalar-flat hypersurfaces in $\Rr^{n+1}$ are locally characterized as extrema of the action $$\mathcal{A}_1 = \int_{\bar M} \bar S_1\, \dd \bar M.$$ In this context, the Jacobi operator $L$ is naturally linked to stability of the hypersurface. For details, we refer the reader to [@Re], [@Ro] and [@BC].
In this section, we are concerned with the stability of the scalar-flat cones and graphs we had defined above. For that, we consider a function $u\in C^2_0(\bar M^*)$. The first and second variation formulae for $\mathcal{A}_1$ are: $$\begin{aligned}
& & \mathcal{A}_1'(0)= 0,\quad \mathcal{A}_1 ''(0) = -\int_{\bar
M^*} u\, Lu \,\mathrm{d}\bar M.\end{aligned}$$ We recall that the Jacobi operator in the last formula is $$\begin{aligned}
Lu =\bar L_1u - 3\bar S_3 u = S_1 t^{1-n} (t^{n-2}u_t)_t+
\frac{1}{t^3}\,(L_1 u(t,\cdot)-3S_3u).\end{aligned}$$ We decompose $u$ in its Fourier coefficients with respect to the eigenfunctions $\{\phi_j\}$ of $-\frac{1}{S_1}\big(L_1-3S_3)$ obtaining $u = \sum_j b_j \phi_j$ with $b_j(0)=b_j(1)=0$ and $$\begin{aligned}
& & L u = \sum_j S_1\,\big( t^{1-n}(t^{n-2}b_j')_t-t^{-3}\mu_j
b_j\big)\,\phi_j.\end{aligned}$$ Since the metric of $\bar M^*$ in spherical coordinates $(t,\theta)$ is written in the form $\mathrm{d}t^2 + t^2
\theta_{ij}\mathrm{d}\theta^i\otimes \mathrm{d}\theta^j$, one has $\dd \bar M = t^{n-1}\dd t \,\dd \theta$, where $\dd \theta$ is the volume form in $M$. Since $b_j(1)=0$, for all $j$, it results that $$\begin{aligned}
\int_{\bar M^*} u \, L u \,\dd \bar M & = & \sum_{j,k}\int^1_0
((t^{n-2}b_j')_t-t^{n-4}\mu_j b_j)b_k \int_{M} \phi_j\,\phi_k
S_1(\theta) \mathrm{d}\theta
\\
& = & \int^1_0 \sum_j((t^{n-2}b_j')_tb_j-t^{n-4}\mu_j
b_j^2) \,\mathrm{d}t\\
& = & -\int^1_0 \sum_j (t^{n-2}(b_j')^2 + t^{n-4}\mu_j
b_j^2)\,\mathrm{d}t.\end{aligned}$$ The first term in the last integral is given by $$\int_{\bar M^*} u_t^2 \bar S_1 \dd \bar M= \int_{\bar M^*} u_t^2
t^{-1}S_1\dd \bar M= \int^1_0 t^{n-2}\sum_j (b_j')^2 \dd t.$$ Denote $\mu_1^-=\max\{-\mu_1,0\}$, where $\mu_1$ is the smallest eigenvalue of the operator $-\frac{1}{S_1}(L_1-3S_3)$. Thus, one obtains $$\begin{aligned}
\label{ineq-1} & & \int_{\bar M^*} u \,L u\,\dd \bar M \le
-\int_{\bar M^*} u_t^2 \bar S_1\dd \bar M + \mu^{-}_1 \int^1_0
t^{n-4} \sum_ j b_j^2\,\mathrm{d}t.\end{aligned}$$ However, one has $$\int_{\bar M^*} u^2 t^{-2}\bar S_1 \dd \bar M= \int_{\bar M^*} u^2
t^{-3}S_1 \,\dd\bar M = \int^1_0 t^{n-4} \sum_j b_j^2 \,\dd t$$ and the expression on the right hand side of (\[ineq-1\]) may be calculated as follows $$\begin{aligned}
\int^1_0 t^{n-4}\sum_j b_j^2 \mathrm{d}t &=& \frac{1}{n-3}\int_0^1
(t^{n-3}\, \sum_j b_j^2)_t\,\mathrm{d}t
-\frac{2}{n-3}\,\int^1_0 t^{n-3}
\sum_j b_j b_j' \dd t\\
& \le & \frac{2}{n-3}\bigg( \int^1_0 t^{n-2}\sum_j (b_j')^2 \dd
t\bigg)^{1/2} \bigg(\int^1_0 t^{n-4}\sum_j b_j^2 \dd t\bigg)^{1/2}.\end{aligned}$$ Therefore, it follows that $$\begin{aligned}
\int_{\bar M^*} u^2 t^{-2}\bar S_1 \dd \bar M & = & \int^1_0
t^{n-4}\sum_j b_j^2 \,\dd t \le \frac{4}{(n-3)^2}\, \int^1_0
t^{n-2}\sum_j
(b_j')^2 \dd t\nonumber\\
& = & \frac{4}{(n-3)^2}\, \int_{\bar M^*} u_t^2 \bar S_1 \dd \bar
M. \label{c-s}\end{aligned}$$ Finally, we conclude that $$\int_{\bar M^*} u \,L u\,\dd \bar M \le
\left(\frac{4\mu_1^-}{(n-3)^2}-1\right)
\int_{\bar M^*} u_t^2 \bar S_1 \,\dd\bar M.$$ Suppose $n\ge 4$ and define $$\mu_{\bar M}:= (1- 4\mu^-_1/ (n-3)^2).$$ We suppose that $\mu_{\bar M}\ge 0$. Hence, it follows from (\[c-s\]) that $$\begin{aligned}
-\int_{\bar M^*} u \,L u\ge \mu_{\bar M} \, \int_{\bar M^*} u_t^2
\bar S_1 \dd \bar M \ge \mu_{\bar M} \frac{(n-3)^2}{4}\int_{\bar
M^*} u^2 t^{-2}\bar S_1 \dd \bar M.\end{aligned}$$ Now, we define the truncated cone $\bar M_{\sigma,\tau}$ as the set of points $t\theta$ in $\bar M^*$ with $0<\sigma < t < \tau\le 1$. Let $\lambda_{\sigma,1}$ be the smallest eigenvalue of the Dirichlet eigenvalue problem $$\begin{aligned}
L u + t^{-2}\bar S_1\lambda u = 0\,\,\, \textrm{on}\,\,\, \bar
M_{\sigma,1}, \quad u = 0 \,\,\,\textrm{on}\,\,\,
\partial \bar M_{\sigma,1}.\end{aligned}$$ Hence, we may characterize $\lambda_{\sigma,1}$ as the Rayleigh quotient $$\lambda_{\sigma,1}= -\inf_{u\in C^1_0(\bar M_{\sigma,1}),\,
u\not\equiv 0} \frac{\int_{\bar M^*} u \, Lu\, \dd \bar M }
{\int_{\bar M^*} \frac{u^2}{t^2} \bar S_1 \,\dd \bar M}.$$ We define $$I :=\inf_{u\in C^1_0(\bar M^*)} \bigg(-\int_{\bar M^*} u \, Lu \, \dd \bar M\bigg)$$ and $$I_+ :=\inf_{u\in C^1_0(\bar M^*),u\not\equiv 0} \frac{-\int_{\bar M^*} u \, Lu \, \dd \bar
M}{\int_{\bar M^*} \frac{u^2}{t^2}\bar S_1 \,\dd \bar M}.$$ Therefore, if $\mu_{\bar M}\ge 0$ (respectively, $\mu_{\bar
\mu}>0$), then $I\ge 0$ and $\inf_{\sigma} \lambda_{\sigma, 1}\ge 0$ (respectively, $I_+>0$ and $\inf_{\sigma} \lambda_{\sigma, 1}> 0$). In the first case, we say that $\bar M^*$ is $1$-stable. In the second case, $\bar M^*$ is said to be strictly $1$-stable.
Thus, we have proved that $\mu_{\bar M}\ge 0$ (respectively, $\mu_{\bar M}>0$) implies that $\bar M^*$ is $1$- stable (respectively, strictly $1$-stable).
Conversely, if $\mu_{\bar M} <0$, then $\bar M^*$ is not $1$-stable. In fact, in this case, we have $\mu_1 < -(n-3)^2/4$. Thus, the root $\gamma_1$ of $\gamma^2 + (n-3)\gamma - \mu_1 =0$ is not real. Moreover, the function $u_1 = \Re(t^{\gamma_1}\phi_1)$ is a Jacobi field, i.e., a solution for $\bar L_1 u - 3\bar S_3 u = 0$. Notice that $u_1(t,\theta)=0$ for all $\theta$ whenever $t^{\gamma_1}$ is a pure imaginary number. This happens if and only if $\ln t \Im \gamma_1 = k\pi/2$, where $k$ is a negative integer. Thus, we choose $\sigma, \tau$ so that $u_1(\sigma,\cdot)=u_1(\tau,\cdot)=0$ and define the test function for the Rayleigh quotient $$w(t,\theta) = u(t,\theta) \,\,\,\textrm{if}\,\,\, \sigma < t <
\tau\quad \textrm {and} \quad w = 0 \,\,\,\textrm{otherwise}.$$ It is clear that $w$ is a piecewise differentiable function which satisfies $$\int_{\bar M^*}\big( \langle \bar T_1 \bar\nabla w, \bar\nabla w
\rangle +3\bar S_3 w \big)\dd \bar M = 0.$$ So, $\lambda_{\sigma/2,1}<0$ since the compact support of $w$ is strictly contained in the truncated cone $\bar
M_{\frac{\sigma}{2},1}$. We conclude that $\inf_{\sigma}
\lambda_{\sigma,1}<0$.
In a similar way, we may prove that if $\mu_{\bar M}= 0$, then $\bar
M^*$ is not strictly $1$-stable.
These results can now be used to prove
If $\bar M^*$ is strictly $1$-stable, then the graph $\bar
M^*_\lambda$ of the function $u_{\lambda}$ given in Theorem 1 is strictly $1$-stable for $\lambda$ sufficiently small.
[*Proof.*]{} Let $\bar S_r(\lambda)$, $1\le r\le n$, denote the elementary symmetric functions of the eigenvalues of the Weingarten map $\bar A(\lambda)$ of $\bar M^*_\lambda$. We also denote $\bar T_1(\lambda)=\bar S_1(\lambda)\textrm{Id}-\bar
A(\lambda)$.
As $\bar S_3(\lambda)$ depends on the Hessian of $u_\lambda$, it follows from the $C^{2,\alpha}$ estimates on $u_\lambda$ given in Proposition 2 that $$\sup_{\lambda}
\frac{1}{\lambda} \sup_{\bar M^*} \frac{1}{t^3} \big(\bar
S_3(\lambda) - S_3\big)<\infty.$$ Consequently, for small $\lambda$, it holds that $$\int_{\bar M^*_\lambda}(\langle \bar T_1(\lambda) \bar\nabla u,
\bar\nabla u\rangle -\bar S_3(\lambda) u^2)\dd \bar M \ge \mu_{\bar
M}/2
>0,$$ for all $u\in C^{1}_0 (\bar M^*_\lambda)$ with $$\int_{\bar M^*_\lambda} \frac{u^2}{ t^2}\bar S_1(\lambda) \,\dd \bar M= 1.$$ This finishes the proof of the theorem.
[999]{}
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Leite, M. L.: [*The tangency principle for hypersurfaces with vanishing curvature functions.*]{} XI Escola de Geometria Diferencial, UFF (2000).
Pacard, F.: [*Connected sum construction in geometry and nonlinear analysis*]{} at http://perso-math.univ-mlv.fr/users/pacard.frank/Lecture-Part-I.pdf.
Pacard, F., Riviere, T.: [*Linear and nonlinear aspects of vortices. The Ginzburg-Landau model.*]{} Birkhauser, Boston, (2000).
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Jorge H. S. de Lira\
Departamento de Matemática\
Universidade Federal do Ceará\
Bloco 914, Campus do Pici\
60455-760, Fortaleza - Ceará, Brasil\
jorge.lira@pq.cnpq.br
Marc Soret\
Laboratoire de Mathématiques et Physique Théorique\
Université de Tours\
Parc de Grandmont, 37200, Tours, France\
Marc.Soret@lmpt.univ-tours.fr
|
---
abstract: |
We discuss the grounded, equipotential ellipse in two-dimensional electrostatics to illustrate different ways of extending the domain of the potential and placing image charges such that homogeneous boundary conditions are satisfied. In particular, we compare and contrast the Kelvin and Sommerfeld image methods.
author:
- |
H Alshal$^{1,2}$[^1], T Curtright$^{2}$[^2], and S Subedi$^{2}$[^3]\
$^{1}$Department of Physics, Cairo University, Giza, 12613, Egypt\
$^{2}$Department of Physics, University of Miami, Coral Gables, FL 33124-8046, USA
date: |
5 December 2018\
\
[Few things are harder to put up with than the annoyance of a good example. - Mark Twain]{}
title: 'Image Charges Re-Imagined'
---
Introduction {#introduction .unnumbered}
------------
When a source charge is placed near a real, grounded conductor, electrical charge flows between the ground and the conductor. In the static limit, for an idealized conductor, the resulting induced charge distribution is entirely on the surface of the conductor. In this ideal static situation the *interior* of the conductor is an equipotential containing no charge, and therefore not very interesting. However, for mathematical expediency, in some cases one can easily *imagine* a distribution of charge located entirely *inside* the conductor, instead of on the surface, which gives exactly the same *exterior* effects as the actual surface charge distribution.
All this is well-known, of course, but it may not be fully appreciated that the imagined distribution of charge within the conductor is *not* uniquely determined.[^4] Perhaps the most interesting aspect of this non-uniqueness lies in the mathematical freedom to choose the interior of an idealized conductor (i.e. the domain of the image charge and its potential) as an extension of the exterior (i.e. the domain of the real source charge and its potential) to be almost *any* imagined manifold, with the only essential restriction being that the image and source domains have in common a boundary, namely, the surface of the ideal conductor.[^5]
This somewhat surprising mathematical freedom can be illustrated by a simple example to be discussed below: The grounded two-dimensional (2D) ellipse. Two image methods, established long ago by [Thomson](http://en.wikipedia.org/wiki/William_Thomson,_1st_Baron_Kelvin) (a.k.a. Lord Kelvin) [@KelvinTait] and somewhat later by [Sommerfeld](https://en.wikipedia.org/wiki/Arnold_Sommerfeld) [@Sommerfeld], will be compared and contrasted. The image domains for these two methods have different geometries, but nevertheless give exactly the same physical results. The Kelvin method has the advantage that the [Green](https://en.wikipedia.org/wiki/George_Green_(mathematician)) function [@Green] is usually easier to extend from the source domain to the image domain. On the other hand, the Sommerfeld method has the advantage that the location of the image is always obvious given the location of the actual source charge.
Kelvin versus Sommerfeld images — a simple illustration {#kelvin-versus-sommerfeld-images-a-simple-illustration .unnumbered}
-------------------------------------------------------
The simplest example to illustrate the method of images is the problem of the grounded plane, or rather, for the purposes of this paper, its 2D analogue, the grounded line. The standard Green function on the entire plane follows from the logarithmic potential, $-\frac{1}{2\pi}\ln\left( \left\vert
\overrightarrow{r}\right\vert /R\right) $, which involves an arbitrary scale $R$. In terms of rectangular Cartesian coordinates on the entire plane, $-\infty<x<+\infty$ and $-\infty<y<+\infty$, with orthogonal unit vectors, $\widehat{x}$ and $\widehat{y}$, and with $\overrightarrow{r}=x~\widehat{x}+y~\widehat{y}$, the standard Green function is then$$g\left( x_{1},y_{1};x_{2},y_{2}\right) =-\frac{1}{4\pi}\ln\left(
\frac{\left( x_{1}-x_{2}\right) ^{2}+\left( y_{1}-y_{2}\right) ^{2}}{R^{2}}\right) \ . \label{g}$$ Here $\left( x_{1},y_{1}\right) $ is the field point and $\left( x_{2},y_{2}\right) $ is the source point. Note the symmetries $g\left( x_{1},y_{1};x_{2},y_{2}\right) =g\left( x_{2},y_{1};x_{1},y_{2}\right) =g\left( x_{1},y_{2};x_{2},y_{1}\right) =g\left( x_{2},y_{2};x_{1},y_{1}\right) $.
This $g$ is [a fundamental solution](https://en.wikipedia.org/wiki/Fundamental_solution) of the inhomogeneous equation$$\left( \frac{\partial^{2}}{\partial x_{1}^{2}}+\frac{\partial^{2}}{\partial
y_{1}^{2}}\right) g\left( x_{1},y_{1};x_{2},y_{2}\right) =-~\delta\left(
x_{1}-x_{2}\right) ~\delta\left( y_{1}-y_{2}\right) \ , \label{gEqn}$$ with a 2D Dirac delta source on the right-hand side (RHS). This Green function is therefore the logarithmic potential produced at the field point by an ideal point charge located at the source point. For a general source charge density on the plane, $\rho$, an electrostatic potential satisfying $$\left( \frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial
y^{2}}\right) \Phi\left( x,y\right) =-\rho\left( x,y\right) \ ,$$ with some implicit boundary conditions, is then given by$$\Phi\left( x,y\right) =\int_{-\infty}^{+\infty}dX\int_{-\infty}^{+\infty
}dY~g\left( x,y;X,Y\right) ~\rho\left( X,Y\right) \ .$$
To ground the line $y=a$, and obtain the Green function as well as a general potential on the half-plane $y>a$, such that both satisfy homogeneous Dirichlet boundary conditions for $y=a$, it suffices just to replace the Green function $g$ with the linear combination$$g_{o}\left( x_{1},y_{1};x_{2},y_{2}\right) =g\left( x_{1},y_{1};x_{2},y_{2}\right) -g\left( x_{1},2a-y_{1};x_{2},y_{2}\right) \ .
\label{Kelvingo}$$ so that $g_{o}\left( x_{1},a;x_{2},y_{2}\right) =0$. This result has a well-known interpretation, sometimes appropriately attributed to Kelvin, but most often with no attribution at all.
The interpretation follows from noting it is also true that $$g_{o}\left( x_{1},y_{1};x_{2},y_{2}\right) =g\left( x_{1},y_{1};x_{2},y_{2}\right) -g\left( x_{1},y_{1};x_{2},2a-y_{2}\right) \ .$$ Thus the first term on the RHS is interpreted as the potential at the field point $\left( x_{1},y_{1}\right) $ due to a point charge source at $\left(
x_{2},y_{2}\right) $ while the second term is interpreted as the potential at the field point due to a negative mirror image point charge source at $\left( x_{2},2a-y_{2}\right) $, the so-called Kelvin image. In this construction the half-plane $y>a$ has been extended to the full plane, including all $y<a$, to allow placement of the Kelvin image in the unphysical region below the grounded line. Consequently, for all $y>a$ the equation (\[gEqn\]) holds for $g_{o}$ as well as for $g$.
The corresponding grounded potential for a general charge density situated in the half-plane $y>a$ is then$$\Phi\left( x,y\right) =\int_{-\infty}^{+\infty}dX\int_{a}^{+\infty}dY~g_{o}\left( x,y;X,Y\right) ~\rho\left( X,Y\right) \ .$$ From (\[Kelvingo\]) it follows immediately that $\Phi\left( x,a\right)
=0$. Moreover, the contributions to $\Phi$ arising from the two terms in $g_{o}$ may then be interpreted respectively as due to the real source density $\rho\left( X,Y\right) $ above the grounded line, and an image source density $-\rho\left( X,2a-Y\right) $ below that line.
However, there are other ways to visualize the image charges. For example, the Euclidean plane may be *folded* along the grounded line to obtain two copies of the half-plane $y>a$, with the negative image charge now located on the second copy of the half-plane at the *same* position as the source point, namely, $\left( x_{2},y_{2}\right) $. This technique of employing a second copy of the physical space is due to Sommerfeld, following in the footsteps of Riemann to construct a branched manifold. The real beauty of Sommerfeld’s technique, in principle, is that doubling the physical space *obviously* works to provide the location of the image charges for all homogeneous boundary condition potential problems in any number of dimensions. But let’s not get ahead of ourselves.
As it happens, for this particularly simple example, there is essentially *no difference* in the two methods. Mostly this is just because the intrinsic geometry of the folded plane is indistinguishable from that of the unfolded plane. Nevertheless, it is instructive to exhibit analytically the parameterization of the folded space to be able to express the Green function in Sommerfeld’s approach. Here this is easily done: Represent the original half-plane by points $\left( x,y\right) =\left( x,a+w\right) $ for $w>0$ and the second copy of the half plane by points $\left( x,y\right) =\left(
x,a-w\right) $ for $w<0$. That is to say, the branched, folded plane is represented by the points $\left( x,y\right) =\left( x,a+\left\vert
w\right\vert \right) $ for $-\infty<w<+\infty$. It is then important to understand that point charges placed at the same $x$ but at different values of $w$ do *not* coincide, even though they may have the same $\left\vert
w\right\vert $. Such points with different $w$ but the same $\left\vert
w\right\vert $ are on opposite branches of the folded, doubled space.
The Green function on both branches of the folded space is now given by$$g\left( x_{1},w_{1};x_{2},w_{2}\right) =-\frac{1}{4\pi}\ln\left(
\frac{\left( x_{1}-x_{2}\right) ^{2}+\left( w_{1}-w_{2}\right) ^{2}}{R^{2}}\right) \ ,$$ for $-\infty<x_{1,2}<+\infty$ and $-\infty<w_{1,2}<+\infty$, and it again provides a fundamental solution of $$\left( \frac{\partial^{2}}{\partial x_{1}^{2}}+\frac{\partial^{2}}{\partial
w_{1}^{2}}\right) g\left( x_{1},w_{1};x_{2},w_{2}\right) =-~\delta\left(
x_{1}-x_{2}\right) ~\delta\left( w_{1}-w_{2}\right) \ .$$ But indeed, for this simple example, this $g$ is *exactly the same expression* as the previous Green function on the unfolded plane. Similarly, grounding the line at $y=a$ is now accomplished by the linear combination$$g_{o}\left( x_{1},w_{1};x_{2},w_{2}\right) =g\left( x_{1},w_{1};x_{2},w_{2}\right) -g\left( x_{1},-w_{1};x_{2},w_{2}\right) \ ,$$ Moreover, the potential on the half-plane $y>a$, for a general $\rho$ distributed on that same half-plane, with the line $y=a$ grounded, is now$$\Phi\left( x,w\right) =\int_{-\infty}^{+\infty}dX\int_{0}^{+\infty}dW~g_{o}\left( x,w;X,W\right) ~\rho\left( X,W\right) \ ,$$ where the field point is $\left( x,y\right) =\left( x,a+w\right) $ for $w>0$. The contributions arising from the two terms in $g_{o}$ may then be interpreted respectively as due to the real source density $\rho\left(
X,W\right) $ above the grounded line, and the image source density $-\rho\left( X,-W\right) $ also above the grounded line, but *on the opposite branch of the folded plane*.
We wish to emphasize that the grounded line example is unique in its simplicity as a 2D image system, since other examples have very different geometries for their Kelvin and Sommerfeld image domains. We consider next a situation where the alternative geometries of the combined source and image manifolds for the Kelvin and Sommerfeld approaches are not so simply related, namely, the grounded 2D ellipse.
Green functions for a 2D ellipse {#green-functions-for-a-2d-ellipse .unnumbered}
--------------------------------
This problem is nicely solved using complex analysis, as has been known since the 19th century (e.g. see the literature cited in [@Duffy]). However, here we use real variables in anticipation of higher dimensional situations. (Two Appendices discuss connections between our choice of real variables and those of the conventional complex plane.) In terms of real elliptic coordinates for the $xy$-plane[^6] as shown in the title page Figure,$$x=a\cosh u\cos v\ ,\ \ \ y=a\sinh u\sin v\ ,\ \ \ 0\leq u\leq\infty
\ ,\ \ \ 0\leq v\leq2\pi\ . \label{EllipticCoords}$$ Remarkably, the standard method to construct the 2D Laplacian Green function as sums of harmonic functions (e.g. see [@AlshalCurtright; @CurtrightEtAl]) now leads to an unusual form for the result. $$G\left( u_{1},v_{1};u_{2},v_{2}\right) =-\frac{1}{4\pi}\left\vert
u_{1}-u_{2}\right\vert -\frac{1}{4\pi}\ln\left( 1+e^{-2\left\vert u_{1}-u_{2}\right\vert }-2e^{-\left\vert u_{1}-u_{2}\right\vert }\cos\left(
v_{1}-v_{2}\right) \right) \ . \label{G}$$ Note that in addition to being $2\pi$-periodic[^7] in each of the $v$s this Green function also has the following symmetries similar to those for $g$ above: $\ G\left(
u_{1},v_{1};u_{2},v_{2}\right) =G\left( u_{2},v_{1};u_{1},v_{2}\right)
=G\left( u_{1},v_{2};u_{2},v_{1}\right) =G\left( u_{2},v_{2};u_{1},v_{1}\right) $. By construction, $G$ is again a fundamental solution of the equation[^8] $$\left( \frac{\partial^{2}}{\partial u_{1}^{2}}+\frac{\partial^{2}}{\partial
v_{1}^{2}}\right) G\left( u_{1},v_{1};u_{2},v_{2}\right) =-~\delta\left(
u_{1}-u_{2}\right) ~\delta\left( v_{1}-v_{2}\right) \ , \label{GEqn}$$ and it incorporates some implicit boundary conditions. For example, all $v$ dependence in $G$ is exponentially suppressed as either $u_{1}$ or $u_{2}$ become infinite, with the other $u$ fixed.
It is interesting to compare (\[G\]) to the more well-known form given in (\[g\]). This is easily done using the elementary identity$$\begin{aligned}
& \left( \cosh u_{1}\cos v_{1}-\cosh u_{2}\cos v_{2}\right) ^{2}+\left(
\sinh u_{1}\sin v_{1}-\sinh u_{2}\sin v_{2}\right) ^{2}\nonumber\\
& =\left( \cosh\left( u_{1}-u_{2}\right) -\cos\left( v_{1}-v_{2}\right)
\right) \left( \cosh\left( u_{1}+u_{2}\right) -\cos\left( v_{1}+v_{2}\right) \right) \ . \label{AnID}$$ Upon converting $x_{1,2}$ and $y_{1,2}$ to the elliptic coordinates in (\[EllipticCoords\]), this identity gives$$\begin{aligned}
g\left( x_{1},y_{1};x_{2},y_{2}\right) & =-\frac{1}{4\pi}\ln\left(
\frac{a^{2}}{R^{2}}\left( \cosh u_{1}\cos v_{1}-\cosh u_{2}\cos v_{2}\right)
^{2}+\left( \sinh u_{1}\sin v_{1}-\sinh u_{2}\sin v_{2}\right) ^{2}\right)
\nonumber\\
& =-\frac{1}{4\pi}\ln\left( 2\cosh\left( u_{1}-u_{2}\right) -2\cos\left(
v_{1}-v_{2}\right) \right) -\frac{1}{4\pi}\ln\left( \frac{a^{2}}{2R^{2}}\left( \cosh\left( u_{1}+u_{2}\right) -\cos\left( v_{1}+v_{2}\right)
\right) \right) \nonumber\\
& =G\left( u_{1},v_{1};u_{2},v_{2}\right) -\frac{1}{4\pi}\ln\left(
\frac{a^{2}}{2R^{2}}\left( \cosh\left( u_{1}+u_{2}\right) -\cos\left(
v_{1}+v_{2}\right) \right) \right)\end{aligned}$$ Therefore, for $u_{1}+u_{2}\neq0$ and real $v_{1}+v_{2}$, the difference $g-G$ is a non-singular, harmonic function, as must be the case for two fundamental solutions of (\[GEqn\]).
### The Kelvin image method {#the-kelvin-image-method .unnumbered}
Characterized generally, albeit rather vaguely, the Kelvin image method makes use of both the interior *and* the exterior of the ellipse, placing source and image charges in opposite regions so as to satisfy boundary conditions. In the elliptic coordinate frame, an obvious construction of a Green function for a grounded ellipse is given by the linear combination$$G_{o}\left( u_{1},v_{1};u_{2},v_{2}\right) =G\left( u_{1},v_{1};u_{2},v_{2}\right) -G\left( u_{1},v_{1};2U-u_{2},v_{2}\right) \label{KelvinGo}$$ where the grounded ellipse consists of points given by $\left( U,v\right) $ for a fixed $U$ and $0\leq v\leq2\pi$. By construction, $G_{o}\left(
u_{1},v_{1};U,v_{2}\right) =0$ for all $v_{2}$. From the symmetry of $G$ it is also true that $G_{o}\left( U,v_{1};u_{2},v_{2}\right) =0$ for all $v_{1}$. Some contour plots of $G_{o}$ are given in Appendix C, for $U=1$ and some representative field points.
For a general distribution of source charge either inside or outside the grounded ellipse, as given by $\rho\left( u,v\right) $, the solution of $$\left( \frac{\partial^{2}}{\partial u^{2}}+\frac{\partial^{2}}{\partial
v^{2}}\right) \Phi\left( u,v\right) =-k~\rho\left( u,v\right)$$ is then reduced to the evaluation of an integral involving $G_{o}$ and $\rho$. In particular, for field points and actual sources outside the grounded ellipse, the electric potential is $$\Phi\left( u_{1},v_{1}\right) =k\int_{U<u_{2}\leq\infty}\int_{0\leq
v_{2}\leq2\pi}G_{o}\left( u_{1},v_{1};u_{2},v_{2}\right) \rho\left(
u_{2},v_{2}\right) du_{2}dv_{2}\ .$$ Here we have introduced $k$ as a 2D analogue of the Coulomb constant.
The first $G$ in (\[KelvinGo\]) is universally interpreted as the potential at field point $\left( u_{1},v_{1}\right) $ produced by a positive unit point source at location $\left( u_{2},v_{2}\right) $. The second $G$ in (\[KelvinGo\]) is similarly interpreted as the potential at field point $\left( u_{1},v_{1}\right) $ produced by another point-like, but in this case negative, *Kelvin image* at location $\left( 2U-u_{2},v_{2}\right)
$. However, for the grounded ellipse construction in (\[KelvinGo\]) there are some interesting — perhaps unexpected — features.
For both field and source points inside the grounded ellipse, such that $0\leq
u_{1},u_{2}\leq U$, the Kelvin image is always outside that ellipse with $U\leq2U-u_{2}\leq2U$, and therefore the image is *never* located at infinity[^9] as long as both $a\neq0$ and $U\neq\infty$. That is to say, to implement an interior Green function construction inside a grounded ellipse at $u=U$, it suffices to use a single point-like Kelvin image that lies between the confocal ellipses at $u=U$ and $u=2U$. As expected, the image is outside the source domain defined by $0\leq u\leq U$. In any case, only *one* copy of the plane $\mathbb{E}_{2}$ is sufficient for the construction of the interior Green function.
On the other hand, for field and source points outside the grounded ellipse, such that $U\leq u_{1},u_{2}\leq\infty$, the Kelvin image is inside that ellipse, with $0\leq2U-u_{2}\leq U$, only so long as the source is not too distant from the grounded ellipse. That is to say, the interior of the original grounded ellipse contains the image only for $u_{2}\leq2U$. But if the source is more distant, with $u_{2}>2U$, the chosen Kelvin image of the point source passes through the line connecting the two foci and moves onto a *second* copy of $\mathbb{E}_{2}$ as also defined by (\[EllipticCoords\]) except with negative $u$. Therefore, for the point-like Kelvin image construction of the complete exterior Green function as expressed in (\[KelvinGo\]), *two* copies of the real plane are required: One for $u>0$ and another for $u<0$. Effectively, the two elliptical foci on the $x$-axis at $x=\pm a$ are connected by a straight line segment that acts as a branch line doorway joining together these two copies of $\mathbb{E}_{2}$.
So, to solve the exterior electric potential problem for a grounded ellipse, when real coordinates are used and point-like Kelvin images are located in an obvious way, a branched manifold is necessarily encountered. To put it another way, the actual, real interior of a grounded 2D elliptical conductor is insufficient to accommodate the location of a single point-like Kelvin image for an exterior point source, when that source is far from the conductor. More interior space is needed! All this is represented graphically in Figure 1.
\
Figure 1: Representative trajectories for exterior sources (orange) and their Kelvin images (green) for a grounded ellipse (red) with $U=3/2$. As a point source moves away from the red ellipse along one of the orange curves, its image moves away from the red ellipse along a corresponding (connected) green curve. A straight line segment between the foci is shown in blue.
Another way to see these features for Kelvin images is through the use of conformal mapping. By mapping a circle onto an ellipse, the standard Kelvin image solution for a grounded circle is mapped onto an image solution for the grounded ellipse. (Please see Appendix B.)
### Mapping an infinite cylinder onto planes {#mapping-an-infinite-cylinder-onto-planes .unnumbered}
What is at work here is the fact that $G$ in (\[G\]) is really a Green function not just for the semi-infinite cylinder, with $u\geq0$, but actually provides a solution to (\[GEqn\]) for the infinite $uv$-cylinder, where $-\infty\leq u\leq+\infty$, along with $0\leq v\leq2\pi$. So no matter where the source is placed on that infinite cylinder, to construct $G_{o}$ such that it vanishes at a fixed value of $u$, there is always room to accommodate a Kelvin point image.
The only open issue is then how to map the infinite $uv$-cylinder onto one or more copies of the $xy$-plane. Sticking with the $x\left( u,v\right) $ and $y\left( u,v\right) $ relations in (\[EllipticCoords\]) gives a map that produces two copies of $\mathbb{E}_{2}$ as represented by the embedding shown in Figure 1[^10] for the case $U=3/2$. The original infinite $uv$-cylinder is flared out by the map onto $x$ and $y$, both for large positive and for large negative $u$, but pinched down to a straight line segment connecting the foci at $x=\pm a$ when $u=0$, with that segment situated below the grounded ellipse at $u=U$ ($=3/2$ in the Figure). This is the geometry that underlies the Kelvin image method as applied here. The pinched line segment has some obviously singular geometric features, but these are not pathological.
On the other hand, there is another clear choice to map the infinite $uv$-cylinder onto planes that gives a different geometry. Rather than pinch the cylinder shut in terms of $x$ and $y$, at $u=0$ or some other value of $u$, the cylinder may be folded around the location of the grounded ellipse so that the submanifold below the fold is just a mirror image of the submanifold above the fold. (Please see Figure 2.[^11]) This leads to the Sommerfeld image method which we describe in detail in the following. The fold also has some obviously singular geometric features, but again these are not pathological.
### The Sommerfeld image method {#the-sommerfeld-image-method .unnumbered}
Consider the same exterior Green function situation using Sommerfeld images. (The history of this alternate method is discussed in [@Eckert].) In this approach, the interior of the ellipse is eliminated, and two copies of the plane outside the grounded ellipse are joined together along the grounded ellipse.
The new parameterization of both copies of the $xy$-plane outside the ellipse with $u=U>0$, again written in terms of real elliptic coordinates, is[^12]$$\begin{aligned}
u & =U+\left\vert w\right\vert \ ,\\
x & =a\cosh\left( U+\left\vert w\right\vert \right) \cos
v\ ,\ \ \ y=a\sinh\left( U+\left\vert w\right\vert \right) \sin
v\ ,\ \ \ -\infty\leq w\leq\infty\ ,\ \ \ 0\leq v\leq2\pi\ .\end{aligned}$$ So, when both field and source points are on the upper branch of the surface, such that $0<w_{1},w_{2}<\infty$, the Green function is$$G\left( w_{1},v_{1};w_{2},v_{2}\right) =-\frac{1}{4\pi}\left\vert
w_{1}-w_{2}\right\vert -\frac{1}{4\pi}\ln\left( 1+e^{-2\left\vert w_{1}-w_{2}\right\vert }-2e^{-\left\vert w_{1}-w_{2}\right\vert }\cos\left(
v_{1}-v_{2}\right) \right) \ .$$ But when the field point is on the upper branch and the source is on the lower branch, albeit with the same convention $0<w_{1},w_{2}<\infty$, the Green function is$$G\left( w_{1},v_{1};-w_{2},v_{2}\right) =-\frac{1}{4\pi}\left( w_{1}+w_{2}\right) -\frac{1}{4\pi}\ln\left( 1+e^{-2\left( w_{1}+w_{2}\right)
}-2e^{-\left( w_{1}+w_{2}\right) }\cos\left( v_{1}-v_{2}\right) \right)
\ .$$ In this approach the exterior Green function for a grounded ellipse is the linear combination$$\begin{aligned}
G_{o}\left( w_{1},v_{1};w_{2},v_{2}\right) & =G\left( w_{1},v_{1};w_{2},v_{2}\right) -G\left( w_{1},v_{1};-w_{2},v_{2}\right)
\label{SommerfeldGo}\\
& =-\frac{1}{4\pi}\left\vert w_{1}-w_{2}\right\vert +\frac{1}{4\pi}\left(
w_{1}+w_{2}\right) -\frac{1}{4\pi}\ln\left( 1+e^{-2\left\vert w_{1}-w_{2}\right\vert }-2e^{-\left\vert w_{1}-w_{2}\right\vert }\cos\left(
v_{1}-v_{2}\right) \right) \nonumber\\
& +\frac{1}{4\pi}\ln\left( 1+e^{-2\left( w_{1}+w_{2}\right) }-2e^{-\left(
w_{1}+w_{2}\right) }\cos\left( v_{1}-v_{2}\right) \right) \ ,\nonumber\end{aligned}$$ assuming that both field and source points are on the upper branch, i.e. $0\leq w_{1},w_{2}\leq\infty$. Otherwise, $G\left( w_{1},v_{1};w_{2},v_{2}\right) =G\left( w_{2},v_{2};w_{1},v_{1}\right) $ and $G_{o}\left(
-w_{1},v_{1};w_{2},v_{2}\right) =-G_{o}\left( w_{1},v_{1};w_{2},v_{2}\right) $.
Remarkably, as the reader may readily verify, the expressions (\[KelvinGo\]) and (\[SommerfeldGo\]) give exactly the same functions on the $xy$-plane when both field point $\left( x_{1},y_{1}\right) $ and source point $\left(
x_{2},y_{2}\right) $ are located outside the grounded ellipse and on the upper $\mathbb{E}_{2}$ branch, despite the differences in the Kelvin and Sommerfeld image locations as evident upon comparing Figure 1 with the following Figure 2.
\
Figure 2: Representative trajectories for exterior sources (orange) and their Sommerfeld images (green) for a grounded ellipse (red), again with $U=3/2$. All $\left( x,y\right) $ points inside the red ellipse are excluded from the 2D manifold in this method.
Visualization of the features in these 3D Figures — especially their differences — may be easier if 2D vertical slices are considered. In Figure 3, the source and image domains along the $y$-axis are shown in green for the Kelvin method and in orange for the Sommerfeld method. Particular choices for point sources and their images are shown as small circles, squares, or diamonds, for an ellipse whose $x=0$ points are shown in red. The source domain is always the same — namely, the planar region outside the grounded ellipse — no matter what image method is under consideration, so the orange and green curves in the Figure are the same for $u>3/2$ or $w>0$.
In Figure 4, the source and image domains along the $x$-axis are shown in green for the Kelvin method and in orange for the Sommerfeld method. As before, particular choices for point sources and their images are shown as small circles, squares, or diamonds, and the $y=0$ points on the ellipse are shown in red. Once again, the source domain is always the same no matter what image method is under consideration, but the image domains differ, depending on how the manifold is extended beyond the source domain.
\
Figure 3: Source and image domains for $x=0$, as solid and dashed curves, respectively.
\
Figure 4: Source and image domains for $y=0$, as solid and dashed curves, respectively.
To summarize, it seems fair to say the image domain is largely determined just by one’s imagination.
Induced charge density {#induced-charge-density .unnumbered}
----------------------
The actual linear charge density induced on the grounded ellipse is proportional to the normal component of the electric field evaluated in the limit where the field point approaches the ellipse. It suffices to consider the density induced by a unit point source outside the ellipse. Then the relevant normal electric field is just $\left. -\partial G_{o}/\partial
u_{1}\right\vert _{u_{1}=U}$ for the Kelvin image method, or $\left.
-\partial G_{o}/\partial w_{1}\right\vert _{w_{1}=0}$ for the Sommerfeld image method. The results are the same, using either method. The situation of interest for the external problem involves a unit source at $u_{2}>U$ or $w_{2}>0$.
In terms of the result for the Sommerfeld method, (\[SommerfeldGo\]), we find the linear charge density$$\lambda\left( v_{1};w_{2},v_{2}\right) =-\left. \frac{\partial}{\partial
w_{1}}G_{o}\left( w_{1},v_{1};w_{2},v_{2}\right) \right\vert _{w_{1}=0,\ w_{2}>0}=\frac{1}{2\pi}\frac{e^{-2w_{2}}-1}{e^{-2w_{2}}-2e^{-w_{2}}\cos\left( v_{1}-v_{2}\right) +1}\ . \label{LinearDensity}$$ $\allowbreak$Note that the total charge induced by a $+1$ source is always $-1$, $$\int_{0}^{2\pi}\lambda\left( v_{1};w_{2},v_{2}\right) dv_{1}=-1\ ,$$ even if the unit source is removed to infinity.[^13] In that infinite limit, the induced charge density becomes constant around the ellipse.$$\lambda\left( v_{1};w_{2},v_{2}\right) \underset{w_{2}\rightarrow
\infty}{\sim}-\frac{1}{2\pi}\ .$$
Plots of the charge density for various selected source distances from the grounded ellipse are straightforward to produce and evince all the expected features when expressed in terms of our chosen elliptic coordinates.
\
Figure 5: $\ \lambda$ as a function of $v=v_{1}-v_{2}$ for various $w_{2}$. Specifically, $w_{2}=1/2$ red, $w_{2}=1$ orange, $w_{2}=2$ sienna, $w_{2}=4$ brown.
A straight line limit {#a-straight-line-limit .unnumbered}
---------------------
A straight line limit of the ellipse is achieved by first setting $v=\pi/2$ in (\[EllipticCoords\]) so that $x\equiv0$, and then letting $a\rightarrow
\infty\ $and$\ u\rightarrow0$ so that $\lim_{a\rightarrow\infty,u\rightarrow
0}\left( a\sinh u\right) =y$ remains finite. The essential idea is that as $a\rightarrow\infty$ the elliptical $\left( u,v\right) $ coordinates near the center of the $x$-axis become just rectangular Cartesian coordinates, $\left( x,y\right) $. This behavior is evident in the title page Figure, even for finite $a$.
That is to say, let $u=y/a$ as $a\rightarrow\infty$ so that$$a\sinh u\rightarrow ay/a=y\ .$$ At the same time, let $U=Y/a$ for $v=\pi/2$. Then $y\left( U,\pi/2\right)
\rightarrow Y$ as $a\rightarrow\infty$. In this limit the Green functions (\[G\]) and (\[KelvinGo\]) for similarly restricted $u$s and $v$s are given by$$\begin{aligned}
G\left( u_{1}=y_{1}/a,v_{1}=\pi/2;u_{2}=y_{2}/a,v_{2}=\pi/2\right) &
=-\frac{1}{4\pi a}\left\vert y_{1}-y_{2}\right\vert -\frac{1}{4\pi}\ln\left(
1+e^{-2\left\vert y_{1}-y_{2}\right\vert /a}-2e^{-\left\vert y_{1}-y_{2}\right\vert /a}\right) \nonumber\\
& \underset{a\rightarrow\infty}{\sim}-\frac{1}{2\pi}\ln\left( \left\vert
y_{1}-y_{2}\right\vert /a\right) +O\left( \frac{1}{a}\right) \ ,\end{aligned}$$ $$G_{o}\left( u_{1}=y_{1}/a,v_{1}=\pi/2;u_{2}=y_{2}/a,v_{2}=\pi/2\right)
\underset{a\rightarrow\infty}{\sim}-\frac{1}{2\pi}\ln\left( \frac{\left\vert
y_{1}-y_{2}\right\vert }{\left\vert y_{1}-2Y+y_{2}\right\vert }\right)
+O\left( \frac{1}{a}\right) \ .$$ Finally then,$$\lim_{a\rightarrow\infty}G_{o}\left( u_{1}=y_{1}/a,v_{1}=\pi/2;u_{2}=y_{2}/a,v_{2}=\pi/2\right) =-\frac{1}{2\pi}\ln\left( \frac{\left\vert
y_{1}-y_{2}\right\vert }{\left\vert y_{1}+y_{2}-2Y\right\vert }\right) \ .$$ But for $y_{1}>Y$ and $y_{2}>Y$, this is precisely the 2D Green function at field point $\left( 0,y_{1}\right) $ for a grounded straight line parallel to the $x$-axis, passing through the point $\left( x,y\right) =\left(
0,Y\right) $, as obtained by placing at the point $\left( 0,2Y-y_{2}\right)
$ a single negative Kelvin point image of a unit point source placed at position $\left( 0,y_{2}\right) $. Of course, in this straight line limit where $x_{1}=x_{2}$ the system is translationally invariant with respect to $x$, so there is no $x$ dependence in the final Green functions.
When $x_{1}\neq x_{2}$ but both are fixed and small, while $a$ becomes infinite, a similar but slightly more tedious limit calculation gives the 2D Green function on the grounded half-plane, namely, $$G_{half-plane}\left( x_{1},y_{1};x_{2},y_{2}\right) =-\frac{1}{2\pi}\ln\left( \frac{\sqrt{\left( x_{1}-x_{2}\right) ^{2}+\left( y_{1}-y_{2}\right) ^{2}}}{\sqrt{\left( x_{1}-x_{2}\right) ^{2}+\left(
y_{1}+y_{2}-2Y\right) ^{2}}}\right) \ .$$ Once again, translational invariance with respect to $x$ accounts for the dependence on only the difference, $x_{1}-x_{2}$. We leave the detailed derivation of $G_{half-plane}$ from $G_{o}$ for the ellipse as an exercise for the reader.[^14]
Discussion {#discussion .unnumbered}
----------
The standard problems involving a grounded circular ring in 2D [@CurtrightEtAl] or grounded spheres in higher dimensions [@AlshalCurtright] can also be easily solved using either the Kelvin or Sommerfeld methods. However, there are many problems where the Kelvin method is very difficult, if not impossible, to implement, but which are directly solvable by the Sommerfeld method. Grounded semi-infinite planes and the circular disk in 3D Euclidean space provide well-studied examples [@Sommerfeld; @Hobson; @Waldmann; @DavisReitz; @DavisReitzAgain; @Duffy].
Beyond these previously solved examples, the grounded ellipsoid in 3D and hyper-ellipsoids in higher dimensions are difficult problems that should be more tractable using Sommerfeld images. Existing image methods applied to these problems are quite involved, and usually require detailed properties of ellipsoidal harmonics [@Dassios]. In fact, extant treatments of the exterior 3D Green function problem for grounded ellipsoids use, in addition to an interior point image, a *continuous* distribution of Kelvin image charge on the surface of an interior confocal ellipsoid [@DassiosAgain; @XueDeng] (also see Sections 7.4.3 and 7.4.4 in [@Dassios]). This non-trivial array of image charges results from requiring that all such charges reside entirely within the physical interior of the ellipsoid, *without* invoking a second copy of $\mathbb{E}_{3}$. In our opinion, these treatments are tantamount to walking on broken glass while bare-footed.
In contrast, the Sommerfeld method applied to a grounded ellipsoid embedded in $N$ Euclidean dimensions only requires a single point image of the point source, in complete parallel to the grounded 2D ellipse treated here, albeit at the cost of introducing a second copy of $\mathbb{E}_{N}$. Therefore, in principle the Sommerfeld method should simplify the analysis required to construct Green functions for such ellipsoids, both conceptually and practically.
**Acknowledgements** It has been our pleasure to reconsider this elementary subject during the Sommerfeld Sesquicentennial year. This work was supported in part by a University of Miami Cooper Fellowship and by a Clark Way Harrison Visiting Professorship at Washington University in Saint Louis.
Appendix A: Complex variables {#appendix-a-complex-variables .unnumbered}
------------------------------
Let$$x+iy=a\left( \cos v\cosh u+i\sin v\sinh u\right) =a\cosh\left( u+iv\right)
\ . \tag{A1}$$ That is to say, $u+iv=\pm\operatorname{arccosh}\left( \frac{x+iy}{a}\right)
+2i\pi k\mid k\in\mathbb{Z}
$. Choose the $+$ solution with $k=0$ so that$$u=\operatorname{Re}\left( \operatorname{arccosh}\left( \frac{x+iy}{a}\right) \right) \ ,\ \ \ v=\operatorname{Im}\left(
\operatorname{arccosh}\left( \frac{x+iy}{a}\right) \right) \ . \tag{A2}$$ Then find$$\begin{aligned}
r^{2} & =x^{2}+y^{2}=a^{2}\left( \cosh^{2}u\ \cos^{2}v+\sinh^{2}u\ \sin
^{2}v\right) \nonumber\\
& =\frac{1}{2}~a^{2}\left( \cosh2u+\cos2v\right) =a^{2}\operatorname{arccosh}\left( \frac{x+iy}{a}\right) \operatorname{arccosh}\left( \frac{x-iy}{a}\right) \ , \tag{A3}$$ as well as$$\begin{aligned}
x^{2}-y^{2} & =a^{2}\left( \cosh^{2}u\ \cos^{2}v-\sinh^{2}u\ \sin
^{2}v\right) =\frac{1}{2}~a^{2}\left( \cosh2u\cos2v+1\right) \ ,\tag{A4}\\
xy & =a^{2}\cosh u\ \cos v\ \sinh u\ \sin v=\frac{1}{4}~a^{2}\sinh
2u\sin2v\ . \tag{A5}$$ In addition find$$\begin{aligned}
\sinh^{2}2u & =\frac{1}{2}~\cosh4u-\frac{1}{2}\nonumber\\
& =\frac{2}{a^{4}}\left( x^{2}+y^{2}\right) ^{2}+\frac{2}{a^{2}}\left(
y^{2}-x^{2}\right) +\frac{2}{a^{4}}\left( x^{2}+y^{2}\right) \sqrt{\left(
\left( x-a\right) ^{2}+y^{2}\right) \left( \left( x+a\right) ^{2}+y^{2}\right) }\ , \tag{A6}$$ along with$$v=\arccos\left( \frac{x/a}{\cosh u}\right) =\arcsin\left( \frac{y/a}{\sinh
u}\right) \ . \tag{A7}$$
Appendix B: Circle $\longleftrightarrow$ ellipse conformal mapping {#appendix-b-circle-longleftrightarrowellipse-conformal-mapping .unnumbered}
-------------------------------------------------------------------
Define a standard ellipse and its fiducial circle by$$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\ ,\ \ \ X^{2}+Y^{2}=\left(
\frac{a+b}{2}\right) ^{2} \tag{B1}$$ Then circles in the complex $Z=X+iY$ plane are mapped to ellipses in the complex $z=x+iy$ plane, and vice versa, by [@Nehari]$$z=Z+\frac{c^{2}}{4Z}\ ,\ \ \ c^{2}=a^{2}-b^{2}\ , \tag{B2}\label{map}$$ By definition for any circle in the $Z$ plane, $R^{2}=\left( X^{2}+Y^{2}\right) =\left\vert Z\right\vert ^{2}$. Expressing $R^{2}$ in terms of $x$ and $y$ as given by the map (\[map\]) then leads to $$1=\frac{R^{2}}{\left( R^{2}+\frac{1}{4}c^{2}\right) ^{2}}~x^{2}+\frac{R^{2}}{\left( R^{2}-\frac{1}{4}c^{2}\right) ^{2}}~y^{2} \tag{B3}$$ This is indeed another ellipse, confocal with the standard ellipse, only now with $$a^{2}=R^{2}\left( 1+\frac{c^{2}}{4R^{2}}\right) ^{2}\ ,\ \ \ b^{2}=R^{2}\left( 1-\frac{c^{2}}{4R^{2}}\right) ^{2}\ ,\ \ \ a^{2}-b^{2}=c^{2}\ .
\tag{B4}$$ The point is, concentric circles centered on the origin of the $Z$-plane are mapped by (\[map\]) to confocal ellipses centered on the origin of the $z$-plane, and vice versa. Moreover, it is obvious and well-known [@Nehari] that the $Z\rightarrow z$ map actually covers the complex $z$-plane twice: Both the interior and the exterior of the fiducial circle cover the $z$-plane under the map.
But now consider the well-known electrostatics method to ground a circle by placing an image charge at a point obtained by inversion of the source location with respect to that grounded circle. Where does the conformal map (\[map\]) take a point $Z$ after it has been inverted with respect to the circle of radius $\frac{1}{2}\left( a+b\right) $? The effect of the inversion is $$X\rightarrow\widetilde{X}=\left( \frac{a+b}{2}\right) ^{2}\frac{X}{R^{2}}\ ,\ \ \ Y\rightarrow\widetilde{Y}=\left( \frac{a+b}{2}\right) ^{2}\frac
{Y}{R^{2}}\ . \tag{B5}$$ That is to say, $$\widetilde{R}^{2}=\widetilde{X}^{2}+\widetilde{Y}^{2}=\left( \frac{a+b}{2}\right) ^{4}\frac{1}{R^{2}}\ ,\ \ \ \widetilde{Z}=\widetilde{X}+i\widetilde{Y}=\frac{1}{R^{2}}\left( \frac{a+b}{2}\right) ^{2}Z\ . \tag{B6}$$ So then, the conformal map of this inverted point gives $$\widetilde{z}=\widetilde{Z}+\frac{c^{2}}{4\widetilde{Z}}=\frac{1}{R^{2}}\left( \frac{a+b}{2}\right) ^{2}\left( Z+\frac{c^{2}}{\frac{4}{R^{4}}\left( \frac{a+b}{2}\right) ^{4}Z}\right) \tag{B7}$$ For example, suppose $a=3$ and $b=1$, then $\left( a+b\right) /2=2$ and $c^{2}=8$. Then$$\begin{aligned}
x & =\left( 1+\frac{c^{2}}{4R^{2}}\right) X\ ,\ \ \ y=\left(
1-\frac{c^{2}}{4R^{2}}\right) Y\nonumber\\
\widetilde{x} & =\left( 1+\frac{c^{2}}{4\widetilde{R}^{2}}\right)
\widetilde{X}\ ,\ \ \ \widetilde{y}=\left( 1-\frac{c^{2}}{4\widetilde{R}^{2}}\right) \widetilde{Y}\tag{B8}\\
\widetilde{X} & =\left( \frac{a+b}{2}\right) ^{2}\frac{X}{R^{2}}\ ,\ \ \ \widetilde{Y}=\left( \frac{a+b}{2}\right) ^{2}\frac{Y}{R^{2}}\nonumber\end{aligned}$$ More specifically, consider$$\begin{aligned}
\left. \left( \widetilde{X},\widetilde{Y}\right) \right\vert
_{a=b=2,X=2.5\cos\theta,Y=2.5\sin\theta} & =\left( 2^{2}\times\frac{1}{2.5}\cos\theta,2^{2}\times\frac{1}{2.5}\sin\theta\right) \tag{B9}\\
\left. \left( \widetilde{x},\widetilde{y}\right) \right\vert
_{a=b=2,X=2.5\cos\theta,Y=2.5\sin\theta} & =\left( \left( 1+\frac
{8}{4\left( \frac{2^{2}}{2.5}\right) ^{2}}\right) 2^{2}\times\frac{1}{2.5}\cos\theta,\left( 1-\frac{8}{4\left( \frac{2^{2}}{2.5}\right) ^{2}}\right) 2^{2}\times\frac{1}{2.5}\sin\theta\right) \nonumber\end{aligned}$$ For other points, see Figures B1 and B2. Upon comparing these two Figures, the various curves are related by the map (\[map\]). Thus the solid or dashed circles shown in Figure B1 map to the solid or dashed ellipses of the same color shown in Figure B2, and vice versa. Also, the light gray straight radial line in Figure B1 maps to the light gray hyperbolic curve in Figure B2, and similarly for other such radial lines.
Various source (solid color curves) and Kelvin image (dashed color curves) charge locations for a grounded circle (shown in black). For one-to-one point source $\leftrightarrow$ point image pairing, only one copy of the plane is needed.
Various source (solid color curves) and Kelvin image (dashed color curves) charge locations for a grounded ellipse (shown in black). For one-to-one point source $\leftrightarrow$ point image pairing, two copies of the plane are now needed.
These Figures reproduce and confirm the explanation in the text that made use of real variables, namely, two copies of the plane are required to ground the ellipse using a single point image for each point source. The image locations shown by the orange and red dashed curves in Figure B2 are actually on the second branch of the doubled plane.
Appendix C: Contour plots of $G_{o}$ {#appendix-c-contour-plots-of-g_o .unnumbered}
-------------------------------------
Consider the grounded ellipse defined by $\left( x,y\right) =\left(
\cos\left( v\right) \cosh\left( 1\right) ,\sin\left( v\right)
\sinh\left( 1\right) \right) $ for $0\leq v\leq2\pi$. Three dimensional contour plots of $G_{o}\geq0$, as functions of the field points on the $xy$-plane, are shown in the following Figures for three representative point source locations, with values near the point source truncated at $G_{o}=0.25$. (For an animated version, with source locations varied for $0\leq v\leq2\pi
$, please see [this URL](http://www.physics.miami.edu/~curtright/GoRotatingFigure.gif).)
[99]{}
H Alshal and T Curtright, Grounded Hyperspheres as Squashed Wormholes [arXiv:1806.03762 \[physics.class-ph\]](https://arxiv.org/abs/1806.03762).
T Curtright, H Alshal, P Baral, S Huang, J Liu, K Tamang, X Zhang, and Y Zhang. The Conducting Ring Viewed as a Wormhole [arXiv:1805.11147 \[physics.class-ph\]](https://arxiv.org/abs/1805.11147).
G Dassios, *[*Ellipsoidal Harmonics*](https://www.amazon.com/Ellipsoidal-Harmonics-Encyclopedia-Mathematics-Applications-ebook/dp/B00D2WQARA/ref=sr_1_1?s=digital-text&ie=UTF8&qid=1533862975&sr=1-1)*, Cambridge University Press (2013) ISBN-13: 978-0521113090.
G Dassios, Directional dependent Green’s function and Kelvin images [Arch. Appl. Mech. 82 (2012) 1325–1335](https://doi.org/10.1007/s00419-012-0669-6).
L C Davis and J R Reitz, Solution to potential problems near a conducting semi-infinite sheet or conducting disc [Am. J. Phys. 39 (1971) 1255-1265](https://doi.org/10.1119/1.1976616).
L C Davis and J R Reitz, Solution of potential problems near the corner of a conductor [J. Math. Phys. 16 (1975) 1219–1226](https://doi.org/10.1063/1.522671).
D G Duffy, *[*Green’s Functions with Applications*](https://www.crcpress.com/Greens-Functions-with-Applications-Second-Edition/Duffy/p/book/9781138894464)*, Second Edition, CRC Press (2017) ISBN-13: 978-1482251029.
M Eckert, *[*Arnold Sommerfeld: Science, Life and Turbulent Times 1868-1951*](https://www.amazon.com/Arnold-Sommerfeld-Science-Turbulent-1868-1951/dp/1461474604/ref=sr_1_7?s=books&ie=UTF8&qid=1524241334&sr=1-7&keywords=sommerfeld)*, Springer-Verlag (2013) ISBN-13: 978-1461474609.
G Green, [*An Essay*](http://en.wikipedia.org/wiki/An_Essay_on_the_Application_of_Mathematical_Analysis_to_the_Theories_of_Electricity_and_Magnetism) *on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism*, Nottingham (1828).
E W Hobson, On Green’s function for a circular disc, with application to electrostatic problems [Trans. Cambridge Philos. Soc. 18 (1900) 277- 291](https://babel.hathitrust.org/cgi/pt?id=coo.31924069328965;view=1up;seq=315).
Z Nehari, *[*Conformal Mapping*](https://www.amazon.com/Conformal-Mapping-Dover-Books-Mathematics-ebook/dp/B00A73A348/ref=sr_1_fkmr0_1?s=digital-text&ie=UTF8&qid=1542954089&sr=1-1-fkmr0&keywords=conformal+transformations+nehari),* Dover Publications (2011), especially pp 269-271, Eqn(4) et seq.
A Sommerfeld, Über verzweigte Potentiale im Raum [Proc. London Math. Soc. (1896) s1-28 (1): 395-429](https://doi.org/10.1112/plms/s1-28.1.395); ibid. 30 (1899) 161.
W Thomson and P G Tait, *Treatise on Natural Philosophy*, [Part I](http://babel.hathitrust.org/cgi/pt?id=wu.89068226117;view=1up;seq=13) & [Part II](http://babel.hathitrust.org/cgi/pt?id=wu.89068225952;view=1up;seq=5), Cambridge University Press (1879 & 1883).
L Waldmann, Zwei Anwendungen der Sommerfeld’schen Methode der verzweigten Potentiale [Physikalische Zeitscrift 38 (1937) 654–663](https://babel.hathitrust.org/cgi/pt/search?id=mdp.39015076063125;view=1up;seq=5;q1=Waldmann;start=1;sz=10;page=search;orient=0)
C Xue and S Deng, Green’s function and image system for the Laplace operator in the prolate spheroidal geometry [AIP Advances 7 (2017) 015024](https://doi.org/10.1063/1.4974156).
[^1]: [halshal@sci.cu.edu.eg]{}
[^2]: [curtright@miami.edu]{}
[^3]: [sushil.subedi04@gmail.com]{}
[^4]: In making this statement, we are not comparing apples to oranges. It is well-known that different boundary conditions, such as Dirichlet and Neumann, require different image charges. However, our statement is correct even when we are dealing with only one set of boundary conditions. In particular, we consider mixed homogeneous Dirichlet and Neumann boundary conditions in this paper.
[^5]: The topology of the extended manifold may also be re-imagined, but here we will not discuss that issue any further.
[^6]: The straight line segment connecting the two elliptical foci on the $x$-axis at $\pm a$ is covered twice using real elliptic coordinates.
[^7]: As a consequence of this $2\pi$-periodicity, $G$ could also be interpreted as the potential for an infinite line of uniformly spaced point charges on the $uv$-plane, i.e. on the covering space for the $\left( u,v\right) $ cylinder defined by (\[EllipticCoords\]). In that case the $\delta\left( v_{1}-v_{2}\right)
$ on the RHS of (\[GEqn\]) would be a [Dirac comb](https://en.wikipedia.org/wiki/Dirac_comb). However, here we are interested in only one copy of the cylinder, so this interpretation is not relevant to the problem at hand.
[^8]: At first sight it may be surprising that (\[GEqn\]) is the equation to be solved, since the elliptic coordinates defined in (\[EllipticCoords\]) involve a non-trivial metric. However, the metric dependence factors out of the invariant Laplacian expressed in terms of those elliptic coordinates. Thus, the covariant equation for the Green function, namely, $\frac{1}{\sqrt{g}}~\partial_{\mu}\left( \sqrt{g}~g^{\mu\nu}\partial_{\nu}G\right) =-\frac{1}{\sqrt{g}}~\delta\left( u_{1}-u_{2}\right)
~\delta\left( v_{1}-v_{2}\right) $, simply reduces to (\[GEqn\]).
[^9]: This differs from a grounded circle in 2D (or sphere in 3D) where the image is located by inversion and *can* move toward infinity as the source moves toward the center of the circle (or sphere). The limit where the ellipse becomes a circle of radius $R$ is achieved here by $R=\lim_{a\rightarrow0}\left[ a\cosh U\right\vert _{U=\ln\left( 2R/a\right)
}=\lim_{a\rightarrow0}\left[ a\sinh U\right\vert _{U=\ln\left( 2R/a\right)
}$. In this limit, only one copy of $\mathbb{E}_{2}$ is sufficient to solve either the interior or the exterior problem using the Kelvin method. See [@CurtrightEtAl] for a thorough discussion of the grounded circular ring in 2D, where the standard Kelvin method is compared to the Sommerfeld method in considerable detail.
[^10]: For the chosen 3D embedding the surface has the appearance of being intrinsically curved, but that is an artefact of the parameterization. That part of the surface either above or below the blue line in Figure 1 corresponds to an open subset of $\mathbb{E}_{2}$.
[^11]: As in the previous Figure, for the chosen 3D embedding the surface has the appearance of being intrinsically curved, but that is again an artefact of the parameterization. That part of the surface either above or below the red ellipse in Figure 2 corresponds to an open subset of $\mathbb{E}_{2}$.
[^12]: If the apparent $du/dw$ slope discontinuity causes anxiety on the part of the reader, one may take instead $u\left( w\right) =\left(
U^{2p}+w^{2p}\right) ^{1/2p}$ for $p>1/2$, again with $-\infty\leq
w\leq\infty$. For example, see [@CurtrightEtAl]. However, the ensuing complications in expressions involving Green functions are not worth making this generalization, in our opinion.
[^13]: This is a peculiarity of the long-range Coulomb potential in 2D — it’s logarithmic! In 3D the charge induced on a grounded ellipsoid by a unit source outside the sphere is not always $-1$, and in fact falls to zero as the source is removed to infinity [@DassiosAgain; @XueDeng]. For a grounded hyper-sphere in $N$ spatial dimensions, it is an interesting exercise to show the induced charge falls as a function of the source distance like $r^{2-N}$ [@AlshalCurtright].
[^14]: Results given in Appendix A may be helpful.
|
---
abstract: 'The performance of principal component analysis (PCA) suffers badly in the presence of outliers. This paper proposes two novel approaches for robust PCA based on semidefinite programming. The first method, *maximum mean absolute deviation rounding* (), seeks directions of large spread in the data while damping the effect of outliers. The second method produces a *low-leverage decomposition* () of the data that attempts to form a low-rank model for the data by separating out corrupted observations. This paper also presents efficient computational methods for solving these SDPs. Numerical experiments confirm the value of these new techniques.'
author:
- Michael McCoy
- 'Joel A. Tropp'
bibliography:
- 'library.bib'
title: |
Two Proposals for Robust PCA\
using Semidefinite Programming
---
Introduction {#sec:introduction}
============
Principal component analysis (PCA), proposed in 1933 by Hotelling [@Hotelling1933], is a common technique for summarizing high-dimensional data. Principal components are designed to identify directions in which the observations vary most. As a consequence, PCA is often used to reduce the dimension of the data.
Statistics based on variance, such as principal components, are highly sensitive to outliers [@Tukey1960]. The literature on robust statistics contains a wide variety of techniques that attempt to correct this shortcoming [@Huber2009]. Unfortunately, many of these approaches are based on intractable optimization problems or lack a principled foundation.
Our focus in this work is to develop new formulations for robust PCA that can be solved efficiently using convex programming algorithms. Our first proposal, which we call *maximum mean absolute deviation rounding* (), exchanges the variance in the definition of PCA with a function less sensitive to outliers known as the mean absolute deviation. Although this formulation leads to a non-convex optimization problem, we demonstrate that it is possible to approximate the optimum by relaxing to a semidefinite program and randomly rounding the solution. This method can be viewed as a specific instance of projection-pursuit PCA [@Li1985].
Our second proposal uses a different semidefinite program to split the input data into the sum of a low-leverage matrix and a matrix of corrupted observations. We refer to this dissection as a *low-leverage decomposition* () of the data. This method is similar in spirit to the rank-sparsity decomposition of Chandrasekaran et al. [@Chandrasekaran2009]. While preparing this manuscript, we learned of an independent investigation into this formulation of robust PCA by Xu et. al.[@Xu2010; @Xu2010a].
We describe algorithms that solve these semidefinite programs efficiently, and we provide numerical experiments that confirm the effectiveness of these new techniques. We begin with a brief overview of our proposals before laying out the details in Sections \[sec:mdr\] and \[sec:lld\].
The Data Model {#sec:datamodel}
--------------
Suppose that we have a family $\{\vec x_i\}_{i=1}^n$ of $n$ observations in $p$ dimensions. We form an $n\times p$ data matrix $\mat X$ whose rows are the observations. The observations are assumed to be centered; that is, $\frac{1}{n}\sum_i \vec x_i \approx
\mathbf{0}$. While our methods do not explicitly require the data to be centered, this hypothesis allows us to interpret principal components as directions of high variance in the data. We discuss practical centering approaches in Section \[sec:experiments\].
Maximizing the Mean Absolute Deviation {#sec:submdr}
--------------------------------------
Our first method is designed to mitigate a source of sensitivity in classical principal component analysis. The top principal component $\vec v_\pca$ is defined as a direction of maximum variance in the data: $$\label{eq:pca}
\vec v_\pca =
\argmax_{\ltwo{\vec v } = 1}
\sum\nolimits_{i=1}^n \abs{\Inner{ \vec x_i , \vec v}}^2.$$ The squared inner products in may lead to outsized influence of outlying points because squaring a large number results in a huge number, which can drag the principal component away from the bulk of the data. We can reduce this effect by replacing the squared inner product with a measure of spread that is less sensitive. We propose the use of the absolute value of the inner product: $$\label{eq:mdpca}
\vec v_{\mathrm{MD}} = \argmax_{\ltwo{\vec v}= 1}
\sum\nolimits_{i=1}^n \abs{\Inner{\vec x_i, \vec v}},$$ where we have added the subscript $\mathrm{MD}$ to indicate that we have exchanged the variance in equation with a measure of spread known as the *mean absolute deviation* (MD) [@Huber2009 p. 2].
This revision results in some complications. The formulation is an eigenvector problem which can be solved efficiently. In contrast, it is -hard to compute $\vec
v_\md$. Nevertheless, we develop an efficient randomized algorithm that provably computes an approximate solution to . We call this approach *maximum mean absolute deviation rounding* ().
Our main result, Theorem \[thm:mdr\], states that, for any failure probability $\delta > 0$ and loss factor $\eps > 0$, our algorithm produces a unit-norm vector $\vec v_{\mdr}$ such that $$\sum\nolimits_{i=1}^n \abs{\Inner{\vec x_i, \vec v_\mdr}} \ge
\sqrt{\frac{2}{\pi}}\left(1 -\eps\right)
\max_{\ltwo{\vec v } = 1} \sum\nolimits_{i=1}^n \abs{\Inner{\vec x_i, \vec v}}.$$ The algorithm requires that we solve one semidefinite program (SDP) whose size is polynomial in the number of observations. Since SDPs are solvable in polynomial time using interior-point methods, our algorithm is tractable in principle. In practice, solving SDPs can be daunting even for moderately sized input data—say, more than 100 observations. To address this issue, we detail a technique of Burer and Monteiro [@Burer2003a; @Burer2004a] that can usually solve the SDP efficiently, and in Section \[sec:experiments\] we provide some numerical evidence that this approach succeeds.
We find additional components by greedily restricting the data to a subspace perpendicular to the previous components and solving again.
This proposal is not without precedent. A more general formulation appears in Huber’s book [@Huber1981 p. 203], and it is now known as *projection-pursuit PCA* (PP-PCA) [@Li1985]. We provide further detail on PP-PCA in Section \[sec:pppca\] and discuss the history of the method in \[sec:pppca\_background\].
A Low-Leverage Decomposition {#sec:lldintro}
----------------------------
Our second proposal stems from a different interpretation of classical principal component analysis. Instead of viewing classical principal components as directions of maximum variance, we can view them as an optimal low-rank model for the data [@Candes2009]. Suppose $\mat
P_\opt$ is a matrix that solves $$\minprog{}{\fronorm{\mat X - \mat P}}{\rank(\mat P) = T.}$$ The dominant principal components of $\mat X$ are given by the $T$ right singular vectors of $\mat P_\opt$ corresponding with the nonzero singular values of $\mat P_\opt$.
With real data, one is often faced with the situation where entire observations are corrupted. If this is the case, we would still like to recover a low-rank model. We can develop as natural formulation for identifying a low-rank model using the well-known rank sparsity [@Fazel2002] and group sparsity [@Rao1998a] heuristics. We propose to decompose the data matrix as $\mat X = \mat
P_\lld + \mat C_\lld$ by solving the semidefinite program $$\label{eq:lldintro}
\minprog{}{\sum_{i} \sigma_i(\mat P) + \gamma \sum_j \ltwo{\vec c_j}}
{\mat P + \mat C = \mat X.}$$ We have written $\sigma_i(\mat P)$ for the $i$th singular value of $\mat P$ and $\vec c_i$ for the $i$th row of $\mat C$.
We view the optimal matrix $\mat P_\lld$ as a surrogate for the low-rank approximation to the uncorrupted data, and the optimal matrix $\mat C_\lld$ as an approximation of the corrupted data. The formulation has an interesting property even when $\mat P_\lld$ is not low-rank or $\mat C_\lld$ is not row-sparse: $\mat P_\lld$ is guaranteed to be a low-leverage set of observations in a sense we make precise in Section \[sec:justlld\]. As a result, we refer to $\mat X = \mat P_\lld + \mat C_\lld$ as a *low-leverage decomposition* (LLD) of the data. We define the dominant components as the right singular vectors of $\mat
P_\lld$.
This optimization problem is similar to the rank-sparsity decomposition problem proposed in [@Chandrasekaran2009]; see also [@Candes2009]. We discuss these ideas at more length in Section \[sec:prevwork\]. As this manuscript was being prepared, we learned of an independent investigation of the program for robust PCA by Xu et. al. [@Xu2010; @Xu2010a] that provides conditions for recovery of the support of the corruption and the row-space of the uncorrupted observations.
Road map {#sec:roadmap}
--------
Sections \[sec:mdr\] and \[sec:lld\] describe our proposals in more detail, including theoretical guarantees and practical algorithms. Section \[sec:prevwork\] offers an overview of previous work on robust PCA, while Section \[sec:experiments\] describes numerical experiments illustrating the performance of our methods in various settings. A technical appendix contains the proofs of supporting results.
Notation {#sec:notation}
--------
We work exclusively with real numbers. The symbols $\prob$ and $\xpcd$ denote probability and expectation, respectively. We use $\subg$ to denote the subgradient map.
Bold capital letters denote matrices while bold lower-case letters denote vectors. We represent the $i$th row of a matrix $\mat A$ by $\vec a_i$ and the $j$th entry of a vector $\vec a$ by $a_j$. The adjoint of a matrix $\mat A$ is written $\mat A^\ad$. When referring to matrix elements, we sometimes use the notation $[\mat A]_{ij}$, and similarly for vectors we use $[\vec a]_i$.
We use the compact convention for the singular value decomposition (SVD) of a matrix: when $\mat A$ is rank $r$, we write its SVD as $\mat A = \mat U \mat \Sigma \mat V^{\ad}$, where $\mat U$ and $\mat
V$ have orthonormal columns, and $\mat \Sigma$ is a non-singular diagonal matrix whose entries are positive and are arranged in weakly decreasing order. The notation $\mat A\ggeq\mat B$ denotes that $\mat
A - \mat B$ is positive semidefinite.
### Norms {#sec:norms}
We denote the $\ell_p$ vector norm as $\norm{\vec u}_p= \left(\sum_i
\abs{u_i}^p\right)^{1/p}$ for $1 \le p < \infty$ and $\norm{\vec
u}_\infty = \max_i \abs{u_i}$. The Frobenius norm of a matrix is defined by $\fronorm{\mat A}^2 = \Inner{\mat A, \mat A} $, where $\Inner{\cdot,\cdot}$ represents the standard inner product. The Moore–Penrose pseudoinverse of a matrix $\mat A$ is denoted $\mat
A^\dagger$.
We define the $\ell_p$ to $\ell_q$ operator norm and its dual respectively by $$\norm{\mat A}_{p \to q} = \sup_{\norm{\vec u}_p = 1} \norm{\mat A
\vec u}_q, \quad \text{ and } \quad \norm{\mat B}_{p \to q}^* = \sup_{\norm{\mat A}_{p
\to q} = 1} \Inner{\mat B, \mat A}.$$ Table \[tab:normsummary\] describes some of the specific operator norms used in this work. We also use the norms $\norm{\mat A}_{2
\to 1}$ and $\norm{\mat A}_{\infty \to 1}$, which lack such simple descriptions; see Sections \[sec:mdpca\_is\_hard\] and \[sec:rounding\].
The operator norm of the adjoint satisfies $\norm{\mat A^\ad}_{q^* \to
p^*} = \norm{\mat A}_{p \to q} $ where $p$ and $q$ satisfy the conjugacy relations $1/p + 1/p^* = 1$ and $1/q + 1/q^* = 1$ with the convention $1/\infty =0$.
Norm Description Description of Dual
-------------------------------- --------------------------------------- --------------------------------------------
$\norm{\mat A}_{2\to 2}$ Maximum singular value of $\mat A$ Sum of the singular values of $\mat A$
$\norm{\mat A}_{2 \to \infty}$ Maximum $\ell_2$ row norm of $\mat A$ Sum of the $\ell_2 $ row norms of $\mat A$
$\norm{\mat A}_{1 \to \infty}$ Maximum absolute entry of $\mat A$ Sum of the absolute entries of $\mat A$
: Summary of the norms used in this work.
\[tab:normsummary\]
Maximum Mean Absolute Deviation Rounding {#sec:mdr}
========================================
Our first method is based on the classical interpretation of the top principal component as the direction of maximum empirical variance in multidimensional data. It has long been recognized that the variance is highly sensitive to outliers in the data [@Tukey1960]. The field of robust statistics has reacted by developing and analyzing robust measures of spread known as robust scales; see [@Huber2009 Ch. 5] or [@Maronna2006 Sec. 2.5]. This literature describes a generic method for determining robust principal components by replacing the variance with a robust measure of scale. Li and Chen [@Li1985] published the first investigation of this under the name *projection-pursuit PCA* (PP-PCA). Our proposal is a specific instance of PP-PCA with the mean absolute deviation scale . We show that this formulation is computationally intractable, but we develop an algorithm that provably approximates its solution. To our knowledge, this is the first rigorous algorithm for PP-PCA with a robust scale.
Scales {#sec:robscales}
------
A *scale* is a function that measures the spread of one-dimensional data [@Huber2009 Ch. 5]. By far, the most common scale is the empirical standard deviation, defined[^1] as $$\std(\vec y) = \left(\sum\nolimits_i y_i^2\right)^{1/2} = \ltwo{\vec y},$$ where we we assume the data $\vec y$ is centered. Of course, the standard deviation is not the only way to measure the spread of the data. An alternative proposal [@Huber2009 p. 2] is the *mean absolute deviation* (MD). For centered data $\vec y$, the MD scale is defined as $$\label{eq:md}
\md(\vec y) = \sum_i \abs{y_i} = \lone{\vec y}.$$ More generally, a scale is a function $S:\Re^n \to \Re$ such that $S(\alpha \vec y) =\abs{\alpha} S(\vec y)$. Scales are typically chosen so that they are less sensitive to outliers than the standard deviation. The robust statistics literature focuses on scales that have a positive breakdown point: the value of the scale cannot be arbitrarily corrupted by nefariously chosen observations, so long as the fraction of bad observations in the entire data set is small. Although the mean absolute deviation has a breakdown point of zero, it exhibits more efficient behavior than the standard deviation under contaminated distributions [@Tukey1960].
### Scales for multivariate data {#sec:highscale}
We extend the definition of scales to multivariate data by considering the scale of the data in a given direction. The projection of the rows of $\mat X$ onto the unit direction $\vec u$ is given by the product $\mat X \vec u$. Note that if $\mat X$ is centered in the sense of Section \[sec:datamodel\], then the projection $\mat X \mat
u$ is also centered by linearity. We define the scale of $\mat X$ in the direction $\vec u$ to be the scale of the projected data $S(\mat X
\vec u)$.
As noted in [@Huber1981], this definition is equivariant under an orthogonal change of basis: for any $\mat Q$ with $\mat Q^\ad \mat Q =
\mathbf{I}$, the scale of $\mat X$ in the direction $\vec u$ is equal to the scale of $\mat X \mat Q^\ad$ in the direction $\mat Q
\vec u$.
Projection-Pursuit PCA {#sec:pppca}
----------------------
Classically, the top principal component is defined as the direction where the empirical standard deviation in the data is largest: $$\label{eq:varianceinv}
\vec v_\pca = \argmax_{\ltwo{\vec v} = 1} \; \std(\mat X \vec v).$$ A natural approach for finding robust components is to replace the standard deviation in with a robust scale $S(\cdot)$, so that the robust component is the direction of maximum robust scale $$\vec v_\pp = \argmax_{\ltwo{\vec v} = 1} S(\mat X \vec v).$$ We define further robust components inductively by adding orthogonality constraints: $$\label{eq:orthRestPPPCA}
\vec v_\pp^{(k)} = \argmax_{\substack{\ltwo{\vec v} = 1 \\
\vec v \perp \vec v_\pp^{(j)} \; \forall\, j < k}}
S(\mat X \vec v).$$ This greedy method of constructing orthogonal components based on robust scales goes by the name projection-pursuit PCA. This scheme was originally proposed by Huber [@Huber1981 p. 203], but was first studied in detail by Li and Chen [@Li1985]. PP-PCA reduces to PCA when the scale is given by the standard deviation due to the variational characterization of eigenvectors by Courant and Fischer.
To implement the PP-PCA method, one only needs a method that finds the first component. We discuss how to enforce the orthogonality constraints in Section \[sec:orthrest\].
PP-PCA with the MD Scale is -Hard {#sec:mdpca_is_hard}
---------------------------------
Finding the top principal component is an eigenvector problem that amounts to computing the direction where the norm $\norm{\cdot}_{2\to
2}$ is achieved. Similarly, PP-PCA with the MD scale amounts to finding a vector that achieves an operator norm. Indeed, the problem $\vec v_{\md} = \argmax_{\ltwo{\vec v}=1} \lone{\mat X \vec v}$ is equivalent to the problem $$\label{eq:mdop}
\text{find } \ltwo{\vec v_\md }=1 \text{ such that }
\lone{\mat X \vec v_\md} = \norm{\mat X}_{2\to 1}.$$ Unfortunately, exchanging the $\ell_2$ norm for the $\ell_1$ norm leads to an -hard computational problem. To see this, we require the following result, which we establish in the Appendix.
\[prop:inf1factor\] For each matrix $\mat X$, the identity $\norm{\mat X}_{2\to 1}^2 =
\norm{\mat X \mat X^\ad}_{\infty\to 1}$ holds.
Rohn [@Rohn2000a] shows that there exists a class of well-conditioned positive matrices $\mathcal{M}$ such that the existence of a polynomial-time algorithm for accurately computing $\norm{\mat M}_{\infty \to 1}$ for all $\mat M \in \mathcal{M}$ implies $\mathsf{P}=\mathsf{NP}$. Since we can factor positive matrices $\mat M = \mat R \mat R^\ad$ in polynomial time using, for example, a Cholesky factorization, the existence of an accurate polynomial-time algorithm that computes $\norm{\mat R}_{2\to 1}^2$ for any matrix $\mat R$ implies that $\mathsf{P}=\mathsf{NP}$.
The observation that Equation is -hard to solve for the specific choice $S(\cdot) = \lone{\cdot}$ has serious implications for existing PP-PCA algorithms. The algorithms available in the literature for PP-PCA [@Croux2007; @Croux2005; @Li1985] are general schemes that claim to work for any choice of scale $S$. As a result, none of these algorithms can provide both accurate and efficient solutions to the PP-PCA problem. This issue is not merely theoretical because these algorithms tend to perform poorly in practice. We discuss this point further in Section \[sec:pppca\_background\].
Approximating the $\ell_2\to \ell_1$ Norm using Randomized Rounding {#sec:rounding}
-------------------------------------------------------------------
Although it is -hard to compute the $\ell_2\to\ell_1$ norm, it is possible to approximate its value efficiently. This fact is a consequence of the little Grothendieck theorem [@Pisier1986a Sec. 5b], but the algorithm depends on ideas of Nesterov [@Nesterov1998], a technique of Burer and Monteiro [@Burer2003a; @Burer2004a], and a new factorization step.
### The semidefinite relaxation of the $\ell_2 \to \ell_1$ norm {#sec:sdprelax}
Before describing our algorithm, we begin by showing how the computation of $\ell_2\to \ell_1$ operator norm can be relaxed to a semidefinite program. First, apply Fact \[prop:inf1factor\] to change the computation of the $\ell_2 \to \ell_1$ norm to the computation of the $\ell_\infty\to\ell_1$ norm: $$\label{eq:2to1->1toinf}
\norm{\mat X}_{2\to 1}^2 = \norm{\mat X\mat X^\ad}_{\infty \to 1}
= \max_{\norm{\vec y}_{\infty} = 1}
\vec y^\ad \mat X \mat X^\ad \vec y.$$ The second identity above follows from the proof of Fact \[prop:inf1factor\]; see also [@Rohn2000a Prop. 1]. Interpreting the quadratic form on the right hand side of as a trace implies that $\norm{\mat
X}_{2\to 1}^2$ is the optimal value of the (non-convex) program $$\label{eq:relax}
\maxprog{}{ \trace(\mat X \mat X^\ad \mat Z) }{ \mat Z = \vec y \vec y^\ad,
\quad [\mat Z]_{ii} = 1 \text{ for all } i. }$$ Relaxing the rank one constraint $\mat Z = \vec y \vec y^\ad$ to a positive-semidefinite constraint $\mat Z \ggeq \mathbf{0}$ leads to the SDP $$\label{eq:relax2}
\maxprog{}{\trace(\mat X \mat X^\ad \mat Z)}
{\mat Z \ggeq \mathbf{0},\quad [\mat Z]_{ii} = 1
\text{ for all } i.}$$ It follows that $\norm{\mat X}_{2\to 1} \le \alpha_\opt$, where $\alpha_\opt^2$ is the optimal value of . Moreover, Grothendieck’s inequality for positive-semidefinite matrices implies that $$\label{eq:alphabdd}
\alpha_\opt^2 \le \frac{\pi}{2} \norm{\mat X \mat X^\ad}_{\infty \to 1},$$ where this inequality is asymptotically the best possible [@Alon2006 Sec. 4.2]. Thus, $\alpha_\opt$ is within a factor of $\sqrt{\pi/2}<1.26$ of the true value of the norm $\norm{\mat X}_{2\to 1}$.
The MDR Algorithm {#sec:mdralg}
-----------------
The fact that equation gives us a good upper bound on the *value* of $\norm{\mat X}_{2\to 1}$ is of secondary importance. We would prefer an approximation for $\vec v_\md$ in , that is, a vector $\vec v_\opt$ with $\ltwo{\vec
v_\opt} = 1$ such that $\norm{\mat X \vec v_\opt} \approx \norm{\mat
X}_{2\to 1}$. We accomplish this goal via a randomized procedure that rounds an optimal solution $\mat Z_\opt$ to back to a vector $\vec v_\opt$. The entire procedure is detailed in Algorithm \[alg:mdr\].
The first step of the algorithm solves the SDP relaxation . In Step \[step:mdr\_random\](a), we draw a random $\vec y \in \{\pm 1\}^n$ with $\xpcd \lone{\mat X \mat
X^\ad \vec y } = 2\alpha_\opt^2/\pi $. This procedure is well understood [@Nesterov1998]. The method in Step \[step:mdr\_random\](b) that we use to compute $\vec v$ from $\vec y$ is novel, and it requires a proof of correctness, which appears in the Appendix. By choosing the best random outcome, Step \[step:mdr\_max\] limits the probability that our method fails to provide a reasonable approximation.
The following theorem describes the behavior of Algorithm \[alg:mdr\].
\[thm:mdr\] Suppose that $\mat X$ is an $n\times p$ matrix, and let $K$ be the number of rounding trials. Let $(\vec v_\opt, \alpha_\opt)$ be the output of Algorithm \[alg:mdr\]. Then $\alpha_\opt \ge \norm{\mat
X}_{2\to 1}$. Moreover, for $\theta < 1$, the inequality $$\label{eq:mdrineq}
\lone{\mat X \vec v_\opt } > \theta \sqrt{\frac{2}{\pi}}
\alpha_\opt$$ holds except with probability $\mathrm{e}^{-2K(1-\theta^2)/\pi}$.
In Theorem \[thm:mdr\], it may be more natural to specify a failure probability $\delta >0$ and approximation loss $\eps = 1-\theta>0$ instead of a repetition number $K$. In this case, simple algebra shows that $\lone{\mat X \vec v_\opt} > (1-\eps)\sqrt{2/\pi}
\norm{\mat X}_{2\to 1}$ except with probability $\delta$, so long as $$K \ge \frac{\pi}{2}\cdot \frac{\log(1/\delta)}{\eps(2-\eps)} =
\Oh\left(\eps^{-1}\log(\delta^{-1})\right).$$ In particular, the choice $K = 94$ implies that $\lone{\mat X \vec v_\opt} > 0.75
\norm{\mat X}_{2\to 1}$ with probability at least $0.999$.
We use the approximation ratio $\rho = \lone{\mat X \vec
v_\opt}/\alpha_\opt $ to measure the quality of the optimal solution in Section \[sec:experiments\]. Although Theorem \[thm:mdr\] only guarantees that we can make $\rho$ as close to $\sqrt{2/\pi} > 0.79$ as we desire, in practice we typically see a $0.95$ approximation ratio or higher. This observation does not indicate that the analysis of the algorithm is loose; it follows directly from [@Alon2006 Sec. 4.2] that this bound is asymptotically tight for a class of examples as $n\to \infty$.
Implementation of Algorithm \[alg:mdr\] {#sec:mdr_imp}
---------------------------------------
For a fixed iteration count $K$, the complexity of Algorithm \[alg:mdr\] is typically dominated by Step \[alg:sdp\]. When applied to , modern interior-point methods are guaranteed to compute the optimal objective value $\alpha_\opt$ and optimal point $\mat Z_\opt$ accurately in polynomial time. The factor $\mat R_\opt$ is determined using a Cholesky factorization of $\mat
Z_\opt$. In practice, interior-point methods are very slow for large-scale problems, so we prefer an algorithm of Burer and Monteiro [@Burer2004a].
The algorithm of Burer and Montiero never forms the semidefinite matrix $\mat Z$; rather it operates directly with the factor $\mat R$. We express the objective function of in terms of $\mat R$ as $\trace( \mat R \mat R^{\ad} \mat X \mat X^{\ad}) =
\fronorm{\mat X^{\ad} \mat R}^2$. The constraints $[\mat Z]_{ii} = 1$ are equivalent to constraints on the rows of $\mat R$ of the form $\ltwo{\vec r_i} = 1$.
We implicitly enforce these row constraints by incorporating them into the objective function as in [@Burer2003a Sec. 4.2]. The resulting unconstrained, nonconvex optimization problem takes the form $$\label{eq:bm_opt}
\maximize_{\mat R} \fronorm{\mat X^{\ad} \mathcal{N}(\mat R)}^2,$$ where $\mathcal{N}(\mat R)$ denotes the operator that normalizes the rows of $\mat R$, that is, $[\mathcal{N}(\mat R)]_{ij} = [\vec r_i]_j
/\ltwo{\vec r_i}$.
We then apply a conjugate gradient algorithm to maximize the unconstrained objective in . Our particular implementation uses the algorithm of Hager and Zhang [@Hager2006], which we have found to work well in our experiments. We refer to our online code for the choice of parameters in this conjugate gradient algorithm [@McCoy2010b].
This factorization technique for solving is advantageous because it reduces the dimension of the problem. The paper [@Burer2004a] shows that restricting $\mat R$ to be an $n\times k$ matrix for $k = \Oh(\sqrt{n})$ suffices to solve this problem exactly. To be precise, when $k = \floor{(1+\sqrt{9 + 8
n})/2}$ any *local* minimum $\mat R_\opt \in \Re^{n \times
k}$ of gives a *global* minimum $\mat
Z_\opt$ of via the map $\mat Z_\opt = \mat R_\opt
\mat R_\opt^{\ad}$, provided a mild technical condition holds.
### Orthogonal Restriction {#sec:orthrest}
Algorithm \[alg:mdr\] only approximates the first principal component in . In order to approximate the $k$th robust principal component for $k>1$, we define a new matrix $\mat
X_k$ by restricting the rows of $\mat X$ to the subspace perpendicular to the span of $\vec v_1,\dotsc,\vec v_{k-1}$. Ignoring numerical stability, we can inductively define $$\label{eq:naiveorth}
\mat X_k = \mat X_{k-1} - \mat X \vec v_{k-1} \vec v_{k-1}^{\ad}
= \mat X \Bigl( \mathbf{I}- \sum\nolimits_{j=1}^{k-1} \vec v_j \vec v_j^\ad\Bigr),$$ which ensures each row of $\mat X$ is orthogonal to the previous components $\vec v_{j}$ for $j<k$. We then apply Algorithm \[alg:mdr\] to the restricted matrix $\mat X_k$ to produce the component $\mat v_k$. Since the output $\vec v_\opt$ of Algorithm \[alg:mdr\] is a linear combination of the rows of the input matrix by Step \[step:mdr\_random\](b), this iterative procedure ensures that $\mat v_k$ is perpendicular to the previous components.
In practice, the implementation can be done using Householder reflections as in [@Croux2005]; see [@Stoer2002] for further background on the implementation of Householder transformations. Householder reflections are more numerically stable than the naïve method . Moreover, they take full advantage of the fact that we are only searching over a $p-k+1$ dimensional subspace by reducing the dimension of $\mat X_{k}$ to $n\times(p-k+1)$.
Extending the Rounding to Multiple Components {#sec:extensions}
---------------------------------------------
We have also attempted to extract a collection of robust components simultaneously by solving a single semidefinite program. That is, we would like to solve the problem $$\label{eq:allatonce}
\maxprog{}{\sum_{i=1}^T \norm{\mat X \vec v_i}_1 }
{ \Inner{\vec v_i,\vec v_j } = \delta_{ij}},$$ where $\delta_{ij}$ is the Kronecker delta function. When $T=1$, equation is equivalent with . When $T>1$, the restriction $\Inner{\vec v_i,\vec v_j} = \delta_{ij}$ ensures that the optimum occurs at an orthogonal set of unit vectors.
We can rephrase this optimization problem by the equivalent quadratically constrained quadratic program $$\label{eq:qcqp}
\maxprog{}{\sum_{i=1}^n \vec w_i^{\ad} \mat X \vec v_i}
{
\diag(\vec w_i \vec w_i^{\ad} ) = 1, \quad
\Inner{\vec v_i,\vec v_j } = \delta_{ij}}$$ The diagonal restrictions on $\vec w_i$ ensure that $\vec w_i \in
\{\pm 1\}^n$ for each $i = 1,\dotsc, n$. The nonconvex problem can be approximated via a semidefinite relaxation proposed in [@Nemirovski2007]. The results of [@So2009] imply that the optimal value of this relaxation is guaranteed to be larger than the optimal value of by no more than a logarithmic factor. The rounding procedure does not produce orthogonal vectors, so we need to apply an additional orthogonalization step to achieve feasibility for . Empirically, we have found that the orthogonalization increases the objective value over the standard rounding, so it appears that there is no loss in applying this procedure.
Unfortunately, this method does not appear to be competitive with the projection pursuit method. The vectors we find by coupling Algorithm \[alg:mdr\] with the orthogonal pursuit of Section \[sec:orthrest\] are feasible for and typically provide a larger objective value than rounding coupled with post-processing orthogonalization. A better rounding procedure for this type of relaxation may prove more effective than the projection-pursuit approach; this is a direction for further research.
The Low-Leverage Decomposition {#sec:lld}
==============================
Our second method is derived from the interpretation of principal component analysis as a matrix approximation problem. When the observations are drawn from a highly correlated family, the singular values of the data matrix $\mat X$ tend to decay rapidly. If this is the case, then the matrix $\mat X$ is well approximated by a low-rank matrix $\mat P$.
It is rare that a large data set can be compiled without error, but it is often the case that the errors only affect a subset of the observations. We can model these errors through a multi-population model. Suppose that the bulk of the observations is well-explained by a low-rank model while the remainder come from another population or are corrupted by measurement noise. A prudent approach to robust principal component analysis would first separate the corrupted data from the uncorrupted data before attempting to recover a low-rank model. When the corrupted rows are unknown, this task may seem daunting.
To accomplish this task, we propose a semidefinite program that decomposes the input $\mat X$ into two matrices: $$\label{eqn:lld}
\minprog{(\mat P,\mat C)}{\nucnorm{\mat P}+\gamma \rsnorm{\mat C}}{
\mat P+ \mat C=\mat X.}$$ The norm $\nucnorm{\mat P}$ is the sum of the singular values of $\mat
P$ and is known to promote low-rank solutions [@Fazel2002], while $\rsnorm{\mat C}$ is the sum of the $\ell_2$ norms of the rows of $\mat C$ and promotes group sparsity [@Rao1998a].
We call the optimal matrix pair $(\mat P_\opt,\mat C_\opt)$ for the problem the *low-leverage decomposition* (LLD) of $\mat X$; we can interpret $\mat C_\opt$ as an identified corruption and $\mat P_\opt$ as a surrogate for the uncorrupted observations. We define our robust components as the right singular vectors of the surrogate matrix $\mat P_\opt$. The detailed procedure appears in Algorithm \[alg:lld\]. We show in Section \[sec:justlld\] that our recovered data matrix $\mat P_\opt$ has the additional property of being a low-leverage set of observations.
The formulation is related to recent proposals [@Candes2009; @Chandrasekaran2009], and we discuss this point more in Section \[sec:cvx\_approach\].
As we were preparing this manuscript, we became aware of the independent work [@Xu2010; @Xu2010a] which also considers for the robust PCA problem. This work shows that, under certain hypotheses, the recovered low-rank data $\mat P_\opt$ has the same row-space as the true data and the corrupted rows are correctly identified.
Low-Leverage by Duality {#sec:justlld}
-----------------------
In this section, we demonstrate that extracts a low-leverage model for the data. This result follows from duality arguments that characterize the optimum of the convex program.
\[lem:lldsubg\] A feasible pair $(\mat P,\mat C)$ is optimal for if and only if there exists a matrix $\mat Q$ such that
\[eqs:dualcert\] $$\begin{aligned}
\phantom{-}\Inner{\mat Q,\mat P} &= \phantom{\gamma}\nucnorm{\mat P},
\quad
\norm{\mat Q}_{2\to 2} \le 1 \label{Qspec}\\
-\Inner{\mat Q,\mat C} &= \gamma \rsnorm{\mat C},
\quad \rmnorm{\mat Q}\le \gamma, \label{Qrow}
\end{aligned}$$
It follows from standard subdifferential conditions that a feasible point $(\mat P,\mat C)$ minimizes the functional in if and only if zero is in the subgradient of $f(\mat P) =
\nucnorm{\mat P}+\gamma\rsnorm{\mat X-\mat P}$. By the additivity of subgradients [@Rockafellar1970 Thm. 23.8], this condition holds if and only if there exists a matrix $\mat Q$ such that the subgradient conditions $\mat Q \in \subg\nucnorm{\mat P}$ and $-\mat
Q \in \subg \gamma \rsnorm{\mat C}$ are in force.
We show that these subgradient conditions are equivalent to . By definition of the subdifferential, $\mat Q
\in \subg \norm{\mat P}_{2\to 2}^*$ if and only if for every perturbation $\mat \Delta$ the subgradient inequality $$\label{eq:subIneq}
\Inner{\mat Q, \mat \Delta} \le
\norm{\mat P+\mat \Delta}_{2\to 2}^* - \norm{\mat P}_{2\to 2}^*$$ holds. Suppose first that holds. Then, for all $\mat
\Delta$, we have $$\Inner{\mat Q, \mat \Delta} =
\Inner{\mat Q, \mat P + \mat \Delta } - \norm{\mat P}_{2\to 2}^*
\le \norm{\mat Q}_{2\to 2} \norm{\mat P + \mat \Delta}_{2\to 2}^*
- \norm{\mat P}_{2\to 2}^* ,$$ where the inequality follows by the definition of dual norms. Since $\norm{\mat Q}\le 1$ by assumption, the subgradient inequality must hold.
It remains to show that the subgradient inequality implies . Taking $\mat
\Delta = \mat P$ in gives $\Inner{\mat Q, \mat P}
\le \norm{\mat P}_{2\to 2}^*$, while $\mat \Delta = - \mat P$ gives the reverse inequality $\Inner{\mat Q, \mat P} \ge \norm{\mat
P}_{2\to 2}^*$. Therefore the subgradient inequality implies $\Inner{\mat Q, \mat P } =
\norm{\mat P}_{2\to 2}^*$.
On the other hand, suppose that $\mat \Delta \ne \mathbf{0}$ satisfies $\Inner{\mat Q, \mat \Delta } = \norm{\mat Q}_{2\to 2}
\norm{\mat \Delta}_{2\to 2}^*$; such a matrix $\mat \Delta$ must always exist in finite dimensions since suprema are attained in the trace definition of norms. Then the subgradient inequality implies $$\norm{\mat Q}_{2\to 2} \norm{\mat \Delta}_{2\to 2}^* \le
\norm{\mat P+\mat \Delta }_{2\to 2}^* -\norm{\mat P}_{2\to 2}^*
\le
\norm{\mat \Delta}$$ where the second inequality follows by the triangle inequality. Since $\mat \Delta\ne \mathbf{0}$, we have shown that the subgradient inequality implies $\norm{\mat Q}_{2\to 2} \le 1$. Hence $\mat Q \in \subg \norm{\mat P}_{2\to 2}^*$ is equivalent to . The equivalence between $-\mat Q \in \subg \gamma
\norm{\mat C}_{2\to \infty}^*$ and relation follows analogously.
Before continuing, we introduce another fact concerning the subgradient of unitarily invariant norms. Let $\mat P = \mat U \mat
\Sigma \mat V^\ad$ be the compact SVD of $\mat P$. It follows from [@Watson1992] that implies $\mat Q = \mat U
\mat V^\ad + \mat W$, where, in particular, $\mat U \mat V^\ad \mat W
= \mathbf{0}$.
### Leverage scores {#sec:levscores}
The *leverage score* of the observation $\mat x_i$ corresponding to the $i$th row of $\mat X$ is given by the number $[\mat H]_{ii}$, where $\mat H=\mat X(\mat X^*\mat X)^\dagger \mat X^*$ is the orthoprojector onto the column space of $\mat X$. We refer to $\mat
H$ as the *hat matrix* in accord with common statistical practice. A large leverage score tends to indicate that the corresponding observation lies outside of the bulk of the data, although it does not necessarily indicate that the point is influential in linear regression. We refer to [@Montgomery2006 Ch. 6] for further discussion of leverage scores.
The following theorem shows that the leverage scores of our decomposition are bounded above by $\gamma^2$, justifying the terminology low-leverage decomposition for the solution of the program .
\[thm:lld\] Suppose $(\mat P_\opt,\mat C_\opt)$ is an optimal point of the program . Then the diagonal elements of the hat matrix $\mat H=\mat P_\opt(\mat P_\opt^\ad\mat P_\opt)^\dagger \mat
P_\opt^\ad$ are bounded above by $\gamma^2$.
From the characterization of the subgradient of unitarily invariant norms [@Watson1992] discussed above, we know that $\mat Q = \mat
U\mat V^{\ad} + \mat W$ with $\mat U \mat V^{\ad} \mat W^{\ad} =
\mathbf{0}$. Thus, $$\mat Q\mat Q^{\ad} =
\mat U\mat U^{\ad} + \mat W\mat W^{\ad} \ggeq \mat U\mat
U^{\ad} = \mat H,$$ where the last equality can be easily checked using the definition of $\mat H$ and the SVD of $\mat P_\opt$. Since the diagonal entries of a positive-semidefinite matrix are nonnegative, this relation implies $[\mat H]_{ii} \le [\mat Q\mat Q^{\ad}]_{ii}$. Recall that the $\ell_2 \to \ell_\infty$ operator norm is the maximum $\ell_2$ row norm of the matrix. Thus relation of Lemma \[lem:lldsubg\] implies that $[\mat Q\mat Q^{\ad}]_{ii}
\le \gamma^2$, which completes the proof.
We can view our proposal as a method of decomposing a data matrix $\mat X$ into a component with a (user-specified!) upper bound on the leverage plus an error term. Moreover, this result gives a statistical interpretation to the regularization parameter $\gamma$ in .
We note that while our program guarantees a low-leverage decomposition, an assumption of suitably small leverage is a technical hypothesis in other works, e.g., [@Candes2009 eq. (1.2)].
The reader should be warned that this method does not necessarily produce a low-leverage solution if we use our program to identify outlying data and then “prune” the rows. That is, suppose $(\mat
P_\opt,\mat C_\opt)$ is an optimal point of and $\vec
c_i = \mathbf{0}$ for row indices $i \in I$. Then the corresponding matrix $\mat X_I= \mat P_I$ *does not* necessarily have leverage scores bounded above by $\gamma^2$.
The Choice of $\gamma$ {#sec:lld_gamma}
-----------------------
In this section, we study how the value of the regularization parameter $\gamma$ affects the properties of the decomposition.
We begin by showing that, when $\gamma \ge 1$, the degenerate solution $(\mat P_\opt,\mat C_\opt) = (\mat X, \mathbf{0})$ minimizes . This claim follows by explicit construction. Let $\mat U \mat \Sigma \mat V^{\ad}$ be the compact SVD of $\mat X$, and define $\mat Q = \mat U \mat V^{\ad}$. Clearly $\Inner{\mat Q, \mat X}= \nucnorm{\mat X}$, so $\mat Q$ satisfies with $\mat P_\opt = \mat X$. By construction, the maximum singular value of $\mat Q$ is bounded above by one. Equivalently, $\mat Q \mat Q^\ad \gleq \mathbf{I}$. This inequality implies $[\mat Q \mat Q^\ad]_{ii}\le 1$. Since the diagonal entries of $\mat Q \mat Q^\ad$ are the squared row norms of $\mat Q$, we have shown that $\norm{\mat Q}_{2\to \infty} \le 1 \le \gamma$. This bound demonstrates that $\mat Q$ satisfies with $\mat C_\opt =
\mathbf{0}$, which certifies optimality of this degenerate solution by Lemma \[lem:lldsubg\].
We now show that the regularization parameter $\gamma$ gives an upper bound on the rank of the optimal $\mat P_\opt$. It is easy to show using the SVD of $\mat P_\opt$ that the trace of the hat matrix $\mat
H$ defined above is equal the rank of $\mat P_\opt$. Since $[\mat
H]_{ii} \le \gamma^2$ by Theorem \[thm:lld\], we must have $$\label{eq:gammarank}
\rank(\mat P_\opt) = \trace(\mat H) \le n \gamma^2.$$ The rank is a positive integer, so $\gamma < 1/\sqrt{n}$ implies that the optimal $\mat P_\opt$ is trivial. Moreover, in order to get $T$ meaningful components in Step \[Alg:eigs\] of Algorithm \[alg:lld\], we require $\rank(\mat P_\opt) \ge T$. Thus, we can limit ourselves to situations where $\gamma \in [\sqrt{T/n},
1]$.
Inequality has implications for the numerical solution of . As we discuss in Section \[sec:lldalg\], the bulk of the computation comes from computing an SVD at each iteration. When the solution of the optimization problem has low rank, the iterates also tend to have low rank. This allows us to save significant computational effort by computing partial singular decompositions at each step. A judicious choice of $\gamma$ can increase the performance of our algorithm immensely. We find that taking $n\gamma^2 \approx T^2$ is a useful heuristic for achieving a rank-$T$ optimal solution, so long as $n \gg
T^2$.
On the other hand, typical statistical data does not show true low-rank behavior even when there are no outliers. Therefore, forcing the optimal decomposition to be low rank typically results in a dense corruption $\mat C_\opt$. This effect may be mitigated somewhat by another formulation we discuss briefly in Section \[sec:lld\_ext\]. In practice we find that setting $\gamma$ somewhat less than $\sqrt{p/n}$, say $\gamma = 0.8\sqrt{p/n}$, provides a very good low-rank model, but it does poorly in the context of outlier identification. We discuss specific parameter choices for our experiments in Section \[sec:experiments\].
Computing the Low-Leverage Decomposition {#sec:lldalg}
----------------------------------------
Although general-purpose semidefinite programming software such as `CVX` [@Grant2010; @Grant2008] can solve small instances of efficiently, the interior-point methods they utilize may be unable to complete even a single iteration of a large-scale problem. This observation indicates that we need to use different methods for large-scale problems.
To solve , we recommend an alternating direction augmented Lagrangian algorithm analogous to the one used in [@Candes2009]; see also [@Lin2009]. The generic form of the method is known as the Augmented Lagrangian Method of Multipliers (ALMM). The augmented Lagrangian for with dual variable $\mat Q$ is given by $$\mathcal{L}_\mu(\mat P,\mat C,\mat Q) = \nucnorm{\mat P} + \gamma
\rsnorm{\mat C} + \Inner{\mat X - \mat P - \mat C, \mat Q} +
\frac{\mu}{2} \fronorm{\mat X - \mat P - \mat C}^2.$$ For an initial starting point $\mat P^0$, we alternately solve $\mat
P^{k+1} = \argmin_{\mat P} \mathcal{L}_\mu(\mat P, \mat C^k, \mat
Q^k)$ and $\mat C^{k+1} =\argmin_{\mat C} \mathcal{L}_\mu(\mat
P^{k+1}, \mat C, \mat Q^k)$. We then update the multiplier by the feasibility gap $\mat Q^{k+1} = \mat Q^k + \mu (\mat X - \mat P^{k+1}
- \mat C^{k+1})$.
The minimizations above have an explicit form in terms of shrinkage operations [@Combettes2006]
\[eqs:shrinks\] $$\begin{aligned}
\mat C^{k+1} &= \shrinkrows\left(\mat X - \mat P^k + \frac{1}{\mu}
\mat Q^k,\mu \gamma \right) \\
\mat P^{k+1} &= \shrinkspec\left(\mat
X - \mat C^{k+1} + \frac{1}{\mu} \mat Q^k,\mu \right),
\end{aligned}$$
where $\shrinkrows(\mat A,\nu)$ soft-thresholds each row $\vec a_i$ of $\mat
A$: $$\shrinkrows(:,\nu): \mat A \longmapsto
\diag([1-\nu/\ltwo{\vec a_i}]_+)\cdot \mat A,$$ where $[x]_+ = \max\{x,0\}$. Similarly $\shrinkspec(\mat A, \nu)$ soft-thresholds the singular values of $\mat A$ $$\shrinkspec(:, \nu): \mat U \mat \Sigma \mat V^\ad \longmapsto \mat U
\left[\mat \Sigma -\nu \mathbf{I}\right]_+
\mat V^{\ad},$$ where the operator $[\cdot]_+$ is applied element-wise. We initialize the algorithm with $\mat P^0 = \mathbf{0}$ and set the parameter $\mu
= np/\rsnorm{\mat X}$. We stop the algorithm when the iterates are nearly feasible, that is, $\bigl\|{\mat X - \mat P^k -\mat
C^k}\bigr\|<10^{-7}\fronorm{\mat X}$.
The main computational difficulty when running this algorithm involves computing the spectral shrinkage operator. When the iterates $\mat
P^k$ are low rank, we can save significant computational effort by performing only partial singular value decompositions [@Lin2009]. We can leverage our analysis in Section \[sec:lld\_gamma\] to ensure that the optimal $\mat P_\opt$ is low rank. Since the algorithmic iterates tend to be low-rank in this case, we can significantly improve the performance of our algorithm by choosing $\gamma$ to limit the rank of the optimal solution. In practice, we have found that one should set the quantity $n\gamma^2$ somewhat larger than the desired rank of the solution, e.g., $n\gamma^2 \approx T^2$ when we desire a rank-$T$ solution.
Extensions for a Noisy Model {#sec:lld_ext}
----------------------------
We note that there is an obvious extension of the when one wants to account for an additional of noise in the model. Suppose that in addition to gross corruptions of certain observations, we would also like to model small corruptions or noise that may be spread throughout the data.
Instead of enforcing the equality $\mat X = \mat P + \mat C$, we allow for some additional slack of the form $\fronorm{\mat X - \mat P - \mat
C} \le \eta$, where $\eta$ is an estimate for the noise level. That is, we solve the problem $$\label{eq:lld_extension}
\minprog{}{\norm{\mat P}_{2\to 2}^* + \gamma \norm{\mat C}_{2 \to \infty}^*}
{ \fronorm{\mat X - \mat P - \mat C} \le \eta}$$ When $\eta=0$, this is equivalent to our proposal for the gross corruption model. Other loss functions are also possible. Note that the Frobenius norm remains invariant under a rotation on the right, which is a feature of that we would like to preserve.
This formulation is also studied in the independent work [@Xu2010; @Xu2010a]. It is shown there that under some technical conditions, the decomposition from results in a decomposition where $\mat P_\opt$ is close to a matrix with the same row-space as the true observations, and the matrix $\mat
C_\opt$ is close to a matrix that correctly identifies the column support of the corruption.
Previous Work {#sec:prevwork}
=============
This section describes previous work on robust formulations of principal component analysis. Convex approaches to robust PCA are unusual, and, as a consequence, many other attempts at robust PCA lack rigorous algorithms. Often, proposals are put forward with a mathematical formulation and only a heuristic algorithm—or an algorithm without a clear mathematical formulation.
In Sections \[sec:pppca\_background\] and \[sec:cvx\_approach\], we describe the two methods in the literature most closely related to our proposals. We then describe in detail an approach for robust PCA recommended by Maronna [@Maronna2005] with which we provide comparisons in Section \[sec:experiments\]. We conclude with a short overview of other robust PCA proposals that have appeared in the literature.
Antecedents for MDR: Projection Pursuit PCA {#sec:pppca_background}
--------------------------------------------
Our proposal is a particular instance of an approach that has come to be known as *projection-pursuit PCA* (PP-PCA), as we discuss in Section \[sec:pppca\]. The theoretical properties of PP-PCA are well understood; see for instance [@Cui2003] and [@Croux2005].
All of the algorithms we have found in the literature for computing PP-PCA are meant to operate with an arbitrary scale. In view of the fact that the PP-PCA problem is -hard, it is unsurprising that the literature appears to contain no PP-PCA algorithms with proofs of correctness and tractability. Indeed, we have been unable to find other work that recognizes that the PP-PCA problem is intractable in a rigorous sense.
The original study of Li and Chen [@Li1985] uses a Monte Carlo approach that was found to be computationally expensive. In theory, even simple Monte Carlo methods (e.g., randomly sampling the unit sphere) can produce arbitrarily good solutions to problem with an arbitrary (continuous) scale. Given the computational hardness of the problem, it is unlikely that Monte Carlo approaches can provide guarantees of computational efficiency.
Current algorithms for PP-PCA rely on heuristics. A popular and fast algorithm for generic projection-pursuit PCA is the finite direction method (FDM) of Croux and Ruiz-Gazen [@Croux2005]. This technique replaces the search over the entire unit sphere $\ltwo{\vec v}=1$ with a finite search over the directions that appear among the observations: $\vec v \in \{\vec x_1/\ltwo{\vec x_i}, \dotsc, \vec x_n
/\ltwo{\vec x_n}\}$. The hope is that directions of large scale are likely to be well approximated by directions appearing in the data. This heuristic to performs poorly when $n$ and $p$ are large because it takes an extremely large number of points to cover a high-dimensional sphere.
A convex approach {#sec:cvx_approach}
-----------------
Recently, a method of Chandrasekaran et al. [@Chandrasekaran2009] has been adapted for robust PCA in [@Candes2009]. This approach attempts to decompose the data matrix into a sum of a low-rank matrix and a sparse matrix via the semidefinite program $$\label{eq:nplone}
\minprog{}{\norm{\mat L}_{2\to 2}^* + \lambda \norm{\mat S}_{1\to
\infty}^*}{\mat L+\mat S = \mat X.}$$ The nuclear norm $\norm{\cdot}_{2\to 2}^*$ promotes low rank and the matrix $\ell_1$ norm $\norm{\cdot}_{1\to \infty}^*$ promotes sparsity. We refer to this method as . The works [@Candes2009; @Chandrasekaran2009] provide conditions under which succeeds in *exactly* recovering a low-rank and sparse component.
This convex approach is principled in the sense that the mathematical formulation is also algorithmically tractable. On the other hand, it lacks an invariance to a change in the observation basis possessed by all other methods we discuss, including standard PCA. That is, applying a rotation $\mat U^\ad \mat U = \mathbf{I}$ to the data $\widehat{\mat X} = \mat X \mat U$ does not result in a similar rotation of the decomposition due to the fact that the norm $\norm{\cdot}_{1\to \infty}^*$ is not invariant under this transformation.
One may argue that this invariance is inconsequential: in real data, the particular choice of coordinates has a meaning and outliers may occur coordinate-wise. This argument is defensible in domain specific examples, such as image data that contain specularities [@Candes2009]. Nevertheless, PCA is intended to locate a coordinate basis that explains data more effectively than the standard basis [@Hotelling1933]. If this is the analytical goal, basis invariance is indeed a requisite property. See Section \[sec:no2\] for an experiment where this lack of orthogonal invariance in appears to produce unnerving results.
Spherical PCA {#sec:sph}
-------------
Another approach, known as spherical principal components () [@Locantore1999], rescales the observations to unit (Euclidean) norm and applies standard PCA to this modified data. To implement the method, we first compute a normalized matrix $\widehat{\mat{X}}$. Each row of $\widehat{ \mat{X}}$ is the normalized version of the corresponding row of the centered data matrix $\mat X$, that is $\widehat{ \vec x}_i = \vec x_i /\ltwo{\vec
x_i}$. Using the row-normalization operator from , we can express the normalized matrix as $\widehat{\mat X} =
\mathcal{N}(\mat X)$.
The robust components are then defined as the standard principal components of the rescaled matrix $\widehat{\mat X}$. Since all of the observations from the normalized matrix $\widehat{\mat X}$ have norm one, there are no large magnitude observations that exert an undue influence on the principal components.
A study by Maronna [@Maronna2005] shows that enjoys good practical performance. The ease of implementation and relatively good behavior of leads Maronna to suggest it as the default choice for robust principal component analysis. As a result, we use as a baseline comparison for the performance of our robust methods in Section \[sec:experiments\].
Other proposals {#sec:otherprop}
---------------
Some of the earliest methods for robust PCA compute approximations of correlation or covariance matrices using robust methods. Gnanadesikan and Kettenring propose direct robust estimation of the covariance matrices through robust estimation of the individual entries [@Gnanadesikan1972]. This may lead to counterintuitive results such as non-positive covariance matrices. An alternative approach explicitly enforces positive matrices as minimizers of a functional such as an $M$-estimator [@Devlin1981]; see also the more recent study [@Croux2000].
A representative example of robust PCA from the machine learning community is the work of De La Torre and Black [@DeLaTorre2003]. They define the robust components as the minimum of a highly non-convex energy function and attempt to minimize this energy function using an iteratively reweighted least-squares algorithm coupled with an annealing step. No theoretical guarantees of correctness for the algorithm are provided.
Another recent approach appears in the paper [@Xu2009] of Xu et al. This algorithm randomly removes observations that appear to have high influence in the current estimate of the principal components. The principal component estimate is computed from the trimmed data. Xu et al. are able to establish strong theoretical properties of their algorithm, including a high breakdown point in the high-dimensional scaling regime where $n\to \infty$ and $n/p \to c >
0$.
Numerical Experiments {#sec:experiments}
=====================
This section provides some numerical examples comparing our proposals with standard PCA and other robust PCA methods in the literature. In Section \[sec:toppc\], we look at the projection of two data sets on the top robust component. Section \[sec:bus\] repeats a multiple-component experiment of Maronna [@Maronna2006] with additional robust methods. Section \[sec:movielens\] contains a larger experiment, where we calculate the first two components of a dense matrix with more than twenty million entries.
All of these experiments and algorithms are implemented with <span style="font-variant:small-caps;">Matlab</span>. Following the principle of reproducible research [@Buckheit1995a], we provide code that reproduces the exact experiments in this work [@McCoy2010b].
Projection onto the top component {#sec:toppc}
---------------------------------
In this section, we study the robust component methods applied to two data sets. The first set is a selection of environmental factors that may affect the concentration of nitrogen dioxide around Oslo, Norway. The second example is constructed from standard iris data. In each case, we examine the spread of the data in the direction of the top robust component.
### Experimental setup
For these experiments, we center the data by removing the Euclidean median from each observation. The Euclidean median $\widehat{\vec
\mu}$ is a robust estimate of the center of the data, and is defined as $$\label{eq:eucmed}
\widehat{\vec \mu}
= \argmin_{\vec \mu} \sum\nolimits_{i=1}^n \ltwo{\vec x_i -\vec \mu}.$$ Maronna [@Maronna2006 Ch. 9 ] gives a method to solve this convex problem for $\widehat{\vec \mu}$.
We project the data onto the top component for each method and compare the performance of the methods by the *interquartile range* (IQR), that is, the distance between the $25$th and $75$th percentile of the projected data.
We apply extract the dominant component from each data set using our methods (and ), other robust methods (and ), and standard PCA. For , we use $K=94$ rounding trials as discussed in Section \[sec:rounding\]. We set the weight parameter $\gamma
= 0.8\sqrt{p/n}$. As recommended in [@Candes2009], we set the parameter $\lambda = 1/\sqrt{n}$ for the first experiment. With the iris data in Section \[sec:iris\], we find that $\lambda =
1/\sqrt{n}$ gives a trivial result: no outliers were identified by . Instead, we use the more favorable choice $\lambda =
0.3/\sqrt{n}$.
### Norwegian nitrogen dioxide data {#sec:no2}
Our data for this experiment consists of 500 observations of eight environmental factors around Oslo, Norway, available on the Statlib archive [@Aldrin2004]. The variables include the log-concentration of nitrogen dioxide (NO$_2$) particles, the number of cars per hour, and the wind speed, as well as several additional factors useful for predicting the concentration of NO$_2$ particles.
We calculate the top component of the data using each method. In Figure \[fig:no2\] we plot the projection of the data onto the direction of these components using a standard box-and-whisker plot. The whiskers extend either $1.5$ times the IQR beyond the edge of the box or to the extreme data point. We consider points that lie beyond the whiskers outliers. We give the percentage of outliers and several order statistics of the data in Table \[tab:NO2\].
Every robust method results in a larger IQR than . The component finds the largest IQR, and the method finds the smallest IQR among the robust methods. Except for , every method identifies a direction with a relatively large number of outliers, which indicates that the data has heavy tails.
The method is unique because it does not identify a direction of large spread *outside* of the middle $50\%$ of the data. We have observed that a random change of the observation basis causes the component to perform similarly to the component. By orthogonal equivariance, the results for methods other than are unchanged by a change in the observation basis. This indicates that the behavior of the results given by the component is due to the lack of orthogonal equivariance.
We note that the approximation ratio for the top component is near optimal at $0.978$.
Method IQR min 25th 75th
-------- -------- --------- --------- -------- ------ ------ ------ -------
MDR $2.57$ $-9.07$ $-1.53$ $1.05$ $10$ $82$ $5$ $00$%
sphPCA $2.53$ $-9.06$ $-1.45$ $1.08$ $10$ $71$ $5$ $60$%
N+L1 $2.38$ $-4.58$ $-1.34$ $1.05$ $2$ $79$ $0$ $00$%
LLD $2.27$ $-9.29$ $-1.27$ $1.00$ $11$ $24$ $7$ $40$%
PCA $1.89$ $-9.51$ $-1.08$ $0.81$ $12$ $18$ $11$ $00$%
: \[tab:NO2\] *Statistics for the projected NO$_{\mathit{2}}$ data.* The last column lists the percentage of points lying outside the whiskers in Figure \[fig:no2\].
### Iris data {#sec:iris}
We use Fisher’s iris data [@Fischer1936] in this experiment. The data contains $60$ observations from three different species of iris: *Iris setosa*, *Iris virginica*, and *Iris versicolor*. Each observation consists of four measurements, namely sepal length, sepal width, petal length, and petal width.
Fifty of the observations come from the *setosa* flowers. We corrupt these observations with $5$ measurements of *Iris virginica* and five measurements of *Iris versicolor*. We hope that robust principal components identify a direction of large spread in the *setosa* bulk of the data. As a baseline comparison, we also calculate the dominant principal component of the *setosa* population without the outlying flowers.
As in Section \[sec:no2\], we project the data onto the direction of the dominant components. These points are plotted in Figure \[fig:iris\]; we distinguish the bulk *setosa* points from the *versicolor* and *virginica* observations. We compute an approximate density of the *setosa* observations by convolving the projected data with a unit volume Gaussian kernel of width $\sigma = 0.2$. Table \[table:iris\] gives some order statistics of the projections.
Method [out]{}
-------------- ----- ------ ------ ------ ------ --------- ----- ------ ----- ------ -----------
LLD $0$ $70$ $-1$ $21$ $-0$ $41$ $0$ $29$ $1$ $14$ $0$.$00$%
*Setosa* PCA $0$ $70$ $-1$ $22$ $-0$ $41$ $0$ $29$ $1$ $14$ $0$.$00$%
sphPCA $0$ $69$ $-1$ $19$ $-0$ $41$ $0$ $28$ $1$ $13$ $0$.$00$%
N+L1 $0$ $66$ $-1$ $16$ $-0$ $40$ $0$ $26$ $1$ $07$ $0$.$00$%
MDR $0$ $37$ $-0$ $79$ $-0$ $24$ $0$ $13$ $0$ $53$ $0$.$00$%
PCA $0$ $19$ $-0$ $60$ $-0$ $15$ $0$ $04$ $0$ $37$ $6$.$00$%
: \[table:iris\] *Order statistics for the projection of the *setosa* data onto the top components.* The last column lists the number of *setosa* points further than 1.5 times IQR left of the 25th percentile or the right of the 75th percentile.
The dominant component of , , and each achieves an IQR at least $3$ times that of . These components do not clearly distinguish among the three populations, indicating that these methods are insensitive to the effect of the outliers. and appear the most effective in this situation; indeed, it appears that and perform as well as *setosa*-only PCA.
Although results in the most modest IQR in the *setosa* among the robust methods, the IQR associated with the component is $1.95$ times the IQR of the *setosa* family along the dominant PCA component. Unlike the other robust methods, the component discriminates among the three distinct populations. While it is clear that *does not* reject the influence of the outliers, balances the influence of outliers and the bulk of the data better than . In this experiment the optimality ratio for is $0.9975$, certifying that the component is essentially the direction of maximum mean deviation in the data.
Regression Surface for Bus Data {#sec:bus}
-------------------------------
In this experiment, we construct a regression surface using multiple components. A point is well described by a surface if its Euclidean distance from the surface is small. The dominant $T$ classical principal components span a $T$-dimensional regression surface such that the sum of the squared distances of the observations to the plane is minimized. We would hope that robust components describe the bulk of the points better than standard components when outliers contaminate the data. We illustrate this behavior with an experiment of Maronna et al. [@Maronna2006 p. 214], which we augment with additional robust methods.
### Experimental setup {#sec:bus_setup}
Our data consists of $p=18 $ geometric features collected from $n=218$ bus silhouettes [@Siebert1987] that we arrange into an $n\times p$ matrix $\mat X$. Following Maronna et al., we remove the $9$th variable from the data and divide the columns of $\mat X$ by their median absolute deviation (MADN), a robust measure of scale defined as $$\MADN(\vec x) = \median(\abs{\vec x - \median(\vec x)}).$$ We then center the observations by their Euclidean median. We compute the top three components using PCA, , , , and . We take the parameter $\gamma = 0.8\sqrt{n/m}$, the parameter $\lambda = \sqrt{1/m}$, and the rounding count of $K=94$.
For each method, we determine the Euclidean distance from each observation to the orthogonal regression plane spanned by the dominant three components. In Figure \[Fig:bus\], we plot the ordered distances to the robust hyperplanes against the ordered distances to the PCA hyperplane.
Since the PCA regression surface minimizes the sum of squared distances to the observations, not all of the observations can lie below the 1:1 line. However, a large number of points below the 1:1 line indicates that a robust regression surface explains the bulk of the data better than the classical surface.
### Discussion
Figure \[Fig:bus\] focuses on the third and fourth quantiles of the data; the first and second quantiles roughly follow the pattern apparent in the third quantile. For clarity, we omit the three most outlying points that would appear in the upper right corner of the figure. Each robust method results in a regression surfaces that explains the data better than PCA for more than $75\%$ of the points. In the third quantile, both and lose their explanatory advantage over PCA. It is not until the after $95\%$ of the data that and provide worse explanations than PCA. is the dominating method through the latter part of the data.
explains the bulk of the data less effectively than the other robust methods, yet the final outlying observations are explained better by than the other methods. This indicates that is more sensitive to outlying points than the other robust methods, but is less sensitive to outliers than standard PCA. The optimality ratios for the first three components are, respectively, $0.99999$, $0.99992$, and $0.97253$, implying that essentially succeeds in PP-PCA with the MD scale for this data.
Finally, we note that changing the parameter to $\lambda =
2\sqrt{1/n}$ results in performance similar to .
Movielens {#sec:movielens}
---------
We finish this section with a larger example: the million-rating movielens data [@movielens]. The data consist of 6040 users rating and 3952 movies, though several movies are replicated. The set contains just over one million ratings. Each rating is between one and five stars, and each user in the data set rated at least 20 movies.
We arrange these responses into an $n=6040$ by $p=3952$ matrix $\mat
X$ whose rows correspond to the users and whose columns correspond to the movies. We set unrated movies to the user’s median rating, and center each user’s ratings by their personal median. As with our other experiments, we center the rows by the Euclidean median, which results in a dense matrix with nearly $24$ million entries.
We then compute the top two components using PCA, , , and . In order to speed up processing for , we set $\gamma =
\sqrt{100/p}$. As discussed in Section \[sec:lldalg\], this choice of $\gamma$ limits the rank of the iterates $\mat P^{(k)}$ in the ALMM algorithm, which allows us to compute a partial SVD at each step. Our choice $\gamma= \sqrt{100/n}$ results in iterates whose rank is roughly $10$; the rank of the optimal point $\mat P_\opt$ is nine.
Each component $\vec v$ represents a direction in movie coordinates. The magnitude entry $[\vec v]_i$ indicates how much $\vec v$ points in the direction of movie $i$. We use these magnitude of the entries in the components to rank the movies. We call movies with large magnitudes “important,” and we call the corresponding entry of the component a movie’s “importance.”
### Discussion {#sec:movielensdiscussion}
Table \[tab:movielens1st\] displays the five most important movies identified by the first standard principal component, along with the importance and rank calculated assigned to these movies by the robust components. Each method agrees that the violent mobster movie *GoodFellas* is the most important film. Indeed, *GoodFellas*, *Army of Darkness*, *A Little Princess*, and *Stand by Me* are ranked in the top five movies by every method. However, PCA ranks *Pushing Hands* much higher than the robust methods.
Movie
------------------- ---------- --- ----------- ---- ---------- ---- ---------- ---
GoodFellas $0.0708$ 1 $0.1016$ 1 $0.1092$ 1 $0.0885$ 1
Army of Darkness $0.0697$ 2 $ 0.0914$ 3 $0.0970$ 4 $0.0832$ 2
A Little Princess $0.0664$ 3 $0.0899$ 4 $0.1028$ 2 $0.0826$ 3
Pushing Hands $0.0657$ 4 $0.0772$ 11 $0.0827$ 10 $0.0745$ 8
Stand by Me $0.0656$ 5 $0.0853$ 5 $0.0896$ 5 $0.0765$ 5
: *Most important movies: second component.*[]{data-label="tab:movielens2nd"}
Movie
---------------------- ----------- --- ----------- ---- ----------- ---- ----------- ----
Nikita $0.0982$ 1 $0.1099$ 2 $0.0959$ 13 $0.1071$ 5
Citizen Kane $0.0945$ 2 $0.1051$ 4 $0.0935$ 15 $0.1104$ 3
Fried Green Tomatoes $0.0917$ 3 $0.0934$ 8 $0.0727$ 35 $0.0944$ 11
Unforgiven $0.0891$ 4 $0.0923$ 10 $0.0877$ 21 $0.0982$ 9
Mommie Dearest $-0.0855$ 5 $-0.1108$ 1 $-0.1522$ 1 $-0.1281$ 1
: *Most important movies: second component.*[]{data-label="tab:movielens2nd"}
In Table \[tab:movielens1st\], each importance has positive sign. For each method, the first component assigns very few movies a negative importance for the first component. This fact comes about because the typical user rating is positive; that is, the sum $\sum_j
[\vec x_i]_j$ is greater than zero for most users.
Table \[tab:movielens2nd\] displays the results for the second components. Each robust component views *Mommie Dearest* as the most important movie, while standard PCA relegates it to fifth place. Neither *Fried Green Tomatoes* nor *Unforgiven* are among the top five movies for the robust methods. With the second component, takes the most dramatic shift away from PCA, with only *Mommie Dearest* making it into the top ten movies.
Of course, rankings are not the whole story. The signs are very consistent between methods. *Mommie Dearest* is negative for every method considered and *Fried Green Tomatoes* is positive. The sign consistency indicates that these components are measuring essentially the same thing.
The magnitude of the importance are also telling. PCA assigns the smallest weight to every movie, with the exception of the second component of . This indicates that the robust methods are willing to assign more importance to discriminating movies.
Acknowledgments {#ack .unnumbered}
===============
The first author would like to thank Alex Gittens, Richard Chen, and Stephen Becker for valuable discussions regarding this work.
Proof of Theorem \[thm:mdr\] {#sec:mdrproof}
============================
This appendix contains the proof of Theorem \[thm:mdr\] that we repeat below as Theorem \[thm:mdrapp\]. We begin with some supporting results. The following result of Alon and Naor [@Alon2006 Sec. 4.2] allows us to bound the expectation of $\lone{\mat X \vec v_\opt}$ below. The essence of this result goes back to a 1953 paper of Grothendieck [@grothendieck1953résumé]; see also the little Grothendieck theorem in [@Pisier1986a Sec. 5b].
\[lemma:alon-naor\] Let $\alpha_\opt^2$ be the value of the optimization problem of Algorithm \[alg:mdr\]. Then $\alpha_\opt^2 \ge \norm{\mat X \mat X^{\ad}}_{\infty\to 1}$. Moreover, let $\vec y^{(k)}$ be one of the vectors generated in Step \[step:mdr\_random\]. Then $\xpcd \ltwo{ \mat X^{\ad} \vec
y^{(k)}}^2 \ge \frac{2}{\pi}\alpha_\opt^2$.
The claim $\alpha_\opt^2 \ge \norm{\mat X \mat X^\ad}_{\infty\to 1}$ also follows from our discussion of the SDP relaxation in Section \[sec:sdprelax\]. We also need the following proposition.
\[prop:inf1factorapp\] For each matrix $\mat X$, the identity $\norm{\mat X\mat
X^{\ad}}_{\infty\to 1} = \norm{\mat X}_{2 \to 1}^2$ holds.
We can express $$\norm{\mat X\mat X^{\ad} }_{\infty \to 1} = \max_{\substack{\norm{\vec
w}_\infty = 1\\ \norm{\vec y}_\infty = 1}} \Inner{\mat X^{\ad}
\vec w,\mat X^{\ad}\vec y}.$$ By the conditions for equality in the Cauchy–Schwarz inequality, it follows that we can take $\vec w = \vec y$ above. Hence $$\norm{\mat X \mat X^{\ad}}_{\infty \to 1}= \norm{\mat X^{\ad}}_{\infty \to
2}^2 = \norm{\mat X}_{1\to 2},$$ where the last equality is a standard fact concerning adjoint operators.
We use the following variant of the Paley–Zygmund integral inequality [@Paley1932] to bound the probability that $\lone{\mat
X \vec v_\opt}$ is less than its expectation.
\[lemma:pz\] Suppose $Z$ is a random variable such that $0\le Z \le C$ for some $C>0$. Then, for any scalar $\theta \in [0,1]$, we have $ \prob(Z >
\theta \xpcd [Z] ) \ge C^{-1}(1-\theta) \xpcd [Z]. $
Split the integral $\xpcd[ Z]$ into two integrals, the first over the region $Z \le \theta \xpcd[ Z]$ and the second over the region $Z >
\theta \xpcd [Z]$. Notice that the former integral is bounded above by $\theta \xpcd [Z]$, while the latter integral is bounded above by $C \prob(Z > \theta \xpcd [Z])$. Simple algebraic manipulation then shows the claim.
We now restate and prove the main Theorem of Section \[sec:mdr\].
\[thm:mdrapp\] Suppose that $\mat X$ is an $n\times p$ matrix, and let $K$ be the number of rounding trials. Let $(\vec v_\opt, \alpha_\opt)$ be the output of Algorithm \[alg:mdr\]. Then $\alpha_\opt \ge \norm{\mat
X}_{2\to 1}$. Moreover, for $\theta \in [0,1]$, the inequality $$\lone{\mat X \vec v_\opt } > \theta \sqrt{\frac{2}{\pi}}
\alpha_\opt$$ holds except with probability $\mathrm{e}^{-2K(1-\theta^2)/\pi}$.
Let $\vec y\in\{\pm 1\}^n$ be a sign vector and define $\vec v =
\mat X^* \vec y / \ltwo{\mat X^{\ad} \vec y}$. Then $$\lone{\mat X \vec v} = \ltwo{\mat X^{\ad}\vec y}^{-1}\max_{\vec w \in \{\pm
1\}^n}\Inner{\vec w,\mat X \mat X^{\ad} \vec y} \ge \ltwo{\mat X^{\ad}
\vec y}$$ where the inequality follows by taking the specific choice $\vec w =
\vec y$. In particular, this relation implies that the vectors $\vec v^{(k)} = \mat X^* \vec y^{(k)}/\ltwo{\mat X^* \vec y^{(k)}}$ generated in Step \[step:mdr\_random\] of Algorithm \[alg:mdr\] satisfy $$\begin{aligned}
\label{eq:key}
\xpcd \bigl\|\mat X\vec v^{(k)}\bigr\|_1^2 \ge \xpcd \bigl\|\mat
X^\ad \vec y^{(k)}\bigr\|_2^2 &\ge \frac{2}{\pi} \alpha_\opt^2,
\end{aligned}$$ where the last inequality follows from the second claim in Lemma \[lemma:alon-naor\].
Since $\|{\vec v^{(k)}}\|_2 = 1$, the quantity $\|{\mat X \vec
v^{(k)}}\|_1^2$ is a positive random variable bounded above by $\norm{\mat X}_{2\to 1}^2$. Therefore, inequality and Lemma \[lemma:pz\] imply that $$\begin{aligned}
\prob\left(\bigl\|\mat X\vec v^{(k)}\bigr\|_1^2 >
\theta^2\cdot \frac{2\alpha_\opt^2}{\pi}\right) &
\ge(1-\theta^2)\frac{2}{\pi}\cdot \left(\frac{\alpha_\opt
}{\norm{\mat X}_{2\to 1}}\right)^2 \ge
\frac{2}{\pi}\cdot(1-\theta^2),
\end{aligned}$$ where we have used the fact that $\alpha_\opt \ge \norm{\mat
X}_{2\to 1}$ by Proposition \[prop:inf1factorapp\] and the first claim of Lemma \[lemma:alon-naor\].
In Step \[step:mdr\_max\] of the algorithm we have chosen $\vec
v_\opt$ to maximize $\|\mat X \vec v_\opt\|_1^2$, so the inequality $\lone{\mat X \vec v_\opt}^2 \le 2(1-\theta^2)/\pi$ holds if and only if $\|\mat X \vec v^{(k)}\|_1\le 2(1-\theta^2)/\pi$ for all $k$. Therefore, the independence of $\vec v^{(k)}$ for $k =
1,\dotsc, K$ implies $$\prob\left(\lone{\mat X\vec v_\opt} \le \theta
\sqrt{\frac{2}{\pi}}\norm{\mat X}_{2\to 1} \right) \le \left( 1 -
\frac{2}{\pi}\cdot(1-\theta^2) \right)^K < \mathrm{e}^{-2 K (1-\theta^2) /\pi},$$ which completes the claim.
[^1]: One usually defines scales so that they are unbiased estimates of the sample standard deviation when the data is drawn from a normal distribution. We are more interested in the direction of maximal scale rather than the value, so we can safely ignore the normalization factor.
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Institut für Theoretische Physik, Universität Zürich,\
Winterthurerstr. 190, 8057 Zürich, Switzerland\
E-mail:
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Aude Gehrmann-De Ridder, Mathias Ritzmann\
Institut für Theoretische Physik, ETH,\
CH-8093 Zürich, Switzerland
title: NNLO antenna subtraction with two hadronic initial states
---
Introduction {#sec:introduction}
============
Jet observables are an important tool for precision studies due to their large production cross sections at high energy colliders. Calculating perturbative higher order corrections to jet production cross sections requires a systematic procedure to extract infrared singularities from real radiation contributions. The latter arise if one or more final state particles become soft or collinear. At the next-to-leading order level (NLO), several systematic and process-independent procedures are available. The two main methods are phase space slicing [@Giele:1991vf; @Giele:1993dj] and subtraction based methods [@Frixione:1995ms; @Catani:1996vz]. All subtraction methods consist in introducing terms that are subtracted from the real radiation part at each phase space point. These subtraction terms approximate the matrix element in all singular limits, and are sufficiently simple to be integrated over the corresponding phase space analytically. After this integration, the infrared divergences of the subtraction terms become explicit and the integrated subtraction terms can be added to the virtual corrections yielding an infrared finite result. Since experimental data of jet observables are reaching an accuracy of a few percent or better, accurate precision studies must rely on theoretical predictions that have the same precision. In some cases, this requires corrections at the next-to-next-to-leading order (NNLO) level in perturbative QCD.\
NNLO calculations of observables with $n$ jets in the final state require several ingredients: the two-loop correction to $n$-parton matrix elements, the one-loop correction to $(n+1)$-parton matrix elements, and the tree-level $(n+2)$-parton matrix elements. For most massless jet observables of phenomenological interest, the two-loop matrix elements have been computed some time ago, while the other two types of matrix elements are usually known from calculations of NLO corrections to $(n+1)$ jet production [@mcfm]. At the NNLO level, $n$ jet observables receive contributions from the one-loop ($n+1)$-parton matrix elements, where one of the involved partons can become unresolved (soft or collinear), as well as from $(n+2)$-parton matrix elements where up to two partons can become simultaneously soft and/or collinear. In order to determine the contribution to NNLO jet observables from these configurations, two-parton subtraction terms have to be constructed. Several NNLO subtraction methods have been proposed in the literature [@Kosower:2002su; @nnlosub2; @nnlosub3; @nnlosub4; @nnlosub5]. Another approach used for NNLO calculations of exclusive observables is sector decomposition [@secdec].\
In ref. [@GehrmannDeRidder:2005cm], an NNLO subtraction method was developed for observables with partons in the final state only, the antenna subtraction method. It constructs the subtraction terms from the so-called antenna functions. The latter describe all unresolved partonic radiation between a hard pair of colour-ordered partons, the radiators. The antenna functions are derived systematically from physical matrix elements and can be integrated over their factorized phase space. At the NLO level, this formalism can handle massless partons in the initial or final states [@Daleo:2006xa], as well as massive fermions in the final state [@ritzmann]. For processes with initial-state hadrons, NNLO antenna subtraction terms have to be constructed for two different cases: only one radiator parton is in the initial state (initial-final antenna) or both radiator partons are in the initial state (initial-initial antenna). Recently, in [@Daleo:2009yj; @Luisoni10], NNLO initial-final antenna functions were derived and integrated over their factorized phase space. The case with two radiators in the initial state is however still outstanding.\
In this contribution, we discuss the derivation of NNLO initial-initial antenna functions. We briefly describe the construction of the subtraction terms and the required phase space transformations and discuss how the phase space integrals for initial-initial antennae can be performed using multi-loop techniques.
Subtraction terms for initial-initial configurations {#sec:definitions}
====================================================
At NNLO, there are two types of contributions to $m$-jet observables that require subtraction: the tree-level $m+2$ parton matrix elements (where one or two partons can become unresolved), and the one-loop $m+1$ parton matrix elements (where one parton can become unresolved). In the tree-level double real radiation case, we can distinguish four different types of unresolved configurations depending on how the unresolved partons are colour connected to the emitting hard partons (see ref. [@GehrmannDeRidder:2005cm] for a detailed description of the four cases). In this contribution, we focus on the case with two colour-connected unresolved partons (colour-connected). This is the only case where new ingredients are needed, namely the four-parton initial-initial antenna functions. The unintegrated ones can be obtained by crossing two partons to the initial state in the corresponding final-final antenna functions, which can be found in [@GehrmannDeRidder:2005cm], and have then to be integrated analytically over the appropriate antenna phase space. The corresponding NNLO antenna subtraction term, to be convoluted with the appropriate parton distribution functions for the initial state partons, for a configuration with the two hard emitters in the initial state (partons $i$ and $l$ with momenta $p_1$ and $p_2$) can be written as: $$\begin{aligned}
\label{eq:sub2b}
{\rm d}\sigma_{NNLO}^{S,\,{\scriptscriptstyle}{colour-connected}}
&=& {\cal N}\sum_{m+2}{\rm d}\Phi_{m+2}(k_{1},\ldots,k_{m+2};p_1,p_2)
\frac{1}{S_{{m+2}}} \nonumber \\
&\times& \Bigg [ \sum_{jk}\;\left( X^0_{il,\,jk}
- X^0_{i,\,jk} \,X^0_{Il,\,K} - X^0_{l,\,kj}\, X^0_{iL,\,J} \right)\nonumber \\
&\times&
|{\cal M}_{m}(K_{1},\ldots,{K}_{L},\ldots,K_{m+2};x_1p_1,x_2p_2)|^2\,
\JET_{m}^{(m)}(K_{1},\ldots,{K}_{L},\ldots,K_{m+2})
\Bigg ].\end{aligned}$$ The subtraction term in eq. (\[eq:sub2b\]) is constructed such that all unresolved limits of the four-parton antenna function $X^0_{il,\,jk}$ are subtracted, so that the resulting subtraction term is active only in its double unresolved limits, which explains the presence of the products of three-parton antennae. The subtraction terms for all the other unresolved configurations can be constructed using tree-level three-parton antenna functions. In eq. (\[eq:sub2b\]), the tree antennae $X^0_{il,\,jk}$, $X^0_{i,\,jk}$ and $X^0_{l,\,jk}$ depend on the original momenta $p_1,\, p_2,\, k_j,\, k_k$, whereas the rest of the antenna functions as well as the jet function $\JET$ and the reduced matrix elements ${\cal M}_m$ depend on the redefined momenta from the phase space mapping, labeled by $I,\,J \dots$. In addition to that, the reduced matrix elements depend on the momentum fractions $x_1$ and $x_2$, which we define later. The normalization factor $\cal N$ includes all QCD-independent factors as well as the dependence on the renormalized QCD coupling $\alpha_s$, $\sum_{m+2}$ denotes the sum over all configurations with $m+2$ partons, ${\rm d}\Phi_{m+2}$ is the phase space for an $(m+2)$-parton final state in $d=4-2\varepsilon$, and finally, $S_{m+2}$ is a symmetry factor for identical partons in the final state. The antenna functions can be integrated analytically, provided we have a suitable factorization of the phase space. The factorization is possible through an appropriate mapping of the original set of momenta. These mappings interpolate between the different soft and collinear limits that the subtraction term regulates. They must satisfy overall momentum conservation and keep the mapped momenta on the mass shell.\
A complete factorisation of the phase space into a convolution of an $m$ particle phase space depending on redefined momenta only, with the phase space of partons $j,\, k$, can be achieved with a Lorentz boost that maps the momentum $q \;=\; p_1+p_2-k_j-k_k$, with $q^2>0$, into the momentum $\tilde{q} \,=\, x_1 p_1 + x_2 p_2 $, where $x_{1,2}$ are fixed in terms of the invariants as follows [@Daleo:2006xa]: $$\begin{aligned}
x_1&=&\left(\frac{s_{12}-s_{j2}-s_{k2}}{s_{12}}
\; \frac{s_{12}-s_{1j}-s_{1k}-s_{j2}-s_{k2}+s_{jk}}{s_{12}-s_{1j}-s_{1k}}
\right)^{\frac{1}{2}}\,,\nonumber\\
x_2&=&\left(\frac{s_{12}-s_{1j}-s_{1k}}{s_{12}}
\; \frac{s_{12}-s_{1j}-s_{1k}-s_{j2}-s_{k2}+s_{jk}}{s_{12}-s_{j2}-s_{k2}}
\right)^{\frac{1}{2}}\,.\end{aligned}$$ These last two definitions guarantee the overall momentum conservation in the mapped momenta and the right soft and collinear behavior. The two momentum fractions satisfy the following limits in double unresolved configurations:
1. $j$ and $k$ soft: $x_1\rightarrow 1$, $x_2\rightarrow 1$,
2. $j$ soft and $k_{k}=z_1p_1$: $x_1\rightarrow 1-z_1$, $x_2\rightarrow 1$,
3. $k_{j}=z_1p_1$ and $k_{k}=z_2p_2$: $x_1\rightarrow 1-z_1$, $x_2\rightarrow 1-z_2$,
4. $k_{j}+k_{k}=z_1p_1$: $x_1\rightarrow 1-z_1$, $x_2\rightarrow 1$,
and all the limits obtained from the ones above by the exchange of $p_1$ with $p_2$ and of $k_j$ with $k_k$. The factorized $(m+2)$-partons phase space into an $m$-partons phase space and an antenna phase space is given by: $$\begin{aligned}
\d\Phi_{m+2}(k_1,\dots,k_{m+2};p_1,p_2)&=&
\d\Phi_{m}(K_1,\dots,K_{j-1},K_{j+1},\dots,K_{k-1},K_{k+1},\dots,K_{m+2};x_1p_1,x_2p_2)
\nonumber\\
&&\times \; {\cal J} \;\delta(q^2-x_1\,x_2\,s_{12})\,
\delta(2\,(x_2p_2-x_1p_1). q)\,
\nonumber\\
&&\times \;[\d k_j]\;[\d k_k]\;\d x_1\;\d x_2\,,
\label{PS}\end{aligned}$$ where $[\d k]\, = \,\d^dk/(2\pi)^{(d-1)}\,\delta^+(k^2)$, and ${\cal J}$ is the Jacobian factor defined by $${\cal J}=s_{12}\,\left(x_1(s_{12}-s_{1j}-s_{1k}) + x_2
(s_{12}-s_{2j}-s_{2k})\right)\,.$$ The next step is to integrate the antenna functions over their factorized phase space.
Calculational approach for the double real radiation case $2 \rightarrow 3$ {#sec:calc}
===========================================================================
All the initial-initial antennae have the scattering kinematics $p_1 + p_2 \to k_j+k_k + q$, where $q$ is the momentum of the outgoing particle, for example the vector boson in a vector boson plus jet process. Double real radiation antenna integrals are derived from squared matrix elements and can be represented by forward scattering diagrams as in the following figure:
{width="50.00000%"} \[fig:cutantennae\]
The two delta functions in eq.( \[PS\]) can be represented as mass-shell conditions of fake particles and are shown in the previous picture as a thick solid line (representing a massive particle with mass $M=x_1\,x_2\,s_{12}$) and a dashed line (representing a massless particle). This allows us to use the optical theorem to transform the initial-initial antenna phase space integrals into cut two-loop box integrals and, therefore, use the methods developed for multi-loop calculations [@Anastasiou:2002yz; @Anastasiou:2003ds]. Up to $8$-propagator integrals with $4$ cut propagators are generated in this way. The calculation of the integrated antennae corresponds here to the evaluation of a reduced set of master integrals. We found $30$ of them, obtained using integration-by-part (IBP) and Lorentz identities, following the Laporta algorithm. We then calculate this small set of integrals using the method of differential equations. The simplest master integral is the two loop box with all the internal lines cut and defined as follows\
$$\begin{aligned}
\label{masterPR1}
&&\PR1 \;\;= \,I(x_1,x_2) \,= \,\int d^d q \,d^dk_j \,d^d k_k\,
\delta^d\left(p_1+p_2-q-k_j-k_k\right) \;
\times \nonumber
\\&& \hspace{4cm}
\delta^+\left(k_j^2\right) \,
\delta^+\left(k_k^2\right)\,\delta^+\left(q^2-M^2\right)
\delta(2\,\left(x_2p_2-x_1p_1).q\right) \,.\end{aligned}$$ As we have discussed in section \[sec:definitions\], the phase space integrals (and therefore the master integrals) have to be studied in four different regions of the phase space depending on the values of $x_1$ and $x_2$, namely:
- $x_1\,\neq\,1$, $x_2\,\neq\,1$, we refer to this region as the hard one
- $x_1\,=\,1$, $x_2\,\neq\,1$, and $x_1\,\neq\,1$, $x_2\,=\,1$, referred to as the collinear region
- $x_1\,=\,1$, $x_2\,=\,1$, is the soft region.
In the hard region, the solution of the system of differential equations yields two-dimensional generalized harmonic polylogarithms. The $\varepsilon$ expansion is needed up to transcendentality $2$. In the collinear regions, additional $1/\varepsilon$ coefficients may be generated and the epsilon expansion is done up to transcendentality $3$, whereas in the soft region additional $1/\varepsilon^2$ coefficients may appear and the expansion in epsilon is pushed to transcendentality $4$. We note however that the calculation of the masters in the soft and collinear regions, although needed with deeper expansions in $\varepsilon$, is simpler than in the hard region, and only one-dimensional harmonic polylogarithms are needed in the collinear regions. In the soft region, a direct calculation is possible giving closed form results in $\varepsilon$ in the form of gamma functions. The boundary conditions for the differential equations are obtained, in most of the cases, by studying the master integrals in one of the collinear limits. Otherwise the soft limit is used.\
In a first step towards the calculation of all the integrated initial-initial antennae for the $2 \rightarrow 3$ tree-level double real radiation case, we have focused on all the crossings of two partons from the following final-final antennae: $B_4^0(q,q',\bar{q}',\bar{q})$, $\tilde{E}_4^0(q,q',\bar{q}',g)$ and $H_4^0(q,\bar{q},q',\bar{q}')$ defined in [@GehrmannDeRidder:2005cm], where the index $4$ refers to four partons. There are $13$ master integrals involved in their calculation, and the ones without irreducible scalar products are shown in Fig. \[fig:mastersHBEt\].
![[]{data-label="fig:mastersHBEt"}](masters_BHEt.eps){width="70.00000%"}
\
Finally, the one-loop $2 \rightarrow 2$ antenna functions do not present any difficulty, since the needed one-loop box integrals are known analytically to all orders in $\varepsilon$ for the final-final configuration [@Kramer:1986sr]. Obtaining the initial-initial contribution using these results requires crossing two legs to the initial state and performing the necessary analytic continuation of the involved hypergeometric functions. No integrals are required in this case.
Outlook
=======
In this contribution, we have discussed the extension of the antenna subtraction formalism to the initial-initial configurations, including the required phase space factorisation and mappings. We have focused on the $2 \rightarrow 3$ tree-level double real radiation contribution. In a first step towards the derivation of the complete set of integrated initial-initial antennae, we considered all the crossings of the subset of $4$-parton antennae: $B_4^0(q,q',\bar{q}',\bar{q})$, $\tilde{E}_4^0(q,q',\bar{q}',g)$ and $H_4^0(q,\bar{q},q',\bar{q}')$. Completing the full set of NNLO antenna functions will allow the construction of subtraction terms needed for the evaluation of jet observables at hadron colliders.
Acknowledgments
===============
This work is supported by the Swiss National Science Foundation (SNF) under contracts 200020-116756/2 and PP002-118864.
|
---
abstract: 'The magnetic structure of BaFe$_2$As$_2$ was completely determined from polycrystalline neutron diffraction measurements soon after the ThCr$_2$Si$_2$-type FeAs-based superconductors were discovered. Both the moment direction and the in-plane antiferromagnetic wavevector are along the longer $a$-axis of the orthorhombic unit cell. There is only one combined magnetostructural transition at $\sim$140 K. However, a later single-crystal neutron diffraction work reported contradicting results. Here we show neutron diffraction results from a clean single crystal sample, grown by a self-flux method, that support the original polycrystalline work.'
address:
- '$^1$ Department of Physics, University of Virginia, Charlottesville, VA 22904, USA'
- '$^2$ NIST Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, MD 20899, USA'
- '$^3$ Dept. of Materials Science and Engineering, University of Maryland, College Park, MD 20742, USA'
- '$^4$ Los Alamos National Laboratory, Los Alamos, NM 87545, USA'
- '$^5$ Hefei National Laboratory for Physical Science at Microscale and Department of Physics, University of Science and Technology of China, Hefei, Anhui 230026, China'
author:
- 'M. Kofu$^1$, Y. Qiu$^{2,3}$, Wei Bao$^4$, S.-H. Lee$^1$, S. Chang$^2$, T. Wu$^5$, G. Wu$^5$ and X. H. Chen$^5$'
title: 'Neutron scattering investigation of the magnetic order in single crystalline BaFe$_2$As$_2$'
---
Introduction
============
Over the last two years, several superconductors have been discovered in fluorine-doped lanthanum oxypnictides LaFePO [@Kamihara2006], LaNiPO [@Watanabe2007] and LaFeAsO [@Kamihara2008]. The common ZrCuSiAs-type (1111) structure is composed of alternating Fe(Ni)As(P) and LaO layers. Band structure calculations indicate that electronic states at the Fermi level are contributed predominately by the transition metal Fe or Ni in these semimetals [@bs07; @A030429; @A031279; @A060750; @A071010; @A103274], thus the importance of the layer formed by the edge-sharing Fe(Ni) pnictide tetrahedra. The same edge-sharing tetrahedral layer is also the central structural component in the previously discovered $Ln$Ni$_2$B$_2$C superconductors [@NiBC_str]. However, a distinguishing feature of the LaFeAsO system is that its superconducting transition temperature $T_C$ increases from 26 K to above 40 K [@A033603; @A033790], and finally reaches about 55 K at optimal F doping [@A042053; @A042105] when La is replaced by magnetic Sm or Ce. Other magnetic lanthanide substitutions have also been shown to result in increased $T_C$ [@A034234; @A034283; @A034384; @A040835; @A042582; @A043727; @A044290; @A060926]. Therefore, an unconventional superconducting mechanism is suspected. Indeed while the FeP and NiP materials with their low $T_C\approx 3$-4 K may be accounted for by conventional electron-phonon interactions, the $T_C$ of the FeAs materials is too high for the phonon mechanism according to theoretical calculations [@A032703]. The first-principle phonon spectrum used in the calculations has since found support from neutron scattering, optical and resonant x-ray scattering measurements [@A051062; @A051321; @A073370; @A073172; @A073968].
While the $Ln$O layer ($Ln$=La, Sm, Ce, Nd, Pr, Gd, Tb or Dy [@Kamihara2008; @A033603; @A033790; @A042053; @A042105; @A034234; @A034283; @A034384; @A040835; @A042582; @A043727; @A044290; @A060926]) provides an excellent opportunity to investigate the interaction between superconductivity and rare-earth magnetism in the $Ln$FeAsO systems, its existence is not necessary for superconductivity. Superconductivity has been discovered in related materials with the ThCr$_2$Si$_2$-type (122) structure, where the $Ln$O layer is replaced by elemental Ba [@A054630], Sr [@A061209; @A061301] or Ca [@A064279] layer, and in Li$_{1-x}$FeAs [@A064688] and Fe$_{1+x}$(Se,Te) [@A072369; @A074775] with the PbO-type (11) structure, which does not contain the intervening layer, but Li [@A072228] or excess Fe [@A092058] occupies an interstitial site. Therefore, the multi-orbital theoretical model based on similar semimetallic electronic states from the common Fe structural layer is likely to capture the essential physics for understanding high $T_C$ superconductivity in these Fe-based materials [@A032740; @A033325; @A033982; @A034346; @A041113; @A044678; @A061933; @A063285; @A101476]. In addition, magnetic fluctuations have been proposed as the bosonic glue for Cooper pair formation. Currently, the extended $s$-wave superconductivity mediated by magnetic fluctuations is favored. Experiments supporting a nodeless superconducting gap have emerged [@A054616; @A070398; @A070419].
Stoichiometric $Ln$FeAsO and $A$Fe$_2$As$_2$ ($A$ = Ba, Sr or Ca) are not superconductors. LaFeAsO experiences a structural transition from tetragonal to orthorhombic symmetry at 150 K, which shows up as a strong anomaly in resistivity, and an antiferromagnetic transition at 137 K [@A040795; @A043569]. For BaFe$_2$As$_2$, the structural and magnetic transitions occur at the same temperature [@A062776]. The magnetic propagation vector is ($\pi,0,\pi$) in terms of the primitive tetragonal magnetic unit cell for both LaFeAsO [@A040795; @A063878] and BaFe$_2$As$_2$ [@A062776], although their crystal structures are different. When La is replaced by magnetic Nd or Pr, the magnetic wavevector changes to ($\pi,0,0$) in the combined Fe and rare-earth magnetic order at low temperature [@A062195; @A074441; @A074872]. For Ce substitutions, a different antiferromagnetic ordering of Ce ions was reported without the refinements provided [@A062528], but the Fe part of the magnetic order is still the same as we reported for the Nd compound and is characterized by ($\pi,0,0$) [@A062195].
The in-plane ($\pi,0$) magnetic wavevector is consistent with the nesting of electron and hole Fermi surfaces, which has been anticipated from band structure theory [@A033236; @A033325; @A033286]. It breaks the tetragonal symmetry of the high temperature structure and is consistent with the orthorhombic distortion at low temperature. The antiparellel moment alignment is determined by neutron diffraction to be along the longer of the in-plane axes, and the parallel alignment along the shorter axis of the orthorhombic unit cell in both NdFeAsO and BaFe$_2$As$_2$ [@A062195; @A062776]. This magnetostriction pattern is opposite to the usual case of single orbital magnetism and is explained by careful calculations taking into account the multi-orbital origin of the antiferromagnetic order [@A042252]. The moment direction has also been determined to be along the longer of the in-plane axes [@A062195; @A062776], and the same magnetostrictive expansion and contraction have been found later in SrFe$_2$As$_2$ [@A070632; @A071077], CaFe$_2$As$_2$ [@A071525] and PrFeAsO [@A074441; @A074872] in poly- and single-crystal studies. However, in a single crystal neutron diffraction study of BaFe$_2$As$_2$ [@A071743], results different from our polycrystalline work [@A062776] have been reported. To clarify the issue, we show single crystal results in section \[sec3\], which are consistent with our previous polycrystalline study.
Experimental details
====================
The single crystal sample of BaFe$_2$As$_2$ was grown using a self-flux method [@A062452]. A distinct feature of single crystals grown this way is that the resistivity shows a sharp drop at the phase transition at $\sim$140 K, similar to results from polycrystalline samples. The single crystal used by Su in their single crystal neutron diffraction work was grown in Sn flux [@A071743]. Not only is the transition temperature much reduced, the material becomes an insulator at low temperature, in contrast to the expected metallic behavior. We conducted single crystal neutron diffraction measurements with the cold neutron triple-axis spectrometer SPINS at NIST Center for Neutron Research. The sample was mounted in a He-filled Al can in a closed cycle refrigerator so that ($h0l$) was in the scattering plane. Neutrons of 5 meV were selected using pyrolytic graphite (002) as both monochromator and analyzer. A cold Be filter was placed in the neutron path to reduce contamination from higher order neutrons. The lattice parameters are $a=5.615$, $b=5.571$ and $c=12.97\AA$ at 12.5 K in the othorhombic structure.
Single crystal diffraction experiments {#sec3}
======================================
The mosaic of our BaFe$_2$As$_2$ single crystal sample is shown in (b). The composition uniformity is indicated by the nice peak in the $\theta-2\theta$ scan in (a). At 12.5 K, the orthorhombic distortion of the crystal structure is indicated by the well resolved (400) and (040) Bragg peaks due to twinning in (c). That $a>b$ is reflected in the shorter reciprocal length of the $|(400)|$ in comparison to $|(040)|$. The fact that only (101), not the twinning (011), peak exists in (d) is consistent with our previous determination of the (101), not (011), as the magnetic propagation vector with the definition of $a>b$ [@A062776]. It is opposite to what was reported by Su for their Sn-flux grown single crystal sample [@A071743]. The $l$ scan in (e), close to the rocking direction, further supports the commensurate assignment of the magnetic propagation vector.
![ (a-b) The $\theta-2\theta$ and rocking scans of the (004) structural Bragg peak. (c) The twinning (400) and (040) structural Bragg peaks measured using $\lambda/2$ neutrons near the (200) position. (d-e) Two perpendicular scans through the (101) magnetic Bragg peaks. (f) The (103) magnetic Bragg peak. The blue symbols represent measurements at 12.5 K, and the red at 150 K. The error bars in all the figures represent the standard deviation in the measurement.[]{data-label="fig1"}](fig1.ps "fig:") -.2 cm
Consistent with our previous data [@A062776], the (103) magnetic Bragg peak in (f) is stronger than that of (101), reflecting the moment orientation factor in the magnetic neutron diffraction cross-section when the moment points along the (100) direction. At 150 K above the simultaneous magnetostructural transition, magnetic Bragg peaks disappear completely as shown in (d) for (101).
While there is no dispute regarding the first order nature of the structural transition in BaFe$_2$As$_2$ [@A062776], the continuous appearance of the magnetic order parameter raises a scenario where the magnetic transition is of second order. If so, the lack of hysteresis in the presumed second order magnetic transition would indicate two phase transitions during a cooling/heating cycle, since obvious hysteresis in the structural transition has been observed, suggesting a similar situation to the double transition case of LaFeAsO [@A043569]. However, a continuous appearance of the magnetic order parameter does not preclude a magnetic transition with first-order hysteresis. This has been shown previously to occur in Ca$_3$Ru$_2$O$_7$ in a wide phase space of the temperature-magnetic field plane [@bao08a], where there exists a lattice contraction associating with a Mott transition. It has also been demonstrated for CaFe$_2$As$_2$ [@A071525], verifying that there is indeed only one simultaneous magnetic and structural transition in the 122 materials. In , the squared magnetic order parameter is shown during the cooling and warming cycle for the BaFe$_2$As$_2$ single crystal sample. The small difference between measurements using the two ramping rates indicates the rate is slow enough. A supercooling of about 20 K was observed, which is twice that observed previously for the polycrystalline sample [@A062776]. This larger hysteresis in the single crystal sample is expected due to larger structural strain to be dissipated in the single crystal at the first order transition.
![Temperature dependence of the magnetic (101) Bragg peak as a measure of the squared magnetic order parameter. The blue symbols were measured during cooling and the red heating. The temperature ramping rate was 2.3 K/min for , and 4.5 K/min for . []{data-label="fig2"}](fig2.ps "fig:") -.2 cm
Discussions
===========
Using a single crystal sample of BaFe$_2$As$_2$ grown by a self-flux method, the original magnetic structure determined using a polycrystal sample [@A062776] is confirmed. The single crystal sample used by Su has very different physical properties, likely due to inclusion of the Sn flux into the crystal. It is not clear whether the Sn inclusion is also responsible for the very different magnetic structure reported by them. Their sample also showed decoupled magnetic and structural transitions, and the orthorhombic distortion survived at high temperature in the heating cycle. This is different from our poly and now also single crystal results. Notice that except for the results reported by Su , magnetic and structural properties of the Ba, Sr and Ca 122 materials are very similar.
The prediction of the ($\pi,0$) in-plane magnetic wavevector from the nesting of quasi-two-dimensional Fermi surface before experiments [@A033236; @A033325; @A033286] certainly has boosted the credential of the spin-density-wave (SDW) mechanism for antiferromagnetism discovered in both the 1111 and 122 materials. The same SDW prediction for Fe$_{1+x}$Te [@A074312], however, differs from our observed magnetic structure characterized by a completely different magnetic propagation vector ($\delta\pi,\delta\pi,\pi$) with $\delta$ tunable from 0.346 to 0.5, namely from incommensurate to commensurate magnetic structure, by excess Fe composition [@A092058]. In addition, the customary energy gap from spin-density-wave order is absent in angle resolved photoemission spectroscopy (ARPES) measurements [@A062627]. A possible cause due to a topological constraint of degenerate orbitals has been advanced theoretically [@A053535]. On the other hand, there has been another theoretical approach from the strong correlation side which explains magnetic order in the Fe based materials from a localized magnetic moment picture [@A042252; @A042480]. Both localized and itinerant theories now exist for Fe$_{1+x}$Te [@A094732; @A103274; @A111294], and incommensurate magnetic order is also possible from either localized or itinerant perspective [@A042252; @A104469]. Nevertheless, the strength of electronic correlations may lie in between the weak and strong correlation limits in the ferrous high $T_C$ superconductors, since electronic band structure measured by ARPES shows certain departures from the LDA band structure [@A062627; @A070398; @A070419; @A072009].
Summary
=======
We have performed a single crystal neutron diffraction study on BaFe$_2$As$_2$. The results are completely consistent with those from our original work using a polycrystalline sample [@A062776]. Despite the appearance of a continuous magnetic order parameter, there is only one combined magnetostructural first-order transition in BaFe$_2$As$_2$. The conflicting results reported by Su using a Sn-flux grown single crystal sample are most likely erroneous to represent intrinsic properties of BaFe$_2$As$_2$.
Work at LANL is supported by U.S. DOE-OS-BES, at USTC by the Natural Science Foundation of China, Ministry of Science and Technology of China (973 Project No: 2006CB601001) and by National Basic Research Program of China (2006CB922005), at UVA by the U.S. DOE through DE- FG02-07ER45384. The SPINS at NIST are partially supported by NSF under Agreement No. DMR-0454672.
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---
abstract: 'We present a weak lensing analysis of the Coma Cluster using the Sloan Digital Sky Survey (SDSS) Data Release Five. Complete imaging of a $\sim200$ square degree region is used to measure the tangential shear of this cluster. The shear is fit to an NFW model and we find a virial radius of $r_{200}=1.99_{-0.22}^{+0.21}\mathrm{h^{-1}Mpc}$ which corresponds to a virial mass of $M_{200}=1.88_{-0.56}^{+0.65}\times10^{15}\mathrm{h^{-1}M_{\odot}}$. We additionally compare our weak lensing measurement to the virial mass derived using dynamical techniques, and find they are in agreement. This is the lowest redshift, largest angle weak lensing measurement of an individual cluster to date.'
author:
- 'Jeffrey M. Kubo, Albert Stebbins, James Annis, Ian P. Dell’Antonio, Huan Lin, Hossein Khiabanian, Joshua A. Frieman'
title: The Mass Of The Coma Cluster From Weak Lensing In The Sloan Digital Sky Survey
---
Introduction
============
Measurements of the mass of the Coma cluster date back to @zwicky33 who first applied the virial theorem to Coma and showed that dark matter dominates the cluster on megaparsec scales. Since then a number of methods have been used to determine the mass of Coma under various assumptions. @kent82 used the galaxy surface density profile to measure the mass under the assumption that the mass traces the galaxy distribution. @the86 later used this dataset to measure the mass with a modified version of the virial theorem. @hughes89 used X-ray data under the assumption that the cluster is in hydrostatic equilibrium. More recently @geller99 used the cluster infall region of Coma to extrapolate the mass out to $10h^{-1}\mathrm{Mpc}$ as well as to determine the virial mass.
Weak lensing has become a powerful tool with which to measure the mass of galaxy clusters [@tyson90]. Here the shape distortion of background galaxies due to a foreground cluster is used to determine the mass of the cluster independent of dynamical assumptions. The shear induced on the background galaxies can be used to probe the mass of the foreground cluster out to large radii. The majority of galaxy clusters studied with weak lensing lie at redshift $z\geq0.2$ [@dahle07]. However lower redshift clusters have been probed using this method, for instance @joffre00 studied Abell 3667 at $z=0.055$. At very low redshifts weak lensing measurements of clusters become difficult since this requires imaging of a large area surrounding the cluster.
The possibility of a weak lensing study of the Coma Cluster $(z=0.0236)$ with the SDSS was first suggested by @gould94 and @stebbins96. They pointed out that since Coma is at such a low redshift the weak lensing signal would be measurable in the SDSS since virtually all background galaxies would be at a much higher redshift than Coma. This is in spite of the shallow imaging in the SDSS which yields a typical background galaxy surface density of $\sim 1$ galaxy $\mathrm{arcmin}^{-2}$ [@sheldon04]. Complete imaging of the entire region surrounding the Coma Cluster was recently completed in the SDSS Data Release Five [@adelman07].
Here we present a weak lensing measurement of the virial mass of the Coma Cluster in the SDSS. Our paper is organized as follows: in $\S \ref{sec:analysis}$ we discuss our weak lensing analysis, in $\S \ref{sec:mass}$ our determination of the mass of Coma is discussed, and in $\S \ref{sec:compare}$ we compare our weak lensing mass estimate to previous measurements of the virial mass of Coma.
Since Coma is at such a low redshift our results do not change significantly for different density parameters. However for completeness we have assumed a standard cosmology with $\Omega_{m}=0.3$ and $\Omega_{\Lambda}=0.7$.
Weak Lensing Analysis {#sec:analysis}
=====================
Data {#sec:data}
----
Data used in our study are obtained from the Sloan Digital Sky Survey (SDSS) [@york00] a large imaging and spectroscopic survey of an 8000 square degree region in the Northern Galactic Cap centered on $\alpha=12^{\mathrm{h}}22^{\mathrm{m}}$, $\delta=32^{\circ}13\arcmin$ (J2000). The SDSS uses a dedicated 2.5m telescope at Apache Point Observatory which images the sky in the $ugriz$ bands [@fukugita96] in a drift scan mode [@stoughton02]. The astrometric calibration of the SDSS is described in @pier03 and the photometric calibration is described in @tucker06 and @hogg01. For our analysis we use the SDSS Data Release Five [@adelman07] which contains complete imaging of the region surrounding Coma.
Object detection and shape measurement are performed using the PHOTO pipeline [@lupton01]. PHOTO measures a large number of parameters for each detected object in the SDSS [@stoughton02], including adaptive moments [@bernstein02] which we use to measure galaxy shapes. In the SDSS adaptive moments are measured using an iterative algorithm which adapts a Gaussian weight function to the size and shape of each galaxy [@sheldon04]. PHOTO convolves each image with the local PSF before detecting objects in order to avoid the selection bias described in @bernstein02. Drift scanning in the SDSS creates a time dependent PSF which leads to a spatial variation in image quality. To model the PSF in the SDSS a Karhunen-Loeve (KL) decomposition is used, described in further detail in @lupton01 and @sheldon04. Camera shear due to drift scanning in the SDSS has been previously shown to be small [@hirata04].
For our weak lensing analysis we use a $\sim200$ square degree region centered on the core of the Coma Cluster. The core of Coma contains two bright cluster galaxies (BCGs) : NGC 4874 & NGC 4889. We take as the center of Coma the galaxy NCG 4874 located at $\alpha=13^{\mathrm{h}}02^{\mathrm{m}}00.2^{\mathrm{s}}$, $\delta=27^{\circ}41^{'}26.6^{''}$ (J2000). This BCG is associated with the majority of the X-ray flux in the core [@vikhlinin01] and the large number of galaxies surrounding this BCG in the optical imaging is indicative of this being the center of the cluster potential (see e.g., @hansen05). We in fact find that the shear amplitude is maximized when we use this BCG as the cluster center.
Source Galaxies {#sec:source}
---------------
To create our source galaxy catalog we use only objects which have been detected in all three $gri$ filters. We eliminate objects which contain saturated pixels, were originally blended with another object, or triggered errors in the measurement of adaptive moments by rejecting objects with the respective PHOTO error flag set. For our study we use only objects which have been classified by PHOTO as galaxies $(\mathrm{type}=3)$. We additionally use only shape measurements from the $r$ band since this filter is the most sensitive and typically contains better seeing [@adelman07]. Galaxies are selected in the $r$ band with extinction corrected model magnitudes [@stoughton02] in the range $18<r<21$ (Figure \[fig:mag\]). Model magnitudes are used here instead of Petrosian magnitudes since these provide a higher S/N [@mandelbaum05]. Galaxies with fainter magnitudes have been used in galaxy-galaxy weak lensing studies in the SDSS [@sheldon04], but here we restrict our sample to the magnitude range where errors in shape measurements are typically small [@hirata04] and galaxy sizes are larger.
In our analysis we use photometric redshifts for our source galaxies. The SDSS photometric redshift pipeline of @csabai03 is used since it completely covers the area of Coma in Data Release Five. Galaxies in our magnitude range are typically at much higher redshift than Coma, but we use photometric redshifts here to aid in the rejection of fainter cluster members. For our study we use only galaxies with photometric redshifts in the range $0.2<z_{\mathrm{phot}}<0.8$ and redshift errors $z_{\mathrm{err}}<0.4$. Galaxies with lower photometric redshifts are not used as the fractional error in redshift rapidly increases and most of the shear signal comes from galaxies at high redshifts compared to Coma.
To correct for the PSF anisotropy and dilution we use the linear PSF correction scheme of @hirata03. This algorithm uses the measured galaxy ellipticity and reconstructed PSF at the position of the object to correct the ellipticity components of each galaxy. We use galaxies with corrected ellipticities $e_{\mathrm{corr}}<1.4$ in our analysis, which is typical for weak lensing studies in the SDSS [@hirata04]. We additionally make a cut on the resolution parameter $(R)$, defined as $$R=1-\frac{M^{\mathrm{PSF}}_{\mathrm{rrcc}}}{M_{\mathrm{rrcc}}}$$ [@bernstein02] where $M_{\mathrm{rrcc}}$ is the size of the galaxy and $M^{\mathrm{PSF}}_{\mathrm{rrcc}}$ is the size of the PSF at the position of the galaxy as calculated by PHOTO. For our analysis we use only galaxies with $R>0.33$, which is equivalent to only using objects $1.5$ times larger than the PSF. We additionally rotate the PSF corrected ellipticity components of each galaxy from image coordinates to the equatorial coordinate system.
Within the annulus described in $\S \ref{sec:shear}$, the total number of galaxies in our source galaxy catalog after applying these cuts is $\sim270,000$.
Shear Measurement {#sec:shear}
-----------------
The shear due to Coma is measured by projecting the PSF corrected ellipticity components of each source galaxy to the tangential frame and binning the galaxies into radial annuli between $0.05\mathrm{h^{-1}Mpc}$ and $10.5\mathrm{h^{-1}Mpc}$. The edge of the inner annulus is chosen to be larger than the Einstein radius of Coma $(\sim30\arcsec)$ to ensure we are outside of the strong lensing regime, and also to avoid contamination from the BCGs. Since our radial annuli extend out to $\sim10\mathrm{h^{-1}Mpc}$ ($\simeq8^{\circ}$), we modify the usual flat sky shear matrix into a curved sky shear matrix, following @castro05. The tangential shear is measured using $$\gamma_{t}=\frac{1}{2\mathcal{R}}\frac{\sum e_{t}}{N}$$ where $\gamma_{t}$ is the tangential shear, $e_{t}$ is the tangential ellipticity, N is the total number of objects in each radial bin, and $\mathcal{R}$ is the shear responsivity described in @bernstein02. We chose not to use any ellipticity error weighting here since our galaxies are selected in a magnitude range where source galaxy ellipticity error is typically small in the SDSS [@hirata04]. We additionally also chose not to weight each source galaxy by the critical surface mass density because all of our source galaxies are at a much higher redshift than Coma and therefore the weights are very nearly equal. In the case of no weights, the shear responsivity is $\mathcal{R}=1-\sigma_{\mathrm{SN}}^{2}$, where $\sigma_{\mathrm{SN}}$ is the shape noise (the width of the intrinsic ellipticity distribution per ellipticity component) [@wittman02]. Here we use a shape noise of $\sigma_{\mathrm{SN}}=0.37$ which was used previously in @hirata04 for the same source galaxy magnitude range.
The resulting shear profile for Coma is shown in Figure \[fig:shear\]. We expect to detect the shear out to $\sim5\mathrm{h^{-1}Mpc}$, and the shear (solid squares) is clearly detected in these inner bins. Also shown in Figure \[fig:shear\] is a null test (open triangles) where each source galaxy is rotated by $45^{\circ}$. The shear signal should dissappear after this rotation if the signal is due to lensing. Errors shown for both quantities in Figure \[fig:shear\] are $1\sigma$ error bars, where $\sigma$ is the standard deviation of the mean in each radial bin.
To statistically test the significance of our measured shear signal we test whether it is consistent with the null shear model ($\gamma_{t}=0$). We find that our measured shear gives a $\chi^{2}=23.33$ for $6$ degrees of freedom for the null model. The probability of obtaining a reduced $\tilde{\chi}^{2}$ greater than this value is $\mathrm{P}(\tilde{\chi}^{2}\geq\tilde{\chi}_{o}^{2})=0.07\%$, and therefore we can reject the null hypothesis. For comparison, fitting the shear from the $45^{\circ}$ test to the null model gives a $\chi^{2}=5.65$ for $6$ degrees of freedom, which is consistent with the null hypothesis.
Blank Fields Test
-----------------
To test that our signal is not affected by any remaining systematic in the survey, we split up the SDSS North into separate, non-overlapping ‘Coma sized’ patches that contained no galaxies used in our Coma analysis and which contained no large sections of missing data. We were able to successfully extract 6 blank fields in DR5 each $\sim 20^{\circ}\times20^{\circ}$ wide. From the center of each patch we probed radially outward to $\sim10\mathrm{h^{-1}Mpc}$ at the redshift of Coma. We applied the same cuts to the source galaxy catalog in each of the blank fields that were used in our Coma analysis in $\S \ref{sec:source}$. The resulting inverse variance weighted average signal over all fields is shown in Figure \[fig:blank\], where we have used the same binning as in our Coma analysis. The tangential shear signal is shown with solid squares and the $45^{\circ}$ component is shown with open triangles. The tangential shear component is consistent with the null model giving a $\chi^{2}=1.84$ for $6$ degrees of freedom. The $45^{\circ}$ component gives a $\chi^{2}=12.58$ for $6$ degrees of freedom which has a probability $\mathrm{P}(\tilde{\chi}^{2}\geq\tilde{\chi}_{o}^{2})=5.0\%$. A model with a small positive shear is slightly favored for the $45^{\circ}$ component, however the null model can only be excluded at the $5\%$ level. The field to field scatter in the blank field shear measurements shows a typical standard deviation in a given radial bin of $\sigma_{\gamma}\sim0.0015$ or a standard deviation of the mean of $\sim 0.0006$, similar in magnitude to the statistical errors that are plotted in Figure \[fig:blank\]. In principle the scatter over the blank fields can provide information on the error due to large scale structure (@hoekstra01; also §\[sec:error\] below), but our determination of the scatter is limited by the finite number of blanks fields.
Mass Model {#sec:mass}
==========
Tangential Shear
----------------
The tangential shear due to a foreground cluster lens is given by $$\gamma_{t}=\frac{\bar{\Sigma}(\leq r)-\Sigma(r)}{\Sigma_{\mathrm{crit}}}$$ where $\bar{\Sigma}(\leq r)$ is the average projected surface mass density interior to $r$, and $\Sigma(r)$ is the projected surface mass density at $r$ [@miralda91]. The magnitude of the shear also depends on the critical surface mass density $\Sigma_{\mathrm{crit}}$ which is determined by $$\Sigma_{\mathrm{crit}}=\frac{c^2}{4\pi G}\frac{D_{s}}{D_{l}D_{ls}}.$$ Here $D_{l}$ and $D_{s}$ are the angular diameter distances from the observer to the lens and source respectively, and $D_{ls}$ is the angular diameter distance between the lens and source.
To compute the critical surface mass density we use for the lens the exact spectroscopic redshift of Coma $z=0.0236$ [@geller99]. We assume here that the peculiar velocity for Coma is zero, which has been shown in several studies [@scodeggio97; @giovanelli97]. Photometric redshifts are used to calculate the source angular diameter distances, and we obtain a critical surface mass density of $26299 \mathrm{M_{\odot} pc^{-2}}$. For comparison this is nearly the value obtained if we assume all of our sources were at a fixed redshift of $z=0.3$.
NFW Profile {#sec:nfw}
-----------
Since the S/N of our data is low $(S/N\sim 5)$ we chose to fit the measured shear profile of Coma to a model. We fit the shear to that expected from a Navarro, Frenk, & White (NFW) profile [@navarro96]. The NFW profile is a “universal profile” found in N-body simulations to fit mass density profiles ranging from galaxies to galaxy clusters. The density of an NFW profile is described by $$\rho(r)=\frac{\delta_{c}\rho_{c}}{(r/r_{s})(1+r/r_{s})^2}$$ where $\delta_{c}$ is the halo overdensity, $r_{s}$ is the scale radius, and $\rho_{c}=3H^{2}(z)/8\pi G$ is the critical density at the redshift of the cluster. The halo overdensity is given by $$\delta_{c}=\frac{200}{3}\frac{c^3}{\mathrm{ln}(1+c)-c/(1+c)}$$ where $c$ is the halo concentration. The NFW model is therefore determined by only two parameters $c$ and $r_{s}$ which are highly correlated. The NFW model can be used to determine the virial radius $r_{200}=cr_{s}$ defined as the radius in which the interior mass density falls to $200\rho_{c}$. The corresponding virial mass $M_{200}$ is given by $$M_{200}=\frac{800\pi}{3}\rho_{c}r_{200}^{3}.$$
The expected shear due to an NFW profile has been worked out in detail by @wright00 and we use their result in our analysis. Because the NFW profile is highly non-linear, we use the Levenberg-Marquardt fitting procedure in @press95 to determine the virial radius and the halo concentration. We find that the values which minimize $\chi^{2}$ are given by $$r_{200}=1.99_{-0.22}^{+0.21}\mathrm{h}^{-1}\mathrm{Mpc}$$ and $$c=3.84_{-1.84}^{+13.16}.$$ The concentration parameter is not well constrained. This fit gives a $\chi^{2}=3.87$ for 4 degrees of freedom which has a $\mathrm{P}(\tilde{\chi}^{2}\geq\tilde{\chi}_{o})=42.4\%$ probability of occuring. This corresponds to a virial mass of $$M_{200}= 1.88_{-0.56}^{+0.65}\times10^{15}\mathrm{M_{\odot}}.$$ Our values of $r_{200}$ and $M_{200}$ remain constant if the number of bins is slightly changed, and therefore our choice of binning does not bias this result. It has been shown previously that large scale structure affects the errors in the fit parameters of weak lensing measurements [@hoekstra03], however this is not considered in our fit errors. We discuss additional sources of systematic error not included in our fit in $\S \ref{sec:error}$.
Discussion {#sec:compare}
==========
We compare our weak lensing mass estimate with three other methods used to determine the virial mass of Coma: (1) The Virial Theorem (2) X-ray and (3) the Infall Region. Our value for $M_{200}$ is consistently higher than the previous measurements, though never by more that $2\sigma$. These comparisons are summarized in Table \[tab:mass\]. We also discuss sources of additional error not included in our mass measurement, as well as the source of any potential bias.
Virial Theorem
--------------
@the86 used a sample of galaxies in Coma along with the virial theorem to measure its mass. Their analysis also accounted for an additional portion of the system that may have been missing in their spectroscopic sample. Relaxing the assumption that light traces mass in the system, they found that the mass varied over a wide range. However, assuming that the mass traces the galaxy distribution they found a mass within $2.7\mathrm{h^{-1}Mpc}$ of $0.95\times10^{15}\mathrm{h^{-1}M_{\odot}}$ with a $15\%$ error in mass. Our weak lensing measurement is consistent with this result to within $\sim1.6\sigma$.
X-ray
-----
Our weak lensing measurement is also consistent with the previous X-ray analysis of @hughes89. This study used the assumption of hydrostatic equilibrium but avoided certain assumptions about the state of the gas and also accounted for many systematic effects. The best fit model was one in which mass traces light, giving a virial mass of $(0.93\pm0.12)\times 10^{15} \mathrm{h^{-1} M_{\odot}}$ within $2.5\mathrm{h^{-1}}\mathrm{Mpc}$ which is within $\sim1.7\sigma$ of our result.
Infall Region
-------------
Using a redshift survey of $\sim 1000$ galaxies @geller99 used the infall region of Coma to determine its mass. In redshift space galaxies falling into the potential well of the cluster are used to determine the amplitude of the redshift caustics (the boundaries in line of sight velocity vs. projected radius) [@diaferio97]. The amplitude of the caustics along with the assumptions of spherical symmetry and hierarchal clustering are directly related to the cluster gravitational potential. Using this technique they find an NFW model fits the mass well giving an estimate for $r_{200}=1.5\mathrm{h^{-1}}\mathrm{Mpc}$ which yields an $M_{200}=0.8\times10^{15}M_{\odot}$. Error bars are not quoted for their virial radius or mass; however, our weak lensing virial mass is in agreement with their central result to within $\sim2.0\sigma$
Additional Error and Bias {#sec:error}
-------------------------
Weak lensing measurements of the mass of clusters, in particular low redshift clusters, suffer from two sources of additional error: (1) distant large scale structure and (2) correlated structure near the cluster [@hoekstra03]. Weak lensing is sensitive to all structure along the sight and it has been shown that any background structure introduces additional error in the weak lensing mass estimate of clusters [@hoekstra01]. Specifically for the case of the Coma Cluster, @hoekstra01 showed that the error due to large scale structure increases with imaging depth, thus limiting the total achievable S/N for Coma to $\sim7$. In our study the imaging depth in SDSS is relatively shallow so the contribution of large scale structure to the total error budget is small. For future deep imaging surveys which will also cover the area of Coma such as PanSTARRS [@kaiser02], the error due to large scale structure will be more significant.
The effect of correlated structure (such as filaments) near a foreground cluster has been previously studied using N-body simulations by @cen97 and @metzler99. Here it was shown that filaments can increase the statistical error in weak lensing mass estimates as well as cause the measured $M_{200}$ to be biased toward higher values. In principle Coma should be an interesting testing ground to study this effect since the neighboring structure is well understood. However in practice there is not a clear method to correct a weak lensing measured value of $M_{200}$ for this effect, and therefore we leave a study of any potential bias in our measurement for a future paper.
Conclusion {#sec:end}
==========
We have measured the weak lensing shear due to the Coma Cluster, currently the lowest redshift $(z=0.0236)$ largest angle weak lensing measurement of an individual cluster to date. Our analysis is performed using the SDSS which is the only imaging survey that covers a large enough area to measure the shear due to Coma. We find the shear can be fit well to an NFW profile and have compared our weak lensing derived mass estimate to dynamical methods which have been used to probe the mass of Coma out to large radius. In particular we have compared our weak lensing mass to the mass derived using the virial theorem [@the86], X-ray data [@hughes89], and the cluster infall region [@geller99]. We find the virial mass of Coma from weak lensing is consistent with the mass derived using these other techniques, though the error from weak lensing is larger.
We thank the Fermilab Clusters Group for useful comments during the course of this work. Funding for the SDSS and SDSS-II has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, the U.S. Department of Energy, the National Aeronautics and Space Administration, the Japanese Monbukagakusho, the Max Planck Society, and the Higher Education Funding Council for England. The SDSS Web Site is http://www.sdss.org/.
The SDSS is managed by the Astrophysical Research Consortium for the Participating Institutions. The Participating Institutions are the American Museum of Natural History, Astrophysical Institute Potsdam, University of Basel, University of Cambridge, Case Western Reserve University, University of Chicago, Drexel University, Fermilab, the Institute for Advanced Study, the Japan Participation Group, Johns Hopkins University, the Joint Institute for Nuclear Astrophysics, the Kavli Institute for Particle Astrophysics and Cosmology, the Korean Scientist Group, the Chinese Academy of Sciences (LAMOST), Los Alamos National Laboratory, the Max-Planck-Institute for Astronomy (MPIA), the Max-Planck-Institute for Astrophysics (MPA), New Mexico State University, Ohio State University, University of Pittsburgh, University of Portsmouth, Princeton University, the United States Naval Observatory, and the University of Washington.
[99]{} Adelman-McCarthy, J., et al. 2007, arXiv:0707.3380 Bernstein, G.M. & Jarvis, M.J. 2002, , 123, 583 Castro, P.G., Heavens, A.F., Kitching, T.D. 2005, Phys. Rev. D, 72, 023516 Cen, R. 1997, , 485, 39 Dahle, H. 2007, preprint (astro-ph/0701598) Diaferio, A., & Geller, M.J. 1997, , 481, 633 Csabai, I., et al. 2003, , 125, 580 Fukugita, M., Ichikawa, T., Gunn, J.E., Doi, M., Shimasaku, K., & Schneider, D.P. 1996, , 111, 1748 Geller, M. J., Diaferio, A., Kurtz, M.J. 1999, , 517L, 23 Giovanelli, R., Haynes, M.P., Herter, T., da Costa, L.N., Wolfram, F., Salzer, J.J., Wegner, G., 1997, , 113, 53 Gould, A. & Villumsen, J. 1994, , 428L, 45 Hansen, S.M., McKay, T.A., Wechsler, R.H., Annis, J., Sheldon, E.S., Kimball, A. 2005, , 633, 122 Hirata, C.M. 2004, , 353, 529 Hirata, C.M. & Seljak, U. 2003, , 343, 459 Hoekstra, H. 2001, A&A, 370, 743 Hoekstra, H. 2003, , 339, 1155 Hogg, D.W., Finkbeiner, D.P., Schlegel, D.J., & Gunn, J.E. 2001, , 122, 2129 Hughes, J.P, 1989, , 337, 21 Joffre, M. et al. 2000, , 534L, 131 Kaiser, N. et al. 2002, SPIE, 4836, 154 Kent, S.M. & Gunn, J.E. 1982, , 87, 945 Lupton, R., Gunn, J., Ivezic, Z., Knapp, G.R. 2001, preprint (astro-ph/0101420) Mandelbaum, R., et al. 2005, , 361, 1287 Metzler, C.A., White, M., Norman, M., Loken, C. 1999, , 520L, 9 Miralda-Escude, J. 1991, , 370, 1 Navarro, J.F., Frenk, C.S., White, S.D. 1996, , 462, 563 Pier, J.R., et al. 2003, , 125, 1559 Press, W.H., Flannery, B.P., Teukolsky, S.A., Vetterling, W.T. 1995, Numerical Recipes in C (2nd ed.; Cambridge:Cambridge Univ. Press) Scodeggio, M., Giovanelli, R., & Haynes, M.P., 1997, , 113, 101 Sheldon, E.S., 2004, , 127, 2544 Stebbins, A., Mckay, T., Frieman, J.A., 1996, in IAU Symp. 173, Astrophysical Applications of Gravitational Lensing, ed.s C.S. Kochanek and J.N. Hewitt (Kluwer Academic, New York), 75 Stoughton, C., et al. 2002, , 123, 485 The, L.S., & White, S.D. 1986, , 92, 1248 Tucker, D.L., et al., 2006, AN, 327, 821 Tyson, J.A., Wenk, R.A., & Valdes, F. 1990, , 349L, 1 Vikhlinin, A., Markevitch, M., Forman, W., Jones, C. 2001, , 555L, 87 Wittman,D. 2002, LNP, 608, 55 Wright, C.O. & Brainerd, T.G. 2000, , 534, 34 York, D.G., et al. 2000, , 120, 1579 Zwicky, F. 1933, Helvetica Phys. Acta, 6, 110
[ccc]{} $1.99_{-0.22}^{+0.21}$ & $1.88_{-0.56}^{+0.65}$ & this work\
$1.5$ & $0.8$ & 1\
$2.5$ & $0.93\pm0.12$ &2\
$2.7$ & $0.95\pm0.15$ &3\
\[tab:mass\]
|
---
abstract: 'Let $G=SL_2({\mathbb{R}})^d$ and $\Gamma=\Gamma_0^d$ with $\Gamma_0$ a lattice in $SL_2({\mathbb{R}})$. Let $S$ be any “curved” submanifold of small codimension of a maximal horospherical subgroup of $G$ relative to an ${\mathbb{R}}$-diagonalizable element $a$ in the diagonal of $G$. Then for $S$ compact our result can be described by saying that $a^n \text{vol}_S$ converges in an effective way to the volume measure of $G/\Gamma$ when $n\to \infty$, with $\text{vol}_S$ the volume measure on $S$.'
address: |
Departamento de Matemáticas\
Universidad Autónoma de Madrid\
Madrid 28049\
Spain
author:
- Adrián Ubis
bibliography:
- 'submanifolds.bib'
title: Effective equidistribution of translates of large submanifolds in semisimple homogeneous spaces
---
[^1]
Introduction and results
========================
Let $G$ be a connected Lie group without compact factors and $\Gamma$ a lattice in $G$. For any $a$ Ad-semisimple element of $G$, we can consider (see [@kleinbock_shah_starkov]) the *expanding horospherical subgroup* relative to $a$ $$U^+ = \{ g\in G: \lim_{n\to \infty} a^{-n} g a^{n} = e \}.$$ An element $u\in G$ is in $U^+$ whenever $d_{G}(a^{-n}u,a^{-n})\to 0$ as $n\to\infty$, where $d_G$ is a fixed right $G$-invariant distance on $G$. This says that the action $u\mapsto a^{-1}u$ contracts regions of $U^+$, so the opposite action $u\mapsto au$ expands regions of $U^+$.
Then, one would expect $a^nVx_0$ to be quite large inside $G/\Gamma$ for any $x_0\in G/\Gamma$ and $V$ open set in $U^+$. In fact, if $U^+$ is maximal the following much stronger equidistribution result is known [@veech; @shah_horospheres] (see [@kleinbock_shah_starkov Theorem 3.7.8]): for any probability measure $\lambda$ on $U^+$ which is absolutely continuous with respect to a Haar measure on $U^+$ we have $a^n\lambda^*\to \mu_G$ where $\lambda^*$ is the image of $\lambda$ onto $G/\Gamma$ under the map $g\mapsto gx_0$ for a fixed $x_0\in G/\Gamma$, and $\mu_G$ is the probability Haar measure on $G/\Gamma$, namely $$\label{horosphere_equidistribution}
\lim_{n\to \infty} \int_{U^+} f(a^n u x_0) \, d\lambda(u)=\int_{G/\Gamma} f(r) \, d\mu_G(r)$$ for any $f\in C_c(G/\Gamma)$. In particular this implies $\overline{\cup_{n} a^n V x_0 } =G/\Gamma$. Actually the result in [@shah_horospheres] is much more general than (\[horosphere\_equidistribution\]).
In [@gorodnik; @shah_icm] N. Shah raised the question of trying to generalize this result to some singular measures $\lambda$ on $U^+$, in particular to find conditions on a submanifold $S$ of $U^+$ which make the probability volume measure $\lambda=\lambda_S$ supported on $S$ satisfy (\[horosphere\_equidistribution\]).
In [@shah_curves] this question was solved for the case $G=SO(d,1)$, where it was shown that (\[horosphere\_equidistribution\]) holds whenever $\gamma:[0,1]\to U^+$ is a real-analytic curve with $\gamma([0,1])x_0$ not contained in any proper subsphere of $G/\Gamma$. This was extended to $C^m$ curves in [@shah_curves_cm].
In this work we intend to study the case $G=SL_2({\mathbb{R}})^d$, $\Gamma=\Gamma_0^d$ with $\Gamma_0$ a lattice in $SL_2({\mathbb{R}})$, $a$ an element in the diagonal of $G$ and $S$ a submanifold of $U^+$ of small codimension not contained in an affine subspace of $U^+$. Our methods will be Fourier-analytic and will give effective rates of decay; on the other hand, in order to prove (\[horosphere\_equidistribution\]) we will need to impose a curvature condition on $S$. I do not know whether the ideas from [@shah_curves] could be applied to this case.
In $G=SL_2({\mathbb{R}})^d$ every semisimple element $a$ in the diagonal of $G$ that generates a maximal horospherical subgroup is conjugate to $$\label{translation_definition}
a_y=(a(y),a(y),\ldots, a(y)) \qquad a(y)=
\begin{pmatrix}
\sqrt y & 0 \\
0 & 1/\sqrt y\\
\end{pmatrix}$$ with $0<y$, so it is enough to study the horospherical subgroup corresponding to that element with $0<y<1$, which is $$\label{horosphere_definition}
U^+=\{u_t: t\in {\mathbb{R}}^d\} \quad u_t=(u(t_1),\ldots, u(t_d)) \quad u(t)=
\begin{pmatrix}
1 & 0 \\
t & 1 \\
\end{pmatrix}.$$ We then have that $U^+$ and ${\mathbb{R}}^d$ are isomorphic Lie groups, so we can think of $S$ as a submanifold of ${\mathbb{R}}^d$. Now we are going to impose on $S$ the following curvature condition, which is an strenghtening of the fact that $S$ is not contained in any proper affine subspace of ${\mathbb{R}}^d$.
Let $S$ be a submanifold of ${\mathbb{R}}^d$ of codimension $n\le d/2$. For any $p\in S$, we shall say that $S$ is *totally curved at* $p$ if the second fundamental form (see [@kobayashi_nomizu]) $$\text{II}_p: T_p S\times T_p S \to (T_p S)^{\perp}$$ satisfies $\text{II}_p(V\times T_p S)=(T_p S)^{\perp}$ for every $V$ subspace of $T_p S$ of dimension $n$. We shall say that $S$ is *totally curved* if the set of points at which $S$ is curved is dense in $S$.
Intuitively this condition is saying that the manifold is curved in every direction of ${\mathbb{R}}^d$. In the case of $S$ being an hypersurface (namely $n=1$), $S$ is totally curved at $p$ precisely when it does not have zero curvature at that point. In general we shall show that for $S$ not to be totally curved its coordinates must satisfy a certain differential equation, so a generic submanifold of ${\mathbb{R}}^d$ will be totally curved. That equation is just $R(p)=0$, where $R$ is the complex resultant of the polynomials $s_j$, $0\le j\le n-1$ defined in Propositon \[totally\_curved\_char\].
An example of the exceptional submanifolds that we want to avoid with our curvature condition is $S=\gamma\times S_1$, with $\gamma$ a curve in ${\mathbb{R}}^2$ and $S_1$ a submanifold of ${\mathbb{R}}^{d-2}$; in order to prove (\[horosphere\_equidistribution\]) for $\lambda=\lambda_S$ we would need to prove it for $\lambda_{\gamma}$ in the case $d=2$, and then we would lose the small codimension condition for $S$. This indicates that our condition on $S$ is a natural one for our context.
As our results are quantitative, we need to control the smoothness of $f$ to measure the decay in (\[horosphere\_equidistribution\]); for that purpose we shall use Sobolev norms
Any $X$ in the Lie algebra $\mathfrak g$ of $G$ acts on $C^{\infty}(G/\Gamma)$ by $Xf(g)=\frac{d}{dt} f (e^{tX}g)|_{t=0}$. Thus, by fixing a basis $\mathfrak B$ of $\mathfrak g$ we can define the $L^{\infty}$ *Sobolev norms* as $$\|f\|_{S^j}=\sum_{\text{deg}(\mathfrak{D}) \le j} \| \mathfrak D f\|_{L^{\infty}(G/\Gamma, \mu_G)}$$ for any $j\ge 0$, where $\mathfrak D$ runs over all the monomials in $\mathfrak B$ of degree at most $j$.
With those definitions, we can already state our main result.
\[main\] Let $\Gamma_0$ be a lattice in $SL_2({\mathbb{R}})$. There exists a constant $\gamma>0$ such that for any $S$ real-analytic totally curved submanifold of $U^+$ of codimension $n<\gamma d$, with $U^+$ as in (\[horosphere\_definition\]), and any $\lambda_S$ probability measure with $C_c^{\infty}$ Radon-Nikodym derivative w.r.t. the volume measure on $S$ we have $$|\int_{U^+} f(a_yux_0) \, d\lambda_S(u) -\int_{G/\Gamma} f \, d\mu_{G} |\ll_{\lambda_S,x_0} y^{c} \|f\|_{S^d}$$ for some $c>0$ and any $f\in C^{\infty}(G/\Gamma)$, $x_0\in G/\Gamma=SL_2({\mathbb{R}})^d/\Gamma_0^d$ and $a_y$ as in (\[translation\_definition\]) , where $c$ just depends on $d$ and $S$.
In the Theorem, the $S^d$ norm can be changed by the $S^1$ norm at the expense of diminishing the constant $c$. As pointed out in [@shah_curves], this result can be seen as a hyperbolic equivalent of the equidistribution in ${\mathbb{R}}^d/{\mathbb Z}^d$ of dilations of a real-analytic submanifold of ${\mathbb{R}}^d$ not contained in any proper affine subspace.
This result implies for $\lambda_S$ with a decay rate of $y^{cn}$. The idea of the proof is to see it as an equidistribution problem in ${\mathbb{R}}^d$, as in [@jones], and to apply Fourier analysis there. Due to the conditions on $S$, it will be possible to show that $\widehat{\lambda}_S$ almost always behaves in a way similar to the Fourier transform of the sphere, and then one can take advantage of it by applying the Weyl-Van der Corput Method together with the exponential mixing of one-parameter homogeneous flows of $G$ on $G/\Gamma$.
Actually, the approach described in the previous paragraph works just for functions $f$ for which there is a quick decay for the mixing. But for $G=SL_2({\mathbb{R}})^d$ this will always be the case except for functions coming from $SL_2({\mathbb{R}})^k$ with $k$ small, and for those the theorem can be directly proven.
In principle, our approach could be applied to any $G/\Gamma$ for which one has enough decay for the mixing, or if one can handle in another way the exceptional functions for which such a decay fails to exist.
Our Fourier-analytic arguments are local, in the sense that we use Stationary Phase to evaluate the Fourier Transform of the submanifold at each point. Perhaps it would be possible to extend our result to submanifolds with weaker curvature conditions by using global Fourier-analytic methods. On the other hand, curvature could be substituted by another kind of condition; for instance, in the case of $S$ being an affine subspace of large dimension, as suggested in [@venkatesh_sparse_arxiv Remark 3.1], Fourier Analysis could be directly used to prove that our equidistribution result is true provided its *primitive dimension* is also large (see Definition \[Primitive manifold\]). Our methods cannot by themselves work for submanifolds of small dimension; the only advance on that difficult area seems to be [@venkatesh_margulis_einsiedler].
Throughout the paper we will use the notation $e(t)=e^{2\pi i t}$ for $t\in{\mathbb{R}}$, $\widehat f(\xi)=\int_{{\mathbb{R}}^d} f(x) e(-\xi x) \, dx$ the Fourier Transform of $f$ in ${\mathbb{R}}^d$ with $\xi x$ the scalar product, and $f\ll g$ meaning $|f|\le C|g|$ for some positive constant $C>0$.
Mixing and consequences
=======================
We are going to deduce the exponential mixing for homogeneous one-parameter flows of $G$ on $G/\Gamma$ from the case $SL_2({\mathbb{R}})$ on $SL_2({\mathbb{R}})/\Gamma_0$. The action $h\Gamma\mapsto gh\Gamma$ of an element $g$ of $G$ on $G/\Gamma$ induces an action of $G$ on $C^{\infty}(G/\Gamma)$ described by $g\cdot f(h\Gamma)= f(gh\Gamma)$. On the other hand, we can consider the inner product in $L^2(G/\Gamma)$ with the Haar measure. With those definitions, the following is a direct consequence of (9.6) in [@venkatesh_sparse] (for a more general result see [@kleinbock_shah_starkov])
\[mixing\] For any $g\in G$ and $f_1,f_2\in C^{\infty}(SL_2({\mathbb{R}})/\Gamma_0)$, with $\int f_2 \, d\mu_{SL_2({\mathbb{R}})}=0$ we have $$\langle g\cdot f_1, f_2 \rangle \ll \|g\|^{-r} \|f_1\|_{S^1} \|f_2\|_{S^1}$$ where $\|g\|$ is the matrix norm of $g$ and $r$ is any number in $(0,1/2]$ for which all nonzero eigenvalues of the hyperbolic Laplacian on $\mathcal H/\Gamma_0$ are bounded from below by $r-r^2$.
Now, we are going to use the previous result in order to prove results similar to Lemmas 3.1 and 9.4 in [@venkatesh_sparse]. Their proofs will also be like the ones there.
\[equidistribution\_horocycles\] For any $x_0\in SL_2({\mathbb{R}})/\Gamma_0$, $f\in C^{\infty}(SL_2({\mathbb{R}})/\Gamma_0)$ with $\int f \, d\mu_{SL_2({\mathbb{R}})}=0$ and $\psi\in C_c^{\infty}({\mathbb{R}})$ we have $$\int_{{\mathbb{R}}} f(a(y)u(t) x_0) \psi(t) \, dt \ll y^{r/3}\|\psi\|_{S^1} \|f\|_{S^1},$$ where the implicit constant depends on $x_0$ and the length of the smallest interval containing the support of $\psi$.
Here the Sobolev norms for $\psi$ and $f$ are the ones in ${\mathbb{R}}$ and $SL_2({\mathbb{R}})/\Gamma_0$ respectively.
The idea of the proof is that $a(y)$ expands the $u(t)$ direction and contracts the rest, so a ball flowed by $a(y)$ transforms essentially into a segment in $U^+$.
Let $\rho\in C_c^{\infty}({\mathbb{R}})$ with $\int_{{\mathbb{R}}} \rho=1$. For $\delta\in (0,1)$ let us consider the measure $$\nu_{\delta}(f)=\int_{{\mathbb{R}}^{3}} f(u(t_1)^t a(e^{t_2}) u(t_3) x_0) \rho(t_1)\rho_{\delta}(t_2)\psi(t_3) \, dt_1 dt_2 dt_3$$ with $\rho_{\delta}(t)=\delta^{-1}\rho(t/\delta)$. We can write $$\nu_{\delta} (a(y)\cdot f)= \int_{{\mathbb{R}}^{3}} f(a(y)u(t_1)^t a(e^{t_2}) u(t_3) x_0) \rho(t_1)\rho_{\delta}(t_2)\psi(t_3) \, dt_1 dt_2 dt_3$$ and since $a(y)u(t_1)^t=u(y t_1 )^t a(y)$ we get that $$\nu_{\delta} (a(y)\cdot f)= \int_{{\mathbb{R}}^{3}} f(u(y t_1)^t a(e^{t_2}) a(y) u(t_3) x_0) \rho(t_1)\rho_{\delta}(t_2)\psi(t_3) \, dt_1 dt_2 dt_3.$$ Now, by the mean value theorem (see [@venkatesh_sparse], before Lemma 2.2) $|f(x_2)-f(x_1)|\ll \|f\|_{S^1} d(x_2,x_1)$, with $d$ a fixed right $SL_2({\mathbb{R}})$-invariant distance on $SL_2({\mathbb{R}})/\Gamma_0$, and then due to the identity $\int_{{\mathbb{R}}}\rho=1$ we have $$\nu_{\delta}(a(y)\cdot f)=I+O(\delta+y)\|f\|_{S^1}\|\psi\|_{S^1}$$ with $I$ the integral in the statement of the lemma. On the other hand, any $g\in SL_2({\mathbb{R}})$ outside a set of measure zero can be uniquely written as $g=u(t_1)^t a(e^{t_2}) u(t_3)$, and then we can write $\nu_{\delta}(f)=\int_{SL_2({\mathbb{R}})} f(g) H_{\delta}(g)\, dg$ for some function $H_{\delta}\in C_c^{\infty}(SL_2({\mathbb{R}}))$ depending on $x_0$, and covering $SL_2({\mathbb{R}})$ by translations of a fundamental domain of $SL_2({\mathbb{R}})/\Gamma_0$ we have $$\nu_{\delta}(f)=\langle f, h_{\delta} \rangle,$$ for some $h_{\delta}\in C_c^{\infty}(SL_2({\mathbb{R}})/\Gamma_0)$ with $\|h_{\delta}\|_{S^1}\ll \delta^{-2}\|\psi\|_{S^1}$. Here the implicit constant depends both on $x_0$ and on the length of the smallest interval containing the support of $\psi$. Thus by Lemma \[mixing\] we have $$\nu_{\delta}(a(y)\cdot f)=\langle a(y)\cdot f, h_{\delta}\rangle \ll y^r \|f\|_{S^1} \delta^{-2} \|\psi\|_{S^1}$$ so by choosing $\delta=y^{r/3}$ we get the result.
\[uncorrelation\_horocycles\_characters\] For $f\in C^{\infty}(SL_2({\mathbb{R}})/\Gamma_0)$ with $\psi\in C_c^{\infty}({\mathbb{R}})$ we have $$\int_{{\mathbb{R}}} f(a(y)u(t) x_0) \psi(t) e(ct) \, dt \ll y^{\frac{r^2}{6+12r}} \|\psi\|_{S^1} \|f\|_{S^1}.$$ where the implicit constant depends on $x_0$ and the length of the smallest interval containing the support of $\psi$.
It is enough to prove the result for $\psi$ supported inside $(-1,1)$. Let $\varphi(t)=f(a(y)u(t)x_0)\psi(t)$. For any $v\in {\mathbb{R}}$ we can write the integral in the statement of the lemma as $$I=\int_{{\mathbb{R}}} \varphi(t) e(ct) dt=\int_{{\mathbb{R}}} \varphi(t+v) e(ct) e(cv) dt$$ so for $y<\delta<1$ and $\rho\in C_c^{\infty}(-1,1)$ with $\int \rho=1$ we have $$I= \int_{{\mathbb{R}}} \rho(s) \int_{{\mathbb{R}}} \varphi(t+\delta s) e(ct) e(c\delta s) dt \, ds$$ and then $$|I|\le \int_{-2}^2 |\int_{{\mathbb{R}}} \varphi(t+\delta s) \rho(s) e(c\delta s) \, ds | \, dt.$$ By Cauchy’s inequality $$|I|^2\le 4 \int_{{\mathbb{R}}} |\int_{{\mathbb{R}}} \varphi(t+\delta s) \rho(s) e(c\delta s) \, ds |^2 \, dt$$ and by expanding the square and interchanging the integrals $$\begin{aligned}
|I|^2 & \le 4 \int_{-1}^1 \int_{-1}^1 | \int_{{\mathbb{R}}} \varphi(t+\delta s_1) \overline\varphi(t+\delta s_2) \, dt | ds_1 ds_2, \\
& \le 4 \int_{-2}^2 | \int_{{\mathbb{R}}} \varphi(t+\delta s) \overline\varphi(t) \, dt | ds.\end{aligned}$$ But $$\int_{{\mathbb{R}}} \varphi(t+w) \overline\varphi(t) \, dt =\int_{{\mathbb{R}}} f(a(y) u(w) u(t) x_0) \overline f(a(y)u(t) x_0) \psi(t+w)\overline\psi(t) \,dt$$ and since $a(y)u(w)=u(w/y)a(y)$ we have $$\int_{{\mathbb{R}}} \varphi(t+w) \overline\varphi(t) \, dt = \int_{{\mathbb{R}}} f_w (a(y)u(t) x_0) \psi_w(t) \, dt$$ with $f_w(g\Gamma)=f(u(w/y)g\Gamma)\overline f(g\Gamma)$ and $\psi_w(t)=\psi(t+w)\psi(t)$. We can write $$f_w(g\Gamma)= f_w^*(g\Gamma)+\int f_w(g\Gamma) d\mu_{SL_2({\mathbb{R}})}(g) = f_w^*(g\Gamma)+ \langle u(w/y)\cdot f, f\rangle$$ with $\int f_w^* \, d\mu_{SL_2({\mathbb{R}})} =0$. Now, by applying Lemmas \[equidistribution\_horocycles\] and \[mixing\], and using that $\|f_w\|_{S^1}\ll \|u(\frac wy)\cdot f\|_{S^1}\|f\|_{S^1}\ll \frac wy\|f\|_{S^1}^2$ and $\|\psi_w\|_{S^1}\ll \|\psi\|_{S^1}^2$ [@venkatesh_sparse Lemma 2.2] we have $$\int_{{\mathbb{R}}} \varphi(t+w) \overline\varphi(t) \, dt \ll [ y^{r/3}(w/y)+ (y/w)^r] \|\psi\|_{S^1}^2 \|f\|_{S^1}^2.$$ Using this bound for $|s|>(y/\delta)^{r/(1+r)}$ and the trivial bound otherwise we have $$\int_{-2}^2 | \int_{{\mathbb{R}}} \varphi(t+\delta s) \overline\varphi(t) \, dt | ds\ll [(y/\delta)^{\frac{r}{1+r}} + y^{r/3} (\delta/y)] \|\psi\|_{S^1}^2 \|f\|_{S^1}^2$$ so by picking $\delta$ such that both summands are equal we get the bound in the statement.
In order to transfer the exponential mixing from $SL_2({\mathbb{R}})$ to $G$ in a simple way, we are going to deal just with functions in $G/\Gamma$ that can be written as products of functions in $SL_2({\mathbb{R}})/\Gamma_0$.
\[factorizable\] We say that a function $f:G/\Gamma\to {\mathbb C}$ is *factorizable* if it can be written as $$f((g_1,\ldots, g_d)\Gamma)=f_1(g_1\Gamma)\ldots f_d(g_d\Gamma)$$ with $f_j:SL_2({\mathbb{R}})/\Gamma_0\to {\mathbb C}$.
\[equidistribution\_horospheres\] Let $f\in C_c^{\infty}(G/\Gamma)$ be a factorizable function with $k$ components having vanishing integral. Then, we have $$\int_{{\mathbb{R}}^d} f(a_yu_tx_0) \psi(t) \, dt \ll (y^{r/3})^k \|\psi\|_{S^{4d}} \|f\|_{S^k}$$ with the implicit constant depending on $x_0$ and the volume of the smallest ball containing the support of $\psi$.
It is enough to prove it with the support of $\psi$ contained in the unit ball. By choosing a fixed $\rho_*\in C_c^{\infty} ((-2,2))$ with $\rho_*=1$ in $(-1,1)$ and by setting $\rho(t)$ equal to $\rho_*(t_1)\ldots \rho_*(t_d)$ we can write $$\label{fourier_inv1}
\psi(t)=\rho(t)\psi(t)=\int_{{\mathbb{R}}^d} \widehat\psi(\xi) \rho_{\xi}(t) \, d\xi \qquad \rho_{\xi}(t)=\rho(t) e(\xi t)$$ By applying Lemma \[equidistribution\_horocycles\] on each factor with vanishing integral we have $$\int_{{\mathbb{R}}^d} f(a_yu_tx_0) \rho_{\xi}(t) \, dt \ll (y^{r/3})^k (1+|\xi|)^k \|f\|_{S^k}$$ so the lemma follows from the bound $\widehat \psi(\xi)\ll_j \|\psi\|_{S^j} (1+|\xi|)^{-j}$.
\[horospheres\_un\_characters\] Let $0<y<1$ and $f\in C_c^{\infty}(G/\Gamma)$ be a factorizable function. Then for every $c=(c_1,\ldots,c_d)\in{\mathbb{R}}^d$ with $|c|\ll y^{-r^2/(24+48r)}$ we have $$\int_{{\mathbb{R}}^d} f(a_yu_t x_0) \psi(t) e(c t)\, dt \ll \tilde c^{-2d} \|\psi\|_{S^{4d}}\max_{l\le d} [y^{\frac{r^2}{12+24r}l} \|f\|_{S^l}]$$ with $\tilde c$ the geometric mean of $(1+|c_1|, \ldots, 1+|c_d|)$, with the implicit constant depending on $x_0$ and the measure of the smallest ball containing the support of $\psi$.
We begin as in the proof of Lemma \[equidistribution\_horospheres\], by writing $\psi$ in terms of $\rho_{\xi}$, using (\[fourier\_inv1\]). Since $f$ is factorizable we have $$\int_{{\mathbb{R}}^d} f(a_y u_t x_0) \rho_{\xi}(t) e(c t)\, dt =\prod_{j\le d} \int_{{\mathbb{R}}} f_j(a(y)u(t) x_{0,j}) \rho_{\xi, j}(t_j) e(c_jt_j) \, dt_j,$$ so writing $f_j$ as a constant plus a function having vanishing integral and applying Lemma \[uncorrelation\_horocycles\_characters\] we have $$\int_{{\mathbb{R}}^d} f(a_yu_t x_0) \rho_{\xi}(t) e(c t)\, dt \ll \prod_{j\le d} ( |\widehat \rho_{\xi,j}(c_j)| \|f_j\|_{S^0} + y^{\frac{r^2}{6+12r}}(1+ |\xi|) \|f_j\|_{S^1} ).$$ Now we use the bound $|\widehat \rho_{\xi,j}(c_j)|\ll (1+|\xi|^2)/(1+ |c_j|^{2})$ and expand the product; in the resulting sum, each term with $l$ factors of the shape $y^{r^2/(6+12r)}(1+ |\xi|) \|f_j\|_{S^1}$ is bounded by $$\tilde c^{-2d} (1+|\xi|)^{2d-l} [y^{\frac{r^2}{6+12r}} (1+|c|^2) ]^l \|f\|_{S^l},$$ where we have used that the product of $d-l$ factors $(1+|c_j|)^{-2}$ is bounded by $(1+|c|)^{2l}/|\tilde c|^{2d}$. We finish by using the bound for $|c|$ and $\widehat\psi(\xi)\ll_j \|\psi\|_{S^j} (1+|\xi|)^{-j}$.
From $G/\Gamma$ to ${\mathbb{R}}^d$
===================================
Our aim in this section is to transform the problem of equidistribution of translates of $S$ in $G/\Gamma$ to a related problem in ${\mathbb{R}}^d$. For that we shall use the mixing results from the previous section; but first we need to show that for equidistribution it is enough to handle factorizable functions. This is proven in the next two lemmas.
\[reduction\_to\_bounded\_support\] Let $0<\delta<1$ and $\nu$ be a Borel probability measure on $G/\Gamma$. If $$\label{factorization_1}
| \int_{G/\Gamma} f\, d\nu -\int_{G/\Gamma} f \, d\mu_G| \gg \delta \|f\|_{S^d}$$ then the same kind of inequality is true changing $f$ by either a factorizable function or a function supported on $B_R=B_{0,R}^d$ for some $R\ll\delta^{-1}$, with $B_{0,R}$ the subset of elements $g\Gamma_0$ in $SL_2({\mathbb{R}})/\Gamma_0$ satisfying $\|g\Gamma_0\|^2\le R$, with $\|g\Gamma_0\|=\min_{\gamma_0\in\Gamma_0}\|g\gamma_0\|$, $\|\cdot\|$ the Frobenius matrix norm.
Pick $\psi\in C_c^{\infty}(SL_2({\mathbb{R}}))$ non-negative with $\int_{SL_2({\mathbb{R}})} \psi(h)\, dh=1$, $dh$ a Haar measure on $SL_2({\mathbb{R}})$, and consider the convolution $$\psi_{0,R}(g\Gamma_0)=\int_{SL_2({\mathbb{R}})} \psi(h) 1_{B_{0,R}}(h^{-1}g\Gamma_0) \, dh.$$ By the multiplicativity of $\|\cdot\|$ one can show that the supports of $\psi_{0,R}$ and $1-\psi_{0,R}$ are contained in $B_{0,c_1R}$ and $B_{0,c_2 R}^c$ respectively, with $c_1,c_2>0$ two absolute constants. On the other hand, since $$\psi_{0,R}(g\Gamma_0)=\int_{SL_2({\mathbb{R}})} \psi(gh^{-1})1_{B_{0,R}}(h\Gamma_0)\, dh$$ we deduce that $\|\psi_{0,R}\|_{S^d}\ll 1$. Then, since $\mu_{SL_2({\mathbb{R}})}(B_{0,R}^c)\ll R^{-1}$ [@iwaniec_book], we can write $
f=f\psi_R+\sum_{i} f \epsilon_i\varrho_i
$ with $i\le 2^d-1$, $\psi_{R}(g\Gamma)=\prod_{j\le d}\psi_{0,R}(g_j\Gamma_0)$, $\epsilon_i=\pm 1$ and $\varrho_i$ non-negative factorizable functions with $\|\varrho_i\|_{S^d}\ll 1$ and supported on a set of $\mu_G$ measure $O(R^{-1})$. By submultiplicativity of the $S^d$ norm and linearity we can change $f$ by either $f\psi_{R}$ or $f\varrho_i$ (for some $i$) in (\[factorization\_1\]). In the former case we are done because $f\psi_{R}$ is supported on $B_{R}$; in the latter, if $R\delta$ is larger than an absolute constant, taking into account the measure of the support of $\varrho_i$ and its non-negativity we can change $f\varrho_i$ by the factorizable function $\|f\|_{S^d} \varrho_i$ in (\[factorization\_1\]).
\[reduction\_to\_factorizable\] If we have (\[factorization\_1\]) then there exists a factorizable $f^*$ such that $$\label{factorization_2}
| \int_{G/\Gamma} f^*\, d\nu - \int_{G/\Gamma} f^* \, d\mu_G | \gg \delta^{4d+1} \|f^*\|_{S^d}.$$
By Lemma \[reduction\_to\_bounded\_support\] we can assume that the $f$ in (\[factorization\_1\]) is supported in $B_R$ for some $R\ll \delta^{-1}$. Now, for every $0<\beta<1$ consider the function $\eta_{0,\beta}\in C_c^{\infty}(SL_2({\mathbb{R}}))$ defined as $
\eta_{0,\beta}(h)=c_{\beta}\phi(d_0(h,I)/\beta)
$ with $\phi\in C_c^{\infty}(1,2)$, $d_0$ a fixed right $SL_2({\mathbb{R}})$ invariant Riemannian distance on $SL_2({\mathbb{R}})$ and $c_{\beta}$ a constant such that $\int_{SL_2({\mathbb{R}})} \eta_{0,\beta}(h) \, dh=1$. Thus $c_{\beta}\asymp \beta^{-3}$ and $\eta_{0,\beta}$ satisfies $\|\eta_{0,\beta}\|_{S^j}\ll \beta^{-3-j}$. Let us define the factorizable function $\eta_{\beta}\in C_c^{\infty}(G)$ defined as $\eta_{\beta}(g)=\prod_{j\le d} \eta_{0,\beta}(g_j)$. We have $\|\eta_{\beta}\|_{S^j}\ll \beta^{-3d-j}$ and it is supported on a ball of radius $O(\beta)$ around the identity in $G$, with the metric $d$ induced from $d_0$. For any $g\in G$, let us consider the function $\eta_{\beta}^g(h)=\eta_{\beta}(gh)$. By invariance of the Haar measure we have $\int_G \eta_{\beta}^g(h)\, dh=1$.
Consider the map $I: C_c(G)\to C_c(\Gamma\setminus G)$ defined as $I(w)(\Gamma h)=\sum_{\gamma\in \Gamma} w(\gamma h)$ and the left $G$ invariant measure $\mu^*$ on $\Gamma\setminus G$ defined by $\int_{\Gamma\setminus G} I(w) \, d\mu^* =\int_G w(h)\, dh$ (see [@raghunathan]). Applying this to $\eta_{\beta}^g$ we have $$\label{factorization_3}
\int_{\Gamma\setminus G} I(\eta_{\beta}^g)(\Gamma h) \, d\mu^*(\Gamma h) =1.$$ One can see that $I(\eta_{\beta}^g)(\Gamma h)$ is invariant under the change $g\mapsto g\gamma$, $\gamma\in \Gamma$, and then we can write $I(\eta_{\beta}^g)(\Gamma h)=\rho_{\beta,\Gamma h}(g\Gamma)$. Then, using (\[factorization\_3\]) into (\[factorization\_1\]) and Fubini we have $$\label{factorization_4}
\int_{\Gamma\setminus G}|\int_{G/\Gamma} f\rho_{\beta,\Gamma h} \, d\nu -\int_{G/\Gamma} f\rho_{\beta,\Gamma h} \, d\mu_G| \, d\mu^*(\Gamma h) \gg \delta \|f\|_{S^d}.$$ Now, if $g\Gamma$ is in the support of $\rho_{\beta,\Gamma h}$ we have $d(g\gamma h, I)\ll \beta$ for some $\gamma\in\Gamma$ which by the mean value theorem (see [@venkatesh_sparse], before Lemma 2.2) $|f(g\Gamma)-f(h^{-1}\Gamma)|\ll \|f\|_{S^1} \beta$. Hence, if $\beta/\delta$ is smaller than an absolute positive constant, from (\[factorization\_3\]), (\[factorization\_4\]) and Hölder inequality we deduce that $$|\int_{G/\Gamma} \rho_{\beta,\Gamma h} \, d\nu - \int_{G/\Gamma} \rho_{\beta,\Gamma h} \, d\mu_G |\gg \delta$$ for some $\Gamma h$ with $\|h^{-1}\Gamma\|^2\ll \delta^{-1}$, taking into account the support of $f$ and the fact that if $g\Gamma$ is in the support of $\rho_{\beta,\Gamma h}$ we have $d(g,h^{-1}\gamma^{-1})\ll 1$ for some $\gamma^{-1}\in \Gamma$. One can check that $\rho_{\beta,\Gamma h}$ is factorizable, so that (\[factorization\_2\]) will follow if we show that $\|\rho_{\beta,\Gamma h}\|_{S^j}\ll \| \eta_{\beta}\|_{S^j}\ll \beta^{-3d-j}$ with $\beta^{-1}\ll \delta^{-1}$. But this in turn follows from showing that in the sum defining $\rho_{\beta,\Gamma h}$ there is just one non-vanishing term. Suppose there were at least two non-vanishing terms. Then $d(g\gamma_1 h, I)\ll \beta$ and $d(g\gamma_2 h,I)\ll \beta$ for $\gamma_1\neq\gamma_2\in\Gamma$, $\|h\|^2\ll \delta^{-1}$. By the right invariance of the metric this implies $d(h^{-1}\gamma h,1)\ll \beta$ for $\gamma=\gamma_2^{-1}\gamma_1\neq 1$, hence $h^{-1}\gamma h=\exp(X)$ with $X\in\mathfrak g$, $\|X\|\ll \beta$. Therefore $\gamma=\exp(hXh^{-1})$ and $\|hXh^{-1}\|\ll \|h\| \|X\| \|h^{-1}\|\ll \beta/\delta $ since $\|h^{-1}\|\ll \|h\|$. Since $\Gamma$ is discrete, this gives a contradiction for $\beta/\delta$ small enough.
We are going to write our problem in $G/\Gamma$ as a problem in ${\mathbb{R}}^d$, in the spirit of [@jones], in order to use Fourier Analysis in ${\mathbb{R}}^d$ afterwards. By fixing a function $\rho\in C_c^{\infty}({\mathbb{R}}^d)$ such that $\rho(x)=1$ whenever $u_x$ is in the support of $\lambda_S$, we have $$\int_{U^+} f(a_yux_0) d\lambda_S (u)= \int_{{\mathbb{R}}^d} F(x) \, d\lambda_S(x)$$ with $F(x)=\rho(x) f(a_yu_x x_0)$. We are going to denote this functional as $$\lambda_S(F)=\int_{{\mathbb{R}}^d} F(x) \, d\lambda_S(x);$$ we will need to study its action on functions $F$ satisfying some special properties.
We shall say that $f:{\mathbb{R}}^d\to {\mathbb C}$ is a $(T,\alpha)$-*singular function* if $\|f\|_{S^0}\le 1$, $\|f\|_{S^1}\le T$, with $$| \int_{{\mathbb{R}}^d} f(x) \psi(x) \, dx |\le T^{-\alpha} \|\psi\|_{S^{4d}}$$ for any $\psi\in C^{\infty}({\mathbb{R}}^d)$ and there exist $k\le d/2$ and $\tilde f:{\mathbb{R}}^k\to {\mathbb C}$ such that $f(x_1,x_2)=\tilde f(x_1) \tilde \rho(x_2)$ for every $(x_1,x_2)\in {\mathbb{R}}^k\times {\mathbb{R}}^{d-k}$, with $\tilde\rho\in C_c^{\infty}({\mathbb{R}}^{d-k})$, $\|\tilde\rho\|_{S^1}\le 1\le \int_{{\mathbb{R}}^{d-k}}\tilde\rho$.
We shall say that $f:{\mathbb{R}}^d\to {\mathbb C}$ is a $(T,\alpha)$-*mixing function* if $\|f\|_{S^0}\le 1$, $\|f\|_{S^1}\le T$, $f$ supported in a translation of the unit ball, with $$\label{horosphere_condition}
\int_{|v|<V} |\int_{{\mathbb{R}}^d} f(x+\frac vT) \overline f(x)\psi(x) \, dx |
\, dv \le V^{d/2} \|\psi\|_{S^{4d}}$$ for every $V<T^{\alpha}$ and $$\label{character_condition}
\int_{|h|<T^{\alpha}} \sup_{|v|<T^{\alpha}}|\int_{{\mathbb{R}}^d} f(x+\frac{v}{T}) \overline f(x) \psi(x)e(hx) \, dx | \, dh \le \|\psi\|_{S^{4d}}$$ for any $\psi\in C^{\infty}({\mathbb{R}}^d)$.
Now, the main result of this section is
\[from\_group\_to\_real\] Let $y,\delta\in (0,1/2)$ and $f_*\in C^{\infty}(G/\Gamma)$ with $$|\int_{U^+} f_*(a_yux_0)\, d\lambda_S(u)-\int_{G/\Gamma} f_* \, d\mu_G|>\delta \|f_*\|_{S^d}.$$ Then we have $$|\lambda_S(f)|\gg\delta^{4d+1}$$ for some function $f$ either $(y^{-1},\alpha)$-singular or $(y^{-1},\alpha)$-mixing for some $\alpha>0$ depending just on $\Gamma_0$.
By Lemma \[reduction\_to\_factorizable\] we have $$\label{proof_1}
|\int_{U^+} f_{**}(a_yux_0)\, d\lambda_S(u)-\int_{G/\Gamma} f_{**} \, d\mu_G|>\delta^{4d+1} \|f_{**}\|_{S^d}$$ with $f_{**}$ a factorizable function. We can write $f_{**}-\int_{G/\Gamma} f_{**}\, d\mu_G$ as a sum of $O(1)$ factorizable functions with vanishing integral and each of the factors either constant or with integral zero, so from (\[proof\_1\]) we deduce that $$|\int_{U^+} \tilde f_*(a_yux_0)\, d\lambda_S(u)|\gg\delta^{4d+1} \|\tilde f_{*}\|_{S^d}$$ with $\tilde f_*$ one those $O(1)$ factorizable functions. By dividing $\tilde f_*$ by its norm we can assume that $\|\tilde f_*\|_{S^d}=1$, so defining $$f(x)=\rho(x)\tilde f_* (a_yu_xx_0)$$ with $\rho\in C_c^{\infty}({\mathbb{R}}^d)$ with $\rho=1$ whenever $u_x$ is in the support of $\lambda_S$, where $\rho(t)=\prod_{j\le d} \rho_0(t_j)$ with $\rho_0\in C_c^{\infty}({\mathbb{R}})$ and $\|\rho_0\|_{S^1}\le 1\le \int_{{\mathbb{R}}}\rho_0$, we have $|\lambda_S(f)|\gg\delta^{4d+1}$ and then it only remains to show that $f$ satisfies the properties in the statement of the proposition. The bound on $\|f\|_{S^1}$ comes just from [@venkatesh_sparse Lemma 2.2].
Let us assume first that at least $d/2$ components in the factorization of $\tilde f_*$ are constant. We consider that the constant components are the last ones. Thus we get that $f$ is an $(y^{-1},\alpha)$-singular function for any $\alpha\le r/3$, since by Lemma \[equidistribution\_horospheres\] we have $$\int_{{\mathbb{R}}^d} f(x) \psi(x) \, dx = \int_{G/\Gamma} \tilde f_*(a_yu_x x_0) \rho(x) \psi(x) \, dx \ll y^{r/3} \|\psi\|_{S^{4d}}.$$ Now, we have to consider the case in which $q\ge d/2$ components of $\tilde f_*$ have integral zero. By splitting $\rho$ into several functions we can assume that $\rho$ is supported in a translation of the unit ball. We have $$\int_{{\mathbb{R}}^d} f(x+vy) \overline f(x) \psi(x) \, dx = \int_{{\mathbb{R}}^d} f_*^v (a_yu_x x_0) \rho(x+vy)\overline \rho(x)\psi(x) \, dx$$ with $f_*^v (g\Gamma)= \tilde f_*(u_v g\Gamma) \overline{\tilde f_*}(g\Gamma)$, so by applying Lemma \[equidistribution\_horospheres\] and [@venkatesh_sparse Lemma 2.2] we have $$\int_{{\mathbb{R}}^d} f(x+vy) \overline f(x) \psi(x) \, dx \ll (y^{r/3})^{q} \|f_*^v\|_{S^{q}}\|\psi\|_{S^{4d}} \ll (y^{r/3} |v|)^{q} \|\psi\|_{S^{4d}}$$ and then condition (\[horosphere\_condition\]) follows for any $\alpha\le r/6$. Moreover $$\int_{{\mathbb{R}}^d} f(x+vy)\overline f(x) \psi(x) e(hx) \, dx=\int_{{\mathbb{R}}^d} f_*^v (a_yu_x x_0) \rho(x+vy)\overline \rho(x)\psi(x) e(hx) \, dx$$ so by Lemma \[horospheres\_un\_characters\] and [@venkatesh_sparse Lemma 2.2] we have $$\int_{{\mathbb{R}}^d} f(x+vy)\overline f(x) \psi(x) e(hx) \, dx\ll \tilde h^{-2d} \|\psi\|_{S^{4d}}$$ for $|v|+|h|^2 \ll y^{-\frac{r^2}{12+24r}}$ and then (\[character\_condition\]) follows for every $\alpha\le r^2/(24+48r)$.
We will dedicate the rest of the paper to prove the following result, which by Proposition \[from\_group\_to\_real\] implies Theorem \[main\]:
\[main\_real\] Let $f$ be either a $(T,\alpha)$-singular or a $(T,\alpha)$-mixing function. Let $S$ be a totally curved real-analytic submanifold of ${\mathbb{R}}^d$ of codimension $n<\alpha d/200$, and $\lambda_S$ a probability measure with $C_c^{\infty}$ Radon-Nikodym derivative w.r.t. the volume measure on $S$. Then we have $$|\lambda_S(f)|\ll T^{-c},$$ with $c>0$ depending just on $S$ and $\alpha$.
Geometric properties of the submanifold
=======================================
We begin by expressing that a manifold is totally curved in different ways.
\[totally\_curved\_char\] Let $S=\varphi((-1,1)^m)$ with $\varphi:(-2,2)^m\to {\mathbb{R}}^d$, $\varphi(t)=(t,w(t))$, $w$ a real-analytic function. $S$ being totally curved at a point $p$ is equivalent to any of the following conditions (with $n=d-m$):
1. For every subspace $V$ of $T_p S$ of dimension $n$ there exist $v_1,\ldots v_n$ in $V$ and tangent fields $X_1,\ldots, X_n$ such that $$\langle T_p S, D_{v_j} X_j(p): j\le n \rangle = {\mathbb{R}}^d$$ with $D_v$ the directional derivative in the direction $v$.
2. For every subspace $V$ of $T_p S$ of dimension $n$ there exist functions $v_1,\ldots v_n$ smooth at $\varphi^{-1}(p)$ with $v_j(\varphi^{-1}(h))\in T_h S$ for $h$ near $p$ and $V=\langle v_j(\varphi^{-1}(p)): j\le n\rangle$ such that $$\langle T_p S, \partial_s v_j(\varphi^{-1}(p)): s\le m, j\le n\rangle ={\mathbb{R}}^d.$$
3. For every nonzero $a\in{\mathbb{R}}^d$ orthogonal to $T_p S$ we have that the dimension of $\ker H_a$ is less than $n$, with $H_a:{\mathbb{R}}^m\to {\mathbb{R}}^m$ the Hessian of $a \varphi$ at $\varphi^{-1}(p)$.
4. For every nonzero $z\in {\mathbb{R}}^n$ we have that the dimension of $\ker H_z$ is less than $n$, with $H_z$ the Hessian of $z w$ at $\varphi^{-1}(p)$.
5. If $ \lambda^m+s_{m-1}(z) \lambda^{m-1}+\ldots+s_1(z)\lambda+s_0(z)$ is the characteristic polynomial of $H_z$, the Hessian of $zw$ at $\varphi^{-1}(p)$, then the system of homogeneous polynomial equations $s_0(z)=s_1(z)=
\ldots =s_{n-1}(z)=0$ does not have a solution $z\in{\mathbb{R}}^n\setminus \{0\}$.
The equivalence with (i) and (ii) comes just from the definition of the second fundamental form by the equation $$D_v X(p) = \nabla_v X(p) + \text{II}_p(v, X(p))$$ with $\nabla_v X\in T_p S$ the covariant derivative on $S$. Moreover (iii), (iv) and (v) are clearly equivalent. Thus we just have to prove that (ii) and (iii) are equivalent.
Let us assume that (ii) is not satisfied. Then there exists $V$ such that $$\langle T_p S, \partial_s v_j(\varphi^{-1}(p)): s\le m, j\le n \rangle \neq {\mathbb{R}}^d.$$ We can write $T_p S=\langle (\partial_s\varphi)(\varphi^{-1}(p)):s\le m\rangle$ and $v_j=\sum_{k\le m} r_j^k \partial_k\varphi$. Then we have $$\partial_s v_j= \sum_{k\le m} \partial_s r_j^k \partial_s \varphi + u_j^s$$ with $u_j^s=\sum_{k\le m} r_j^k \partial_s\partial_k \varphi$ so $$\langle T_p S, u_j^s: s\le m, j\le n \rangle \neq {\mathbb{R}}^d.$$ But then there exists $a\neq 0$ in $(T_p S)^{\perp}$ orthogonal to $$\sum_{s\le m} x_s u_j^s = \sum_{s,k\le m} x_s r_j^k \partial_s \partial_k \varphi$$ for every $x_s\in {\mathbb{R}}$ and $j\le n$. Thus $$0=a\sum_{s,k\le m} x_s r_j^k \partial_s \partial_k \varphi =\sum_{s,k\le m} x_s r_j^k \partial_s \partial_k (a\varphi)=x^t H_a r_j$$ with $x=(x_s)_{s\le m}$ and $r_j=(r_j^k)_{k\le m}$ in ${\mathbb{R}}^m$. This implies that $
H_a r_j=0
$ for every $j\le n$. That $\dim V=n$ implies that $r_j$ are independent so $\dim\ker H_a\ge n$, and then (iii) is not satisfied. We can clearly reverse our reasoning to show that if (iii) is satisfied so is (ii).
As we said in the introduction, being totally curved rules out submanifolds of the type $S_1\times S_2$. Now we are going to prove that, and we begin by formalizing the kind of submanifolds that we want to avoid.
\[Primitive manifold\] Let $S$ be a submanifold of ${\mathbb{R}}^d$ . For any $p\in S$, we define its *primitive dimension at $p$* as the maximum $k\in{\mathbb N}$ for which the restriction of $S$ to any $k$ components of ${\mathbb{R}}^d$ is a manifold of dimension $k$.
In terms of a parametrization of $S$, $\varphi=(\varphi_1,\ldots, \varphi_d)$, the primitive dimension is the maximal $k$ such that $\{\nabla \varphi_{i_j} \}_{j=1}^k$ are linearly independent for every $\{i_1,\ldots, i_k\}\subset \{1,\ldots ,d \}$. Clearly, if $S_1$ is a submanifold of ${\mathbb{R}}^k$ of dimension smaller than $k$, then the primitive dimension of $S=S_1\times S_2$ is less than $k$.
\[degeneracy\] Let $m\ge 2n \ge 2$. Let $S$ be a submanifold of ${\mathbb{R}}^d$ of dimension $m$, codimension $n$ and primitive dimension at most $m-n$ for every point in a neighborhood of $p$, with $S=\varphi((-1,1)^m)$, $\varphi:(-2,2)^m\to {\mathbb{R}}^d$ real-analytic and $\varphi(t)=(t,w(t))$. Then, perhaps after rearranging some components, for some $1\le h\le n$ we have $$\partial_j w_r=\sum_{i\le h-1} b_j^i \partial_i w_r \qquad h\le j\le h+n-1$$ for every $r\le h$, with $b_j^i$ real-analytic functions in a neighborhood of $p$.
By definition of primitive dimension there exist $m-n+1$ components whose gradients are linearly dependent. Let us assume that exactly $h\le n$ of those components correspond to the last $n$ ones. Then, after rearranging them we can assume that $$\dim \langle \partial \varphi_{n+h}, \partial\varphi_{n+h+1}, \ldots, \partial\varphi_{m+h}\rangle <m-n+1$$ for some $1\le h\le n$ in a neighborhood of $p$. Since $\varphi(t)=(t,w(t))$ we see that the matrix $(\partial \varphi_j)_{n\le j\le m}$ has the shape $$\begin{pmatrix}
0 & D \\
I & * \\
\end{pmatrix}$$ with $I$ the identity matrix of dimension $m+1-n-h$ and $$D=(\partial_j w_r)_{j\le n+h-1, r\le h}.$$ Then the rank of $D$ must be less than $h$. We can assume that the rank of $D$ equals a constant $0\le s<h$ in a neighborhood of $\varphi^{-1}(p)$. If $s=0$ then the result follows with $b_j^i=0$. Otherwise there exists some $s\times s$ non-zero minor and then, after possibly rearranging the components, the result follows from Cramer’s rule.
Now we shall prove that if $S$ is totally curved then its primitive dimension is larger than $m-n$.
\[curved\_implies\_primitive\] Let $S$ be totally curved at $p$, with $S=\varphi((-1,1)^m)$, $\varphi:(-2,2)^m\to{\mathbb{R}}^d$ real-analytic and $\varphi(t)=(t,w(t))$, where $m+n=d$. Then $S$ has primitive dimension larger than $m-n$ for every point in some neighborhood of $p$.
Suppose $S$ has primitive dimension at most $m-n$ for a sequence of points converging to $p$. We must then prove that $S$ is not totally curved in a neighborhood of $p$. Since $S$ is real-analytic, looking into the proof of Lemma \[degeneracy\] we see that its primitive dimension must be at most $m-n$ in a neighborhood of $p$. Thus by Lemma \[degeneracy\] we have $$\label{EQ1}
\partial_j w_r=\sum_{i\le h-1} b_j^i \partial_i w_r \qquad h\le j\le h+n-1 \quad r\le h.$$ We consider the tangent fields $$\label{EQ2}
v_j=\partial_j \varphi - \sum_{i\le h-1} b_j^i \partial_i \varphi \qquad h\le j\le h+n-1.$$ We are going to see that the $v_j(p)$ generate a vector subspace of $T_p S$ of dimension $n$ and that $$\label{equ3}
Q=\langle T_p S, \partial_s v_j: s\le m, h\le j\le h+n-1\rangle \neq {\mathbb{R}}^d,$$ so by Proposition \[totally\_curved\_char\] (ii) the result follows. We have $$\partial_l \varphi =(0,\ldots, 0, 1,0, \ldots ,0, \partial_l w_1, \ldots, \partial_l w_n)$$ $$\partial_s\partial_l \varphi =(0, \ldots, 0, 0, 0, \ldots, 0, \partial_s\partial_l w_1,\ldots, \partial_s\partial_l w_n)$$ for every $s,l\le m$, with the $1$ in the $l$th position. Then $$v_j=(-b_j^1,\ldots, -b_j^{h-1}, 0, \ldots ,0, 1,\ldots)$$ with 1 in the $j$th position, which shows that $v_j$, $h\le n\le h+n-1$ are linearly independent. Differentiating we get that $$\partial_s v_j= u_s^j- \sum_{i\le h-1} \partial_s b_j^i \partial_i\varphi$$ with $$u_s^j=\partial_s\partial_j\varphi - \sum_{i\le h-1} b_j^i \partial_s\partial_i \varphi$$ so $$Q=\langle T_p S, u_s^j: s\le m, h\le j\le h+n-1\rangle.$$ By differentiating (\[EQ1\]), for any $j,s$ we have $$u_s^j-\sum_{i\le h-1} (\partial_s b_j^i) g_i \in 0\times {\mathbb{R}}^{n-h}$$ with $g_i=(0,0,\ldots, 0, \partial_i w_1, \ldots, \partial_i w_h,0, \ldots 0)$. Thus $$u_s^j\in \langle 0\times{\mathbb{R}}^{n-h}, g_i: i\le h-1 \rangle$$ which is a vector space of dimension at most $n-1$, hence (\[equ3\]) follows.
Singular case
=============
In this section we want to prove Theorem \[main\_real\] for singular functions. We start by defining a local version of $\lambda_S$. Let us fix a $C_c^{\infty}({\mathbb{R}}^d)$ function $\psi$ with $\int_{{\mathbb{R}}^d} \psi=1$ and support contained in the unit ball of ${\mathbb{R}}^d$. For any $x_0\in {\mathbb{R}}^d$ and any $0<\beta<1/2$ we define the measure $$\lambda_{S,x_0,\beta}(f)=\int_{{\mathbb{R}}^d} f(x) \frac{1}{\beta^m} \psi(\frac{x-x_0}{\beta}) \, d\lambda_S(x).$$ By compacity of the support of $\lambda_S$ we have that $\lambda_{S,x_0,\beta}({\mathbb{R}}^d)$ is bounded independently of $x_0$ and $\beta$. If $\sigma_S$ is the volume measure on $S$ we have the following localization result.
\[localization\_in\_space\] Let $0<\delta,\beta<1$ with $\beta\ll\delta$. Let $f:{\mathbb{R}}^d\to {\mathbb{R}}$ with $\|f\|_{L^{\infty}}\le 1$ and $|\lambda_S(f)|>\delta$. Then the set of $x_1$ in $S$ for which there exists an $x_0\in {\mathbb{R}}^d$ at distance $O(\beta)$ with $|\lambda_{S,x_0,\beta}(f)|\gg\delta$ has $\sigma_S$-measure $\gg \delta$ (with the implicit constants depending just on $S$).
Since $\int_{{\mathbb{R}}^d}\psi =1$, by Fubini we have $$\int_{{\mathbb{R}}^d} \lambda_{S,x,\beta}(f) \, dx =\beta^n \lambda_S(f).$$ Let $G$ be the set of points $x_1$ in ${\mathbb{R}}^d$ for which there exists a point $x_0$ at distance smaller than $2\beta$ with $|\lambda_{S,x_0,\beta}(f)|>c\delta$. Then, since the support of $g(x)=\lambda_{S,x,\beta}(f)$ is contained in a set of Lebesgue measure $O(\beta^n)$, we have $$\delta< |\lambda_S (f)|\ll \beta^{-n} m(G) + \beta^{-n} c\delta \beta^n$$ so for $c$ small enough we have $
m(G)\gg \delta \beta^n.
$ Now, $G\subset G'$, with $G'$ the set of points at distance at most $4\beta$ from $G\cap S$. This is so because $\lambda_{S,x_0,\beta}(f)\neq 0$ implies that $x_0$ is at distance less than $ \beta$ from $S$. For the same reason $G$ is a union of balls of radius $ 2\beta$ and center at distance at most $\beta$ from $S$, so we can deduce that $$m(G')\ll \beta^n \sigma_S(G\cap S)$$ which implies $\sigma_S(G\cap S)\gg \delta$ and the result follows.
In what follows we will need to control the size of the set in which a real-analytic function is small. A particular case of Corollary 1 in [@garofalo_garrett] says that if $u$ is a real-analytic function in the ball $B_{1+\delta_1+\delta_2}(0)\subset{\mathbb{R}}^k$ (with $\delta_1,\delta_2>0$ fixed) then there exist $p>1$ and $A>0$ such that $$\label{garrofalo}
\int_{B_{\delta_1/4}(y)} |u| \, dx \, \, ( \int_{B_{\delta_1/4}(y)} |u|^{-1/(p-1)} \, dx)^{p-1} \le A$$ for every $y\in B_1(0)$. From there we deduce
\[sublevel\_set\_lemma\] Let $u$ be a real-analytic non-zero function on $B_{1+h}(0)\subset{\mathbb{R}}^k$ for some $h>0$. There exists $D=D(u)\ge 1$ such that for $0<\delta<1$ we have $$|\{ x\in B_1(0): |u(x)|<\delta\} | \ll \delta^{1/D}.$$
We can cover $B_1(0)$ with a finite number of balls $B_j$ of radius $R=h/8$ centered inside $B_1(0)$. By applying (\[garrofalo\]) with $\delta_1=\delta_2=h/2$ we get $$\int_{B_j} |u| \, dx \, \, ( \int_{B_j} |u|^{-1/(p-1)} \, dx)^{p-1} \le A$$ for each of them. Since $u$ is non-zero, for every $j$ it cannot be zero for every point in $B_j$, and then $\int_{B_j} |u| \, dx\neq 0$ which implies $$\int_{B_j} |u|^{-1/(p-1)} \, dx \ll 1$$ and then $$\int_{B_1(0)} |u|^{-1/(p-1)} \, dx \ll 1$$ from which the result follows with $D=p-1$.
Finally we can prove our result in the singular case.
\[main\_singular\] Let $n\le d/4$. For every $(T,\alpha)$-singular function $f$ we have $$\lambda_S(f)\ll T^{-c}$$ for some $c$ depending just on $\alpha$ and $S$.
We begin by picking $\delta=T^{-\epsilon}$ with $\epsilon>0$ a small constant. Let us suppose that $
|\lambda_S(f)|\gg \delta;
$ we shall see that we arrive at a contradiction. By applying Proposition \[localization\_in\_space\] we have $
|\lambda_{S,x_0,\beta}(f)| \gg \delta
$ for some $x_0$ at distance $O(\beta)$ from $x_1$, for every $x_1$ in a subset $G$ of $S$ of $\sigma_S$-measure $\gg \delta$. Since the support of $\lambda_S$ is compact, we can further assume that this set is contained in $\varphi((-1,1)^m)$, with $\varphi:(-1,1)^m\to S$ some parametrization of $S$, with $\varphi$ a real analytic function on $(-2,2)^m$.
Since $\sigma_S(G)\gg \delta\gg \beta$ we can even assume that for every $x_1\in G$ the support of $\psi_{x_0,\beta}(t)=\beta^{-m}\psi((\varphi(t)-x_0)/\beta)$ is contained in $(-1,1)^m$, and then we have $$\lambda_{S,x_0,\beta}(f)=\int_{{\mathbb{R}}^m} f(\varphi(t)) \psi_{x_0,\beta}(t) R(t) \, dt,$$ for $R(t)$ some $C_c^{\infty}$ function depending just on $\lambda_S$. Since $f$ is singular, for some $k\le d/2$ we can write $f(x_1,x_2)=\tilde f(x_1)\tilde \rho(x_2)$ for $(x_1,x_2)\in{\mathbb{R}}^k\times {\mathbb{R}}^{d-k}$. We have $R(t)=R(t_1)+O(\beta)$ for any $t$ in the support of $\psi_{x_0,\beta}$, with $t_1=\varphi^{-1}(x_1)$, and the same happens for $\tilde\rho(\varphi)$, so $$\tilde\rho(x_1) \int_{{\mathbb{R}}^m} \tilde f(\varphi_1(t))\psi_{x_0,\beta}(t)\, dt\gg \delta,$$ with $\varphi(t)=(\varphi_1(t),\varphi_2(t))\in{\mathbb{R}}^k\times{\mathbb{R}}^{d-k}$. On the other hand, since $d/2\le m-n$ and $S$ is totally curved by Proposition \[curved\_implies\_primitive\] we have that its primitive dimension is larger than $k$ throughout some open subset $U$ of $\varphi((-1,1)^m)$. Thus $\varphi_1(U)$ is a manifold of dimension $k$, which implies that for some variables $t_{i_1},\ldots, t_{i_k}$, $i_j\le m$, the Jacobian $J(t)$ of $\varphi_1$ with respect to them is a nonzero real-analytic function on $(-2,2)^m$. Assume they are the first $k$ variables. Let us write $t=(t',t'')$, $t'\in {\mathbb{R}}^k$, with $J=\det \partial\varphi_1/\partial t'$. By Lemma \[sublevel\_set\_lemma\] there exists $D>1$ such that $$|\{t\in(-1,1)^m: |J(t)|<\gamma^D\delta^D\}|\ll
\gamma\delta$$ for any small constant $0<\gamma<1$. Since $\sigma_S(G)\gg \delta$ this means that for some $x_1\in G$ we have $|J(t_1)|\gg \delta^D$. Then, for $\beta\ll \delta^D$ we have $J(t)=J(t_1)+O(\beta)\neq 0$, so by the change of variables $(t',t'')\mapsto (\varphi_1,t'')$ we obtain that $$\tilde\rho(x_1) \int_{{\mathbb{R}}^{m-k}} \int_{{\mathbb{R}}^k} \tilde f(\varphi_1) \psi_{\beta,x_0}(t) \,\frac{d\varphi_1}{|J(t)|} \, dt''\gg \delta$$ with $t=t(\varphi_1,t'')$ its inverse. We have $J(t)=J(t_1) (1+O(\beta/\delta^D))$ in the support of $\psi_{\beta,x_0}$, so by picking $\beta= \delta^{2D}$ we can write $$\tilde \rho(x_1) \int_{{\mathbb{R}}^k} \tilde f(\varphi_1) \tilde\psi_{\beta,x_0}(\varphi_1) \, d\varphi_1 \gg \delta^{D+1}$$ with $$\tilde\psi_{\beta,x_0}(\varphi_1)=\int_{{\mathbb{R}}^{m-k}} \psi_{\beta,x_0}(t) dt''.$$ Since $\|\tilde\rho\|_{S^1}\le 1 \le \int_{{\mathbb{R}}^{d-k}} \tilde \rho$, it follows that $$\int_{{\mathbb{R}}^d} f(x) \eta(x) \,dx \gg \delta^{D+1}$$ with $\eta(x_1,x_2)=\tilde\psi_{\beta,x_0}(x_1)$. Using implicit differentiation we can obtain the bound $\|\eta\|_{S^l}\ll \beta^{-2l}$ for every $l\ge 0$, and since $f$ is $(T,\alpha)$-primitive we have $$T^{-\alpha} \beta^{-8d}\gg \delta^{D+1}$$ which gives a contradiction for any $\epsilon<\alpha/18dD$, with $\delta=T^{-\epsilon}$, since $\beta=\delta^{2D}$ and $D>1$.
Mixing case: Low and high frequencies
=====================================
In the next sections we shall show that $\lambda_S(f)$ is small for $f$ a mixing function. Taking into account our treatment of singular functions in the previous section, this will complete our proof of Theorem \[main\].
We will handle this case by using Fourier Analysis in ${\mathbb{R}}^d$. We begin by recalling the Fourier transform of the measure $\mu=\lambda_{S,x_0,\beta}$: $$\widehat \mu (\xi)=\mu(e(-\xi\cdot))=\int_{{\mathbb{R}}^d} e(-\xi t) \, d\mu(t) \qquad \xi \in {\mathbb{R}}^d.$$ In this context we have Plancherel Theorem [@hormander Theorem 7.1.14], $$\label{plancherel}
\int_{{\mathbb{R}}^d} f(x) \, d\mu(x) = \int_{{\mathbb{R}}^d} \widehat f(\xi) \overline{\widehat\mu(\xi) }\, d\xi \qquad f\in C_c^{\infty}({\mathbb{R}}^d)$$ which can be seen as coming from Fourier expansion of $f$ plus linearity of $\mu$. Since $f$ will have a derivative of size $T$, it is convenient to split the frequencies into the following ranges $$\label{frequency_splitting}
\mu=\mu^l+\mu^m+\mu^h$$ where $
\mu^*(f)=\int_{{\mathbb{R}}^d} \widehat f(\xi) \overline{\widehat \mu(\xi)} \eta_*(\xi)\, d\xi
$, for $*=l, m, h$ and $$\eta_l(r)=\eta(\frac{r}{\rho T}) \qquad \eta_m(r)=\eta(\frac{r}{T/\rho})-\eta(\frac{r}{\rho T}) \qquad \eta_h(r)= 1-\eta(\frac{r}{T/\rho}),$$ with $\rho\in (0,1)$ and $\eta$ a fixed function in $C_c^{\infty}({\mathbb{R}}^d)$ with $\eta(r)=1$ for any $|r|<1$ and $\eta(r)=0$ for any $|r|>2$. So $\mu^l$, $\mu^m$ and $\mu^h$ take care of the low, midrange and high frequencies respectively.
To handle $\lambda_{S,x_0,\beta}^l$ we are going to use the decay in average of $\widehat\lambda_{S,x_0,\beta}$, which is well known for the Fourier transform of a submanifold. This is a particular case of Theorem 7.1.26 in [@hormander]:
\[fourier\_l2\_decay\] For any $\beta<1$ and $K\ge 1$ we have $$\|1_{[1,2]}(|\cdot|)\widehat\lambda_{S,x_0,\beta}(K\cdot)\|_{L^2({\mathbb{R}}^d)} \ll (\beta K)^{-m/2} .$$
Now the main result of this section is
\[main\_low\_high\] Let $f$ be a $(T,\alpha)$-mixing function. Then we have $$|\lambda_{S,x_0,\beta}^l(f)|+|\lambda_{S,x_0,\beta}^h(f)|\ll (\rho/\beta^2)^{d/4} T^{n} + \rho$$ for any $\rho>T^{-\alpha/2}$, where $\rho$ comes from the splitting (\[frequency\_splitting\]).
We can write $$\lambda_{S,x_0,\beta}^h (f)= \lambda_{S,x_0,\beta}(f)-\int_{{\mathbb{R}}^d} \widehat{f}(\xi) \eta_{T/\rho}(\xi) \overline{\widehat\lambda_{S,x_0,\beta}(\xi)}\, d\xi$$ with $\eta_{U}(r)=\eta(r/U)$. We can write $$\widehat f(\xi)\eta_{T/\rho}(\xi)= \widehat f(\xi)\widehat{ \check{\eta}}_{T/\rho}(\xi) =\widehat{f*\check\eta_{T/\rho}}(\xi)$$ where $*$ indicates the convolution and $\check\eta_U(x)$ is the inverse Fourier Transform of $\eta_U$. By (\[plancherel\]) we have $$\lambda_{S,x_0,\beta}^h (f)= \lambda_{S,x_0,\beta}(f-f*\check\eta_{T/\rho}).$$ By definition $1=\eta_U(0)=\int_{{\mathbb{R}}^d} \check\eta_U(y)\, dy$ and then $$f(x)-f*\check\eta_U(x)=\int_{{\mathbb{R}}^d} [f(x)-f(x-y)] \check\eta_U(y) \, dy.$$ Since $|f(x)-f(x-y)|\le \|f\|_{S^1} |y|\le T|y|$ and $\check\eta_U(x)=U^d \check\eta(Ux)$, by a change of variables we have $$| f(x)-f*\check\eta_U(x)| \le \frac TU \int_{{\mathbb{R}}^d} |y| |\check\eta(y)| \, dy \ll T/U$$ hence $
\lambda_{S,x_0,\beta}^h(f)\ll \rho.
$ On the other hand, since $\eta$ is supported on the ball of radius 2 we have $$\lambda_{S,x_0,\beta}^l(f)=\int_{{\mathbb{R}}^d} \widehat f(\xi)\eta_{\rho T}(\xi) \overline{\widehat\lambda_{S,x_0,\beta}(\xi)} 1_{[0,2\rho T]}(|\xi|) \, d\xi$$ and then reasoning as before and using Cauchy-Schwarz we have $$|\lambda_{S,x_0,\beta}^l(f)|\le \| \widehat{f*\check\eta_{\rho T}}\|_{L^ 2} \|\widehat\lambda_{S,x_0,\beta} 1_{[0,2\rho T]}(|\cdot|)\|_{L^2}$$ so by Lemma \[fourier\_l2\_decay\] and Plancherel we have $$|\lambda_{S,x_0,\beta}^l(f)|\ll \beta^{-m/2} (\rho T)^{n/2} \|f*\check\eta_{\rho T}\|_{L^2}.$$ Moreover $$\|f*\check\eta_{\rho T}\|_{L^2}^2=\int_{{\mathbb{R}}^d\times {\mathbb{R}}^d} \check\eta_{\rho T}(y)\overline{\check\eta}_{\rho T}(y') \int_{{\mathbb{R}}^d} f(x-y) \overline f(x-y')\, dx \, dy\, dy'$$ so by using the properties of the Fourier Transform and changing variables we have $$\|f*\check\eta_{\rho T}\|_{L^2}^2=\int_{{\mathbb{R}}^d} w(y) \int_{{\mathbb{R}}^d} f(x-\frac{y}{T}) \overline f(x) \, dx \, dy$$ with $$w(y)=\rho^d \int_{{\mathbb{R}}^d} \check\eta(\rho y+y') \overline{\check\eta}(y')\, dy'.$$ By the decay of the Fourier transform we have $w(y)\ll \rho^d \min(1,|\rho y|^{-1/|o(1)|})$ so that applying (\[horosphere\_condition\]) for $|y|<\rho^{-1}\epsilon^{-1}$ and the bound $\|f\|_{S^0}\le 1$ and the compact support of $f$ we have $$\|f*\check\eta_{\rho T}\|_{L^2}^2\ll \rho^d (\rho^{-1}\epsilon^{-1})^{d/2}+ \epsilon^{1/|o(1)|}\ll \epsilon^{-d/2}\rho^{d/2}+\epsilon^{1/o(1)},$$ and choosing $\epsilon=\rho^{n/d}$ we get that $\lambda_{S,x_0,\beta}^l(f)\ll \beta^{-m/2} \rho^{d/4} T^{n/2}$.
A Sublevel set estimate
========================
In order to control the midrange frequencies in the decomposition (\[frequency\_splitting\]), we will use both the shape and the decay of the Fourier transform of $\lambda_{S,x_0,\beta}$. This Fourier transform can behave badly if $S$ is not totally curved at $x_0$, or if $x_0$ is near to such a point. In order to distinguish the points for which the Fourier transform behaves nicely we are going to quantify the concept of being totally curved.
Let $p\in S$ and fix a real-analytic parametrization $\varphi=\varphi_p:U\to S$ in $S$ with $\varphi^{-1}(p)\in U\subset {\mathbb{R}}^m$. For every $a\in (T_pS)^{\perp}$ we consider $H_a$, the Hessian of $a \varphi$. $H_a$ has $m$ eigenvalues (counted with multiplicities), and if we order their absolute values we get a sequence $
0\le \beta_1\le \beta_2\le \ldots \le \beta_m.
$ We define the functions $e_{a,\varphi}(p)=\beta_n$ and $$\label{eigenvalue_n_hessian}
e_S(p)=\inf\{e_{a,\varphi}(p): a\in (T_pS)^{\perp}, |a|=1\}.$$ With them we give the following definition.
\[curved points\] Let $0<\delta<1$. We say that $p\in S$ is a *$\delta$-curved point* if $e_S(p)>\delta$.
The fact that $p$ is a $\delta$-curved point actually depends on the chosen parametrization $\varphi_p$. Afterwards this will not cause any problem since by compacity we will just need a finite number of parametrizations to cover the support of $\lambda_S$.
By Proposition \[totally\_curved\_char\], we have that $S$ is not totally curved at $p$ if and only if $p$ is a $0$-curved point. We shall be able to control $\lambda_{S,x_0,\beta}^{m}$ for $x_0$ a $\delta$-curved point for a suitable $\delta$. Thus, we need to show that the set of points that are not $\delta$-curved is negligible, namely $$\sigma_S(\{p\in {\textrm{supp }}\lambda_S: e_S(p)<\delta\})<\delta^{1/c}$$ for some $c>0$. This is a sublevel set estimate for the function $e_S$; this kind of estimates play an important role in some problems of Fourier Analysis [@carbery]. Our case is quite peculiar due to the fact that *every* $a\in (T_p S)^{\perp}$ is involved in the definition of $e_S(p)$. In order to prove such an estimate we will use that $S$ is totally curved and the *Real Nullstellensatz*, that characterizes when several polynomials have the same real root:
Let $lst=(n,d_1,d_2,\ldots, d_n)$ be a list of positive integers; define $m_i$ to be $\binom{n+d_i-1}{n-1}$ and $$m''=m_1+\ldots +m_n.$$ We consider $f_1(c,z), \ldots , f_n(c,z)$ the generic homogeneous polynomials in the variables $z=(z_1,\ldots ,z_n)$ of degrees $d_1,\ldots ,d_n$, with $c$ the generic coefficients, $c\in {\mathbb{R}}^{m''}$. We define the set $W_{lst}({\mathbb{R}})$ of $c\in {\mathbb{R}}^{m''}$ such that the system of equations $$\label{equ6}
f_1(c,z)=f_2(c,z)=\ldots =f_n(c,z)=0$$ has no solution $z\in{\mathbb{R}}^n$, $z\neq 0$. We have that $W_{lst}({\mathbb{R}})$ is a semialgebraic set and there exists [@gonzalez_lombardi Theorem C] an algebraic identity $$\label{positivestellensatz}
p_1(c) |z|^{2s}+\sum_{j\ge 2} p_j(c) a_j(c,z)^2 + \sum_{i=1}^n f_i(c,z) b_i(c,z)^2 = 0,$$ with $s$ a positive integer, $p_j$ semipolynomial functions, and $a_j, b_i$ polynomials in $z$ with coefficients semipolynomial functions, such that $p_1>0$ and $p_j\ge 0$ on $W_{lst}({\mathbb{R}})$ for every $j\ge 2$ (a semipolynomial function is a function built using polynomials and the absolute value several times).
So $p_1$ is a kind of real resultant, because if there is no non-zero solution for the system (\[equ6\]) then $p_1>0$.
\[sublevel\] Let $S$ be a totally curved submanifold of ${\mathbb{R}}^d$ of dimension $m$. Then there exists $c_S>1$ such that the $\sigma_S$ measure of set of points $p$ in the support of $\lambda_S$ which are not $\delta$-curved is $O_{\lambda_S}(\delta^{1/c_S})$.
By compacity of ${\textrm{supp }}\lambda_S$ it is enough to assume that $S=\varphi((-1,1)^m)$ with $\varphi(t)=(t,w(t))$ and $w:(-2,2)^m\to {\mathbb{R}}^n$ a real-analytic function and that $\varphi$ is the parametrization chosen to define $e_S(p)$ for every $p$. Moreover we can substitute $e_S(p)$ by $$e_S^*(p)=\inf\{e_{(0,z),\varphi}(p): z\in{\mathbb{R}}^n, |z|=1\}$$ The characteristic polynomial of the matrix $H_{(0,z)}(t)$ can be written as $$\lambda^m+s_{m-1}\lambda^{m-1}+ \ldots + s_1 \lambda +s_0.$$ Now, $s_j$ is a homogeneous polynomial of degree $m-j$ in $z$, and then we can write it as $$s_{j-1}(t,z)=f_j(\varrho(t),z) \qquad 1\le j\le m$$ with $f_1(c,z), \ldots, f_n(c,z)$ the generic homogeneous polynomials of degrees $m, m-1,\ldots, m-n+1,$ each component of $\varrho$ being a polynomial in $ \partial_i\partial_j w(t)$, so a real-analytic function. By (\[positivestellensatz\]), for every $t\in (-1,1)^m$ we have $$\label{resultant_equation}
p_1(\varrho(t)) |z|^{2s}+\sum_{j\ge 2} p_j(\varrho(t)) a_j(\varrho(t),z)^2 + \sum_{i=1}^n s_{i-1}(t,z) b_i(\varrho(t),z)^2 = 0.$$ We can write $$(-1,1)^m=\varrho^{-1}(W)\cup \varrho^{-1}(W^c),$$ with $W=W_{lst}({\mathbb{R}})$ for $lst=(n,m,m-1,\ldots,m-n+1)$. If $t\in \varrho^{-1}(W^c)$ then the dimension of $\ker H_z$ is at least $n$. Hence, since $S$ is totally curved, Proposition \[totally\_curved\_char\] implies that there cannot exist an open set $U\subset \varrho^{-1}(W^c)$. But writing $\varrho(v)=\Phi(D^2 w(v))$, with $\Phi$ a polynomial, we have that $\Phi^{-1}(W)$ and $\Phi^{-1}(W^c)$ are semialgebraic sets and $$\varrho^{-1}(W^c)= \{t\in (-1,1)^m : D^2 w\in \Phi^{-1}(W^c)\}=$$ $$=\bigcup_j \{t: q_j(D^2 w(t))>0\} \cup \bigcup_k \{t: r_k(D^2 w(t))=0\}$$ with $q_j$ and $r_k$ polynomials, and since the set concerning the $q_j$ is open it must be void, and then $$\varrho^{-1}(W^c)=\bigcup_k \{t: r_k(D^2 w(t))=0\},$$ with $r_k(D^2 w)$ non-zero as a real-analytic function. Thus $\varrho^{-1}(W^c)$ is a set of measure zero. On the other hand, since $p_1$ is a semipolynomial and $p_1>0$ on $\varrho^{-1}(W)$, we can split $\varrho^{-1}(W)$ into a disjoint and finite union of sets $V_j$, and on each of them we have $p_j(\Phi(D^2 w(t)))=h_j(D^2 w(t))$, with $h_j$ a polynomial. Clearly $h_j(D^2 w)$ can be extended to a real analytic non-zero function on $(-2,2)^m$ and we have $$m \{t\in(-1,1)^m: p_1(\varrho(t))<\delta \} \le \sum_j m\{t\in(-1,1)^m: |h_j(D^2 w)(t)|<\delta\}.$$ Then, by applying Lemma \[sublevel\_set\_lemma\] to the functions $h_j(D^2 w)$ we have $$\label{equ5}
m(\{t: p_1(\varrho(t))<\delta \})<\delta^{1/c}.$$ Let us consider a fixed $t\in \varrho^{-1}(W)$ with $p_1(\varrho(t))\ge \delta$. Since $p_j(\rho(t))\ge 0$ for every $j\le m$, by (\[resultant\_equation\]) for every $|z|=1$ we have $$-s_{i-1}(t,z)\gg \delta$$ for some $i\le n$. But by the definition of $s_{i-1}(t,z)$ as a symmetric polynomial in terms of the roots of the characteristic polynomial, we get that $$| \lambda_1\lambda_2\ldots \lambda_{m-(i-1)}| \gg \delta$$ for some $\lambda_j$, $j\le m-(i-1)$ eigenvalues of $H_{(0,z)}$. But, since all eigenvalues are $O(1)$ we get that $$|\lambda_j|\gg \delta$$ for every $j\le m-(i-1)$. Thus if $t\in\varrho^{-1}(W)$ with $p_1(\varrho(t))\ge \delta$ then $e_S^*(\varphi(t))\gg \delta$, so the result follows from (\[equ5\]).
This result implies that it is enough to control $\lambda_{S,x_0,\beta}(f)$ for $x_0$ near curved points in order to control $\lambda_S(f)$.
\[local\_to\_global\] Let $f:{\mathbb{R}}^d\to{\mathbb{R}}$ with $\|f\|_{L^{\infty}}\le 1$. If $\lambda_{S,x_0,\beta}(f)\ll \epsilon$ for every point $x_0$ at distance $O(\beta)$ from some $\gamma$-curved point of $S$, then $$\lambda_S(f)\ll \epsilon+\beta+\gamma^{1/c_S},$$ with $c_S>0$ a constant depending on $S$.
We can assume that $\epsilon\ge \beta$, because otherwise we could change $\epsilon$ by $\beta$. By Proposition \[sublevel\] we have $\lambda_{S,x_0,\beta}(f)\ll \epsilon$ for every point at distance $O(\beta)$ from a subset $E\subset {\textrm{supp }}\lambda_S$, with $\sigma_S({\textrm{supp }}\lambda_S\setminus E)\ll \gamma^{1/c_S}$. Thus if we were to have $\lambda_S(f)\gg \delta$, with $\delta=c(\epsilon+\beta+\gamma^{1/c_S})$, $c$ a large constant, we would get a contradiction by applying Proposition \[localization\_in\_space\].
Fourier transform of the submanifold
====================================
Here we want to study $\widehat\lambda_{S,x_0,\beta}$ at each frequency $\xi$, when $x_0$ is at distance $O(\beta)$ from a $\beta$-curved point on $S$. We would like to proceed by performing stationary phase, but we have the problem that we do not control all the eigenvalues of the Hessian of the phase function in the oscillatory integral. By the properties of the point $x_0$ however, we know that most eigenvalues are large, so we shall first separate the small eigenvalues and then use stationary phase with the big ones. We begin by diagonalizing a quadratic form with non-constant coefficients in the case in which the diagonal dominates over the rest of coefficients.
\[diagonalization\] Let $$F(x)=\sum_{i\le l} \lambda_i x_i^2 +2\delta \gamma\sum_{i\le j\le l} x_i x_j \phi_{ij}(x)$$ with $0<\delta<1$, $\phi_{ij}(x)$ real analytic functions on $[0,1]^l$, and $\gamma\le |\lambda_l|\le \ldots \le |\lambda_1|$, $\lambda_j\in{\mathbb{R}}$. For $\delta$ small enough (depending just on the functions $\phi_{ij}$) there is a real-analytic change of variable $y=\psi(x)$ such that $$F(x)=\sum_{i\le l} \lambda_i y_i^2$$ and $\det D\psi(x)=1+O(\delta)$.
We can assume $\gamma=1$. We begin by $x_1$, writing $$F(x)=(\lambda_1+2 \delta\phi_{11}(x))x_1^2 + 2 x_1 \delta \sum_{1<j} x_j\phi_{1j}(x) +\ldots$$ and then $$F(x)= (\lambda_1+2\delta \phi_{11}(x)) [ x_1 +\delta\frac{ \sum_{1<j} x_j \phi_{1j}(x)}{\lambda_1+2\delta \phi_{11}(x)}]^2 - \frac{ (\delta\sum_{1<j}x_j \phi_{1j}(x))^2}{\lambda_1+2 \delta\phi_{11}(x)}+\ldots$$ so we start with the change $$x_1^*=x_1 +\delta\frac{ \sum_{1<j} x_j \phi_{1j}(x)}{\lambda_1+2\delta \phi_{11}(x)}$$ that for $\delta$ small enough satisfies $$\frac{\partial x_1^*}{\partial x_1}= 1 + O(\delta), \qquad \frac{\partial x_1^*}{\partial x_j}= O(\delta),$$ for any $j\ge 2$. Thus $$F=(\lambda_1+\delta \phi_{11}(x)) (x_1^*)^2 + \sum_{2\le i\le l}\lambda_i x_i^2+ 2\delta\sum_{2\le i\le j\le l} x_i x_j \phi_{ij}^*(x)$$ with $$\phi_{ij}^*=\phi_{ij} - \frac{\delta^2}{\lambda_1+2\delta \phi_{11}(x)} \phi_{1i} \phi_{1j}.$$ Continuing like that with $x_2,x_3,\ldots$, we arrive at $$F=\sum_i (\lambda_i + \delta f_i(x^*)) (x_i^*)^2 =\sum_i \lambda_i(1 + \frac{\delta}{\lambda_i} f_i(x_*)) (x_i^*)^2$$ with $\partial x_i^*/\partial x_i=1+O(\delta)$ and $\partial x_i^*/\partial x_j= O(\delta)$ for any $i\neq j$, and $f_i(x^*)$ are real-analytic functions on $[0,1]^l$ with bounded derivatives. We finish with the change $$y_i= x_i^*\sqrt{1+\frac{\delta}{\lambda_i} f_i(x^*)}= x_i^* + \frac{\delta}{\lambda_i} \tilde f_i(x^*)$$ whose determinant is a smooth function near 1, and then $$F=\sum_i \lambda_i y_i^2$$ with $\det Dy/Dx=1+O(\delta)$.
Now we give our result for the Fourier transform of the manifold, in which we can see the resemblance with the case of the sphere.
\[stationary\_phase\] Let $\xi_0\in {\mathbb{R}}^d$ with $|\xi_0|>1$. There exist constants $c'<1<c$ depending just on $S$ such that if $x_0\in {\mathbb{R}}^d$ is a point at distance $O(\beta)$ from a $c\beta$-curved point of $S$, with $0<\beta<c'$, then for every $\xi\in{\mathbb{R}}^d$ with $|\xi-\xi_0|<\beta|\xi_0|$ we have $$\widehat\lambda_{S,x_0,\beta}(\xi)=(|\xi_0|\beta^3)^{-\frac{m-n}2}\int_{{\mathbb{R}}^n} \Phi_r(\tilde\xi)e(|\xi_0|\Phi_r^*(\tilde\xi))\, dr + O((|\xi_0|\beta^3)^{-\frac 1{|o(1)|}}),$$ with $\tilde\xi=\xi/|\xi_0|$, $\Phi_r,\Phi_r^*$ depending on $S, x_0$ and $\beta$ but with uniformly bounded derivatives and $\Phi_{\cdot}(\cdot)$ with uniformly bounded support on ${\mathbb{R}}^{2d}$.
We can assume that near $x_0$ the manifold $S$ is parametrized as $(t,w(t))$, with $w:(-1,1)^m\to {\mathbb{R}}^n$. Then $ \widehat\lambda_{S,x_0,\beta}(\xi)$ can be written as $$\widehat\lambda_{S,x_0,\beta}(\xi)=\beta^{-m}\int_{{\mathbb{R}}^m} e(-\xi_1 t-\xi_2 w(t)) \, \psi_*(\frac{t-t_0}{\beta}) \, dt$$ with $\xi=(\xi_1,\xi_2)$ and $\psi_*(s)=\psi(s+c_1, \frac{w(t_0+\beta s)-w(t_0)}{\beta}+c_2) g(t_0+\beta s)$, where $(t_0,w(t_0))$ is the $c\beta$-curved point on $S$ at distance $O(\beta)$ from $x_0$, $c_1,c_2$ bounded constants, and $g$ a $C_c^{\infty}$ function coming from $\lambda_S$. Thus we have that $\psi_*$ is $C^{\infty}$ function with uniformly bounded support and derivatives. By the change $t\mapsto \beta(t+t_0)$ we have $$\widehat\lambda_{S,x_0,\beta}(\xi)=\int_{{\mathbb{R}}^m} \psi_*(t) e(|\xi_0|(-\tilde\xi_1 (t_0+\beta t)-\tilde\xi_2 w(t_0+\beta t))) \, \, dt,$$ with $\tilde\xi=(\tilde\xi_1,\tilde\xi_2)$. Let $\tilde\xi_0=(\tilde\xi_{0,1},\tilde\xi_{0,2})$. If $|\tilde\xi_{0,2}|<k$ for some constant $0<k<1$ depending just on $w$, integrating by parts with respect to some variable we have $\widehat\lambda_{S,x_0,\beta}\ll (\beta|\xi_0|)^{-1/|o(1)|}$, hence we can assume that $|\tilde\xi_{0,2}|\gg 1$. Let $g_0$ be an orthogonal matrix such that the matrix $g_0^t D^2(\tilde\xi_{0,2}w)(t_0)g_0$ is diagonal. Then since $(t_0,w(t_0))$ is a $c\beta$-curved point on $S$, by the change $t\mapsto g_0 t$ we get that $$\widehat\lambda_{S,x_0,\beta}(\xi)=e_{\xi}\int_{{\mathbb{R}}^m} \tilde\psi(t) e(|\xi_0|\sum_{j\le m} \lambda_j \beta^2 t_j^2-s_j\beta t_j) e(|\xi_0|\beta^3 W(\tilde\xi,t)) \, dt$$ with $e_{\xi}=e(|\xi_0|\Phi^*(\tilde\xi))$, $\Phi^*$ satisfying the properties required for $\Phi^*_r$ in the statement (for $r$ fixed), $\tilde\psi$ satifying the same properties as $\psi_*$, $\lambda=(\lambda_j)_{j\le m}$ constant, $\lambda=O(1)$, with $|\lambda_j|\ge c\beta$ for every $j\le m-n$, $W$ real-analytic on $(-2,2)^d\times{\textrm{supp }}\tilde \psi$ with uniformly bounded coefficients and terms of degree at least 2 in $t$, and $(s_j)_{j\le m}=(\tilde\xi_1+\tilde\xi_2 Dw(t_0) )g_0$. By separating the variables $r=(t_{m-n+1},\ldots t_m)$ we can write $$\widehat\lambda_{S,x_0,\beta}(\xi)=\int_{{\mathbb{R}}^n} e(|\xi_0|\Phi_{r,1}^*(\tilde\xi)) \, \int_{{\mathbb{R}}^{m-n}} \psi_r(t) e(|\xi_0|f_r(\tilde\xi, t)) \, dt \, dr$$ with $\psi_r$ and $\Phi_{r,1}^*$ satisfying the same properties as $\tilde\psi$ and $\Phi_r^*$ respectively, $$f_r(\tilde\xi,t)=\beta^3 W_r(\tilde\xi,t)+\sum_{j\le m-n} \lambda_j \beta^2 t_j^2-s_j \beta t_j.$$ If $|s_j|\gg \beta|\lambda_j|$ for some $j\le m-n$ then integrating by parts we have $\widehat\lambda_{S,x_0,\beta}(\xi)\ll (\beta^3|\xi_0|)^{-1/|o(1)|}$. Thus, we can assume $s_j/\beta\lambda_j\ll 1$ for every $j\le m-n$. The equation $Df_r(\tilde\xi, t)=0$ can be written as $$t_j= \frac{s_j(\tilde\xi)}{2\lambda_j\beta}-\frac{\beta}{2\lambda_j} \frac{\partial W_{r}}{\partial t_j}(\tilde\xi,t) \qquad j\le m-n;$$ we have $|\beta/2\lambda_j|\le 1/2c$, and then for $c$ large enough that equation has a unique solution $t=\varrho_r(\tilde\xi)$, with $\varrho_r$ a real-analytic function with uniformly bounded coefficients on the support of $\psi_r$. By the change $t\mapsto t+\varrho_r(\tilde\xi)$ we have $$\widehat\lambda_{S,x_0,\beta}(\xi)=\int_{{\mathbb{R}}^n} e(|\xi_0|\Phi_{r,2}(\tilde\xi)) \, \int_{{\mathbb{R}}^{m-n}} \psi_{r,2}(t,\tilde\xi) e(|\xi_0|f_r^*(\tilde\xi,t)) \, dt \, dr$$ with $$f_r^*(\tilde\xi,t)= \beta^3 W_r^*(\tilde\xi,t)+\sum_{j\le m-n}\lambda_j \beta^2 t_j^2,$$ $W_r^*$ real-analytic with uniformly bounded coefficients and terms of degree at least 2 in $t$. By applying the change from Lemma \[diagonalization\] we have $$\widehat\lambda_{S,x_0,\beta}(\xi)=\int_{{\mathbb{R}}^n} e(|\xi_0|\Phi_{r,3}(\tilde\xi)) \, \int_{{\mathbb{R}}^{m-n}} \psi_{r,3}(t,\tilde\xi) e(\beta^2|\xi_0|\sum_{j\le m-n} \lambda_j t_j^2) \, dt \, dr.$$ Finally, the result follows from the following fact: if $g(t,x)\in C_c^{\infty}({\mathbb{R}}\times{\mathbb{R}}^l)$ is a function bounded support and derivatives, then for any $\lambda>1$ we can write $$\int_{{\mathbb{R}}} g(t,x) e(\lambda t^2)\, dt = \lambda^{-1/2} G_{\lambda}(x)$$ where $G_{\lambda}\in C_c^{\infty}({\mathbb{R}}^l)$ is a function with bounded support and derivatives, with those bounds depending just on the related ones for $g$ (not on $\lambda$). To prove this we just need to perform the change $t\mapsto t/\sqrt{\lambda}$, consider the splitting $$\int_{{\mathbb{R}}} g(\frac t{\sqrt \lambda}, x) e(t^2) \, dt= \int_{|t|\le 1} g(\frac t{\sqrt \lambda}, x) e(t^2) \, dt + \int_{|t|>1} g(\frac t{\sqrt \lambda}, x) e(t^2) \, dt$$ and integrate by parts twice in the second integral.
Mixing case: Midrange frequencies
=================================
We begin by noticing what we need in order to control the midrange frequencies, after using the information about the Fourier Transform of $S$ obtained in the previous section.
\[midrange\_bound\_oscillatory\] Let $S$ be totally curved and $f\in C_c^{\infty}({\mathbb{R}}^d)$. Let $x_0$ be a point at distance $O(\beta)$ from a $c\beta$-curved point on $S$, with $0<\beta<1$. There exist $\Phi,\Phi^*\in C_c^{\infty}({\mathbb{R}}^d)$ with uniformly bounded derivatives and support, and $\rho T\ll U \ll T/\rho$ such that $$\lambda_{S,x_0,\beta}^m(f)\ll U^n \beta^{-3d} \log(1/\rho) [ I_U + O((\beta^3 U)^{-1/|o(1)|})]$$ with $$I_U=\frac{1}{\sqrt{ U^d}}\int_{{\mathbb{R}}^d} \Phi(\frac{\xi}{U}) \widehat f(\xi) e(x\xi) e(U\Phi^*(\frac{\xi}{U})) \, d\xi.$$ for some $x\in{\mathbb{R}}^d$.
We can split the frequencies into $O(\beta^{-d}\log(1/\rho))$ pieces, so that $$\lambda_{S,x_0,\beta}^m(f)\ll \beta^{-d}\log(1/\rho) \int_{{\mathbb{R}}^d} \eta_*(\frac{\xi-\xi_0}{\beta |\xi_0|}) \widehat f(\xi)\widehat\lambda_{S,x_0,\beta}(\xi) \, d\xi$$ with $\eta_* \in C_c^{\infty}$ independent from $\beta,\rho$ and $T$, $\xi_0$ some frequency in the range $\rho T\ll |\xi_0|\ll T/\rho$. By applying Proposition \[stationary\_phase\], interchanging the integrals and using the compacity of the support of $\Phi_{\cdot}(\cdot)$ we have $$\lambda_{S,x_0,\beta}^m(f)\ll U^n \beta^{-3d} \log(1/\rho) [ \tilde I_U + O((\beta^3 U)^{-1/|o(1)|})]$$ with $U=|\xi_0|$ and $$\tilde I_U=\frac{1}{\sqrt{ U^d}}\int_{{\mathbb{R}}^d} \eta_*(\frac{\xi-\xi_0}{\beta U}) \Phi(\frac{\xi}{U}) \widehat f(\xi) e(U\Phi^*(\frac{\xi}{U})) \, d\xi.$$ Now, by using the formula $\eta_*(v)=\int_{{\mathbb{R}}^d} \widehat\eta_*(x)e(vx)\, dx$, interchanging the order of the integrals and applying the trivial Hölder inequality we are done, since $\int_{{\mathbb{R}}^d} |\widehat\eta_*(x)|\, dx<\infty$.
We could try to control the oscillatory integral appearing in the statement of the previous proposition by applying the Fourier Transform (Plancherel Theorem), but in doing so we would come back essentially to our original definition for $\lambda_{S,x_0,\beta}$, and then we would get just the trivial bound $O(1)$. This problem comes from the fact that the range of frequencies $\xi$ and the growth of the exponential’s phase are both very large; we are going to see that both problems can be addressed by using Weyl-Van der Corput inequality before applying the Fourier Transform, and we just have to pay by having to control $\widehat f_s$ instead of $\widehat f$, with $f_s(x)=f(x+s)\overline f(x)$.
\[van\_der\_corput\] Let $f\in C_c^{\infty}({\mathbb{R}}^d)$ with $\|f\|_{L^{\infty}}\le 1$. Let $1\le U$ and $\Phi^{**}(\xi)=\Phi^*(\xi)+y\xi$ with $y\in{\mathbb{R}}^d$ and $\Phi,\Phi^*\in C_c^{\infty}({\mathbb{R}}^d)$. For every $V\le U$ and $0<\delta<1$ we have $$| \int_{{\mathbb{R}}^d} \Phi(\frac{\xi}{U}) \widehat f(\xi) e(U\Phi^{**}(\frac{\xi}{U})) \, \frac{d\xi}{\sqrt{U^d}} |^2 \ll \delta^{-\frac d2} \int_{{\mathbb{R}}^d} |\tilde\Phi(\frac{\xi}V) \widehat f_{s_{\xi}}(\xi) | \, \frac{d\xi}{\sqrt{ V^d}} + h_{\delta}$$ for some function $s$, with $s(\xi)=s_{\xi}$ bounded by $V/U\delta$, with $f_s(x)=f(x+s)\overline f(x)$, $\tilde\Phi\in C_c^{\infty}({\mathbb{R}}^d)$ and $h_{\delta}\ll V^d O(1/\delta)^{-1/|o(1)|}$. The function $\tilde\Phi$ does not depend on anything, and the constants implicit in the bounds are independent of $y$ and depend just on bounds for the derivatives and supports of $\Phi$ and $\Phi^*$.
Let us call $I$ to the integral we want to bound. Then, we have $$I=\int_{{\mathbb{R}}^d} \varrho(\frac{\zeta}V) I \, d\frac{\zeta}V$$ for any $\varrho\in C_c^{\infty}$ real, even and supported in the unit ball with $\int_{{\mathbb{R}}^d}\varrho=1$, with $$I=\frac{1}{\sqrt{U^d}}\int_{{\mathbb{R}}^d} \widehat f(\xi+\zeta) E^U(\xi+\zeta) \, d\xi$$ for every $\zeta\in{\mathbb{R}}^d$, with $E^U(\xi)=E(\xi/U)$, $E(\xi)=\Phi(\xi) e(U(y+\Phi^*(\xi)))$. Interchanging the integrals and applying Cauchy’s inequality, taking into account the support of $\Phi$ and $\varrho$, we have $$|I|^2 \ll \int_{{\mathbb{R}}^d} |\int_{{\mathbb{R}}^d} \varrho(\frac{\zeta}V) \widehat f(\xi+\zeta) E^U(\xi+\zeta) \, d\frac{\zeta}V |^2 \, d\xi.$$ Expanding the square and making a change of variables we have $$|I|^2\ll \int \varrho*\varrho (\frac{\zeta}V) [\int_{{\mathbb{R}}^d} (\widehat f)_{\zeta}(\xi) \overline{ (\overline E^U)_{\zeta}(\xi)} \, d\xi ] \, d\frac{\zeta}V$$ with $g_{\zeta}(\xi)=g(\xi+\zeta)\overline g(\xi)$. Since $\widehat{(\widehat f)_{\zeta}}(x)=\widehat{\overline f_x}(\zeta)$ and $\widehat{(\overline E^U)_{\zeta}}(x)=U^d \widehat{\overline E_{\zeta/U}}(Ux)$, by Plancherel and the change of variable $x\mapsto x/U$ we have $$\int_{{\mathbb{R}}^d} (\widehat f)_{\zeta}(\xi)\overline{ (\overline E^U)_{\zeta}(\xi)} \, d\xi = \int_{{\mathbb{R}}^d} \widehat{\overline f_{x/U}}(\zeta) \overline{\widehat{\overline E_{\zeta/U}}(x)} \, dx.$$ We have that $E_{\zeta/U}$ has compact support and its $jth$ derivative is bounded by $O((1+|\zeta|)^j)$, hence $\widehat{E_{\zeta/U}}(x)\ll_j (x/(1+|\zeta|))^{-j}$. Since $\zeta\ll V$, splitting the integral into two parts we have $$\int_{{\mathbb{R}}^d} \widehat{\overline f_{x/U}}(\zeta) \overline{\widehat{\overline E_{\zeta/U}}(x)} \, dx \ll |\widehat{\overline f_{s_{\zeta}}}(\zeta)| \int_{|x|<V/\delta} |\widehat{\overline E_{\zeta/U}}(x)| \, dx + h_{\delta}$$ for some $|s_{\zeta}|\le V/U\delta$. We finish by applying Cauchy’s inequality followed by Plancherel, since $E_{\zeta/U}$ is bounded and compactly supported.
We are finally ready to control $\lambda_{S,x_0,\beta}^m (f)$ for $f$ a mixing function and $x_0$ a curved point.
\[main\_midrange\] Let $S$ be totally curved and $f$ a $(T,\alpha)$-mixing function. Let $0<\beta<1$. There exists $c>1$ depending just on $S$ such that if $x_0$ is at distance $\beta$ from a $c\beta$-curved point of $S$ then we have $$\lambda_{S,x_0,\beta}^m(f) \ll T^{n-\frac{\alpha}2 d} \beta^{-4d}\rho^{-3d/2} [1+R]$$ with $R\ll T^{\alpha d}\min(\beta^{-1},\beta^3 \rho T)^{-1/|o(1)|}$.
Apply Proposition \[midrange\_bound\_oscillatory\] followed by Proposition \[van\_der\_corput\]. Since $f$ is $(T,\alpha)$-mixing the resulting integral can be bounded for any $V\le \delta \rho T^{\alpha}$, by (\[character\_condition\]) (in order to apply (\[character\_condition\]) to the integral coming from the bound in Proposition \[van\_der\_corput\] we only need to use the support of $\tilde\Phi$). Thus, picking $\delta=\beta$ and $V=\beta\rho T^{\alpha}$, since $\rho T<U<T/\rho$ we get that $$\lambda_{S,x_0,\beta}^m(f)\ll T^n \beta^{-3d} \rho^{-n}\log(1/\rho) [ \beta^{-d}\rho^{-d/2} T^{-\frac{\alpha}2 d} + R]$$ with $R\ll T^{\alpha d}\min(\beta^{-1},\beta^3 \rho T)^{-1/|o(1)|}$.
\[main\_local\] Let $S$ be totally curved and $f$ a $(T,\alpha)$-mixing function. If $x_0$ is a point at distance $O(\beta)$ from a $c\beta$-curved point of $S$ we have $$\lambda_{S,x_0,\beta}(f)\ll \beta^{1/8}$$ for any $\beta$ in the range $T^{-\frac{\alpha}{20}+\frac 1{10}\frac nd} < \beta < T^{-8\frac nd}$.
From Propositions \[main\_low\_high\] and \[main\_midrange\], by picking $\rho=\beta^3$, we get that $$\lambda_{S,x_0,\beta}(f)\ll \beta^3 +\beta^{d/4} T^{n} + T^{n-\frac{\alpha}2 d} \beta^{-9d} [1+R]$$ with $R\ll T^{\alpha d} \min(\beta^{-1},\beta^6 T)^{-1/|o(1)|}$, for every $x_0$ at distance $O(\beta)$ from a $c\beta$-curved point of $S$, with $\beta>T^{-\alpha/6}$. The result follows in the range chosen for $\beta$.
Theorem \[main\_real\] finally follows from Propositions \[main\_local\] and \[local\_to\_global\], together with Proposition \[main\_singular\].
Acknowledgements {#acknowledgements .unnumbered}
================
I worked on this paper as a result of an invitation by E. Lindenstrauss to stay for four months at The Einstein Institute of Mathematics, at the Hebrew University of Jerusalem. I would like to thank him for his help with the problem as well as the people at the Institute for the great conditions to work there.
I would also like to thank F. Chamizo for his comments on the paper, which improved it substantially.
[^1]:
|
---
author:
- 'L. Leśniak[^1]'
- 'R. Kamiński'
- 'A. Furman'
- 'J.-P. Dedonder'
- 'B. Loiseau'
title: Dalitz plot studies of decays
---
Introduction {#intro}
============
Dalitz-plot time dependent amplitude analyses have been recently performed [@BaBar; @Belle] for the $CP$ self conjugate $D^0$ meson decays into $K^0_S \pi^+ \pi^-$. These studies have allowed a direct measure of the $D^0$-$\bar D^0$ mixing parameters, the knowledge of which could show the presence of new physics contribution beyond the standard model. Studies of $B^{\pm}\rightarrow D^{(*)} K^{\pm}, D\rightarrow K^0_S \pi^+ \pi^-$ decays, in which the interference between $D^0$ and $\bar D^0$ mesons was used to measure the Cabibbo-Kobayashi-Maskawa angle $\gamma$, have also been accomplished [@BaBar2; @Belle2]. A good understanding of the final state interactions between mesons in the $D^0$ decays into $K^0_S \pi^+ \pi^-$ is essential in order to reduce errors of the $D^0-\bar D^0$ mixing parameters and the measured values of the angle $\gamma$. A very rich resonance spectrum seen in the Dalitz plots is a direct signal of the complexity of the strong meson interactions. Using these high statistics data theoretical models of the decay amplitudes can be tested.
The experimental analyses [@BaBar; @Belle] relied mainly on application of an isobar model. In such a model one can accomodate many resonances coupled to interacting pairs of mesons. However, one should stress that the corresponding decay amplitudes are not unitary: unitarity is not preserved in the three-body decay channels and it is also violated in the two-body subchannels. Within the isobar model it is particularly difficult to distinguish the $S$-wave amplitudes from the non-resonant background terms. Their interference is often very strong which means that some two-body branching fractions, extracted from the data, could be unreliable. The isobar model is flexible but it has many free parameters (at least two fitted parameters for each amplitude component). For example, the Belle Collaboration has used 40 fitted parameters in ref. [@Belle] and the Babar Collaboration 43 free parameters in ref. [@BaBar].
The construction of unitary three-body strong interaction amplitudes in a wide range of meson-meson effective masses is difficult. Therefore, as a first step, we incorporate two-body unitarity in our model of the $D$-decay amplitudes with final state interactions in the $K^0_S \pi^{\pm}$ $S$- and $P$-waves and in the $\pi^+\pi^-$ $S$-wave. The branching fraction corresponding to a sum of these amplitudes is close to 80 % of the total branching fraction of the $D\rightarrow K^0_S \pi^+ \pi^-$ decay.
Decay amplitudes {#sec:1}
================
The $D^0$ decays into $K^0_S \pi^+ \pi^-$ are analysed in the framework of the quasi-two body factorization approach. The decay amplitude consists of the following 27 non-zero parts: seven *allowed* tree amplitudes, six doubly Cabibbo *suppressed* tree amplitudes, fourteen *annihilation* (W-exchange) amplitudes (7 allowed and 7 Cabibbo suppressed). The allowed amplitudes are generated by the quark $c \rightarrow s u \bar d$ transitions and the doubly Cabibbo suppressed amplitudes correspond to $c \rightarrow d u \bar s$ decays. Seven partial wave amplitudes are considered. We count $S$, $P$ and $D$ waves in the $K^0_S \pi^{\pm}$ and the $\pi^+ \pi^-$ subsystems and separately the $P$-wave amplitude for the $G$- parity violating $\omega \rightarrow \pi^+ \pi^-$ transition. There are numerous resonances contributing to different partial wave amplitudes. For example, in the $K^0_S \pi^-$ $S$-wave subchannel one can list $K_0^*(800)^-$ or $\kappa^-$ and $K_0^*(1430)^-$ resonances, in the $P$-wave one $K^*(892)^-$, $K_1(1410)^-$ and $K^*(1680)^-$ and in the $D$-wave the $K_2^*(1430)^-$ resonance. The same resonances but with the opposite charge are active in the $K^0_S \pi^+$ subchannels. In the $\pi^+ \pi^-$ subchannels one can enumerate the $f_0(500)$ or $\sigma$, $f_0(980)$ and $f_0(1400)$ scalars, the $\rho(770)$, $\rho(1450)$ and $\omega(782)$ vector states and the $f_2(1270)$ tensor resonance.
The main part of the weak decay effective Hamiltonian is proportional to the following operator: $$\label{op}
O=\frac{G_F}{\sqrt2}V^*_{cs} V_{ud} j_1 j_2,$$ where $G_F$ is the Fermi coupling constant, $V_{cs}$ and $V_{ud}$ are the Cabibbo-Kobayashi-Maskawa matrix elements, the two quark currents are defined as $j_1=(\bar s c)_{V-A}$ and $j_2=(\bar u d)_{V-A}$, the symbol $V-A$ means that the axial current is subtracted from the vector current. In the quasi-two body factorization approximation one can write: $$\label{fact}
\langle K^0_S \pi^-\pi^+|j_1 j_2|D^0 \rangle \cong
\langle K^0_S \pi^-|j_1|D^0 \rangle \langle \pi^+|j_2|0 \rangle
+\langle \pi^- \pi^+|j'_1|D^0 \rangle \langle K^0_S|j'_2|0 \rangle
+\langle 0|j'_1|D^0 \rangle \langle K^0_S \pi^-\pi^+|j'_2|0 \rangle,$$ where $|0 \rangle$ denotes the vacuum state and $j'_1=(\bar u c)_{V-A}$, $j'_2=(\bar s d)_{V-A}$ are the quark currents obtained from the $j_1, j_2$ currents by the Fierz transformation. This form of the transition matrix element enables one to introduce the pion $f_{\pi}$, kaon $f_K$ and $D^0$ $f_D$ decay constants. The corresponding expressions are: $$\label{fff}
\langle \pi^+|j_2|0 \rangle =if_{\pi} p_{\pi},
~~~~~~\langle K^0_S|j'_2|0 \rangle =if_K p_K,
~~~~~~\langle 0|j_1'|D^0 \rangle =-if_D p_D,$$ where $p_{\pi}$, $p_K$ and $p_D$ are the pion, kaon and $D$ meson four-momenta, respectively.
Transition matrix elements {#trans}
--------------------------
The expression of the transition matrix elements can be simplified if we assume that the two strongly interacting mesons $h_2$ and $h_3$ of momenta $p_2$ and $p_3$ form a resonance $R$ in the final state. Then one can write: $$\label{hhD}
\langle h_2(p_2)h_3(p_3)|j~|D^0(p_D)\rangle = G_{Rh_2h_3}(s_{23})\langle R(p_2+p_3)|j~|D^0(p_D)\rangle,$$ where $G_{Rh_2h_3}(s_{23})$ with $s_{23}=(p_2+p_3)^2$ is the vertex function describing the $R$ decay into $h_2$ and $h_3$ mesons. For example, let us consider the reaction $D^0(p_D) \rightarrow \pi^+(p_1) K^0_S(p_2) \pi^-(p_3)$. If in the intermediate state the $K^*(892)^-$ resonance is formed and decays into a $\bar K^0\pi^-$ pair then the transition form factor $A_0^{DK^{*-}}(m_{\pi}^2)$ appears in the matrix element $$\label{RjD}
\langle R(p_2+p_3)|j~|D^0(p_D)\rangle=-2im_{K^*}\frac{\epsilon^* \cdot p_D}{p_1^2}~p_1
~A_0^{DK^{*-}}(m_{\pi}^2) ~~ +3~~other~~ terms.$$ Here $j=(\bar s c)_{V-A}$, $\epsilon$ is the $K^*(892)^-$ polarization vector, $m_{K^*}$ is the $K^*$ mass and $p_D=p_1+p_2+p_3$. The “$3~~ other~~ terms$” do not contribute to the transition amplitude in eq. (\[fact\]). The vertex function $G_{K^{*-}\bar K^0\pi^-}$ can be expressed in terms of the kaon-pion transition vector form factor $F_1^{\bar K^0\pi^-}(s_{23})$ $$\label{GK*}
G_{K^{*-}\bar K^0\pi^-}(s_{23})=\epsilon \cdot(p_2-p_3)\frac{1}{m_{K^*}f_{K^*}}F_1^{\bar K^0\pi^-}
(s_{23}).$$ In the above equation $f_{K^*}$ is the $K^*$ decay constant. Assuming isospin symmetry one can equate the transition vector form factor $F_1^{\bar K^0\pi^-}(s_{23})$ to the charged kaon to charged pion transition vector form factor $F_1^{K^-\pi^+}(s_{23})$ calculated in ref. [@PR09].
The third term of eq. (\[fact\]) can also be simplified if the two hadrons, for example $h_2$ and $h_3$, interact via a resonant state $R$. Then, similarly to eq. (\[RjD\]) we write $$\label{hhhj0}
\langle h_1(p_1)h_2(p_2)h_3(p_3)|j'|0\rangle=G_{Rh_2h_3}(s_{23})\langle h_1(p_1)R(p_2+p_3)|j'|0\rangle.$$ If, for example, $h_1=\bar K^0$, $R=f_0\rightarrow \pi^+\pi^-$ and $j'=(\bar s d)_{V-A}$ then the matrix element reads $$\label{Kf0}
\langle \bar K^0(p_1)f_0(p_2+p_3)|j'|0\rangle=-i\frac{m^2_{K^0}-s_{23}}{p_D^2}p_D
F_0^{\bar K^0f_0}(m_D^2)~~~~ +~~~~second~~~ term,$$ where $F_0^{\bar K^0f_0}(m_D^2)$ is the kaon to $f_0$ scalar transition form factor. The vertex function for the $f_0$ decay into $\pi^+\pi^-$ can be parametrized as $$\label{f0pipi}
G_{f_0\pi^+\pi^-}(s_{23})=\chi_2 F^{\pi+\pi-}_0(s_{23}),$$ where $\chi_2$ is a constant and $F^{\pi+\pi-}_0(s_{23})$ is the pion scalar form factor. Its functional form is taken from our $B^{\pm} \rightarrow \pi^{\pm} \pi^{\mp}\pi^{\pm}$ decay study [@APP11]. It preserves unitarity and groups together three scalar-isoscalar resonances $f_0(500)$, $f_0(980)$ and $f_0(1400)$.
Examples of allowed transition amplitudes with $K^0_S\pi^-$ final state interactions
------------------------------------------------------------------------------------
Using the assumptions introduced in subsection \[trans\] one can derive formulae for the decay amplitudes in which the $K^0_S\pi^-$ final state interactions are explicitely included. The $S$-wave amplitude reads: $$\label{S}
A_S=-\frac{G_F}{2}\Lambda_1 a_1 f_{\pi} (m_D^2-m^2_{\pi})~ F_0^{D K_0^{*-}}(m^2_{\pi})
~F_0^{\bar K^0\pi^-}(m^2_-).$$ Here, $\Lambda_1=V_{cs}^*V_{ud}$, $a_1$ is the effective Wilson coefficient, $F_0^{D K_0^{*-}}(m^2_{\pi})$ is the scalar $D^0$ to $K_0^{*-}$ transition form factor and $F_0^{\bar K^0\pi^-}(m_-^2)$ is the scalar $\bar K^0$ to $\pi^-$ transition form factor which depends on the $K^0_S\pi^-$ effective mass squared $m^2_-$. The latter form factor can be taken equal to the $K^-$ to $\pi^+$ transition scalar form factor $F_0^{K^-\pi^+}(m^2_{\pi})$ which has been evaluated in the study of $B \rightarrow K \pi^+\pi^-$ decays [@PR09]. In this form factor both $K^*_0(800)$ and $K^*_0(1430)$ resonances are included in a unitary way. The expression for the $P$-wave amplitude is: $$\label{P}
A_P=-\frac{G_F}{2}\Lambda_1 a_1 \frac{f_{\pi}}{f_{K^*}}[m_0^2-m_+^2+\frac{(m_D^2-m_{\pi}^2)
(m_K^2-m_{\pi}^2)}{m_-^2}]~A_0^{DK^{*-}}(m_{\pi}^2)~F_1^{\bar K^0\pi^-}(m^2_-),$$ where $m^2_+$ is the $K^0_S\pi^+$ effective mass squared and $m^2_0$ denotes the $\pi^+ \pi^-$ effective mass squared. The $D$-wave decay amplitude depends on the mass $m_{K_2^*}$ and the width $\Gamma_{K_2^*}$ of the resonance $K_2^*(1430)$: $$\label{D}
A_D = -\frac{G_F}{2}\Lambda_1 a_1 f_{\pi} F^{DK_2^*}(m^2_-)\frac{g_{K_2^*K_S^0\pi}
D(m_+^2,m_-^2)}{m^2_{K_2^*}-m_-^2-im_{K_2^*}\Gamma_{K_2^*}}.$$ Here, $F^{DK_2^*}(m^2_-)$ is a combination of three types of the $D$ to $K_2^*(1430)^-$ transition form factors, $g_{K_2^*K_S^0\pi}$ is the decay coupling constant and the $ D(m_+^2,m_-^2)$ is the $D$-wave angular distribution function.
Derivation of other decay amplitudes proceeds in a quite similar way as shown above for the $A_S$, $A_P$ and $A_D$ amplitudes. One can notice that in our amplitude model the meson transition form factors play a very important role.
Results {#res}
=======
The theoretical model outlined in the previous section has 28 free real parameters, most of them are unknown complex values of the meson-meson transition form factors appearing in the fourteen W- exchange amplitudes. The scalar and vector kaon-pion and scalar pion form factors are constrained using unitarity, analyticity and chiral symmetry. Fig. 1 shows that the model preliminary results are in fair agreement with the Belle Collaboration data. Also the total braching fraction is well reproduced [@PDG].
\[rys\]
Acknowledgements
================
This work has been supported in part by the IN2P3-Polish Laboratories Convention (project No 08-127).
P. del Amo Sanchez *et al.* (BABAR Collaboration), Phys. Rev. Lett. **105**, (2010) 081803\
and arXiv: hep-ex 1004.5053v3 L. M. Zhang, *et al.* (Belle Collaboration), Phys. Rev. Lett. **99**, (2007) 131803 P. del Amo Sanchez *et al.* (BABAR Collaboration), Phys. Rev. Lett. **105**, (2010) 121801 A. Poluektov *et al.* (Belle Collaboration), Phys. Rev. **D 81**, (2010) 112002 B. El-Bennich *et al.*, Phys. Rev. **D 79**, (2009) 094005 J.-P. Dedonder *et al.*, Acta Physica Polonica **42**, (2011) 2013 K. Nakamura *et al.* (Particle Data Group), Journal of Physics G **37**, (2010) 075021
[^1]:
|
---
abstract: |
The microscopic basis for the stability of itinerant ferromagnetism in correlated electron systems is examined. To this end several routes to ferromagnetism are explored, using both rigorous methods valid in arbitrary spatial dimensions, as well as Quantum Monte Carlo investigations in the limit of infinite dimensions (dynamical mean-field theory). In particular we discuss the qualitative and quantitative importance of (i) the direct Heisenberg exchange coupling, (ii) band degeneracy plus Hund’s rule coupling, and (iii) a high spectral density near the band edges caused by an appropriate lattice structure and/or kinetic energy of the electrons. We furnish evidence of the stability of itinerant ferromagnetism in the pure Hubbard model for appropriate lattices at electronic densities not too close to half-filling and large enough $U$. Already a weak direct exchange interaction, as well as band degeneracy, is found to reduce the critical value of $U$ above which ferromagnetism becomes stable considerably. Using similar numerical techniques the Hubbard model with an easy axis is studied to explain metamagnetism in strongly anisotropic antiferromagnets from a unifying microscopic point of view. \
\
[71.27.+a,75.10.Lp]{}
address: |
$^1$ Theoretische Physik III, Elektronische Korrelationen und Magnetismus, Universität Augsburg, D-86135 Augsburg, Germany\
$^2$ Department of Physics, University of Illinois at Urbana-Champaign, 1110 West Green Street, Urbana, IL 61801-3080, USA
author:
- 'D. Vollhardt$^1$, N. Blümer$^2$, K. Held$^1$, M. Kollar$^1$, J. Schlipf$^1$, and M. Ulmke$^1$'
title: 'Non-perturbative approaches to magnetism in strongly correlated electron systems'
---
Introduction
============
Even after several decades of research the microscopic foundations of itinerant ferromagnetism are not sufficiently understood. Indeed, in contrast to other collective electronic phenomena such as antiferromagnetism or conventional superconductivity there exists a remarkable gap between theory and experiment in this field. This has mainly to do with the fact that itinerant ferromagnetism is a quantum-mechanical [*strong-coupling*]{} phenomenon whose explanation requires the application of [*non-perturbative*]{} techniques. Thus it belongs into the class of the most difficult many-body problems in condensed-matter physics. Significant progress was made in this field in the last few years due to the development and application of several new analytic and numerical approaches. It is the purpose of this paper to present and discuss some of these new, exciting results.
In Sec. 2 we recapitulate the derivation of a general lattice model for correlated electrons starting from a continuum model, and discuss the various truncation steps which eventually lead to the Hubbard model. In particular, the implications of the Lieb-Mattis theorem on the impossibility of ferromagnetism in these truncated models in $d=1$ dimension are analyzed. In Sec. 3 several microscopic mechanisms favoring ferromagnetism are discussed. New and very recent results concerning ferromagnetism in Hubbard-type models obtained by various non-perturbative methods are presented and put into perspective. Metamagnetic phase transitions and the need for the application of non-perturbative techniques for these investigations are the subject of Sec. 4. Finally, a conclusion is presented in Sec. 5.
Electronic Correlations and Magnetism
=====================================
General lattice model
---------------------
Within the occupation number formalism the Hamiltonian for electrons with spin $ \sigma $ interacting via a [*spin-independent*]{} interaction $ V^{ee}
({\bf r} - {\bf r\, '})$ in the presence of an ionic lattice potential $
V^{ion} ({\bf r})$ has the form [@Lieb62; @Hubbard63etc] $\hat{H} = \hat{H_0} + \hat{H}_{\mbox{\scriptsize\em int}}$, where $$\begin{aligned}
\hat{H_0} &=& \sum_{\sigma} \int d^{3}r \hat{\psi}^{+}_{\sigma} ({\bf r})
\left[ - \frac{\hbar^2}{2m} \Delta + V^{ion} ({\bf r}) \right]
\hat{\psi}_\sigma ({\bf r} ) \label{h0}\\
\hat{H}_{\mbox{\scriptsize\em int}} &=&
\frac{1}{2} \sum_{\sigma \sigma'} \int d^{3}r \int
d^{3}r\, ' V^{ee} ({\bf r} - {\bf r\,'}) \hat{n}_{\sigma}
({\bf r} ) \hat{n}_{\sigma'} ({\bf r} \,') \, . \label{hint}\end{aligned}$$ Here $\hat{\psi}_{\sigma} ({\bf r} ) , \hat{\psi}^{+}_{\sigma} ({\bf r} )$ are the usual field operators and $\hat{n}_{\sigma} ({\bf r}) = \hat{\psi}^{+}_{\sigma} ({\bf r} )
\hat{\psi}_{\sigma} ({\bf r} )$ is the local density. We note that the interaction term is diagonal in the space variables $ {\bf r}, {\bf r}\, ', $ i.e. it depends only on the (operator-valued) densities of the electrons at site $ {\bf r}, {\bf r}\, '$ which interact via $V^{ee} ({\bf r} - {\bf r}\, '). $ The lattice potential entering the non-interacting part (\[h0\]) leads to a splitting of the parabolic dispersion into infinitely many bands which we enumerate by the index $ \alpha. $ The non-interacting problem is then characterized by the Bloch wave functions $ \phi_{\alpha
{\bf k}} ({\bf r})$ and the band energies $ \epsilon_{\alpha {\bf k}}. $ We may introduce Wannier functions localized at site $ {\bf R_i} $ by $$\chi_{\alpha i}({\bf r} ) = \frac{1}{\sqrt{L}} \sum_{{\bf k}} e^{-i
{\bf k}\cdot{\bf R_i}} \ \phi_{\alpha{\bf k}}({\bf r})\, , \label{wann}$$ where $L$ is the number of lattice sites, and thus construct creation and annihilation operators $\hat{c}^{+}_
{\alpha i \sigma}, \hat{c}^{\phantom +}_{\alpha i \sigma}$ for electrons with spin $\sigma$ in the band $\alpha$ at site ${\bf R_i}$ as $$\begin{aligned}
\hat{c}^{+}_{\alpha i \sigma} & = &\int d^{3}r \ \chi_{\alpha i} ({\bf r})\
\hat{\psi}^{+}_{\sigma} ({\bf r}) \nonumber \\ \longleftrightarrow
\hat{\psi}^{+}_{\sigma} ({\bf r}) & = & \sum_{i\alpha}\ \chi^{*}_{\alpha i}
({\bf r})\hat{c}^{+}_{\alpha i \sigma} \, . \label{creation}\end{aligned}$$ Thereby the Hamiltonian may be written in lattice representation as [@Hubbard63etc] $$\begin{aligned}
\hat{H} & = &\sum_{\alpha i j \sigma} t_{\alpha i j} \hat{c}^{+}_{\alpha i
\sigma} \ \hat{c}^{\phantom +}_{\alpha j \sigma} \nonumber \\ +
& & \frac{1}{2} \sum_{\alpha \beta \gamma \delta}
\sum_{i j m n} \sum_{\sigma \sigma'} {\cal V}^{\alpha\beta\gamma\delta}_{i
j m n} \ \hat{c}^{+}_{\alpha i \sigma} \ \hat{c}^{+}_{\beta j \sigma'} \
\hat{c}^{\phantom +}_{\delta n \sigma'} \ \hat{c}^{\phantom +}_{\gamma m
\sigma}
\, , \label{ham}\end{aligned}$$ where the matrix elements are given by $$\begin{aligned}
t_{\alpha i j} &=& \int d^{3}r \ \chi^{*}_{\alpha i} ({\bf r}) \left[ -
\frac{\hbar^2}{2m} \Delta + V^{ion} ({\bf r}) \right] \chi_{\alpha i} ({\bf r})
\label{matr1}\\
{\cal V}^{\alpha\beta\gamma\delta}_{i j m n} &=& \int d^{3}r \int d^{3}r\, '
V^{ee} ({\bf r} - {\bf r}\, ' ) \nonumber \\ & & \chi^{*}_{\alpha i} ({\bf r}) \
\chi^{*}_{\beta j} ({\bf r}\, ') \ \chi_{\delta n} ({\bf r}\, ') \
\chi_{\gamma m} ({\bf r}) \, . \label{matr2}\end{aligned}$$ We note that in contrast to the field-operator representation defined in the continuum, the Wannier representation does [*not*]{} lead to a site-diagonal form of the electron-electron interaction, i.e. the interaction does not only depend on the [*densities*]{} $\hat{n}_{i \sigma} = \hat{c}^+_{i \sigma}
\hat{c}^{\phantom +}_{i \sigma}$ but contains explicit off-diagonal contributions which will be discussed later.
One-band models
---------------
The Hamiltonian (\[ham\]) is too general to be tractable in dimensions $d > 1$. Hence it has to be simplified using physically motivated truncations [@Hubbard63etc]. In particular, if the Fermi surface lies within a single conduction band, and if this band is well separated from the other bands and the interaction is not too strong, it may be justified to restrict the discussion to a [*single*]{} band ($ \alpha = \beta = \gamma = \delta = 1$). In this case (\[ham\]) reduces to $$\begin{aligned}
\hat{H}_{\mbox{\scriptsize\em 1-band}} & = &
\sum_{ij\sigma} t_{ij} \hat{c}^+_{i\sigma} \hat{c}^{\phantom +}_{j\sigma}
\nonumber \\
& & + \frac{1}{2} \sum_{ijmn} \sum_{\sigma \sigma'} {\cal V}_{ijmn}
\hat{c}^+_{i\sigma} \hat{c}^+_{j\sigma'} \hat{c}^{\phantom +}_{n\sigma'}
\hat{c}^{\phantom +}_{m\sigma} \, .
\label{1band}\end{aligned}$$ For most purposes this single-band Hamiltonian is still too complicated. Taking into account the weak overlap between neighboring orbitals in a tight-binding description one may expect that the overlap between nearest-neighbors is most important. Hence the site-indices in (\[1band\]) are restricted to nearest-neighbor positions. In the interaction this leaves us with a purely local contribution ${\cal V}_{iiii} = U$, the Hubbard term, and the four nearest-neighbor contributions ${\cal V}_{ijij} = V$, ${\cal V}_{iiij} = X$, ${\cal V}_{ijji} = F$, and ${\cal V}_{iijj} = F'$, which are [*off*]{}-diagonal in the site indices. The remaining one-band, nearest-neighbor Hamiltonian has the form [@Hubbard63etc; @Caron68; @Kivelson87etc; @Gammel88etc; @Baeriswyl88; @Hirsch89etc; @Painelli89; @Strack94] $$\hat{H}^{\mbox{\scriptsize\em NN}}_{\mbox{\scriptsize\em 1-band}} =
\hat{H}_{\mbox{\scriptsize\em Hub}} +
\hat{V}^{\mbox{\scriptsize\em NN}}_{\mbox{\scriptsize\em 1-band}}
\label{nn}$$ where $$\hat{H}_{\mbox{\scriptsize\em Hub}} = -t \sum_{\langle i,j\rangle ,\sigma}
(\hat{c}^+_{i\sigma} \hat{c}^{\phantom +}_{j\sigma} + {\rm h.c.})
+ U \sum_{i} \hat{n}_{i\uparrow} \hat{n}_{i\downarrow} \label{hub}$$ is the Hubbard model and $$\begin{aligned}
\hat{V}^{\mbox{\scriptsize\em NN}}_{\mbox{\scriptsize\em 1-band}} & = &
\sum_{\langle i,j\rangle} \ \lbrack
V \hat{n}_i \hat {n}_j \nonumber\\
& & + X \sum_{\sigma} (\hat{c}^+_{i\sigma}
\hat{c}^{\phantom +}_{j\sigma} + {\rm h.c.})
(\hat {n}_{i\, -\sigma}+\hat {n}_{j \, -\sigma}) \nonumber\\
& & + F{'} (\hat{c}^+_{i\uparrow} \hat{c}^+_{i\downarrow}
\hat{c}^{\phantom +}_{j\downarrow}
\hat{c}^{\phantom +}_{j\uparrow}+ {\rm h.c.})\nonumber\\
& & -2 F ({\bf {\hat{S}}}_{i} {\bf {\hat{S}}}_{j}
+\frac {1}{4}\hat {n}_{i}\hat {n}_{j})\rbrack \label{vnn} \end{aligned}$$ is the contribution of nearest-neighbor interactions. Here $\hat{n}_{i} = \sum_{\sigma} \hat{n}_{i\sigma}$ and ${\bf {\hat{S}}}_{i}=1/2 \sum_{\sigma\sigma'} \hat{c}^+_{i\sigma}
\tau^{\phantom +}_{\sigma\sigma'} \hat{c}^{\phantom +}_{i\sigma'}$, where $\tau$ denotes the vector of Pauli matrices. In (\[vnn\]) the $V$-term describes a density-density interaction, the $X$-term is a bond-charge–site-charge interaction (“density-dependent hopping”), the $F'$-term describes the hopping of local pairs consisting of an up and a down electron and, finally, the $F$-term corresponds to the [*direct*]{} Heisenberg exchange which is generically ferromagnetic in nature. The occupation number formalism illustrates particularly clearly that a spin-independent interaction plus the Pauli principle is able to lead to a mutual orientation of spins. Of all interactions in [(\[nn\])]{} the Hubbard interaction $U$ is certainly the strongest. Hence, in a final truncation step one may try to neglect even the nearest-neighbor interactions and retain only the on-site interaction $U$. This leaves us with the Hubbard model [(\[hub\])]{}, the simplest correlation model for lattice electrons [@Hubbard63etc; @Gutzwiller63etc; @Kanamori63].
### The Hubbard model
The Hubbard model was originally introduced in an attempt to understand itinerant ferromagnetism in $3d$-transition metals [@Hubbard63etc; @Gutzwiller63etc; @Kanamori63]. The expectation was that in this model ferromagnetism would arise naturally since in a polarized state the electrons do not interact at all. However, it soon became clear that in a ferromagnetic state the [*[kinetic]{}*]{} energy is also reduced. This makes the stability of ferromagnetism in the Hubbard model a particularly delicate problem. Indeed, the kinetic energy with nearest-neighbor hopping usually favors [*[anti]{}*]{}ferromagnetism. At half-filling $(n = 1)$ and on bipartite lattices antiferromagnetism is a generic effect since it appears both at weak coupling (Hartree-Fock or Slater mean-field theory) and strong coupling (Anderson’s “superexchange” mechanism). Hence it arises naturally in any perturbational approach and, in particular, is tractable by renormalization group methods [@Shankar94]. By contrast, ferromagnetism is a non-trivial strong-coupling phenomenon which cannot be investigated by any standard perturbation theory.
The above discussion shows that, to understand the microscopic origin of itinerant ferromagnetism, non-perturbative techniques are required. Unfortunately, there are not many approaches of this type available; rigorous mathematical methods (for recent reviews see ref. [@Mielke93; @Lieb95; @Strack95]), large-scale numerical methods [@Gammel88etc; @Macedo92; @Hirsch94etc; @Hlubina96etc; @Daul97etc], and variational approaches [@Shastry90; @Fazekas90; @MuellerHartmann93; @Oles84; @Fulde95; @Buenemann96etc] are the ones most frequently used.
As to rigorous results about ferromagnetism in the Hubbard model the Lieb-Mattis theorem [@Lieb62] of 1962 is one of the most famous. It proves that for spin- and velocity-independent forces between electrons ferromagnetism cannot occur in one spatial dimension. This theorem applies to the general Hamiltonian (\[ham\]) with infinitely many bands. On the other hand the general [*single-*]{}band model (\[1band\]) is [*not*]{} covered by the theorem unless (a) the interaction matrix element ${\cal V}_{ijmn} $ is site-diagonal such that the interaction depends only on densities ${\hat n}_{i}$, and (b) the hopping and interaction does not extend beyond nearest neighbors. Hence the theorem applies to the model (\[nn\]) with $U , V \;{\ne}\; 0$ but $X,F,F'=0$, and thereby also to the Hubbard model $(V = 0)$. We note that any critique of the single-band model (\[nn\]) in view of the fact that it can lead to ferromagnetism in $d = 1$ in contrast to the Lieb-Mattis theorem would apply even more so to the Hubbard model since the latter is a particularly special single-band model. In other words: the fact that the Lieb-Mattis theorem applies to the Hubbard model but *not *to the *general *single-band model (\[nn\]) does not make the Hubbard model a “more physical” model than (\[nn\]); after all it is only a special case of (\[nn\]).****
Another well-known theorem, that by Nagaoka [@Nagaoka66] of 1966, provides explicit, albeit highly idealized, conditions under which ferromagnetism [*is*]{} stable in the Hubbard model with nearest-neighbor hopping. It proves that for $U = \infty$ the microscopic degeneracy of the ground state at half-filling (number of electrons $N$ = number of lattice sites $L$) is lifted by a single hole, i.e. when one electron is removed $(N = L - 1)$. In this case a saturated ferromagnetic ground state is stable for any value of the hopping $t$ on simple cubic and bcc lattices, and for $t < 0$ on fcc and hcp lattices. For the Nagaoka mechanism to work the lattice needs to contain loops along which the holes can move. Once the hole moves, the maximal overlap between the initial and final state clearly occurs in a [*ferromagnetic*]{} configuration. The problem is that Nagoaka’s proof does not even extend to two holes, that a single hole is thermodynamically irrelevant, and that the limit of $U=\infty$ is highly unrealistic.
Microscopic Mechanisms Favoring Ferromagnetism
==============================================
The Hubbard interaction is the result of an extreme truncation of the interaction in the general Hamiltonian (\[ham\]). All interactions beyond the purely local part (e.g. nearest-neighbor density-density interactions, direct exchange, band degeneracy and the associated Hund’s rule couplings) are totally neglected. The Hubbard interaction is therefore very unspecific — it does not depend on the lattice at all and hence not on the spatial dimension. The lattice structure only enters via the kinetic energy. Therefore the stability of ferromagnetism in the Hubbard model can be expected to depend in a sensitive way on the precise form of the kinetic energy [@Mielke93; @Lieb95; @Nagaoka66; @Tasaki95; @Penc96; @Fazekas96etc]. Strategies to find the essential “kick” for ferromagnetism in the Hubbard model and more general models should then proceed in different directions: one may (i) keep interactions beyond the Hubbard-$U$ (in particular the direct exchange term $F$ in (\[nn\])), (ii) keep band degeneracy and Hund’s rule couplings, or (iii) find an appropriate kinetic energy and lattice structure. We will now discuss several recent results obtained along these lines.
The importance of the Heisenberg exchange interaction
-----------------------------------------------------
The Heisenberg exchange interaction, caused by direct quantum-mechanical exchange of electrons at nearest-neighbor positions (the $F$-Term in (\[vnn\]) with $F>0$), favors the alignment of the electronic spins and hence supports ferromagnetism in a straightforward way [@Hirsch89etc; @Strack94; @Strack95]. However, since this interaction is rather weak (Hubbard [@Hubbard63etc] estimated $F\sim\frac{1}{40}$ eV for $3d$-metals, such that $F\ll U$) it cannot be the [*sole*]{} origin of itinerant ferromagnetism in systems like Fe, Co, Ni. Nevertheless it may be qualitatively important, since it may well give a correlated system with more or less strong ferromagnetic tendencies the ultimate push and trigger ferromagnetism in spite of its smallness. It is therefore unjustified to neglect the exchange interaction for merely [ *quantitative*]{} reasons. This becomes particularly clear in the limit of large $U$ (with $U\gg|t|$, $|V|$, $|X|$, $F$, $F'$) close to $n=1$, when the one-band model (\[nn\]) can be transformed into an effective $t$-$J$-model [@Gammel88etc; @Strack94etc] $$\begin{aligned}
\label{extendedtj}
\hat{H}^{\mbox{\scriptsize\em NN}}
_{\mbox{\scriptsize\em 1-band,$tJ$}}
&=&-t\!\!\sum_{\langle i,j\rangle,\sigma}
\!\!\hat{P}(\hat{c}^+_{i\sigma}
\hat{c}^{\phantom +}_{j\sigma}
+\mbox{h.c.})\hat{P}
+J \sum_{\langle i,j\rangle}\hat{\bf S}_{i}\hat{\bf S}_{j}
\end{aligned}$$ where $\hat{P}$ projects onto the subspace without doubly occupied sites. The effective exchange coupling $$\label{effectivej}
J=\frac{4t^2}{U}\Big[\Big(1-\frac{X}{t}\Big)^2-\frac{FU}{2t^2}\Big]$$ has an antiferromagnetic part, due to Anderson’s superexchange but modified by the $X$-term, and a ferromagnetic part, due to the direct Heisenberg exchange. Hence for large enough Heisenberg exchange $F$ and/or Hubbard repulsion $U$ the exchange becomes effectively ferromagnetic. This effect is completely neglected in the Hubbard model where even in the Nagaoka-limit ($U=\infty$) the dimensionless parameter $FU/t^2$ is kept zero!
### Generalization of Nagaoka’s theorem
If the Heisenberg exchange coupling is taken into account, it is possible to generalize Nagaoka’s theorem to $U<\infty$ [@Kollar96]. We start with the Hamiltonian $\hat{H}^{\mbox{\scriptsize\em
NN}}_{\mbox{\scriptsize\em 1-band}}$, (\[nn\]). This Hamiltonian, like the Hubbard Hamiltonian, commutes with the total spin $\hat{\bf
S}=\sum_i\hat{\bf S}_i$. The eigenvalues of $\hat{\bf S}^2$ are denoted by $S(S+1)$. We are concerned with saturated ferromagnetic states with one hole below half-filling, i.e. with largest possible eigenvalue $S_{\mbox{\scriptsize\em max}}\equiv
N/2=(L-1)/2$. There are $2S_{\mbox{\scriptsize\em
max}}+1=L$ such states with the same energy eigenvalue. It can be shown that the ground states of $\hat{H}_{\mbox{\scriptsize\em NN}}$ with one hole (i. e. $N=L-1$) have maximum total spin $S=S_{\mbox{\scriptsize\em max}}=(L-1)/2$ and are non-degenerate (apart from the above-mentioned $(2S_{\mbox{\scriptsize\em max}}+1)$-fold spin degeneracy) in the following cases [@Kollar96]:
Case 1: On any lattice, if $F>0$, $t\leq0$ and (a) $X\neq t$ and $U>U_c^{(1)}$, or (b) $X=t$ and $U\geq U_c^{(2)}$.
Case 2: On lattices with loops, if $X=t<0$, $F=0$, and $U>U_c^{(2)}$.
In both cases $t>0$ is allowed if the lattice is bipartite.
These results are summarized in Table \[crtab\]. The constants $U_c^{(1)}$ and $U_c^{(2)}$ are given by $$\begin{aligned}
\label{bounds1}
U_{c}^{(1)}&=&Z\Big(2|t|+\Big|V-F-2|t|\Big|
\nonumber\\
&+&{\frac{(X-t)^2}{F}}+
\Big|F'-{\frac{(X-t)^2}{F}}\Big|\Big)\,,\\
\label{bounds2}
U_{c}^{(2)}&=&Z\Big(2|t|+
\Big|V-{\frac{F}{2}}-2|t|\Big|+|F'|\Big)
\end{aligned}$$ where $Z$ is the number of nearest neighbors. Hence, if $F>0$ ferromagnetic ground states are stable on any lattice for $U$ larger than a [*finite*]{} critical value. For $F\to0^+$ we have $U_c^{(1)}\to\infty$, thus yielding Nagaoka’s condition for the pure Hubbard model. This shows once more that the Heisenberg interaction $F$, which is neglected in the Hubbard model, provides an obvious mechanism for stabilizing ferromagnetic ground states at finite $U$. Note that since $X$ and $t$ are expected to be of the same order of magnitude, the sensitive dependence on $F$, due to the term $(X-t)^2/F$, may cancel from $U_c^{(1)}$. The dependence of $U_c$ on $t,V,F$ is depicted in Fig. \[uvf\]. The case $X=t$ is special, since in this case the stability of ferromagnetism can be achieved either by $F>0$, or by $F\geq0$ and $t<0$ if the lattice has loops. Note that the case $F>0$ is [*not connected*]{} to the case $F=0$ by a limiting procedure, since only in the latter case the lattice is required to have loops.
The critical couplings $U_c^{(1)}$ and $U_c^{(2)}$ are sums of terms, each of which corresponds to a typical energy scale. This means that the on-site interaction $U$ has to be larger than the energy describing the paramagnetic state (bandwidth $\sim Z|t|$), as well as the threshold energies for the onset of a charge-density wave or phase separation ($\sim Z|V|$), $\eta$-pairing superconductivity [@deBoer95a] ($\sim
Z|F'|$), and a spin-density wave ($\sim(X-t)^2/F$). However, these terms do not enter separately, but appear in combinations, i. e. the effects interfere, as should be expected.
The above conditions are [*sufficient*]{} conditions. The occurrence of ground states with maximum spin outside the above parameter region is not ruled out. As in Nagaoka’s theorem for the pure Hubbard model, the ferromagnetic ground state is an itinerant state with non-zero kinetic energy, but the proof of its stability cannot yet be extended to doping beyond a single hole.
### Magnetic phase diagram within the dynamical mean-field theory
Even for the simplest electronic correlation model, the Hubbard model, exact solutions are not available in $d=2,3$ dimensions, and numerical methods — whether exact diagonalizations or Quantum Monte Carlo (QMC) techniques — are limited by the smallness of the systems that can be studied. Hence one would like to construct, at least, a thermodynamically consistent mean-field theory which is valid also at strong coupling. Such a (non-perturbative) approximation is provided by the exact solution of a model in $d=\infty$. It is now known that in the limit $d\to\infty$ [@Metzner89etc] one obtains a [ *dynamical*]{} mean-field theory for Hubbard-type models where the spatial dependencies become local, but all quantum fluctuations of the $d$-dimensional model are included [@MuellerHartmann89etc; @Janis91; @Georges92a; @Jarrell92; @Janis92a]. In fact the problem is equivalent to an Anderson impurity model complemented by a self-consistency condition [@Georges92a; @Jarrell92], leading to dynamical mean-field equations which can be solved numerically, e.g. within a finite-temperature QMC scheme [@Hirsch86]; for reviews see refs. [@Pruschke95; @Georges96]. We employed this numerical approach (for details see ref. [@Ulmke95a]) to investigate the influence of the direct exchange interaction $F$ on the stabilization of ferromagnetism in the Hubbard model, using the one-band Hamiltonian (\[nn\])-(\[vnn\]) with $X=F'=0$ [@Bluemer97etc]. To take the limit $d\to\infty$ the couplings in (\[nn\])-(\[vnn\]) have to be scaled appropriately [@Metzner89etc; @MuellerHartmann89etc], i.e. $$\label{scale}
t=\frac{t^*}{\sqrt{Z}},\ F=\frac{F^*}{Z},\ V=\frac{V^*}{Z}.$$ In the following we set $t^*\equiv 1$. In the limit $d\to\infty$ the $V^*$-term [@MuellerHartmann89etc] and $F^*$-term reduce to their Hartree-contributions. Hence their influence is that of a generalized, i.e. spin- and site-dependent, chemical potential. In the homogeneous phase the spin- and site-dependent terms vanish. This implies that the nearest-neighbor interactions $F^*$ and $V^*$ become important only in the symmetry-[*broken*]{} phase. Consequently the susceptibilities of the model (\[nn\])-(\[vnn\]) with $X=F'=0$ may be calculated from the pure Hubbard model [@Bluemer97etc]; this simplifies the matter considerably.
The phase boundaries between the paramagnetic, antiferromagnetic and ferromagnetic phases may then be calculated in principle as follows [@Bluemer97etc]: (i) QMC simulations are performed in the homogeneous phase of the pure Hubbard model for given $U$, temperature $T$, filling $n$, and number of Matsubara frequencies $\Lambda$; (ii) for arbitrary values of $F^*$, $V^*$ the appropriate susceptibilities are calculated; (iii) the inverse susceptibilities are extrapolated to $\Lambda\to\infty$, if they are negative the homogeneous phase is found to be unstable; (iv) to obtain ground state properties the calculated quantities have to be extrapolated to $T\to 0$. The results of these calculations for $n=1$, $T=0$ and a semi-elliptic DOS are collected in Fig. \[phaseT0\], where the exchange coupling $F^*$ is plotted versus the Hubbard interaction $U$. We neglect the density-density terms in (\[vnn\]), since they become important only for $V^*-F^*/2>U$ [@vanDongen91etc; @Bluemer97etc]. The solid line marks the phase boundary to the ferromagnetic phase. As expected the critical value of $F^*$, $F^*_c$, decreases with $U$: it depends on $U$ as $(F^*_c(0)-F^*_c(U))\propto U$ for small $U$ (Hartree-Fock limit) and as $F^*_c(U)\propto 1/U$ for large $U$ (Heisenberg limit). At $U=12$ the value of $F^*$ necessary to induce ferromagnetism is seen to be as small as $F^*_c\sim 0.1$. This shows how important even a weak exchange coupling $F$ is for the stability of ferromagnetism. For $U>4$ there may well be a direct transition between the antiferromagnetic and the ferromagnetic phases.
The shape of the DOS and the band filling $n$ are very important factors concerning the stability of ferromagnetism, as will be discussed below (Sec. 3.3). Indeed, a [*symmetric*]{} DOS and filling $n=1$, as used in the above calculation, are especially disadvantageous for ferromagnetism, partly because antiferromagnetism will be the generic magnetic order in this case.
Band degeneracy and Hund’s rule coupling
----------------------------------------
Another important route to ferromagnetism may be taken by considering more than one energy band, namely by starting from $M>1$ Wannier (or tight-binding) orbitals. It is known from atomic magnetism that there are ferromagnetic couplings between electrons on the same atom leading to Hund’s rules. These on-site “Hund’s rule couplings” express the fact that by putting electrons in a maximum spin state an atomic exchange energy may be gained by the following mechanism. A spin wave function with maximum spin is always symmetric. Therefore, for the total wave function to be antisymmetric, the coordinate wave function must be antisymmetric. This reduces the probability for electrons to come close to each other which in turn lowers the Coulomb energy between them. This consideration establishes that in general there will be ferromagnetic on-site interactions [*even in a bulk system.*]{} Whether the presence of these terms is sufficient for ferromagnetism to appear in the ground state is, however, strongly dependent on their relative strength compared with the kinetic energy, on the lattice structure, electron density, etc.
Let us therefore take a closer look at the terms in the Hamiltonian (\[ham\]) in the case of $M$ relevant bands. In this case we have to retain the band index $\alpha=1,\ldots M$. Now there exist important on-site interactions even beyond the Hubbard interaction $U={\cal
V}_{iiii}^{\alpha\alpha\alpha\alpha}$, namely the following couplings that are off-diagonal in the [*band*]{} indices, and are hence only present for $M>1$ bands: density-density interaction $V_0={\cal
V}_{iiii}^{\alpha\beta\alpha\beta}$, direct exchange interaction $F_0={\cal
V}_{iiii}^{\alpha\beta\beta\alpha}$, and hopping of double occupancies $F_0'={\cal
V}_{iiii}^{\alpha\alpha\beta\beta}$. For simplicity we assume the orbitals to be equivalent, i.e. $U$ is the same for all orbitals $\alpha$, and $V_0$, $F_0$, $F_0'$ each have a fixed value for all pairs of orbitals $\alpha$, $\beta$. Furthermore it should be noted that for equivalent orbitals these parameters are not independent, but are related by $U=V_0+2F_0$, $F_0=F_0'$ [@Oles83]. In addition to these Hund’s rule couplings there are still the inter-site terms, namely the hopping $t^{\alpha}_{ij}$ which takes place only between like orbitals (this follows from the general derivation above), and the next-neighbor interactions $V_1={\cal
V}_{ijij}^{\alpha\alpha\alpha\alpha}$, $X_1={\cal
V}_{iiij}^{\alpha\alpha\alpha\alpha}$, $F_1={\cal
V}_{ijji}^{\alpha\alpha\alpha\alpha}$, $F_1'={\cal
V}_{iijj}^{\alpha\alpha\alpha\alpha}$. For simplicity, these next-neighbor couplings are assumed to be diagonal in the band indices, i.e. they act only between like orbitals on neighboring sites. Finally, since we are dealing with equivalent orbitals, we assume that the next-neighbor parameters $t$, $X_1$, $F_1$, $F_1'$ each have the same value for all bands $\alpha$.
The resulting multi-band Hamiltonian then reads $$\label{mbham}
\hat{H}_{\mbox{\scriptsize\em M-band}}
^{\mbox{\scriptsize\em NN}}=
\sum_{\alpha=1}^{M}
\hat{H}^{\mbox{\scriptsize\em NN}}
_{\mbox{\scriptsize\em 1-band,$\alpha$}}
+\hat{H}_{\mbox{\scriptsize\em interband}}$$ where $$\begin{aligned}
\label{alphn}
\hat{H}^{\mbox{\scriptsize\em NN}}
_{\mbox{\scriptsize\em 1-band,$\alpha$}}
&=&
-t \sum_{\langle i,j\rangle,\sigma}
(\hat{c}^+_{i\alpha\sigma}
\hat{c}^{\phantom +}_{j\alpha\sigma}
+\mbox{h.c.})
\nonumber\\
&+&U \sum_{i\sigma}
\hat{n}_{i\alpha\downarrow}
\hat{n}_{i\alpha\uparrow}
+\sum_{\langle i,j\rangle}
\Big[
V_1 \hat{n}_{i\alpha}\hat{n}_{j\alpha}
\nonumber\\
&+&X_1 \sum_\sigma(\hat{c}^+_{i\alpha\sigma}
\hat{c}^{\phantom +}_{j\alpha\sigma}
+\mbox{h.c.})
(\hat{n}_{i\alpha-\sigma}+\hat{n}_{j\alpha-\sigma})
\nonumber\\
&+&F_1'(\hat{c}^+_{i\alpha\uparrow}
\hat{c}^+_{i\alpha\downarrow}
\hat{c}^{\phantom +}_{j\alpha\downarrow}
\hat{c}^{\phantom +}_{j\alpha\uparrow}
+\mbox{h.c.})
\nonumber\\
&-&2F_1 (\hat{\bf S}_{i\alpha}\hat{\bf S}_{j\alpha}
+\frac{1}{4}\hat{n}_{i\alpha}\hat{n}_{j\alpha})
\Big]
\end{aligned}$$ is a straightforward generalization of the one-band Hamiltonian (\[nn\]) to more than one band, and $$\begin{aligned}
\label{atham}
\hat{H}_{\mbox{\scriptsize\em interband}}&=&
\sum_{i;\alpha<\beta}\Big[
V_0 \hat{n}_{i\alpha}\hat{n}_{i\beta}
\nonumber\\
&-&2F_0 (\hat{\bf S}_{i\alpha}\hat{\bf S}_{i\beta}
+\frac{1}{4}\hat{n}_{i\alpha}\hat{n}_{i\beta})
\nonumber\\
&+&F_0' (\hat{c}^+_{i\alpha\uparrow}
\hat{c}^+_{i\alpha\downarrow}
\hat{c}^{\phantom +}_{i\beta\downarrow}
\hat{c}^{\phantom +}_{i\beta\uparrow}
+\mbox{h.c.})\Big]\,.
\end{aligned}$$ Several of the processes contained in the Hamiltonian (\[mbham\]) are illustrated in Fig. \[orb\].
The physical picture of bulk ferromagnetism, originally put forward by Slater, is the following. If the on-site Hund’s rule couplings are strong enough, they lead to an independent ferromagnetic alignment of spins on each atom. In this situation the kinetic energy plays a decisive role since it can serve to communicate the spin alignment across the solid. Indeed, if the alignment on neighboring atoms were different, the hopping of electrons from one atom to the next would generate on-site Hund’s rule interactions and thus increase the energy. These interactions can only be avoided if the spin alignment on neighboring atoms is the same, implying [ *global*]{} ferromagnetism. This mechanism for the stabilization of ferromagnetic order works all the better the larger the number $M$ of orbitals, i. e. bands, is onto which an electron can hop; it does not work for a single band ($M=1$). Therefore one should expect band degeneracy to favor ferromagnetism. This very qualitative picture is indeed found in Stoner mean-field theory [@Cyrot73]: Since the non-interacting DOS is proportional to the number of degenerate bands, the critical interaction decreases as $U_c^{\mbox{\scriptsize\em Stoner}}(M)\sim 1/M$. On the other hand, few [*exact*]{} results are known in the case of degenerate bands. For example, for a one-dimensional chain with two orbitals ($M=2$) and infinite on-site Coulomb interactions it has been shown that the ground state is ferromagnetic for $N=L+1$ electrons [@Lacroix76]. This statement has been extended to $L+1\leq
N\leq2L-1$, i.e. up to one electron less than half-filling [@Kubo82]. The important point is that on a one-dimensional chain with only one orbital and hopping between nearest neighbors, Nagaoka’s theorem is not applicable, since the lattice does not have loops. In that case the ground state is degenerate with respect to the total spin $S$. On the other hand, if there are two orbitals the loop property is restored, and the ferromagnetic states with maximum spin become the only ground states.
Furthermore, the following rigorous result can be established for the Hamiltonian (\[mbham\]) at half-filling [@Kollar97etc]. For $N=ML$ electrons, the ground states of $\hat{H}_{\mbox{\scriptsize\em
M-band}}$ have maximum spin $S=S_{\mbox{\scriptsize\em max}}=ML/2$ if $$2V_0\geq F_0\geq\frac{U_c^{(1,2)}}{1+M/2}
\;\;\;\mbox{and}\;\;\;F_1>0$$ where $U_c^{(1,2)}$ are the critical values for $U$ in the [*single*]{} band system, as given in (\[bounds1\])-(\[bounds2\]). The meaning of these bounds is the following. The requirement that $2V_0\geq F_0$ and $F_0$ be greater than a certain threshold leads to an alignment of the electronic spins on an isolated atom. On the other hand, ferromagnetism within each band is brought about by the next-neighbor exchange $F_1>0$ and the Hubbard interaction $U$ larger than a threshold related to $U_c^{(1,2)}$. The combination of these two effects (using the fact that $U=V_0+2F_0$ for equivalent bands) leads to a critical value for the Hund’s rule coupling $F_0$, which indeed becomes lower as the number of bands increases.
While this result contains some ideas of Slater’s picture, it does not explain the itinerant aspects of multi-band ferromagnetism, since at half-filling the ferromagnetic ground states are insulating. So far, this result can only be modified to apply also to Nagaoka’s case (one hole, $N=ML-1$) [@Kollar97etc].
Kinetic energy and lattice structure
------------------------------------
The stability of ferromagnetism is intimately linked with the structure of the underlying lattice and the kinetic energy (i.e. the hopping) of the electrons [@Hirsch94etc; @Hlubina96etc; @Daul97etc; @Shastry90; @Fazekas90; @MuellerHartmann93; @Nagaoka66; @Tasaki95; @Penc96; @Fazekas96etc]. This is supported by several facts: (a) Nagaoka’s proof of ferromagnetism in the Hubbard model for a single hole at $U=\infty$ [@Nagaoka66] depends on the existence of closed loops along which the hole can move, and (b) on bipartite lattices $anti$ferromagnetism is the generic magnetic state making it hard for ferromagnetism to become stable. Hence $non$-bipartite lattices with loops (or with a kinetic energy involving hopping between nearest $and$ next-nearest neighbors sites effectively leading to a motion on loops) should be expected to support ferromagnetism because the competing antiferromagnetic tendencies are severely weakened, and because the corresponding DOS of non-interacting electrons is asymmetric and thus has a peak at a non-symmetric position. Indeed, a peak at one of the band edges as in the case of the fcc lattice is favorable for ferromagnetism [@Gutzwiller63etc; @Macedo92; @Hirsch94etc; @Shastry90; @Fazekas90; @MuellerHartmann93]. This is supported by the observation [@MuellerHartmann93] that Co and Ni, having non-bipartite hcp and fcc lattice structure, respectively, show a full magnetization while bcc-Fe has only a partial magnetization.
### A model density of states
To gain insight into why a DOS with a peak (or more precisely with a high spectral density) at the band edge, e.g. at the lower band edge for $n<1$, may be favorable for ferromagnetism, we study non-interacting electrons with the following model DOS: $$N^0(E) = \frac{1}{\Delta} \left(1+\frac{1}{3}A^2+\frac{2A}{\Delta}E\right)$$ where $\Delta$ is the width of the band and $A$ parameterizes the asymmetry. The lower band edge is $-(A+3)\Delta/6$ such that the first moment of $N^0(E)$ vanishes. For $A=-1 (+1)$ the DOS has a triangular shape with the peak at the lower (upper) edge, while for $A=0$ it is flat. We wish to calculate the energy difference $\delta \epsilon$ between the fully polarized and the paramagnetic state $$\delta \epsilon \equiv \epsilon_{\mbox{\scriptsize\em ferro}}(n,A)-
\epsilon_{\mbox{\scriptsize\em para}}(n,A)$$ as a function of $A$ and the band filling. It is easy to confirm that for all $n$ $\delta \epsilon$ is lowest for $A=-1$. The reason is this: In the paramagnetic state $N$ non-interacting electrons ($N/2$ electrons with spin up and down each) fill the lowest $N/2$ $\vec k$-states up to an energy $E_F^{\mbox{\scriptsize\em para}}$, while in the ferromagnetic state the $N$ singly occupied $\vec k$-states below $E_F^{\mbox{\scriptsize\em ferro}}>E_F^{\mbox{\scriptsize\em para}}$ are filled, i.e. the $N/2$ $\vec k$-states with energy above $E_F^{\mbox{\scriptsize\em para}}$ are also occupied. The higher the DOS is at the lower edge the less the additional $N/2$ $\vec k$-states are forced into high-energy states, i.e. the lower $E_F^{\mbox{\scriptsize\em ferro}}$ will be. Then $\delta \epsilon$ is kept at a minimum.
### The Hubbard model on fcc type lattices
As explained in the previous section, a lattice structure which gives a high DOS at low energies is expected to be favorable for ferromagnetism (at $n<1$) because this situation reduces the loss in kinetic energy. The question remains: is a strongly peaked DOS *sufficient *to induce ferromagnetic order in the single band Hubbard model without additional interactions?**
In this section we discuss the results of a QMC-investigation of the stability of ferromagnetism in the single band Hubbard model in the limit of infinite dimensions.
Based on the considerations in Sec. 3.3.1, the fcc lattice is expected to be a good candidate for ferromagnetism because of its high (divergent) DOS at the lower band edge. The fcc-lattice can be generalized to higher dimensions in different ways [@MuellerHartmann91etc; @Uhrig96]. Here we use the definition of an fcc lattice as the set of all points with integer cubic coordinates summing up to an even integer [@MuellerHartmann91etc]. It is a non-bipartite Bravais lattice for any dimension $d>2$. For $d=2$ it is identical to the square lattice. Nearest neighbors are connected by *two different *unit vectors on a simple hypercubic (hc) lattice. The coordination number is hence $Z=2d(d-1)$. With the proper scaling of the hopping term, (\[scale\]), the non-interacting DOS of the generalized fcc lattice can be calculated in $d=\infty$ [@MuellerHartmann91etc] as: $$N^0_{\mbox{\scriptsize\em gfcc}}(E) = e^{-(1+\sqrt{2} E)/2}/\sqrt{\pi (1+\sqrt{2} E)}
\label{glferro1}$$ which has a strong square-root divergency at the lower band edge, $-1/\sqrt{2}$, and no upper band edge. Since we choose a *positive *hopping integral $(-t>0)$, $N^0_{\mbox{\scriptsize\em gfcc}}(E)$ might be regarded as the DOS of holes rather than electrons. A full polarization of holes would hence correspond to a *maximal *(though not full) polarization in a more than half-filled band.******
While the three dimensional fcc lattice has no square-root but only a logarithmic divergency at the band edge it is worth mentioning that a square-root divergency arises on the fcc lattice in any dimension if there is an additional next nearest neighbor hopping of the size $t'=t/2$ between sites that are linked by two unit vectors in the *same *direction on the hc lattice. The energy dispersion and the DOS of this so-called “half-hypercubic” (hh) lattice [@Uhrig96] can easily be obtained from the hc lattice as: $$\begin{aligned}
\epsilon_{hh}(k) & = & \frac{t}{2t_{hc}} \epsilon_{hc}^2(k) - 3t
\label{glferro2} \\
N^0_{hh}(E)
& = & \frac{2t_{hc}^2}{t} \frac{N^0_{hc}(\sqrt{E+3t})}{\sqrt{E+3t}} \; .
\label{glferro3}
\end{aligned}$$ In the limit $d\to\infty$, the hh and fcc lattices become equivalent.**
The fact that the fcc lattice provides a good ‘environment’ for ferromagnetism has also been supported by variational studies of the stability of the Nagaoka state [@Shastry90; @MuellerHartmann93]. Variational calculations provide limits for the critical density $n_c$ and the critical interaction $U_c$ where saturated ferromagnetism becomes unstable: While for the hc lattice in $d=\infty$ the stability regime shrinks to the point $U_c=\infty$, $\delta_c=0$ [@Fazekas90], for the fcc DOS (\[glferro1\]) there is a critical line $U_c(n)$ with $U_c(0)=0$ and $U_c(1)=\infty$ [@MuellerHartmann95etc]. The Nagaoka state is always unstable in the case of electron doping $(n>1)$.
Recently, these variational boundaries were qualitatively confirmed by Uhrig’s calculation of the exact single spin-flip energy of the Nagaoka state in $d=\infty$ [@Uhrig96]. While on the hh lattice $U_c$ vanishes at low densities, $U_c$ remains finite for all densities in the case of the “laminated” lattice which is a different generalization of the fcc lattice without a divergent DOS at the band edge. His results, too, emphasize the subtle dependence of the stability of ferromagnetism on the lattice structure.
Antiferromagnetism is not expected on the fcc-lattice in high dimensions because the difference of the numbers of not frustrated bonds and frustrated bonds is only of the order of $d$ resulting in an effective field of the order of $t^2 d\propto 1/d$ [@MuellerHartmann95etc]. Even in $d=3$ antiferromagnetism is frustrated and is expected only very close to half-filling.
To detect a ferromagnetic instability we calculated the temperature dependence of the uniform static susceptibility, $\chi_F$, from the two-particle correlation functions [@Ulmke95a]. At an intermediate interaction strength of $U=4$ we found the ferromagnetic response to be strongest around quarter filling ($n\simeq 0.5$). $\chi_F$ obeys a Curie-Weiss law (Fig. \[figferro1\]) and the Curie temperature $T_c$ can safely be extrapolated from the zero of $\chi_F^{-1}$ to a value of $T_c=0.051(2)$ at $n=0.58$. Below $T_c$ the magnetization $m$ grows rapidly, reaching more than 80% of the fully polarized value ($m_{max}=n=0.58$) at the lowest temperature which is only 30% below $T_c$. The three data points $m(T)$ (Fig. \[figferro1\]) are consistent with a Brillouin function with the same critical temperature of $T_c=0.05$ and an extrapolated full polarization at $T=0$. A saturated ground state magnetization is also consistent with the single spin-flip energy of the fully polarized state which is positive at the present parameter values [@Uhrig96].
Translated to the three dimensional fcc lattice with $Z=12$ nearest neighbors and a bandwidth of $W=16t$, the critical temperature becomes $T_c(3D)\approx 0.011 W$ [@U4comment]. Thus, despite the oversimplifications of the single band Hubbard model, the resulting Curie temperature has a realistic order of magnitude of 500-800K for typical values of $W$ around 5 eV.
In order to answer the question if the system is still metallic we also calculated the single particle spectrum [@Ulmke96etc]. We find that the system is metallic since both spectra have a finite value at the Fermi level.
We conclude that within the dynamical mean-field theory a DOS with sufficiently large spectral weight at low energies is able to induce itinerant ferromagnetism in the single band Hubbard model. Detailed investigations of the dependence of ferromagnetism on the electronic density, the interaction strength, and in particular the extension to more realistic densities of states (e.g. fcc lattice in $d=3$) is under progress and will be presented elsewhere [@Ulmke96etc].
In the $d=\infty$ limit the dynamics of the correlated system is fully taken into account, while spatial fluctuations are suppressed [@MuellerHartmann89etc; @Janis91; @Georges92a; @Jarrell92; @Janis92a]. One might therefore suspect that the stability of ferromagnetism is somehow $over$estimated in this approach (in particular since the rivaling antiferromagnetism is completely absent on the fcc lattice). However, most recently similar results were reported for the Hubbard model with nearest and next-nearest neighbor hopping $t$ and $t'$, respectively, in dimensions $d=1$ [@Daul97etc] and $d=2$ [@Hlubina96etc], which are consistent with the results in $d=\infty$. Hence the existence of itinerant ferromagnetism in the pure Hubbard model with a suitable kinetic energy seems to be established at last.
Metamagnetic phase transitions
==============================
An issue related to the stability of (itinerant) ferromagnetism in the Hubbard model is the question concerning metamagnetism [@Becquerel39] in this and other models. Here it is an external magnetic field $H$ which helps to suppress the (not necessary long-range) antiferromagnetic correlations in the system and thereby induces a pronounced transition from a state with low magnetization to one with high magnetization. At the metamagnetic transition the magnetization curve $m(H)$ shows an up-turn such that the susceptibility $\chi(H)=\partial m / \partial H$ has some kind of maximum. This feature serves as a convenient general definition for “metamagnetism”. Metamagnetic transitions were first observed in [ strongly anisotropic antiferromagnets]{} of which FeCl$_2$ and Dy$_3$Al$_5$O$_{12}$ (DAG) are well-studied prototypes [@Stryjewski77etc]. These materials are insulators where the valence electrons are localized at the Fe and Dy ions, respectively. The arising local moments order [antiferromagnetically]{} and are [ strongly anisotropic]{} in the sense that they are constrained to lie along an easy axis ${\mathbf e}$. In this case a spin-flop transition in an external magnetic field ${\mathbf H} \parallel {\mathbf e}$ cannot occur. Apart from the above materials there are also conducting systems that most probably belong to this class, e.g. the conductors UA$_{1-x}$B$_x$ (where $\mathrm A =P$, $\mathrm As$; $\mathrm B = S$, $\mathrm Se$) [@Stryjewski77etc], SmMn$_2$Ge$_2$ [@Brabers94] and TbRh$_{2-x}$Ir$_x$Si$_2$ [@Ivanov95].
Hitherto completely different theories are employed to describe metamagnetic phase transitions in these different, strongly anisotropic antiferromagnets. Investigations of localized systems are usually based on the Ising model, where more than one interaction has to be introduced to describe the experimentally observed first order phase transitions [@Lawrie84etc]. With antiferromagnetic coupling $J$ between the $Z$ nearest-neighbor (NN) spins and a ferromagnetic coupling $J'$ between the $Z'$ next-nearest-neighbors (NNN) one obtains $$H_{\mbox{\scriptsize\em Ising}} =
J \sum_{NN} S_iS_j - J' \sum_{NNN} S_i S_j - 2 H \sum_i S_i \; .
\label{isi}$$ In Weiss mean-field theory this model shows two different types of phase diagrams depending on the parameter $R \equiv Z'J'/(ZJ)$ [@Kincaid74etc]. For $R>3/5$ the first and second order phase transition line join smoothly at the same point, producing a tricritical point (TCP), while for $R<3/5$ there is no common endpoint (see Fig. \[scheme\]). However the scenario of Fig. \[scheme\]b was not found when evaluating (\[isi\]) beyond mean-field theory [@Herrmann93etc].
For [*itinerant electron metamagnetism*]{} (IEM) Moriya and Usami [@Moriya77] proposed a Landau theory, where the parameters have to be deduced from microscopic models. The idea is to calculate first the independent electron band structure and to introduce the Coulomb interaction within the random phase approximation.
It is our purpose to investigate the origin of metamagnetism in strongly anisotropic antiferromagnets from a microscopic, quantum-mechanical point of view, and to describe different kinds of metamagnets (i.e. metallic [*and*]{} insulating, band-like [*and*]{} localized systems) qualitatively within a single model. To this end we study the Hubbard model (\[hub\]) with the additional constraint that the antiferromagnetic magnetization ${\mathbf m}_{st}$ lies [*parallel*]{} to the external magnetic field ${\mathbf H}$ [@Giesekus93etc]. In this way the existence of an [ easy axis]{} ${\mathbf e}$ along which ${\mathbf H}$ is directed, such that ${\mathbf e} \parallel
{\mathbf m}_{st} \parallel {\mathbf H}$, is incorporated in a natural way. By this approach, both kinetic energy and Coulomb interaction are captured microscopically, whereas the relativistic corrections (responsible for the easy axis) are not. This procedure is justified since the relativistic corrections are of ${\cal O}(10^{-2}$ eV) and are thus small compared to the kinetic and Coulomb energy which are of ${\cal O}(1$eV). Therefore the existence of an anisotropy axis ${\mathbf e}$ and the correlations described by the Hubbard model are quite unrelated. Note, that the existing Ising and IEM theories do not treat the kinetic energy and Coulomb interaction microscopically, but within an effective model.
A perturbative treatment of the Hubbard model with easy axis in the weak and strong coupling limit shows that the appearance of a tri- or multicritical point is a delicate matter, since neither of these two limits is able to describe a change of the transition from first to second order [@Held96]. Apparently the entire transition scenario depends sensitively on the value of the electronic on-site interaction $U$. To study this point in greater detail we have to go to intermediate coupling. In this non-perturbative regime we employ again QMC simulations to calculate the magnetization $m(H)$ and the staggered magnetization $m_{st}(H)$ of the Hubbard model (\[hub\]) in $d=\infty$ [@Held96]. As the results do not much depend on the precise form of the density of states we choose $N^0(\epsilon) = [(2t^*)^2 - \epsilon^2 ]^{1/2}/(2 \pi t^{*2})$, setting $t^* \equiv 1$ in the following. All calculations are performed at half-filling.
The results for $m(H)$ and $m_{st}(H)$ are used to construct the $H-T$ phase diagram at $U=4$ (Fig. \[hvst\]). It displays all the features of Fig. \[scheme\]b. In particular, the first order line continues [*into*]{} the ordered phase, separating two different AF phases: AF$_I$ (where $ m \simeq 0$) and AF$_{II}$ (where $m > 0$). The position of its endpoint cannot, at present, be determined accurately (dotted line). This phase diagram is surprisingly similar to the experimental phase diagram of FeBr$_2$ [@Azevedo95etc].
A change in $U$ and the filling $n$ will affect the phase diagram quantitatively and qualitatively. These results will be reported elsewhere [@Held97etc].
Conclusion
==========
In the last few years, and especially most recently, considerable progress was made in our understanding of the microscopic origin of itinerant ferromagnetism. These results were obtained on the basis of well-defined lattice models of correlated electrons, of which the one-band Hubbard model is a particularly important ingredient, by applying new, non-perturbative techniques, ranging from rigorous to large-scale numerical methods. There exists convincing evidence now that on appropriate, non-artificial lattices, or for an appropriate kinetic energy, itinerant ferromagnetism is stable even in the pure Hubbard model, for electronic densities not too close to half-filling and large enough $U$. Important ingredients are:\
(i) lattices with loops (or a kinetic energy allowing for motion on loops, e.g. with $t,t'$ hopping) such that the Nagaoka mechanism works and antiferromagnetism is suppressed,\
(ii) a large spectral weight near the band edge. (We note that this condition goes far beyond the mean-field Stoner criterion for ferromagnetism, $UN(0)=1,$ where $N(0)$ is the DOS at the Fermi energy.)
The direct exchange interaction, as well as band degeneracy, will strongly reduce the critical value of $U$ above which ferromagnetism becomes stable.
Furthermore, using the dynamical mean-field theory to solve the Hubbard model with easy-axis in the intermediate coupling regime, it appears to be possible to describe various, different forms of metamagnetism within a single microscopic theory.
The development and application of controlled, non-perturbative techniques will continue to be of particular importance for the investigation of correlated electron systems.
Acknowledgement {#acknowledgement .unnumbered}
===============
One of us (DV) thanks Elliott Lieb for very useful discussions. In its early stages this work was supported in part by the Deutsche Forschungsgemeinschaft under SFB 341.
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[lll]{} 1a&$\!\!F>0$, $X\neq t$, $U>U_c^{(1)}$
----------
any $t$
$t\leq0$
----------
-------------------
bipartite lattice
non-bip. lattice
-------------------
\
1b&$\!\!F>0$, $X=t$, $U\geq U_c^{(2)}$
----------
any $t$
$t\leq0$
----------
-------------------
bipartite lattice
non-bip. lattice
-------------------
\
2&$\!\!F=0$, $X=t$, $U>U_c^{(2)}$
----------
$t\neq0$
$t<0$
----------
-----------------------
bipartite, with loops
non-bip., with loops
-----------------------
\
|
---
abstract: 'We consider an experimentally realizable scheme for manipulating quantum states using a general superposition of products of field annihilation ($\hat{a}$) and creation ($\hat{a}^\dag$) operators of the type ($s \hat{a}\hat{a}^\dag+ t \hat{a}^\dag \hat{a}$), with $s^2 + t^2 = 1$. Such an operation, when applied on states with classical features, is shown to introduce strong nonclassicality. We quantify the generated degree of nonclassicality by the negative volume of Wigner distribution in the phase space and investigate two other observable nonclassical features, sub-Poissonian statistics and squeezing. We find that the operation introduces negativity in the Wigner distribution of an input coherent state and changes the Gaussianity of an input thermal state. This provides the possibility of engineering quantum states with specific nonclassical features.'
address: |
$^1$School of Physical Sciences, Jawaharlal Nehru University, New Delhi 110067, India\
$^2$School of Natural Sciences, Shiv Nadar University, Gautam Budh Nagar, UP 203207, India
author:
- 'Arpita Chatterjee$^1$, Himadri Shekhar Dhar$^1$ and Rupamanjari Ghosh$^{1,2}$'
title: Nonclassical properties of states engineered by superpositions of quantum operations on classical states
---
(Some figures in this article are in colour only in the electronic version)
[*Keywords*]{}: quantum state engineering, nonclassical states, volume of negativity, sub-Poissonian statistics, squeezing
Introduction {#sec1}
============
In recent years, the generation and manipulation of nonclassical states of the electromagnetic field have gained much importance in active research in quantum optics and quantum information theory [@pieter; @braunstein05], in the context of physical realization of quantum tasks, protocols and communications [@zeilinger98; @bouwmeester98; @bouwmeester00; @nielsen00] using continuous quantum variables. Various methods of manipulation at the single-photon level have been suggested for the preparation of nonclassical states of the optical field, based on operations of photon addition ($\hat{a}^\dag$) [@zavatta04] and subtraction ($\hat{a}$) [@wenger04] on a classical field. Agarwal and Tara [@agarwal91] first proposed an $m$-photon-added scheme to create a nonclassical state from any classical state. Zavatta *et al* [@zavatta05] demonstrated a single photon-added coherent state by homodyne tomography technology. A remarkable development has been the experimental realization of a general scheme by Zavatta *et al* [@zavatta09], based on single-photon interference, for implementing superpositions of distinct quantum operations. Hu *et al* [@hu10] have recently investigated the nonclassical properties of the field states generated by subtracting any number of photons from a squeezed thermal state. Lee and Nha [@lee10] have studied the action of an elementary coherent superposition of $\hat{a}$ and $\hat{a}^\dag$ on continuous variable systems.
We wish to consider a general superposition of the two product (SUP) operations, $s\hat{a}\hat{a}^\dag + t\hat{a}^\dag \hat{a}$ on a classical state, where $s$ and $t$ are scalars with $s=\sqrt{1-t^2}$. This operation can be realized experimentally under suitable modification of the interference set-up proposed by Kim *et al* [@kim08_2]. The basic unit for photon addition is a twin-photon source based on the nonlinear optical process of parametric down-conversion [@PDC]. The nonclassicality in the SUP operated states can be quantified and analyzed using quasiprobability distributions in the phase space. The scalars $s$ and $t$ act as control parameters for manipulation of the nonclassical character of the output state.
We observe that the SUP operation introduces nonclassicality in the classical *coherent* state and there is a finite negativity in the Wigner distribution which is analyzed for different scalar parameters. The nonclassical features of the SUP operated coherent state can be further analyzed using observable features such as squeezing and sub-Poissonian statistics. In case of input *thermal* state, no negativity of the Wigner function is observed. The SUP operation introduces non-Gaussianity in the classical thermal state. The nonclassical characteristics of the input thermal state is highlighted by its sub-Poissonian distribution and squeezing effect. This paper is organized as follows. We begin by outlining a practical SUP operation scheme in section \[sec2\]. Various nonclassicality indicators used in our study are defined in section \[sec3\]. In section \[sec4\], we present the results on SUP operated coherent and thermal states, in terms of the Wigner distribution function, a general operator-ordering parametrized quasiprobability function, Mandel’s $Q$ parameter and also the quadrature squeezing parameter. The last section contains a summary of our main results.
SUP operation scheme {#sec2}
====================
The generation of the desired quantum operation, $s\hat{a}\hat{a}^\dag + t\hat{a}^\dag \hat{a}$, where $\hat{a}$ and $\hat{a}^\dag$ are the annihilation and the creation operators, respectively, and $s$ and $t$ are scalars with $s=\sqrt{1-t^2}$, involves proper sequencing of photon subtraction ($\hat{a}$) and photon addition ($\hat{a}^\dag$) operators [@zavatta09; @kim08_2], and then coherent superposition of the ordered products by removing which-path information between them.
The schematic of the experimental proposal for the generation of the quantum operation $s(\hat{a}\hat{a}^\dag)+t(\hat{a}^\dag \hat{a})$ is shown in figure \[fig1\]. High transmissivity beam-splitters B1 and B2 are used for photon subtraction. When an input field is incident upon a high transmissivity beam-splitter, with the other input in a vacuum mode, detection of a photon in the photo-detector implies that a photon has been subtracted ($\hat{a}$) from the incident state. The parametric down-converter (PDC) is used to add photons. A PDC produces twin photons into two different modes. If an incident field is passed through a PDC, with the other input in a vacuum state, the detection of a photon in the detector would imply that a photon (the undetected twin) has been added ($\hat{a}^\dag$) to the incident field. The operation of this optical scheme is dependent on the photo-detectors, P1, P2 and P3, which detect the success of the addition or subtraction process in an optical path. M is a highly reflective mirror. The variable transmissivity beam-splitter B3 is used to generate the desired superposition of the product states by removing the which-path information.
As shown in the schematic in figure \[fig1\], the input state $|\psi\rangle_{\mathrm{in}}$ is incident upon B1. The subtraction of a photon at B1 will lead to a photon-subtracted state along path I and a photon along path II after reflection from M. In the absence of B3, the photon can be detected at P2. No detection at P2 would ensure that no subtraction has taken place at B1. The input field then proceeds to the PDC. The detection of a photon at P1, along path III, confirms the addition of a photon due to parametric down-conversion. Hence the simultaneous detection of a photon at P1 and P2 (in the absence of B3) ensures the operation $\hat{a}\hat{a}^\dag$. The photon-added field then proceeds to B2. The subtraction of a photon at B2 leads to a photon subtracted state along path I and a photon along path IV, which would be detected by P3 in the absence of B3. Hence detection at P1 and P3 (in the absence of B3), with no detection at P2 [@note], ensures the operation $\hat{a}^\dag \hat{a}$. No detection at P3 would ensure that no subtraction has taken place at B2. The final output state can either be $\hat{a}\hat{a}^\dag$ or $ \hat{a}^\dag \hat{a}$ depending on the detection at P2 or P3 respectively. Hence, the two paths of detection II and IV are the two product operation indicators. The beam-splitter B3 removes this path information, and produces a superposition of the two paths and hence of the two operations.
The generation of the superposed product states can be shown mathematically using standard operators for the various paths involved in the scheme. If $|\psi\rangle_{\mathrm{in}}$ (mode I) is incident upon a high transmissivity beam-splitter B1 (transmissivity, $t_1$ $\simeq$ 1), with the other input in vacuum mode (mode II), we obtain $$\hat{B}_{B1}|\psi\rangle_{\mathrm{in,I}}|0\rangle_{\mathrm{II}} \simeq (1-\frac{r_1^*}{t_1}\hat{a}_{\mathrm{I}} \hat{a}^\dag_{\mathrm{II}})|\psi_{\mathrm{in}}, 0\rangle_{\mathrm{I,II}} .$$ The state is now incident upon a PDC with a small coupling constant $g$ and the other input in vacuum mode (III). The resulting operation can be written as $$\begin{aligned}
&& (1-g \hat{a}_{\mathrm{I}} \hat{a}^\dag_{\mathrm{III}})\hat{B}_{B1}|\psi_{\mathrm{in}}, 0, 0\rangle_{\mathrm{I,II,III}} \nonumber \\ && \simeq (1 - g \hat{a}_{\mathrm{I}} \hat{a}^\dag_{\mathrm{III}} - \frac{r_1^*}{t_1} \hat{a}_{\mathrm{I}} \hat{a}^\dag_\mathrm{{II}} + g\frac{r_1^*}{t_1} \hat{a}_{\mathrm{I}} \hat{a}^\dag_{\mathrm{I}} \hat{a}^\dag_{\mathrm{II}} \hat{a}^\dag_{\mathrm{III})}|\psi_{\mathrm{in}},0,0\rangle_{\mathrm{I,II,III}} .\end{aligned}$$ The photon addition occurs only when a photon is created in mode III at the PDC. Hence the state producing one photon, detected by P1, corresponds to $$(-g \hat{a}^\dag_{\mathrm{I}} + g\frac{r^*_1}{t_1} \hat{a}^\dag_{\mathrm{I}} \hat{a}_{\mathrm{I}} \hat{a}^\dag_{\mathrm{II}})|\psi_{\mathrm{in}},0\rangle_{\mathrm{I,II}}$$ The state is then incident on the second high transmissivity beam-splitter B2 (transmissivity, $t_2$ $\simeq$ 1), with the other input in vacuum mode (IV). The operation leads to $$\begin{aligned}
&&(1-\frac{r_2^*}{t_2}\hat{a}_{\mathrm{I}} \hat{a}^\dag_{\mathrm{IV}})(-g \hat{a}^\dag_{\mathrm{I}} + g \frac{r^*_1}{t_1} \hat{a}^\dag_{\mathrm{I}} \hat{a}_{\mathrm{I}} \hat{a}^\dag_\mathrm{{II}})~|\psi_{\mathrm{in}},0, 0\rangle_{\mathrm{I,II,IV}} \nonumber \\
&& \simeq (-g \hat{a}^\dag_{\mathrm{I}} + g \frac{r^*_1}{t_1} \hat{a}^\dag_{\mathrm{I}} \hat{a}_{\mathrm{I}} \hat{a}^\dag_{\mathrm{II}} - g \frac{r_2^*}{t_2}\hat{a}_{\mathrm{I}} \hat{a}^\dag_{\mathrm{I}} \hat{a}^\dag_{\mathrm{IV}} - g \frac{r_1^*}{t_1}\frac{r_2^*}{t_2}\hat{a}_{\mathrm{I}} \hat{a}^\dag_{\mathrm{I}} \hat{a}_{\mathrm{I}} \hat{a}^\dag_{\mathrm{II}} \hat{a}^\dag_{\mathrm{IV}})|\psi_{\mathrm{in}},0, 0\rangle_{\mathrm{I,II,IV}}.\end{aligned}$$ In the absence of beam-splitter B3, photon detection of mode II at P2 (along with detection at P1 and no detection at P3) leads to the state $(g\frac{r^*_1}{t_1} \hat{a}^\dag_{\mathrm{I}} \hat{a}_{\mathrm{I}}) |\psi\rangle_{\mathrm{in}}$ and photon detection of mode IV at P3 (along with detection at P1 and no detection at P2) leads to the state $(- g \frac{r_2^*}{t_2}\hat{a}_{\mathrm{I}} \hat{a}^\dag_{\mathrm{I}} |\psi\rangle_{\mathrm{in}})$. Hence we can obtain the product state $\hat{a}^\dag_{\mathrm{I}} \hat{a}_{\mathrm{I}}$ ($\hat{a}_{\mathrm{I}} \hat{a}^\dag_{\mathrm{I}}$) using consecutive photon addition (subtraction) and subtraction (addition). Finally, we can use the beam-splitter B3 with transmissivity $t_3$ and reflectivity $r_3$ to produce the superposition state. The operation of the beam-splitter can be represented by the transformations $\acute{b} = t_3 b + r_3 c$, and $\acute{c} = t_3^*c - r_3^*b$, where $b$ and $c$ ($\acute{b}$ and $\acute{c}$) are the input (output) modes of the beam-splitter. Using the above relations, the superposition states we obtain are: $$(g t_3 \frac{r^*_1}{t_1} \hat{a}^\dag_{\mathrm{I}} \hat{a}_{\mathrm{I}} - r_3 g \frac{r_2^*}{t_2} \hat{a}_{\mathrm{I}} \hat{a}^\dag_{\mathrm{I}}) |\psi\rangle_{\mathrm{in}} ,$$ $$(-g r_3^* \frac{r^*_1}{t_1} \hat{a}^\dag_{\mathrm{I}} \hat{a}_{\mathrm{I}} - t_3^* g \frac{r_2^*}{t_2}\hat{a}_{\mathrm{I}} \hat{a}^\dag_{\mathrm{I}}) |\psi\rangle_{\mathrm{in}} ,$$ which can be conveniently cast in the general form $(s\hat{a}\hat{a}^\dag+t\hat{a}^\dag \hat{a})|\psi\rangle_{\mathrm{in}}$.
Nonclassicality indicators {#sec3}
==========================
*Wigner function*:The nonclassicality of a quantum state can be studied in terms of its phase-space distribution characterized by the Wigner distribution. For a quantum state $\hat{\rho}$, the Wigner function of the system is defined in terms of the coherent state basis [@scully97] as $$W(\beta, \beta^*) = \frac{2}{\pi^2}e^{2|\beta|^2} \int d^2\gamma{\langle-\gamma|\hat{\rho}|\gamma\rangle e^{-2(\beta^*\gamma-\beta\gamma^*)}},$$ where $|\gamma\rangle=\exp(-|\gamma|^2/2+\gamma \hat{a}^\dag)|0\rangle$ is a coherent state. By using the relation [@abramowitz72] $$\sum_{n=k}^\infty n_{C_k}\,y^{n-k} = (1-y)^{-k-1},$$ the Wigner function can be expressed in series form as [@moyacessa93] $$W(\beta, \beta^*) = \frac{2}{\pi} \sum_{k=0}^\infty (-1)^k \langle \beta,k|\hat{\rho}|\beta,k\rangle ,
\label{eq7}$$ where $|\beta,k\rangle$ is the usual displaced number state.
The partial negative value of the Wigner function is a one-sided condition for the nonclassicality of the related state [@wang64], in the sense that one cannot conclude the state is classical when the Wigner function is positive everywhere. For example, the Wigner function of the squeezed state is Gaussian and positive everywhere but it is a well-known nonclassical state. For a classical state, a necessary but not sufficient condition is the positivity of the Wigner function. Hence a state with a negative region in the phase-space distribution is essentially nonclassical.
We may consider a generalized distribution function, viz. a parametrized quasiprobability function $\mho^{(F)}(\beta)$ describing a field state $\hat{\rho}$, defined as [@cahill] $$\begin{aligned}
\label{s}
\mho^{(F)}(\beta) \equiv \frac{1}{\pi}\mathrm{Tr}\{\hat{\rho}\hat{T}^{(F)}(\beta)\},\end{aligned}$$ where the operator $\hat{T}^{(F)}(\beta)$ is given by $\hat{T}^{(F)}(\beta) = \frac{1}{\pi}\int \exp(\beta \xi^*-\beta^*\xi)\hat{D}^{(F)}(\xi)d^2\xi$, with $\hat{D}^{(F)}(\xi) = e^{F|\xi|^2/2}\hat{D}(\xi)$ and $\hat{D}(\xi) = e^{\xi {\hat{a}}^\dag-\xi^* \hat{a}}$. The function $\mho^{(F)}(\beta)$ can be rewritten in the number-state basis as $\mho^{(F)}(\beta) = \frac{1}{\pi}\sum_{n,m}\rho(n,m)\langle n|\hat{T}^{(F)}(\beta)|m\rangle$, where the matrix elements of the operator $\hat{T}^{(F)}(\beta)$ are given by $$\begin{aligned}
\nonumber
\label{s1}
\langle n|\hat{T}^{(F)}(\beta)|m\rangle & = & \left(\frac{n!}{m!}\right)^{1/2}\left(\frac{2}{1-F}\right)^{m-n+1}\left(\frac{F+1}{F-1}\right)^n{\beta^*}^{m-n}\\
& & \times\exp\left(-\frac{2|\beta|^2}{1-F}\right)L_n^{m-n}\left(\frac{4|\beta|^2}{1-F^2}\right),\end{aligned}$$ in terms of the associated Laguerre polynomials $L_n^{m-n}(x)$. The above equation gives explicitly the $F$-dependence of $\mho^{(F)}(\beta)$. For the special values of $F = 1, 0$ and $-1$, $\mho^{(F)}(\beta)$ becomes the Glauber-Sudarshan $P$ [@GS], the Wigner $W$ and the Husimi $Q$ [@Husimi] functions, respectively. The negativity of $\mho^{(F)}(\beta)$ for any value of the parameter $F$ indicates nonclassical nature of the state.
The nonclassical nature of a positive Wigner function can be determined using other features of the state, such as sub-Poissonian statistics and quadrature squeezing. These features, discussed later, can be attributed to the negative values which arise due to the dispersion of normally-ordered observables but are not captured by the Wigner function [@semenov]. In such cases, the nonclassicality is often manifested by the negativity of the $F$-parametrized distribution. *Negative Volume*: A good indicator of nonclassicality of quantum states was defined by Kenfack *et al* [@kenfack04]. It measures the volume of the integrated negative part of the Wigner function as $$V = \int\int d^2\beta |W(\beta, \beta^*)|-1.
\label{nv}$$ By definition, this quantity $V$ is equal to zero only when the state under consideration has non-negative Wigner function. For a classical system, the Wigner distribution is positive, and the integration $\int\int d^2\beta |W(\beta, \beta^*)|$=1. For a negative Wigner function of a quantum state, the absolute value of the Wigner function can be calculated, and the above integration can be evaluated numerically.
*Sub-Poissonian statistics: Mandel’s Q parameter*: The quantum character of a field can be demonstrated either in measurements of time intervals $\tau$ between detected photons demonstrating antibunching, or in photon counting measurements yielding sub-Poissonian statistics. The condition for sub-Poissonian photon statistics is given by $\langle (\Delta \hat{n})^{2} \rangle - \langle \hat{n} \rangle < 0$, which makes the normalized second-order intensity correlation function, $\gamma (0) < 1$. The states with sub-Poissonian statistics have no classical description. To determine the photon statistics of a single-mode radiation field, we consider Mandel’s $Q$ parameter defined by [@mandel79] $$Q \equiv \frac{\langle \hat{a}^{\dag 2} \hat{a}^2 \rangle - \langle \hat{a}^\dag \hat{a}\rangle^2}{\langle \hat{a}^\dag \hat{a} \rangle} .$$ $Q=0$ stands for Poissonian photon statistics. $Q<0$ corresponds to the case of sub-Poissonian distribution. This means that a nonclassical state often shows negative $Q$ values. *Squeezing*: The quadrature squeezing of a field can be used to study its nonclassical properties. To analyze the squeezing properties of the radiation field, we introduce two hermitian quadrature operators $$\hat{X}=\hat{a}+\hat{a}^\dag,~~~~~~\hat{Y}=-i(\hat{a}-\hat{a}^\dag).$$ These two quadrature operators satisfy the commutation relation $[\hat{X}, \hat{Y}]=2i$, and, as a result, the uncertainty relation $(\Delta \hat{X})^2(\Delta \hat{Y})^2\geq 1$. A state is said to be squeezed if either $(\Delta \hat{X})^2$ or $(\Delta \hat{Y})^2$ is less than its coherent state value. To review the principle quadrature squeezing [@luks88], we define an appropriate quadrature operator [@wang03] $$\begin{aligned}
\hat{X}_\theta = \hat{X}\cos\theta+\hat{Y}\sin\theta = \hat{a}e^{-i\theta}+\hat{a}^\dag e^{i\theta}.\end{aligned}$$ The squeezing of $\hat{X}_\theta$ is characterized by the condition $\langle:(\Delta \hat{X}_\theta)^2:\rangle<0$, where the double dots denote the normal ordering of operators. After expanding the terms in $\langle:(\Delta \hat{X}_\theta)^2:\rangle$ and minimizing its value over the whole angle $\theta$, one gets [@lee10] $$\begin{aligned}
S_{\mathrm{opt}} & = & \langle:(\Delta \hat{X}_\theta)^2:\rangle_{\mathrm{min}} \nonumber \\
& = & -2|\langle \hat{a}^{\dag 2}\rangle-\langle \hat{a}^\dag\rangle^2|+2\langle \hat{a}^\dag \hat{a}\rangle-2|\langle \hat{a}^\dag \rangle|^2 .
\label{eq14}\end{aligned}$$ The nonclassical states correspond to the negative values of $S_{\mathrm{opt}}$, $-1\leq S_{\mathrm{opt}}<0$.
The negativity of the $Q$ function and squeezing are not necessary conditions for identifying nonclassical regimes of quantum states but are sufficient ones. In different regimes of the scalars $s$ and $t$, the nonclassicality of the SUP operated states are exhibited via these indicators. As mentioned earlier, the $Q$ function and the squeezing parameter $S$ can be negative and hence nonclassical for states that have a positive Wigner function, as shown in [@semenov; @cessa; @janszky96]. Conversely, there are also instances where states with partial negative Wigner functions have positive $Q$ functions [@OC].
Results {#sec4}
=======
SUP operated coherent state
---------------------------
Let the density matrix of an arbitrary quantum input state of the single-mode radiation field be $$\hat{\rho}_{\mathrm{in}} = \sum_{m=0}^\infty \sum_{n=0}^\infty \rho(m, n)|m\rangle\langle n|.$$ The output state, produced by applying the SUP operator ($s\hat{a}\hat{a}^\dag + t\hat{a}^\dag \hat{a}$) on $\hat{\rho}_{\mathrm{in}}$, is given by $$\hat{\rho}_{\mathrm{out}} = \frac{1}{N}[s(\hat{a}\hat{a}^\dag)+t(\hat{a}^\dag \hat{a})]\hat{\rho}_{\mathrm{in}}[s(\hat{a}\hat{a}^\dag)+t(\hat{a}^\dag \hat{a})],
\label{eq2}$$ where $N$ is the normalization constant.
If we consider our input single-mode radiation field to be in a coherent state, $$|\alpha\rangle =\exp\left(\frac{-|\alpha|^2}{2}\right)\sum_{n=0}^{\infty} \frac{\alpha^{n}}{\sqrt{n!}}|n\rangle ,$$ where $|\alpha|^2$ is the average photon number, the SUP operated coherent state (SOCS) is given by $$\hat{\rho}_{\mathrm{coh}} = N_1^{-1}[s(\hat{a}\hat{a}^\dag)+t(\hat{a}^\dag \hat{a})]|\alpha\rangle\langle\alpha|[s(\hat{a}\hat{a}^\dag)+t(\hat{a}^\dag \hat{a})] ,
\label{eq3}$$ where $N_1 = s^2+(s+t)(3s+t)|\alpha|^2+(s+t)^2|\alpha|^4$ is the normalization constant.
To analyze the nonclassicality of the SOCS, we need to obtain the phase-space distribution of the density matrix in terms of the Wigner function. The expression for the Wigner function in the series form is given in (\[eq7\]). For SOCS density matrix, the displaced number state expectation value is given by the relation $$\begin{aligned}
& & \langle\beta,k|(s\hat{a}\hat{a}^\dag+t\hat{a}^\dag \hat{a})|\alpha\rangle \nonumber \\ & & = \langle k|D^\dag(\beta)(s\hat{a}\hat{a}^\dag+t\hat{a}^\dag
\hat{a})|\alpha\rangle \nonumber \\ & & = \langle k|\{s(\hat{a}+\beta)(\hat{a}^\dag+\beta^*)+ t(\hat{a}^\dag+\beta^*)(\hat{a}+\beta)\}D^\dag(\beta)|\alpha \rangle .
\label{eq8}\end{aligned}$$ Substituting the above expression (\[eq8\]) into the general expression (\[eq7\]), the Wigner function for SOCS is obtained as $$\begin{aligned}
W_{\mathrm{SOCS}}(\beta, \beta^*) & = & W_{\mathrm{coh}}(\beta, \beta^*) N_1^{-1}\big[{M_1}^2+2(s+t)M_1(\alpha^*\beta+\alpha\beta^*) \nonumber \\
& & +(s+t)^2|\alpha|^2(4|\beta|^2-1)\big] ,\end{aligned}$$ where $W_{\mathrm{coh}}(\beta, \beta^*)=\frac{2}{\pi} e^{-2|\beta-\alpha|^2}$ is the Wigner function of the input coherent state $|\alpha\rangle$,$M_1=s-(s+t)|\alpha|^2$, $N_1 = s^2+(s+t)(3s+t)|\alpha|^2+(s+t)^2|\alpha|^4$.
Figure \[fig2\] represents the Wigner distribution, $W_{\mathrm{SOCS}}$, in the phase space for fixed value of $|\alpha|$ and different values of the scalar parameters $s$ and $t$. The Wigner distribution is plotted as a function of Re$(\beta)$ and Im$(\beta)$ in the phase space. The plots on the right of figure \[fig2\] are the 2D plots of the Wigner distribution varying with Re$(\beta)$ (Im$(\beta)$=0). The negative dip of the Wigner distribution increases with $t$. It is clear that performing the SUP operation transforms a purely classical coherent state to a nonclassical one in terms of the negativity of the Wigner distribution [@comment].
Let us analyze the area of negativity of the Wigner function, $W_{\mathrm{SOCS}}(\beta, \beta^*)$, for selected scalar parameters. For $t=1$, the Wigner distribution is given by the relation $$\begin{aligned}
W_{\mathrm{SOCS}}(\beta, \beta^*) & = & \frac{8}{\pi}(a^2+b^2) e^{-2\{(x-a)^2+(y-b)^2\}} \nonumber \\
& & \times \left[\left(x-\frac{a}{2}\right)^2+\left(y-\frac{b} {2}\right)^2-\frac{1}{4} \right] ,\end{aligned}$$ where $\beta=x+iy$, $\alpha=a+ib$. The negative region is represented by $\left\{\left(x-\frac{a}{2}\right)^2+\left(y-\frac{b}{2}\right)^2-\frac{1}{4}\right\}<0$, bounded by a circle of radius 1/2 and centered at $\left( \frac{a}{2}, \frac{b}{2} \right)$.
For $t=1/\sqrt{2}$, the Wigner distribution is $$\begin{aligned}
W_{\mathrm{SOCS}}(\beta, \beta^*) & = &
W_{\mathrm{coh}}(\beta, \beta^*) s^2\big[1-8|\alpha|^2+16|\alpha|^2|\beta|^2+4|\alpha|^4 \nonumber \\
& & +4(1-2|\alpha|^2)(\alpha^*\beta+\alpha\beta^*)\big].\end{aligned}$$ Thus the area of negativity is bounded by a circular region $\left\{x-\left(\frac{a}{2}-\frac{a}{4(a^2+b^2)}\right)\right\}^2+\left\{y-\left(\frac{b}{2}
-\frac{b}{4(a^2+b^2)}\right)\right\}^2 = \frac{1}{4}$.
For $t=0$, the Wigner function is $$\begin{aligned}
W_{\mathrm{SOCS}}(\beta, \beta^*) & = & W_{\mathrm{coh}}(\beta, \beta^*) s^2\big[1-3|\alpha|^2+4|\alpha|^2|\beta|^2+|\alpha|^4 \nonumber \\
& & +2(1-|\alpha|^2)(\alpha^*\beta+\alpha\beta^*)\big] ,\end{aligned}$$ and the corresponding negative region is again within a circle, $\left\{x-\left(\frac{a}{2}-\frac{a}{2(a^2+b^2)}\right)\right\}^2+\left\{y-\left(\frac{b}{2} -\frac{b}{2(a^2+b^2)}\right)\right\}^2 = \frac{1}{4}$.
Therefore, in all the cases, the negative region becomes a circle of radius $1/2$, i.e. the negative area is independent of the choice of the scalar $t$. The nonclassical nature of the SOCS is evident from the negative region of the Wigner function; however, the degree of quantumness cannot be quantified for different scalar parameters.
The nonclassicality of SOCS can also be investigated using the $F$-parametrized quasiprobability function. For the SOCS density matrix (\[eq3\]), using the relation for $\mho^{(F)}$ (\[s\]), we obtain $$\begin{aligned}
\nonumber
& &\mho^{(F)}_{\mathrm{coh}}(\beta) = \frac{1}{\pi}N_1^{-1}\left[s^2 \langle\alpha|\hat{T}^{(F)}(\beta)|\alpha\rangle+s(s+t)\alpha^* e^{-|\alpha|^2}\sum_n\sqrt{n+1}\frac{(|\alpha|^2)^n}{n!}\right.\\\nonumber
& & \left.\langle n+1|\hat{T}^{(F)}(\beta)|n\rangle+s(s+t)\alpha e^{-|\alpha|^2}\sum_n\sqrt{n+1}\frac{(|\alpha|^2)^n}{n!}\langle n|\hat{T}^{(F)}(\beta)|n+1\rangle\right.\\\nonumber
& & \left.+(s+t)^2|\alpha|^2 e^{-|\alpha|^2}\sum_n(n+1)\frac{(|\alpha|^2)^n}{n!}\langle n+1|\hat{T}^{(F)}(\beta)
|n+1\rangle\right],\end{aligned}$$ where $\langle\alpha|\hat{T}^{(F)}(\beta)|\alpha\rangle = \left(\frac{2}{1-F}\right)\exp\left[-\left(\frac{2}{1-F}\right)|\beta-\alpha|^2\right]$ and $\langle n|\hat{T}^{(F)}(\beta)|m\rangle$ is given by (\[s1\]).
![$F$-parametrized quasiprobability function $\mho^{(F)}_{\mathrm{coh}}(\beta)$ of a coherent state $|\alpha\rangle$ after the operation $(s\hat{a}\hat{a}^\dag+t\hat{a}^\dag \hat{a})$, where $s=\sqrt{1-t^2}$, as a function of $F$ for $|\beta| = 0.8$, $|\alpha|= 0.4$ and $t = 0.5$.[]{data-label="pars"}](fig2_s-fn_coh.n1.eps)
In figure (\[pars\]) we observe the behavior of the $F$-parametrized function for different values of the parameter $F$. The function is negative around $F$=0, which is consistent with our observed nonclassicality of the Wigner function.\
The nonclassical nature of SOCS can also be captured by its negative volume (\[nv\]).
![Negative volume as a function of $t$ for a coherent state input with $|\alpha| =$ 0.2 (continuous), 0.4 (dashed) and 0.6 (dot-dashed).[]{data-label="fig3"}](fig1_volm_coh.eps){width="8cm"}
Figure \[fig3\] clearly shows that, for fixed $t$, the negative volume $V_{\mathrm{SOCS}}$ increases with the amplitude $|\alpha|$ of the input coherent field. $V_{\mathrm{SOCS}}$ first increases and then decreases with increasing $t$. The range of $t$ for which SOCS shows nonclassicality is dependent on the value of $|\alpha|$.
The sub-Poissonian statistics of SOCS can be established by using the following: $$Q_{\mathrm{SOCS}} = -\frac{|\alpha|^2}{K_1}N_1^{-1}\left\{K_1(K_1-1)-(s+t)^2 (2|\alpha|^2+3) - 2s(s+t) \right\} ,$$ where $K_1=\left[(s+t)|\alpha|^2+(2s+t)\right]^2+(s+t)^2|\alpha|^2$.
In order to see the variation of the $Q$ parameter with $|\alpha|$ (coherent field), we plot the $Q$ function against the parameter $t$ in figure \[fig4\]. $Q$ exhibits the sub-Poissonian character for the coherent input state and increases its negativity as $|\alpha|$ increases. But at $t=1$, the $Q$ parameter suddenly changes its characteristics to indicate super-Poissonian distribution.
The squeezing parameter for SOCS is calculated to yield the following: $$S_{\mathrm{SOCS}} = 2N_1^{-1}(s+t)^2|\alpha|^2 .
\label{sqz}$$
From equation (\[sqz\]), we can see that $S_{\mathrm{SOCS}}$ is positive for all $t$ and $|\alpha|$. The superposed product (SUP) operation cannot inject squeezing property into the coherent state character (see figure \[fig5\]). In general, the nonclassicality indicators can be compared to observe the signature of quantumness introduced by the SUP operation on the input coherent field. We observe from the phase space distribution, the negative region of the Wigner distribution is an indicator of nonclassicality of the state but cannot quantify the degree of nonclassicality. Another indicator of nonclassicality, viz. negative volume of the Wigner function is able to quantitatively classify the nonclassicality. The negative volume decreases and the $Q$ parameter increases when $t$ becomes close to 1. The squeezing parameter $S$ does not exhibit any nonclassicality and hence fails as an indicator. A comparison of the indicators is shown in figure \[fig6\].
![A comparison of the different nonclassical indicators for SOCS as a function of the scalar parameter $t$ with $|\alpha|=0.4$.[]{data-label="fig6"}](combined_fig1_coh.eps){width="8cm"}
The generated nonclassical SOCS states can prove to be useful in a wide variety of tasks and applications, with optical components effecting Gaussian processes being readily available in the laboratory. The experimental and theoretical developments on continuous-variable quantum information processes in the Gaussian realm can be found in a recent review [@RMPloyd].
SUP operated thermal state
--------------------------
If we consider an input single-mode thermal field with frequency $\omega$ and at absolute temperature $T$, with the Fock state representation $$\hat{\rho}_{\mathrm{in}}=\frac{1}{(1+\bar{n})}\sum_n\left(\frac{\bar{n}}{1+\bar{n}}\right)^n|n\rangle\langle n| ,$$ where $\bar{n}=[e^{\hbar \omega/kT}-1]^{-1}$ is the average photon number, $k$ being the Boltzmann constant, the resulting SUP operated thermal state (SOTS) is given by $$\hat{\rho}_{\mathrm{th}} = N_2^{-1}
[s(\hat{a}\hat{a}^\dag)+t(\hat{a}^\dag \hat{a})]\hat{\rho}_{\mathrm{in}}[s(\hat{a}\hat{a}^\dag)+t(\hat{a}^\dag \hat{a})] ,
\label{therm}$$ where $N_2 = s^2(1+\bar{n})(1+2\bar{n})+4st\bar{n}(1+\bar{n})+t^2\bar{n}(1+2\bar{n})$ is the normalization constant.
Using the series expression (\[eq7\]) for the Wigner function, the phase space distribution for the thermal state is $$W_{\mathrm{SOTS}}(\beta, \beta^*) = W_{\mathrm{th}}(\beta, \beta^*)N_2^{-1}\big[(M_2+s)^2+(s+t)M_2\big] ,$$ where $W_{\mathrm{th}}(\beta, \beta^*)=\frac{2} {\pi}\frac{1}{(1+2\bar{n})}e^{-\frac{2|\beta|^2}{1+2\bar{n}}}$ is the Wigner function of the input thermal state, $M_2=\frac{4\bar{n}(1+\bar{n})}{(1+2\bar{n})^2}(s+t)|\beta|^2$, $N_2 = s^2(1+\bar{n})(1+2\bar{n})+4st\bar{n}(1+\bar{n})+t^2\bar{n}(1+2\bar{n})$.
In figure \[fig7\], we plot the Wigner distribution $W_{\mathrm{SOTS}}(\beta, \beta^*)$ as a function of $t$ for fixed $\bar{n}=0.2$. Unlike the case of input coherent state, $W_{\mathrm{SOTS}}$ has no negative region but the Wigner function does not remain Gaussian. Hence, $s(\hat{a}\hat{a}^\dag)+t(\hat{a}^\dag \hat{a})$ generates a non-Gaussian state from the Gaussian thermal state. The nonclassical nature of SOTS cannot be captured by the Wigner function, which remains positive. However this is not a necessary condition for the nonclassicality of SOTS. The nonclassicality of the state can be investigated using the $F$-paramterized quasiprobability function (\[s\]). For the SUP operated thermal state, using the expression for the output density matrix (\[therm\]), we obtain $$\begin{aligned}
\nonumber
\mho^{(F)}_{\mathrm{th}}(\beta) & = & \frac{1}{\pi}N_2^{-1}\left(\frac{1}{1+\bar{n}}\right)\left(\frac{2}{1-F}\right)\exp\left(-\frac{2|\beta|^2}{1-F}\right)\\
& & \times\sum_n [s+(s+t)n]^2\left\{\left(\frac{\bar{n}}{1+\bar{n}}\right)\left(\frac{F+1}{F-1}\right)\right\}^n
L_n\left(\frac{4|\beta|^2}{1-F^2}\right) .\end{aligned}$$
![$F$-parametrized quasiprobability function $\mho^{(F)}_{\mathrm{th}}(\beta)$ of a thermal state after the operation $(s\hat{a}\hat{a}^\dag+t\hat{a}^\dag \hat{a})$, where $s=\sqrt{1-t^2}$, as a function of $F$ for $|\beta| = 0.5$, $\bar{n} = 0.2$ and $t = 0.9$.[]{data-label="fig7.1"}](fig3_s-fn_thm.n1.eps){width="7.9cm"}
In figure \[fig7.1\], we plot $\mho^{(F)}_{\mathrm{th}}(\beta)$ as a function of $F$. For $t = 0.9$, the $F$-parametrized quasiprobability is positive for $F < 0$ and becomes negative for $F > 0.3$. The function is always positive at $F = 0$ matching with non-negative Wigner function for SOTS. $\mho^{(F)}_{\mathrm{th}}(\beta)$ becomes highly negative as $F \rightarrow 1$. Hence, the nonclassicality of SOTS can be evidenced by the negativity of the normally ordered Glauber-Sudarshan $P$ function. Similar behavior is also observed for other values of the parameter $t$.
To further check the nonclassical effects of SUP operations on input thermal state, we study the sub-Poissonian statistics of SOTS. Mandel’s $Q$ parameter for SOTS is found to be $$Q_{\mathrm{SOTS}} = -\frac{\bar{n}}{K_2}N_2^{-1}\left\{ K_2(K_2-3)-6(\bar{n}+1)^2(s+t)^2+t^2 \right\} ,$$ where $K_2=2(\bar{n}+1)(s+t)\left[3\bar{n}(s+t)+2s\right]+t^2$.
![Mandel’s $Q$ parameter as a function of $t$ and with $\bar{n} =$ 0.1 (continuous), 0.2 (dashed) and 0.3 (dot-dashed), for an input thermal state.[]{data-label="fig8"}](fig1_Q_thm.eps){width="8cm"}
We plot the $Q$ parameter against $t$ for different values of $\bar{n}$ (thermal field) in figure \[fig8\]. $Q$ exhibits sub-Poissonian character for the input thermal state, and increases its negativity as $\bar{n}$ increases. But at $t=1$, the $Q$ parameter suddenly changes its characteristic to mark super-Poissonian statistics. Positive $Q$ parameter values are also observed in the negative range of $t$. We emphasize that though the SUP operated thermal field has no negative Wigner function, it displays sub-Poissonian property. Hence, Mandel’s $Q$ parameter is a good indicator of the nonclassicality of the SUP operated thermal states.
The nonclassical nature of SOTS can also be analyzed by studying the squeezing parameter, which is calculated in terms of the average photon number $\bar{n}$, $$S_{\mathrm{SOTS}} = 2N_2^{-1}\bar{n}\left[2\bar{n}(5s+3t)(s+t)+(2s+t)^2\right].$$
From figure \[fig9\], it is clear that the squeezing parameter $S_{\mathrm{SOTS}}$ goes negative. Hence, the SUP operator can introduce the squeezing property into the input thermal state character. Hence we observe here that the Wigner function fails to indicate any nonclassicality for SOTS. Mandel’s $Q$ parameter goes negative for some values of the control $t$. Unlike the input coherent state, SOTS shows squeezing property, and hence $S_{\mathrm{SOTS}}$ is a good measure. A comparison of the nonclassical indicators is shown in figure \[com\_thm\]. The reason for the success of indicators such as the $Q$ parameter and the squeezing parameter $S$ over the non-negative Wigner function could be the negative values of normally ordered observables which are not realized by the Wigner function [@semenov]. As shown by the $F$-parametrized distribution function (figure \[fig7.1\]), such observables are determined by the Glauber-Sudarshan $P$ function. There are other examples of generated states that have non-negative Wigner functions and yet exhibit nonclassical features indicated by the $Q$ parameter [@cessa; @janszky96].
![A comparison of the different nonclassical indicators for SOTS as a function of the scalar parameter $t$ with $\bar{n}=0.2$.[]{data-label="com_thm"}](combined_fig1_thm.eps){width="8cm"}
Non-Gaussian states are an important tool in quantum information processing and allow for applications of quantum algorithms which cannot be done using nonclassical Gaussian states. Nonclassical states with non-Gaussian Wigner function, as generated in the operation, have been used in the design for an efficient and universal quantum computation device [@Lloyd]. Non-Gaussian states have applications also in other quantum tasks [@Science; @CC].
Conclusions {#sec5}
===========
The generation and manipulation of nonclassical field states is of great importance from the perspective of quantum tasks and information processing. We have shown here that in continuous variable systems, a nonclassical state can be generated from an input classical state by the use of a general superposition of two product operations of the type $s(\hat{a}\hat{a}^\dag)+t(\hat{a}^\dag \hat{a}) = s + (s+t)\hat{a}^\dag \hat{a}$. The nonclassical property has been analyzed using the phase-space distribution of the generated state, its photon statistics and also the quadrature squeezing parameter.
The Wigner function is used to study the phase-space properties. The negativity of the Wigner distribution is a sufficient condition for the nonclassicality, and we have checked the area of negativity for different values of the scalar parameter $t$. For input coherent states, there is a distinct negativity in the Wigner function of the SUP operated output state. The negative volume for the SUP operated coherent state is shown to decrease with increasing $t$. Hence the negativity of the output state can be manipulated using the control parameter. For input thermal states, the Wigner distribution is positive for all values of $t$. However, the SUP operated thermal state is non-Gaussian in nature. Thus the SUP operation can be used to generate non-Gaussian or nonclassical state for quantum operations on classical coherent or thermal fields. For the input coherent and thermal fields, we have checked for the negativity of the $F$-parametrized quasiprobability function as the operator ordering parameter $F$ is varied.
The photon statistics of the output state is another important indicator of nonclassicality. Field states with the classically most random photon distribution has a Poissonian character, in which the variance of the distribution is equal to the mean. Hence, any field with a sub-Poissonian photon distribution is essentially nonclassical. The sub-Poissonian statistics can be quantified using Mandel’s $Q$ parameter. We have observed that the $Q$ parameter is negative (sub-Poissonian) for SUP operated coherent states for most values of the scalar parameter $t$ and is positive for values close to $t=$ 1. For SUP operated thermal states, the $Q$ parameter exhibits both super- and sub-Poissonian statistics based on the values of the control parameter $t$. Hence, the photon statistics of the output states can be controlled using $t$, which in turn generates the necessary nonclassicality.
Another important indicator of nonclassicality is the squeezing property of the output field state. The SOCS does not but the SOTS does exhibit squeezing property and hence the nonclassicality of the SOTS is well-described by the squeezing property. The importance of generating nonclassical states using SUP operations is the fact that such operations can be realized experimentally using photon addition ($\hat{a}^\dag$) and subtraction ($\hat{a}$) properties of the input continuous variable states of the input field. The experimental generation of photon added and subtracted states in the laboratories using parametric down converters and controlled beam-splitters makes the physical realization of SUP operations. Hence one can generate nonclassical states from classical distributions that can be suitably manipulated for specific quantum tasks with specific negativity of phase space distribution or sub-Poissonian statistics. It can also generate requisite non-Gaussianity from Gaussian states. These output states can thus have a wide range of useful applications.
From the perspective of quantum information applications in continuous variable regime, the generated output states can be used for various tasks that can be realized and measured in the laboratory. As mentioned earlier, Gaussian nonclassical states prove to be more useful than discrete quantum states in practical applications of various information protocols. On the other hand, states with non-Gaussian Wigner function can be used in the conceptual design of universal quantum computers and to implement quantum algorithms.
AC thanks National Board of Higher Mathematics, DAE, Government of India, and HSD thanks University Grants Commission, India, for financial support.
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abstract: 'In forthcoming years, the Internet of Things (IoT) will connect billions of smart devices generating and uploading a deluge of data to the cloud. If successfully extracted, the knowledge buried in the data can significantly improve the quality of life and foster economic growth. However, a critical bottleneck for realising the efficient IoT is the pressure it puts on the existing communication infrastructures, requiring transfer of enormous data volumes. Aiming at addressing this problem, we propose a novel architecture dubbed Condense (refigurable knowleg acquisitio ystms), which integrates the IoT-communication infrastructure into data analysis. This is achieved via the generic concept of network function computation: Instead of merely transferring data from the IoT sources to the cloud, the communication infrastructure should actively participate in the data analysis by carefully designed en-route processing. We define the Condense architecture, its basic layers, and the interactions among its constituent modules. Further, from the implementation side, we describe how Condense can be integrated into the 3rd Generation Partnership Project (3GPP) Machine Type Communications (MTC) architecture, as well as the prospects of making it a practically viable technology in a short time frame, relying on Network Function Virtualization (NFV) and Software Defined Networking (SDN). Finally, from the theoretical side, we survey the relevant literature on computing “atomic” functions in both analog and digital domains, as well as on function decomposition over networks, highlighting challenges, insights, and future directions for exploiting these techniques within practical 3GPP MTC architecture.'
author:
- 'Dejan Vukobratovic, Dusan Jakovetic, Vitaly Skachek, Dragana Bajovic, Dino Sejdinovic, Gunes Karabulut Kurt, Camilla Hollanti, and Ingo Fischer[^1][^2][^3][^4] [^5] [^6] [^7] [^8] [^9]'
title: 'CONDENSE: A Reconfigurable Knowledge Acquisition Architecture for Future 5G IoT'
---
Internet of Things (IoT), Big Data, Network Coding, Network Function Computation, Machine learning, Wireless communications.
Introduction
============
A deluge of data is being generated by an ever-increasing number of devices that indiscriminately collect, process and upload data to the cloud. An estimated 20 to 40 billion devices will be connected to the Internet by 2020 as part of the Internet of Things (IoT) [@Siemens]. IoT has the ambition to interconnect smart devices across cities, vehicles, appliances, connecting industries, retail and healthcare domains, thus becoming a dominant fuel for the emerging Big Data revolution [@Time]. IoT is considered as one of the key technologies to globally improve the quality of life, economic growth, and employment, with the European Union market value expected to exceed one trillion Euros in 2020 [@EU]. However, a critical bottleneck for the IoT vision is the pressure it puts on the existing communication infrastructures, by requiring transfer of enormous amounts of data. By 2020 IoT data will exceed 4.4 ZB (zettabytes) amounting to 10$\%$ of the global “digital universe” (compared to 2$\%$ in 2013) [@ZDNet]. Therefore, a sustainable solution for IoT and cloud integration is one of the main challenges for contemporary communications technologies.
The state-of-the-art in IoT/cloud integration assumes uploading and storing all the raw data generated by IoT devices to the cloud. The IoT data is subsequently processed by cloud-based data analysis that aims to extract useful knowledge [@IoTCloud]. For majority of applications, this approach is inefficient since there is typically a large amount of redundancy in the collected data. As a preprocessing step prior to data analysis, projections to a much lower-dimensional space are often employed, essentially discarding large portions of data. With the growth of IoT traffic, the approach where communications and data analysis are separated will become unsustainable, necessitating a fundamental redesign of IoT communications.
In this work, we propose a generic and reconfigurable IoT architecture capable of adapting the IoT data transfer to the subsequent data analysis. We refer to the proposed architecture as Condense (refigurable knowleg acquisitio ystms). Instead of merely transferring data, the proposed architecture provides an active and reconfigurable service leveraged by the data analysis process. We identify a common generic interface between data communication and data analysis: *the function computation*, and we distinguish it as a core Condense technology. Instead of communicating a stream of data units from the IoT devices to the cloud, the proposed IoT architecture processes the data units *en-route* through a carefully designed process to deliver a stream of network function evaluations stored in the cloud. In other words, the Condense architecture does not transfer all the raw data across the communications infrastructure, but only what is needed from the perspective of the current application at hand.
To illustrate the idea with a toy example, consider a number of sensors which constitute a fire alarm system, e.g., [@Giridhar05], [@Giridhar]. Therein, we might only be interested in the maximal temperature across the sensed field, and not in the full sensors readings vector. Therefore, it suffices to deliver to the relevant cloud application only an evaluation of the maximum function applied over the sensors readings vector; Condense realizes this maximum function as a composition of “atomic” functions implemented across the communications infrastructure.
We describe how to implement the proposed approach explained above in the concrete third generation partnership project (3GPP) Machine Type Communications (MTC) architecture [@Taleb2012]. The 3GPP MTC service is expected to contribute a dominant share of the IoT traffic via the upcoming fifth generation (5G) mobile cellular systems, thus providing an ideal setup for the demonstration of Condense concepts. We enhance the 3GPP MTC architecture with the network function computation (NFC) – a novel envisioned MTC-NFC service. We define the layered Condense architecture comprised of three layers: i) atomic function computation layer, ii) network function computation layer, and iii) application layer, and we map these layers onto the 3GPP MTC architecture. In the lowermost atomic function computation (AFC) layer, carefully selected atomic modules perform local function computations over the input data. The network function computation layer orchestrates the collection of AFC modules into the global network-wide NFC functionality, thus evaluating non-trivial functions of the input data as a coordinated composition of AFCs. Furthermore, the NFC layer provides a flexible and reconfigurable MTC-NFC service to the topmost application layer, where cloud-based data analysis applications directly exploit the outputs of the NFC layer. Throughout the system description, we provide a review of the theoretical foundations that justify the proposed architecture and point to the tools for the system design and analysis. Finally, we detail practical viability of incorporating NFC services within 3GPP MTC service, relying on emerging concepts of Network Function Virtualization (NFV) [@etsiNFV], [@Han2015] and Software Defined Networking (SDN) [@Lantz2010], [@Kreutz2015]; this upgrade is, thanks to the current uptake of the SDN/NFV concepts, achievable within a short time frame.
This paper is somewhat complementary with respect to other works that consider architectures for 5G IoT communications. For example, reference [@Condoluci] focuses on machine-type multicast services to ensure end-to-end reliability, low latency and low energy consumption of MTC traffic (including both up and downlinks). Reference [@Palattella] provides a detailed analysis of integration of 5G technologies for the future global IoT, both from technological and standardization aspects. However, while existing works consider making communication of the MTC-generated *raw data* efficient, here we aim to improve the overall system efficiency through communicating over the network infrastructure only the application-requested *functions over data*. In other words, this paper describes how we can potentially exploit decades of research on function computation and function decomposition over networks within the concrete, practical and realizable knowledge acquisition system for the IoT-generated data. In particular, we review the main results on realizing (atomic) function computation in the analog (wireless and optical) and digital domains, as well as on function evaluation and decomposition over networks, including the work on sensor fusion, e.g., [@Giridhar05; @Giridhar], network coding for computing [@AFKZ; @Kowshik2012][-]{} [@Dougherty], and neural networks [@Li2014][-]{} [@Srivastava2014]. While this paper does not provide novel contributions to these fields, it identifies and discusses main challenges in applying them within the practical 3GPP MTC architecture, and it points to interesting future research directions.
**Paper organization**. The rest of the paper is organized as follows. In Sec. [II]{}, we review the state-of-the-art 3GPP MTC architecture, briefly present SDN/NFV concepts, and give notational conventions. In Sec. [III]{}, we introduce the novel layered Condense architecture that, through the rest of the paper, we integrate into the 3GPP MTC architecture. In Sec. [IV]{}, we describe the atomic function computation layer that defines the basic building block of the architecture, distinguishing between the analog (or *in-channel*) AFC and digital (*in-node*) AFC modules. The theoretical fundamentals and practical aspects of the NFC layer are presented in Sec. [V]{}. In Sec. [VI]{}, the interaction between the application layer and the NFC layer is discussed, where several application layer examples are presented in detail. Further implementation issues are discussed in Sec. [VII]{}, and the paper is concluded in Sec. [VIII]{}.
Background and Preliminaries
============================
Subsection [II-A]{} reviews the current 3GPP MTC architecture, Subsection [II-B]{} gives background on software defined networking (SDN) and network function virtualization (NFV), while Subsection [II-C]{} defines notation used throughout the rest of the paper.
The 3GPP MTC Architecture
-------------------------
Machine Type Communications (MTC) is an European Telecommunications Standards Institute (ETSI)-defined architecture that enables participating devices to send data to each other or to a set of servers [@Taleb2012]. While ETSI is responsible for defining the generic MTC architecture, specific issues related with mobile cellular networks are addressed in 3GPP standardization [@3gppmtc]. 3GPP MTC is first included in Release [10]{} and will evolve beyond current 3GPP Long Term Evolution (LTE)/LTE-Advanced Releases into the 5G system [@mtc5g].
Fig. \[3GPP-MTC\] illustrates the 3GPP MTC architecture. It consists of: i) the MTC device domain containing MTC devices that access MTC service to send and/or receive data, ii) the network domain containing network elements that transfer the MTC device data, and iii) the MTC application domain containing MTC applications running on MTC servers. MTC devices access the network via Radio Access Network (RAN) elements: base stations (eNB: eNodeB) and small cells (HeNB: Home-eNodeB). Packet data flows follow the Evolved Packet Core (EPC) elements: HeNB Gateway (HeNB-GW), Service Gateway (S-GW) and Packet Gateway (P-GW), until they reach either a mobile operator MTC server or a third party MTC server via the Internet. In this work, we address MTC device data processing and focus on the data plane while ignoring the control plane of the 3GPP architecture.
Abstracted to its essence, the current 3GPP MTC approach in the context of IoT/cloud integration is represented by three layers (Fig. 1). The MTC device domain, or data layer, contains billions of devices that generate data while interacting with the environment. The network domain, or communication layer, provides mere data transfer services to the data layer by essentially uploading the generated data to the cloud in its entirety. The application domain, or application layer, contains data centres running MTC servers which provide storage and processing capabilities. MTC applications running in data centres enable, e.g., machine learning algorithms to extract knowledge from the collected data. In this paper, we challenge this 3GPP MTC layered structure and propose a novel Condense layered architecture described in Sec. [III]{}.
Software Defined Networking (SDN) and Network Function Virtualization (NFV)
---------------------------------------------------------------------------
SDN and NFV are novel concepts in networking research that increase network flexibility and enable fast implementation of new services and architectures. Both SDN and NFV are under a current consideration for future integration in the 3GPP cellular architecture [@sdncell], [@3gppnfv]. Although not yet part of the 3GPP architecture, in Fig. \[3GPP-MTC\], we present main NFV/SDN management entities: the NFV manager and the SDN controller, as they will be useful for the description of the Condense architecture.
SDN is a novel network architecture that decouples the control plane from the data plane [@Lantz2010], [@sdnieee]. This is achieved by centralizing the traffic flow control where a central entity, called the SDN controller, manages the physical data forwarding process in SDN-enabled network nodes. The SDN controller remotely manages network nodes and flexibly controls traffic flows at various flow granularities. In other words, the SDN controller can easily (re)define forwarding rules for data flows passing through an SDN network node. Using SDN, various network services are able to quickly re-route their data flows, adapting the resulting (virtual) network topology to their needs.
NFV is another recent trend in networking where, instead of running various network functions (e.g., firewalls, NAT servers, load balancers, etc.) on dedicated network nodes, the network hardware is virtualized to support software-based implementations of network functions [@Han2015]. This makes network functions easy to instantiate anywhere across the network when needed. Multiple instances of network functions are jointly administered and orchestrated by the centralized NFV management.
NFV and SDN are complementary concepts that jointly provide flexible and efficient service chaining: a sequence of data processing tasks performed at different network nodes [@Li2015]. The NFV manager has the capability to actually instantiate the targeted (atomic) function computations at each node in the network. Similarly, SDN has the power to steer data flows and hence establish a desired (virtual) network topology which supports the desired network-wide computation. This feature will be fundamental for a fast implementation and deployment of the Condense architecture, as detailed in the rest of the paper. For more details about SDN/NFV concepts in 3GPP networks, we refer the interested reader to [@sdncell].
Notational preliminaries
------------------------
Throughout, we use bold symbols to denote vectors, where $\ell$-th entry of a vector $\mathbf{x}$ of length $L$ is denoted by $x[\ell]$, $\ell=1,...,L$. We denote by $\mathbb R$ the set of real numbers, and by ${\mathbb R}^Q$ the $Q$-dimensional real coordinate space. A finite field is denoted by $\mathbb F$, a finite alphabet (finite discrete set) by $\mathbb A$, and by ${\mathbb A}^Q$ the set of $Q$-dimensional vectors with the entries from $\mathbb A$. Symbol $|\cdot|$ denotes the cardinality of a set. We deal with vectors $\mathbf x \in {\mathbb R}^Q$, $\mathbf x \in {\mathbb A}^Q$, and also $\mathbf x \in {\mathbb F}^Q$, and it is clear from context which of the three cases is in force. Also, addition and multiplication over both $\mathbb R$ and $\mathbb F$ are denoted in a standard way – respectively as $+$ and $\cdot$ (or the multiplication symbol is simply omitted), and again the context clarifies which operation is actually applied.
We frequently consider a directed acyclic graph $\mathcal{G} = \left( \mathcal V, \mathcal E\right)$, where $\mathcal V$ denotes the set of nodes, and $\mathcal E$ the set of directed edges (arcs). An arc from node $u$ to node $v$ is denoted by $u \rightarrow v$. Set $\mathcal V = \mathcal S \cup \mathcal A \cup \mathcal D$, where $\mathcal S$, $\mathcal A$, and $\mathcal D$ denote, respectively, the set of source nodes, atomic nodes, and destination nodes. We let $\mathcal S \cap \mathcal D = \varnothing$. We also introduce $N = |\mathcal S|$, $M = |\mathcal A|$, and $R = |\mathcal D|$. As we will see further ahead, source nodes correspond to MTC devices ($N$ data generators), atomic nodes correspond to the 3GPP communication infrastructure nodes which implement atomic functions ($M$ atomic nodes), and destination nodes are MTC servers in data centers which are to receive the desired function computation results ($R$ destination nodes). We index an arbitrary node in $\mathcal S$ by $s$, and similarly we write $a \in \mathcal A$, and $d \in \mathcal D$. When we do not intend to make a distinction among $\mathcal S, \mathcal A$, and $\mathcal D$, we index an arbitrary node by $v \in \mathcal V$. For each node $v \in \mathcal V$, we denote by $\mathcal V^{(v)}_{\mathrm{in}}$ its in-neighborhood, i.e., the set of nodes $v^\prime$ in $\mathcal V$ such that the arc $v^\prime \rightarrow v$ exists. Analogously, $\mathcal V^{(v)}_{\mathrm{out}}$ denotes the node $v$’s out-neighborhood. As we frequently deal with in-neighborhoods, we will simply write $\mathcal V^{(v)} \equiv \mathcal V^{(v)}_{\mathrm{in}}$. We call the in-degree of $v$ the cardinality of $\mathcal V^{(v)}_{\mathrm{in}}$, and we analogously define the out-degree. Although not required by the theory considered ahead, just for simplicity of notation and presentation, all sections except Section V consider the special case where $\mathcal G$ is a directed rooted tree with $N$ sources and a single destination. Pictorially, we visualize $\mathcal G$ as having the source nodes $\mathcal S$ at the bottom, and the destination node $d$ at the top (see Sec. V, Fig. \[NFCgraph\], right-hand side). In the case of a directed rooted tree graph $\mathcal G$, the leaf nodes’ set of $v$ coincides with its in-neighborhood $\mathcal{V}^{(v)}$, and all nodes except the destination nodes have the out-degree one, the destination node having the out-degree zero.
We index (vector) quantities associated with sources $s \in \mathcal S$ through subscripts, i.e., $\mathbf{x}_s$ is the source $s$’s vector. When considering a generic directed acyclic graph $\mathcal G$ (Section V), we associate to each arc $u \rightarrow v$ a vector quantity $\mathbf{x}^{(u \rightarrow v)}$. With directed rooted trees (Sections III, IV, and VI), each node (except the destination node) has the out-degree one; hence, for simplicity, we then use node-wise (as opposed to edge-wise) notation, i.e., we index quantity $\mathbf{x}^{(u \rightarrow v)}$ as $\mathbf{x}^{(u)}$. Note that this notation is sufficient as, with directed rooted trees, there is only a single arc outgoing a (non-destination) node. When needed, time instances are denoted by $t=1,2,...,T$; a vector associated with source $s$ and time $t$ is denoted by $\mathbf{x}_{s,t}$; similarly, we use $\mathbf{x}^{(v)}_t$ for non-source nodes.
CONDENSE Architecture: IoT/Cloud Integration for 5G
===================================================
In this section, we present the Condense architecture that upgrades the 3GPP MTC architecture with the concept of network function computation (NFC). NFC creates a novel role 3GPP MTC service should offer: instead of communicating raw data, it should deliver function computations over the data, providing for a novel MTC-NFC service. The NFC design should be generic, flexible and reconfigurable to meet the needs of increasing number of MTC applications that extract knowledge from MTC data. For most applications, indiscriminate collection of MTC data is extremely wasteful and MTC-NFC service may dramatically reduce MTC traffic while preserving operational efficiency of MTC applications.
The Condense architecture challenges the conventional division into data, communications and application layer (Sec. 2A). Instead, we propose a novel architecture consisting of: i) atomic function computation (AFC) layer, ii) network function computation (NFC) layer, and iii) application layer. In this section, we provide a high-level modular description of the architecture by carefully defining its basic building blocks (modules). In the following three sections, we delve into details of each layer and provide both theoretical justifications and implementation discussion that motivated this work.
CONDENSE Architecture: Modules and Layers
-----------------------------------------
The Condense architecture is presented in Fig. \[MTC-NFC\]. It consists of an interconnected collection of basic building blocks called AFC modules. Each AFC module evaluates an (atomic) function over the input data packets and delivers an output data packet representing the atomic function evaluation. A generic AFC module may have multiple input and multiple output interfaces, each output interface representing a different AFC over the input data. The collection of interconnected and jointly orchestrated AFC modules delivers a network function computation over the source data packets. The resulting NFC evaluations are the input to application layer MTC server application.
![Condense MTC-NFC architecture.[]{data-label="MTC-NFC"}](Figure_2.pdf){width="3.4in"}
Let us assume that an MTC network contains $N$ MTC devices representing the set of source modules (or source nodes) $\mathcal{S}$. Source node $s \in \mathcal{S}$ produces a message $\mathbf{x}_{s}=(x_{s}[1],x_{s}[2],\ldots,x_{s}[L])$ containing $L$ symbols from a given alphabet $\mathbb A$. The message $\mathbf{x}_{s}$ is transmitted at an output interface of the source module $s$. For simplicity, we assume that every source module has a single output interface.
In addition to the source nodes, the MTC network contains $M$ AFC modules (or AFC nodes) representing the set $\mathcal{A}$. An arbitrary AFC node $a \in \mathcal{A}$ has $P$ input and $Q$ output interfaces. For simplicity, unless otherwise stated, we will assume single-output AFC modules, i.e., $Q=1$. At input interfaces, the AFC node $a$ receives the set of input data packets $\{\mathbf{x}^{(b)}\}_{b \in \mathcal V^{(a)}}$, while at the output interface, it delivers the output data packet $\mathbf{x}^{(a)}$. AFC node $a$ associates an atomic function $g^{(a)}$ to the output interface, where $\mathbf{x}^{(a)}=g^{(a)}(\{\mathbf{x}^{(b)}\}_{b \in \mathcal V^{(a)}})$. Finally, the MTC network contains $R$ MTC servers (or destination nodes) representing the set of destination nodes $\mathcal D$.
The source nodes $\mathcal{S}$, AFC nodes $\mathcal{A}$ and destination nodes $\mathcal D$ are interconnected into an NFC graph $\mathcal{G}=(\mathcal{V}=\mathcal{S}\cup\mathcal{A}\cup\mathcal{D},\mathcal{E}),$ where $\mathcal{V}$ is the set of nodes (modules) and $\mathcal{E} \subseteq {\mathcal V} \times {\mathcal V}$ is the set of edges, i.e., connections between modules. For simplicity, unless otherwise stated, we restrict our attention to directed rooted trees (also called in-trees), where each edge is oriented towards the root node[^10]. Source nodes $\mathcal{S}$ represent leaves of $\mathcal{G}$. The set of all edges in the graph is completely determined by the set of child nodes of all AFC and destination nodes. We let $\mathcal{V}^{(v)}$ denote the set of child nodes of an arbitrary node $v$. The collection of sets $\{\mathcal{V}^{(v)}\}_{v \in \mathcal{V}}$ fully describes the set of connections between modules.
Finally, we introduce the control elements: *topology processor* and *function processor*, that organize AFC modules into a global NFC evaluator. Based on the MTC server application requirements, the topology and function processors reconfigure the AFC modules to provide a requested MTC-NFC service. In particular, the function processor decomposes a required global NFC into a composition of local AFCs and configures each AFC module accordingly. In other words, based on the requested global network function $f(\mathbf{x}_1,\mathbf{x}_2,\ldots,\mathbf{x}_N)$, the function processor defines a set of atomic functions $\{g^{(a)}\}_{a \in \mathcal{A}}$ and configures the respective AFC modules. Similarly, by defining the graph $\mathcal{G}$ via the set $\{\mathcal{V}^{(v)}\}_{v \in \mathcal{V}}$ and by configuring each AFC node accordingly, the topology processor will interconnect AFC modules into a directed graph of MTC data flows. The topology and function processor are key NFC layer entities. They manage, connect and orchestrate the AFC layer entities, i.e., source modules and AFC modules.
CONDENSE Architecture: Implementation
-------------------------------------
The above described abstract Condense architecture can be mapped onto the 3GPP MTC architecture. We present initial insights here, while details are left for the following sections.
The AFC layer is composed of AFC modules that evaluate atomic functions. Examples of atomic functions suitable for AFC implementations are the addition, modulo addition, maximum/minimum, norm, histogram, linear combination, threshold functions, etc. Atomic functions can be evaluated straightforwardly in the digital domain using digital processing in network nodes. In addition to that, atomic functions could be realized by exploiting superposition of signals in the analog domain. Thus, we consider two types of AFC modules: i) Analog-domain AFC (A-AFC), and ii) Digital-domain AFC (D-AFC) modules.
An A-AFC, also referred to as an in-channel AFC, harnesses interference in a wireless channel or signal combining in an optical channel to perform atomic function evaluations. An example of the technology that can be easily integrated as an A-AFC module is the Physical Layer Network Coding (PLNC) [@plnc], [@plncweb], where the corresponding atomic function is finite field addition, e.g., bit-wise modulo 2 sum in the case of the binary field.
A D-AFC, also referred to as in-node AFC, evaluates atomic functions in the digital domain using, e.g., reconfigurable hardware-based modules in the context of SDN-based implementation [@sdnieee]. Alternatively, they can also be implemented using software-based virtual network functions in the context of a NFV-based implementation [@Han2015]. An example of the technology that can be easily integrated as a D-AFC module is the packet-level Random Linear Network Coding (RLNC) [@fragouli2006], [@practicalNC]. RLNC is a mature technology in terms of optimized software implementations (see, e.g., [@kodo]) and it evaluates linear combinations over finite fields as atomic functions. We note that it has been recently proposed and demonstrated within the SDN/NFV framework [@rlncsdn], [@rlncnfv].
The NFC layer can be naturally implemented within the SDN/NFV architecture. In particular, the topology processor naturally fits as an SDN application running on top of the SDN controller within the SDN architecture. In addition, the function processor role may be set within an NFV manager entity, e.g., taking the role of the NFV orchestrator. Using the SDN/NFV framework, MTC-NFC service can be quickly set and flexibly reconfigured according to requests arriving from a diverse set of MTC applications.
Atomic Function Computation Layer
=================================
In this Section, we discuss theoretical and implementation aspects of realizing atomic functions within AFC modules. Subsection [IV]{}-A discusses the AFC modules operating in the analog domain, while Subsection [IV]{}-B considers digital domain AFCs.
Analog-domain Atomic Function Computation (A-AFC)
-------------------------------------------------
**Wireless-domain A-AFC: Theory**. An A-AFC module’s functionality of computing functions over the incoming packets is based on harnessing interference, i.e., the superposition property of wireless channels. We survey the relevant literature on such function computation over wireless channels, finalizing the subsection with presenting current theoretical and technological capabilities.
The idea of harnessing interference for computation is investigated in terms of a joint source-channel communication scheme in [@Gastpar2003], targeting to exploit multiple access channel characteristics to obtain optimal estimation of a target parameter from noisy sensor readings. Extensions of the analog joint source-channel communication are further investigated in the literature, see e.g., [@ex1; @ex2; @ex3; @ex4]. Following the impact of network coding ideas across the networking research, reference [@plnc] proposes the concept of PLNC to increase throughput of wireless channels; PLNC essentially performs specific A-AFC computations (finite field arithmetics) in a simple two-way relay channel scenario. Computation of linear functions or, more precisely, random linear combinations of the transmitted messages over multiple access channels (MAC) has been considered in [@Nazer2007] and extended in [@Nazer2011]; therein, the authors propose the compute-and-forward (CF) transmission scheme for computing linear functions at the relays, who attempt to decode the received random message combinations (the randomness is induced by the fading channel coefficients) to integer combinations, which hence become lattice points in the original code lattice. After this, the relays forward the lattice points to the destination, who can then solve for the original messages provided that the received system of equations is invertible.
Finally, reference [@main] addresses non-linear function computation over wireless channels (see also [@Nazer2007]). While it is intuitive that a linear combination of packets (signals) from multiple sources can be obtained through a direct exploitation of interference, more general, non-linear functions can also be computed through introducing a non-linear (pre-)processing of packets prior to entering the wireless medium, and their (post-)processing after the pre-processed signals have been superimposed in the wireless channel.
Following [@main], we now describe in more detail how this non-linear function computation works – and hence how the A-AFC modules (in principle) operate. Assume that length-$L$ source node data packets $\mathbf{x}_{s}=(x_{s}[1],x_{s}[2],\ldots,x_{s}[L]), s \in \mathcal V^{(a)} \subseteq \mathcal{S}$, arrive at the input interfaces of an AFC node $a$. The packets are first pre-processed by the source node (MTC device) through a pre-processing function $\varphi_s(\mathbf{x}_{s})$. The result is the transmitted symbol sequence $\mathbf{y}_{s}=(y_{s}[1],y_{s}[2],\ldots,y_{s}[L])$, where ${y}_{s}[\ell]=\varphi_s ({x}_{s}[\ell]), \ell=1,2,\ldots,L$. Assuming a block-fading wireless channel model for narrowband signals, the received sequence can be modelled as $\textbf{r}^{(a)}=\left( r^{(a)}[1],r^{(a)}[2],\ldots,r^{(a)}[L] \right)$, where: $$\begin{aligned}
r^{(a)}[\ell] = \sum_{s \in \mathcal{V}^{(a)}} h_{s} \cdot y_{s}[\ell], \quad \ell=1,2,\ldots,L.\end{aligned}$$ At the destination, a post-processing function $\psi(\textbf{r}^{(a)})$ is used to obtain $\mathbf{x}^{(a)}$, where $x^{(a)}[\ell] = \psi(r^{(a)}[\ell])$. Therefore, symbol-wise, the A-AFC module $a$ realizes computation of the following (possibly non-linear) function: $$\begin{aligned}
\label{eqn-2-nomog}
g^{(a)}\left( \{x_s[\ell]\}_{s \in \mathcal{V}^{(a)}}\right)
= \psi\left( \sum_{s \in \mathcal{V}^{(a)}} h_{s} \,\varphi_s ({x}_{s}[\ell]) \right).\end{aligned}$$ The class of functions computable via A-AFC modules, i.e., which are of form , are called nomographic, and they include important functions such as the arithmetic mean and the Euclidean norm [@main].
Fig. \[WD-A-AFC\] illustrates an A-AFC: its position in the real-world system (left), its representation as an A-AFC module (central), and as part of the NFC graph (right).
**Wireless-domain A-AFC: Implementation**. The above framework implies that an A-AFC module is physically spread across all input devices and the output device connected to the A-AFC module, as illustrated in Fig. \[WD-A-AFC-2\]. At the input devices (e.g., MTC devices), an appropriate input A-AFC digital interface needs to be defined that accepts input data packets and implements pre-processing function $\varphi(\cdot)$ before the signal is transmitted into the channel. Similarly, at the output device, e.g., small base station (HeNB), an appropriate output A-AFC digital interface needs to be defined that delivers output data packets after the signal received from the channel is post-processed using $\psi(\cdot)$. We also note that, although above we assume input nodes to A-AFC module are source nodes (MTC devices), wireless-domain A-AFC module can be part of the wireless backhaul network, e.g., connecting several HeNBs to the eNB.
**Challenges and Future Directions**. In the current state-of-the-art, A-AFC is investigated in the context of joint computation and communications in wireless sensor networks. Current research works are limited in terms of the computed functions, such as addition, multiplication, norm, arithmetic/geometric means, and are also limited in scope, as they are only targeting the wireless – and not optical – communication links. Design and implementation of generic A-AFC in wireless setting which is adaptive to the channel conditions remains an open problem. Note that any such design should take practical implementation aspects into account, including channel estimation errors, timing and frequency offsets and quantization issues. Furthermore, the link qualities between network nodes, including adaptive schemes that select the computation nodes according to the robustness of communications between links, need to be considered to improve the reliability of the computed function outputs.
**Optical-domain A-AFC: Discussion**. If we consider the PLNC example, it is clear that wireless domain A-AFC modules are close to become a commercially available technology (see, e.g., [@plncweb]). The question that naturally arises is whether A-AFC modules can be implemented in optical channels within optical access networks such as passive optical networks (PON). This would further increase the richness of AFC layer and bring novel AFC modules into the MTC-NFC network. Here, we briefly comment on the status of optical-domain function computation.
Recent works analyzed applicability of network coding of data packets within PONs in some simple scenarios [@Miller2010][@NCPON]. However, in contrast to the above vision of A-AFC modules, in these works signals are not “in-channel” combined, rather, network coding is done at the end-nodes, in the digital domain.
Information processing in the photonic domain has been envisioned in the 1970s. But implementations of digital optical computing could not keep pace with the development of electronic computing. Nevertheless, with advances in technology, the role of optics in advanced computing has been receiving reawakened interest [@Caulfield]. Moreover, unconventional computing techniques, in particular reservoir computing (RC), find more and more interest and are being implemented in different photonic hardware. RC is a neuro-inspired concept for designing, learning, and analysing recurrent neural networks – neural networks where, unlike the most popular feed-forward neural networks, the interconnection network of neurons possesses cycles (feedback loops). A consequence of the presence of loops is, as pointed out in [@Lukosevicius], that recurrent neural networks can process and account for temporal information at their input. A recent breakthrough was a drastic simplification of the information-processing concept of reservoir computing (RC) in terms of hardware requirements [@Appeltant]. The appeal of RC therefore resides not only in its simple learning, but moreover in the fact that it enables simple hardware implementations. Complex networks can be replaced by a single or a few photonic hardware nodes with delayed feedback loops [@Larger2012], [@Paquot2012], [@Brunner]. Different tasks, including spoken digit recognition, nonlinear time series prediction and channel equalization have been performed with excellent performance, speed and high energy efficiency [@Brunner], [@Vandoorne]. Beyond these first successes, meanwhile, using simple hardware, learning approaches including RC, extreme learning machines and back-propagation learning of recurrent neural networks have been demonstrated [@Hermans2015], illustrating the flexibility and potential of this approach.
Digital-domain Atomic Function Computation (D-AFC)
--------------------------------------------------
D-AFC modules evaluate atomic functions in the digital domain, within the network nodes such as base stations (eNB or HeNB) and core network gateways (HeNB-GW, S-GW, P-GW). Although digital-domain in-node processing offers many possibilities for D-AFC implementation, here we address two possible options suitable for the SDN and NFV architectures.
The first option for D-AFC are reconfigurable hardware-based Field Programmable Gate Array (FPGA) platforms. FPGA platforms are frequently used in combination with high-speed networking equipment to perform various work-intensive and high-throughput demanding functions over data packets such as packet filtering [@netfpga]. FPGAs are either integrated in network nodes as co-processing units, or can be easily attached as external units to network nodes via high-speed network interfaces. FPGAs offer flexible and reconfigurable high-throughput implementations of various linear or non-linear atomic functions. For example, implementing random linear combinations over input data packets in network nodes – as part of RLNC – is considered in several recent works [@rlncfpga1], [@rlncfpga2]. D-AFC implementations via FPGA platforms offer seamless integration in SDN concepts, because SDN data flows can be easily filtered and fed into either internal or external FPGA units. Depending on the application, FPGAs achieve speed increase over general processing units by factor of tens to hundreds. Note also that FPGAs can be reprogrammed and reconfigured in short time intervals (order of minutes).
The second possibility for efficient D-AFC is to use software-based implementations in high-level programming languages that run on general processing units, either in network nodes or externally on dedicated general-purpose servers [@kodo]. This approach offers full flexibility for atomic function evaluation at the price of lower data processing throughput as compared with the FPGA approach. An example of a D-AFC implementation of random linear combinations over incoming data packets in the context of RLNC is given in [@rlncsdn], [@rlncnfv]. Software-based D-AFC implementations can be easily and remotely instantiated across the network nodes in a virtualized environment following NFV concepts.
Fig. \[D-AFC\] illustrates a D-AFC: its position in the real-world system (left), its representation as an D-AFC module (center), and as part of the NFC graph (right).
Network Function Computation Layer
==================================
The NFC layer is responsible for configuring the Condense topology and assigning the appropriate atomic functions across the AFC modules, such that a desired network-wide function computation is realized. Subsection [V-A]{} discusses theoretical aspects (capabilities and limitations) of computing functions over networks, surveying the relevant literature on sensor fusion and network coding for computing. Subsection [V-B]{} describes a possible implementation of NFC functionalities within the 3GPP MTC system, through a more detailed view of SDN/NFV modules, i.e., the function and topology processors.
Theoretical Aspects of NFC Layer
--------------------------------
The need for mathematical theory of function computation in networks is advocated in [@Giridhar05], [@Giridhar]. The authors discuss various challenges in sensor networks, and argue that computation of functions in a sensor network could lead to a lower data overhead, as well as to a reduced data traffic. For our toy example, in the fire alarm sensor network, we are only interested in the measurements of the highest temperature in the set of sensors. Alternatively, in monitoring temperature range in a green house, we might only be interested in the measurements of the average temperature from the set of sensors. Therefore, for various practical applications, it would be beneficial if the network node would be able to perform basic (atomic) computation, which in the context of the whole network could lead to computation of more sophisticated functions in the destination nodes.
This subsection elaborates on the mathematical tools behind the realization of the Condense NFC layer. There is a number of works studying function computation over a network, which are available in the literature. The relevant work includes those in contexts of sensor fusion, network coding for computing, and neural networks. The two former work threads are discussed here, while the latter is discussed in Subsection [VI-C]{}. Hereafter, we mostly follow the framework defined in [@AFKZ; @Kowshik2012], adapting notation to our needs here.
**Mathematical settings**. Consider a finite directed acyclic graph $\mathcal G = (\mathcal V, \mathcal E)$, consisting of $M$ AFC nodes belonging to set $\mathcal{A}$, a set of $N$ sources (MTC devices) $\mathcal{S}$, and a set of $R$ destinations $\mathcal{D}$, such that $\mathcal{S} \cap \mathcal{D} = \varnothing$. The network uses a finite alphabet $\mathbb A$, called *network alphabet*. Each source $s$ generates $K$ random symbols $\sigma_s[1], \sigma_s[2],\ldots,\sigma_s[K] \in \mathbb A$. Here, we say that the source symbol $\sigma_s[k]$ belongs to the $k$-th generation of the source symbols.
We assume that each packet sent over a network link is a vector of length $L$ over $\mathbb A$. Suppose that each of the $R$ destination nodes requests computation of a (vector-valued) function $f$ of the incoming MTC device vectors ${\boldsymbol \sigma}_s,$ $s=1,...,N$. The target vector function is of the form $f: {\mathbb A}^{N\cdot K} \rightarrow {\mathbb B}^K$, where $\mathbb B $ is a function alphabet, and each component function $f: {\mathbb A}^{N} \rightarrow {\mathbb B}$ is of the same form, applied to each source’s $k$-th symbol, $k=1,...,K$. More precisely, we wish to compute $f \left( \sigma_1[k],...,\sigma_N[k]\right)$, $k=1,...,K$.
With each arc $a \rightarrow v$ outgoing an AFC node $a \in \mathcal{A}$, we associate the atomic function $g^{(a \rightarrow v)} \left( \cdot \right)$, which takes the $|\mathcal{V}^{(a)}|$ length-$L$ incoming vectors $\mathbf{x}^{(u \rightarrow a)}$, $u \in {\mathcal V}^{(a)}$, and produces the length-$L$ outgoing vector $\mathbf{x}^{(a \rightarrow v)}$, i.e.: $$\mathbf{x}^{(a \rightarrow v)} = g^{(a \rightarrow v)} \left( \{\mathbf{x}^{(u \rightarrow a)}\}_{u \in \mathcal{V}^{(a)}}\right).$$
Similarly, with each arc $s \rightarrow v$ outgoing a source node $s \in \mathcal{S}$, the atomic function $g^{(s \rightarrow v)} \left( \cdot \right)$ takes the $|\mathcal{V}^{(s)}|$ length-$L$ incoming vectors $\mathbf{x}^{(u \rightarrow s)}$, $u \in {\mathcal V}^{(s)}$, as well as the $K$ generated symbols $\boldsymbol{\sigma}_s$ = $(\sigma_s[1],...,\sigma_s[K])$, and produces the length-$L$ outgoing vector $\mathbf{x}^{(s \rightarrow v)}$, i.e.: $$\mathbf{x}^{(s \rightarrow v)} = g^{(s \rightarrow v)} \left( \{\mathbf{x}^{(u \rightarrow s)}\}_{u \in \mathcal{V}^{(s)}};\,\boldsymbol{\sigma}_s\right).$$ (Note that we consider here the most general case in which a source node does not have to lie on the “bottom-most” level of the network, i.e., it can also have some incoming edges.) We refer here to both $g^{(a \rightarrow v)}$’s and $g^{(s \rightarrow v)}$’s as *encoding functions*.
Finally, a destination node $d \in \mathcal D$ takes its $|\mathcal{V}^{(d)}|$ incoming length-$L$ messages and performs decoding, i.e., it produces the vector of function evaluation estimates $\widehat{\mathbf{f}}^{(d)} = \left(\widehat{f}^{(d)}[1],...,\widehat{f}^{(d)}[K]\right)$, as follows: $$\widehat{\mathbf{f}}^{(d)} = \Psi^{(d)}\left( \{\mathbf{x}^{(u \rightarrow d)}\}_{u \in \mathcal{V}^{(d)}}\right),$$ where $\Psi^{(d)}(\cdot)$ is the destination node $d$’s function. Note that $\Psi^{(d)}(\cdot)$ recovers back the $K$-dimensional vector from the $L$-dimensional incoming quantities (where $L>K$), and it is therefore referred to as a decoding function.
We say that the destination $d \in \mathcal D$ *computes* the function $f \; : \; {\mathbb A}^N \rightarrow {\mathbb B}$, if for every generation $k \in \{1,...,K\}$, it holds that: $$\widehat{f}^{(d)}[k] = f \left( \sigma_1[k],...,\sigma_N[k]\right).$$ Further, we say that the problem of computing $f$ is *solvable* if there exist atomic functions $g^{(s \rightarrow v)}\left( \cdot\right)$, $g^{(a \rightarrow v)}\left( \cdot\right)$ across all arcs in $\mathcal E$ and decoding functions $\Psi^{(d)} \left( \cdot \right)$, $d =1,...,R$, such that $f$ is computed at all destinations $d \in \mathcal D$ (that is, their corresponding composition computes $f$ at all destinations).
**Connection to network coding and beyond**. The reader can observe that the problem of network coding [@Ahlswede] is a special case of the function computation problem with $L = K = 1$, where the target function $f$ is an identity function: $f(\sigma_1, \sigma_2, \cdots, \sigma_N ) = (\sigma_1, \sigma_2, \cdots, \sigma_N ).$ In particular, in linear network coding, the alphabet $\mathbb A$ is taken as a finite field $\mathbb F$, the function alphabet $\mathbb B$ is ${\mathbb{F}}^N$, and all encoding functions $g^{(a \rightarrow v)}$, $g^{(s \rightarrow v)}$ and decoding functions $\Psi^{(d)}$ are linear mappings over ${\mathbb{F}}$. The case of linear network coding is relatively well understood. In particular, it is known that the problem of computing $f$ is solvable if and only if each of the minimum cuts between all the sources and any destination has capacity of at least $N$ [@KM]. In Subsection [VI-A]{}, we provide further details on this special case.
For non-linear network coding, the universal criteria for network coding problem solvability are not fully understood. It is known, for example, that for the case where each sink requests a subset of the original messages, there exist networks, which are not solvable by using linear functions $g^{(a \rightarrow v)}(\cdot)$, $g^{(s \rightarrow v)}(\cdot)$ and $\Psi^{(d)}(\cdot)$, yet they can be solved by using non-linear functions (see, for example, [@Dougherty]).
In order to understand the fundamental limits on solvability of the general function computation problem, the authors of [@AFKZ] define what they term the computing capacity of a network as follows: $$\begin{gathered}
{\mathsf{C}}(\mathcal G,f) = \\
\sup \left\{ \frac{K}{L} \; : \; \mbox{computing $f$ in $\mathcal G$ is solvable } \right\} \; .\end{gathered}$$ They derive a general min-cut type upper bound on the computing capacity, as well as a number of more specific lower bounds. In particular, special classes of functions, such as symmetric functions, divisible functions and exponential functions, are considered therein (see [@Dougherty] for more detail). It should be mentioned that the considered classes of functions are rather restricted, and that they possess various symmetry properties. The problem turns out to be very difficult, however, for more general, i.e., less restricted, classes of functions.
Another related work is [@AF], where a set-up with linear functions $g^{(a \rightarrow v)}(\cdot)$, $g^{(s \rightarrow v)}(\cdot)$ and $\Psi^{(d)}(\cdot)$ and general linear target function $f$ is considered. The authors are able to characterize some classes of functions, for which the cut-set bound gives sufficient condition for solvability, and for which it does not.
**Other results**. In [@Giridhar05], the function computation rate is defined and lower bounds on such rate are obtained for various simple classes of functions. Recently, in [@ST], information-theoretic bounds on the function computation rate were obtained for a special case, when network is a directed rooted tree, and the set of sinks contains only the root of the tree, and the source symbols satisfy a certain Markov criterion.
A number of works study computation of sum in the network. It is shown in [@RaiDey] that if each sink requests a sum of the source symbols, the linear coding may not be sufficient in networks where non-linear coding can be sufficient. Other related works include [@RL; @Kannan; @Kowshik; @Shah; @Lalitha].
There is a significant number of works related to *secure* function computation available in the literature. However, usually, the main focus of these works is different. We leave that topic outside of the scope of this paper.
**Challenges and research directions**. Research on network function computation is still in its infancy and general theoretic foundations are yet to be developed. Here, we identify several challenges with network function computation relevant for Condense architecture. First, it is important to consider the issue of solvability when the encoding and decoding functions $g^{(a \rightarrow v)}(\cdot)$, $g^{(s \rightarrow v)}(\cdot)$ and $\Psi^{(d)}(\cdot)$ are restricted to certain classes dictated by the underlying physical domain. For instance, A-AFC modules operating in the wireless domain are currently restricted to a certain class of functions (see Subsection [IV-A]{}), while, clearly, D-AFCs operating in the digital domain have significantly more powerful capabilities. Second, it interesting to study NFC in simpler cases, when the network topology is restricted to special classes of graphs, for example rooted trees, directed forests, rings, and others. Third, under the above defined constraints on the AFC capabilities, the question that arises is how well we can approximate a desired function, even if solvability is impossible. Finally, practical and efficient ways for *actual constructions* of $g^{(a \rightarrow v)}(\cdot)$, $g^{(s \rightarrow v)}(\cdot)$ and $\Psi^{(d)}(\cdot)$, as opposed to existence-type results, are fundamental for Condense implementation.
Implementation Aspects of NFC Layer
-----------------------------------
In practical terms, the NFC layer should deal with control and management tasks of establishing and maintaining an NFC graph of AFC modules for a given service request, as sketched in Fig. \[NFCgraph\]. In our vision, the main control modules that define the NFC layer functionality are: the function processor (FP) and the topology processor (FP) (see Fig. \[MTC-NFC\]). Both modules can be seamlessly integrated in the SDN/NFV architecture.
The TP module organizes the MTC data flows and sends configuration instructions via the SDN control plane. In abstract terms, for all nodes in a directed rooted tree (or directed acyclic graph), TP needs to provide the set of child nodes $\{\mathcal{V}^{(v)}\}_{v \in \mathcal{V}}$ from which to accept MTC data flows, and to identify the exact MTC data flows that will be filtered for each output flow, if there are multiple output flows.
Based on the MTC server application requests and the configured topology, FP processes the global function request, and, based on the available library of AFC modules, it generates the set of atomic functions to be used: $\{g^{(a)}\}_{a \in \mathcal{A}}$. Note that, as described before, AFC modules may be: i) A-AFC modules, ii) hardware-based D-AFC modules, and ii) software-based D-AFC modules. A-AFC modules (e.g., PLNC module) need more complex instantiation control as they spread over several physical nodes and involve configuration of input and output interfaces and pre/post-processing functions (Sec. IVA). Hardware-based D-AFC modules (e.g., internal/external FPGA modules within or attached to network elements) require SDN-based control of MTC data flows that will filter selected flows and direct them through the D-AFC module. Finally, the most flexible case of software-based D-AFC modules (e.g., software-based modules in virtual machines running over the virtualized hardware in network elements or external servers) is a library of AFC implementations where each atomic function from the library can be remotely instantiated via the NFV control. Overall, FP needs to know the list of available AFC resources in the entire NFC network in order to optimize the set of instantiated AFC modules.
In terms of realization, we use the standard proposal for NFV/SDN complementary coexistence [@Li2015] where the FP module can be implemented as an NFV architecture block called the NFV orchestrator. The TP module can be implemented as an SDN application. Besides communicating directly, both FP and TP modules, observed as SDN applications, approach the SDN control plane via the SDN northbound interface. Based on the FP/TP inputs, the SDN control plane will configure physical devices (network nodes) via the southbound interface. This NFV/SDN based control of Condense is illustrated in Fig. [7]{}.
Application Layer
=================
In this section, we present three examples of applications at the application layer of the Condense architecture: data recovery through network coding, minimizing population risk via a stochastic gradient method, and binary classification via neural networks[^11]. The purpose of these examples is two-fold. First, they demonstrate that a wide range of applications can be handled via the Condense architecture. Second, they show that Condense is compatible with widely-adopted concepts in learning and communications, such as random linear network coding, stochastic gradient methods, and neural networks.
The three examples are also complementary from the perspective of the workload required by the FP module. With the first example (data recovery via network coding), the function of interest is decomposed into mutually uncoordinated atomic (random) linear functions, and hence no central intervention by the function controller is required, nor is the inter-AFC modules coordination needed as long as atomic functions are concerned. With the third example (binary classification via neural networks), the desired network-wide function is realized through a distributed coordination of the involved AFC modules. Finally, with the second example (minimizing population risk via a stochastic gradient method), the most generic case requires the intervention of the central FP module, in order that the desired network-wide function be decomposed and computed.
Data recovery through network coding
------------------------------------
A special case of an application task with the Condense architecture is to deliver the *raw data* to the data center of interest. This corresponds to a trivial, identity function over the input data as a goal of the overall network function computation. However, this is not achieved through simply forwarding the raw data to the data center, but through the usage of network coding. In other words, atomic functions are not identity functions but random linear combinations over the input data. While such solution may not reduce the total communication cost with respect to the conventional (forwarding) solution, this solution is significantly more flexible, robust and reliable, e.g., [@fragouli2006], [@practicalNC]. As recently noted, it can be flexibly implemented within the context of network coded cloud storage [@RLNCcloud].
We follow the standard presentation of linear network coding, e.g., [@fragoulibook], adapting it to our setting. For ease of presentation, we assume here that graph $\mathcal G$ is a directed rooted tree. Therein, the destination node is the root of the tree. Suppose that each of the $N$ available MTC devices has a packet $\mathbf{x}_s$ consisting of $L$ symbols, each symbol belonging to a finite field $\mathbb F$. We adopt the finite field framework as it is typical with network coding. The goal is to deliver the whole packet vector $\mathbf{x}=(\mathbf{x}_1,...,\mathbf{x}_N)$ to the data center (the destination node $d$). With Condense, this is achieved as follows. Each atomic node $a$ generates the message pair $\left( \mathbf{x}^{(a)}, \mathbf{c}^{(a)} \right)$ (to be sent to the parent node) based on the received messages from its child nodes $\left( \mathbf{x}^{(b)}, \mathbf{c}^{(b)} \right)$, where $b \in {\mathcal V}^{(a)}$, and we recall that $\mathcal{V}^{(a)}$ is the set of child nodes of node $a$. As we will see, the quantity $\mathbf{x}^{(a)} \in {\mathbb F}^L$ is by construction a linear combination of the (subset of) MTC devices’ packets $\mathbf{x}_s \in {\mathbb F}^L$, $s=1,...,N$. The quantity $\mathbf{c}^{(a)} = (c^{(a)}[1],...,c^{(a)}[N]) \in {\mathbb F}^N$ stacks the corresponding weighting (or coding) coefficients; that is: $$\label{eqn-net-cod-1}
\mathbf{x}^{(a)} = \sum_{s=1}^N c^{(a)}[s]\,\,\mathbf{x}_s.$$ Now, having received $\left( \mathbf{x}^{(b)}, \mathbf{c}^{(b)} \right)$, $b \in {\mathcal V}^{(a)}$, node $a$ computes $\mathbf{x}^{(a)}$ using random linear network coding approach. It first generates new random (local) coding coefficients ${e}^{(a)}[b] \in \mathbb F$, $b \in {\mathcal V}^{(a)}$, uniformly from $\mathbb F$, and independently of the received messages. Then, it forms $x^{(a)} $ as: $$x^{(a)} = \sum_{b \in \mathcal{V}^{(a)}} e^{(a)}[b] \,\,\mathbf{x}^{(b)}.$$ Once $\mathbf{x}^{(a)}$ has been computed, node $a$ also has to compute the global coding coefficients $\mathbf{c}^{(a)}$ *with respect to the MTC packets* $\mathbf{x}_s$, $s=1,...,N$, as per . It can be shown that: $$c^{(a)}[s] = \sum_{b \in \mathcal{V}^{(a)}} e^{(a)}[b] \,\,c^{(b)}[s],\,s=1,...,N.$$ For the end-leaf (MTC device) nodes $s$, we clearly have that $c^{(s)}[s]=1,$ and $c^{(s)}[u] =0$, $u \neq s$. Once the destination (root) node $d$ receives all its incoming messages, it has available a random linear combination of the MTC’s packets $\mathbf{x}_1,...,\mathbf{x}_N$: $$\mathbf{x}^{(d),1} = \sum_{s=1}^N c^{(d),1}[s]\,\,\mathbf{x}_s,$$ and the corresponding global coding coefficients vector $\mathbf{c}^{(d),1} = \left( c^{(d),1}[1],...,c^{(d),1}[N] \right)$. Afterwards, the whole process described above is repeated sequentially $N^\prime -1$ times, such that the data center obtains $N^\prime-1$ additional pairs $\left( \mathbf{x}^{(d),k}, \mathbf{c}^{(d),k}\right)$, $k=2,...,N^\prime$. It can be shown that, as long as $N^\prime$ is slightly larger than $N$, MTC data vector $\mathbf{x}=\left(\mathbf{x}_1,...,\mathbf{x}_N\right)$ can be recovered with high probability through solving the linear system of equations with unknowns $\mathbf{x}_s$: $$\mathbf{x}^{(d),k} = \sum_{s=1}^N c^{(d),k}[s]\,\,\mathbf{x}_s,\,k=1,...,N^\prime.$$ Note that, for this application example, each atomic function is *linear*. Moreover, there is no requirement on the *coordination* of the atomic functions which correspond to different atomic nodes, as they are generated randomly and mutually independently [@Ho2006]. Hence, this application does not require a centralized control by the FP module.
Finally, when certain a priori knowledge on $\mathbf{x}$ is available (e.g., sparsity, i.e., many of the packets $\mathbf{x}_s$ are the zero $L$-tuples of symbols from $\mathbb F$), then the recovery probability close to one can be achieved even when the number of linear combinations $N^\prime$ at the MTC server is significantly smaller than $N$. Omitting details, this can be in principle achieved using the theories of compressed sensing and sparse recovery, e.g., [@Candes], [@Hayashi].
Statistical estimation and learning
-----------------------------------
A dominant trend in current machine learning research are algorithms that scale to large datasets and are amenable to modern distributed processing systems. Machine learning systems are widely deployed in architectures with a large number of processing units at different physical locations and communication is becoming a resource that is taking the center stage in the algorithm design considerations [@Jordan2015; @Zhang2013; @Agarwal2011; @Boyd2011; @Shamir2014].
Typically, the task of interest (parameter estimation, prediction, etc.) is performed through solving an optimization problem of minimizing a risk function [@Wasserman2004]. In most widely used models of interest, which include logistic regression and neural networks, this optimization needs to be performed numerically using gradient descent methods and is simply based on successive computation of the gradient of the loss function of interest. In large datasets (of size $T$), obtaining the full gradient comes with a prohibitive computational (a linear computational cost in $T$ *per iteration* of gradient descent cannot be afforded) as well as a prohibitive communication cost (due to the need to access *all* training examples even though they may be, and typically are, stored at different physical locations). For these reasons, stochastic gradient methods are the norm – they typically access only a small number of data points at a time, giving an unbiased estimate to the gradient of the loss function needed to update the parameter values – and have enjoyed tremendous popularity and success in practice [@Bottou2011; @Yousefian2012; @NiuArxiv; @Bottou2010].
Most existing works assume that the data has already been collected and transmitted through the communication architecture and is available at request. That is, typically the data is first transmitted in its raw form from the MTC devices to the data center, and only afterwards a learning (optimization) algorithm is executed.
In contrast, the Condense architecture integrates the learning task into the communication infrastructure. That is, a learning task is seen as a sequence of oracle calls to a certain network function computation, and the role of the NFC layer is to provide these function computations at the data center’s processing unit (destination node) and only the computed value (e.g., of the gradient to the loss function) is being communicated. This way, Condense will generically embed various learning algorithms into the actual 3GPP MTC communication infrastructure.
We now dive into more details and exemplify learning over the proposed Condense system with the estimation of an unknown parameter vector $\mathbf{w}^\star \in {\mathbb R}^Q$ through the minimization of population risk. Specifically, we consider stochastic gradient-type methods to minimize the risk. To begin, consider a directed rooted tree NFC graph $\mathcal G$, and assume that there are $N$ MTC devices which generate samples $\mathbf{x}_t \in {\mathbb R}^{L\cdot N}$ over time instants $t=0,1,2,...$, drawn i.i.d. in time from a distribution $ \mathbf{X} = (\mathbf{X}_1,...,\mathbf{X}_N)\sim {\mathbb P}$, defined over ${\mathbb R}^{L\,N}$. Here, $\mathbf{x}_t = (\mathbf{x}_{1,t},...,\mathbf{x}_{N,t})$, where $\mathbf{x}_{s,t} \in {\mathbb R}^L$ is a sample of $\mathbf{X}_s$ generated by the MTC device $s$. The goal is to learn the parameter vector $\mathbf{w}^\star \in {\mathbb R}^Q$ that minimizes the population risk: $$\label{eqn-app-layer-loss}
\mathbb E_X \left[ \phi(\mathbf w; \mathbf X) \right],$$ where expectation is well-defined for each $\mathbf w \in {\mathbb R}^Q$, and, for each $\mathbf x \in {\mathbb R}^{L\cdot N}$, function $\phi(\cdot;\mathbf x):\,{\mathbb R}^Q \rightarrow \mathbb R$ is differentiable. In the rest of this subsection, we specialize the approach on a single but illustrative example of Consensus; more elaborate examples such as logistic regression are relevant but not included here for brevity.
**Example: Consensus – computing the global average; e.g., [@Consensus; @DeGroot1974; @Kar2009]**. When $w \in {\mathbb R}$, $\mathbf x = (x_1,...,x_N) \in {\mathbb R}^N$ (each MTC device generates scalar data), and $\phi(w,\mathbf x) = \frac{1}{N}\sum_{s=1}^N (x_s-w)^2$, then solving corresponds to finding $\frac{1}{N}\sum_{s=1}^N \mathbb E[X_s]$. E.g., when MTC devices are pollution sensors at different locations in a city, this corresponds to finding the city-wide average pollution.
**Conventional 3GPP MTC solution**. Consider first the conventional 3GPP MTC system, where samples $\mathbf{x}_t$, $t=0,1,2,...$, arrive (through the communication layer) to a processing unit at the data center at time instants $t=0,1,2,...$ *in their raw form*. (We ignore here the communication delays.) Upon reception of each new sample $\mathbf{x}_t$, the processing unit (destination node $d$) performs a stochastic gradient update to improve its estimate $\mathbf{w}^{(t)} \in {\mathbb R}^Q$ of $w^\star$: $$\label{eqn-app-layer-stoch-grad}
\mathbf{w}^{(t+1)} = \mathbf{w}^{(t)} - \eta_t \, \nabla \phi\left( \mathbf{w}^{(t)};\,\mathbf{x}_t\right),$$ where $\nabla \phi\left( \mathbf{w}^{(t)};\,\mathbf{x}_t\right)$ is the gradient of $\phi\left( \cdot ;\,\mathbf{x}_t\right)$ at $\mathbf{w}^{(t)}$, and $\eta_t$ is the step-size (learning rate). For the consensus example with learning rate $\eta_t = 1/(t+1)$, it can be shown that update takes the particularly simple form: $$\label{eqn-app-layer-stoch-grad-cons}
{w}^{(t+1)} = \frac{t}{t+1} {w}^{(t)} + \frac{1}{t+1}\left(\frac{1}{N}\sum_{s=1}^N x_{s,t}\right).$$ Note that, with the conventional 3GPP MTC architecture, data samples $\mathbf{x}_t$ are transmitted to the destination node $d$ for processing in their entirety, i.e., the communication infrastructure acts only as a routing network (forwarder) of the data. **Condense solution**. In contrast with the conventional solution, with the Condense architecture the raw data sample $\mathbf{x}_t$ is not transmitted to the data center and is hence not available at the corresponding processing unit. Instead, update is implemented as follows. Given the current estimate $\mathbf{w}^{(t)}$, the processing unit (destination node $d$) defines function $f_t(\cdot):=\nabla \phi\left(\mathbf{w}^{(t)};\cdot\right)$. Subsequently, it sends the request to the function processor to perform the decomposition of $f_t(\cdot)$ over the NFC layer. The function processor performs the required decomposition of $f_t(\cdot)$ into atomic functions and remotely installs the corresponding obtained atomic function at each atomic node (eNB, HeNB, etc.) of the topology.[^12] Once the required atomic functions are ready, the sample $\mathbf{x}_t$ starts travelling up the graph $\mathcal G$, and upon the completion of evaluation of all intermediate atomic functions, the value $f_t(\mathbf{x}_t)$ becomes available at the data center. This in turn means that the processing unit can finalize update . Specifically, with the consensus example in , function $f_t(\cdot)$ takes a particularly simple form of the average: $f_t(x)= f(x) = \frac{1}{N} \sum_{s=1}^N x_s$, and it is independent of $ {w}^{(t)}$ and of $t$. There exist many simple and efficient methods to decompose[^13] the computation of the average, e.g., [@Aggregation], and hence algorithm can be implemented very efficiently within the Condense architecture.
**Challenges, insights and research directions**. We close this subsection by discussing several challenges which arise when embedding learning algorithms in the Condense architecture. Such challenges are manyfold but are nonetheless already a reality in machine learning practice. First, data arrives in an asynchronous, delayed, and irregular fashion, and it is often noisy. Condense actually embraces this reality and puts the learning task at the center stage: the desired function of the data is of interest, not the data itself. Secondly, it is often the case that, depending on the infrastructure, interface and functionality constraints of the network computation layer, approximations of the desired function computations (as opposed to exact computations) will need to be employed. For instance, function $f_t(\cdot):=\nabla \phi\left(\mathbf{w}^{(t)};\cdot\right)$ in the example above may only be computable approximately in general. The quality of such an approximation leads to trading-off statistical efficiency of the learning procedure with the accuracy of the network function computation, and the analyses of such trade-offs will be an important research topic. Finally, from a more practical perspective, an important issue is to ensure interoperability with the existing distributed processing paradigms (e.g., Graphlab [@Graphlab] and Hadoop [@Hadoop]).
Neural networks
---------------
With modern large scale applications of neural networks, the number of parameters to be learned (neuron’s weights) can be excessively large, like, e.g., with deep neural networks [@Bengio2015]. As such, storage of the parameters themselves should be distributed, and their updates also include a large communication cost that needs to be managed [@Li2014]. However, neural networks can be naturally embedded into the Condense architecture – somewhat similarly to the related work on distributed training for deep learning [@Dean2012] – as detailed next.
Specifically, consider the example of binary classification of the MTC devices’ generated data. At each time instant $t$, $N$ MTC devices generate data vector $\mathbf{x}_t = \left( \mathbf{x}_{1,t},...,\mathbf{x}_{N,t} \right) \in {\mathbb R}^{N\cdot L}$, where each device generates an $L$-dimensional vector $\mathbf{x}_{i,t}$. Each data vector $\mathbf{x}_t$ is associated with its class label $y_t \in \{-1,1\}$. A binary classifier $F:\,{\mathbb R}^{N\cdot L} \rightarrow \{-1,1\}$ takes a data sample $\mathbf{x}_t$ and generates an estimate $F(\mathbf{x}_t)$ of its class label $y_t$. Classifier $F$ is “learned” from the available training data $\left( \mathbf{x}_t,\mathbf{y}_t\right)$, $t =0,1,...,T$, where $T$ is the *learning period*. In other words, once the learning period is completed and $F$ is learned, then the *prediction period* is initiated, and for each new data sample $\mathbf{x}_t$, $t>T$, classifier $F$ generates an estimate of $y_t$. For example, $\mathbf{x}_t$ can correspond to measurements of pressure, temperature, vibration, acoustic, and other sensors in a large industrial plant within a time period $t$; $y_t=1$ can correspond to the “nominal” plant operation, while $y_t=-1$ to the “non-nominal” operation, defined for example as the operation where energy efficiency or greenness standards are not fully satisfied. We consider neural network-based classifiers $F$ embedded in the Condense architecture. Therein, the classifier function $F$ is a composition of the neuron functions associated with each AFC module (node). We consider a Condense rooted tree graph $\mathcal G$ with $N$ sources and one destination node $d$, however, here we assume $\mathcal G$ is *undirected* as we will need to pass messages upwards and downwards. For convenience, as it is common with neural networks, we organize all nodes in $\mathcal G$ (source, atomic, and the destination node) in levels $\ell = 1,2,...,\mathcal L$, such that the leaves of nodes at the first level ($\ell=1$) are the MTC devices (sources in $\mathcal S$), while the data center’s processing unit (the destination node $d$) corresponds to $\ell = \mathcal L$. Then, all nodes (sources, atomic nodes, and the destination node) are indexed through the index pair $(\ell,m)$, where $\ell=0,1,...,\mathcal L$ is the level number and $m$ is the order number of a node within its own level, $m=1,...,\mathcal{N}_{\ell}$. Here, $\mathcal{N}_{\ell}$ denotes the number of nodes at the $\ell$-th level.
**Prediction**. We first consider the *prediction period* $t >T$, assuming that the learning period is completed. This corresponds to actually executing the application task of classification, through evaluating the network function $F$ at a data sample $\mathbf{x}_t$. (As we will see ahead, the learning period corresponds to *learning* function $F$, which essentially parallels the task of how a desired network function is decomposed across the AFC modules into the appropriate atomic functions.) Each node $(\ell,m)$ is assigned a weight vector $\mathbf{w}^{(\ell,m)}$ (obtained within the learning period), whose length equals the number of its associated leaf nodes. Denote by ${x}^{(\ell,m)}_t$ the output (also referred to as activity) of node $(\ell,m)$ associated with the data sample $\mathbf{x}_t$, $t > T$, to be computed based on the incoming activities $\mathbf{x}^{(\ell-1,q)}_t$ from the adjacent lower level nodes $(\ell-1,q)$. Also, denote by $\mathbf{x}^{(\ell-1)}_t$ the vector that stacks all the $\mathbf{x}^{(\ell-1,q)}_t$’s at the level $\ell-1$. Then, ${x}^{(\ell,m)}_t$ is calculated by: $$\label{eqn-NN-1}
{x}^{(\ell,m)}_t = \mathcal{U} \left( (\mathbf{w}^{(\ell,m)})^\top \mathbf{x}^{(\ell-1)}_t \right),$$ where $z \in \mathbb R \mapsto \mathcal{U}(z) = \frac{1}{1+\mathrm{exp}(z)}$ is the logistic unit function. Therefore, with neural networks, the atomic function $g^{(\ell,m)}(\cdot)$ associated with each AFC module (node) $(\ell,m)$ is a composition of 1) the linear map parameterized with its weight vector $\mathbf{w}^{(\ell,m)}$; and 2) the logistic unit function.
**Learning**. The learning period corresponds to learning function $F$, i.e., learning the weight vectors $\mathbf{w}^{(\ell,m)}$ of each AFC module. Differently from the example of minimizing a generic population risk in Subsection [VI-B]{}, here learning $F$ (learning atomic functions of ATC modules) can be done in a distributed way, without the involvement of the FP module. The learning is distributed in the sense that it involves passing messages in the “upward” direction (from the MTC devices towards the data center) and the “downward” direction (from the data center towards the MTC devices) along the Condense architecture (graph $\mathcal G$).
Specifically, we assume that weight vectors $\mathbf{w}^{(\ell,m)}$ are learned by minimizing the log-loss $J \left( \{ \mathbf{w}^{(\ell,m)}\} \right)$ via a stochastic gradient descent (back-propagation) algorithm, wherein one upward/downward pass corresponds to a single training data sample $\mathbf{x}_t$, $t\leq T$. We now proceed with detailing both the upward and the downward pass [@Ripley1996]. We assume that, before initiating the pass, $\mathbf{x}_t$ is available at the bottom-most layer (MTC devices), while label $y_t$ is available at the data center (destination node $d$). This is reasonable to assume as the label’s data size per $t$ is insignificant (here it is just one bit) and can be delivered to the data center, e.g., by forwarding (conventional) means through the 3GPP MTC system. **Upward pass**. Each node $\left(\ell,m\right)$ computes the gradient of its activity with respect to its weights as well as with respect to the incoming activities: $$\frac{\partial {x}_t^{(\ell,m)}} {\partial \mathbf{w}^{(\ell,m)}}= {x}_t^{(\ell,m)} \left( 1-{ x}_t^{(\ell,m)} \right) \mathbf{x}_t^{(\ell-1)}.$$
$$\frac{\partial { x}_t^{(\ell,m)} }{\partial \mathbf{x}_t^{(\ell-1)} }={ x}_t^{(\ell,m)} \left(1-{ x}_t^{(\ell,m)} \right) \mathbf{w}^{(\ell,m)}.$$
At this point, node $(\ell,m)$ stores tuple $\left(t,\frac{\partial {x}_t^{(\ell,m)}} {\partial \mathbf{w}^{(\ell,m)}},\frac{\partial {x}_t^{(\ell,m)} }{\partial \mathbf{x}_t^{(\ell-1)} }\right)$ (these are local gradients, needed for weight update in the downward pass.). **Downward pass**. Label $y_{t} $ has been received at the data center’s processing node. Now gradients of loss function $J$ are backpropagated. Having obtained $\frac{\partial J}{\partial {x}_t^{(\ell,m)}}$, each node $(\ell,m)$ sends to its lower layer neighbour $(\ell-1,k)$ the message consisting of $\left(t,\delta_{t}^{\ell}(m\to k)\right)$ (which we refer to here as gradient contribution), where $$\begin{aligned}
\delta_{t}^{\ell}(m\to k) &=& \frac{\partial J}{\partial {x}_t^{(\ell,m)}} \frac{\partial {x}_{t}^{(\ell,m)}}
{\partial \mathbf{x}_{t}^{(\ell-1,k)}} \\
&=& \frac{\partial J} {\partial {x}_{t}^{(\ell,m)}} {x}_{t}^{(\ell,m)}
\left(1-{x}_{t}^{(\ell,m)} \right) \mathbf{x}_{t}^{(\ell-1)}.\end{aligned}$$ Node $(\ell-1,k)$ now can compute $$\frac{\partial J}{\partial {x}_{t}^{(\ell-1,k)}}=\sum_{m}\delta_{t}^{\ell}(m\to k).$$ This is instantiated at the top layer: $$\frac{\partial J}{\partial {x}_{t}^{(\mathcal L)}}=-\frac{y_{t}}{ {x}_{t}^{(\mathcal L)}}+\frac{1-y_{t}}{1- {x}_{t}^{(\mathcal L)}}.$$ Moreover, after sending $\delta_{t}^{\ell}(m\to k)$, node $(\ell,m)$ updates its weights with stochastic gradient update and step-size $\eta_t$: $$\begin{aligned}
\mathbf{w}^{(\ell,m)} \leftarrow \mathbf{w}^{(\ell,m)}-\eta_t\,
\frac{\partial J}{\partial {x}_{t}^{(\ell,m)}} {x}_{t}^{(\ell,m)} \left(1-{x}_{t}^{(\ell,m)}\right) \mathbf{w}^{(\ell,m)},\end{aligned}$$ and removes “local gradients” from the memory.
**Challenges, insights and research directions**. We close this subsection with several challenges and practical considerations which arise when embedding neural networks into the Condense architecture.
The first challenge is on implementing the required AFC modules (atomic functions) in the analog domain. These modules are typically linear combinations plus nonlinearities (sigmoids, rectified linear units). Secondly, even when they are implemented in the digital domain, an interesting question is to study the effects of propagation of the quantization error across the Condense architecture.
Next, network topology and busy nodes will dictate that not all nodes see the activities corresponding to the $t$-th example. This is just like the dropout method [@Srivastava2014] which deliberately “switches off” neurons randomly during each learning stage. Dropout is a hugely successful method for learning regularization as it prevents overfitting by weight co-adaptation and demonstrates that the learning process can be inherently robust to the node failures, echoing the overall case against the learning and network layer separation, which presumes all data to be available on request at all times. Moreover, many activities will not be sent to all the nodes in the upper layer. In this case, $\frac{\partial {x}_{t}^{(\ell,m)}}{\partial {w}^{(\ell, m)}[k]}=0$, so weights will not be affected. In this case, corresponding gradients do not need to be stored, nor does the downward pass need to happen. More problematic is the situation in which upward pass has happened but downward pass fails at some point, i.e., some of the $\delta_{t}^{\ell}(m\to k)$ are not received at $(\ell-1,k)$. This injects additional noise to the gradient. Studying the effect of this noise is an interesting research topic.
We finally provide some insights on the communication and computational costs. Each AFC module (node) broadcasts one real number per training data example: its activity (together with data example index $t$), in the upward pass, and one real number per example, per receiver: gradient contribution $\delta_{t}^{\ell}(m\to k)$ (together with index $t$), in the downward pass. Thus, each node broadcasts to upper layers and sends specific messages to specific nodes in bottom layers. Upward pass happens whenever a new input is obtained, while downward pass whenever a new output is obtained. Due to this asynchrony, gradient updates might be out of date – therefore, each node could purge local gradients for outdated examples.
Other implementation aspects
============================
This Section briefly discusses some aspects of the Condense architecture not considered in earlier Sections.
**Size of NFC graph $\mathcal G$**. We first discuss a typical size of an NFC graph. Referring to Figure 6 and assuming a directed rooted tree graph, it can typically have depth (number of layers) around $5-7$. Regarding the number of source nodes (MTC devices – lower most layer), it is estimated that the number of MTC devices per macro-cell eNB will be in the range of $10^3-10^5$. The number of small cells per macro cell is in the range of $10^1-10^2$, which makes the number of MTC devices per small cell approximately $10^2-10^3$. Assuming a $30\mathrm{km} \times 30\mathrm{km}$ city area and a $100\mathrm{m} \times 100\mathrm{m}$ coverage of a small cell, we can have a total of $10^4-10^5$ small cells within a city-wide Condense network. Therefore, in a city-wide Condense network, we may have $10^7-10^8$ MTC devices (number of nodes at the lower-most layer), and on the order of $10^4-10^5$ nodes at the (base station) layer above. The number of nodes at the upper layers going further upwards is lower and is few tens or less. In summary, a typical city-wide Condense rooted tree network may have a total of $10^7-10^8$ nodes, it has a large “width” and a moderate “depth”. This goes relatively well in line with the supporting theory; e.g., neural networks are considered deep with depths of order $7$ or so, while arbitrary functions can be well-approximated even with shallow neural networks. Of course, the graph size can be virtually adjusted according to current application needs both horizontally (to adjust width) and vertically (to adjust depth) through implementing multiple (virtual) nodes within a single physical device.
**Synchronization**. We initially assess that synchronization may not be a major issue with realizing Condense. This is because, actually, synchronization is critical only with implementing analog atomic functions, e.g., within a single HeNB module. Network-wide orchestration of atomic functions may be successfully achieved through the control mechanisms of SDN and NFV, as well as through the usage of *buffering* at the upper Condense layers (HeNB-GW, S-GW, and P-GW), to compensate for delays and asynchrony.
**Features and pros** **Future directions**
----------------------------------- ----------------- ------------------------------------------- --------------------------------------------------
- Reconfigurable architecture; - Analog: wireless and optical domains; - Implementation challenges: channel
- Novel service of computing - Digital: FPGA/software; estimation, timing and frequency
functions over MTC-data; **AFC layer** - Theory: Nomographic functions offsets and quantization issues;
- Three layers: atomic, (analog wireless) and Reservoir - Development of standardized A-AFC
network and application; computing (analog optical) and D-AFC modules;
- Two control elements: topology - Topology processor: NFV orchestrator; - Actual constructions
and function processor; - Function processor: SDN application; of function decompositions;
- Can be integrated within **NFC layer** - Theory: Network coding for computing, - Decomposability (solvability) under restricted
3GPP MTC architecture; sensor fusion and neural networks function classes and network topologies;
- Can exploit theories of - Development of function and topology
sensor fusion, network coding and processor SDN/NFV modules;
computation and neural networks; - Implementation examples: RLNC, neural - Asynchronous, delayed and irregular
- Can be customized for variety **Application** networks and stochastic gradient descent; arrival of data;
of MTC applications; **layer** - Theory: neural networks, statistical - Inexact network function computation;
learning and prediction - Interoperability with existing data
analytics platforms;
**Communication, computational, and storage costs**. We now discuss reductions of communication costs (per application task) of Condense with respect to the conventional (forwarding) 3GPP MTC solution. How much communications is saved depends largely on the application at hand. For very simple tasks (functions), like, e.g., computing maximum or global average, it is easy to see that the savings can be very high. In contrast, for forwarding (computing the identity function), the savings may not be achieved (but the reliability is improved through random linear network coding). Also, overall communication savings depend on the overhead incurred by the signalling from the topology and function processors to the AFC modules (in order to orchestrate the topology, perform function decomposition and instal the appropriate atomic functions, etc.) However, this overhead is projected to eventually become small, as, upon a significant development of the technology, atomic function libraries and NFC decompositions will be pre-installed. Further, it is clear that Condense requires additional storage and computational functionalities at network nodes, when compared with current 3GPP MTC systems. However, this is a reasonable assumption for the modules (eNBs, GWs, etc.) of 3GPP MTC systems due to upcoming trends in mobile edge computing [@MEC].
**Data privacy and data loss**. Condense naturally improves upon privacy of IoT systems, as the data center (except when computing the identity function) does not receive the data in its raw from. Finally, if certain IoT-generated data has to be stored in a cloud data center in its raw form so as to ensure its long lifetime, Condense supports this functionality through identity functions. However, it is natural to expect that this request is, on average across all IoT data sources and all applications, only occasionally imposed, rendering significant communication savings overall.
Finally, Table \[Table\_1\] provides a summary of the proposed architecture. The table briefly indicates main points presented in this paper in terms of the advantages of the proposed architecture, relevant theoretical and implementation aspects, and main future research directions.
Conclusions
===========
In this paper, we proposed a novel architecture for knowledge acquisition of IoT-generated data within the 3GPP MTC (machine type communications) systems, which we refer to as Condense. The Condense architecture introduces a novel service within 3GPP MTC systems – computing linear and non-linear functions over the data generated by MTC devices. This service brings about the possibility that the underlying communication infrastructure communicates only the desired function of the MTC-generated data (as required by the given application at hand), and not the raw data in its entirety. This transformational approach has the potential to dramatically reduce the pressure on the 3GPP MTC communication infrastructure.
The paper provides contributions along two main directions. First, from the architectural side, we describe in detail how the function computation service can be realized within 3GPP MTC systems. Second, from the theoretical side, we survey the relevant literature on the possibilities of realizing “atomic” functions in both analog and digital domains, as well as on the theories and techniques for function decomposition over networks, including the literature on sensor fusion, network coding for computing, and neural networks. The paper discusses challenges, provides insights, and identifies future research directions for implementing function computation and function decomposition within practical 3GPP MTC systems.
Acknowledgment {#acknowledgment .unnumbered}
==============
The authors thank the following researchers for valuable help in developing the Condense concept: J. Coon, R. Vicente, M. Greferath, O. Gnilke, R. Freij-Hollanti, A. Vazquez Castro, V. Crnojevic, G. Chatzikostas, C. Mirasso, P. Colet, and M. C. Soriano. The authors would also like to acknowledge valuable support for collaboration through the EU COST IC 1104 Action.
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http://www.etsi.org/technologies-clusters/technologies/mobile-edge-computing
[^1]: D. Vukobratovic is with the Department of Power, Electronics and Communications Engineering, University of Novi Sad, Serbia, e-mail: dejanv@uns.ac.rs.
[^2]: D. Jakovetic is with the BioSense Institute, Novi Sad, Serbia, and with the Department of Mathematics and Informatics, Faculty of Sciences, University of Novi Sad, Serbia, email: djakovet@uns.ac.rs.
[^3]: V. Skachek is with the Institute of Computer Science, University of Tartu, Estonia, email: vitaly.skachek@ut.ee.
[^4]: D. Bajovic is with the BioSense Institute, Novi Sad, Serbia, and with the Department of Power, Electronics and Communication Engineering, University of Novi Sad, Serbia, email: dbajovic@uns.ac.rs.
[^5]: D. Sejdinovic is with the Department of Statistics, University of Oxford, UK, email: dino.sejdinovic@stats.ox.ac.uk.
[^6]: G. Karabulut Kurt is with the Department of Electronics and Communication Engineering, Istanbul Technical University, Turkey, email: gkurt@itu.edu.tr.
[^7]: C. Hollanti is with the Department of Mathematics and Systems Analysis, Aalto University, Finland, email: camilla.hollanti@aalto.fi.
[^8]: I. Fischer is with the Institute for Cross-Disciplinary Physics and Complex Systems (UIB-CSIC), Spain, email: ingo@ifisc.uib-csic.es.
[^9]: D. Vukobratovic is financially supported by Rep. of Serbia TR III 44003 grant. V. Skachek is supported in part by the grant PUT405 from the Estonian Research Council. G. Karabulut Kurt is supported by TUBITAK Grant 113E294. C. Hollanti is financially supported by the Academy of Finland grants \#276031, \#282938 and \#283262.
[^10]: We note that this restriction is for simplicity of presentation only; extension to directed acyclic graphs is straightforward and will be required in Sec. V.
[^11]: Strictly speaking, learning a neural network can be considered a special case of a stochastic gradient method (with a non-convex loss function). We present it here as a distinct subsection as we consider the implementation where the neural network weight parameters are distributed across the Condense network.
[^12]: We assume that performing decomposition of $f_t(\cdot)$ and the installation of the atomic functions across the AFC layer is completed prior to the initiation of flow of sample $\mathbf{x}_t$ “onwards” through the NFC topology. In other words, the time required for the latter process is sufficiently smaller than the time intervals of generation of data samples $\mathbf{x}_t$.
[^13]: Strictly speaking, $f_t(x) = \frac{1}{N} \sum_{s=1}^N x_s$ is not defined for the consensus example here as the gradient of $\phi(w,\cdot)$ at $x$, but it is defined as the additive term in (6) which is dependent upon $\mathbf{x}_t$. The gradient actually equals $\frac{1}{N}\sum_{s=1}^N (w - x_s)$; applying (5) to this gradient form with $\eta_t=1/(t+1)$ yields (6).
|
---
abstract: 'We prove an existence result for the Poisson equation on non-compact Riemannian manifolds satisfying weighted Poincaré inequalities outside compact sets. Our result applies to a large class of manifolds including, for instance, all non-parabolic manifolds with minimal positive Green’s function vanishing at infinity. On the source function we assume a sharp pointwise decay depending on the weight appearing in the Poincaré inequality and on the behavior of the Ricci curvature at infinity. We do not require any curvature or spectral assumptions on the manifold.'
address:
- 'Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy'
- 'Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy'
- 'Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy'
author:
- Giovanni Catino
- 'Dario D. Monticelli'
- Fabio Punzo
title: |
The Poisson equation\
on Riemannian manifolds with\
weighted Poincaré inequality at infinity
---
Introduction
============
The existence of solutions to the Poisson equation $$\Delta u = f$$ on a complete Riemannian manifold $(M, g)$, for a given function $f$ on $M$, is a classical problem which has been the object of deep interest in the literature. Malgrange [@mal] obtained solvability of the Poisson equation for any smooth function $f$ with compact support, as a consequence of the existence of a Green’s function for $-\Delta$ on every complete Riemannian manifold. Under integrability assumptions on $f$, existence of solutions have been established by Strichartz [@str] and Ni-Shi-Tam [@nst Theorem 3.2] (see also [@ni Lemma 2.3]). Moreover, in the same paper, the authors proved an existence result for the Poisson problem on manifolds with non-negative Ricci curvature under a sharp integral assumption involving suitable averages of $f$. This condition in particular is satisfied if $$|f(x)|\leq \frac{C}{\big(1+r(x)\big)^{\alpha}}$$ for some $C>0$ and $\alpha>2$, where $r(x):=\operatorname{dist}(x,p)$ is the distance function of any $x\in M$ from a fixed reference point $p\in M$. In fact, they proved a more general result where the decay rate of $f$ is just assumed to be of order $1+\varepsilon$. Note that this result is sharp on the flat space ${\mathbb{R}}^{n}$.
From now on let us consider solutions $u$ of the Poisson equation $\Delta u=f$ which can be represented as $$u(x)=\int_{M} G(x,y)f(y)\,dy\,,$$ where $G(x,y)$ is a Green’s function of $-\Delta$ on $M$ (see Section \[sect-prel\] for further details). Muntenau-Sesum [@ms] addressed the case of manifolds with positive spectrum, i.e. $\lambda_1(M)>0$, and Ricci curvature bounded from below, obtaining existence of solutions under the pointwise decay assumption $$|f(x)|\leq \frac{C}{\big(1+r(x)\big)^{\alpha}}$$ for some $C>0$ and $\alpha>1$. Note that this result is sharp on ${\mathbb{H}}^{n}$. Their proof relies on very precise integral estimates on the minimal positive Green’s function, which are inspired by the work of Li-Wang [@liwa1].
In [@cmp] the authors generalized the result in [@ms], obtaining existence of solutions on manifolds with positive essential spectrum, i.e. $\lambda_1^{\text{ess}}(M)>0$, for source functions $f$ satisfying $$\sum_{m=1}^{\infty}\frac{\theta_{R}(m+1)-\theta_{R}(m)}{\lambda_{1}\left(M\setminus B_{m}(p)\right)}\sup_{M\setminus B_m(p)}|f| < \infty,$$ for any $R>0$, where $\theta_{R}(m)$ is a function related to a lower bound on the Ricci curvature, locally on geodesic balls with center $p$ and radius $2R+m$. In particular, the authors showed in [@cmp Corollary 1.3] existence of solutions on Cartan-Hadamard manifolds with strictly negative Ricci curvature, whenever $$-C\big(1+r(x)\big)^{\gamma_{1}} \leq {\mathrm{Ric}}\leq -\frac{1}{C}\big(1+r(x)\big)^{\gamma_{2}} ,\quad |f (x)| \leq \frac{C}{\big(1+r(x)\big)^{\alpha}},$$ for some $C>0$ and $\gamma_{1},\gamma_{2}\geq 0$ with $\alpha>1+\frac{\gamma_{1}}{2}-\gamma_{2}$.
Observe that the results in [@ms] and [@cmp] cannot be used whenever the Ricci curvature tends to zero at infinity fast enough (see [@jpw]) since, in this case, one has $\lambda_1^{\text{ess}}(M)=0$ (and so $\lambda_1(M)=0$). In particular the case of ${\mathbb{R}}^n$ is not covered. On the other hand, the result in [@nst] does not apply on manifolds with negative curvature. The purpose of our paper is to obtain a general result which includes, as special cases, both manifolds with strictly negative curvature and manifolds with Ricci curvature vanishing at infinity. Moreover, our result is sharp on spherically symmetric manifolds, and in particular on ${\mathbb{R}}^n$ and ${\mathbb{H}}^n$.
Note that the condition $\lambda_1(M)>0$ is equivalent to the validity of the Poincaré inequality $$\lambda_1(M)\int_M u^2\, dV \leq \int_M |\nabla u|^2\,dV$$ for any $u\in C^\infty_c(M)$. On the other hand, one has positive essential spectrum if and only if, for some compact subset $K\subset M$, one has $\lambda_1(M \setminus K)>0$ and $$\lambda_1(M \setminus K)\int_M u^2\, dV \leq \int_M |\nabla u|^2\,dV$$ for any $u\in C^\infty_c(M\setminus K)$. Generalizing the previous inequalities, one says that $(M,g)$ satisfies a [*weighted Poincaré inequality*]{} with a non-negative weight function $\rho$ if $$\label{wpi2}
\int_M \rho \,v^2\, dV \leq \int_M |\nabla v|^2 \,dV$$ for every $v\in C^\infty_c(M)$. If for any $R\geq R_0>0$ there exists a non-negative function $\rho_R$ such that holds for every $v\in C^\infty_c(M\setminus B_R(p))$ and for $\rho\equiv\rho_R$, we say that $(M,g)$ satisfies a [*weighted Poincaré inequality at infinity*]{}. In addition, inspired by [@liwa1], we say that $(M,g)$ satisfies the property $\left(\mathcal{P}^{\infty}_{\rho_R}\right)$ if a weighted Poincaré inequality at infinity holds for the family of weights $\rho_R$ and the conformal $\rho_R$-metric defined by $$g_{\rho_R} := \rho_R\, g$$ is complete for every $R\geq R_0$. The validity of a weighted Poincaré inequality on some classes of manifolds has been investigated in the literature. It is well known that on ${\mathbb{R}}^n$ inequality holds with $\rho(x)=\frac{(n-2)^2}{4}\frac{1}{r^2(x)}$. It is also called [*Hardy inequality*]{}. More in general, it holds on every Cartan-Hadamard manifold with $\rho(x)=\frac{C}{r^2(x)}$, for some $C>0$ (see [@car] and [@gri] for some refinement of this result).
In order to state our main results, we need to introduce a (increasing) function $\omega(s)$ related to the value of the Ricci curvature on the annulus $B_{\frac{5}{4}s}(p)\setminus B_{\frac{3}{4}s}(p)$ (see for the precise definition). In this paper we prove the following result.
\[teo1\] Let $(M,g)$ be a complete non-compact Riemannian manifold satisfying the property $\left(\mathcal{P}^\infty_{\rho_R}\right)$ and let $f$ be a locally Hölder continuous function on $M$. If $$\sum_{m}^{\infty}\Big(\omega(m+1)-\omega(m)+1\Big)\sup_{M\setminus B_m(p)}\left|\frac{f}{\rho_m}\right| < \infty,$$ then the Poisson equation $$\Delta u=f \quad\hbox{in } M$$ admits a classical solution $u$.
Assume that $\lambda_1^{\text{ess}}(M)>0$ and $${\mathrm{Ric}}\geq -C\big(1+r(x)\big)^{\gamma}$$ for some $\gamma\geq 0$. Then it is direct to see that $$\omega(m+1)-\omega(m)\sim C\Big(\theta_{R}(m+1)-\theta_{R}(m)\Big) \sim C m^{\frac{\gamma}{2}}$$ for every $R>0$ and the property $\left(\mathcal{P}^\infty_{\rho_R}\right)$ holds for every $R$ with $\rho_R(x)=\lambda_1(M\setminus B_R(p))$. Thus $$\Big(\omega(m+1)-\omega(m)+1\Big)\sup_{M\setminus B_m(p)}\left|\frac{f}{\rho_m}\right| \sim C\, \frac{\theta_{R}(m+1)-\theta_{R}(m)}{\lambda_{1}\left(M\setminus B_{m}(p)\right)}\sup_{M\setminus B_m(p)}|f| \,,$$ therefore our result is in accordance with those in [@ms] and [@cmp].
We recall that by [@liwa1 Corollary 1.4, Lemma 1.5] the validity of a weighted Poincaré inequality on $M$ implies the non-parabolicity of the manifold; on the contrary, if $(M,g)$ is non-parabolic, then a weighted Poincaré inequality holds on $M$, with weight $$\rho(x):=\frac{|\nabla G(p,x)|^2}{4 G^2(p,x)},$$ where $G$ is the minimal positive Green’s function on $(M,g)$. Exploiting this result, using similar techniques as in Theorem \[teo1\], we obtain the following refined result on complete non-compact non-parabolic manifolds.
\[teo2\] Let $(M,g)$ be a complete non-compact non-parabolic Riemannian manifold with minimal positive Green’s function $G$. Let $\rho(x)=\frac{|\nabla G(p,x)|^2}{4 G^2(p,x)}$ and let $f$ be a locally Hölder continuous function on $M$. If $$\sum_{m}^{\infty}\Big(\omega(m+1)-\omega(m)\Big)\sup_{M\setminus B_m(p)}\left|\frac{f}{\rho}\right| < \infty,$$ then the Poisson equation $$\Delta u=f \quad\hbox{in } M$$ admits a classical solution $u$.
We explicitly observe that in Theorem \[teo2\] the completeness of the conformal metric $g_\rho=\rho g$ is not required. As it was observed in [@liwa1], the completeness of $g_\rho$ would hold if $G(p,x)\to 0$ as $r(x)\to \infty$, a condition that we do not need to assume here.
It is well-known that ${\mathbb{R}}^n$ is a non-parabolic manifold if $n\geq3$, with minimal positive Green’s function $G(x,y)=\frac{c_n}{|x-y|^{n-2}}$ for some positive constant $c_n$. Moreover the weighted Poincaré – Hardy’s inequality holds on ${\mathbb{R}}^n$ with $$\rho(x)=\frac{|\nabla G(0,x)|^2}{4 G^2(0,x)}=\frac{(n-2)^2}{4}\frac{1}{|x|^2}.$$ In this case, using the definition of the function $\omega(s)$, it is easy to see that $$\omega(m+1)-\omega(m)\sim C \log \left(1+\frac{1}{m}\right) \sim \frac{C}{m}\,.$$ Hence we can apply Theorem \[teo2\], with $$\Big(\omega(m+1)-\omega(m)\Big)\sup_{M\setminus B_m(p)}\left|\frac{f}{\rho_m}\right| \sim C\, m\, \sup_{M\setminus B_m(p)}\left|f\right|$$ and the convergence of the series follows, whenever $|f(x)|\leq C/(1+r(x))^\alpha$ for some $\alpha>2$. This condition is optimal, as it can be easily verified by explicit computations.
In general, concerning Cartan-Hadamard manifolds, by using Theorem \[teo1\] we improve [@cmp Corollary 1.3] allowing the Ricci curvature to approach zero at infinity.
\[cor-2\] Let $(M,g)$ be a Cartan-Hadamard manifold and let $f$ be a locally Hölder continuous, bounded function on $M$. If $$-C\big(1+r(x)\big)^{\gamma_{1}} \leq {\mathrm{Ric}}\leq -\frac{1}{C}\big(1+r(x)\big)^{\gamma_{2}} ,\quad |f (x)| \leq \frac{C}{\big(1+r(x)\big)^{\alpha}},$$ for some $C\geq 1$, $\gamma_1,\gamma_2\in{\mathbb{R}}$, $\gamma_{1}\geq\gamma_{2}$, $\gamma_1\geq0$ and $\alpha$ satisfying $$\alpha >
\begin{cases}
1+\frac{\gamma_1}{2}-\gamma_2 &\quad\hbox{if } \gamma_2\geq-2 \\
3+\frac{\gamma_1}{2} &\quad\hbox{if } \gamma_2< -2 \end{cases}$$ then the Poisson equation $$\Delta u=f \quad\hbox{in } M$$ admits a classical solution $u$.
\[rem-rot\] In the special case $\gamma_{1}=\gamma_{2}=\gamma\geq 0$ the condition on $\alpha$ in the previous corollary becomes $$\alpha >
\begin{cases}
1-\frac{\gamma}{2}&\quad\hbox{if } \gamma\geq-2 \\
2 &\quad\hbox{if } \gamma< -2\,.
\end{cases}$$ In particular in $(M,g)$ is the standard hyperbolic space ${\mathbb{H}}^n$, then $\gamma=0$. Thus we need that $\alpha>1$ and this condition is sharp as observed above. We will consider also the case $\gamma<0$ in the Subsection \[ssu\] on model manifolds.
The paper is organized as follows: in Section \[sect-prel\] we collect some preliminary results and we define precisely the function $\omega$; in Section \[sec-grad\] we prove a refined local gradient estimates for positive harmonic functions; in Section \[sec-est\] we prove key estimates on the positive minimal Green’s function $G(x,y)$ of a non-parabolic manifold, by means of the property $\left(\mathcal{P}^{\infty}_{\rho_R}\right)$; in Section \[sec-proofs\] we prove Theorem \[teo1\]; finally in Section \[sec-ex\] we prove Corollary \[cor-2\] and show the optimality of the assumption in Theorem \[teo2\] for rotationally symmetric manifolds.
Finally we note that some results concerning the Poisson equation on some manifolds satisfying a weighted Poincaré inequality have been very recently obtained in [@msw2]. However their assumptions and results apparently are completely different to ours.
Preliminaries {#sect-prel}
=============
Let $(M,g)$ be a complete non-compact $n$-dimensional Riemannian manifold. For any $x\in M$ and $R>0$, we denote by $B_{R}(x)$ the geodesic ball of radius R with centre $x$ and let ${\mathrm{Vol}}(B_{R}(x))$ be its volume. We denote by ${\mathrm{Ric}}$ the Ricci curvature of $g$. For any $x \in M$, let $\mu(x)$ be the smallest eigenvalue of ${\mathrm{Ric}}$ at $x$. Thus, for any $V\in T_{x}M$ with $|V|=1$, ${\mathrm{Ric}}(V,V)(x) \geq \mu(x)$ and we have $\mu(x)\geq -\omega(r(x))$ for some $\omega\in C([0,\infty))$, $\omega\geq 0$. Hence, for any $x\in M$, we have $$\label{eq3}
{\mathrm{Ric}}(V,V)(x) \geq -(n-1) \frac{\varphi''(r(x))}{\varphi(r(x))},$$ for some $\varphi\in C^{\infty}((0,\infty))\cap C^{1}([0,\infty))$ with $\varphi(0)=0$ and $\varphi'(0)=1$. Note that $\varphi,\varphi',\varphi''$ are positive in $(0,\infty)$. We set $$K_R(x):=\sup_{y\in B_{r(x)+R}\setminus B_{r(x)-R}}\frac{\varphi''(r(y))}{\varphi(r(y))}$$ for $r(x)>R>1$; $$I_R(x):=\begin{cases}
\sqrt{K_R(x)}\coth\left(\sqrt{K_R(x)} R/2\right)&\text{if }\,K_R(x)>0 \\
\frac{2}{R} &\text{if }\,K_R(x)=0;
\end{cases}$$ $$\begin{aligned}
\label{defQ}
Q_{R}(x):=\max\left\{K_R(x), \frac{I_R(x)}{R}, \frac{1}{R^2}\right\}.\end{aligned}$$ Note that $Q_{R}(x)\equiv Q_{R}(r(x))$. For any $z\in M$, let $\gamma$ be the minimal geodesic connecting $p$ to $z$. We define the function $$\label{eq127}
\omega(z)=\omega(r(z)):=\int_a^{r(z)} \sqrt{Q_{\frac{r((\gamma(s))}{4}}(r(\gamma(s))}\,ds,$$ for a given $a>0$. Note that $t\mapsto\omega(t)$ is increasing and so invertible.
Under $\eqref{eq3}$, we know that $$\label{eq6}
{\mathrm{Vol}}(B_{R}(p)) \leq C \int_{0}^{R}\varphi^{n-1}(\xi)\,d\xi.$$ Moreover, let $\operatorname{Cut}(p)$ be the [*cut locus*]{} of $p\in M$.
It is known that every complete Riemannian manifold admits a Green’s function (see [@mal]), i.e. a smooth function defined in $(M\times M)\setminus \{(x,y)\in M\times M:\,x=y\} $ such that $G(x,y)=G(y,x)$ and $\Delta_{y} G(x,y)=-\delta_{x}(y)$. We say that $(M,g)$ is non-parabolic if there exists a minimal positive Green’s function $G(x,y)$ on $(M,g)$, and parabolic otherwise.
We say that $(M,g)$ satisfies a [*weighted Poincaré inequality*]{} with a non-negative weight function $\rho$ if $$\label{wpi}
\int_M \rho \,v^2\, dV \leq \int_M |\nabla v|^2 \,dV$$ for every $v\in C^\infty_c(M)$. If for any $R\geq R_0>0$ there exists a non-negative function $\rho_R$ such that holds for every $v\in C^\infty_c(M\setminus B_R(p))$ and for $\rho\equiv\rho_R$, we say that $(M,g)$ satisfies a [*weighted Poincaré inequality at infinity*]{}. In addition, inspired by [@liwa1], we say that $(M,g)$ satisfies the property $\left(\mathcal{P}^{\infty}_{\rho_R}\right)$ if a weighted Poincaré inequality at infinity holds for the family of weights $\rho_R$ and the conformal $\rho_R$-metric defined by $$g_\rho := \rho_R\, g$$ is complete. With this metric we consider the $\rho$-distance function $$r_\rho (x,y)=\inf_{\gamma} \, l_\rho (\gamma)$$ where the infimum of the lengths is taken over all curves joining $x$ and $y$, with respect to the metric $g_\rho$. For a fixed point $p\in M$, we denote by $$r_\rho(x) = r_\rho (p,x).$$ Note that $|\nabla r_\rho (x)|^2 = \rho(x)$. Finally, we denote by $$B^\rho_R(p)=\{x \in M: r_\rho(x)\leq R\}.$$
Let $\lambda_{1}(M)$ be the bottom of the $L^{2}$-spectrum of $-\Delta$. It is known that $\lambda_{1}(M)\in[0,+\infty)$ and it is given by the variational formula $$\lambda_{1}(M) = \inf_{v\in C^{\infty}_{c}(M)}\frac{\int_{M}|\nabla v|^{2}\,dV}{\int_{M}v^{2}\,dV}\,.$$ If $\lambda_{1}(M)>0$, then $(M,g)$ is non-parabolic (see [@gri1 Proposition 10.1]). Whenever $(M,g)$ is non-parabolic, let $G_{R}(x,y)$ be the Green’s function of $-\Delta$ in $B_{R}(z)$ satisfying zero Dirichlet boundary conditions on $\partial B_{R}(z)$, for some $z\in M$. We have that $R\mapsto G_{R}(x,y)$ is increasing and, for any $x,y\in M$, $$\label{eq9}
G(x,y) = \lim_{R\to\infty} G_{R}(x,y),$$ locally uniformly in $(M\times M)\setminus \{(x,y)\in M\times M:\,x=y\} $. We define $\lambda_{1}(\Omega)$, with $\Omega$ an open subset of $M$, to be the first eigenvalue of $-\Delta$ in $\Omega$ with zero Dirichlet boundary conditions. It is well known that $\lambda_{1}(\Omega)$ is decreasing with respect to the inclusion of subsets. In particular $R\mapsto\lambda_{1}(B_{R}(x))$ is decreasing and $\lambda_{1}(B_{R}(x))\to \lambda_{1}(M)$ as $R\to\infty$.
For any $x\in M$, for any $s>0$ and for any $0\leq a < b\leq +\infty$, we define $$\begin{aligned}
\mathcal{L}_{x}(s) &:= \{y \in M\,:\,G(x,y)=s \},\\
\mathcal{L}_{x}(a,b)&:= \{y \in M\,:\, a< G(x,y)< b \}.\end{aligned}$$
Local gradient estimate for harmonic functions {#sec-grad}
==============================================
In this section we improve [@cmp Lemma 3.1]. We set
$$k_R(z):=\sup_{B_R(z)}\frac{\varphi''(r(y))}{\varphi(r(y))}$$ for $z\in M$ and $R>0$; $$i_R(z):=\begin{cases}
\sqrt{k_R}\coth\left(\sqrt{k_R(z)} R/2\right)&\text{if }\,k_R(z)>0 \\
\frac{2}{R} &\text{if }\,k_R(z)=0.
\end{cases}$$
\[lemma00\] Let $R>0$ and $z\in M$. Let $u\in C^{2}(B_{R}(z))$ be a positive harmonic function in $B_{R}(z)$. Then $$|\nabla u(\xi)| \leq C \sqrt{\max\left\{k_R(z), \frac{i_R(z)}{R}, \frac{1}{R^2}\right\}}\, u(\xi)\quad\text{for any}\quad \xi\in B_{R/2}(z),$$ for some positive constant $C>0$.
Following the classical argument of Yau, let $v:=\log u$. Then $$\Delta v = - |\nabla v|^{2} .$$ Let $\eta(\xi)=\eta(d(\xi))$, with $d(\xi):=\operatorname{dist}(\xi,z)$, a smooth cutoff function such that $\eta(\xi)\equiv 1$ on $B_{R/2}(z)$, with support in $B_{R}(z)$, $0\leq \eta\leq 1$ and $$-\frac{4}{R}\leq \frac{\eta'}{\eta^{1/2}} \leq 0 \quad\text{and}\quad \frac{|\eta''|}{\eta} \leq \frac{8}{R^{2}}.$$ Let $w=\eta^{2}|\nabla v|^{2}$. Then $$\begin{aligned}
\frac12 \Delta w &= \frac12 \eta^{2} \Delta |\nabla v|^{2} + \frac12 |\nabla v|^{2} \Delta \eta^{2} + \langle \nabla |\nabla v|^{2},\nabla \eta^{2}\rangle.\end{aligned}$$ Then, from classical Bochner-Weitzenböch formula and Newton inequality, one has $$\begin{aligned}
\frac12 \Delta |\nabla v|^{2} & = |\nabla^{2} v|^{2} + {\mathrm{Ric}}(\nabla v,\nabla v) + \langle \nabla v,\nabla \Delta v\rangle \\
&\geq \frac1n (\Delta v)^{2} - (n-1) \frac{\varphi''}{\varphi} |\nabla v|^{2} - \langle \nabla |\nabla v|^{2},\nabla v\rangle \\
&= \frac1n |\nabla v|^{4} - (n-1) \frac{\varphi''}{\varphi} |\nabla v|^{2} - \langle \nabla |\nabla v|^{2},\nabla v\rangle .\end{aligned}$$ Moreover, by Laplacian comparison, since ${\mathrm{Ric}}\geq -(n-1)k_R(z)$ in $B_R(z)$, we have $$\begin{aligned}
\frac12 \Delta \eta^{2} &= \eta \eta' \Delta \rho + \eta \eta'' + (\eta')^{2} \\
&\geq (n-1)i_R(z)\eta\eta' + \eta \eta''+ (\eta')^{2}\\
&\geq -\frac{4}{R} \left((n-1)i_R(z)+\frac{2}{R}\right)\eta\end{aligned}$$ pointwise in $B_{R}(z)\setminus (\{z\}\cup \operatorname{Cut}(z))$ and weakly on $B_{R}(z)$. Thus, $$\begin{aligned}
\frac12 \Delta w &\geq \frac1n \frac{w^{2}}{\eta^{2}}-(n-1)\frac{\varphi''}{\varphi}w - \frac{4}{R}\left((n-1)i_R(z)+\frac{2}{R}\right)\frac{w}{\eta} \\
&-4\frac{|\eta'|^{2}}{\eta^{2}}w + \frac{2}{\eta}\langle \nabla w,\nabla \eta\rangle-\langle \nabla w,\nabla v\rangle + \frac{2}{\eta}\langle \nabla v,\nabla \eta \rangle w \\
&\geq \frac1n \frac{w^{2}}{\eta^{2}}-(n-1)\frac{\varphi''}{\varphi}w - \frac{4}{R}\left((n-1)i_R(z)+\frac{2}{R}\right)\frac{w}{\eta} \\
& + \frac{2}{\eta}\langle \nabla w,\nabla \eta\rangle-\langle \nabla w,\nabla v\rangle - \frac{64}{R^{2}}\frac{ w}{\eta}-\frac{8}{R}\frac{ w^{3/2}}{\eta^{3/2}}\\
&\geq \frac{1}{2n} \frac{w^{2}}{\eta^{2}}-(n-1)\frac{\varphi''}{\varphi}w - \frac{4}{R}\left((n-1)i_R(z)+\frac{18+8n}{R}\right)\frac{w}{\eta} \\
& + \frac{2}{\eta}\langle \nabla w,\nabla \eta\rangle-\langle \nabla w,\nabla v\rangle.\end{aligned}$$ Let $q$ be a maximum point of $w$ in $\overline{B}_{R}(z)$. Since $w\equiv0$ on $\partial B_{R}(z)$, we have $q\in B_{R}(z)$. First assume $q\notin \operatorname{Cut}(z)$. At $q$, we obtain $$\begin{aligned}
0 &\geq \left[\frac{1}{2n} w - (n-1)\frac{\varphi''}{\varphi}-\frac{4}{R}\Big((n-1)i_R(z)+\frac{18+8n}{R}\Big)\right]w.\end{aligned}$$ So $$w(q)\leq 2n(n-1)\frac{\varphi''\big(r(q)\big)}{\varphi\big(r(q)\big)}+\frac{8n(n-1)}{R}i_R(z)+\frac{144n+64n^2}{R^2}.$$ Thus, for any $\xi \in B_{R/2}(z)$, $$\begin{aligned}
|\nabla v(\xi)|^{2}&\leq 2n(n-1)\frac{\varphi''\big(r(q)\big)}{\varphi\big(r(q)\big)}+\frac{8n(n-1)}{R}i_R(z)+\frac{144n+64n^2}{R^2}\\
&\leq 2n(n-1)k_R(z)+\frac{8n(n-1)}{R}i_R(z)+\frac{144n+64n^2}{R^2}\end{aligned}$$ We get $$\frac{|\nabla u(\xi)|}{u(\xi)}=|\nabla v(\xi)| \leq C \sqrt{\max\left\{k_R(z), \frac{i_R(z)}{R}, \frac{1}{R^2}\right\}}.$$ for some positive constant $C>0$. By standard Calabi trick (see [@cal; @cy]), the same estimate can be obtained when $q\in \operatorname{Cut}(z)$. This concludes the proof of the lemma.
As a corollary we have the following
\[lemma0\] Let $(M,g)$ be non-parabolic. If $r(z)>R>0$, then $$|\nabla G(p,z)| \leq C \sqrt{Q_{R}(z)}\, G(p,z),$$ for some positive constant $C>0$.
Green’s function estimates {#sec-est}
==========================
Pointwise estimate
------------------
\[lemma1\] Let $(M,g)$ be non-parabolic and let $a>0$ and $y\in M\setminus B_{a}(p)$. Then $$A^{-1} \exp \left(-B\, \omega(y)\right) \leq G(p,y) \leq A \exp \left(B\, \omega(y)\right),$$ with $A:=\max\{ \max_{\partial B_a(p)}G(p,\cdot), \left(\min_{\partial B_a(p)}G(p,\cdot)\right)^{-1}\}$ and $B=2n(n-1)$.
Let $y\in M\setminus \overline{B_{a}(p)}$ with $a> 0$ and consider the minimal geodesic $\gamma$ joining $p$ to $y$ and let $y_{0}\in\partial B_{a}(p)$ be a point of intersection of $\gamma$ with $\partial B_{a}(p)$. Since $G(p,\cdot)$ is harmonic in $B_{r(z)/4}(z)$, for every $z\in \gamma$ with $r(z)\geq a$, by Lemma \[lemma0\] we get $$|\nabla G(p,z)| \leq C \sqrt{Q_{r(z)/4}(z)}\,G(p,z) .$$ We have $$\begin{aligned}
G(p,y)&=G(p,y_0)+\int_{a}^{r(y)}\langle \nabla G(p,\gamma(s)), \dot{\gamma}(s)\rangle \,ds \\
&\leq G(p,y_0) + C\int_{a}^{r(y)} \sqrt{Q_{\frac{r(\gamma(s))}{4}}\big(r(\gamma(s))\big)} G(p,\gamma(s)) \,ds.\end{aligned}$$ By Gronwall inequality, $$G(p,y) \leq G(p,y_0) \exp\left(C\int_{a}^{r(y)} \sqrt{Q_{\frac{r(\gamma(s))}{4}}\big(r(\gamma(s))\big)}\,ds\right)\leq A \exp\left(B\,\omega(y)\right),$$ with $A:=\max\{ \max_{\partial B_a(p)}G(p,\cdot), \left(\min_{\partial B_a(p)}G(p,\cdot)\right)^{-1}\}$ and $B=2n(n-1)$. Similarly, $$G(p,y) \geq A^{-1} \exp\left(-B\,\omega(y)\right).$$
\[remark101\] We also note that $$\mathcal{L}_{p}\left(A \exp \left(B\, \omega(a)\right),\infty\right) \subset B_{a}(p).$$ In fact, let $y\in M\setminus B_a(p)$ and take $j>r(y)$. Since $G_{j}(p,y)\leq G(p,y)$ and $G_{j}(p,\cdot)\equiv 0$ on $\partial B_{j}(p)$, by Lemma \[lemma1\], we have $$G_{j}(p,y)\leq A \exp \left(B \omega(a)\right)\quad\text{on}\quad \partial\left(B_{j}(p)\setminus B_{a}(p)\right);$$ note that the right hand side is independent of $y$. Since $y\mapsto G_{j}(x,y)$ is harmonic in $B_{j}(p)\setminus B_{a}(p)$, by maximum principle, $$G_{j}(p,y)\leq A \exp \left(B \omega(a)\right)\quad\text{in}\quad B_{j}(p)\setminus B_{a}(p).$$ Sending $j\to\infty$, by , we obtain $$G(p,y)\leq A \exp \left(B \omega(a)\right)\quad\text{in}\quad M\setminus B_{a}(p),$$ and the claim follows.
Auxiliary estimates
-------------------
\[lemma2\] Let $(M,g)$ be non-parabolic. For any $s>0$, there holds $$\int_{\mathcal{L}_{p}(s)}|\nabla G(p,y)|\,dA(y) = 1$$ where $dA(y)$ is the $(n-1)$-dimensional Hausdorff measure on $\mathcal{L}_{x}(s)$. As a consequence, by the co-area formula, for any $0<a<b$, there holds $$\int_{\mathcal{L}_{p}(a,b)}\frac{|\nabla G(p,y)|^2}{G(p,y)}\,dy = \log\left(\frac{b}{a}\right) \,.$$
For the proof see [@ms]. Moreover, we get the following weighted integrability property for the Green’s function.
\[lemmastoc\] Assume that $(M,g)$ satisfies the property $\left(\mathcal{P}^\infty_{\rho_R}\right)$. Fix $m\geq R_0$. Then, for any $R_1>0$ such that $B_m(p)\subset B^{\rho_m}_{R_1}(p)$, one has $$\int_{M\setminus B^{\rho_m}_{2R_1}(p)} \rho_m(y)\,|G(p,y)|^2\,dy < \infty \,.$$
Note that $B_m(p)\subset B^{\rho_m}_{R_1}(p)$ for every $R_1$ large enough.
In order to simplify the notation, let $\rho\equiv \rho_m$. Fix $R_1>0$ such that $B_m(p)\subset B^\rho_{R_1}(p)$ and let $\phi$ be defined as $$\phi(x):=\begin{cases}
0 & \textrm{on } B^\rho_{R_1}(p) \\
\frac{r_\rho(x)-R_1}{R_1} & \textrm{on } B^\rho_{2R_1}(p)\setminus B^\rho_{R_1}(p)\\
1 & \textrm{on } M\setminus B^\rho_{2R_1}(p) \,.
\end{cases}$$ Let $R>2R_1$ and $G^{\rho}_{R}(p,y)$ be the Green’s function of $-\Delta$ in $B^{\rho}_{R}(p)$ satisfying zero Dirichlet boundary conditions on $\partial B^{\rho}_{R}(p)$. Following the proof in [@liwa1], since $G^{\rho}_R$ is harmonic in $B^{\rho}_{R}(p)$, one has $$\begin{aligned}
\int_{B^{\rho}_R(p)}|\nabla \left(\phi \,G^{\rho}_R\right)|^2\,dV &= \int_{B^{\rho}_R(p)}|\nabla \phi|^2 \left(G^{\rho}_R\right)^2\,dV + \int_{B^{\rho}_R(p)}|\nabla G^{\rho}_R |^2 \phi^2\,dV\\
&+ 2 \int_{B^{\rho}_R(p)}\langle \nabla \phi, \nabla G^{\rho}_R \rangle \phi G^{\rho}_R \,dV \\
&= \int_{B^{\rho}_R(p)}|\nabla \phi|^2 \left(G^{\rho}_R\right)^2\,dV + \frac{1}{2}\int_{B^{\rho}_R(p)}\Delta\left(G^{\rho}_R\right)^2 \phi^2\,dV\\
&+ 2 \int_{B^{\rho}_R(p)}\langle \nabla \phi, \nabla G^{\rho}_R \rangle \phi G^{\rho}_R \,dV \\
&= \int_{B^{\rho}_R(p)}|\nabla \phi|^2 \left(G^{\rho}_R\right)^2\,dV\end{aligned}$$ where the last equality follows by integration by parts and the fact that $G^{\rho}_{R}(p,y)$ vanishes on $\partial B^{\rho}_{R}(p)$. Hence, the weighted Poincaré inequality yields $$\begin{aligned}
\int_{M\setminus B^\rho_{R_1}(p)} \rho \,\left(G^{\rho}_R\right)^2\phi^2\,dV \leq \int_{B^{\rho}_R(p)}|\nabla \left(\phi \,G^{\rho}_R\right)|^2\,dV \leq \frac{1}{R_1^2}\int_{B^{\rho}_{2R_1}(p)\setminus B^{\rho}_{R_1}(p)}\rho\,\left(G^{\rho}_R\right)^2\,dV\end{aligned}$$ Letting $R\rightarrow \infty$, by Fatou’s lemma and uniform convergence of $G_R^\rho \rightarrow G$ on compact subsets, we get $$\int_{M\setminus B^\rho_{2R_1}(p)} \rho \,G^2\,dV \leq \frac{1}{R_1^2}\int_{B^{\rho}_{2R_1}(p)\setminus B^{\rho}_{R_1}(p)}\rho\, G^{2}\,dV$$ and the thesis follows.
We expect a decay estimate similar to the one in [@liwa1 Theorem 2.1]. However we leave out this refinement since it is not necessary in our arguments.
Integral estimates on level sets
--------------------------------
We begin by noting that, using Remark \[remark101\] and the fact that $G(p,\cdot)\in L^1_{\text{loc}}(M)$ one has the following integral estimate on large level sets.
\[lemma3\] Let $(M,g)$ be non-parabolic. Choose $A,B$ as in Lemma \[lemma1\]. Then $$\begin{aligned}
\int_{\mathcal{L}_{p}\left(A \exp \left(B\, \omega(a)\right),\infty\right)} &G(p,y)\,dy <\infty.\end{aligned}$$
For intermediate levels sets, we get the following key inequality.
\[claim2\] Assume that $(M,g)$ satisfies the property $\left(\mathcal{P}^\infty_{\rho_R}\right)$. Then, there exists a positive constant $C$ such that, for any function $f$ and any $0<\delta<1$, $\varepsilon >0$ satisfying $\mathcal{L}_p \left(\frac{\delta\varepsilon}{2},2\varepsilon\right) \subset M \setminus B_m(p)$ for some $m>R_0$, one has $$\left|\int_{\mathcal{L}_{p}(\delta \varepsilon, \varepsilon)} G(p,y)\,f(y)\,dy \right| \leq C \left(-\log\delta +1\right) \sup_{\mathcal{L}_{p}(\delta \varepsilon, \varepsilon)} \left|\frac{f}{\rho_m}\right|\,.$$
We follow the general argument in [@liwa1] and [@ms]; however some relevant differences are in order, due to the use of the property $\left(\mathcal{P}^\infty_{\rho_R}\right)$. Let $\phi:=\chi \psi$ with $$\chi(y):=\begin{cases}
\frac{1}{\log 2} \log \left(\frac{2 G(p,y)}{\delta \epsilon}\right) & \textrm{on } \mathcal{L}_p \left(\frac{\delta\varepsilon}{2},\delta\varepsilon\right)\\
1 & \textrm{on } \mathcal{L}_p \left(\delta\varepsilon,\varepsilon\right)\\
\frac{1}{\log 2} \log \left(\frac{2 \varepsilon}{G(p,y)}\right) & \textrm{on } \mathcal{L}_p \left(\varepsilon,2\varepsilon\right) \\
0 & \textrm{elsewhere}
\end{cases}$$ and for any fixed $R>0$ $$\psi(y):=\begin{cases}
1 & \textrm{on } B^{\rho_m}_{R}(p) \\
R+1-r_{\rho_m}(y) & \textrm{on } B^{\rho_m}_{R+1}(p)\setminus B^{\rho_m}_{R}(p)\\
0 & \textrm{on } M\setminus B^{\rho_m}_{R+1}(p) \,.
\end{cases}$$ By the weighted Poincaré inequality at infinity we get $$\begin{aligned}
\left|\int_{\mathcal{L}_{p}(\delta \varepsilon, \varepsilon)\cap B^{\rho_m}_{R}(p)} G(p,y)\,f(y)\,dy \right| &\leq \int_{\mathcal{L}_{p}(\delta \varepsilon, \varepsilon)\cap B^{\rho_m}_{R}(p)} G(p,y)\,|f(y)|\,dy \\
&\leq \sup_{\mathcal{L}_{p}(\delta \varepsilon, \varepsilon)\cap B^{\rho_m}_{R}(p)} \left|\frac{f}{\rho_m}\right| \, \int_{M} \rho_m(y)\,G(p,y) \phi^2(y)\,dy \\
&\leq \sup_{\mathcal{L}_{p}(\delta \varepsilon, \varepsilon)\cap B^{\rho_m}_{R}(p)} \left|\frac{f}{\rho_m}\right| \, \int_{M} \left|\nabla \left( \sqrt{G(p,y)} \phi(y)\right)\right|^2\,dy \,.\end{aligned}$$ We estimate $$\begin{aligned}
\int_{M} \left|\nabla \left( \sqrt{G(p,y)} \phi(y)\right)\right|^2\,dy &\leq \frac{1}{2} \int_{\mathcal{L}_{p}(\frac{\delta \varepsilon}{2}, 2\varepsilon)} \frac{|\nabla G(p,y)|^2}{G(p,y)}\,dy + 2 \int_M G(p,y)|\nabla \phi|^2 \,dy \\
&= C(-\log\delta+1) + 2 \int_M G(p,y)|\nabla \phi|^2 \,dy\end{aligned}$$ where we used Lemma \[lemma2\] in the last equality. On the other hand $$\begin{aligned}
\int_M G(p,y)|\nabla \phi|^2 \,dy &\leq 2 \int_M G(p,y)|\nabla \chi|^2 \psi^2 \,dy + 2 \int_M G(p,y)|\nabla \psi |^2 \chi^2 \,dy \\
&\leq 2(\log 2)^2 \int_{\mathcal{L}_{p}(\frac{\delta \varepsilon}{2}, 2\varepsilon)} \frac{|\nabla G(p,y)|^2}{G(p,y)}\,dy \\
&\quad\ + 2 \int_{B^\rho_{R+1}(p)\setminus B^\rho_{R}(p)} \rho_m(y) \,G(p,y) \chi^2 \,dy \\
&\leq C(-\log\delta+1)+ \frac{4}{\delta\varepsilon} \int_{B^{\rho_m}_{R+1}(p)\setminus B^{\rho_m}_{R}(p)} \rho_m(y) \,G^2(p,y) \,dy \,.\end{aligned}$$ Now we let $R\rightarrow \infty$ and use Lemma \[lemmastoc\]. The thesis now follows.
In the special case when $M$ is non-parabolic with positive minimal Green’s function $G$ and with weight $\rho(x)=\frac{|\nabla G(p,x)|^2}{4 G^2(p,x)}$, we have the following refinement of Proposition \[claim2\].
\[claim3\] Assume that $(M,g)$ is non-parabolic with positive minimal Green’s function $G$ and with weight $\rho(x)=\frac{|\nabla G(p,x)|^2}{4 G^2(p,x)}$. Then there exists a positive constant $C$ such that for any function $f$ and any $0<\delta<1$, $\varepsilon >0$ one has $$\left|\int_{\mathcal{L}_{p}(\delta \varepsilon, \varepsilon)} G(p,y)\,f(y)\,dy \right| \leq C \left(-\log\delta \right) \sup_{\mathcal{L}_{p}(\delta \varepsilon, \varepsilon)} \left|\frac{f}{\rho}\right|\,.$$
We have $$\begin{aligned}
\left|\int_{\mathcal{L}_{p}(\delta \varepsilon, \varepsilon)} G(p,y)\,f(y)\,dy \right| &\leq\sup_{\mathcal{L}_{p}(\delta \varepsilon, \varepsilon)} \left|\frac{f}{\rho}\right| \left(\int_{\mathcal{L}_{p}(\delta \varepsilon, \varepsilon)} G(p,y)\,\rho(y)\,dy\right)\\
&=\frac{1}{4}\sup_{\mathcal{L}_{p}(\delta \varepsilon, \varepsilon)} \left|\frac{f}{\rho}\right| \left(\int_{\mathcal{L}_{p}(\delta \varepsilon, \varepsilon)} \frac{|\nabla G(p,y)|^2}{G(p,y)}\,dy\right)\\
&=\frac{1}{4}\left(-\log\delta \right)\sup_{\mathcal{L}_{p}(\delta \varepsilon, \varepsilon)} \left|\frac{f}{\rho}\right|,
\end{aligned}$$ where we have used Lemma \[lemma2\] in the last equality.
Proof of Theorem \[teo1\] {#sec-proofs}
=========================
In order to prove Theorem \[teo1\], we will show that $$|u(x)|=\left| \int_{M}G(x,y)f(y)\,dy \right| \leq v(x),$$ with $v\in C^{0}(M)$. We divide the proof in two parts, we first consider the case when $(M,g)$ is non-parabolic and then the case when it is parabolic.
[**Case 1:**]{} [*$(M,g)$ non-parabolic.*]{}
By assumption, $(M,g)$ satisfies $\left(\mathcal{P}_{\rho_R}^\infty\right)$. Let $x\in M$ and choose $R=R(x)>R_0$ large enough so that $x\in B_R (p)$. One has
$$\begin{aligned}
\left|\int_M G(x,y)\,f(y)\, dy\right| &\leq \left| \int_{B_R(p)} G(x,y)\,f(y)\,dy \right|+\left|\int_{M\setminus B_R(p)} G(x,y)\,f(y)\,dy\right|\\
&\leq C_1(x) + \int_{M\setminus B_R(p)} G(x,y)\,|f(y)|\,dy\end{aligned}$$
since $G(x,\cdot)\in L^1_{\text{loc}}(M)$. Hence, by Harnack’s inequality, we have $$\begin{aligned}
\label{eq501}
\left|\int_M G(x,y)\,f(y)\, dy\right| &\leq C_1(x) + C_2(x)\int_{M\setminus B_R(p)} G(p,y)\,|f(y)|\,dy \\ \nonumber
&\leq C_1(x) + C_2(x)\int_{M} G(p,y)\,|f(y)|\,dy \,,\end{aligned}$$ where $C_2(x)$ can be chosen as the constant in the Harnack’s inequality for the ball $B_{r(x)+1}(p)$. Then we estimate
$$\begin{aligned}
\int_{M}G(p,y)\,|f(y)|\,dy &= \int_{\mathcal{L}_{p}\left(0, \,A \exp \left(B\, \omega(a)\right)\right)} G(p,y)\,|f(y)|\,dy \\
&\,\,\,+ \int_{\mathcal{L}_{p}\left(A \exp \left(B\, \omega(a)\right),\infty\right)} G(p,y)\,|f(y)|\,dy \,.\end{aligned}$$
By Proposition \[lemma3\], Remark \[remark101\] we get $$\begin{aligned}
\label{eq128}
\int_{M}G(p,y)\,|f(y)|\,dy &\leq \int_{\mathcal{L}_{p}\left(0, A \exp \left(B\, \omega(a)\right)\right)} G(p,y)\,|f(y)|\,dy + C_3(a)\end{aligned}$$ for some positive constant $C_3(a)$. To estimate the first integral, we observe that, for any $m_{0}=m_{0}(x)\geq a$ one has $$\begin{aligned}
\label{eq129}
\int_{\mathcal{L}_{p}\left(0, \,A \exp \left(B\, \omega(a)\right)\right)} &G(x,y)\,|f(y)|\,dy = \int_{\mathcal{L}_{p}\left(0, \,(2A)^{-1}\exp(-B\omega(m_{0}))\right)}G(x,y)\,|f(y)|\,dy \nonumber \\
&\quad+ \int_{\mathcal{L}_{p}\left((2A)^{-1}\exp(-B\omega(m_{0})),\,A \exp \left(B\, \omega(a)\right)\right)}G(x,y)\,|f(y)|\,dy \,.\end{aligned}$$ We need the following lemma.
\[lemma5\] Choose $A,B$ as in Lemma \[lemma1\]. For any $m\geq m_0\geq a$ one has $$\label{eq400}\mathcal{L}_{p}\left(0, A^{-1}\exp(-B\omega(m))\right) \subset M \setminus B_m(p).$$
Since $m_{0}\geq a$, by Remark \[remark101\] imply $$\label{eq300}
\mathcal{L}_{p}\left(0, A^{-1}\exp(-B\omega(m_{0}))\right) \subset\mathcal{L}_{p}\left(0, A^{-1} \exp \left(-B\, \omega(a)\right)\right)\subset M\setminus B_{a}(p) .$$ If $$z\in \mathcal{L}_{p}\left(0, A^{-1}\exp(-B\omega(m))\right) \subset M\setminus B_{a}(p) \,,$$ then by Lemma \[lemma1\] $$A^{-1}\exp(-B\omega(m)) \geq G(p,z) \geq A^{-1}\exp(-B\omega(z)) \,.$$ Thus, $$\omega(z)\geq \omega(m)$$ and, by monotonicity of $\omega$, we obtain $r(z)\geq m$.
In particular, we get $$\mathcal{L}_{p}\left(0, (2A)^{-1}\exp(-B\omega(m_{0}))\right) \subset \mathcal{L}_{p}\left(0, A^{-1}\exp(-B\omega(m_{0}))\right) \subset M\setminus B_{m_0}(p).$$ Thus, $$\mathcal{L}_{p}\left((2A)^{-1}\exp(-B\omega(m_{0})),\,A \exp \left(B\, \omega(a)\right)\right) \subset B_{m_{0}}(p)$$ Then, since $G(x,\cdot)\in L^1_{\text{loc}}(M)$, we get $$\begin{aligned}
\label{eq130}
\int_{\mathcal{L}_{p}\left((2A)^{-1}\exp(-B\omega(m_{0})),\,A \exp \left(B\, \omega(a)\right)\right)}G(x,y)\,|f(y)|\,dy \leq C_4(a,m_0).\end{aligned}$$ Now, for any $m\geq m_{0}$, let $$\label{11}
\varepsilon:=(2A)^{-1}\exp(-B\omega(m)),\quad\quad\delta:=\exp(B\omega(m)-B\omega(m+1)).$$ By Lemma \[lemma5\], $$\mathcal{L}_p(0,2\varepsilon) \subset M\setminus B_m(p).$$ Hence we can apply Proposition \[claim2\] obtaining $$\begin{aligned}
\label{eq201}
&\int_{\mathcal{L}_{p}\left(0, (2A)^{-1}\exp(-B\omega(m_{0}))\right)}G(x,y)\,|f(y)|\,dy \\\nonumber
&= \sum_{m\geq m_{0}} \int_{\mathcal{L}_{p}\left((2A)^{-1}\exp(-B\omega(m+1)), (2A)^{-1}\exp(-B\omega(m))\right)}G(x,y)\,|f(y)|\,dy \\\nonumber
&\leq C \sum_{m\geq m_{0}}^{\infty}\left(\omega(m+1)-\omega(m)+1\right)\sup_{\mathcal{L}_{p}\left((2A)^{-1}\exp(-B\omega(m+1)), (2A)^{-1}\exp(-B\omega(m))\right)}\left|\frac{f}{\rho_m}\right|\\ \nonumber
&\leq C \sum_{m\geq m_{0}}^{\infty}\left(\omega(m+1)-\omega(m)+1\right)\sup_{\mathcal{L}_{p}\left(0, A^{-1}\exp(-B\omega(m))\right)}\left|\frac{f}{\rho_m}\right|\\ \nonumber
&\leq C \sum_{m\geq m_{0}}^{\infty}\left(\omega(m+1)-\omega(m)+1\right)\sup_{M \setminus B_m(p)}\left|\frac{f}{\rho_m}\right| <\infty \,,\end{aligned}$$ where in the last inequality we used Lemma \[lemma5\]. The proof of Theorem \[teo1\] is complete in this case.
[**Case 2:**]{} [*$(M,g)$ parabolic.*]{}
Let $G(x,y)$ be a Green’s function on $M$ (which is positive inside a certain ball, and negative outside). Fix any $R>0$ and let $\rho \equiv\rho_{R_0}$. Note that, arguing as in the proof of , it is sufficient to estimate $$\begin{aligned}
\int_{M}|G(p,y)||f(y)|\,dy &= \int_{M\setminus B^\rho_{R}(p)}|G(p,y)||f(y)|\,dy + \int_{B^\rho_{R}(p)}|G(p,y)||f(y)|\,dy \\
&\leq \int_{M\setminus B^\rho_{R}(p)}|G(p,y)||f(y)|\,dy+ C,\end{aligned}$$ since $G(p,\cdot)\in L^{1}_{\rm{loc}}(M)$ and $f$ is locally bounded. We have that $$M\setminus B^\rho_{R}(p) = \bigcup_{i=1}^{N} E_{i},$$ where each $E_{i}$ is an end with respect to $B^\rho_{R}(p)$. Note that every end $E_{i}$ is parabolic. In fact, if at least one end $E_{i}$ is non-parabolic, then $(M,g)$ is non-parabolic (see [@li] for a nice overview), but we are in the case that $(M,g)$ is parabolic. Since every $E_{i}$ is parabolic, every $E_{i}$ has finite weighted volume (see [@liwa2]), i.e. $$\int_{E_i} \rho\,dy < \infty \,.$$ Now choose $R$ large enough so that we can apply Lemma \[lemmastoc\] obtaining $$\begin{aligned}
&\int_{M\setminus B^\rho_{R}(p)}|G(p,y)||f(y)|\,dy \\
&\qquad\leq \left(\int_{M\setminus B^\rho_{R}(p)}\rho (y)|G(p,y)|^{2}\,dy\right)^{\frac{1}{2}}\left(\int_{M\setminus B^\rho_{R}(p)}\rho(y)\left(\frac{|f(y)|}{\rho(y)}\right)^{2}\,dy\right)^{\frac{1}{2}}\\
&\qquad\leq C\, \sup_{M\setminus B_{R_{0}}(p)} \left|\frac{f}{\rho}\right| \int_{M\setminus B^\rho_{R}(p)}\rho\,dy <\infty\,.\end{aligned}$$ This concludes the proof of Theorem \[teo1\].
We start as in the proof of Theorem \[teo1\] using , , and . Then, similar to , using Proposition \[claim3\], we obtain $$\begin{aligned}
&\int_{\mathcal{L}_{p}\left(0, (2A)^{-1}\exp(-B\omega(m_{0}))\right)}G(x,y)\,|f(y)|\,dy \\\nonumber
&= \sum_{m\geq m_{0}} \int_{\mathcal{L}_{p}\left((2A)^{-1}\exp(-B\omega(m+1)), (2A)^{-1}\exp(-B\omega(m))\right)}G(x,y)\,|f(y)|\,dy \\
&\leq C \sum_{m\geq m_{0}}^{\infty}\left(\omega(m+1)-\omega(m)\right)\sup_{M \setminus B_m(p)}\left|\frac{f}{\rho}\right| <\infty \,,\end{aligned}$$ Then $$\left| \int_{M}G(x,y)f(y)\,dy \right| <\infty$$ and the proof of Theorem \[teo2\] is complete.
Cartan-Hadamard and model manifolds {#sec-ex}
===================================
We consider Cartan-Hadamard manifolds, i.e. complete, non-compact, simply connected Riemannian manifolds with non-positive sectional curvatures everywhere. Observe that on Cartan-Hadamard manifolds the cut locus of any point $p$ is empty. Hence, for any $x\in M\setminus \{p\}$ one can define its polar coordinates with pole at $p$, namely $r(x) = \operatorname{dist}(x, p)$ and $\theta\in \mathbb S^{n-1}$. We have $$\textrm{meas}\big(\partial B_{r}(p)\big)\,=\, \int_{\mathbb S^{n-1}}A(r, \theta) \, d\theta^1d \theta^2 \ldots d\theta^{n-1}\,,$$ for a specific positive function $A$ which is related to the metric tensor [@gri1 Sect. 3]. Moreover, it is direct to see that the Laplace-Beltrami operator in polar coordinates has the form $$\Delta \,=\, \frac{\partial^2}{\partial r^2} + m(r, \theta) \, \frac{\partial}{\partial r} + \Delta_{\theta} \, ,$$ where $m(r, \theta):=\frac{\partial }{\partial r}(\log A)$ and $ \Delta_{\theta} $ is the Laplace-Beltrami operator on $\partial B_{r}(p)$. We have $$m(r,\theta) =\Delta r(x).$$
Let $$\mathcal A:=\left\{f\in C^\infty((0,\infty))\cap C^1([0,\infty)): \, f'(0)=1, \, f(0)=0, \, f>0 \ \textrm{in}\;\, (0,\infty)\right\} .$$ We say that $(M,g)$ is a rotationally symmetric manifold or a model manifold if the Riemannian metric is given by $$\label{e2}
g \,=\, dr^2+\varphi(r)^2 \, d\theta^2,$$ where $d\theta^2$ is the standard metric on $\mathbb S^{n-1}$ and $\varphi\in \mathcal A$. In this case, $$\Delta \,=\, \frac{\partial^2}{\partial r^2} + (n-1) \, \frac{\varphi'}{\varphi} \, \frac{\partial}{\partial r} + \frac1{\varphi^2} \, \Delta_{\mathbb S^{n-1}} \, .$$ Note that $\varphi(r)=r$ corresponds to $M=\mathbb R^n$, while $\varphi(r)=\sinh r$ corresponds to $ M=\mathbb H^n $, namely the $n$-dimensional hyperbolic space. The Ricci curvature in the radial direction is given by $${\mathrm{Ric}}( \nabla r, \nabla r) (x) = -(n-1)\frac{\varphi''(r(x))}{\varphi(r(x))}.$$
Cartan-Hadamard manifolds
-------------------------
Concerning the validity of the property $\left(\mathcal{P}_\rho^\infty\right)$ on a Cartan-Hadamard manifold we have the following result.
\[lemma-peso\] Let $(M,g)$ be a Cartan-Hadamard manifold with $${\mathrm{Ric}}( \nabla r, \nabla r) (x)\leq -C\big(1+r(x)\big)^{\gamma}$$ for some $\gamma\in {\mathbb{R}}$, $C>0$ and any $x\in M\setminus\{p\}$. Then $(M,g)$ satisfies the property $\left(\mathcal{P}_{\rho_{R}}^\infty\right)$ with $$\rho_R(x) = \begin{cases}
C'\, r(x)^{\gamma} &\quad\hbox{if } \gamma\geq -2 \\
C'\, r(x)^{-2} &\quad\hbox{if } \gamma < -2 \end{cases}$$ for all $R>0$ large enough and some $C'>0$.
As it will be clear from the proof, we have a weighted Poincaré inequality on $M$ if $\gamma \leq 0$ and a the weighted Poincaré inequality for functions with compact support in $M\setminus B_1(p)$ if $\gamma>0$.
We can find $\varphi\in \mathcal{A}$ given by $$\label{15}
\varphi(r)=
\begin{cases}
\exp\big(B\,r^{1+\frac{\gamma}{2}}\big) &\quad\hbox{if } \gamma>-2 \\
r^\delta &\quad\hbox{if } \gamma=-2 \\
r &\quad\hbox{if } \gamma<-2
\end{cases}$$ for $r$ large enough, $B>0$ small, $\delta=\delta(C)>1$ such that ${\mathrm{Ric}}( \nabla r, \nabla r) (x) \leq -\frac{\varphi''(r(x))}{\varphi(r(x))}$. By the Laplacian comparison in a strong form, which is valid only on Cartan-Hadamard manifolds (see [@xin Theorem 2.15]), one has $$\Delta r(x) \geq
\begin{cases}
C\, r(x)^{\gamma/2} &\quad\hbox{if } \gamma\geq-2 \\
C r(x)^{-1} &\quad\hbox{if } \gamma<-2 \,.
\end{cases}$$ Suppose $\gamma\leq 0$ and let $\alpha:=\max\{\gamma,-2\}\leq 0$. For any $u\in C^\infty_c (M)$, since $|\nabla r|^2=1$, we have $$\begin{aligned}
&C \int_M r(y)^\alpha \,u(y)^2\,dy\\
&\qquad\leq \int_M u(y)^2 r(y)^{\alpha/2} \Delta r (y)\,dy \\
&\qquad= -2 \int_M \langle \nabla u, \nabla r\rangle u(y) r(y)^{\alpha/2}\,dy + \frac{\alpha}{2} \int_M u(y)^2 r(y)^{\alpha/2-1} |\nabla r(y)|^2\,dy \\
&\qquad\leq 2 \int_M |u(y)| |\nabla u(y)| r(y)^{\alpha/2}\,dy\\
&\qquad\leq \frac{C}{2} \int_M r(y)^\alpha \,u(y)^2\,dy + \frac{2}{C} \int_M |\nabla u(y)|^2\,dy \,.\end{aligned}$$ Thus $$\int_M r(y)^\alpha \,u(y)^2\,dy \leq \frac{4}{C^2} \int_M |\nabla u(y)|^2\,dy$$ and the weighted Poincaré inequality on $M$ follows in this case.
Suppose now $\gamma >0$. By a Barta-type argument (see e.g. [@gri2 Theorem 11.17]), $$\lambda_1(M\setminus B_R(p)) \geq [C R^{\frac{\gamma}{2}}]^2 \quad \textrm{in}\;\; M\setminus B_R(p)\,.$$ Thus, the Poincaré inequality reads $$\begin{aligned}
\label{poineq}
C R^\gamma \int_M u(y)^2\,dy \leq \int_M |\nabla u(y)|^2\,dy\end{aligned}$$ for any $u$ with compact support in $M\setminus B_R(p)$. Now let $R>1$ and, for every $k\in\mathbb{N}$, define the cutoff functions $$\varphi_k(x):=\begin{cases} r(x)-k+1, &r(x)\in[k-1,k)\\ k+1-r(x), &r(x)\in[k,k+1)\\ 0 &\text{otherwise}.\end{cases}$$ Note that $|\nabla \varphi_k|\leq 1$ and for all $x\in M\setminus B_1(p)$, $\sum_k \varphi_k =1$ and $x\in \operatorname{supp}\varphi_k$ at most for two integers $k$. If $\operatorname{supp} u \subset M \setminus B_1(p)$, we have $$\begin{aligned}
\int_M r(y)^\gamma \,u(y)^2\,dy &= \int_M r(y)^\gamma \,\left(\sum_k \varphi_k (y) u(y)\right)^2\,dy \\
&\leq 2\sum_k \int_M r(y)^\gamma \,\varphi_k (y)^2 u(y)^2\,dy \\
&\leq C\sum_k (k-1)^\gamma \int_M \varphi_k (y)^2 u(y)^2\,dy \\
&\leq C\sum_k \int_M |\nabla\left(\varphi_k (y) u(y)\right)|^2\,dy,\end{aligned}$$ where in the last passage we used with $R=k-1$. Thus $$\begin{aligned}
\int_M r(y)^\gamma \,u(y)^2\,dy &\leq C\sum_k \left(\int_M u(y)^2|\nabla \varphi_k (y)|^2\,dy+\int_M \varphi_k(y)^2|\nabla u(y)|^2\,dy\right)\\
&\leq C\int_M u(y)^2\,dy+C\int_M |\nabla u(y)|^2\,dy\\
&\leq C\int_M |\nabla u(y)|^2\,dy,\end{aligned}$$ where in the last passage we used with $R=1$. Hence the weighted Poincaré inequality holds for functions with support in $M\setminus B_1(p)$.
Finally, the completeness of the metric $g_{\rho_R}:= {\rho_R}\, g$ follows. In fact, for any curve $\eta(s)$ parametrized by arclength with $0\leq s \leq T$, the length of $\eta$ with respect tp $g_{\rho_R}$ is given by $$\int_\eta \sqrt{{\rho_R}}\,ds \to \infty \quad\hbox{as } T\to \infty \,.$$
Let us write some estimates which will be useful both in the proof of Corollary \[cor-2\] and in the last Subsection \[ssu\]. Choose $\varphi\in\mathcal{A}$ as in with $\gamma=\gamma_1$ obtaining $$\frac{\varphi'(r(x))}{\varphi(r(x))}=\begin{cases} C\,r(x)^{\gamma_1/2} &\quad\hbox{if } \gamma_1\geq -2 \\ C\,r(x)^{-1} &\quad\hbox{if } \gamma_1< -2 \end{cases}$$ and $$\frac{\varphi''(r(x))}{\varphi(r(x))} = \begin{cases} C\,r(x)^{\gamma_1}+C' r(x)^{\gamma_1/2-1} &\quad\hbox{if } \gamma_1\geq -2 \\ 0 &\quad\hbox{if } \gamma_1<-2\, \end{cases}$$ for $r(x)>R>1$. A simple computation shows that, for $R=r(x)/4$, one has $$K_R(x) = \begin{cases} C\, r(x)^{\gamma_1/2} &\quad\hbox{if } \gamma_1\geq -2 \\ 0 &\quad\hbox{if } \gamma_1<-2\,, \end{cases}$$ $$\frac{I_R(x)}{R} = \begin{cases} C\, r(x)^{\gamma_1/2-1}\coth\left(C'r(x)^{\gamma_1/2+1}\right) &\quad\hbox{if } \gamma_1\geq -2 \\ \frac{2}{r(x)^2} &\quad\hbox{if } \gamma_1<-2\, \end{cases}$$ and $$Q_R(x) = \begin{cases} C\, r(x)^{\gamma_1} &\quad\hbox{if } \gamma_1\geq -2 \\ \frac{2}{r(x)^2} &\quad\hbox{if } \gamma_1<-2\,. \end{cases}$$ Thus $$\omega(r) = \begin{cases} C\, r^{\gamma_1/2+1} &\quad\hbox{if } \gamma_1\geq -2 \\ C \log r &\quad\hbox{if } \gamma_1<-2\,, \end{cases}$$ and, as $m\to\infty$, $$\label{asdf}
\omega(m+1)-\omega(m) \sim\begin{cases} C\, m^{\gamma_1/2} &\quad\hbox{if } \gamma_1\geq -2 \\ C m^{-1} &\quad\hbox{if } \gamma_1<-2\,. \end{cases}$$ On the other hand, using Lemma \[lemma-peso\] with $\gamma=\gamma_2$, we get the estimate $$\sup_{M\setminus B_m(p)} \frac{1}{\rho_m} \leq \begin{cases} C\,m^{-\gamma_2} &\quad\hbox{if }\gamma_2\geq -2 \\
C\, m^{2} &\quad\hbox{if } \gamma_2 < -2 \end{cases}\,.$$
For $\gamma_1\geq \gamma_2$ and $\gamma_1\geq 0$, we get $$\sum_{m}^{\infty}\Big(\omega(m+1)-\omega(m)+1\Big)\sup_{M\setminus B_m(p)}\left|\frac{f}{\rho_m}\right| \leq \begin{cases} C \sum_{m}^{\infty} \,m^{\gamma_1/2-\gamma_2-\alpha} &\quad\hbox{if }\gamma_2\geq -2 \\
C \sum_{m}^{\infty}\, m^{2+\gamma_1/2-\alpha} &\quad\hbox{if } \gamma_2< -2. \end{cases}$$ and the thesis immediately follows.
Optimality on rotationally symmetric manifolds {#ssu}
----------------------------------------------
We show that the assumptions in Theorem \[teo2\] are sharp on model manifolds. Let $(M,g)$ be a rotationally symmetric manifold with $\varphi\in\mathcal{A}$ defined as in for any $r>1$. One has $$\int_{M}G(x,y)f(y)\,dy<\infty \quad\quad\hbox{for any }\, x \in M \quad \Longleftrightarrow \quad \int_{M}G(p,y)f(y) \,dy<\infty .$$ Hence a solution of $\Delta u = f$ in $M$ exists if and only if $$u(p)=\int_{0}^{\infty}\left(\int_{r}^{\infty}\frac{1}{\varphi(t)^{n-1}}dt\right)f(r)\,\varphi(r)^{n-1}\,dr <\infty.$$
[*Case 1:*]{} $\gamma>-2$. With our choice of $\varphi$, by the change of variable $s=t^{1+\frac{\gamma}{2}}$, it is easily seen that, for any $r>0$ sufficiently large $$\label{asd}
\int_{r}^{\infty}\frac{1}{\varphi(t)^{n-1}}dt \sim C r^{-\frac{\gamma}{2}}\exp\left(-(n-1)r^{1+\frac{\gamma}{2}}\right).$$ Hence $$\begin{aligned}
\frac 1{C} \int_{1}^{\infty} & r^{-\frac{\gamma}{2}}\exp\left(-(n-1)r^{1+\frac{\gamma}{2}}\right) \frac{1}{\big(1+r\big)^{\alpha}}\exp\left((n-1)r^{1+\frac{\gamma}{2}}\right)\,dr \leq |u(p)|\\&\leq C \int_{1}^{\infty} r^{-\frac{\gamma}{2}}\exp\left(-(n-1)r^{1+\frac{\gamma}{2}}\right) \frac{1}{\big(1+r\big)^{\alpha}}\exp\left((n-1)r^{1+\frac{\gamma}{2}}\right)\,dr\end{aligned}$$ Therefore, $$\begin{aligned}
\frac 1 C\int_{1}^{\infty}\frac{1}{r^{\alpha+\frac{\gamma}{2}}}\,dr &\leq |u(p)|\leq C \int_{1}^{\infty}\frac{1}{r^{\alpha+\frac{\gamma}{2}}}\,dr\,.\end{aligned}$$ This yields that $$|u(p)|<\infty \quad \textrm{
if and only if} \quad
\alpha>1-\frac{\gamma}{2}.$$
On the other hand, a direct computation, using , shows that $$\rho(x)=\frac{|\nabla G(p,x)|^2}{4G^2(p,x)} \sim C r(x)^{\gamma}\,.$$ Furthermore, from , the assumption of Theorem \[teo2\] is satisfied if and only if $$\alpha>1-\frac{\gamma}{2},$$ and the optimality follows in this case.
[*Case 2:*]{} $\gamma=-2$. We have, $$\label{qwe}
\int_{r}^{\infty}\frac{1}{\varphi(t)^{n-1}}dt = C\, r^{-\delta(n-1)+1}\,.$$ Thus $$\begin{aligned}
\frac 1{C} \int_{1}^{\infty} r^{-\delta(n-1)+1}\frac{1}{\big(1+r\big)^{\alpha}}\,r^{\delta(n-1)}\,dr \leq |u(p)|\leq C \int_{1}^{\infty} r^{-\delta(n-1)+1}\frac{1}{\big(1+r\big)^{\alpha}}\,r^{\delta(n-1)}\,dr\end{aligned}$$ Therefore, $$\begin{aligned}
\frac 1 C\int_{1}^{\infty}\frac{1}{r^{\alpha-1}}\,dr &\leq |u(p)|\leq C \int_{1}^{\infty}\frac{1}{r^{\alpha-1}}\,dr\,,\end{aligned}$$ and $$|u(p)|<\infty \quad \textrm{
if and only if} \quad
\alpha>2.$$
On the other hand, a direct computation, using , shows that $$\rho(x)=\frac{|\nabla G(p,x)|^2}{4G^2(p,x)} \sim C r(x)^{-2}\,.$$ Furthermore, from , the assumption of Theorem \[teo2\] is satisfied if and only if $$\alpha>2,$$ and the optimality follows in this case.
[*Case 3:*]{} $\gamma<-2$. We have, $$\label{zxc}
\int_{r}^{\infty}\frac{1}{\varphi(t)^{n-1}}dt = C\, r^{2-n}\,.$$ Thus $$\begin{aligned}
\frac 1{C} \int_{1}^{\infty} r^{2-n}\frac{1}{\big(1+r\big)^{\alpha}}\,r^{n-1}\,dr \leq |u(p)|\leq C \int_{1}^{\infty} r^{2-n}\frac{1}{\big(1+r\big)^{\alpha}}\,r^{n-1}\,dr\end{aligned}$$ Therefore, $$\begin{aligned}
\frac 1 C\int_{1}^{\infty}\frac{1}{r^{\alpha-1}}\,dr &\leq |u(p)|\leq C \int_{1}^{\infty}\frac{1}{r^{\alpha-1}}\,dr\,,\end{aligned}$$ and $$|u(p)|<\infty \quad \textrm{
if and only if} \quad
\alpha>2.$$
On the other hand, a direct computation, using , shows that $$\rho(x)=\frac{|\nabla G(p,x)|^2}{4G^2(p,x)} \sim C r(x)^{-2}\,.$$ Furthermore, from , the assumption of Theorem \[teo2\] is satisfied if and only if $$\alpha>2,$$ and the optimality follows in this last case.
The authors are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). The first two authors are supported by the PRIN project “Variational methods, with applications to problems in mathematical physics and geometry”.
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abstract: 'The inclusive distributions of gluons and pions are calculated with absolute normalization for high-energy nucleon-nucleon collisions. The results for several unintegrated gluon distributions from the literature are compared. The gluon distribution proposed recently by Kharzeev and Levin based on the idea of gluon saturation is tested against DIS data from HERA. We find huge differences in both rapidity and transverse momentum distributions of gluons and pions in nucleon-nucleon collisions for different models of unintegrated gluon distributions. The approximations used recently in the literature are discussed. The Karzeev-Levin gluon distribution gives extremely good description of momentum distribution of charged hadrons at midrapidities. Contrary to a recent claim in the literature, we find that the gluonic mechanism discussed does not describe the inclusive spectra of charged particles in the fragmentation region, i.e. in the region of large $|y|$ for any unintegrated gluon distribution from the literature.'
---
[**From unintegrated gluon distributions to particle production in nucleon-nucleon collisions at RHIC energies**]{} [^1]
[A. Szczurek $^{1,2}$]{}
$^{1}$ [*Institute of Nuclear Physics\
PL-31-342 Cracow, Poland\
*]{} $^{2}$ [*University of Rzeszów\
PL-35-959 Rzeszów, Poland\
*]{}
Introduction
============
The recent results from RHIC (see e.g. [@RHIC]) have attracted renewed interest in better understanding the dynamics of particle production, not only in nuclear collisions.
Quite different approaches [@thermal; @PHOBOS; @KL01] have been used to describe the particle spectra from the nuclear collisions [@PHOBOS]. The thermal models do not make a direct link to nucleon-nucleon collisions. In contrast, in dual parton approaches (DPM) the nucleon-nucleon collisions are the basic ingredients of nuclear collisions. Somewhat extreme model in Ref.[@KL01] with an educated guess for unintegrated gluon distribution describes surprisingly well the whole charged particle rapidity distribution by means of gluonic mechanisms only. Such a gluonic mechanism would lead to the identical production of positively and negatively charged hadrons. The recent results of the BRAHMS experiment [@BRAHMS] put into question the successful description of Ref.[@KL01] and show that the DPM type approaches seems more correct. In the light of the BRAHMS experiment is becomes obvious that the large rapidity regions have more complicated flavour structure. The pure gluonic mechanisms, if at all, can be dominant only at midrapidities although the charged kaons [@BRAHMS] show that even this is doubtful. Similarly also the thermal models have difficulties to describe the (pseudo)rapidity dependence of particle to antiparticle ratios [@BRAHMS] and have to limit to the midrapidity only. In principle, the dynamics in nucleus-nucleus collision is fairly complicated and requires a separate analysis. In the following I concentrate only on nucleon-nucleon collisions – the basic ingredients of the nucleus-nucleus collisions.
On the microscopic level the approach of Kharzeev and Levin [@KL01] is based on the gluon-gluon fusion. The gluon-gluon fusion is expected to be the dominant process at midrapidities and at asymptotically large energies. It is not clear how large the energy should be to validate this thesis. The physics in the fragmentation region is somewhat different. It was suggested long ago [@DH77] that pions in the fragmentation region are correlated with the valence quark distributions in hadrons.
The standard hadronization approaches are based rather on the 2 $\rightarrow$ 2 partonic subprocesses which constitute only a part of the dynamics. The perturbative component of these hybrid models has a flavour structure as dictated by the quark/antiquark distributions. On the other hand, the flavour structure of the remaining soft component is not so explicit. Furthermore the partition into “soft” and “hard” components is somewhat arbitrary, being to some extend rather an artifact of a natural failure in applying the (2 $\rightarrow$ 2) pQCD at low transverse momenta of hadrons than a clear border of the two regions.
In this paper I discuss the relation between unintegrated gluon distributions in hadrons and the inclusive momentum distribution of particles produced in hadronic collisions. The results obtained with different unintegrated gluon distributions presented recently in the literature are shown and compared. In the present study I limit to the nucleon-nucleon collisions only and leave the nucleus-nucleus collisions for a separate analysis.
Photon-nucleon cross section at high energies
=============================================
It became a standard in recent years to first describe the HERA data and only then to test the resulting gluon distributions in other processes. We try to follow this reasonable methodology also for jet and particle production.
It is known that the LO total $\gamma^* N$ cross section can be written in the form $$\sigma_{tot}^{\gamma^* N} = \sum_q \int dz \int d^2 \rho
\; | \Psi_{\gamma^* \rightarrow q \bar q}(Q,z,\rho) |^2
\cdot \sigma_{(q \bar q) N}(x,\rho) \; .
\label{gamma_N}$$ In this paper we take the so-called quark-antiquark photon wave function of the perturbative form [@NZ90]. As usual, in order to correct the photon wave function for large dipole sizes (nonperturbative region) we introduce an effective quark/antiquark mass ($m_{eff} = m_0$). The dipole-nucleon cross section can be parametrized or calculated from the unintegrated gluon distribution $$\begin{aligned}
\sigma_{(q \bar q) N}(x,\rho)
&=&
\frac{4 \pi}{3} \int \frac{d^2 \kappa_t}{\kappa_t^2}
\left[1 - \exp(i\vec{\kappa_t} \vec{\rho}) \right]
\alpha_s {\cal F}(x,\kappa_t^2) \nonumber \\
&=& \frac{4 \pi^2}{3} \int \frac{d \kappa_t^2}{\kappa_t^2}
\left[ 1 - J_0(\kappa_t \rho) \right]
\alpha_s {\cal F}(x,\kappa_t^2) \; .
\label{Fourier_transform}\end{aligned}$$ In the equation above the running coupling constant is fixed constant or is frozen according to an analytic prescription [@SS97]. In the next section we shall compare the dipole-nucleon cross sections calculated from different unintegrated gluon distributions.
Unintegrated gluon distributions
================================
Search for the unintegrated gluon distribution in the nucleon was a subject of active both theoretical and phenomenological research in recent years. Still at present the unintegrated gluon distributions are rather poorly known. The main reason of the difficulties is the fact that the unintegrated gluon distribution is a quantity which depends on at least two variables ($x$ and $\kappa^2$) in a nontrivial and a priori unknown way. Another difficulty is in an unambiguous separation of perturbative and nonperturbative regions. In general different phenomena test the unintegrated gluon distribution in different corners of the phase space. Therefore it is not surprising that different gluon distributions found in the literature, extracted from the analyses of different phenomena, differ among themselves considerably [@small_x_collaboration]. In this section I collect and briefly discuss gluon distributions used in the present calculation of the jet and particle production. There are two different conventions of introducing unintegrated gluon distributions in the literature. The resulting quantities are denoted as $f$ (dimenionless quantity) and ${\cal F}$ (with dimension 1/GeV$^2$). We shall keep this notation throughout the present paper.
BFKL gluon distribution
-----------------------
At very low $x$ the unintegrated gluon distributions are believed to fulfil BFKL equation [@BFKL] (see also [@KMSR90]). After some simplifications [@AKMS94] the BFKL equation reads $$-x \frac{\partial f(x, q_t^2)}{\partial x} =
\frac{\alpha_s N_c}{\pi} q_t^2
\int_0^{\infty} \frac{dq_{1t}^2}{q_{1t}^2}
\left[
\frac{f(x,q_{1t}^2) - f(x,q_t^2)}{|q_t^2 - q_{1t}^2|}
+ \frac{f(x,q_t^2)}{\sqrt{q_t^4+4 q_{1t}^4}}
\right] \; .
\label{BFKL_equation}$$ The homogeneous BFKL equation can be solved numerically [@AKMS94]. Here in the practical applications we shall use a simple parametrization for the solution [@ELR96] $$f(x,\kappa_t^2) = \frac{C}{x^{\lambda}}
\left(\frac{\kappa_t^2}{q_0^2}\right)^{1/2}
\frac{{\tilde \phi}_0}{\sqrt{ 2 \pi \lambda''\ln(1/x)}}
\exp\left[ - \frac{\ln^2(\kappa_t^2/{\bar q}^2)}{r2\lambda ''\ln(1/x)}
\right]
\label{ELR_parametrization}$$ In the above expression $\lambda = 4 {\bar \alpha}_s \ln2$, $\lambda''$ = 28 ${ \bar \alpha}_s \zeta(3)$, ${\bar \alpha}_s
= 3 \alpha_s/\pi, \zeta(3)$ = 1.202. The remaining parameters were adjusted in [@ELR96] to reproduce with a satisfactory accuracy the gluon distribution which was obtained in [@AKMS94] as the numerical solution of the BFKL equation. It was found that ${\bar q} = q_0$ = 1, $C {\bar \phi_0}$ = 1.19 and $r$ = 0.15 [@ELR96].
Golec-Biernat-Wüsthoff gluon distribution
-----------------------------------------
Another parametrization of gluon distribution in the proton can be obtained based on the Golec-Biernat-Wüsthoff parametrization of the dipole-nucleon cross section with parameters fitted to the HERA data [@GBW]. The resulting gluon distribution reads [@GBW_glue]: $$\alpha_s {\cal F}(x,\kappa_t^2) =
\frac{3 \sigma_0}{4 \pi^2} R_0^2(x) \kappa_t^2 \exp(-R_0^2(x)
\kappa_t^2) \; ,
\label{GBW_glue}$$ where $$R_0(x) = \frac{1}{GeV} \left( \frac{x}{x_0} \right)^{\lambda/2}
\; .$$ From their fit to the data: $\sigma_0$ = 29.12 mb, $x_0$ = 0.41 $\cdot$ 10$^{-4}$, $\lambda$ = 0.277 [@GBW]. In order to determine the gluon distribution needed in calculating jet and particle production we shall take $\alpha_s$ = 0.2.
Kharzeev-Levin gluon distribution
---------------------------------
Another parametrization, also based on the idea of gluon saturation, was proposed recently in [@KL01]. In contrast to the GBW approach [@GBW], where the dipole-nucleon cross section is parametrized, in the Karzeev-Levin approach it is the gluon distribution which is parametrized. In the following we shall consider the most simplified functional form: $$\begin{aligned}
{\cal F}(x,\kappa^2) =
\begin{cases}
f_0 & \text{if} \; \kappa^2 < Q_s^2 , \\
f_0 \cdot \frac{Q_s^2}{\kappa^2} & \text{if} \; \kappa^2 > Q_s^2 .
\end{cases}
\label{KL_glue}\end{aligned}$$ The saturation momentum $Q_s$ is parametrized exactly as in the GBW model $Q_s^2(x) =$ 1 GeV$^2$ $\cdot \left( \frac{x_0}{x}
\right)^{\lambda}$. It was claimed in [@KL01] that the gluon distribution like (\[KL\_glue\]) leads to a good description of the recent RHIC rapidity distributions. It is interesting to check its performance for the deep inelastic scattering at low Bjorken $x$.
In the following the normalization constant $f_0$ is adjusted to roughly describe the HERA data. We find $f_0$ = 170 mb. The quality of the fit is shown in Fig. \[fig:HERA\] for $Q^2$ = 0.25, 5, 10 GeV$^2$.
![ *The cross section $\sigma_{tot}^{\gamma^* p}$ as a function of the center of mass energy $W$ for $Q^2$ = 0.25 GeV$^2$, $Q^2$ = 5 GeV$^2$ and $Q^2$ = 10 GeV$^2$. The results obtained with the KL gluon distribution ($m_0$ = 0.15/0.10 GeV) are shown by the solid and dashed lines. Experimental data were taken from [@HERA_data]. \[fig:HERA\]*](sig_w.eps){width="7cm"}
In this fit the running coupling constant frozen according to [@SS97] was used. The result at low photon virtuality ($Q^2$ = 0.25 GeV$^2$) depends also on the value of the quark/antiquark effective mass. In the calculation in Fig. \[fig:HERA\] $m_0$ = 0.15 GeV (solid) and $m_0$ = 0.10 GeV (dashed) was used. It can be inferred from the figure that the (virtual) photon-proton cross section at large virtuality ($Q^2$ = 5,10 GeV$^2$) is in practice independent of the effective quark mass. This allows to fix the gluon normalization constant $f_0$.
In order to better visualize the difference to the GBW model, in Fig. \[fig:dip\_nuc\] I compare the dipole-nucleon
![ *The dipole-nucleon cross section as a function of the transverse dipole size $\rho$ for the GBW (dashed) and KL (dotted) unintegrated gluon distributions. \[fig:dip\_nuc\]*](sdip_rho.eps){width="7cm"}
cross sections in both parametrizations for $x=10^{-2}$, 10$^{-3}$, 10$^{-4}$. In the GBW approach the dipole-nucleon cross section saturates at large dipole size. In contrast to the GBW parametrization, the dipole-nucleon cross section calculated according to Eq.(\[Fourier\_transform\]) based on the KL gluon distribution (\[KL\_glue\]) grows slowly with the dipole size $\rho$.
Kimber-Martin-Ryskin gluon distribution
---------------------------------------
The unintegrated gluon distribution can be obtained even when the integrated gluon distribution fulfils standard DGLAP evolution equation. At very small $x$ $${\cal F}(x,\kappa^2) = \frac{\partial}{\partial Q^2}
\left[ x g(x,Q^2) \right] |_{Q^2 = \kappa^2} \; .
\label{derivative}$$ This prescription breaks at larger values of $x$ when the derivative of the gluon distribution becomes negative. This may be somewhat improved by introducing a Sudakov form factor $T_g(\kappa^2,\mu^2)$. Then the unintegrated gluon distribution reads [@KMR]: $${\cal F}(x,\kappa^2,\mu^2) =
\frac{\partial}{\partial Q^2}
\left[
T(Q^2,\mu^2) x g(x,Q^2) \right] |_{Q^2 = \kappa^2}
\; .
\label{KMR}$$
Resumming virtual contributions to DGLAP equation, the unintegrated parton distributions can be written as [@KMR] $$f_a(x,\kappa^2,\mu^2) = T_a(\kappa^2,\mu^2)
\cdot \frac{\alpha_s(\kappa^2)}{2 \pi}
\sum_{a'} \int_x^{1-\delta} P_{aa'}(z) \left(\frac{x}{z} \right)
a'\left(\frac{x}{z},\kappa^2\right) dz \; .
\label{KMR_master}$$ Specializing to the gluon distribution the Sudakov form factor reads as $$T_g(\kappa^2,\mu^2) =
\exp\left(
-\int_{\kappa^2}^{\mu^2} \frac{d p^2}{p^2} \frac{\alpha_s(p^2)}{2 \pi}
\int_{0}^{1-\delta} dz z \left[P_{gg}(z)+\sum_q P_{qg}(z) \right]
\right) \; .
\label{Sudakov}$$ The Sudakov form factor introduces a dependence on a second scale $\mu^2$. It is reasonable to assume that the unintegrated gluon density given by (\[KMR\_master\]) starts only for $\kappa_t^2 > \kappa_{t0}^2$ [@KMS97]. At lower $\kappa_t^2$ an extrapolation is needed. In our case of particle distributions the results are sensitive to rather low $\kappa$. Because of this, a use of the GRV integrated gluon distribution [@GRV95; @GRV98] in (\[KMR\_master\]) seems more adequate than any other PDF. Following Ref.[@MT02] $\kappa_{t0}^2$ = 0.5 GeV$^2$ is taken as the lowest value where the unintegrated gluon distribution is calculated from Eq.(\[KMR\_master\]). Below it is assumed $${\cal F}(x,\kappa^2) = f(x,\kappa^2)/\kappa^2 =
f(x,\kappa_0^2)/\kappa_0^2 \; .
\label{low_kappa}$$
The choice of $\mu^2$ in our case of jet (particle) production is not completely obvious. In the present analysis $\mu^2 = p_t^2$ is assumed, where $p_t$ is transverse momentum of the produced gluon ($\equiv$ jet). In accord with the interpretation of the Sudakov form factor as a survival probability we assume that if transverse momentum of the produced gluon is smaller than the transverse momentum of the last gluon of the ladder ($p_t < \kappa_{1}$ or $p_t < \kappa_{2}$, see next section) then the corresponding Sudakov form factor is set to 1, i.e. we do not allow for any enhancement. If $T_g$ in Eq.(\[KMR\_master\]) is ignored we shall denote the corresponding gluon distribution as $f_{DGLAP}$ or ${\cal F}_{DGLAP}$ and call it DGLAP gluon distribution for brevity.
Blümlein gluon distribution
---------------------------
In the approach of Blümlein [@Blue95] the $\kappa_t^2$ dependent gluon distribution satisfying the BFKL equation can be represented as the convolution of the integrated gluon density $x g(x,\mu^2)$ and a universal function $\cal{B}$ $${\cal F}(x,\kappa_t^2,\mu^2) = \int_x^1 {\cal B}(z,\kappa_t^2,\mu^2)
\frac{x}{z} g(\frac{x}{z},\mu^2) dz \; .
\label{bluemlein_convo}$$ The universal function ${\cal B}(x,\kappa_t^2,\mu^2)$ can be represented as a series [@Blue95]. The first term of the expansion describes BFKL dynamics in the double-logarithmic approximation: $${\cal B}(z,\kappa_t^2,\mu^2) =
\begin{cases}
\frac{{\bar \alpha}_s}{z \kappa_t^2}
J_0(2 \sqrt{{\bar \alpha_s} \log(1/z) \log(\mu^2/\kappa_t^2)})
& \text{if} \; \kappa_t^2 < \mu^2 \\
\frac{{\bar \alpha}_s}{z \kappa_t^2}
I_0(2 \sqrt{{\bar \alpha_s} \log(1/z) \log(\kappa_t^2/\mu^2)})
& \text{if} \; \kappa_t^2 > \mu^2 ,
\end{cases}
\label{B_function}$$ where ${\bar \alpha_s} = 3 \alpha_s/\pi$.
In DIS there is a natural choice of the scale $\mu^2$. The choice of the scale is not so obvious in the case considered in the present paper. I shall argue that in practice the dependence on that scale is very weak. In the following I shall use the integrated gluon distribution in Eq.(\[bluemlein\_convo\]) from Ref. [@GRV95].
Inclusive gluon production
==========================
Before we go to particle production in the next section, let us consider the first step of the process – production of partons. At high energies gluons are the most abundantly produced partons in hadron-hadron collisions. Also gluons are responsible for their production.
At sufficiently high energy the cross section for inclusive gluon production in $h_1 + h_2 \rightarrow g$ can be written in terms of the unintegrated gluon distributions “in” both colliding hadrons: $$\frac{d \sigma}{dy d^2 p_t} = \frac{16 N_c}{N_c^2 - 1}
\frac{1}{p_t^2}
\int
\alpha_s(\Omega^2)
{\cal F}_1(x_1,\kappa_1^2) {\cal F}_2(x_2,\kappa_2^2)
\delta(\vec{\kappa}_1+\vec{\kappa}_2 - \vec{p}_t)
\; d^2 \kappa_1 d^2 \kappa_2 \; .
\label{inclusive_glue0}$$ In the equation above $f_1$ and $f_2$ are unintegrated gluon distributions in hadron $h_1$ and $h_2$, respectively. The longitudinal momentum fractions are fixed by kinematics: $x_{1/2} = \frac{p_t}{\sqrt{s}} \cdot \exp(\pm y)$. Generally the smaller jet (parton) momenta $p_t$, the smaller $x_{1/2}$ come into play. The argument of the running coupling constant is taken as $\Omega^2 = \max(\kappa_1^2,\kappa_2^2,p_t^2)$. The formula (\[inclusive\_glue0\]) above was first written by Gribov, Levin and Ryskin [@GLR81] (see also [@LL94]) and used later e.g. in [@ELR96]. As discussed in Ref.[@GM97] the normalization of the cross section in some previous works was not always correct. Making use of the $\delta$ function (momentum conservation) one can simplify (\[inclusive\_glue0\]) to the integral $$\frac{d \sigma}{dy d^2 p_t} = \frac{16 N_c}{N_c^2 - 1}
\frac{1}{p_t^2} {\frac{1}{4}}
\int
\alpha_s(\Omega^2)
{\cal F}_1\left(x_1,\left( \frac{\vec{p}_t + \vec{q}_t}{2} \right) \right)
{\cal F}_2\left(x_2,\left( \frac{\vec{p}_t - \vec{q}_t}{2} \right) \right)
d^2 q_t
\; ,
\label{inclusive_glue1}$$ where $\vec{q}_t = \vec{\kappa}_1 - \vec{\kappa}_2$ was introduced. The factor 1/4 is the jacobian of transformation from ($\vec{\kappa}_1$, $\vec{\kappa}_2$) to ($\vec{p}_t$, $\vec{q}_t$). The integral above is a two-dimensional integral over $d^2 q_t$, i.e. over $q_t dq_t d\phi$, where $\phi$ is the azimuthal angle between $q_t$ and $p_t$. The original integral (\[inclusive\_glue1\]) can be written as $$\frac{d \sigma}{dy d^2 p_t} = \int \; I(\phi) \; d \phi \; ,
\label{I_phi_def}$$ where $$I(\phi) =
\frac{4 N_c}{N_c^2 - 1}
\frac{1}{p_t^2}
\int
\alpha_s(\Omega^2)
{\cal F}_1\left(x_1,\kappa_1^2\right)
{\cal F}_2\left(x_2,\kappa_2^2\right)
\; q_t d q_t \; .
\label{I_phi_explicit}$$
![ *The intrinsic azimuthal correlations for different unintegrated gluon distributions: GBW (dashed), KL (solid), BFKL (dotted), Blümlein (thick dash-dotted), DGLAP (thick dashed) and KMR (thick solid) at W = 200 GeV. \[fig:phi\_correl\]*](phi.eps){width="8cm"}
In Fig. \[fig:phi\_correl\], I show the intrinsic angular correlation function $I(\phi)$ for different models of unintegrated gluon distributions for a RHIC energy $W$ = 200 GeV. In this calculation y=0 and $p_t$ = 1 GeV was taken. Quite a different pattern is obtained for different unintegrated gluon distributions. The $\phi$-distribution is flat for the KL, BFKL, DGLAP and KMR gluon distributions. The most pronounced structure is obtained with the Blümlein gluon distribution [@Blue95](GRV95, $\mu^2$ = 10 GeV$^2$). It was checked that the Blümlein (GRV95) gluon distribution is not very sensitive to the choice of the second scale $\mu^2$. The $\phi$ dependence at $y \ne$ 0 also strongly depends on the unintegrated gluon distribution.
It was suggested in [@KL01] that the integral (\[inclusive\_glue1\]) may be approximated by the formula $$\frac{d \sigma}{dy d^2 p_t} = \frac{4 N_c \alpha_s}{N_c^2 - 1}
\frac{1}{p_t^2}
\int \left[ {\cal F}_1(x_1,{p}_t^2) {\cal F}_2(x_2,{q}_t^2) +
{\cal F}_1(x_1,{q}_t^2) {\cal F}_2(x_2,{p}_t^2) \right]
d q_t^2 \; .
\label{inclusive_glue2}$$
![ *A comparison of the gluon rapidity distributions obtained from the exact (\[inclusive\_glue1\]) (thick lines) and approximate (\[inclusive\_glue2\]) (thin lines) formula for different models of unintegrated gluon distributions at W = 200 GeV. \[fig:approx1\]*](y_ex_ap.eps){width="8cm"}
![*A comparison of the gluon transverse momentum distributions obtained from the exact (\[inclusive\_glue1\]) (thick lines) and approximate (\[inclusive\_glue2\]) (thin lines) formula for different models of unintegrated gluon distributions at W = 200 GeV. \[fig:approx2\]*](pt_ex_ap.eps){width="8cm"}
In Fig.\[fig:approx1\] (rapidity distribution) and Fig.\[fig:approx2\] (transverse momentum distribution) I compare the results using the exact Eq.(\[inclusive\_glue1\]) and the approximate Eq.(\[inclusive\_glue2\]) formulae for different models of unintegrated gluon distributions. In Fig.\[fig:approx1\] the integration over $p_t >$ 0.5 GeV is performed while in Fig.\[fig:approx2\] -1 $ < y < $ 1. In both cases $\alpha_s$ was fixed at 0.2. As can be seen by inspection of the figures the use of the approximate formula is quantitatively justified for the KL, BFKL gluon distributions and not justified for the GBW one.
![*Inclusive gluon rapidity distribution ($p_t >$ 0.5 GeV) at W = 200 GeV for different models of unintegrated gluon distributions. \[fig:glu\_y\]* ](jet_y.eps){width="8cm"}
In Fig.\[fig:glu\_y\] I compare the cross section $\frac{d \sigma}{dy}(y)$ for different models of unintegrated gluon distributions. In this calculation $p_t <$ 0.5 was assumed. The rapidity distribution of gluons are rather different for different gluon PDF. Average values of $x_1$ and $x_2$ obtained with different gluon distribution with the $p_t$ interval chosen are shown in Fig.\[fig:x1\_x2\]. The following general observations can be made. Average value $<x_1>$ and $<x_2>$ only weakly depend on the model of unintegrated gluon distribution. For $y \sim$ 0 at the RHIC energy W = 200 GeV one tests unintegrated gluon distributions at $x_g$ = 10$^{-3}$ - 10$^{-2}$. This is the region known already from the HERA kinematics. When $|y|$ grows one tests more and more asymmetric (in $x_1$ and $x_2$) configurations. For large $|y|$ either $x_1$ is extremely small ($x_1 <$ 10$^{-4}$) and $x_2 \rightarrow$ 1 or $x_1 \rightarrow$ 1 and $x_2$ is extremely small ($x_2 <$ 10$^{-4}$). These are regions of gluon momentum fraction where the unintegrated gluon PDF is rather poorly known. The approximation used in obtaining unintegrated gluon distributions are valid certainly only for $x <$ 0.1. In order to extrapolate the gluon distribution to $x_g \rightarrow$ 1 I multiply the gluon distributions from the previous section by a factor $(1-x_g)^n$, where n = 5-7.
![*The average value of $x_1$ and $x_2$ for $p_t >$ 0.5 GeV and at W = 200 GeV. Lines corresponding to different unintegrated gluon PDF are identical as in the previous figure. \[fig:x1\_x2\]* ](jet_x1x2.eps){width="8cm"}
In the approach considered in the present paper (for details see next section) the production of particles is sensitive to rather small gluon (called equivalently jet despite of the small transverse momentum) transverse momenta.
![*Inclusive gluon transverse momentum distribution (-1 $< y <$ 1) at W = 200 GeV for different models of unintegrated gluon distributions: BFKL (dotted), GBW (dashed), KL (solid), Blümlein (thick dash-dotted) and DGLAP (thick dashed). \[fig:glu\_pt\]* ](jet_pt.eps){width="8cm"}
In Fig.\[fig:glu\_pt\] I plot $\frac{d \sigma}{dp_t}(p_t)$ in the low $p_t$ region. In these calculations the gluon rapidity was integrated in the interval -1 $< y <$ 1. The results obtained with different models for unintegrated gluon distributions differ considerably. The transverse momentum distribution obtained with the GBW gluon density is much steeper than the distribution for any other gluon density. The inclusion of DGLAP evolution as in [@BGK02] would probably change the situation. In the case of the Blümlein gluon distribution the transverse momentum spectrum has a natural low-$p_t$ cut-off if the scale $\mu^2 = p_t^2$ is chosen. If similar prescription of the scale is used for calculating gluon transverse momentum distribution with KMR method the DGLAP and KMR results are almost identical. Contrary to the claim in [@KL01] the result obtained with the GBW and KL gluon distributions differ considerably.
The rapidity and pseudorapidity distributions of partons (massless particles) are identical. The situation changes when massive particles are produced in the final state via fragmentation. Below we discuss how to take into account the unknown hadronization process with the help of phenomenological fragmentation functions.
From gluon to particle distributions
====================================
In Ref.[@KL01] it was assumed, based on the concept of local parton-hadron duality, that the rapidity distribution of particles is identical to the rapidity distribution of gluons. This seems to be a very severe assumption and for massive particles this idea must lead to incorrect results, especially in the fragmentation region. This approach leads to e.g. (massive) particles with rapidities ($y_h$) beyond the allowed kinematical region ($y_{h,min},y_{h,max}$). Furthermore in [@KL01] the normalization of rapidity distributions was fitted to the experimental charged particle rapidity distributions. In our opinion, the good description of the charged particle distribution in the full range of rapidity in Ref. [@KL01] is due to these simplifications rather than due to the underlying dynamics.
In the present approach I follow a different, yet simple, approach which makes use of phenomenological fragmentation functions (see e.g.[@W00; @EH02]). For our present exploratory study it seems sufficient to assume that the emitted hadron, mostly pion, is collinear to the gluon direction ($\theta_h = \theta_g$). This is equivalent to $\eta_h = \eta_g = y_g$, where $\eta_h$ and $\eta_g$ are hadron and gluon pseudorapitity, respectively.
In experiments a good identification of particles is not always achieved which makes impossible to determine the rapidity of a particle. The practice then is to measure pseudorapidity. The rapidity of a given type of hadrons ($y_h$) with a mass $m_h$ can be obtained from the pseudorapidity as $$y_h = \frac{1}{2}
\left[
\frac{\sqrt{\frac{m_h^2+p_{t,h}^2}{p_{t,h}^2} + \sinh^2\eta_h } + \sinh\eta_h }
{\sqrt{\frac{m_h^2+p_{t,h}^2}{p_{t,h}^2} + \sinh^2\eta_h } - \sinh\eta_h }
\right] \; .
\label{yh_etay}$$ The collinearity of partons and particles leads to the following relation between rapidity of the gluon and hadron $$y_g = \mathrm{arsinh} \left( \frac{m_{t,h}}{p_{t,h}} \sinh y_h \right)
\; ,
\label{yg_yh}$$ where the transverse mass $m_{t,h} = \sqrt{m_h^2 + p_{t,h}^2}$. In order to introduce phenomenological fragmentation functions one has to define a new kinematical variable. In accord with $e^+e^-$ and $e p$ collisions I define a standard auxiliary quantity $z$ by the equation $E_h = z E_g$. This leads to the following relation between transverse momenta of the gluon and hadron $$p_{t,g} = \frac{p_{t,h}}{z} J(m_{t,h},y_h) \; ,
\label{ptg_pth}$$ where $$J(m_{t,h},y_h) =
\left( 1 - \frac{m_h^2}{m_{t,h}^2 \cosh^2 y_h} \right)^{-1/2} \; .
\label{J}$$ Now we can write the single particle distribution in terms of the gluon distribution from the last section as follows $$\begin{aligned}
\frac{d \sigma (\eta_h, p_{t,h})}{d \eta_h d^2 p_{t,h}} =
\int d y_g d^2 p_{t,g} \int
dz \; D_{g \rightarrow h}(z,\mu_D^2) \\ \nonumber
\delta(y_g - \eta_h) \;
\delta^2\left(\vec{p}_{t,h} - \frac{z \vec{p}_{t,g}}{J}\right)
\cdot \frac{d \sigma (y_g, p_{t,g})}{d y_g d^2 p_{t,g}} \; .
\label{from_gluons_to_particles}\end{aligned}$$ Making use of the $\delta$ functions we can write the single particle spectrum as $$\frac{d \sigma(\eta_h,p_{t,h})}{d \eta_h d^2 p_{t,h}} =
\int_{z_{min}}^{z_{max}} dz
\frac{J^2 D_{g \rightarrow h}(z, \mu_D^2)}{z^2}
\frac{d \sigma(y_g,p_{t,g})}{d y_g d^2 p_{t,g}}
\Bigg\vert_{y_g = \eta_h \atop p_{t,g} = J p_{t,h}/z} \; .
\label{single_particle_spectrum}$$ Experimentally instead of the two-dimensional spectrum (\[single\_particle\_spectrum\]) one determines rather one-dimensional spectra in either $\eta_h$ or $p_{t,h}$.
The one-dimensional pseudorapidity distribution can be obtained by integration over hadron transverse momenta $$\frac{d \sigma(\eta_h)}{d \eta_h} =
\int d^2 p_{t,h} \;
\frac{d \sigma(\eta_h,p_{t,h})}{d \eta_h d^2 p_{t,h}} \; .
\label{eta_had_distribution}$$
![*Charged-pion pseudorapidity distribution at W = 200 GeV for the KL unintegrated gluon distribution for different parametrizations of fragmentation functions. In this calculation $p_{t,h} >$ 0.2 GeV. The experimental data of the UA5 collaboration are taken from [@UA5_exp]. \[fig:eta\_ff\]* ](eta_ff.eps){width="8cm"}
Stable particles [^2] are produced directly in the fragmentation process or are decay products of other unstable particles. There are a few global analyses of fragmentation function in the literature up to next-to-leading order [@BKK95; @KKP00; @Kretzer00; @BFGW01]. In the present calculation I shall use only leading order fragmentation functions from [@BKK95; @KKP00]. One should remember, however, that both $e^+ e^-$ and $e p$ collisions do not allow to uniquely determine $D_{g \rightarrow h}$ fragmentation functions. In order to test sensitivity of our results to these, in my opinion, not quite well known objects I shall use also simple functional forms: $D_{g\rightarrow h}(z) = 2 \frac{1-z}{z}$ (model I) or $D_{g\rightarrow h}(z) = 3 \frac{(1-z)^2}{z}$ (model II) with the factors in front adjusted to conserve momentum sum rule. When charged particles are measured only, then to a good approximation it is sufficient to multiply the fragmentation functions above by a factor 2/3.
![*Charged-pion pseudrapidity distribution at W = 200 GeV for different models of unintegrated gluon distributions. In this calculation $p_{t,h} >$ 0.2 GeV. The experimental data of the UA5 collaboration are taken from [@UA5_exp]. \[fig:eta\_glue\]* ](eta_glue.eps){width="8cm"}
In Fig.\[fig:eta\_ff\] I compare pseudorapidity distribution of charged pions at W = 200 GeV calculated with the KL gluon distribution and different parametrizations of fragmentation functions. For the BKK1995 [@BKK95] and for the KKP2000 [@KKP00] fragmentation functions the factorization scale was set to $\mu_D^2 = p_{t,g}^2$, except for $p_{t,g} <$ 1 GeV, where it was frozen at $\mu_D^2 =$ 1 GeV$^2$. For reference shown are also experimental data for charged particles measured by the UA5 collaboration at CERN [@UA5_exp]. The results only weakly depend on the choice of the $g \rightarrow \pi$ fragmentation function. It is worth stressing that the theoretical cross section at $\eta_h \approx$ 0 is almost consistent with the experimental one. However, the shapes of theoretical and experimental pseudorapidity distributions differ significantly. It seems there is a room for different mechanisms typical for fragmentation regions. The specificity of these regions will be discussed elsewhere.
Let us analyze now how the results for pseudorapidity distributions depend on the choice of the unintegrated gluon distribution. In Fig.\[fig:eta\_glue\] I compare pseudorapidity distribution of charged pions for different models of unintegrated gluon distributions. In this calculation the Binnewies-Kniehl-Kramer fragmentation function [@BKK95] has been used. The conclusions inferred above stay true also here. Having in view a dramatically steep $p_{t,g}$ distribution in Fig.\[fig:glu\_pt\] it is rather surprising that the normalization of the spectra at midrapidities comes roughly correct, although very is a tendency to an overestimation for some gluon distributions. This can be due to the fit to DIS data, where the resolved photon component has been neglected. If the resolved photon component is explicitly included [@PS03] then the normalization of the dipole-nucleon component (dipole-nucleon cross section or unintegrated gluon distribution) must be reduced.
![*Average values of $\left< p_t^2 \right>$ (solid), $\left< \kappa_1^2 \right>$ (dashed) and $\left< \kappa_2^2 \right>$ (dotted) as a function of pion pseudorapidity \[fig:ave\_pt\_eta\]* ](ave_pt_eta.eps){width="8cm"}
What are typical transverse momenta of gluons involved in the calculations is shown in Fig.\[fig:ave\_pt\_eta\]. In this calculation we have used the KL unintegrated gluon distribution and the BKK $g \rightarrow \pi$ fragmentation functions [@BKK95]. We observe a maximum of the transverse momentum squared of the produced gluon at $\eta_h \approx$ 0. In our implementation of fragmentation ($p_{t,h}^2 \ll p_{t,g}^2$) one tests relatively large $p_{t,g}^2$. While at midrapidities $ \left< p_t^2 \right> >
\left< \kappa_1^2 \right>, \left< \kappa_2^2 \right> $, when going to the fragmentation regions the relation reverses. In the whole range of pseudorapidity one tests on average $ \kappa_1^2, \kappa_2^2 \sim$ 1 GeV$^2$. One should remember, however, that at the same time $\left<x_1\right>$ and $\left<x_2\right>$ change dramatically when going from midrapidities to the fragmentation region.
![*The pseudorapidity distribution of protons (solid), antiprotons (dashed) and the difference of the spectra of $\pi^+$ and $\pi^-$ (dash-dotted) in the proton-proton collision at W = 200 GeV obtained with the code HIJING [@HIJING]. The thick solid line corresponds to the sum of these three contributions. The experimental data of the UA5 collaboration are taken from [@UA5_exp]. \[fig:HIJING\]* ](eta_hijing.eps){width="8cm"}
In contrast to Ref.[@KL01], where the whole pseudorapidity distribution, including fragmentation regions, has been well described in an approach similar to the one presented here, in the present paper pions produced from the fragmentation of gluons in the $gg \rightarrow g$ mechanism populate only midrapidity region, leaving room for other mechanisms in the fragmentation regions. These mechanisms involve quark/antiquark degrees of freedom or leading protons among others. In Fig.\[fig:HIJING\] I show the pseudorapidity spectra of protons, antiprotons and the difference $d \sigma/d \eta_{\pi^+} - d \sigma/d \eta_{\pi^-}$ obtained with the code HIJING [@HIJING] (see also [@HIJING_had]). The difference of the proton-antiproton spectra gives an idea of leading particle contribution. Both protons from deeply inelastic events as well as protons from diffraction dissociation (single diffraction) have been included. The difference of the positively and negatively charged pions gives the lower limit on the $\pi^+ - \pi^-$ asymmetric mechanisms not taken into account in the Kharzeev-Levin approach. The sum of the three contributions (thick solid) gives then lower limit on the missing contributions. It is of the similar size as the missing contributions in Fig.\[fig:eta\_ff\] and Fig.\[fig:eta\_glue\]. This strongly suggests that the agreement of the result of the $gg \rightarrow g$ approach with the PHOBOS distributions [@PHOBOS] in Ref.[@KL01] in the true fragmentation region is rather due to approximations made in [@KL01] than due to correctness of the reaction mechanism. In principle, this can be verified experimentally at RHIC by measuring the $\pi^+ / \pi^-$ ratio in proton-proton scattering as a function of (pseudo)rapidity in possibly broad range. It seems that the BRAHMS experiment, for instance, can do it even with the existing apparatus.
![*Transverse momentum distributions of charged pions at W = 200 GeV for the KL gluon distribution and different fragmentation functions. The experimental data of the UA1 collaboration are taken from [@UA1_exp]. \[fig:pt\_ff\]* ](pt_ff.eps){width="8cm"}
The transverse momentum distribution of charged hadrons is shown in Fig.\[fig:pt\_ff\] together with experimental data of the UA1 collaboration at CERN from Ref.[@UA1_exp]. In this calculation the KL gluon distribution has been used. It is not completely clear to me how the experimental data in [@UA1_exp] should be interpreted. [^3] I assume that the experimental data should be interpreted as: $$X = \int \frac{d \sigma}{d \eta_h d^2 p_t} d \eta_h
\; / \; \int d \eta_h
\; .
\label{UA1_interpretation}$$ We have taken $\eta_h \in$ (-2.5,2.5). The simple hadronization functions, called model I and II above, correctly fit low $p_{t,h}$ data and fail in the large $p_{t,h}$ region. This is due to lack of QCD evolution [@FF_evolution]. The results obtained with fragmentation functions from [@BKK95; @KKP00] which include DGLAP evolution, extremely well describe the large $p_{t,h}$ data. Having in mind the ambiguity of the experimental data interpretation, the KL gluon distribution does a fairly good job.
In Fig.\[fig:pt\_glue\] I compare the theoretical transverse momentum distributions of charged pions obtained with different gluon distributions with the UA1 collaboration data [@UA1_exp]. The best agreement is obtained with the Karzeev-Levin gluon distribution. The distribution with the GBW model is much too steep in comparison to experimental data. This is probably due to neglecting QCD evolution.
![*Transverse momentum distributions of charged pions at W = 200 GeV for BKK1995 fragmentation function and different models of unintegrated gluon distributions. The experimental data of the UA1 collaboration are taken from [@UA1_exp]. \[fig:pt\_glue\]* ](pt_glue.eps){width="8cm"}
Conclusions
===========
I have calculated the inclusive distributions of gluons and associated charged pions in the nucleon-nucleon collisions through the $g g \rightarrow g$ mechanism in the $k_t$-factorization approach. The results for several unintegrated gluon distributions proposed recently in the literature have been compared. The results, especially transverse momentum distributions, obtained with different models of unintegrated gluon distributions differ considerably.
A special attention has been devoted to the gluon distribution proposed recently by Kharzeev and Levin to describe charged particle production in relativistic heavy-ion collisions. In the first step I have tested the gluon distribution in electron deep-inelastic scattering at small Bjorken $x$. A rather good description of the HERA data can be obtained by adjusting a normalization constant. In the next step so-fixed gluon distribution has been used to calculate (pseudo)rapidity and transverse momentum distribution of gluonic jets and charged particles.
Huge differences in both rapidity and transverse momentum distributions of gluons and pions for different models of unintegrated gluon distributions have been found.
Some approximations used recently in the literature have been discussed. Contrary to a recent claim in Ref.[@KL01], we have found that the gluonic mechanism discussed does not describe the inclusive spectra of charged particles in the fragmentation region, i.e. in the region of large (pseudo)rapidities for any unintegrated gluon distribution from the literature. Clearly the gluonic mechanism is not the only one and other mechanisms (see e.g.[@W00; @EH02] ) neglected in [@KL01] must be added. Some of them have been estimated with the help of the HIJING code, giving a right order of magnitude for the missing strength.
Since the mechanism considered is not complete, it is not possible at present to precisely verify different models of unintegrated gluon distributions. The existing gluon distributions lead to the contributions which almost exhaust the strength at midrapidities and leave room for other mechanisms in the fragmentation regions. It seems that a measurement of transverse momentum distributions of particles at RHIC should be helpful to test better different unintegrated gluon distributions. A good identification of particles is required to verify the other mechanisms.
In contrast to standard integrated gluon distributions, the extraction of unintegrated gluon distribution from experimental data seems a rather difficult task. At present, one can rather test different unintegrated gluon distributions based on different models existing in the literature. In the present analysis I have discussed whether the production of particles can provide some information on unintegrated gluon distributions in the nucleon. There are many other reactions where this is possible, to mention here only heavy quark or jet production in $e p$ and $p p$ collisions. Going to more exclusive measurements seems indispensable. An example is a careful study of jet correlation in photon-proton [@SNSS01] and nucleon-nucleon [@LO00] collisions. In my opinion, we are at the beginning of the long way to extract gluon or more generally parton unintegrated distributions.
[**Acknowledgements**]{} I am indebted to Jan Kwieciński for several discussions on different subjects concerning high-energy physics, for his willingness to share his knowledge, for his optimism and friendly attitude. The discussion with Andrzej Budzanowski is kindly acknowledged. The pseudorapidity distributions of particles from the code HIJING have been obtained from Piotr Paw[ł]{}owski. I thank Leszek Motyka for providing me with some FORTRAN routines.
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[^1]: To appear in the special issue of Acta Physica Polonica to celebrate the 65th Birthday of Prof. Jan Kwieciński
[^2]: Here by stable particles we mean the particles registered in detectors
[^3]: The notion of the invariant cross section in [@UA1_exp] is contradictory to the lack of particle identification there.
|
---
abstract: |
Blockchain is a distributed database that keeps a chronologically-growing list (*chain*) of records (*blocks*) secure from tampering and revision. While computerisation has changed the nature of a ledger from clay tables in the old days to digital records in modern days, blockchain technology is the first true innovation in record keeping that could potentially revolutionise the basic principles of information keeping. In this note, we provide a brief self-contained introduction to how the blockchain works.\
[**Keywords:**]{} Data Security, Information Storage, Distributed Ledger, Blockchain, Bitcoin, Cryptocurrency
address: 'Record Currency Management, Windsor, UK'
author:
- 'J. H. Witte'
date: |
November 2016\
Record Currency Management, Windsor, UK
title: |
[The Blockchain: A Gentle\
Four Page Introduction]{}
---
Introduction
============
A *record* is any piece of evidence about the past – especially an account kept in writing or some other resilient form – which facilitates the accurate preservation of information through time. Traditionally, a specific and regularly updated record in paper or book form is referred to as a *ledger*.
For any two individuals engaging in an information-dependent correspondence or exchange (as is almost always the case in business), record keeping presents a risk. Unless
- both parties are happy to rely on the records of a central authorised entity,
- or one of the parties is happy to rely fully on the record keeping of the other party (which is rather rare in business),
careful comparison and matching of two independently held sets of records is the only possible solution.
Blockchain technology promises a distributed database that keeps a continuously-growing list (*chain*) of records (*blocks*) secured from tampering and revision, making comparison and matching of any separately held records unnecessary. If successful, blockchain could therefore substantially facilitate (and therefore accelerate) almost every multi-party decision-making process, including business, governmental, and private.
For example, the avoidance of past tampering of a transaction trail ensures that no new money has been unlawfully created. For cash, this is done through the use of bank notes that cannot (or are very hard to) be forged, and for electronic accounts, this is done by only allowing trusted participants (normally banks) to store balances. Evidently, it is central authorised entities that keep a ledger and prevent our financial systems from falling into chaos. Blockchain technology proposes to use a distributed (and therefore democratic) confirmation process of financial transactions instead.
Technical Prerequisites
=======================
A blockchain is designed based on two well-established mathematical ideas.
Public Key Encryption
---------------------
Modern cryptography relies on the use of a *public* and a *private* key. Alice can share her public key freely with anyone (and maybe even publish it on her website), while her private key is known only to her. If Bob encrypts a message using Alice’s public key, then no one except for Alice with her private key can ever decrypt the message – as long as Alice keeps her private key safe, the communication between the two is unbreakable.
Hash Keys
---------
A *hash key* is any fully-defined function which takes an alpha-numeric sequence (i.e., a string of letters and numbers) of arbitrary length and reduces it to one of predefined finite length. Even though, in theory, this means that duplication (i.e. more than one input resulting in the same output) is possible, for modern hash keys this can be assumed to be extremely unlikely.
One notable feature of hash keys is that they are (almost) impossible to invert. For example, if Bob is an internet provider who stores only the hash keys of user passwords (rather than the passwords themselves), than he can check Alice’s user-login correctly without even knowing Alice’s actual password.
Bitcoin: The Beginnings
=======================
In his 2008 paper *Bitcoin: A Peer-to-Peer Electronic Cash System*, a person calling themselves Satoshi Nakamoto (the name is a pseudonym and the real author is unknown) laid the foundation for the algorithmic network behind the crypto-currency *bitcoin*. Even though the bitcoin itself continues to function and circulate successfully, Nakamoto’s ideas behind the creation of the bitcoin have since exceeded the original application by far and are now known independently as the *blockchain*.
Nakamoto created the bitcoin by inventing the following architecture.
- We have a distributed network of participating agents all of whom store a copy of the blockchain. The blockchain (the *distributed ledger*) is a record of all *transactions* to date. (This is very important: the blockchain stores transactions, not balances.)
- Participating agents also hold bitcoin wallets, which represent ownership of any received bitcoin payments. More specifically, every bitcoin transaction is represented in the blockchain as the public key of a cryptographic public/private key pair, while the transaction’s private key is held in someones bitcoin wallet. Anyone in possession of the blockchain can use a transaction’s public key to encode messages which only the transaction’s rightful owner can decrypt correctly.
For example, if Alice tells Bob that she owns 30 bitcoins (which she earlier received in a transaction from Charlie), then Bob (or anyone else) can instantly confirm Alice’s claim to ownership by asking encrypted questions – unless Alice truly owns the private key she claims to have, she cannot reply correctly.
Up to this point, the described design creates a *static* distributed ledger. But imagine that now Alice wants to buy Bob’s bike for 30 bitcoins, but that Bob is only willing to hand over the bike once both of them can independently confirm Alice’s payment to Bob from their copies of the blockchain. To make the transaction possible, we have to extend and update the blockchain.
- To initiate the transfer of 30 bitcoins to Bob, Alice requests a new public key from Bob and then sends her proposed transaction to all agents in the network.
- For Alice’s proposed transaction to be approved, one agent in the network now needs to
- confirm that Alice is the rightful owner of the 30 bitcoins she intends to transfer (see ii. above),
- and create a new hash key encoding both, the last previous hash key as well as the new transaction information.
Unfortunately, if it was known in advance which agent was going to confirm the transaction, then Alice could secretly collaborate with this agent and therefore manipulate the blockchain through an illegitimate extension.
To enforce a random choice of the confirming agent in the distributed network, the blockchain rules additionally require the newly generated hash key in b) to have a certain structure. For example, it may have to have a certain number of zeros at the beginning or end (or another similar specification). The distributed agents are then allowed to append the alpha-numeric string which is to be hashed by a personally chosen component called a *nonce*. A confirmation is accepted as soon as one agent in the network manages to propose a nonce which results in a hash key of the required format.
The competition in which all agents in the network simultaneously try to find a suitable nonce to complete the given task is referred to as *mining*. The requirements of the new hash key can be chosen such that mining takes a certain average time (e.g., ten minutes) even for very fast computers. As long as a large number of participating agents in the distributed network have the computational ability to find the new hash key within the expected time, it is unlikely that one particular agent will be able to always dominate (and therefore manipulate) the process.
Adding this detail to point iv. above, the more specifically phrased item b) becomes the following.
- - and create a new hash key *of specified format* encoding the combined string of *three* items, the last previous hash key, the new transaction information, and a newly-mined nonce.
In the original bitcoin design, the completion of b)\* is rewarded with a payment of (newly created) bitcoin, to ensure that participation in the confirmation process is desireable for a large number of distributed agents (and hence the name *mining*).
Suppose that a random agent called Charlie is the first to successfully confirm Alice’s proposed transaction of 30 bitcoins. Charlie then immediately broadcasts the new hash key together with Alice’s new transaction to everyone in the distributed network – a new *block* is being added to the blockchain (i.e., the distributed ledger is updated). As soon as his synchronised version of the blockchain contains Alice’s confirmed bitcoin transfer, Bob can hand over his bike to Alice and complete the sale of the underlying physical good.
Blockchain: Electronic Transaction Without Intermediary
=======================================================
A certain technological complexity (e.g., as ensuring instant synchronisation of information across a large distributed network) aside, the blockchain design allows the creation of a secure distributed database for almost any type of information.
Three particular properties of the blockchain have to be kept in mind.
Network Size
------------
It is crucial to have a large distributed network, which means agents have to be incentivised to participate in the mining process. In many cases, this may mean offering a small *fee* for every successfully confirmed transaction (or newly created block).
Blockchain Depth
----------------
All agents participating in the distributed network will always update their version of the blockchain to the longest prevailing one. This means that if someone managed to manipulate (i.e., *break*) the latest added block to re-route Alice’s transfer after Bob has already seen it and then integrate it in a longer blockchain, then Bob might find himself handing over his bike even though his 30 bitcoins have actually been sent elsewhere.
The repeated hashing resulting from the addition of new blocks to the blockchain means that past transactions become more impenetrable as they sink deeper and deeper into the blockchain. If Bob, before handing over his bike, waits for a number of more blocks (the current standard is *six*) to be added after his own transaction, than he can be certain that his receipt of money has been logged as a permanent and unchangeable part of the blockchain.
51$\%$ Attack
-------------
While the name *51$\%$ attack* is misleading (as the number 51 is of no particular relevance), this refers to the risk of having a maliciously intended agent or group of agents who dominate the aggregate computational power available in the distributed network – they could, therefore, manipulate the addition of new blocks by consistently leading the confirmation process. In practice, the true impact of this is smaller than widely believed, as it primarily requires waiting for more blockchain depth before moving physical goods or making further money transfers – creating a speed rather than a security problem.
Theft
-----
The blockchain technology does not prevent theft of property. Especially for the anonymous design of the bitcoin, where an individual’s money is stored as a collection of private keys in a bitcoin wallet, property can be stolen if the electronic wallet is accessed illegally and bitcoins are spent.
[9]{} Satoshi Nakamoto, *Bitcoin: A Peer-to-Peer Electronic Cash System*, originally published on mailinglist *metzdowd.com* in January 2009, now available on https://bitcoin.org/bitcoin.pdf
[**DISCLAIMER**]{}
The views expressed are those of the author and do not reflect the official policy or position of Record Currency Management.
|
---
abstract: 'We show that the two-time physics model leads to a mechanical system with Dirac brackets consistent with the Snyder noncommutative space. An Euclidean version of this space is also obtained and it is shown that both spaces have a dual system describing a particle in a curved space.'
author:
- 'Juan M. Romero'
- Adolfo Zamora
title: 'Snyder noncommutative space-time from two-time physics'
---
Introduction
============
Inspired by a conformal field theory, R. Marnelius [@Mr:gnus] built a classical mechanics model having the conformal group as the global symmetry and the symplectic group $S_{P}(2)$ as the local one. This model has interesting unusual properties. One of them is that it must have two time coordinates; that is why it is normally called the two-time physics (2T) model. By imposing different gauge conditions on this, one can obtain systems such as the relativistic particle with mass and the massless free particle in the $AdS$ space-time. In this sense the 2T model can be used as a toy model for unification. Supersymmetric extensions of the 2T model can be found in Ref. [@super:gnus]. Recently, I. Bars and co-workers reinvented the 2T model in string theory [@Ibars:gnus] and carried out several extensions in different contexts (see Refs. [@Bars:gnus; @Ibars1:gnus] and references therein). Ref. [@Otros:gnus] also deals with the same problem. In another work, M. Montesinos, C. Rovelli and T. Thiemann proposed a classical mechanics model simulating the gauge structure of general relativity. In this, the gauge group is $SL(2,R)$ [@merced:gnus] and, since $SL(2,R)$ is isomorphic to $S_{P}(2)$, this model is analogous to the 2T. As it is, the 2T model has several interesting properties one would like to see in a realistic model.\
From different results in string theory [@witten:gnus], the possibility that the space-time at short distances is noncommutative has been extensively studied recently. R. Snyder [@Snyder:gnus] investigated these ideas first and built a noncommutative Lorentz invariant discrete space-time: the so called Snyder space. Contrarily to the noncommutative spaces from string theory, in Snyder space the noncommutativity depends on the space-time. After the work of Kontsevich [@kontsevich:gnus], Snyder-like spaces in the sense of noncommutativity have attracted great attention. Snyder space is also interesting because it can be mapped to the $k$-Minkowski space-time [@kowalski:gnus]. This space-time is a realization of the “Doubly Special Relativity” theory, which is a new proposal to deal with quantum gravity phenomena [@amelino:gnus]. An important result from loop quantum gravity, in addition, is that it leads to discrete geometric quantities [@Livine:gnus], and in this sense the discreteness of Snyder space becomes also attractive.
We show in this investigation how, by imposing an alternative gauge condition on the 2T model, one gets to a mechanical system with Dirac brackets consistent with the commutation rules of the Snyder noncommutative space. Using other gauge conditions, we also show that an Euclidean version of the Snyder space can be obtained. Then, by exploiting the symmetries of the Hamiltonian, we conclude that each system has a dual. For the Snyder space the dual system is the massless particle in the $AdS$ space, but for the Euclidean Snyder it is the non-linear sigma model in one dimension.\
The work in this paper is organized as follows. In Section $2$ a brief introduction to the 2T model is provided. Then, in Section $3$ the gauge conditions to get to the Snyder space are given. The analogous conditions, but to obtain the Euclidean Snyder space are determined in Section $4$. In Section $5$ we show that both of these spaces have a dual system; and finally in Section $6$ we summarize our results.\
The 2T model
============
Let us first review some properties of the 2T action and its symmetries.
2T action
---------
The action for the 2T model is defined as the Hamiltonian action $$S=\int_{\tau_{1}}^{\tau_{2}} d\tau \bigg [\dot X\cdot P-
\left(\lambda^{1}\frac{1}{2}P^{2}+\lambda^{2}X\cdot P+
\lambda^{3}\frac{1}{2}X^{2}\right)\bigg ]\,,\label{eq:2s}$$ with the Hamiltonian given by $$H_{2T}=\left(\lambda^{1}
\frac{1}{2}P^{2}+\lambda^{2}X\cdot P+\lambda^{3}
\frac{1}{2}X^{2}\right)\,, \label{eq:H2T}$$ where $\lambda^{1},\lambda^{2},\lambda^{3}$ are Lagrange multipliers. From this, one can obtain the equations of motion $$\begin{aligned}
&\dot X^{M} &\!\!\!=\lambda^{1}P^{M}+\lambda^{2}X^{M}\,,\label{eq:igo1}\\
&\dot P^{M} &\!\!\!=-\lambda^{2}P^{M}-\lambda^{3}X^{M}\,,\label{eq:igo2}\\
&P^{2} &\!\!\!\approx X^{2}\approx X\cdot P\approx 0\,,
\label{eq:igo3}\end{aligned}$$ where the symbol of weak equivalence $(\approx)$ has been used in the sense of Dirac [@D1:gnus; @Te:gnus]. Now, by defining $$\phi_{1}=\frac{1}{2}P^{2}\,,\quad
\phi_{2}=X\cdot P\,,\quad\phi_{3}=\frac{1}{2}X^{2}\,, \label{eq:2gen}$$ and considering that the Poisson brackets are given by: $\{X_{M},P_{N}\}=\eta_{MN}$, and zero otherwise, with $\eta_{MN}$ being a flat metric, it can be seen that the algebra $$\{\phi_{2},\phi_{3}\}=-2\phi_{3}\,,\quad
\{\phi_{2},\phi_{1}\}=2\phi_{1}\,,\quad
\{\phi_{1},\phi_{3}\}=-\phi_{2}\,, \label{eq:Merced}$$ holds. That is, all three constraints are first class. Eq. (\[eq:Merced\]) represents the Lie algebra of the $S_{P}(2)$ group which is formed by the $2\times 2$ matrices with determinant one. If one redefines variables as $$H_{1}=\phi_{1}\,, \quad H_{2}=-\phi_{3}\,, \quad D=\phi_{2}\,,$$ the Lie algebra of the $SL(R,2)$ is obtained. This has been already proposed as a toy model simulating the gauge group of general relativity [@merced:gnus].\
Now, if we consider the Euclidean or Minkowski metrics as the background space, the surface defined by Eq. (\[eq:igo3\]) is trivial. Therefore, the simplest metric giving a non-trivial surface is the flat metric with two time coordinates. Throughout this work we will assume this metric only. If the configuration space has dimensionality $D=d+2$, a flat metric $\eta_{MN}$ with signature $${\rm sig}(\eta )=(-,-,+,\cdots,+)\,, \label{eq:signa}$$ must be used. The coordinates of the phase space can be taken as $$\begin{aligned}
&X^{M}&\!\!\!=(X^{0\prime},X^{1\prime}, X^{0}, X^{i})\,, \nonumber\\
&P^{M}&\!\!\!=(P^{0\prime},P^{1\prime}, P^{0},P^{i})\,,\quad (i=1,\dots,d-1)\,,
\label{eq:coor}\end{aligned}$$ where the zeroes are associated with the time coordinates.\
In principle the phase space of the system has $2(d+2)$ independent coordinates. However, as there are three first-class constraints, six degrees of freedom must be subtracted. Therefore, there are $2(d-1)$ effective degrees of freedom and so the configuration space has $(d-1)$ independent coordinates.
Symmetries
----------
The equations of motion (\[eq:igo1\]) and (\[eq:igo2\]) can be rewritten as $$\frac{d}{dt}\left(
\begin{array}{c}
X^{M} \\
P^{M} \\
\end{array} \right)=
A(t)
\left(
\begin{array}{c}
X^{M} \\
P^{M} \\
\end{array} \right)\,,
\label{eq:pao}$$ with $
A(t)=\left(
\begin{array}{cc}
\lambda^{2} & \lambda^{1}\\
-\lambda^{3} & -\lambda^{2}
\end{array} \right)\,.
$ By performing a gauge transformation with an arbitrary matrix of $Sp(2)$, $$\left(
\begin{array}{c}
\bar X^{M} \\
\bar P^{M} \\
\end{array} \right)=
U(t)
\left(
\begin{array}{c}
X^{M} \\
P^{M} \\
\end{array} \right)\,,\label{eq:ramona}$$ where $
U(t)=\left(
\begin{array}{cc}
a & b\\
c & d
\end{array} \right)
,\quad ad-bc=1\,,
$ one gets to the transformed equations of motion $$\frac{d}{dt}\left(
\begin{array}{c}
\bar X^{M} \\
\bar P^{M} \\
\end{array} \right)=
\bar A(t)
\left(
\begin{array}{c}
\bar X^{M} \\
\bar P^{M} \\
\end{array} \right)\,,$$ where $$\bar A(t)=U(t)A(t)U(t)^{-1}-U(t)\frac{dU(t)^{-1}}{dt}\,. \label{eq:abel}$$ It can be easily seen that $A(t)$ transforms as a connection under the gauge transformation $U(t)$ and that the equations of motion (\[eq:pao\]) are invariant under this gauge transformation as well.\
Now, the action in Eq. (\[eq:2s\]), when rewritten in terms of the transformed variables, takes the form $$\begin{aligned}
S &\!\!\!=\!\!\!&\int_{\tau_{1}}^{\tau_{2}} d\tau
\bigg [\dot X\cdot P-\left(\lambda^{1}
\frac{1}{2}P^{2}+\lambda^{2}X\cdot P+\lambda^{3}
\frac{1}{2}X^{2}\right)\bigg ]\nonumber\\
&\!\!\!=\!\!\!&\int_{\tau_{1}}^{\tau_{2}} d\tau \bigg
[\frac{d \bar X}{d\tau}\cdot \bar P-\left(\bar\lambda^{1}
\frac{1}{2}\bar P^{2}+\bar\lambda^{2}\bar X\cdot \bar P+\bar\lambda^{3}
\frac{1}{2}\bar X^{2}\right)\nonumber\\
& &{} + \frac{d}{d\tau}\left(-ab \frac{1}{2}\bar P^{2}+bc \bar X\cdot
\bar P-dc\frac{1}{2}\bar X^{2}\right)
\bigg ]\,, \label{eq:elisa}\end{aligned}$$ where $\bar \lambda^{i}$ is given by Eq. (\[eq:abel\]). Thus, up to a boundary term, the action in Eq. (\[eq:2s\]) is invariant under the gauge transformations (\[eq:ramona\]) and (\[eq:abel\]). On the other hand, the quantities $X\cdot P$, $ X^{2}$, $ P^{2}$ and $\dot X\cdot P$ are clearly invariant under global transformations $\Lambda$ that satisfy $$\Lambda^{T}\eta\Lambda =\eta\,,$$ with the signature $\eta$ defined in Eq. (\[eq:signa\]). Thus, the action in Eq. (\[eq:2s\]) is invariant under global transformations of $SO(2,d)$. It can be shown that in phase space the generators of this symmetry are $$L^{MN}=X^{M} P^{N}-X^{N} P^{M}\,,\label{eq:gero}$$ which satisfy the conformal algebra [@Maldacena:gnus] and are conserved quantities. Moreover, they satisfy $\{L^{MN},\phi_{i}\}=0$, i.e. they are gauge invariant.\
Snyder space
============
Let us now consider the gauge conditions to get the Snyder space $$P_{1^{\prime}}=L={\rm const.}\,, \qquad X_{1^{\prime}}=0\,.\label{eq:s1}$$ Substituting them into the equations of motion (\[eq:igo1\]) and (\[eq:igo2\]) we obtain $$\lambda^{2}=\lambda^{1}=0\,.$$ By using Eq. (\[eq:igo3\]) it can be seen that the independent reduced equations of motion are $$\begin{aligned}
&\dot X^{\mu}&\!\!\!=0\,,\\
&\dot P_{\mu}&\!\!\!=-\lambda^{3}X_{\mu}\,,\\
&\phi_{3}&\!\!\!=\frac{1}{2}G_{\mu\nu}X^{\mu}X^{\nu}\approx 0,\quad
G_{\mu\nu}= \left(\eta_{\mu\nu}-
\frac{P_{\mu}P_{\nu}}{P_{\alpha}P^{\alpha}+L^{2}}\right),\end{aligned}$$ with the dependent variables being $$P_{0^{\prime}}=\sqrt{P_{\mu}P^{\mu}+L^{2}}\,,\quad
X_{0^{\prime}}=\frac{P_{\mu}X^{\mu}}{\sqrt{P_{\mu}P^{\mu}+L^{2}}}\,.$$ After performing an integration by parts and substituting the dependent variables into Eq. (\[eq:2s\]) one obtains $$S=\int d\tau \left[-G_{\mu\nu}X^{\mu}\dot P^{\nu}-\frac{\lambda^{3}}{2}
G_{\mu\nu}X^{\mu}X^{\nu}\right]\,.$$ To quantize this system with the canonical formalism, the Dirac brackets [@D1:gnus; @Te:gnus] must be constructed. In this process the Dirac brackets are replaced by commutators. Now, let us consider $$\begin{aligned}
&\chi_{1}&\!\!\!=P_{1^{\prime}}-L\,,\\
&\chi_{2}&\!\!\!=X_{1^{\prime}}\,,\\
&\chi_{3}&\!\!\!=P\cdot X\,,\\
&\chi_{4}&\!\!\!=\frac{1}{2}P^{2}\,,\\
&\phi&\!\!\!=\frac{1}{2}X^{2}\,. \label{eq:phi}\end{aligned}$$ A straightforward calculation shows that Eq. (\[eq:phi\]) is a first-class constraint while the others are second class. For the later ones we find $$C_{\alpha\beta}\approx\{\chi_{\alpha},\chi_{\beta}\} \approx \left(
\begin{array}{rrrr}
0 & -1 & -L & \quad 0 \\
1 & 0 & 0 & L \\
L & 0 & 0 & 0 \\
0 & -L & 0 & 0
\end{array}
\right)\,, \label{eq:Definida}$$ from which, $$C^{\alpha\beta} \approx -\frac{1}{L}\left(
\begin{array}{rrrr}
0 & 0 & -1 & 0 \\
0 & 0 & 0 & 1 \\
1 & 0 & 0 & - \frac{1}{L} \\
0 & -1 & \frac{1}{L} & 0
\end{array}
\right)\,.$$ In general, given two functions $A$ and $B$ in phase space, the Dirac brackets are defined as $$\{A, B\}^{*}=\{A,B\}-\{A,\chi_{\alpha}\}C^{\alpha\beta}\{\chi_{\beta},B\}\,.$$ In particular, for the phase space coordinates $$\begin{aligned}
&\{X_{\mu}\,,X_{\nu}\}^{*}&\!\!\!=\frac{1}{L^{2}}\left(X_{\mu}P_{\nu}-
X_{\nu}P_{\mu}\right)\,,\\
&\{X_{\mu}\,,P_{\nu}\}^{*}&\!\!\!=\eta_{\mu\nu}+\frac{1}{L^{2}}P_{\mu}P_{\nu}\,,\\
&\{P_{\mu}\,,P_{\nu}\}^{*}&\!\!\!=0\,.\end{aligned}$$ These Dirac brackets are the classic version of the commutation rules of Snyder space [@Snyder:gnus]. Therefore, after quantizing the system we have the Snyder space as the background.
Now, by defining ${\bf X_{\mu}}=G_{\mu\nu}X^{\nu}$, the pair $({\bf X_{\mu}}, P_{\mu})$ satisfies the Dirac brackets $$\{{\bf X}_{\mu},{\bf X}_{\nu}\}^{*}=0\,,\quad
\{{\bf X}_{\mu},P_{\nu}\}^{*}=\eta_{\mu\nu}\,,\quad
\{P_{\mu},P_{\nu}\}^{*}=0\,.$$ That is, with the variables $({\bf X_{\mu}}, P_{\mu})$, the usual Poisson brackets are obtained. Nevertheless, at the quantum level the definition of ${\bf X_{\mu}}$ is ambiguous.
Euclidean Snyder space
======================
Other gauge conditions from which a noncommutative space can be obtained are $$\chi_{1}=P_{0}-1=0\,, \qquad \chi_{2}=X_{0}=0\,. \label{eq:coco}$$ For these the independent equations of motion are $$\begin{aligned}
&\dot X^{i}&\!\!\!=0\,,\\
&\dot P^{i}&\!\!\!=-\lambda^{3} X^{i}\,,\\
&\phi_{3}&\!\!\!=g_{ij}X^{i}X^{j}\approx 0\,, \quad
g_{ij}=\left(\delta_{ij}+\frac{P_{i}P_{j}}{1-P_{k}P^{k}}\right)\,, \end{aligned}$$ and, as can be easily seen, the second-class constraints are given by $$\begin{aligned}
&\chi_{1}&\!\!\!=P_{0}-1\,,\\
&\chi_{2}&\!\!\!=X_{0}\,,\\
&\chi_{3}&\!\!\!=P\cdot X\,,\\
&\chi_{4}&\!\!\!=\frac{1}{2}P^{2}\,.\end{aligned}$$ From a straightforward calculation it can be observed that in this case the matrix $C_{\alpha\beta}\approx\{\chi_{\alpha},\chi_{\beta}\}$ is minus the matrix in Eq. (\[eq:Definida\]) with $L=1$. Using this, we find for the phase space coordinates $$\begin{aligned}
&\{X_{i}\,,X_{j}\}^{*}&\!\!\!=-\left (X_{i}P_{j}-X_{j}P_{i}\right)\,,\\
&\{X_{i}\,,P_{j}\}^{*}&\!\!\!=\delta_{ij}-P_{i}P_{j}\,,\\
&\{P_{i}\,,P_{j}\}^{*}&\!\!\!=0\,.\end{aligned}$$ Thus, after quantizing the reduced system, a noncommutative space in the coordinates is obtained.\
By defining the variable ${\bf X}_{i}=g_{ij}X^{j}$, it can be seen that the pair $({\bf X}_{i}, P_{j})$ satisfies $$\{{\bf X}_{i},{\bf X}_{j}\}^{*}=0\,,\quad
\{{\bf X}_{i},P_{j}\}^{*}=\delta_{ij}\,,\quad
\{P_{i},P_{j}\}^{*}=0\,.$$ Now, as the only gauge transformations permitted are of the type of Eq. (\[eq:ramona\]), there is no gauge transformation which takes the Snyder space to this system. In this sense they are different physical systems.
$O(d+1)$ non-linear sigma model in one dimension
================================================
It can be seen that the Hamiltonian $H_{2T}$ from Eq. (\[eq:H2T\]) is invariant under the transformations $$(X^{M}, P^{M})\to (P^{M}, X^{M})\,,\quad
(\lambda^{1},\lambda^{2},\lambda^{3})\to
(\lambda^{3},\lambda^{2},\lambda^{1})\,.$$ This symmetry implies that if we impose gauge conditions and then the $X$s and $P$s are swapped, we obtain analogous reduced systems. Notice, however, that the physical interpretation of each one is different. As an example, by performing this swap in the gauge conditions of the Euclidean Snyder space from Eq. (\[eq:coco\]), one obtains $$\chi_{1}=X_{0}-1=0 \qquad{\rm and}\qquad \chi_{2}=P_{0}=0\,.
\label{eq:nopasa}$$ In this case the independent reduced equations of motion are $$\begin{aligned}
&\dot X^{i}&\!\!\!=\lambda^{1}P^{i}\,, \qquad (i=1,\dots,d)\label{eq:perez}\\
&\dot P^{i}&\!\!\!=0\,,\label{eq:monarres}\\
&\phi_{1}&\!\!\!=\tilde g_{ij}P^{i}P^{j}\approx 0, \quad \tilde g_{ij}=\left(\delta_{ij} +
\frac{X_i X_j}{1-X_k X^k}\right), \label{eq:monarres1}\end{aligned}$$ with dependent variables given by $$X^{0'}=\sqrt{X^{i}X_{i}-1}\,,\quad
P^{0'}=\frac{(P^{i}X_{i})}{\sqrt{X^{i}X_{i}-1}}\,.
\label{eq:america2}$$ Now, by rewriting Eq. (\[eq:2s\]) in terms of the independent variables we obtain $$S= \int _{\tau_{1}} ^{\tau_{2}} d\tau \left[\tilde g_{ij}\dot X^{i}
P^{j}-\frac{\lambda^{1}}{2}\tilde g_{ij} P^{i}P^{j}\right]\, .
\label{eq:tardesy}$$ From this expression one gets to the equations of motion (\[eq:perez\])–(\[eq:monarres1\]). Now, substituting Eq. (\[eq:perez\]) into Eq. (\[eq:tardesy\]) and eliminating $P_{i}$ as a dynamic variable we get $$S=\frac{1}{2}\int _{\tau_{1}} ^{\tau_{2}} d\tau \frac{\tilde g_{ij}\dot
X^{i}\dot X^{j}}{\lambda^{1}}\,.\label{eq:mesias}$$ Eq. (\[eq:mesias\]) could be interpreted as the action of a massless free particle in a space with metric $\tilde g_{ij}$, but non-relativistic massless particles are not natural. In a better interpretation, for $\lambda^{1}=1$, this equation represents the action of the $O(d+1)$ non-linear sigma model in one dimension [@Zin:gnus].\
The Dirac brackets for the phase space coordinates, in this case, are $$\begin{aligned}
&\{X_{i}\,,X_{j}\}^{*}&\!\!\!=0\,, \label{eq:LL}\\
&\{X_{i}\,,P_{j}\}^{*}&\!\!\!=\delta_{ij}-X_{i}X_{j}\,,\\
&\{P_{i}\,,P_{j}\}^{*}&\!\!\!=- \left(X_{i}P_{j}-X_{j}P_{i}\right)\,
\label{eq:LLL}.\end{aligned}$$ However, using the coordinates $(X_{i}, \bar{\bf P}_{i}=\tilde g_{ij}P^{j})$ one gets to $$\{X_{i}\,,X_{j}\}^{*}=0\,, \quad
\{X^{i}\,,{\bf \bar P}_{j}\}^{*}=\delta^{i}_{j}\,,\quad
\{{\bf \bar P}_{j}\,,{\bf \bar P}_{j}\}^{*}=0\, \label{eq:rech}.$$ In terms of the variables $(X_{i}, \bar{\bf P}_{i})$, the action of Eq. (\[eq:tardesy\]) becomes $$S= \int _{\tau_{1}} ^{\tau_{2}} d\tau \left[\dot X^{i}
{\bf P}_{i}-\frac{\lambda^{1}}{2}\tilde g^{ij} {\bf P}_{i}{\bf P}_{j}
\right]\, .$$ Thus, this system can be thought of: either a particle in an Euclidean metric with a deformed Poisson structure, Eqs. (\[eq:LL\])–(\[eq:LLL\]), or as a particle in the metric $\tilde g_{ij}$ with the standard Poisson structure, Eq. (\[eq:rech\]). A similar interpretation can be given to the systems presented in sections $3$ and $4$.\
By performing the change of variables $(X,P)\to (P,X)$ in the gauge conditions for the Snyder space, Eq. (\[eq:s1\]), one gets to the gauge conditions $$X_{1^{\prime}}=L={\rm const.}\,, \qquad P_{1^{\prime}}=0\,.
\label{eq:s11}$$ In Refs. [@Mr:gnus] and [@Ibars1:gnus] it is shown that, using the conditions from Eq. (\[eq:s11\]), the massless particle in the $AdS$ space is obtained. This can also be easily verified by repeating the calculation using the conditions of Eq. (\[eq:nopasa\]).\
It is remarkable that in the 2T model both dynamics in noncommutative spaces have as dual a dynamics in a curved space-time.
Summary
=======
In this work we study a mechanical system with two times and gauge freedom called the two-time physics. It is shown that considering a particular gauge one gets a mechanical system with Dirac brackets consistent with the commutation rules of the Snyder noncommutative space. Using other gauge conditions an Euclidean version of the Snyder space is obtained. By exploiting a symmetry of the Hamiltonian we show that these noncommutative systems have a dual system. For the Snyder space, the dual is a massless particle in the $AdS$ space, while for the Euclidean Snyder the dual is the non-linear sigma model in one dimension.
The authors would like to thank J. D. Vergara for discussions.
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|
---
abstract: 'Monotonicity is a simple yet significant qualitative characteristic. We consider the problem of segmenting an array in up to $K$ segments. We want segments to be as monotonic as possible and to alternate signs. We propose a quality metric for this problem, present an optimal linear time algorithm based on novel formalism, and compare experimentally its performance to a linear time top-down regression algorithm. We show that our algorithm is faster and more accurate. Applications include pattern recognition and qualitative modeling.'
author:
- |
Daniel Lemire\
University of Quebec at Montreal (UQAM)\
100 Sherbrooke West\
Montréal, Qc, Canada, H2X 3P2\
lemire@ondelette.com\
- |
Martin Brooks, Yuhong Yan\
National Research Council of Canada\
1200 Montreal Road\
Ottawa, ON, Canada, K1A 0R6\
First.Last@nrc.gc.ca\
bibliography:
- '../MBR/modelabstraction.bib'
- '../MBR/yuhong.bib'
- 'metric1d.bib'
title: 'An Optimal Linear Time Algorithm for Quasi-Monotonic Segmentation'
---
Introduction
============
Monotonicity is one of the most natural and important qualitative properties for sequences of data points. It is easy to determine where the values are strictly going up or down, but we only want to identify significant monotonicity. For example, the drop from 2 to 1.9 in the array $0,1,2,1.9, 3,4$ might not be significant and might even be noise-related. The quasi-monotonic segmentation problem is to determine where the data is approximatively increasing or decreasing.
We present a metric for the quasi-monotonic segmentation problem called the Optimal Monotonic Approximation Function Error (OMAFE); this metric differs from previously introduced OPMAFE metric [@YLBIJCAI05] since it applies to all segmentations and not just “extremal” segmentations. We formalize the novel concept of a maximal [$\ast$-pair]{} and shows that it can be used to define a unique labelling of the extrema leading to an optimal segmentation algorithm. We also present an optimal linear time algorithm to solve the quasi-monotonic segmentation problem given a segment budget together with an experimental comparison to quantify the benefits of our algorithm.
Monotonicity Error Metric (OMAFE)
=================================
Suppose $n$ samples noted $F: D=\{x_,\ldots, x_n\} \rightarrow \mathbb{R}$ with $x_1 < x_2 < \ldots x_n$. We define, $F_{|[a,b]}$ as the restriction of $F$ over $D\cap [a,b]$. We seek the best monotonic (increasing or decreasing) function $f:\mathbb{R}\rightarrow \mathbb{R}$ approximating $F$. Let $\Omega_{\uparrow}$ (resp. $\Omega_{\downarrow}$) be the set of all monotonic increasing (resp. decreasing) functions. The **Optimal Monotonic Approximation Function Error (OMAFE)** is $\min_{f \in \Omega}
\max_{x\in D} \vert f(x) - F(x)\vert$ where $\Omega$ is either $\Omega_{\uparrow}$ or $\Omega_{\downarrow}$.
The segmentation of a set $D$ is a sequence $S = X_1 ,\dots, X_K$ of intervals in $\mathbb{R}$ with $[\min
D,\max D] = \bigcup_i X_i $ such that $\max X_i =\min X_ {i+1} \in D$ and $X_i \cap X_j= \emptyset$ for $j \neq i+1, i, i-1$. Alternatively, we can define a segmentation from the set of points $X_i \cap X_
{i+1}=\{y_{i+1}\}$, $y_1= \min X_1$, and $y_{K+1}=\max X_K$. Given $F: \{x_1,\ldots, x_n\} \rightarrow \mathbb{R}$ and a segmentation, the Optimal Monotonic Approximation Function Error (OMAFE) of the segmentation is $\max _i \textrm{OMAFE}(F_{|X_i})$ where the monotonicity type (increasing or decreasing) of the segment $X_i$ is determined by the sign of $F(\max X_i)-F(\min X_i)$. Whenever $F(\max X_i)=F(\min X_i)$, we say the segment has no direction and the best monotonic approximation is just the flat function having value $(\max F_{|X_i} - \min F_{|X_i})/2$. The error is computed over each interval independently; optimal monotonic approximation functions are not required to agree at $\max X_i =\min X_ {i+1}$. Segmentations should alternate between increasing and decreasing, otherwise sequences such as $0,2,1,0,2$ can be segmented as two increasing segments $0,2,1$ and $1,0,2$: we consider it is natural to aggregate segments with the same monotonicity.
We solve for the best monotonic function as follows. If we seek the best monotonic increasing function, we first define $\overline{f}_{\uparrow} (x) = \max \{ F(y): y \leq x \}$ (the maximum of all previous values) and $\underline{f}_{\uparrow} (x) = \min \{ F(y): y \geq x \}$ (the minimum of all values to come). If we seek the best monotonic decreasing function, we define $\overline{f}_{\downarrow} (x) = \max \{ F(y): y \geq x \}$ (the maximum of all values to come) and $\underline{f}_{\downarrow} (x) =\min \{ F(y): y \leq x \}$ (the minimum of all previous values). These functions, which can be computed in linear time, are all we need to solve for the best approximation function as shown by the next theorem which is a well-known result [@Ubhaya1974].
Given $F: D=\{x_1,\ldots, x_n\} \rightarrow \mathbb{R}$, a best monotonic increasing approximation function to $F$ is $f_{\uparrow}= (\overline{f}_{\uparrow} + \underline{f}_{\uparrow})/2$ and a best monotonic decreasing approximation function is $f_{\downarrow}=(\overline{f}_{\downarrow} +
\underline{f}_{\downarrow})/2$. The corresponding error (OMAFE) is $\max_{x\in D} (\vert
\overline{f}_{\uparrow}(x) - \underline{f}_{\uparrow}(x) \vert)/2$ or $\max_{x\in D}
(\vert \overline{f}_{\downarrow}(x) - \underline{f}_{\downarrow}(x) \vert)/2$ respectively.
A Scale-Based Algorithm for Quasi-Monotonic Segmentation
========================================================
We use the following proposition to prove that the segmentations we generate are optimal (see Theorem \[thm:validlabelling\]).
\[stupidresult2\] A segmentation $y_1,\ldots,y_{K+1}$ of $F: D=\{x_1,\ldots,x_n\} \rightarrow
\mathbb{R}$ with alternating monotonicity has a minimal OMAFE $\epsilon$ for a number of alternating segments $K$ if
A. $F(y_i)=\max F([y_{i-1},y_{i+1}])$ or $F(y_i)=\min F([y_{i-1},y_{i+1}])$ for $i=2,\ldots,K$;
B. in all intervals $[y_i,y_{i+1}]$ for $i=1,\ldots,K$, there exists $z_1, z_2$ such that $\vert F(z_2)-F(z_1)\vert > 2\epsilon$.
For simplicity, we assume $F$ has no consecutive equal values, i.e. $F(x_i) \ne F(x_{i+1})$ for $i = 1,\dots, n-1$; our algorithms assume all but one of consecutive equal values values have been removed. We say $x_i$ is a maximum if $i \ne 1$ implies $F(x_i) > F(x_{i-1})$ and if $i \ne n$ implies $F(x_i) > F(x_{i+1})$. Minima are defined similarly.
Our mathematical approach is based on the concept of $\delta$-pair:
The tuple $x,y$ ($x<y \in D$) is a $\delta$-pair (or a pair of scale $\delta$) for $F$ if $|F(y)-F(x)| \geq \delta$ and for all $z \in D$, $x<z<y$ implies $|F(z)-F(x)|< \delta$ and $|F(y)-F(z)|< \delta$. A $\delta$-pair’s *direction* is *increasing* or *decreasing* according to whether $F(y)>F(x)$ or $F(y)<F(x)$.
$\delta$-Pairs having opposite directions cannot overlap but they may share an end point. $\delta$-Pairs of the same direction may overlap, but may not be nested. We use the term “[$\ast$-pair]{}” to indicate a $\delta$-pair having an unspecified $\delta$. We say that a [$\ast$-pair]{} is significant at scale $\delta$ if it is of scale $\delta'$ for $\delta' \geq \delta$.
We define $\delta$-monotonicity as follows:
Let $X$ be an interval, $F$ is $\delta$-*monotonic* on $X$ if all $\delta$-pairs in $X$ have the same direction; $F$ is *strictly* $\delta$-monotonic when there exists at least one such $\delta$-pair. In this case:
- $F$ is $\delta$-*increasing* on $X$ if $X$ contains an increasing $\delta$-pair.
- $F$ is $\delta$-*decreasing* on $X$ if $X$ contains a decreasing $\delta$-pair.
A $\delta$-monotonic interval $X$ satisfies $\textrm{OMAFE}(X)<\delta/2$. We say that a [$\ast$-pair]{} $x,y$ is **maximal** if whenever $z_1,z_2$ is a [$\ast$-pair]{} of a larger scale in the same direction containing $x,y$, then there exists a [$\ast$-pair]{} $w_1,w_2$ of an opposite direction contained in $z_1,z_2$ and containing $x,y$. For example, the sequence $1,3,2,4$ has 2 maximal [$\ast$-pair]{}s: $1,4$ and $3,2$. Maximal [$\ast$-pair]{}s of opposite direction may share a common point, whereas maximal [$\ast$-pair]{}s of the same direction may not. Maximal [$\ast$-pair]{}s cannot overlap, meaning that it cannot be the case that exactly one end point of a maximal [$\ast$-pair]{} lies strictly between the end points of another maximal [$\ast$-pair]{}; either neither point lies strictly between or both do. In the case that both do, we say that the one maximal [$\ast$-pair]{} properly contains the other. All [$\ast$-pair]{}s must be contained in a maximal [$\ast$-pair]{}.
The smallest maximal [$\ast$-pair]{} containing a [$\ast$-pair]{} must be of the same direction.
Suppose a [$\ast$-pair]{} is immediately contained in a maximal [$\ast$-pair]{} $W$. Suppose $W$ is not in the same direction, then within $W$, seek the largest [$\ast$-pair]{} in the same direction as $P$ and containing $P$, then it must be a maximal [$\ast$-pair]{} in $D$ since maximal [$\ast$-pair]{}s of different directions cannot overlap.
The first and second point of a maximal [$\ast$-pair]{} are extrema and the reverse is true as well as shown by the next lemma.
\[prop:either\] Every extremum is either the first or second point of a maximal [$\ast$-pair]{}.
The case $x=x_1$ or $x=x_n$ follows by inspection. Otherwise, $x$ is the end point of a left and a right [$\ast$-pair]{}. Each [$\ast$-pair]{} must immediately belong to a maximal [$\ast$-pair]{} of same direction: a [$\ast$-pair]{} $P$ is contained in a maximal [$\ast$-pair]{} $M$ of same direction and there is no maximal [$\ast$-pair]{} $M'$ of opposite direction such that $P\subset M'\subset M$. Let $M^l$ and $M^r$ be the maximal [$\ast$-pair]{}s immediately containing the left and right [$\ast$-pair]{} of $x$. Suppose neither $M^l$ and $M^r$ have $x$ as a end point. Suppose $M^l \subset M^r$, then the right [$\ast$-pair]{} is not immediately contained in $M^r$, a contradiction. The result follows by symmetry.
Our approach is to label each extremum in $F$ with a scale parameter $\delta$ saying that this extremum is “significant” at scale $\delta$ and below. Our intuition is that by picking extrema at scale $\delta$, we should have a segmentation having error less than $\delta/2$.
\[def:optscalelabel\]The scale labelling of an extremum $x$ is the maximum of the scales of the maximal [$\ast$-pair]{}s for which it is an end point.
For example, given the sequence $1,3,2,4$ with 2 maximal [$\ast$-pair]{}s ($1,4$ and $3,2$), we would give the following labels in order $3,1,1,3$.
Given $\delta > 0$, a [*maximal alternating sequence of $\delta$-extrema*]{} $Y = y_1 \dots y_{K+1}$ is a sequence of extrema each having scale label at least $\delta$, having alternating types (maximum/minimum), and such that there exists no sequence properly containing $Y$ having these same properties. From $Y$ we define a [*maximal alternating $\delta$-segmentation*]{} of $D$ by segmenting at the points $x_1, y_2 \dots y_K, x_n$.
\[thm:validlabelling\] Given $\delta > 0$, let $P = S_1 \dots S_K$ be a maximal alternating $\delta$-segmentation derived from maximal alternating sequence $y_1 \dots y_{K+1}$ of $\delta$-extrema. Then any alternating segmentation $Q$ having OMAFE($Q$) $<$ OMAFE($P$) has at least $K+1$ segments.
We show that conditions A and B of Proposition \[stupidresult2\] are satisfied with $\epsilon =$ OMAFE($P$).
First we show that each segment $S_i$ is $\delta$-monotone; from this we conclude that $\textrm{OMAFE}(P) < \delta/2$. Intervals $[x_1, y_1]$ and $[y_K, x_n]$ contain no maximal [$\ast$-pair]{}s of scale $\delta$ or larger, and therefore contain no [$\ast$-pair]{}s of scale $\delta$ or larger. Similarly, no $[y_i, y_{i+1}]$ contains an opposite-direction significant [$\ast$-pair]{}.
Condition A: Follows from $\delta$-monotonicity of each $S_i$ and maximal [$\ast$-pair]{}s not overlapping.
Condition B: We show that $\vert F(y_{i+1})-F(y_i) \vert \geq \delta > 2 \times \textrm{OMAFE}(P)$. If $i=1$, then $y_i$ must begin an maximal [$\ast$-pair]{}, and the maximal [$\ast$-pair]{} must end with $y_{i+1}$ since maximal [$\ast$-pair]{}s cannot overlap. The case $i+1=k$ is similar. Otherwise, since maximal [$\ast$-pair]{}s cannot overlap, each $y_i, y_{i+1}$ is either a maximal [$\ast$-pair]{} of scale $\delta$ or larger or there exist indices $j$ and $k$, $j < i$ and $k > i+1$ such that $y_j , y_i$ is a maximal [$\ast$-pair]{} of scale at least $\delta$, and $y_{i+1}, y_k$ is a maximal [$\ast$-pair]{} of scale at least $\delta$. These two maximal [$\ast$-pair]{}s have the same direction, and that this is opposite to the direction of $[y_i y_{i+1}]$. Now suppose $\vert F(y_i) - F(y_{i+1})| < \delta$. Then $y_j, y_k$ is a [$\ast$-pair]{} properly containing $y_j$, $y_i$ and $y_{i+1}$, $y_k$. But neither $y_j , y_i$ nor $y_{i+1}, y_k$ can be properly contained in a [$\ast$-pair]{} of opposite direction lying within $y_j, y_k$, thus contradicting their maximality and proving the claim.
Sequences of extrema labelled at least $\delta$ are generally not maximal alternating. For example the sequence $0,10,9,10,0$ is scale labelled $10,10,1,10,10$. However, a simple relabelling of certain extrema can make them maximal alternating. Consider two same-sense extrema $z_1 < z_2$ such that lying between them there exists no extremum having scale at least as large as the minimum of the two extrema’s scales. We must have $F(z_1) = F(z_2)$, since otherwise the point upon which $F$ has the lesser value could not be the endpoint of a maximal [$\ast$-pair]{}. This is the only situation which causes choice when constructing a maximal alternating sequence of $\delta$-extrema. To eliminate this choice, replace the scale label on $z_1$ with the largest scale of the opposite-sense extrema lying between them.
Computing a Scale Labelling Efficiently
---------------------------------------
Algorithm \[algo:computedelta\] (next page) produces a scale labelling in linear time. Extrema from the original data are visited in order, and they alternate (maxima/minima) since we only pick one of the values when there are repeated values (such as $1,1,1$).
The algorithm has a main loop (lines 5 to 12) where it labels extrema as it identifies extremal [$\ast$-pair]{}s, and stack the extrema it cannot immediately label. At all times, the stack (line 3) contains minima and maxima in **strictly** increasing and decreasing order respectively. Also at all times, the last two extrema at the bottom of the stack are the absolute maximum and absolute minimum (found so far). Observe that we can only label an extrema as we find new extremal [$\ast$-pair]{}s (lines 7, 10, and 14).
- If the stack is empty or contains only one extremum, we simply add the new extremum (line 12).
- If there are only 2 extrema $z_1,z_2$ in the stack and we found either a new absolute maximum or new absolute minimum ($z_3$), we can pop and label the oldest one ($z_1$) (lines 9, 10, and 11) because the old pair ($z_1,z_2$) forms a maximal [$\ast$-pair]{} and thus must be bounded by extrema having at least the same scale while the oldest value ($z_1$) doesn’t belong to a larger maximal [$\ast$-pair]{}. Otherwise, if there are only 2 extrema $z_1,z_2$ in the stack and the new extrema $z_3$ satisfies $z_3 \in (\min(z_1,z_2), \max(z_1,z_2))$, then we add it to the stack since no labelling is possible yet.
- While the stack contains more than 2 extrema (lines 6, 7 and 8), we consider the last three points on the stack ($s_3,s_2,s_1$) where $s_1$ is the last point added. Let $z$ be the value of the new extrema. If $z \in (\min(s_1,s_2), \max(s_1,s_2))$, then it is simply added to the stack since we cannot yet label any of these points; we exit the while loop. Otherwise, we have a new maximum (resp. minimum) exceeding (resp. lower) or matching the previous one on stack, and hence $s_1,s_2$ is a maximal [$\ast$-pair]{}. If $z\neq s_2$, then $s_3, z$ is a maximal [$\ast$-pair]{} and thus, $s_2$ cannot be the end of a maximal [$\ast$-pair]{} and $s_1$ cannot be the beginning of one, hence both $s_2$ and $s_1$ are labelled. If $z=s_2$ then we have successive maxima or minima and the same labelling as $z\neq s_2$ applies.
During the “unstacking” (lines 13 and following), we visit a sequence of minima and maxima forming increasingly larger maximal [$\ast$-pair]{}s.
**INPUT:** an array $d$ containing the $y$ values indexed from $0$ to $n-1$, repeated consecutive values have been removed **OUTPUT:** a scale labelling for all extrema $S\leftarrow$ empty stack, First($S$) is the value on top, Second($S$) is the second value **define** $\delta(d,S)=\vert d_{\textrm{First}(S)}-d_{\textrm{Second}(S)}\vert$ label First($S$) and Second($S$) with $\delta(d,S)$ pop stack $S$ twice label Second($S$) with $\delta(d,S)$ remove Second($S$) from stack $S$ stack $e$ to $S$ label First($S$) with $\delta(d,S)$ pop stack $S$ label First($S$) and Second($S$) with $\delta(d,S)$
[Once the labelling is complete, we find $K+2$ extrema having largest scale in time $O(n K)$ using $O(K)$ memory, then we remove all extrema having the same scale as the smallest scale in these $K+2$ extrema (removing at least one), we replace the first and the last extrema by $0$ and $n-1$ respectively. The result is an optimal segmentation having at most $K$ segments.]{}
Experimental Results and Comparison to Top-Down Linear Spline {#experimental}
=============================================================
We compare our optimal $O(n K)$ algorithm with the top-down linear spline algorithm [@keogh04] which runs in $O(n K^2)$ time. It successively segments the data starting with only one segment, each time picking the segment with the worse linear regression error and finding the best segmentation point; the linear regression is not continuous from one segment to the other. The regression error can be computed in constant time if one has precomputed the range moments [@LemireCASCON2002]. We run through the segments and aggregate consecutive segments having the same sign where the sign of a segment $[y_k,y_{k+1}]$ is defined by $F(y_{k+1})-F(y_{k})$.
Data Source
-----------
We used samples from the MIT-BIH Arrhythmia Database [@PhysioNet]. These ECG recordings used a sampling rate of 360 samples per second per channel with 11-bit resolution. We keep 4000 samples (11 seconds) and about 14 pulses, and we do no preprocessing such as baseline correction. We can estimate that a typical pulse has about 5 “easily” identifiable monotonic segments. Hence, out of 14 pulses, we can estimate that there are about 70 significant monotonic segments, some of which match the domain-specific markers (reference points P, Q, R, S, and T). A qualitative description of such data is useful for pattern matching applications.
Results
-------
We implemented both the scale-based segmentation algorithm and the $L_2$ norm top-down linear spline approximation algorithm in Python(version 2.3). Each run was repeated 3 times and [we observed that the scale-based segmentation implementation is faster than the top-down linear spline approximation implementation by a factor of 10.]{}
![\[omafevsk\]Results from experiments over ECG data: the optimal algorithm is considerably more accurate.](100_OMAFEvsK){width="1.00\columnwidth"}
The OMAFE with respect to the maximal number of segments ($K$) is given in Fig. \[omafevsk\][.]{} By counting on about 5 monotonic segments per pulse with a total of 14 pulses, there should about 70 monotonic segments in the 4000 samples under consideration. We see that the decrease in OMAFE with the addition of new segments starts to level off between 50 and 70 segments as predicted.
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---
abstract: 'Searching for reaction pathways describing rare events in large systems presents a long-standing challenge in chemistry and physics. Incorrectly computed reaction pathways result in the degeneracy of microscopic configurations and inability to sample hidden energy barriers. To this aim, we present a general enhanced sampling method to find multiple diverse reaction pathways of ligand unbinding through non-convex optimization of a loss function describing ligand-protein interactions. The method successfully overcomes large energy barriers using an adaptive bias potential, and constructs possible reaction pathways along transient tunnels without the initial guesses of intermediate or final states, requiring crystallographic information only. We examine the method on the T4 lysozyme L99A mutant which is often used as a model system to study ligand binding to proteins, provide a previously unknown reaction pathway, and show that using the bias potential and the tunnel widths it is possible to capture heterogeneity of the unbinding mechanisms between the found transient protein tunnels.'
author:
- Jakub Rydzewski
- Omar Valsson
bibliography:
- 'lb99.bib'
title: Finding Multiple Reaction Pathways of Ligand Unbinding
---
Molecular dynamics (MD) simulations provide sufficient temporal and spatial resolution to study physical processes. Unfortunately, MD fails to reach high energy barriers ($\gg k_{\mathrm{B}}T$) that dictate mechanisms and kinetics of rare events. Transport in heterogeneous media, such as ligand unbinding, cannot be simulated directly, and even biased MD methods often fail to find possible reaction pathways along complex transient tunnels of proteins that form spontaneously during dynamics [@rydzewski2017ligand; @rydzewski2017rare; @bruce2018new]. Although many general purpose methods have been developed to sample rare events [@grubmuller1995predicting; @voter1997hyperdynamics; @laio2002escaping], finding multiple reaction pathways of ligand unbinding is especially difficult. Also, experimental methods used currently to quantify ligand binding, e.g., time-resolved crystallography and xenon binding focus primarily on gaseous species, providing indirect evidence for the migration of larger ligands, which makes most details of reaction pathways unresolved.
The main computational limitations that render the reconstruction of reaction pathways for ligand unbinding difficult stem from accounting for internal topological features of proteins (e.g., tunnels), which is related to the degree of coupling between protein dynamics and ligand conformational states. The structural flexibility of protein tunnels allows proteins to facilitate binding by adapting to binding partners along possibly multiple pathways to the binding site. This intrinsic dynamics poses a severe challenge to straightforward biased MD methods that have been used to sample reaction pathways in ligand unbinding [@ludemann2000substrates; @laio2002escaping; @miao2015gaussian; @wang2016mapping]. Typically, such methods either approximate reaction pathways by linear Cartesian coordinates [@heymann2000dynamic], or probe protein tunnels randomly [@ludemann2000substrates; @kokh2018estimation].
An additional and ubiquitous obstacle in describing ligand unbinding is the overestimation of energy barriers, and thus, the underestimation of exponentially dependent kinetic rates arising sampling crude reaction pathways. In other words, an inadequate initial guess of reaction pathways leads to false thermodynamics and kinetics. Another problem which is related to the degeneracy of microscopic configurations originating from inability to capture intrinsic degrees of freedom, which is likely to shadow hidden energy barriers [@schneider2017stochastic; @zhang2018unfolding]. As emphasized by Elber and Gibson [@elber2008toward], sampling should not overestimate preference to more direct and geometrically shorter reaction pathways. Producing and exploring multiple reaction pathways of a complex system remains a huge challenge [@valsson2016enhancing].
In this Letter, we consider a specific part of ligand binding/unbinding problem that is very relevant and not yet fully solved [@ribeiro2018kinetics]. To our knowledge, this is the first work to show that sampling multiple transient ligand tunnels in proteins leads to heterogeneous mechanisms of unbinding between the sampled reaction pathways. We present a general enhanced sampling MD method to find multiple diverse reaction pathways of ligand unbinding along transient protein tunnels. The method does not require many parameters and and does not require initial guesses of intermediate states [@heymann2008pathways; @templeton2017rock], which is a major challenge for existing methods. Its only prerequisite is the knowledge of the initial bound state, without requiring the initial reactive trajectory. The method also takes into account protein dynamics, which is important to observe transient tunnels.
![Sampling of ligand unbinding pathways using the presented biased MD method. As an example, the unbinding of benzene from T4 lysozyme L99A along a reaction pathway is shown. The unbinding is initiated from the bound state (X-ray binding site) of the T4L-benzene complex, and ends once the ligand reaches solvent. (a) The cross-section through X-ray structure of T4L shows no apparent tunnels for benzene to leave the protein, which means that the protein must undergo structural changes to open possible exits. (b) A reaction pathway characterizing atomistically the unbinding along the transient exit tunnel is identified locally during the MD simulations. (c) Namely, to determine the $(k+1)$th intermediate, the conformations of benzene are sampled in the neighborhood of the $k$th intermediate (constrained by the sampling radius). Then, from the sampled ligand conformations the optimal direction of biasing is calculated by selecting the ligand conformation which has the lowest loss function score.[]{data-label="fig:0"}](fig0.pdf){width="0.7\columnwidth"}
To estimate ligand-protein interaction we introduce the concept of a loss function. The method minimizes a loss function $s({\bf x}, {\bf y})$ during MD simulations of a $3X$ set of ligand coordinates ${\bf x}\equiv(x_1, \dots,
x_{3X})$ and a $3Y$ set of protein coordinates ${\bf y}\equiv(y_1, \dots,
y_{3Y})$, where $X$ and $Y$ are the numbers of ligand and protein atoms, respectively. To this aim, we propose the loss function must fulfill three important criteria, e.g., (i) describe physical interactions in a ligand-protein system, (ii) tend to infinity as the ligand moves too close to the protein, and (iii) decrease as the ligand unbinds from the protein; (ii) prohibits the method from sampling ligand configurations that clash with a protein, and (iii) provides a coarse estimate of how ligand conformations are buried within a protein tunnel.
For a schematic depiction of the method, see Fig. \[fig:0\]. The method follows a procedure: (i) it finds a minimum of the chosen loss function in the neighborhood of the current position of the ligand, and, (ii) the position of the ligand is biased in the direction of the localized minimum of the loss function. The minimization is repeated during the MD simulation every $\Delta t$ MD steps, and the biasing is performed until a new solution in the neighborhood of the current position is calculated. In what follows, we explain in detail the above general outline.
We start off by describing the loss function and the minimization procedure which provides a possible ligand configuration sampled in the proximity of the current ligand conformation from the MD simulation, which corresponds to the lowest loss function score, and by explaining how the neighborhood is defined for such an optimization problem. Next, we move on to the adaptive biasing procedure which enforces the ligand conformation to dissociate toward the conformation selected by the minimization procedure. The method is then summarized, and used to model the benzene unbinding reaction pathways from the T4 lysozyme L99A mutant that is often used as a model system to study ligand unbinding processes.
*Loss Function*.—Because empty space in proteins and its intrinsic fluctuations constitute a key feature of tunnels [@rydzewski2017ligand; @bruce2018new], we use a coarse physical model for ligand-protein interactions, which accounts for steric effects only. We motivated our decision by the simplicity of this approach. For the $i$th pair of ligand-protein atoms, we define a partial loss function as $\frac{\text{e}^{-r_i}}{r_i}$, where the rescaled distance between the atoms is given by $r_i=\lambda\|x_k-y_l\|$. The $\lambda$ constant sets length scale in the loss function, and is equal to 1 when using ångströms, i.e., $\lambda=1~\text{\AA}^{-1}$. Hence, we used the loss function of the following form: $$\label{eq:1}
s=\sum_{i=1}^{P_l}\frac{\text{e}^{-r_i}}{r_i},$$ where $P_l$ is the number of ligand-protein atom pairs in the local neighborhood of the ligand (see Supporting Information for details). The sum over all pairs meets the criteria of the loss function for ligand unbinding presented here. The aim of the proposed method is to efficiently sample the configurational space of the ligand-protein complex, and optimize Eq. \[eq:1\] during MD simulations, so that the reconstructed reaction pathways of ligand unbinding minimize the loss function along multiple tunnels. We also allow the ligand to be flexible during the unbinding simulations. In this method, many MD simulations are required to sample multiple reaction pathways.
*Minimization*.—Such an optimization problem can be solved by any method suitable for non-convex loss functions [@hansmann2002global; @rydzewski2015memetic; @rydzewski2018conformational]. Here, for the sake of simplicity the minimization of the loss function is performed using simulated annealing [@kirkpatrick1983optimization]. To this end, the method checks if a randomly chosen neighboring position of the ligand ${\bf x}'$ is preferred in terms of the loss function. The neighbor is selected as a next solution according to the Metropolis-Hastings algorithm [@metropolis1953equation] with the Boltzmann factors (we omit protein coordinates ${\bf y}$ only in notation): $$p=\begin{cases}
\text{e}^{-\beta_j\big(s({\bf x}')-s({\bf x})\big)} & \text{if $s({\bf x}')>s({\bf x})$},\\
1 & \text{otherwise},
\end{cases}$$ where $\beta_j=1/T_j$ is a parameter introduced to decrease the probability of acceptance of a worse solution as the minimization scheme proceeds. $T_j$ is reduced according to the recursive formula, $T_j=kT_{j-1}$, where $j$ is the iteration number during the optimization phase, to promote convergence to an optimum [@kirkpatrick1984optimization]. The minimization procedure is reiterated to find an optimal solution. For details concerning the parameters for simulated annealing, see Supporting Information.
Next, we describe how the neighborhood is defined in our method. The minimization procedure needs constraints to optimize the loss function locally (in the current neighborhood). In our method, intermediate ligand unbinding states are searched for sequentially to get an optimal transition between the X-ray structure and the unbound state. A global minimization of the loss function without a specific definition of the neighborhood would identify only the final state of ligand unbinding. A naive approach [@rydzewski2015memetic; @rydzewski2016machine; @rydzewski2017thermodynamics] is to sample ligand conformations constrained to a sphere with a constant radius, and positioned at the center of mass of the ligand, but this requires an estimate of the radius, which is clearly system-dependent and should changed as protein dynamics is simulated. To alleviate this issue, we take the sampling radius equal to the minimal distance between the ligand-protein atom pairs, e.g., $r_s=\min_{i}r_i$. By doing so, the method dynamically adjusts the conformational space available for the sampling. We underline that the protein neighborhood of the ligand changes as the ligand dissociates during the simulation, and so does the number of ligand-protein atom pairs $P_l$, which makes the identification of the next minimum possible.
*Adaptive Biasing*.—Once the optimal ligand-protein conformation ${\bf x}'=\min_{\bf x}s({\bf x})$ is calculated in the minimization scheme, the conformation of the ligand is biased in the direction of ${\bf x}'$ along transient protein tunnels. This stage is performed by biasing the conformation of the ligand using an adaptive harmonic potential: $$\label{eq:2}
V({\bf x})=\alpha\left(v\Delta
t-({\bf x}-{\bf x}'_{i-1})\cdot\frac{{\bf x}'_i-{\bf x}'_{i-1}}{\|{\bf
x}'_i-{\bf x}'_{i-1}\|}\right)^2,$$ where ${\bf x}'_i$ is the $i$th optimal solution, $v$ is the biasing rate, $\Delta
t$ is the MD time between subsequent loss function minimizations, and $\alpha$ is the force constant. The bias potential (Eq. \[eq:2\]) is a generalization of the harmonic biasing potential introduced by Heymann and Grubmüller [@heymann2000dynamic] to curvilinear reaction pathways. The biasing potential from Ref. uses a constant direction of biasing, but in Eq. \[eq:2\] this direction is approximated as the normalized difference between the subsequent minima of the loss function. In contrast to this method, several recently introduced approaches used a constant bias to sample complex reaction pathways [@rydzewski2015memetic; @rydzewski2016machine; @kokh2018estimation]. The bias potential shown by Eq. \[eq:2\] is adaptive, and dependent on the optimal reaction pathways calculated by minimizing Eq. \[eq:1\].
The harmonic bias potential used in this study is selected to be simple as the potential energy in MD simulations already includes bonded terms for interaction of atoms that are linked by covalent bonds, and nonbonded terms that describe long-range electrostatics and van der Waals forces. Clearly, the bias potential should not be expected to be quantitative as a method to calculate energy barriers along the reaction pathways, but it was employed to enforce the process of ligand unbinding with a constant velocity as in Ref. . The bias potential, however, may serve as a means to shorten the time-scale of ligand unbinding, and as a qualitative measure to estimate relative differences of bias between the reaction pathways.
Our enhanced sampling method is outlined as follows:
1. Initialize the MD simulation,
2. Sample ligand conformations within the protein tunnel using constraints defined as the minimal distance between the ligand-protein atom pairs,
3. Minimize the loss function using a non-convex optimization algorithm and set the biasing direction toward the found minimum,
4. Bias the ligand conformation using Eq. \[eq:2\] during $\Delta t$ steps of the MD simulation,
5. Repeat the steps 2–4 during the MD simulation until the loss function reaches zero,
6. Stop the MD simulation,
which concludes the introduction of the method components, e.g., loss function, minimization, and adaptive biasing, needed to sample ligand unbinding reaction pathways.
*Unbinding Benzene from T4 Lysozyme L99A*.—We illustrate the method on T4 lysozyme L99A (T4L) with bound benzene, which is considered as a model system to study ligand unbinding from proteins. In this example, 300 10-ns trajectories were run to reconstruct the reaction pathways of benzene unbinding from the protein. We used the biasing rate $v=0.02$ Å/ps with the force constant in the stiff-spring regime [@heymann2000dynamic], $\alpha=3.6$ kcal/(molÅ). The optimal position of the ligand was recalculated by minimizing the loss function every $\Delta t=200$ ps. We found that for lower biasing rates the method is unable to find the reaction pathways in the desired span of 10 ns for a single simulation. This is, however, only a technical nuisance that can be overcome by sampling longer MD trajectories. The method is implemented in the official Plumed-2.5 repository [@tribello2014plumed] which is available on Github [@maze] and described in Ref. . For details concerning the model of T4L with bound benzene and the MD simulations, see Supporting Information.
![Reaction pathways of benzene unbinding from T4L. Only the T4L C-terminal domain is depicted, but the complete protein was used in all simulations. The crystallographic bound conformation of benzene is shown. Benzene conformations sampled during the MD simulations are biased by the adaptive bias potential to find multiple exits of T4L via the reaction pathways. The reaction pathways are named pwa-e, which corresponds to the T4L tunnels indicated by helices, i.e., D/F/G tells that the unbinding pathway is located near the D, F, and G helices. \[fig:1\]](fig1.pdf){width="0.7\columnwidth"}
We directly compared our results with reaction pathways found in previous studies. The method identified five reaction pathways for benzene exit from the binding cavity buried in T4L. These reaction pathways correspond to five tunnels of T4L, named pwa–D/F/G (tunnel through helices D, F, and G), pwb–C/D, pwc–F/G/H, pwd–H/J, and pwe–D/G (Fig. \[fig:1\]). The reconstructed reaction pathways pwa–d are mostly in agreement with a recent study by Nunes-Alves et al. [@nunes2018escape] in which the reaction pathways of benzene unbinding were sampled using temperature-accelerated MD simulations [@nunes2018escape]. Other studies also reported pwa [@miao2015gaussian] and pwc [@wang2016mapping]. To our knowledge, the benzene unbinding via pwe is first identified in this study.
Apart from the work of Nunes-Alves et al. [@nunes2018escape], other studies found only one reaction pathway, probably because of the employed biased MD methods. Biased MD methods employed by Wang et al. [@wang2016mapping] and Miao et al. [@miao2015gaussian] may limit the search in configuration space to a most optimal solution. Wang et al. used metadynamics [@barducci2008well] to bias a reaction pathway identified initially by self-penalty walk [@nowak1991reaction], which agrees with the observation that such methods strongly rely on the initial guess of a pathway [@passerone2001action; @lee2017finding]. Thus, it may not be possible to identify all possible reaction pathways that exist in the form of transient sparse tunnels in the studied ligand-protein complex. Interestingly, the reaction pathways identified here agree mostly with exit tunnels retrieved by Nunes-Alves et al. [@nunes2018escape], where MD simulations with elevated temperature were used to overcome large energy barriers along reaction pathways and increase the probability of the rare event. In both temperature-accelerated MD [@abrams2010large] and the method presented in this article there is no need for initial guess of trajectories, which clearly improves sampling of diverse pathways.
pathway tunnel no. trajectories unbinding time \[ns\] $r_s$ \[Å\]
--------- -------- ------------------ ----------------------- --------------- --
pwa D/F/G 65 $3.37\pm0.01$ $2.37\pm0.01$
pwb C/D 82 $2.61\pm0.07$ $2.31\pm0.01$
pwc F/G/H 34 $2.68\pm0.09$ $2.41\pm0.02$
pwd H/J 27 $2.29\pm0.08$ $2.36\pm0.03$
pwe D/G 92 $2.45\pm0.11$ $2.34\pm0.01$
: Reaction pathways of benzene unbinding from T4L. Quantities describing the reaction pathways were calculated from an ensemble of trajectories for the identified exits. These quantities include the number of trajectories (out of 300) that proceed through each pathway, the mean of the distribution of unbinding times that it takes for benzene to unbind in the biased simulations, and its standard deviation, and the average radius of each identified tunnel $r_s$ and its standard deviation. Errors were estimated by a bootstrapping procedure (see Supporting Information).[]{data-label="tab:1"}
The detailed characteristics of the reaction pathways for benzene unbinding from T4L are shown in Tab. I. We found an additional reaction pathway that, to the best of our knowledge, was not identified previously. This pathway corresponds to the benzene unbinding along the T4L tunnel between helices D and G. The method is able to provide the atomistic characterization of unbinding pathways. If, however, one is interested in knowing estimates of the energy barriers along unbinding pathways, a postprocessing procedure is needed to analyze the data. For instance, one way to further understand the results is to look at the averaged bias potential $V(s)$. We followed this approach, and average the bias potential along each pathway projecting it on the loss function (Fig. \[fig:3\]) using the following relation: $$V(s)=\bigg\langle\delta(s-s({\bf x}))V({\bf x})\bigg\rangle_p,$$ where the average $\left<\cdot\right>_p$ is taken over all trajectories classified as a particular reaction pathway $p$, and $s$ is the loss function defined in Eq. \[eq:1\]. This way we were able to identify energy bottlenecks in tunnels indicated by high values of the averaged bias potential $V(s)$ or the sparsity of conformational space available for sampling. Treating the loss function as a collective variable, although may be not intuitive, provides a simple formula to check which transient tunnels are biased the most. Despite roughly the same level of the bias along each reaction pathway (Fig. \[fig:3\]), it is clear that the pathways employ different mechanisms of unbinding, without the need to reconstruct free energies. This is underlined by the bias barriers along pwd and pwe, and a rather smooth decrease of the bias along pwa, pwb, and pwc. Moreover, it is perhaps possible to explain the bias barriers by inspecting the average radius of each tunnel which is used as the sampling radius $r_s$. For instance, pwd and pwe have $r_s$ at about 2.36 Å and 2.34 Å, respectively, and the highest barriers among pathways. This is an indication that the reaction pathways are heterogeneous with respect to each other, and their specific atomistic mechanism of unbinding would not be obvious by calculating averages of the full ensemble of the unbinding trajectories, without decomposition into classes first.
As recently underlined in Ref. , multiple pathways for benzene escape out of the T4L crystallographic binding site exist if one of the end states consists of multiple substrates [@bhatt2011beyond]. Interestingly, the reaction pathway via F/G/H tunnel identified by Fehler et al. [@feher2019mechanisms] is argued to be the most probable in their study. Our results show that this result may be due to the highest tunnel width in comparison to the other pathways (Tab. \[tab:1\]). As it is shown in Tab. \[tab:1\], the sampled trajectories that yield the same reaction pathways (same tunnels) are similar to each other as it is underlined by the small standard deviations of the unbinding time and the sampling radius estimated using bootstrapping (see Supporting Information).
![Averaged bias potentials $V(s)$ from all the simulations that took a specific pathway projected along the reconstructed reaction pathways. Here, we used the loss function $s$ to project the bias potential to depict in what stage of unbinding the bias is higher. The high value of the loss function indicates that the ligand is bound (b) to the T4L matrix (in the X-ray structure the loss function reaches about 4.5), whereas the low value an unbound (u) state (at end of MD simulations the loss function decreases to 0). As can be seen, the characterization of the reaction pathways is heterogeneous between the different classes, showing different mechanisms of the benzene unbinding, and indicates different bias potential barriers for the reaction pathways close to one another, for instance, pwa and pwe near the D helix. \[fig:3\]](fig3.pdf){width="\columnwidth"}
It should be noted that the method lends itself to use as an optimal initial guess of reaction pathways in other biasing MD methods to estimate thermodynamic and kinetic quantities, i.e., metadynamics [@laio2002escaping; @tiwary2013metadynamics] or variationally enhanced sampling [@valsson2014variational]. We point out that computing reaction pathways for the T4L-benzene complex is not needed when calculating the mean-first-passage times of binding and unbinding as shown by Wang [@wang2018frequency], however, it is important in estimating how the mechanisms of binding varies between the calculated reaction pathways, including free energies and conformational changes. Recently, it was shown that some protein-ligand systems can exhibit pathway hopping [@rydzewski2018kinetics; @lotz2018unbiased], and the method presented here can be used to quantify this process.
We note that the reaction pathways of ligand unbinding sampled using the method presented here diverge to diverse suboptimal basins. This is the feature that enables sampling multiple heterogeneous reaction pathways and allows to overcome the problem of the intrinsic dynamics of protein tunnels. This is due to the used sampling which is constrained by the protein structure to provide a local minimum. Also, the probability of selecting a new solution given by the Metropolis-Hastings algorithm is important for the heterogeneity of the reaction pathways. The method searches for an optimal ligand conformation locally to extend the current reaction pathway step by step. This way, the method is able to sample multiple possible unbinding pathways, which for a rare event as with ligand unbinding is necessary to explore configurational space of tunnels exhaustively.
In conclusion, we have presented a general method for finding reaction pathways of ligand unbinding, starting only from available crystallographic information. The method does not need any prerequisite guesses of intermediate states. The introduced approach uses an adaptive bias to drive the ligand to unbind from the fluctuating protein, in the direction effectively calculated by minimizing a simple loss function. The methods adapts to transient tunnels of proteins by estimating the configurational space from which it samples plausible ligand conformations (i.e., it can be also used to determine the tunnel widths). We think the method should be applicable to proteins in which prominent structural motions on a larger scale are important for ligand unbinding (e.g., trypsin [@tiwary2015kinetics]).
Various enhanced sampling techniques have been tested for characterization of rare events and long-timescale dynamics. The method proposed here was suitable to sample a rare conformational event such as benzene escape that occurs on the millisecond timescale experimentally. We provided a rigorous method to find possible reaction pathways, which can be used as a initial reference trajectory to reconstruct thermodynamic and kinetic data. Overall, our results from studies of ligand unbinding from T4L suggests that the method presented here can improve the reconstruction of reaction pathways along transient tunnels, and serve as an optimal choice for other biasing methods, limiting overestimation of hidden free energy barriers. With some adaptations the method can be also used to study other transport processes, e.g., diffusion through a membrane.
[**Note.**]{} At submission stage we became aware of Ref. in which the benzene unbinding pathways from the T4L protein are also studied. Capelli et al. found various reaction pathways that they classified into eight reaction pathways using a different criteria than we use here. In particular, the benzene unbinding pathways marked by Capelli et al. as C, F, G, and H [@capelli2019exhaustive] are subclasses of pwc, while other four are the same as ones we have identified here.
Supplementary Material {#supplementary-material .unnumbered}
======================
See supplementary material for: model of T4 lysozyme L99A; MD simulations; loss function; neighborhoodfor the loss function; minimization procedure; adaptive biasing to a loss function minimum; classification of the reaction pathways; biased unbinding times; software.
We thank H. Grubmüller, W. Nowak, and M. Parrinello for useful discussions, and Tristan Bereau and Claudio Perego for critically reading the manuscript. This work was supported by the National Science Centre, Poland (grant 2016/20/T/ST3/00488, and 2016/23/B/ST4/01770). The MD simulations were computed using facilities of Interdisciplinary Centre of Modern Technologies, Nicolaus Copernicus University, Poland.
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---
abstract: 'Recent numerical simulations of globular clusters (GCs) have shown that stellar-mass black holes (BHs) play a fundamental role in driving cluster evolution and shaping their present-day structure. Rapidly mass-segregating to the center of GCs, BHs act as a dynamical energy source via repeated super-elastic scattering, delaying onset of core collapse and limiting mass segregation for visible stars. While recent discoveries of BH candidates in Galactic and extragalactic GCs have further piqued interest in BH-mediated cluster dynamics, numerical models show that even if significant BH populations remain in today’s GCs, they are typically in configurations that are not directly detectable. We demonstrated in @Weatherford2018 that an anti-correlation between a suitable measure of mass segregation ($\Delta$) in observable stellar populations and the number of retained BHs in GC models can be applied to indirectly probe BH populations in real GCs. Here, we estimate the number and total mass of BHs presently retained in 50 Milky Way GCs from the ACS Globular Cluster Survey by measuring $\Delta$ between populations of main sequence stars, using correlations found between $\Delta$ and BH retention in the `CMC` Cluster Catalog models. We demonstrate that the range in $\Delta$’s distribution from our models matches that for observed GCs to a remarkable degree. Our results further provide the narrowest constraints to-date on the retained BH populations in the GCs analyzed. Of these 50 GCs, we identify NGCs 2808, 5927, 5986, 6101, and 6205 to presently contain especially large BH populations, each with total BH mass exceeding $10^3\,{{\rm{M_\odot}}}$.'
author:
- 'Newlin C. Weatherford'
- Sourav Chatterjee
- Kyle Kremer
- 'Frederic A. Rasio'
bibliography:
- 'bhsurvey.bib'
title: 'A Dynamical Survey of Stellar-Mass Black Holes in 50 Milky Way Globular Clusters'
---
Introduction {#intro}
============
Our understanding of stellar-mass black hole (BH) populations in globular clusters (GCs) has rapidly improved since the turn of the century. To date, five BH candidates have been detected in Milky Way GCs via X-ray and radio observations: two in M22 [@Strader2012], plus one each in M62 [@Chomiuk2013], 47 Tuc [@Miller-Jones2015; @Bahramian2017], and M10 [@Shishkovsky2018]. More recently, three BHs in detached binaries have been reported in NGC 3201, the first to be identified using radial velocity measurements [@Giesers2018; @Giesers2019]. Additional candidates have been spotted in extragalactic GCs [e.g., @Maccarone2007; @Irwin2010]. These observations not only indicate that *some* GCs presently retain populations of BHs, but the lack of any particular observable trend in the GCs hosting BH candidates further suggests that present-day BH retention may be common to *most* GCs in the Milky Way (MWGCs). Such observational evidence complements a number of recent computational simulations which show that realistic clusters can retain up to thousands of BHs late in their lifetimes [e.g., @Morscher2015]. Both observations and simulations feed a growing theoretical understanding of the dynamical importance of BHs in clusters; it is now clear that BHs play a significant role in driving long-term cluster evolution and shaping the present-day structure of GCs [@Merritt2004; @Mackey2007; @Mackey2008; @BreenHeggie2013; @Peuten2016; @Wang2016; @Weatherford2018; @ArcaSedda2018; @Kremer2018b; @Zochi2019; @Kremer2019a; @Antonini2019; @Kremer2019d].
The dynamical importance of BHs in GCs is reflected in their ability to explain the bimodal distribution in core radii distinguishing so-called ‘core-collapsed’ clusters from non-core-collapsed clusters. A convincing explanation for this bimodality, specifically why most GCs are *not* core-collapsed despite their short relaxation times, has challenged stellar dynamicists for decades. However, recent work by [@Kremer2019a; @Kremer2019d] has shown that cluster models naturally reproduce the range of observed cluster properties (such as core radius) when their initial size is varied within a narrow range consistent with the measured radii of young clusters in the local universe [@PortegiesZwart2010]. The missing piece in the explanation is simply the BHs, which guide a young cluster’s evolution to manifest present-day structural features. In this picture, most clusters retain a dynamically-significant number of BHs through to the present. As the BHs mass-segregate to the cluster core, they provide enough energy to passing stars in scattering interactions (via two body relaxation) to support the cluster against gravothermal collapse, at least until their ejection from the cluster [@Mackey2008]. For an in-depth discussion of this ‘BH burning’ process, see @Kremer2019d. Clusters born with high central densities extract the BH-driven dynamical energy faster, eventually ejecting nearly all BHs. With the ensuing reduction in dynamical energy through BH burning, the BH-poor clusters rapidly contract to the observed core-collapsed state.
Despite these advances to our understanding of BH dynamics among the cluster modeling community, observationally inferring the presence of a stellar-mass BH subsystem (BHS) in the core of a GC remains difficult. Contrary to expectations, results from $N$-body simulations suggest that the number of mass-transferring BH binaries in a GC does not correlate with the total number of BHs in the GC at the time [@Chatterjee2017b; @Kremer2018a]. Since the majority of candidate BHs in GCs come from this mass-transferring channel, the observations to-date are of little use in constraining the overall number and mass of BHs presently retained in clusters. Several groups have suggested that the existence of a BHS in a GC can be indirectly inferred from structural features, such as a large core radius and low central density [e.g., @Merritt2004; @Hurley2007; @Morscher2015; @Chatterjee2017a; @Askar2017a; @ArcaSedda2018]. However, the interpretation of such features is ambiguous; the cluster could be puffy due to BH dynamics-mediated energy production or simply because it was born puffy (equivalently, with a long initial relaxation time). Others have suggested that radial variation in the present-day stellar mass function slope may reveal the presence of a BHS [e.g., @Webb2016; @Webb2017]. The challenge here is that obtaining enough coverage of a real GC to measure its mass function over a wide range in radial position requires consolidating observations from different space- and ground-based instruments.
Due to the above ambiguities in interpreting a GC’s large-scale structural features and the observational difficulties in finding its mass-function slope, we recently introduced a new approach to predict the BH content in GCs using mass segregation among visible stars from different mass ranges [@Weatherford2018 W1 hereafter]. In a journey towards energy equipartition, the heavier objects in a star cluster give kinetic energy to passing lighter objects through scattering interactions (two body relaxation), eventually depositing the most massive objects (i.e the BHs) at the cluster’s center, with increasingly lighter stars distributed further and further away, on average [e.g., @BinneyTremaine1987; @HeggieHut2003]. The most massive stars mass-segregate closest to the central BH population over time, thereby undergoing, on average, closer and more frequent scattering interactions with the BHs than do less massive stars distributed further away. While BH burning drives all non-BHs further from the cluster center, the heavier objects receive proportionally more energy through this process. Hence, increasing the number (total mass) of BHs decreases the radial ‘gap’ between the distributions of higher-mass and lower-mass stars (i.e. mass segregation). As a result, the presence of a significant population of BHs in a cluster’s core acts to quench mass segregation [e.g., @Mackey2008; @Alessandrini2016], an effect which can be physically quantified by comparing the relative locations of stars from different mass ranges.
Low levels of mass segregation were first used to infer the existence of an intermediate-mass BH (IMBH) at the center of a GC over a decade ago [@Baumgardt2004; @Trenti2007]. More recently, [@Pasquato2016] used such a measure to place upper limits on the mass of potential IMBHs in MWGCs, while [@Peuten2016] suggested the lack of mass segregation between blue stragglers and stars near the main sequence turnoff in NGC 6101 may be due to an undetected BH population. W1, however, was the first study to use mass segregation to predict the number of stellar-mass BHs retained in specific MWGCs (47 Tuc, M10, and M22). In this study, we will improve upon the method first presented in W1 and apply it to predict the number and total mass of stellar-mass BH populations (${{N_{\rm{BH}}}}$ and ${{M_{\rm{BH}}}}$, respectively) in 50 MWGCs from the ACS Globular Cluster Survey catalog [@Sarajedini2007].
We describe our models and how they are ‘observed’ in . In , we define the stellar populations used to quantify mass segregation ($\Delta$), describe how we measure $\Delta$ in real MWGCs from the ACS Globular Cluster Survey [@Sarajedini2007], and detail the steps necessary to accurately compare $\Delta$ measured in our models to $\Delta$ measured in observed clusters. We present our own present-day ${{N_{\rm{BH}}}}$ and ${{M_{\rm{BH}}}}$ predictions for 50 MWGCs in , discuss how they support our BH burning model in , and finally compare the predictions to previous results (most notably from the MOCCA collaboration) in . Finally, in , we summarize all key findings and discuss a few potential wider interpretations of our results regarding primordial mass segregation and IMBHs hosted in MWGCs.
Numerical Models {#S:models}
================
In this paper, we use the large grid of 148 cluster simulations presented in the `CMC` Cluster Catalog [@Kremer2019d]. These simulations were computed using the latest version of our [[Hénon]{}]{}-type [@Henon1971a; @Henon1971b] Cluster Monte Carlo code (`CMC`). `CMC` has been developed and rigorously tested over the last two decades [@Joshi2000; @Joshi2001; @Fregeau2003; @Fregeau2007; @Chatterjee2010; @Umbreit2012; @Pattabiraman2013; @Chatterjee2013]. For the most recent updates and validation of `CMC` see @Morscher2015 [@Rodriguez2016; @Rodriguez2018; @Kremer2019d].
As described in @Kremer2019d, the model set covers roughly the full parameter space spanned by the MWGCs, with the range of variations motivated by observational constraints from high-mass young star clusters, thought to be similar in properties to GC progenitors [e.g., @Scheepmaker2007; @Chatterjee2010]. Four initial parameters are varied in the model set: the total number of particles ($N=2\times10^5$, $4\times10^5$, $8\times10^5$, and $1.6\times10^6$), the initial cluster virial radius ($r_v/{{\rm{pc}}}=0.5,\,1,\,2,\,4$), the metallicity (${{\rm{Z}}}/{{\rm{Z}}}_\odot=0.01,\,0.1,\,1$), and the Galactocentric distance ($R_{\rm{gc}}/{{\rm{kpc}}}=2,\,8,\,20$) assuming a Milky Way-like galactic potential [e.g., @Dehnen1998]. This gives us a $4\times4\times3\times3$ grid of 144 simulations. We also run four additional simulations with $N=3.2\times10^6$ particles to characterize the most massive clusters in the Milky Way. For these, we fix the Galactocentric distance to $R_{\rm{gc}}/{{\rm{kpc}}}=20$ while varying metallicity (${{\rm{Z}}}/{{\rm{Z}}}_\odot=0.01,\,1$) and virial radius ($r_v/{{\rm{pc}}}=1,\,2$). Finally, note that we exclude a handful of simulations which disrupted before reaching $13\,{{\rm{Gyr}}}$ in age [described in @Kremer2019d] to ensure that our results are not affected by clusters close to disruption – at that point, the assumption of spherical symmetry in `CMC` is incorrect. In total, we use $118$ simulations, each with a unique combination of initial properties.
In all simulations, the positions and velocities of single stars and center of mass of binaries are drawn from a King profile with concentration $w_0=5$ [@King1966]. Stellar masses (primary mass in case of a binary) are drawn from the initial mass function (IMF) given in @Kroupa2001 between $0.08$ and $150\,{{\rm{M_\odot}}}$. Binaries are assigned by randomly choosing $N\times f_b$ stars independent of radial position and mass and assigning a secondary adopting a uniform mass ratio ($q$) between $0.08/m_p$ and $1$, where $m_p$ denotes the primary mass. Binary orbital periods are drawn from a distribution flat in log scale with bounds from near contact to the hard-soft boundary and binary eccentricities are drawn from a thermal distribution. We include all relevant physical processes, such as two body relaxation, strong binary-mediated scattering, and galactic tides using the prescriptions outlined in @Kremer2019d.
Single and binary stellar evolution are followed using the `SSE` and `BSE` packages [@Hurley2000; @Hurley2002], updated to include our latest understanding on stellar winds [e.g., @Vink2001; @Belczynski2010] and BH formation physics [e.g., @Belczynski2002; @Fryer2012]. Neutron stars (NS) are given natal kicks drawn from a Maxwellian with $\sigma=265\,{\rm{km\,s^{-1}}}$. The maximum NS mass is fixed at $3\,{{\rm{M_\odot}}}$. The BH mass spectrum (any remnant above the maximum NS mass is considered a BH) depends on the metallicities and pre-collapse mass [@Fryer2012]. Natal kicks for BHs are given based on results from @Belczynski2002 [@Fryer2012]. Namely, a velocity is first drawn from a Maxwellian with $\sigma=265\,{\rm{km\,s^{-1}}}$ and is then scaled down based on the metallicity-dependent fallback of mass ejected due to supernova. These prescriptions lead to $\sim10^{-3}N$ retention of BHs immediately after they form. More detailed descriptions and justifications are given in past work [e.g., @Morscher2015; @Wang2016; @Askar2017a]. However, note that the primary results in this work do not depend on the exact prescriptions for BH natal kicks, provided that a dynamically significant BH population remains in the cluster post-supernova. These results are expected to depend indirectly on the BH birth mass function, via modest differences it may create in the average stellar mass of the cluster at late times.
‘Observing’ Model Clusters {#S:observing_cluster_models}
--------------------------
`CMC` periodically outputs dynamical and stellar properties of all single and binary stars including the luminosity ($L$), temperature ($T$), and radial positions. Assuming spherical symmetry, we project the radial positions of all single and binary stars in two dimensions to create sky-projected snapshots of models at different times. In line with the typical age range of MWGCs, we use all snapshots ($7,355$ total or $\sim60$ per simulation) corresponding to ages between $9$ and $13\,{{\rm{Gyr}}}$.
For each single star we calculate the temperature $T$ from the luminosity $L$ and the stellar radius $R$ (given by `BSE`) assuming a black-body. We treat binaries as unresolved sources, assigning the combined luminosity $L=L_1+L_2$ and an effective temperature given by the $L$-weighted mean (Eq. 1; W1).
We account for statistical fluctuations by performing $10$ realizations of 2D projections for each snapshot selected as above by varying the random seed. For each realization of the 2D-projected snapshots we calculate the core radius (${{r_{c,\rm{obs}}}}$) and the central surface luminosity density (${{\Sigma_{c,\rm{obs}}}}$) by fitting an analytic approximation of the King model [Eq. 18; @King1962] to the cumulative luminosity profile [e.g., @Chatterjee2017a]. We also calculate the half-light radius (${{r_{\rm{hl}}}}$) as the sky-projected distance from the center within which half of the total cluster light is emitted.
Mass Segregation in Models and Observed Clusters {#S:model_results}
================================================
In general, quantifying $\Delta$ in a star cluster requires comparison between the radial distributions of multiple stellar populations sufficiently different in their average masses [e.g., @Goldsbury2013]. While stellar mass is not directly measured in real clusters, stellar luminosity is, and can be used as a proxy for mass, especially for main sequence (MS) stars [e.g., @Hansen1994]. As in W1, we anchor our population definitions to the location of the MS turnoff (MSTO), the most prominent feature on a color-magnitude or Hertzsprung-Russel diagram (). Defining the MSTO at $L={{L_{\rm{TO}}}}$, where the MS stars (excluding blue stragglers) exhibit the highest temperature, population bounds are then established as fractions of ${{L_{\rm{TO}}}}$. While these details are unchanged from W1, we have upgraded the specific population choices used to measure $\Delta$.
In W1, we sought to maximize the signal strength in $\Delta$ by choosing two populations with characteristic masses (luminosities) as different as possible while still ensuring that the lighter population is bright enough to be easily observable in real MWGCs. In addition, both populations must contain large enough numbers of stars to limit statistical scatter. Under these constraints, we chose a high-mass population containing all stars with $L>{{L_{\rm{TO}}}}$ and a low-mass population consisting of MS stars with ${{L_{\rm{TO}}}}/125\leq L\leq {{L_{\rm{TO}}}}/25$. While these population choices ( and in ) maximized the magnitude of $\Delta$ while ensuring relatively large observable population sizes, reducing statistical scatter compared to other choices in previous studies [e.g., blue stragglers; @Peuten2016; @Alessandrini2016], they were not free from drawbacks [@deVita2019]. Specifically, contains far fewer stars than any of the three MS populations, introducing higher statistical scatter than strictly necessary. Furthermore, while populations must be chosen in a way that their typical stellar masses are sufficiently different to observe mass segregation, an overly extreme mass (luminosity) difference can cause the populations to suffer from vast discrepancies in observational incompleteness, in which the dim stars are washed out by any bright neighbors. As shown in W1, difference in the radially-dependent incompleteness between populations can introduce significant uncertainty in the measured $\Delta$, and as a result, the inferred number of BHs.
As an example, the medians for and masses are $0.82\,{{\rm{M_\odot}}}$ and $0.75\,{{\rm{M_\odot}}}$, respectively, barely different (). Meanwhile, the stellar luminosity in spans about 3 orders of magnitude. As a result, independent of what we choose for the other population, if is included in the $\Delta$ calculation, then the difference in incompleteness becomes severe, resulting in increased uncertainty. In contrast, the three MS populations (, , ) differ much more significantly in their median mass with comparatively small variation in their typical luminosities. In this work, we therefore use these three MS populations to compute $\Delta$ and ignore .
Quantifying Mass Segregation {#S:delta_def}
----------------------------
Having chosen three distinct MS populations, we compute the mass segregation, $\Delta$, between any pair of them using both parameters introduced in W1. The first, ${{\Delta_{r50}}}^{ij}$, is the difference in median cluster-centric distance between ${{\tt{Pop}}}i$ and ${{\tt{Pop}}}j$. The second, ${{\Delta_A}}^{ij}$ is the difference in area under the two populations’ cumulative radial distributions. In both cases, the cluster-centric radial distances used are sky-projected and normalized by the cluster’s sky-projected half-light radius to make $\Delta$ unitless. Mathematical expressions and graphical representations of these mass segregation parameters are given in Section 2.3 of W1.
$\Delta$ vs ${{N_{\rm{BH}}}}/{{N_{\rm{cluster}}}}$ (${{M_{\rm{BH}}}}/{{M_{\rm{cluster}}}}$): Effects of cluster properties {#S:model_trend}
--------------------------------------------------------------------------------------------------------------------------
As introduced in W1, there exists a strong anti-correlation between the ratio of numbers of BHs to all stars retained in a cluster (${{N_{\rm{BH}}}}$/${{N_{\rm{cluster}}}}$) and the cluster’s measured mass segregation (${{\Delta_{r50}}}^{ij}$ and ${{\Delta_A}}^{ij}$). As explained earlier, the anti-correlation is due to BH burning, in which BHs mass-segregate to the cluster’s core and provide energy to passing stars through two body relaxation, pushing those stars farther out into the cluster. On average, the most massive stars gain proportionally more energy from BH burning since they are distributed closer to the BH-core than less massive stars. Hence, clusters with more numerous (more massive) BH populations in their core display reduced mass segregation.
In , we show the $\Delta$-${{N_{\rm{BH}}}}/{{N_{\rm{cluster}}}}$ anti-correlation across all $7,355$ model snapshots with ages between $9$-$13\,{{\rm{Gyr}}}$, colored by metallicity and using a standard radial limit of ${{r_{\rm{lim}}}}= {{r_{\rm{hl}}}}$. (i.e. Only stars within the model clusters’ half-light radii are used when measuring $\Delta$ for this figure, a constraint motivated by field-of-view limits when observing real clusters; see ). In the top panel, ${{\Delta_{r50}}}^{24}$ (${{\Delta_{r50}}}$ between and ) is used for $\Delta$, while the lower panel uses ${{\Delta_A}}^{24}$. Uncertainty bars represent the standard deviation across the 10 randomized 2D projections (‘views’) of each cluster snapshot.
Though not shown, plots of ${{M_{\rm{BH}}}}/{{M_{\rm{cluster}}}}$ versus $\Delta$ are practically indistinguishable from , except with a y-axis range of $\log({{M_{\rm{BH}}}}/{{M_{\rm{cluster}}}}) \in [-1.3,-5.3]$. Other pairings of the four populations in to measure $\Delta$ also result in very similar anti-correlations (though wider spread is indeed apparent whenever is used for the reasons elaborated earlier). With both more models and much fuller coverage of the space of initial cluster parameters characterizing MWGCs ($N$, $r_v$, ${{\rm{Z}}}$, and $R_{\rm{gc}}$), the anti-correlation extends to larger mass segregation and to an order-of-magnitude lower ${{N_{\rm{BH}}}}/{{N_{\rm{cluster}}}}$ than in W1. The metallicity dependence of the trend is also more explicit. The higher the ${{\rm{Z}}}$, the lower the mass of the BHs produced, so higher-${{\rm{Z}}}$ clusters need higher ${{N_{\rm{BH}}}}/{{N_{\rm{cluster}}}}$ to quench $\Delta$ to the same degree as a lower-${{\rm{Z}}}$ cluster.
Other parameters contribute less visibly to the spread in the trend, primarily through their impact on dynamical age, which increases from upper left to lower right along the trend. Specifically, a detailed model-to-model examination reveals that virial radius ($r_v$) has the largest impact on a snapshot’s location along the trend at any given physical time. Clusters with lower initial $r_v$ relax faster, making them dynamically older at late times than GCs with higher $r_v$. Since $\Delta$ correlates with and ${{N_{\rm{BH}}}}$ anti-correlates with dynamical age, the models with lowest $r_v$ appear at the bottom right of each panel in . Initial $N$ also affects the relaxation timescale of a cluster. Thus, the least massive clusters are also dynamically the oldest at the same physical time. These low-mass clusters tend to be at the bottom right. Similarly, all else being fixed, as a cluster gets older, it moves down and right along the trend, albeit to a lesser degree than movement from $N$ or $r_v$ variation, since the age range used here is narrow ($9$-$13\,{{\rm{Gyr}}}$) compared to lifetimes of typical GCs. For an average model, ${{N_{\rm{BH}}}}/{{N_{\rm{cluster}}}}$ drops by 0.5 dex between the $9$ and $13\,{{\rm{Gyr}}}$ snapshots. Finally, increasing Galactocentric distance ($R_{\rm{gc}}$) slightly increases $\Delta$ but has little impact on ${{N_{\rm{BH}}}}/{{N_{\rm{cluster}}}}$, shifting snapshots left-to-right in the figure. This occurs because clusters farther from the Galactic center experience lower tidal forces, increasing the cluster’s tidal radius (boundary) and making it harder for stars to escape the cluster. As will be discussed in the next section, limiting the radial extent of the stellar populations used to measure $\Delta$ decreases $\Delta$.
Measuring Mass Segregation in Observed Clusters {#S:observational_results}
-----------------------------------------------
To measure $\Delta$ in real clusters, we use the ACS Survey for MWGCs [@Sarajedini2007]. Compiled using the wide-field channel of the *Hubble Space Telescope’s* Advanced Camera for Surveys (ACS), this resource catalogs stars within the central $4\arcmin\times4\arcmin$ of 71 MWGCs and exists as an online database of stellar coordinates and calibrated photometry in the ACS VEGA-mag system [@Sirianni2005]. Construction of the database, which may be accessed publicly at <http://www.astro.ufl.edu/~ata/public_hstgc>, is fully detailed by [@Anderson2008].
Using the observed stellar data, we construct the same four turnoff-anchored populations as described above for the models. The exact procedure used for constructing observed populations is fully described and illustrated in Section 4 of W1 (with the addition of and ). A couple important steps are worth highlighting, however. First, since the ACS field-of-view (FOV) is a rhombus covering only the central-most region of each GC, using raw ACS stellar data to construct the populations will introduce a radial bias in observed $\Delta$ when comparing to $\Delta$ in the models, which have effectively unlimited FOVs. We therefore define a radial limit (${{r_{\rm{lim}}}}$) for each observed cluster as the radius of the largest circle inscribed in the FOV. For the MWGCs we analyze, ${{r_{\rm{lim}}}}\in [0.52,3.48]\times {{r_{\rm{hl}}}}$. (A total of 23 different values of ${{r_{\rm{lim}}}}$ are applied depending on the specific GC being analyzed, a fine enough grid to ensure that in all cases, the adopted ${{r_{\rm{lim}}}}$ is never too far from the actual ${{r_{\rm{lim}}}}$ observed.) For each GC, we then compute the observed ${{\Delta_{r50}}}$ and ${{\Delta_A}}$ between each pair of the three MS populations, including only stars within the GC’s radial limit. Similarly, when later applying the model $\Delta-{{N_{\rm{BH}}}},{{M_{\rm{BH}}}}$ correlations to predict ${{N_{\rm{BH}}}}$ and ${{M_{\rm{BH}}}}$ in a real GC, we utilize model data that has been radially-limited to match the observed ${{r_{\rm{lim}}}}$. Second, we found in W1 that observational incompleteness significantly impacts observed $\Delta$ measurements. Correcting for incompleteness (exactly as in Section 4 of W1) is even more critical in this broader survey of MWGCs, as there are more extreme examples with low completeness, especially core-collapsed clusters in which dim stars even relatively far from the cluster center are almost entirely washed out by the brightest members. Even in non-core-collapsed clusters, changes in $\Delta$ of order 50% are common after correcting for incompleteness. As discussed in Section 4.3 of W1, this correction step introduces an uncertainty on the $\Delta$ measurement for an observed cluster. We also find that the resulting probability density functions (PDFs) for observed $\Delta$ are Gaussian in shape with a typical $1\sigma$ uncertainty of order $10\%$ or less.
In W1, we limited our analysis to 47 Tuc, M 10, and M 22 – all known to contain candidate stellar-mass BHs [e.g., @Strader2012; @Shishkovsky2018; @Miller-Jones2015; @Bahramian2017]. In this full survey, we predict ${{N_{\rm{BH}}}}$ and ${{M_{\rm{BH}}}}$ for a total of 50 of the ACS survey’s 71 MWGCs. We do not analyze 21 GCs from the ACS catalog for several reasons. Eight (IC04499, PAL2, PAL15, PYXIS00, RUPR106, and NGCs 0362, 6426, 7006) are excluded because the catalog does not include the necessary information (artificial star files) for performing incompleteness corrections. Three MWGCs (NGCs 6362, 6388, 6441) are excluded because their artificial star data are incomplete, two ($\omega$ Cen and NGC 6121) are excluded because their FOVs do not extend to at least 0.5 ${{r_{\rm{hl}}}}$, and one (NGC 6496) is excluded because its FOV is half-size and triangular rather than rhomboidal. The remaining seven of the survey’s non-NGC clusters (ARP2, E3, LYNGA7, PAL1, PAL2, TERZAN7, and TERZAN8) are excluded because of their general status as outliers relative to the bulk of the MWGCs and the limited coverage by our models of their (lower-right) region of the mass vs. $r_c/{{r_{\rm{hl}}}}$ parameter space, seen in . In this figure, we compare the cluster properties of the selected 50 ACS Survey clusters to the full population of MWGCs [taken from @Baumgardt2018] as well as the models from the `CMC` Cluster Catalog. The figure shows that both the `CMC` models and the selected ACS GCs cover a very similar parameter space, providing confidence in our analysis. In addition, the analyzed clusters span roughly the entire parameter space for all MWGCs, indicating that results from this study are likely representative of the entire population of MWGCs.
Comparing Models to Observations {#S:mseg_comp}
--------------------------------
Because W1 examined only three clusters, we were unable to rigorously compare the $\Delta$ distributions from our models to the corresponding distributions measured in real MWGCs. Now, however, with a representative collection of 50 MWGCs, a general comparison of the observed vs. modeled $\Delta$ distributions is possible. Indeed, such a step is essential to establish that the anti-correlation between BH retention and mass segregation in our models can be reliably applied to predict BH retention in real clusters. This determination is firmly established by , in which the observed $\Delta$ distributions from the 50 real GCs (solid curves) closely match the $\Delta$ distributions across all model snapshots. Note that the tightest radial limit of $0.52\,{{r_{\rm{hl}}}}$ (from a particularly narrow view of M22) is uniformly applied to all model snapshots and observed clusters, for this comparison only. Though this restricts information from stars with $r>{{r_{\rm{lim}}}}$, especially for more distant clusters that are more completely imaged in the ACS Survey, it is necessary to compare all clusters on equal footing, given $\Delta$’s dependence on the applied radial limit. While the (thousands of) model snapshots are numerous enough to be finely binned (shaded histograms), the $\Delta$ distributions for the observed clusters are best represented by equally-weighted and normalized sums of the Gaussian PDFs reflecting the uncertainties on their $\Delta$ measurements (solid curves). Note that the modes and $1\sigma$ uncertainty bars on these measurements are also listed individually for each cluster in (${{\Delta_{r50}}}^{24}$) and (${{\Delta_A}}^{24}$) of the Appendix.
The $\Delta$ distributions from our models and the MWGCs are remarkably similar, not only in range but also in rough shape. This is especially noteworthy considering how strongly the magnitude of $\Delta$ depends on the imposed radial limit and the incompleteness correction. For example, the tight limit of $0.52\,{{r_{\rm{hl}}}}$ in reduces the typical, unlimited value of $\Delta$ by a full order of magnitude from $\Delta \sim 0.1$ to $\Delta \sim 0.01$. Such a comparatively close match between the model and observed $\Delta$ distributions therefore provides strong evidence that our `CMC` models at $9$-$13\,{{\rm{Gyr}}}$ accurately capture the state of mass segregation in MWGCs. Furthermore, this similarity is achieved without having specifically tuned the models to match observed mass segregation; instead, the match derives simply from the observationally-motivated grid of initial conditions we have chosen. The match also demonstrates that our main sequence population-based method of measuring mass segregation is both highly robust and adaptable to significant field-of-view limitations applicable to many MWGCs.
Finally, it is worth pointing out that while the $\Delta$ distribution from the models appears strongly bimodal (even tetramodal on closer examination), this is simply due to the shape of the model grid. The four different initial virial radii divide the model set into four subsets with different initial relaxation times and accordingly divergent levels of mass segregation at late times. Variations in initial $N$ and snapshot age smooth out the resulting spectrum in $\Delta$, but the discreteness of the model grid should *not* be mistaken for a fundamental physical phenomenon. In turn, however, the observed $\Delta$ distribution exhibits a similarly strong peak to the model distribution at $\Delta \sim 0.02$. This specific value is unimportant, as it is dependent on the radial limit, but the peak’s existence does appear to be a statistically significant feature of the true mass segregation distribution for MWGCs. This peak is representative of dynamically young clusters that have yet to undergo core-collapse and retain many BHs. The tail in the distribution is also likely a true feature, representative of dynamically older clusters that have depleted most of their BHs. Together, these features likely point to a birth size distribution for MWGCs, possibly with recent contamination from external sources of more dynamically-evolved GCs, like the nearby dwarf galaxies [e.g., @Mackey2004].
Predicting the Number and Mass of Retained Black Holes in Observed GCs {#S:nbh_predictions}
======================================================================
We now derive PDFs for ${{N_{\rm{BH}}}}/{{N_{\rm{cluster}}}}$ and ${{M_{\rm{BH}}}}/{{M_{\rm{cluster}}}}$ retained in all 50 of the MWGCs analyzed, as inferred from the appropriate radially-limited model sets and measured ${{\Delta_{r50}}}$ or ${{\Delta_A}}$ (hereon referred to jointly as just $\Delta$). Unlike in W1, however, we have multiple different measurements of $\Delta$ for each cluster – one for each pairing of the three MS populations (i.e. $\Delta^{23}$, $\Delta^{24}$, $\Delta^{34}$). In order to combine the measurements into a single prediction, we use trivariate Gaussian kernel density estimates (KDEs) of the $\Delta^{23}-\Delta^{34}-{{N_{\rm{BH}}}}/{{N_{\rm{cluster}}}}$ ($\Delta^{23}-\Delta^{34}-{{M_{\rm{BH}}}}/{{M_{\rm{cluster}}}}$) space, using only the models with ${{\rm{Z}}}$ closest to each cluster’s observed metallicity [@Harris1996 2010 edition]. Models are deemed sufficiently close in metallicity on the basis of simple logarithmic binning: BH predictions for GCs with ${{\rm{Z}}}/{{\rm{Z}}}_\odot < 0.033$ are based only on the $0.01\,{{\rm{Z}}}_\odot$ models, while predictions for clusters with $0.033 < {{\rm{Z}}}/{{\rm{Z}}}_\odot < 0.067$ use both the $0.01$ and $0.1\,{{\rm{Z}}}_\odot$ models. Predictions for GCs with $0.067 < {{\rm{Z}}}/{{\rm{Z}}}_\odot < 0.133$ incorporate just the $0.1\,{{\rm{Z}}}_\odot$ models, etc. In all cases, we exclude $\Delta^{24}$ as a fourth axis in the KDE because it is simply the sum of $\Delta^{23}$ and $\Delta^{34}$ and hence is not an independent additional axis. These trivariate distributions are then used to infer the expected number (total mass) of retained BHs in each GC using the following procedure.
For each GC, we evaluate the above 3D PDF (from the *models*) on a grid of points spanning the $3\sigma$ confidence intervals (CIs) of the *observed* $\Delta^{23}$ and $\Delta^{34}$, and from ${{N_{\rm{BH}}}}/{{N_{\rm{cluster}}}}$ (${{M_{\rm{BH}}}}/{{M_{\rm{cluster}}}}$) = 0 to twice the maximum ${{N_{\rm{BH}}}}/{{N_{\rm{cluster}}}}$ (${{M_{\rm{BH}}}}/{{M_{\rm{cluster}}}}$) seen in our models. The sample points are spaced evenly in linear scale along all three axes – $\Delta^{23}$, $\Delta^{34}$, and ${{N_{\rm{BH}}}}/{{N_{\rm{cluster}}}}$ (${{M_{\rm{BH}}}}/{{M_{\rm{cluster}}}}$) – with a respective grid size of $15\times15\times1001$ sample points. This resolution is high enough to ensure that an order of magnitude resolution increase along each axis changes the final mode, $1\sigma$ and $2\sigma$ CIs on ${{N_{\rm{BH}}}}/{{N_{\rm{cluster}}}}$ (${{M_{\rm{BH}}}}/{{M_{\rm{cluster}}}}$) in the third significant digit, at the most. In this grid form, the 3D PDF from the models is then convolved with both 1D Gaussian PDFs describing $\Delta^{23}$ and $\Delta^{34}$ observed in the GC. The resulting convolution is numerically integrated along the $\Delta^{23}$ and $\Delta^{34}$ axes using Simpson’s rule and normalized to obtain the final 1D PDFs for ${{N_{\rm{BH}}}}/{{N_{\rm{cluster}}}}$ (${{M_{\rm{BH}}}}/{{M_{\rm{cluster}}}}$). These distributions (filled, solid curves) are exemplified for the case of NGC 6254 (M10) in , based on both ${{\Delta_{r50}}}$ (red) and ${{\Delta_A}}$ (blue). Note that these final PDFs are equivalent to those shown in Figure 10 of W1, just with a different KDE formulation from the one used in that paper (namely, the addition of an extra axis to the KDE and convolution of the raw KDE with the observed $\Delta$ PDFs rather than Monte Carlo sampling). Versions of for all 50 MWGCs analyzed in this study are available in the online journal. The corresponding modes, $1\sigma$, and $2\sigma$ confidence intervals on ${{N_{\rm{BH}}}}/{{N_{\rm{cluster}}}}$ (${{M_{\rm{BH}}}}/{{M_{\rm{cluster}}}}$) for each GC are reported in for predictions based on ${{\Delta_{r50}}}$. For the nearly identical results based instead on ${{\Delta_A}}$, see in the Appendix. Because ${{\Delta_{r50}}}$ is a mathematically simpler parameter to calculate, we recommend using it in the future rather than ${{\Delta_A}}$, but we recognize that some observers seem to prefer ${{\Delta_A}}$ [e.g., @Alessandrini2016].
[lccccc|rrrrr|rrrrr]{} NGC 0104 (47Tuc) & 0.55 & 1.77 & 779 & 1000 & $0.062 \pm 0.009$ & 0 & 0.41 & **2.75** & 6.87 & 12.1 & 0 & 13 & **117** & 302 & 555\
NGC 0288 & 0.77 & 2.39 & 116 & 87 & $0.015 \pm 0.002$ & 1.15 & 6.17 & **11.3** & 16.4 & 22.5 & 39 & 263 & **512** & 766 & 1048\
NGC 1261 & 2.45 & 2.12 & 167 & 225 & $0.035 \pm 0.014$ & 1.05 & 6.0 & **11.7** & 18.1 & 24.3 & 27 & 249 & **506** & 771 & 1043\
NGC 1851 & 3.48 & 2.02 & 302 & 367 & $0.104 \pm 0.030$ & 0 & 0.5 & **3.22** & 7.75 & 14.0 & 0 & 19 & **139** & 344 & 624\
NGC 2298 & 1.70 & 0.46 & 12 & 57 & $0.032 \pm 0.007$ & 0 & 1.09 & **4.43** & 8.84 & 14.2 & 0 & 34 & **177** & 401 & 690\
NGC 2808 & 2.25 & 1.64 & 742 & 975 & $0.058 \pm 0.013$ & 0 & 1.84 & **5.87** & 10.3 & 15.1 & 0 & 56 & **212** & 408 & 631\
NGC 3201 & 0.57 & 2.4 & 149 & 163 & $0.013 \pm 0.003$ & 0 & 2.24 & **13.7** & 27.2 & 62.8 & 0 & 67 & **602** & 1246 & 3265\
NGC 4147 & 3.48 & 1.51 & 33 & 50 & $0.042 \pm 0.015$ & 0 & 0.65 & **3.55** & 8.44 & 14.3 & 0 & 28 & **166** & 411 & 713\
NGC 4590 (M68) & 1.15 & 2.02 & 123 & 152 & $0.014 \pm 0.003$ & 0 & 2.57 & **6.81** & 11.1 & 15.4 & 0 & 84 & **275** & 488 & 722\
NGC 4833 & 0.73 & 0.84 & 247 & 317 & $0.014 \pm 0.002$ & 0 & 4.51 & **12.6** & 21.3 & 42.3 & 0 & 164 & **547** & 962 & 2090\
NGC 5024 (M53) & 1.29 & 1.59 & 380 & 521 & $0.043 \pm 0.010$ & 0 & 2.19 & **6.69** & 11.5 & 16.8 & 0 & 78 & **277** & 515 & 787\
NGC 5053 & 0.68 & 1.66 & 57 & 87 & $0.012 \pm 0.001$ & 12.1 & 17.3 & **48.7** & 61.1 & 91.9 & 521 & 759 & **2443** & 3052 & 4588\
NGC 5272 (M3) & 0.77 & 1.56 & 394 & 610 & $0.047 \pm 0.009$ & 0 & 0.48 & **3.22** & 8.07 & 14.2 & 0 & 18 & **149** & 390 & 707\
NGC 5286 & 2.25 & 1.41 & 401 & 536 & $0.099 \pm 0.019$ & 0 & 0.23 & **2.53** & 6.75 & 12.5 & 0 & 10 & **123** & 332 & 622\
NGC 5466 & 0.77 & 1.13 & 46 & 106 & $0.001 \pm 0.002$ & 5.1 & 11.4 & **20.3** & 42.3 & 73.5 & 153 & 431 & **928** & 2001 & 3816\
NGC 5904 (M5) & 1.00 & 1.52 & 372 & 572 & $0.040 \pm 0.008$ & 0 & 1.9 & **6.1** & 11.0 & 16.7 & 0 & 67 & **243** & 463 & 724\
NGC 5927 & 1.62 & 2.61 & 354 & 228 & $0.015 \pm 0.011$ & 3.43 & 9.74 & **17.4** & 24.4 & 38.5 & 106 & 373 & **706** & 1038 & 1726\
NGC 5986 & 1.70 & 2.45 & 301 & 406 & $0.021 \pm 0.018$ & 0.35 & 7.21 & **14.1** & 21.2 & 33.0 & 0 & 285 & **613** & 955 & 1523\
NGC 6093 (M80) & 2.89 & 1.43 & 249 & 335 & $0.120 \pm 0.016$ & 0 & 0.44 & **3.03** & 7.73 & 13.7 & 0 & 18 & **143** & 373 & 672\
NGC 6101 & 1.62 & 3.0 & 127 & 102 & $0.002 \pm 0.003$ & 29.4 & 40.9 & **49.1** & 74.7 & 93.0 & 1376 & 1966 & **2402** & 3918 & 4630\
NGC 6144 & 1.00 & 0.54 & 45 & 94 & $0.018 \pm 0.003$ & 1.27 & 7.82 & **14.7** & 21.8 & 39.2 & 0 & 317 & **659** & 1008 & 1888\
NGC 6171 (M107) & 1.00 & 2.16 & 87 & 121 & $0.018 \pm 0.004$ & 0.03 & 5.23 & **13.2** & 24.8 & 44.2 & 0 & 207 & **589** & 1096 & 2078\
NGC 6205 (M13) & 1.00 & 2.61 & 453 & 450 & $0.021 \pm 0.006$ & 0 & 6.73 & **14.1** & 21.6 & 38.1 & 0 & 260 & **615** & 984 & 1864\
NGC 6218 (M12) & 0.99 & 1.27 & 87 & 144 & $0.016 \pm 0.003$ & 0 & 6.13 & **12.6** & 20.5 & 37.3 & 0 & 269 & **588** & 950 & 1742\
NGC 6254 (M10) & 0.89 & 1.94 & 184 & 168 & $0.022 \pm 0.003$ & 0 & 3.14 & **8.05** & 13.2 & 18.8 & 0 & 112 & **338** & 584 & 876\
NGC 6304 & 1.29 & 1.37 & 277 & 142 & $0.061 \pm 0.025$ & 0 & 2.78 & **12.6** & 18.5 & 27.3 & 0 & 82 & **262** & 455 & 655\
NGC 6341 (M92) & 1.70 & 1.81 & 268 & 329 & $0.077 \pm 0.023$ & 0 & 0.65 & **3.47** & 8.28 & 14.0 & 0 & 26 & **161** & 402 & 700\
NGC 6352 & 0.85 & 2.47 & 94 & 66 & $0.028 \pm 0.004$ & 0 & 2.78 & **7.53** & 12.5 & 20.6 & 0 & 104 & **318** & 549 & 933\
NGC 6366 & 0.57 & 2.34 & 47 & 34 & $0.015 \pm 0.003$ & 0 & 1.88 & **6.58** & 11.8 & 22.2 & 0 & 61 & **267** & 514 & 1027\
NGC 6397 & 0.61 & 2.18 & 89 & 78 & $0.068 \pm 0.004$ & 0 & 0 & **1.5** & 4.26 & 8.86 & 0 & 0 & **81** & 230 & 474\
NGC 6535 & 1.99 & 4.8 & 20 & 14 & $0.062 \pm 0.015$ & 0 & 0.21 & **2.61** & 7.07 & 13.2 & 0 & 8 & **122** & 334 & 627\
NGC 6541 & 1.62 & 1.42 & 277 & 438 & $0.081 \pm 0.020$ & 0 & 0.58 & **3.32** & 8.01 & 13.8 & 0 & 23 & **155** & 391 & 689\
NGC 6584 & 2.45 & 1.12 & 91 & 204 & $0.038 \pm 0.018$ & 0 & 1.82 & **6.08** & 10.9 & 16.0 & 0 & 67 & **255** & 491 & 757\
NGC 6624 & 1.99 & 1.02 & 73 & 169 & $0.147 \pm 0.051$ & 0 & 0.0 & **0.25** & 0.76 & 1.72 & 0 & 0 & **8** & 27 & 60\
NGC 6637 (M69) & 1.99 & - & 200\* & 195 & $0.061 \pm 0.026$ & 0 & 6.29 & **14.6** & 20.9 & 30.8 & 0 & 239 & **577** & 866 & 1364\
NGC 6652 & 3.48 & - & 96\* & 79 & $0.090 \pm 0.032$ & 0 & 0.4 & **2.61** & 6.63 & 11.6 & 0 & 14 & **112** & 292 & 525\
NGC 6656 (M22) & 0.52 & 2.15 & 416 & 430 & $0.026 \pm 0.002$ & 0 & 1.13 & **6.61** & 13.3 & 37.8 & 0 & 39 & **303** & 624 & 1956\
NGC 6681 (M70) & 2.45 & 2.0 & 113 & 121 & $0.080 \pm 0.026$ & 0 & 1.48 & **5.94** & 11.6 & 19.2 & 0 & 58 & **256** & 534 & 898\
NGC 6715 (M54) & 2.25 & 2.04 & 1410 & 1680 & $0.104 \pm 0.009$ & 0 & 0.21 & **2.38** & 6.35 & 11.8 & 0 & 9 & **117** & 313 & 587\
NGC 6717 (Pal9) & 2.45 & - & 22\* & 31 & $0.064 \pm 0.020$ & 0 & 0.13 & **2.17** & 5.81 & 11.0 & 0 & 6 & **106** & 288 & 546\
NGC 6723 & 1.15 & 1.77 & 157 & 232 & $0.012 \pm 0.005$ & 0.48 & 7.73 & **19.0** & 29.3 & 60.1 & 0 & 313 & **792** & 1297 & 2884\
NGC 6752 & 0.91 & 2.17 & 239 & 211 & $0.069 \pm 0.013$ & 0 & 0.06 & **2.09** & 5.73 & 11.3 & 0 & 3 & **107** & 297 & 583\
NGC 6779 (M56) & 1.62 & 1.58 & 281 & 157 & $0.029 \pm 0.007$ & 0.36 & 4.2 & **9.03** & 13.9 & 18.3 & 0 & 153 & **380** & 610 & 829\
NGC 6809 (M55) & 0.61 & 2.38 & 188 & 182 & $0.010 \pm 0.002$ & 1.59 & 7.78 & **18.3** & 41.7 & 72.9 & 0 & 247 & **823** & 1946 & 3739\
NGC 6838 (M71) & 1.00 & 2.76 & 49 & 30 & $0.015 \pm 0.004$ & 0.87 & 6.2 & **17.3** & 31.3 & 61.6 & 0 & 243 & **740** & 1400 & 2946\
NGC 6934 & 2.45 & 1.76 & 117 & 163 & $0.060 \pm 0.024$ & 0 & 1.3 & **4.98** & 9.62 & 15.1 & 0 & 47 & **199** & 414 & 661\
NGC 6981 (M72) & 1.70 & - & 63\* & 112 & $0.005 \pm 0.004$ & 4.29 & 13.4 & **21.4** & 34.6 & 48.2 & 152 & 529 & **908** & 1473 & 2768\
NGC 7078 (M15) & 1.70 & 1.15 & 453 & 811 & $0.111 \pm 0.009$ & 0 & 0.27 & **2.55** & 6.71 & 12.4 & 0 & 12 & **126** & 336 & 620\
NGC 7089 (M2) & 1.70 & 1.62 & 582 & 700 & $0.109 \pm 0.012$ & 0 & 0.23 & **2.55** & 6.79 & 12.5 & 0 & 10 & **124** & 334 & 624\
NGC 7099 (M30) & 1.70 & 1.85 & 133 & 163 & $0.081 \pm 0.017$ & 0 & 0.02 & **1.94** & 5.31 & 10.5 & 0 & 1 & **98** & 269 & 537\
\[T:raw\_results\_dr\]
[l|rrrrr|rrrrr]{} NGC 0104 (47Tuc) & 0 & 6 & **43** & 107 & 189 & 0 & 101 & **911** & 2353 & 4323\
NGC 0288 & 3 & 14 & **26** & 38 & 52 & 45 & 305 & **594** & 889 & 1216\
NGC 1261 & 4 & 20 & **39** & 60 & 81 & 45 & 416 & **845** & 1288 & 1742\
NGC 1851 & 0 & 3 & **19** & 47 & 85 & 0 & 57 & **420** & 1039 & 1884\
NGC 2298 & 0 & 0 & **1** & 2 & 3 & 0 & 4 & **21** & 47 & 80\
NGC 2808 & 0 & 27 & **87** & 153 & 224 & 0 & 416 & **1573** & 3027 & 4682\
NGC 3201 & 0 & 7 & **41** & 81 & 187 & 0 & 100 & **897** & 1857 & 4865\
NGC 4147 & 0 & 0 & **2** & 6 & 9 & 0 & 9 & **55** & 135 & 235\
NGC 4590 (M68) & 0 & 6 & **17** & 27 & 38 & 0 & 103 & **338** & 600 & 888\
NGC 4833 & 0 & 22 & **62** & 105 & 209 & 0 & 405 & **1351** & 2376 & 5162\
NGC 5024 (M53) & 0 & 17 & **51** & 87 & 128 & 0 & 296 & **1053** & 1957 & 2991\
NGC 5053 & 14 & 20 & **55** & 69 & 104 & 295 & 430 & **1383** & 1727 & 2597\
NGC 5272 (M3) & 0 & 4 & **25** & 64 & 112 & 0 & 71 & **587** & 1537 & 2786\
NGC 5286 & 0 & 2 & **20** & 54 & 100 & 0 & 40 & **493** & 1331 & 2494\
NGC 5466 & 5 & 10 & **19** & 39 & 67 & 70 & 197 & **423** & 912 & 1740\
NGC 5904 (M5) & 0 & 14 & **45** & 82 & 124 & 0 & 249 & **904** & 1722 & 2693\
NGC 5927 & 24 & 69 & **123** & 173 & 273 & 375 & 1320 & **2499** & 3675 & 6110\
NGC 5986 & 2 & 43 & **85** & 128 & 199 & 0 & 858 & **1845** & 2875 & 4584\
NGC 6093 (M80) & 0 & 2 & **15** & 38 & 68 & 0 & 45 & **356** & 929 & 1673\
NGC 6101 & 75 & 104 & **125** & 190 & 236 & 1748 & 2497 & **3051** & 4976 & 5880\
NGC 6144 & 1 & 7 & **13** & 20 & 36 & 0 & 144 & **299** & 457 & 855\
NGC 6171 (M107) & 0 & 9 & **23** & 43 & 77 & 0 & 180 & **512** & 954 & 1808\
NGC 6205 (M13) & 0 & 61 & **128** & 196 & 345 & 0 & 1178 & **2786** & 4458 & 8444\
NGC 6218 (M12) & 0 & 11 & **22** & 35 & 65 & 0 & 233 & **509** & 822 & 1507\
NGC 6254 (M10) & 0 & 12 & **30** & 49 & 69 & 0 & 206 & **622** & 1075 & 1612\
NGC 6304 & 0 & 15 & **70** & 102 & 151 & 0 & 227 & **726** & 1260 & 1814\
NGC 6341 (M92) & 0 & 3 & **19** & 44 & 75 & 0 & 70 & **431** & 1077 & 1876\
NGC 6352 & 0 & 5 & **14** & 23 & 39 & 0 & 98 & **298** & 515 & 875\
NGC 6366 & 0 & 2 & **6** & 11 & 21 & 0 & 29 & **126** & 243 & 486\
NGC 6397 & 0 & 0 & **3** & 8 & 16 & 0 & 0 & **72** & 204 & 421\
NGC 6535 & 0 & 0 & **1** & 3 & 5 & 0 & 2 & **24** & 67 & 125\
NGC 6541 & 0 & 3 & **18** & 44 & 76 & 0 & 64 & **429** & 1083 & 1909\
NGC 6584 & 0 & 3 & **11** & 20 & 29 & 0 & 61 & **231** & 445 & 687\
NGC 6624 & 0 & 0 & **0** & 1 & 3 & 0 & 0 & **6** & 20 & 44\
NGC 6637 (M69) & 0 & 25 & **58** & 84 & 123 & 0 & 478 & **1154** & 1732 & 2728\
NGC 6652 & 0 & 1 & **5** & 13 & 22 & 0 & 13 & **107** & 279 & 501\
NGC 6656 (M22) & 0 & 9 & **55** & 111 & 314 & 0 & 162 & **1260** & 2596 & 8137\
NGC 6681 (M70) & 0 & 3 & **13** & 26 & 43 & 0 & 66 & **289** & 603 & 1015\
NGC 6715 (M54) & 0 & 6 & **67** & 179 & 333 & 0 & 127 & **1650** & 4413 & 8277\
NGC 6717 (Pal9) & 0 & 0 & **1** & 3 & 5 & 0 & 1 & **23** & 63 & 120\
NGC 6723 & 2 & 24 & **60** & 92 & 189 & 0 & 491 & **1243** & 2036 & 4528\
NGC 6752 & 0 & 0 & **10** & 27 & 54 & 0 & 7 & **256** & 710 & 1393\
NGC 6779 (M56) & 2 & 24 & **51** & 78 & 103 & 0 & 430 & **1068** & 1714 & 2329\
NGC 6809 (M55) & 6 & 29 & **69** & 157 & 274 & 0 & 464 & **1547** & 3658 & 7029\
NGC 6838 (M71) & 1 & 6 & **17** & 31 & 60 & 0 & 119 & **363** & 687 & 1446\
NGC 6934 & 0 & 3 & **12** & 23 & 35 & 0 & 55 & **233** & 484 & 773\
NGC 6981 (M72) & 5 & 17 & **27** & 44 & 61 & 96 & 334 & **573** & 929 & 1747\
NGC 7078 (M15) & 0 & 2 & **23** & 61 & 112 & 0 & 54 & **571** & 1522 & 2809\
NGC 7089 (M2) & 0 & 3 & **30** & 79 & 146 & 0 & 58 & **722** & 1944 & 3632\
NGC 7099 (M30) & 0 & 0 & **5** & 14 & 28 & 0 & 1 & **130** & 358 & 714\
\[T:bh\_dr\_Baumgardt\]
To obtain final ${{N_{\rm{BH}}}}$ and ${{M_{\rm{BH}}}}$ estimates (not normalized by total cluster mass or star count) we assume an average stellar mass of $0.5\,{{\rm{M_\odot}}}$ and therefore multiply ${{N_{\rm{BH}}}}/{{N_{\rm{cluster}}}}$ (${{M_{\rm{BH}}}}/{{M_{\rm{cluster}}}}$) by twice (once) ${{M_{\rm{cluster}}}}$. We utilize total cluster mass estimates (see columns 4-5 of ) based on scaled-up $N$-body simulations [@Baumgardt2018; @Mandushev1991] as well as values computed from the integrated V-band magnitudes in @Harris1996 [2010 edition], assuming a uniform cluster mass-to-light ($M/L$) ratio of two. While the former mass values (henceforth, ‘Baumgardt/Mandushev’) are not purely observational, introducing modeling uncertainties, they *do* account for variation in the cluster $M/L$ ratio, which can differ significantly from the standard value of two in some GCs (see column 3 of ). Meanwhile, the latter estimates (henceforth, ‘Harris’) are purely observational but do not account for variation in the $M/L$ ratio. Among the 50 GCs analyzed, the Harris mass values are only about 25% higher, on average, than those from Baumgardt/Mandushev. However, the difference exceeds a factor of two for a few clusters.
Given the above trade-off between observational purity versus a variable $M/L$ ratio, we present ${{N_{\rm{BH}}}}$ and ${{M_{\rm{BH}}}}$ predictions based on both mass estimates in and Tables \[T:bh\_dA\_Baumgardt\]-\[T:bh\_dA\_Harris\] of the Appendix. Each table contains the modes, $1\sigma$ and $2\sigma$ CIs on ${{N_{\rm{BH}}}}$ and ${{M_{\rm{BH}}}}$ for each of the 50 GCs, but are based on slightly different measures of mass segregation (${{\Delta_{r50}}}$ versus ${{\Delta_A}}$) and different GC mass estimates (Baumgaurdt/Mandushev versus Harris). For simplicity, since the differences between the four sets of predictions are generally quite minor, the rest of our discussion will focus only on the ${{N_{\rm{BH}}}}$ and ${{M_{\rm{BH}}}}$ predictions in , which is based on ${{\Delta_{r50}}}$ and the Baumgardt/Mandushev cluster masses.
As a visual aid, shows the modes and $1\sigma$ CIs for ${{N_{\rm{BH}}}}$ (top panel) and ${{M_{\rm{BH}}}}$ (bottom panel), using the Baumgardt/Mandushev masses. Excepting the case of NGC 6624 – the most mass-segregated cluster in our sample – the minimal effect of the choice of ${{\Delta_{r50}}}$ (red) versus ${{\Delta_A}}$ (blue) is evident. Furthermore, in the majority of the MWGCs analyzed ($36/50$), observed mass segregation suggests the GC retains a relatively small BH subsystem consisting of fewer than $50$ BHs with a combined mass less than $10^3\,{{\rm{M_\odot}}}$. While the combined evidence from the surveyed clusters overwhelmingly indicates the commonality of modest BH populations in GCs, we cannot even rule out $0$ BHs at 95% confidence in all but 13 of those clusters. However, our survey does pinpoint a few GCs that are likely to host a BH subsystem with ${{N_{\rm{BH}}}}> 80$ (${{M_{\rm{BH}}}}> 1,500\,{{\rm{M_\odot}}}$), specifically NGCs 2808, 5927, 5986, 6101, and 6205. The existence of a BHS in the latter three GCs has previously been suggested by @ArcaSedda2018 – and even earlier by @Peuten2016 in the case of NGC 6101 – while NGCs 2808 and 5927 are both new BHS candidates.
The Role of Black holes in Cluster Core Evolution {#S:corecollapse}
=================================================
As described in , the evolution of a cluster, and specifically the cluster’s core structure, is tied to stellar-mass BH dynamics. When a large number of BHs are retained, the energy generated through BH burning is sufficient to delay the onset of core collapse. As the number of retained BHs decreases, so too does the cluster’s core radius ($r_c$), until ultimately, the core collapses completely. This connection between core structure and ${{N_{\rm{BH}}}}$ has been pointed out by a number of recent studies [e.g., @Mackey2007; @Mackey2008; @Kremer2018a; @Askar2018; @Kremer2019d].
In , we plot $r_c/r_{\rm{hl}}$ [taken from @Harris1996 2010 edition] versus the number (${{N_{\rm{BH}}}}$, left panel) and total mass (${{M_{\rm{BH}}}}$, right panel) of retained BHs for each of the 50 GCs analyzed –– in respective units of the star count (${{N_{\rm{cluster}}}}$) and total cluster mass (${{M_{\rm{cluster}}}}$). We show ${{N_{\rm{BH}}}}/{{N_{\rm{cluster}}}}$ (${{M_{\rm{BH}}}}/{{M_{\rm{cluster}}}}$) values – with $1\sigma$ confidence intervals – predicted using both the ${{\Delta_{r50}}}$ (red) and ${{\Delta_A}}$ (blue) mass segregation parameters. As expected, shows that ${{N_{\rm{BH}}}}/{{N_{\rm{cluster}}}}$ and ${{M_{\rm{BH}}}}/{{M_{\rm{cluster}}}}$ correlate prominently with $r_c/{{r_{\rm{hl}}}}$. Those clusters predicted to harbor the largest BH populations tend to have larger observed values of $r_c/{{r_{\rm{hl}}}}$, further validating the connection between core evolution and BH dynamics suggested in previous works. For additional detail on this point, see especially Figure 3 of @Kremer2019d, which shows how the number (total mass), and cumulative radial distributions of BHs vary with core radius across our models. In general, nearly $100\%$ of BHs retained in our models at late times reside within the cluster’s core radius.
Finally, energy flow in systems with negative heat capacity – like GCs – causes the core to contract to smaller $r_c$ and the halo to expand to larger ${{r_{\rm{hl}}}}$ [e.g., @HeggieHut2003]. So, clusters naturally evolve toward smaller values of $r_c/{{r_{\rm{hl}}}}$, which can therefore be used as a proxy for a cluster’s dynamical age. Thus, among GCs with similar total masses, those with fewer BHs retained and smaller cores at present are dynamically older (i.e., have evolved for more relaxation times) compared to GCs with more retained BHs and larger cores. In principle, this relation may be leveraged to place additional constraints upon BH populations, for example, in coordination with other tracers of cluster dynamical ages [e.g., blue stragglers; @Ferraro2012]. We reserve a detailed examination of the connection between cluster age and BH dynamics for a later study.
Comparison with Previous Results {#S:comparison}
================================
Our primary finding is that many MWGCs contain non-negligible BH populations at present. However, the number and total mass of BHs in these populations are less than predicted in previous analyses (with some exceptions). We here discuss our predictions in relation to those previous findings, both from models and XRB observations. We especially examine the discrepancy between our results and those of [@Askar2018], currently the only other set of ${{N_{\rm{BH}}}}$ and ${{M_{\rm{BH}}}}$ predictions across multiple GCs.
Before comparing with results from other groups, however, it is first important to check for consistency between our new, fully developed ${{N_{\rm{BH}}}}$ predictions and our trial predictions presented in W1 for the MWGCs 47 Tuc, M10 and M22. As discussed in the preceding sections, the three primary differences between the old and new methods are the choice of populations used to quantify mass segregation, the details of the KDE formulation, and the estimated masses of observed GCs. Looking only at ${{\Delta_{r50}}}$ and re-scaling the old results using the new GC masses (), the new (old) ${{N_{\rm{BH}}}}$ predictions for these respective clusters are: $43^{+64}_{-37}$ ($21^{+57}_{-17}$), $30^{+19}_{-18}$ ($44^{+26}_{-22}$), and $55^{+56}_{-46}$ ($70^{+72}_{-48}$). The shifts (up for 47 Tuc, down for M10 and M22) are well within the $1\sigma$ uncertainty of the predictions. As expected, the new methodology yields results consistent with W1.
Comparison to MOCCA {#S:MOCCA}
-------------------
Concurrently with the publication of W1, the creators of the *MOCCA-Survey Database I* – another large set of Monte Carlo cluster models similar to those produced by `CMC` – developed and continue to use an alternate probe of the BH content in GCs [@ArcaSedda2018; @Askar2018; @ArcaSedda2019]. Both their methodology (discussed below) and results are quite different from our mass segregation approach. As evidenced in , we predict lower ${{N_{\rm{BH}}}}$ in 16 of the 18 GCs surveyed that the MOCCA team shortlisted as likely BHS hosts, with significantly lower ${{N_{\rm{BH}}}}$ (non-overlapping $1\sigma$ CIs) in 9-10 such cases (depending on choice of $\Delta$).
In particularly striking examples, we rule out more than $70$ BHs (${{M_{\rm{BH}}}}>1800$) to 95% confidence in NGCs 0288 and 5466, both of which were predicted to have over $170$ BHs (${{M_{\rm{BH}}}}>2000$) in the MOCCA survey. Finally, the MOCCA survey’s estimates generally have significantly higher uncertainties, on average. Given these discrepancies, it is essential to more deeply examine the methodology behind the MOCCA results and the potential benefits or drawbacks of their methods relative to our own.
First, [@ArcaSedda2018] find a set of scaling relations between key properties of the BH subsystem that mass-segregates to the core of a GC. Specifically, they define $R_{\rm{BHS}}$ as the cluster-centric distance within which half of the total mass is in BHs (the other half of the mass is contained in stars). BHs within distance $R_{\rm{BHS}}$ from the cluster center count as members of the subsystem, which then typically contains around 60% of the total number of BHs in the models. The authors correlate $R_{\rm{BHS}}$ with number ($N_{\rm{BHS}}$) and total mass ($M_{\rm{BHS}}$) of BHs in the subsystem, and anti-correlate these three quantities with the associated BH mass density $\rho_{\rm{BHS}} = M_{\rm{BHS}}/R_{\rm{BHS}}^3$. Finally, they establish a tight model correlation between $\rho_{\rm{BHS}}$ and GC average surface luminosity $L/r_{\rm{hl,obs}}^2$, which they apply in a companion paper [@Askar2018] to short-list 29 MWGCs with sizable BH subsystems, using observed V-band magnitudes and half-light radii from the Harris catalog [@Harris1996 2010 edition]. Currently, the authors are utilizing a very similar method to identify MWGCs that potentially host an IMBH [@ArcaSedda2019].
Applying the above definitions to our own model set results in similar correlations, but a closer examination of the method reveals several issues. The most critical concern is statistical. Whereas we use a *non-parametric* KDE to directly relate our observables ($\Delta^{23}$, $\Delta^{34}$) to ${{N_{\rm{BH}}}}/{{N_{\rm{cluster}}}}$ (or ${{M_{\rm{BH}}}}/{{M_{\rm{cluster}}}}$), [@Askar2018] indirectly chain 5 separate correlations together, each with their own assumed parametric form, to relate their observable ($L_V/{{r_{\rm{hl}}}}^2$) to ${{N_{\rm{BH}}}}$. Specifically, linear curve-fits in log-log scale are applied in each of the 5 steps along the following chain: $L_V/{{r_{\rm{hl}}}}^2$ to $\rho_{\rm{BHS}}$ to $R_{\rm{BHS}}$ to $M_{\rm{BHS}}$ to $N_{\rm{BHS}}$ to ${{N_{\rm{BH}}}}$ (i.e. $N_{\rm{BH-ALL}}$, the total number of BHs in the cluster). The latter four of these power-law relations are shown in (a-d) for both our own model set and the MOCCA data. Chaining curve fits like this, especially in log-log scale, amplifies any deviations from a perfect trend, boosting uncertainty and often obscuring any potential biases. This can easily be seen by plotting the observable directly against the final predicted value. In our case, we skip the first step in the chain ($L_V/{{r_{\rm{hl}}}}^2$ to $\rho_{\rm{BHS}}$) and simply plot $\rho_{\rm{BHS}}$ vs ${{N_{\rm{BH}}}}$ in the bottom panel of , bypassing the intermediate variables $R_{\rm{BHS}}$, $M_{\rm{BHS}}$, and $N_{\rm{BHS}}$. When plotted directly like this, it is evident that there is only a very weak anti-correlation between $\rho_{\rm{BHS}}$ and ${{N_{\rm{BH}}}}$ (black curve). After propagating error on MOCCA’s own chained curve fits (which results in spurious correlation; gray curve) the $1\sigma$ confidence interval on predicted ${{N_{\rm{BH}}}}$ spans nearly 2 orders of magnitude for any given value of $\rho_{\rm{BHS}}$. This largely explains why the MOCCA ${{N_{\rm{BH}}}}$ and ${{M_{\rm{BH}}}}$ predictions are more uncertain, on average.
In addition, as mentioned in Section 2.6 of [@Askar2018], models without ${{N_{\rm{BH}}}}> 15$ at $12\,{{\rm{Gyr}}}$ were excluded from their analysis. Of these excluded models, most incorporated high BH natal kicks, but $\sim 40$ utilized the standard mass fallback prescription [@Belczynski2002] and had *similar values of the observable* ($L_V/{{r_{\rm{hl}}}}^2$) to the $163$ included models, despite having significantly *fewer* BHs (${{N_{\rm{BH}}}}< 15$). This fact indicates that $L_V/{{r_{\rm{hl}}}}^2$ is not actually a strong predictor of the BH content in GCs, supporting our findings in the bottom panel of . At the very least, excluding the 20% of models with lowest ${{N_{\rm{BH}}}}$ would naturally cause the MOCCA team to over-predict ${{N_{\rm{BH}}}}$ and ${{M_{\rm{BH}}}}$, partially explaining why our analysis generally yields significantly lower predictions.
Ultimately, we find that quantities like $\Delta$ that parameterize mass segregation are a more reliable predictor of the total mass and number of BHs inside a GC than the $L_V/{{r_{\rm{hl}}}}^2$-BHS correlations used in the MOCCA survey [@ArcaSedda2018; @Askar2018; @ArcaSedda2019]. In addition to providing narrower constraints on the BH content in GCs than in any prior studies, our analysis suggests while some BH retention is common to many GCs, fewer are retained – generally less than $50$ – than has typically been suggested previously. In addition to this general point, we discuss findings of particular interest for specific MWGCs in the following subsections.
47 Tuc {#S:47Tuc}
------
As one of the nearest and therefore most well-studied GCs, 47 Tuc (NGC 0104) is an important cluster for benchmarking. The cluster’s mass of around $10^6\,{{\rm{M_\odot}}}$ () is near the maximum of our model space at $13\,{{\rm{Gyr}}}$, but its Galactocentric distance and metallicity are well within the model bounds [@Harris1996 2010 edition]. In W1, our ${{N_{\rm{BH}}}}$ predictions for 47 Tuc were limited by a dearth of models with high mass segregation. Now, without such a limitation, we predict the cluster retains more BHs, around 40 totalling 900 ${{\rm{M_\odot}}}$. This is in line with a contemporary effort to model 47 Tuc specifically that predicts a relatively small BHS in the cluster [@Henault-Brunet2019]. However, at 95% confidence, we can neither exclude zero BHs nor a large population of up to $\sim200$ BHs totalling 4,300 ${{\rm{M_\odot}}}$.
NGC 3201 {#S:NGC3201}
--------
Recently, @Giesers2018 reported a stellar-mass BH in the cluster NGC 3201. They inferred the BH’s presence from the large radial velocity variations ($\sim 100$ km/s) of an apparently lone main sequence star, thereby presumed to be orbiting a compact remnant. This detection – along with two more recent ones [@Giesers2019] – made NGC 3201 the fifth MWGC known to host a stellar-mass BH. Shortly thereafter, [@Kremer2018b] used `CMC` to model the cluster, reporting that it likely retains $>200$ stellar-mass BHs at present, an estimate that was revised down to ${{N_{\rm{BH}}}}= 120 \pm 10$ in a follow-up using updated BH formation physics [@Kremer2019a]. This revised prediction is in line with the MOCCA team’s estimate, ${{N_{\rm{BH}}}}= 114^{+60}_{-35}$ [@Askar2018], but mass segregation predicts an even lower number: ${{N_{\rm{BH}}}}= 41^{+40}_{-34}$.
NGC 6101 {#S:NGC6101}
--------
Of the 50 MWGCs surveyed, NGC 6101 is the least mass-segregated and is by far the best candidate in which to find a large number of BHs. To 95% confidence, we estimate it contains $75-236$ BHs with a combined mass of $1,750-5,900\,{{\rm{M_\odot}}}$. Most likely, it contains $\sim 125$ BHs totalling $\sim 3,000\,{{\rm{M_\odot}}}$. This conclusion is supported by a growing body of evidence from other sources. [@Dalessandro2015] were the first to draw attention to this GC’s unusually low mass segregation, finding no evidence for the phenomenon based on three different measures: the radial distribution of blue stragglers, that of MS binaries, and the luminosity function. Following this finding, [@Peuten2016] and [@Webb2017] explored the anti-correlation between ${{N_{\rm{BH}}}}$ and mass segregation in $N$-body simulations to demonstrate that the cluster may contain a large population of BHs. [@BaumgardtSollima2017] disputed these suggestions because their estimates of NGC 6101’s mass-function slope indicated mass segregation after all. However, given that this rebuttal relies on the same ACS dataset as applied in this study and because their results similarly suggest that NGC 6101 has one of the lowest levels of mass segregation among MWGCs, we find no contradiction to our conclusions; NGC 6101 is very likely to host a robust population of stellar-mass BHs. This determination is further supported by the findings of [@Askar2018] discussed above.
While NGC 6101’s low mass segregation is likely due to the presence of a BHS in its core, we caution that NGC 6101 is not exactly a run-of-the-mill MWGC. The cluster is among the oldest and most metal-poor in the Galaxy, with an age of 13+ ${{\rm{Gyr}}}$ [@Dotter2010] and a metallicity ${{\rm{Z}}}\leq 0.01\,{{\rm{Z}}}_{\odot}$ [@Harris1996 2010 edition]. Generally, these factors would suggest a smaller BH population, but NGC 6101 also orbits the Galactic center on an unusual retrograde motion that may reflect a prior association with the Canis Major dwarf galaxy [@Martin2004]. However, since Martin et al. propose a similar association for NGCs 1851, 2298, 3201, and 6934, all of which we find to be significantly more mass-segregated, it seems unlikely that this abnormal origin could explain NGC 6101’s low mass segregation.
NGC 6535 {#S:NGC6535}
--------
NGC 6535 is unusual in that it’s relatively old but has a high mass-to-light ratio in the range 5 [@Baumgardt2018] to 11 [@Zaritsky2014]. [@Halford2015] found that its observed mass-function has a positive slope – indicating a high loss-rate of low-mass stars and making its high $M/L$ ratio even more puzzling. Given NGC 6535’s small Galactocentric distance of $3.9$ kpc [@Harris1996 2010 edition], it is likely that increased tidal stripping of low-mass stars near the Galactic center is responsible for the positive mass-function slope. However, [@Halford2015] found no evidence that clusters near the Galactic center with similarly top-heavy mass functions had artificially inflated mass estimates, raising the possibility that some dark mass may be responsible for NGC 6535’s high $M/L$ ratio. Recently, [@Askar2017b] demonstrated that $N$-body simulations of clusters containing an IMBH or BHS were able to fit the photometric and kinematic properties of NGC 6535, but later concluded the cluster contains neither a significant BHS nor an IMBH [@Askar2018; @ArcaSedda2019]. Since we rule out more than $130\,{{\rm{M_\odot}}}$ of BHs in NGC 6535 to 95% confidence, the mystery of the apparently missing mass in this cluster remains an open question.
NGC 6624 {#S:NGC6624}
--------
[@Perera2017] reported the possible presence of an IMBH in NGC 6624 based on timing observations of a millisecond pulsar near the projected cluster center. Their timing analysis indicated the presence of an IMBH with mass in the range 7,500 to 10,000 ${{\rm{M_\odot}}}$, even up to 60,000 ${{\rm{M_\odot}}}$. This finding was disputed by [@Gieles2018], who demonstrated that dynamical models without an IMBH produce maximum accelerations at the pulsar’s position comparable to its observed line-of-sight acceleration. Recently, [@Baumgardt2019] similarly found that their $N$-body models without an IMBH could provide excellent fits to the observed velocity dispersion and surface brightness profiles (VDPs and SDPs) in NGC 6624. Their cluster models with an IMBH indicated that an IMBH in NGC 6624 with mass $>$1,000 ${{\rm{M_\odot}}}$ was incompatible with the cluster’s observed VDP and SBP. Meanwhile, based on data from HST and ATCA, [@Tremou2018] found that all radio emissions observed from NGC 6624 are consistent with being from a known ultra-compact X-ray binary in the cluster’s core. Their radio observations place a $3\sigma$ upper limit on the cluster’s possible IMBH mass of 1,550 ${{\rm{M_\odot}}}$. Although we have yet to explore how much difference an IMBH has on quenching $\Delta$ compared to a BHS, our results support the latter three studies; we find to 95% confidence that there are no more than $\sim400\,{{\rm{M_\odot}}}$ of BHs in NGC 6624 (using Baumgardt’s cluster mass, otherwise $<\sim900\,{{\rm{M_\odot}}}$ of BHs using Harris’ cluster mass). Indeed, NGC 6624 is the most mass-segregated cluster in our sample, suggesting that it may in fact be one of the MWGCs *least* likely to host an IMBH or significant BHS.
M54 {#S:M54}
---
Thought to be a MWGC for over two centuries, the cluster M54 (NGC 6715) is now known to be coincident with the center of the Sagittarius Dwarf Galaxy [e.g., @Monaco2005], perhaps even as the galaxy’s original nucleus [@Layden2000]. While M54’s metallicity is well-covered by our model parameter space, its effective Galactocentric distance is unreliable because our models assume a MW-like potential for tidal boundary calculations. Its approximate mass is also at the extreme upper end of the model space (). Therefore, with some reservation, despite M54’s highly mass-segregated present state, we predict a significant number of BHs remain in the cluster at present, with $67^{+112}_{-61}$ BHs totalling around $1,650^{+2,763}_{-1,523}$ ${{\rm{M_\odot}}}$. This prediction is consistent with the $3\sigma$ upper limit on a single accreting IMBH of $<3,000\,{{\rm{M_\odot}}}$ imposed by VLA radio observations [@Tremou2018].
Summary & Discussion {#S:summary}
====================
Summary
-------
In this paper, we have demonstrated that mass segregation is a truly robust indicator of the BH content in GCs. We briefly summarize our key findings below.
1. We demonstrated that, overall, the `CMC` Cluster Catalog models yield mass segregation ($\Delta$) values which closely match the observed distribution in $\Delta$ among real MWGCs (see ). This provides strong evidence that our models capture the state of mass segregation in realistic MWGCs, complementing the results of @Kremer2019d.
2. By using $\Delta$ as a predictive parameter, we have constrained the total number and mass in stellar-mass BHs contained in 50 MWGCs more tightly than any prior studies.
3. We find that 35 of the 50 GCs studied retain more than 20 BHs at present and 8 retain more than 80 BHs. These predictions indicate that present-day BH retention is common to many MWGCs, though to a lesser extent than suggested in competing analyses, [e.g., @Askar2018].
4. Specifically, we have identified NGCs 2808, 5927, 5986, 6101, and 6205 to contain especially large BH populations, each with total BH mass exceeding $10^3\,{{\rm{M_\odot}}}$. These clusters may serve as ideal observational targets for BH candidate searches.
5. We also explored in detail the advantages and disadvantages of our statistical methods compared to other similar analyses in the literature.
Discussion and Future Work
--------------------------
Here, we predict smaller BH populations in a few GCs compared to our previous analyses which also utilized `CMC` models [e.g., @Kremer2019a]. The exact number of BHs is highly uncertain (indeed, this is reflected by the uncertainty bars in and all the tables). Hence, discrepancy between these results and those of our previous work – which implemented entirely different methods based on fitting surface brightness and velocity dispersion profiles to predict ${{N_{\rm{BH}}}}$ – is unsurprising. Critically, as shown in , the overall connection between cluster core evolution and BH dynamics put forward in previous work [@Mackey2008; @Kremer2018b; @Kremer2019a] is confirmed. This further validates the significant role BHs play in GC evolution.
There are a couple of more speculative conclusions hinted at by our results which are worth mentioning briefly, but require additional study. First, it is tempting to extrapolate our predictions of total BH mass in GCs to place upper limits on the masses of possible intermediate-mass black holes (IMBHs) in those clusters. Indeed, $N$-body simulations have shown that an IMBH of mass $>1\%$ of its host GC’s overall mass should significantly quench mass segregation – even among only visible giants and MS stars [e.g., @Gill2008; @Pasquato2016]. The generally significant mass segregation we measure in the 50 GCs studied – representative of the MW as a whole – therefore suggests that IMBHs with mass $>$1,000 ${{\rm{M_\odot}}}$ are rare in MW clusters. However, firmer constraints would require testing beyond the scope of this study, specifically on how similar the dynamical impact of a single IMBH is to that of a stellar-mass BH population with identical total mass. Is it a one-to-one relation, or does a, for example, $1,000\,{{\rm{M_\odot}}}$ IMBH perhaps have a much weaker effect on mass segregation than a population of a hundred $10\,{{\rm{M_\odot}}}$ BHs? For now, the prospect of IMBHs in GCs is still best analyzed through direct observations in the X-ray and radio bands, as well as via the accelerations of luminous stars within the IMBH’s ‘influence radius,’ but further study may be able to extend our constraints on stellar-mass BH populations to IMBHs in GCs.
Second, it has been suggested that clusters were born already mass-segregated to a degree, a property termed ‘primordial’ mass segregation [e.g., @Baumgardt2008]. Our models assume clusters have no primordial mass segregation. Hence, the close match between $\Delta$ in our models and the $\Delta$ distribution observed in the MWGCs (see ) demonstrates that our models do not need to start off with some degree of mass segregation to match real clusters. This finding could suggest that primordial mass segregation is minimal or non-existent in the MWGCs, but such a conclusion is tenuous since primordial mass segregation is likely to be washed out at the present day after many relaxation times. Further consideration of the late-time effects of primordial mass segregation on presently observable $\Delta$ is necessary to make any further conclusions on this matter.
Finally, although mass segregation has been shown here to be a strong indicator of BH populations in clusters, recent analyses have shown that many other observables, including millisecond pulsars [@Ye2019], blue stragglers [@Kremer2019d], and cluster surface brightness and velocity dispersion profiles [e.g., @Mackey2008; @Kremer2018b], may also correlate with BH dynamics and thus may also serve as indicators of retained BH numbers. In order to pin down more precisely the true number of BHs retained in specific clusters, all of these observables should be leveraged in tandem. We intend to pursue such analysis further in future works.
We thank Mario Spera for detailed comments on the manuscript and Claire Ye, Nicholas Rui, and Giacomo Fragione for useful discussions throughout the preparation of this work. This work was supported by NSF Grant AST-1716762 and through the computational resources and staff contributions provided for the [Quest]{} high performance computing facility at Northwestern University. [Quest]{} is jointly supported by the Office of the Provost, the Office for Research, and Northwestern University Information Technology. K.K. acknowledges support by the National Science Foundation Graduate Research Fellowship Program under Grant No. DGE-1324585. S.C. acknowledges support from NASA through Chandra Award Number TM5-16004X issued by the Chandra X-ray Observatory Center (operated by the Smithsonian Astrophysical Observatory for and on behalf of NASA under contract NAS8-03060).
[lccccc|rrrrr|rrrrr]{}\[b!\] NGC 0104 (47Tuc) & 0.55 & 1.77 & 779 & 1000 & $0.052 \pm 0.006$ & 0 & 0.41 & **2.67** & 6.8 & 11.9 & 0 & 16 & **121** & 313 & 558\
NGC 0288 & 0.77 & 2.39 & 116 & 87 & $0.008 \pm 0.001$ & 2.24 & 9.93 & **18.2** & 26.6 & 46.9 & 44 & 408 & **797** & 1202 & 2219\
NGC 1261 & 2.45 & 2.12 & 167 & 225 & $0.020 \pm 0.010$ & 1.25 & 6.02 & **11.6** & 18.1 & 23.6 & 40 & 253 & **507** & 764 & 1004\
NGC 1851 & 3.48 & 2.02 & 302 & 367 & $0.093 \pm 0.032$ & 0 & 0.52 & **3.11** & 7.61 & 13.2 & 0 & 19 & **134** & 336 & 590\
NGC 2298 & 1.70 & 0.46 & 12 & 57 & $0.022 \pm 0.003$ & 0 & 1.63 & **5.41** & 9.68 & 14.5 & 0 & 53 & **215** & 432 & 696\
NGC 2808 & 2.25 & 1.64 & 742 & 975 & $0.067 \pm 0.009$ & 0 & 1.3 & **4.74** & 8.87 & 13.4 & 0 & 39 & **169** & 358 & 578\
NGC 3201 & 0.57 & 2.4 & 149 & 163 & $0.012 \pm 0.001$ & 0 & 1.8 & **11.3** & 22.1 & 54.9 & 0 & 53 & **494** & 1010 & 2810\
NGC 4147 & 3.48 & 1.51 & 33 & 50 & $0.043 \pm 0.009$ & 0 & 0.44 & **3.09** & 7.88 & 14.0 & 0 & 20 & **151** & 391 & 698\
NGC 4590 (M68) & 1.15 & 2.02 & 123 & 152 & $0.014 \pm 0.002$ & 0.12 & 3.49 & **7.97** & 12.5 & 16.8 & 0 & 119 & **329** & 547 & 772\
NGC 4833 & 0.73 & 0.84 & 247 & 317 & $0.011 \pm 0.001$ & 0 & 4.47 & **11.9** & 19.7 & 38.5 & 0 & 165 & **515** & 888 & 1864\
NGC 5024 (M53) & 1.29 & 1.59 & 380 & 521 & $0.034 \pm 0.006$ & 0 & 1.4 & **5.12** & 9.72 & 15.0 & 0 & 46 & **205** & 439 & 714\
NGC 5053 & 0.68 & 1.66 & 57 & 87 & $0.005 \pm 0.001$ & 7.3 & 12.6 & **22.4** & 52.0 & 83.6 & 291 & 497 & **1037** & 2546 & 4276\
NGC 5272 (M3) & 0.77 & 1.56 & 394 & 610 & $0.036 \pm 0.006$ & 0 & 0.42 & **3.12** & 8.01 & 14.3 & 0 & 17 & **148** & 392 & 716\
NGC 5286 & 2.25 & 1.41 & 401 & 536 & $0.099 \pm 0.012$ & 0 & 0.02 & **2.01** & 5.5 & 10.9 & 0 & 1 & **100** & 275 & 548\
NGC 5466 & 0.77 & 1.13 & 46 & 106 & $0.004 \pm 0.001$ & 9.64 & 15.0 & **41.7** & 54.1 & 83.2 & 400 & 610 & **1115** & 2664 & 4261\
NGC 5904 (M5) & 1.00 & 1.52 & 372 & 572 & $0.033 \pm 0.005$ & 0 & 1.27 & **4.77** & 9.28 & 14.5 & 0 & 43 & **183** & 387 & 626\
NGC 5927 & 1.62 & 2.61 & 354 & 228 & $0.016 \pm 0.006$ & 2.62 & 9.62 & **16.4** & 22.4 & 32.9 & 78 & 375 & **662** & 943 & 1447\
NGC 5986 & 1.70 & 2.45 & 301 & 406 & $0.019 \pm 0.011$ & 0.96 & 7.67 & **14.2** & 20.9 & 29.6 & 4 & 310 & **624** & 934 & 1314\
NGC 6093 (M80) & 2.89 & 1.43 & 249 & 335 & $0.111 \pm 0.012$ & 0 & 0.27 & **2.59** & 6.82 & 12.5 & 0 & 11 & **124** & 333 & 618\
NGC 6101 & 1.62 & 3.0 & 127 & 102 & $0.003 \pm 0.002$ & 35.8 & 44.6 & **50.9** & 76.6 & 94.7 & 1665 & 2211 & **2518** & 4043 & 4693\
NGC 6144 & 1.00 & 0.54 & 45 & 94 & $0.011 \pm 0.002$ & 1.28 & 8.01 & **13.4** & 18.9 & 25.9 & 30 & 357 & **632** & 909 & 1253\
NGC 6171 (M107) & 1.00 & 2.16 & 87 & 121 & $0.015 \pm 0.002$ & 1.18 & 7.22 & **17.4** & 27.2 & 45.7 & 13 & 283 & **721** & 1195 & 2106\
NGC 6205 (M13) & 1.00 & 2.61 & 453 & 450 & $0.013 \pm 0.004$ & 0 & 6.59 & **13.2** & 19.9 & 32.7 & 0 & 259 & **577** & 901 & 1553\
NGC 6218 (M12) & 0.99 & 1.27 & 87 & 144 & $0.011 \pm 0.002$ & 0 & 6.58 & **12.7** & 20.0 & 35.6 & 0 & 286 & **583** & 912 & 1523\
NGC 6254 (M10) & 0.89 & 1.94 & 184 & 168 & $0.018 \pm 0.002$ & 0 & 3.01 & **7.67** & 12.6 & 17.7 & 0 & 107 & **324** & 559 & 827\
NGC 6304 & 1.29 & 1.37 & 277 & 142 & $0.050 \pm 0.015$ & 0 & 3.23 & **12.3** & 17.3 & 24.1 & 0 & 83 & **254** & 422 & 591\
NGC 6341 (M92) & 1.70 & 1.81 & 268 & 329 & $0.066 \pm 0.015$ & 0 & 0.52 & **3.18** & 7.84 & 13.7 & 0 & 20 & **149** & 379 & 681\
NGC 6352 & 0.85 & 2.47 & 94 & 66 & $0.017 \pm 0.002$ & 0 & 2.91 & **7.88** & 13.7 & 21.1 & 0 & 102 & **317** & 563 & 924\
NGC 6366 & 0.57 & 2.34 & 47 & 34 & $0.015 \pm 0.001$ & 0 & 1.37 & **5.07** & 9.38 & 16.7 & 0 & 39 & **190** & 393 & 785\
NGC 6397 & 0.61 & 2.18 & 89 & 78 & $0.059 \pm 0.003$ & 0 & 0 & **0.61** & 1.8 & 4.06 & 0 & 0 & **34** & 100 & 226\
NGC 6535 & 1.99 & 4.8 & 20 & 14 & $0.037 \pm 0.006$ & 0 & 2.45 & **7.11** & 11.3 & 15.4 & 0 & 72 & **264** & 478 & 713\
NGC 6541 & 1.62 & 1.42 & 277 & 438 & $0.068 \pm 0.013$ & 0 & 0.5 & **3.11** & 7.71 & 13.5 & 0 & 21 & **149** & 380 & 677\
NGC 6584 & 2.45 & 1.12 & 91 & 204 & $0.025 \pm 0.011$ & 0 & 0.94 & **4.12** & 8.8 & 14.2 & 0 & 33 & **174** & 407 & 690\
NGC 6624 & 1.99 & 1.02 & 73 & 169 & $0.125 \pm 0.038$ & 0.7 & 19.6 & **23.2** & 26.8 & 31.1 & 73 & 193 & **305** & 416 & 525\
NGC 6637 (M69) & 1.99 & - & 200\* & 195 & $0.046 \pm 0.017$ & 0.3 & 5.39 & **11.8** & 17.3 & 22.7 & 0 & 205 & **472** & 705 & 951\
NGC 6652 & 3.48 & - & 96\* & 79 & $0.077 \pm 0.023$ & 0 & 0.51 & **2.75** & 6.68 & 11.3 & 0 & 17 & **115** & 292 & 511\
NGC 6656 (M22) & 0.52 & 2.15 & 416 & 430 & $0.018 \pm 0.001$ & 0 & 1.02 & **7.11** & 14.5 & 40.9 & 0 & 29 & **324** & 675 & 2104\
NGC 6681 (M70) & 2.45 & 2.0 & 113 & 121 & $0.075 \pm 0.016$ & 0 & 1.21 & **5.02** & 10.1 & 16.3 & 0 & 47 & **216** & 466 & 770\
NGC 6715 (M54) & 2.25 & 2.04 & 1410 & 1680 & $0.074 \pm 0.007$ & 0 & 0 & **1.44** & 3.95 & 8.36 & 0 & 0 & **71** & 198 & 420\
NGC 6717 (Pal9) & 2.45 & - & 22\* & 31 & $0.047 \pm 0.011$ & 0 & 0.15 & **2.13** & 5.67 & 10.7 & 0 & 6 & **104** & 277 & 528\
NGC 6723 & 1.15 & 1.77 & 157 & 232 & $0.010 \pm 0.003$ & 1.69 & 9.44 & **19.9** & 29.4 & 59.5 & 34 & 382 & **828** & 1299 & 2853\
NGC 6752 & 0.91 & 2.17 & 239 & 211 & $0.054 \pm 0.007$ & 0 & 0.02 & **1.96** & 5.35 & 10.6 & 0 & 1 & **100** & 277 & 548\
NGC 6779 (M56) & 1.62 & 1.58 & 281 & 157 & $0.019 \pm 0.004$ & 0.65 & 4.81 & **9.8** & 14.7 & 19.4 & 5 & 176 & **411** & 646 & 865\
NGC 6809 (M55) & 0.61 & 2.38 & 188 & 182 & $0.009 \pm 0.001$ & 1.9 & 8.51 & **18.6** & 43.5 & 73.8 & 16 & 284 & **837** & 2057 & 3786\
NGC 6838 (M71) & 1.00 & 2.76 & 49 & 30 & $0.008 \pm 0.002$ & 9.36 & 16.4 & **23.0** & 44.7 & 69.7 & 365 & 664 & **990** & 2038 & 3315\
NGC 6934 & 2.45 & 1.76 & 117 & 163 & $0.052 \pm 0.016$ & 0 & 1.38 & **5.18** & 10.1 & 15.7 & 0 & 54 & **214** & 433 & 681\
NGC 6981 (M72) & 1.70 & - & 63\* & 112 & $0.003 \pm 0.003$ & 6.02 & 13.1 & **22.3** & 31.4 & 45.5 & 235 & 547 & **957** & 1388 & 2103\
NGC 7078 (M15) & 1.70 & 1.15 & 453 & 811 & $0.102 \pm 0.006$ & 0 & 0 & **1.82** & 4.98 & 10.0 & 0 & 0 & **92** & 252 & 507\
NGC 7089 (M2) & 1.70 & 1.62 & 582 & 700 & $0.101 \pm 0.008$ & 0 & 0 & **1.92** & 5.29 & 10.6 & 0 & 0 & **89** & 248 & 502\
NGC 7099 (M30) & 1.70 & 1.85 & 133 & 163 & $0.067 \pm 0.011$ & 0 & 0 & **1.67** & 4.6 & 9.39 & 0 & 0 & **84** & 235 & 481\
\[T:raw\_results\_dA\]
[l|rrrrr|rrrrr]{} NGC 0104 (47Tuc) & 0 & 6 & **42** & 106 & 185 & 0 & 125 & **943** & 2438 & 4347\
NGC 0288 & 5 & 23 & **42** & 62 & 109 & 51 & 473 & **925** & 1394 & 2574\
NGC 1261 & 4 & 20 & **39** & 60 & 79 & 67 & 423 & **847** & 1276 & 1677\
NGC 1851 & 0 & 3 & **19** & 46 & 80 & 0 & 57 & **405** & 1015 & 1782\
NGC 2298 & 0 & 0 & **1** & 2 & 3 & 0 & 6 & **25** & 50 & 81\
NGC 2808 & 0 & 19 & **70** & 132 & 199 & 0 & 289 & **1254** & 2656 & 4289\
NGC 3201 & 0 & 5 & **34** & 66 & 164 & 0 & 79 & **736** & 1505 & 4187\
NGC 4147 & 0 & 0 & **2** & 5 & 9 & 0 & 7 & **50** & 129 & 230\
NGC 4590 (M68) & 0 & 9 & **20** & 31 & 41 & 0 & 146 & **405** & 673 & 950\
NGC 4833 & 0 & 22 & **59** & 97 & 190 & 0 & 408 & **1272** & 2193 & 4604\
NGC 5024 (M53) & 0 & 11 & **39** & 74 & 114 & 0 & 175 & **779** & 1668 & 2713\
NGC 5053 & 8 & 14 & **25** & 59 & 95 & 165 & 281 & **587** & 1441 & 2420\
NGC 5272 (M3) & 0 & 3 & **25** & 63 & 113 & 0 & 67 & **583** & 1544 & 2821\
NGC 5286 & 0 & 0 & **16** & 44 & 87 & 0 & 4 & **401** & 1103 & 2197\
NGC 5466 & 9 & 14 & **38** & 49 & 76 & 182 & 278 & **508** & 1215 & 1943\
NGC 5904 (M5) & 0 & 9 & **35** & 69 & 108 & 0 & 160 & **681** & 1440 & 2329\
NGC 5927 & 19 & 68 & **116** & 159 & 233 & 276 & 1328 & **2343** & 3338 & 5122\
NGC 5986 & 6 & 46 & **85** & 126 & 178 & 12 & 933 & **1878** & 2811 & 3955\
NGC 6093 (M80) & 0 & 1 & **13** & 34 & 62 & 0 & 27 & **309** & 829 & 1539\
NGC 6101 & 91 & 113 & **129** & 195 & 241 & 2115 & 2808 & **3198** & 5135 & 5960\
NGC 6144 & 1 & 7 & **12** & 17 & 23 & 14 & 162 & **286** & 412 & 568\
NGC 6171 (M107) & 2 & 13 & **30** & 47 & 80 & 11 & 246 & **627** & 1040 & 1832\
NGC 6205 (M13) & 0 & 60 & **120** & 180 & 296 & 0 & 1173 & **2614** & 4082 & 7035\
NGC 6218 (M12) & 0 & 11 & **22** & 35 & 62 & 0 & 247 & **504** & 789 & 1317\
NGC 6254 (M10) & 0 & 11 & **28** & 46 & 65 & 0 & 197 & **596** & 1029 & 1522\
NGC 6304 & 0 & 18 & **68** & 96 & 134 & 0 & 230 & **704** & 1169 & 1637\
NGC 6341 (M92) & 0 & 3 & **17** & 42 & 73 & 0 & 54 & **399** & 1016 & 1825\
NGC 6352 & 0 & 5 & **15** & 26 & 40 & 0 & 96 & **297** & 528 & 867\
NGC 6366 & 0 & 1 & **5** & 9 & 16 & 0 & 18 & **90** & 186 & 371\
NGC 6397 & 0 & 0 & **1** & 3 & 7 & 0 & 0 & **30** & 89 & 201\
NGC 6535 & 0 & 1 & **3** & 5 & 6 & 0 & 14 & **53** & 96 & 143\
NGC 6541 & 0 & 3 & **17** & 43 & 75 & 0 & 58 & **413** & 1053 & 1875\
NGC 6584 & 0 & 2 & **7** & 16 & 26 & 0 & 30 & **158** & 369 & 626\
NGC 6624 & 1 & 29 & **34** & 39 & 45 & 53 & 141 & **223** & 304 & 384\
NGC 6637 (M69) & 1 & 22 & **47** & 69 & 91 & 0 & 410 & **944** & 1410 & 1902\
NGC 6652 & 0 & 1 & **5** & 13 & 22 & 0 & 16 & **110** & 279 & 488\
NGC 6656 (M22) & 0 & 8 & **59** & 121 & 340 & 0 & 121 & **1348** & 2808 & 8753\
NGC 6681 (M70) & 0 & 3 & **11** & 23 & 37 & 0 & 53 & **244** & 527 & 870\
NGC 6715 (M54) & 0 & 0 & **41** & 111 & 236 & 0 & 0 & **1001** & 2792 & 5922\
NGC 6717 (Pal9) & 0 & 0 & **1** & 2 & 5 & 0 & 1 & **23** & 61 & 116\
NGC 6723 & 5 & 30 & **62** & 92 & 187 & 53 & 600 & **1300** & 2039 & 4479\
NGC 6752 & 0 & 0 & **9** & 26 & 51 & 0 & 2 & **239** & 662 & 1310\
NGC 6779 (M56) & 4 & 27 & **55** & 83 & 109 & 14 & 495 & **1155** & 1815 & 2431\
NGC 6809 (M55) & 7 & 32 & **70** & 164 & 277 & 30 & 534 & **1574** & 3867 & 7118\
NGC 6838 (M71) & 9 & 16 & **23** & 44 & 68 & 179 & 326 & **486** & 1001 & 1628\
NGC 6934 & 0 & 3 & **12** & 24 & 37 & 0 & 63 & **250** & 507 & 797\
NGC 6981 (M72) & 8 & 17 & **28** & 40 & 57 & 148 & 345 & **604** & 876 & 1327\
NGC 7078 (M15) & 0 & 0 & **16** & 45 & 91 & 0 & 0 & **417** & 1142 & 2297\
NGC 7089 (M2) & 0 & 0 & **22** & 62 & 123 & 0 & 0 & **518** & 1443 & 2922\
NGC 7099 (M30) & 0 & 0 & **4** & 12 & 25 & 0 & 0 & **112** & 313 & 640\
\[T:bh\_dA\_Baumgardt\]
[l|rrrrr|rrrrr]{} NGC 0104 (47Tuc) & 0 & 8 & **55** & 137 & 242 & 0 & 130 & **1170** & 3020 & 5550\
NGC 0288 & 2 & 11 & **20** & 28 & 39 & 34 & 228 & **444** & 664 & 909\
NGC 1261 & 5 & 27 & **53** & 81 & 109 & 61 & 560 & **1139** & 1735 & 2347\
NGC 1851 & 0 & 4 & **24** & 57 & 103 & 0 & 70 & **510** & 1262 & 2290\
NGC 2298 & 0 & 1 & **5** & 10 & 16 & 0 & 19 & **101** & 229 & 395\
NGC 2808 & 0 & 36 & **114** & 201 & 294 & 0 & 546 & **2067** & 3978 & 6152\
NGC 3201 & 0 & 7 & **45** & 89 & 205 & 0 & 109 & **981** & 2031 & 5322\
NGC 4147 & 0 & 1 & **4** & 8 & 14 & 0 & 14 & **83** & 206 & 358\
NGC 4590 (M68) & 0 & 8 & **21** & 34 & 47 & 0 & 128 & **418** & 742 & 1097\
NGC 4833 & 0 & 29 & **80** & 135 & 268 & 0 & 520 & **1734** & 3050 & 6625\
NGC 5024 (M53) & 0 & 23 & **70** & 120 & 175 & 0 & 406 & **1443** & 2683 & 4100\
NGC 5053 & 21 & 30 & **84** & 106 & 159 & 451 & 657 & **2113** & 2640 & 3969\
NGC 5272 (M3) & 0 & 6 & **39** & 98 & 173 & 0 & 110 & **909** & 2379 & 4313\
NGC 5286 & 0 & 2 & **27** & 72 & 134 & 0 & 54 & **659** & 1780 & 3334\
NGC 5466 & 11 & 24 & **43** & 90 & 156 & 162 & 457 & **984** & 2121 & 4045\
NGC 5904 (M5) & 0 & 22 & **70** & 126 & 191 & 0 & 383 & **1390** & 2648 & 4141\
NGC 5927 & 16 & 44 & **79** & 111 & 176 & 242 & 850 & **1610** & 2367 & 3935\
NGC 5986 & 3 & 59 & **114** & 172 & 268 & 0 & 1157 & **2489** & 3877 & 6183\
NGC 6093 (M80) & 0 & 3 & **20** & 52 & 92 & 0 & 60 & **479** & 1250 & 2251\
NGC 6101 & 60 & 83 & **100** & 152 & 190 & 1404 & 2005 & **2450** & 3996 & 4723\
NGC 6144 & 2 & 15 & **28** & 41 & 74 & 0 & 298 & **619** & 948 & 1775\
NGC 6171 (M107) & 0 & 13 & **32** & 60 & 107 & 0 & 250 & **713** & 1326 & 2514\
NGC 6205 (M13) & 0 & 61 & **127** & 194 & 343 & 0 & 1170 & **2768** & 4428 & 8388\
NGC 6218 (M12) & 0 & 18 & **36** & 59 & 107 & 0 & 387 & **847** & 1368 & 2508\
NGC 6254 (M10) & 0 & 11 & **27** & 44 & 63 & 0 & 188 & **568** & 981 & 1472\
NGC 6304 & 0 & 8 & **36** & 53 & 78 & 0 & 116 & **372** & 646 & 930\
NGC 6341 (M92) & 0 & 4 & **23** & 54 & 92 & 0 & 86 & **530** & 1323 & 2303\
NGC 6352 & 0 & 4 & **10** & 17 & 27 & 0 & 69 & **211** & 363 & 618\
NGC 6366 & 0 & 1 & **4** & 8 & 15 & 0 & 21 & **90** & 174 & 347\
NGC 6397 & 0 & 0 & **2** & 7 & 14 & 0 & 0 & **63** & 178 & 367\
NGC 6535 & 0 & 0 & **1** & 2 & 4 & 0 & 1 & **17** & 45 & 85\
NGC 6541 & 0 & 5 & **29** & 70 & 121 & 0 & 101 & **679** & 1713 & 3018\
NGC 6584 & 0 & 7 & **25** & 44 & 65 & 0 & 137 & **520** & 1002 & 1544\
NGC 6624 & 0 & 0 & **1** & 3 & 6 & 0 & 0 & **14** & 46 & 101\
NGC 6637 (M69) & 0 & 25 & **57** & 82 & 120 & 0 & 466 & **1125** & 1689 & 2660\
NGC 6652 & 0 & 1 & **4** & 10 & 18 & 0 & 11 & **88** & 230 & 414\
NGC 6656 (M22) & 0 & 10 & **57** & 114 & 325 & 0 & 168 & **1303** & 2683 & 8411\
NGC 6681 (M70) & 0 & 4 & **14** & 28 & 46 & 0 & 70 & **310** & 646 & 1087\
NGC 6715 (M54) & 0 & 7 & **80** & 213 & 396 & 0 & 151 & **1966** & 5258 & 9862\
NGC 6717 (Pal9) & 0 & 0 & **1** & 4 & 7 & 0 & 2 & **33** & 90 & 171\
NGC 6723 & 2 & 36 & **88** & 136 & 279 & 0 & 726 & **1837** & 3009 & 6691\
NGC 6752 & 0 & 0 & **9** & 24 & 48 & 0 & 6 & **226** & 627 & 1230\
NGC 6779 (M56) & 1 & 13 & **28** & 44 & 57 & 0 & 240 & **597** & 958 & 1302\
NGC 6809 (M55) & 6 & 28 & **67** & 152 & 265 & 0 & 450 & **1498** & 3542 & 6805\
NGC 6838 (M71) & 1 & 4 & **10** & 19 & 37 & 0 & 73 & **222** & 420 & 884\
NGC 6934 & 0 & 4 & **16** & 31 & 49 & 0 & 77 & **324** & 675 & 1077\
NGC 6981 (M72) & 10 & 30 & **48** & 78 & 108 & 170 & 592 & **1017** & 1650 & 3100\
NGC 7078 (M15) & 0 & 4 & **41** & 109 & 201 & 0 & 97 & **1022** & 2725 & 5028\
NGC 7089 (M2) & 0 & 3 & **36** & 95 & 175 & 0 & 70 & **868** & 2338 & 4368\
NGC 7099 (M30) & 0 & 0 & **6** & 17 & 34 & 0 & 2 & **160** & 438 & 875\
\[T:bh\_dr\_Harris\]
[l|rrrrr|rrrrr]{} NGC 0104 (47Tuc) & 0 & 8 & **53** & 136 & 238 & 0 & 160 & **1210** & 3130 & 5580\
NGC 0288 & 4 & 17 & **32** & 46 & 81 & 38 & 354 & **691** & 1042 & 1924\
NGC 1261 & 6 & 27 & **52** & 81 & 106 & 90 & 569 & **1141** & 1719 & 2259\
NGC 1851 & 0 & 4 & **23** & 56 & 97 & 0 & 70 & **492** & 1233 & 2165\
NGC 2298 & 0 & 2 & **6** & 11 & 17 & 0 & 30 & **123** & 247 & 398\
NGC 2808 & 0 & 25 & **92** & 173 & 261 & 0 & 380 & **1648** & 3491 & 5636\
NGC 3201 & 0 & 6 & **37** & 72 & 179 & 0 & 86 & **805** & 1646 & 4580\
NGC 4147 & 0 & 0 & **3** & 8 & 14 & 0 & 10 & **76** & 196 & 350\
NGC 4590 (M68) & 0 & 11 & **24** & 38 & 51 & 0 & 181 & **500** & 831 & 1173\
NGC 4833 & 0 & 28 & **75** & 125 & 244 & 0 & 523 & **1633** & 2815 & 5909\
NGC 5024 (M53) & 0 & 15 & **53** & 101 & 156 & 0 & 240 & **1068** & 2287 & 3720\
NGC 5053 & 13 & 22 & **39** & 90 & 145 & 252 & 430 & **897** & 2202 & 3699\
NGC 5272 (M3) & 0 & 5 & **38** & 98 & 174 & 0 & 104 & **903** & 2391 & 4368\
NGC 5286 & 0 & 0 & **22** & 59 & 117 & 0 & 5 & **536** & 1474 & 2937\
NGC 5466 & 20 & 32 & **88** & 115 & 176 & 424 & 647 & **1182** & 2824 & 4517\
NGC 5904 (M5) & 0 & 15 & **55** & 106 & 166 & 0 & 246 & **1047** & 2214 & 3581\
NGC 5927 & 12 & 44 & **75** & 102 & 150 & 178 & 855 & **1509** & 2150 & 3299\
NGC 5986 & 8 & 62 & **115** & 170 & 240 & 16 & 1259 & **2533** & 3792 & 5335\
NGC 6093 (M80) & 0 & 2 & **17** & 46 & 84 & 0 & 37 & **415** & 1116 & 2070\
NGC 6101 & 73 & 91 & **104** & 156 & 193 & 1698 & 2255 & **2568** & 4124 & 4787\
NGC 6144 & 2 & 15 & **25** & 36 & 49 & 28 & 336 & **594** & 854 & 1178\
NGC 6171 (M107) & 3 & 17 & **42** & 66 & 111 & 16 & 342 & **872** & 1446 & 2548\
NGC 6205 (M13) & 0 & 59 & **119** & 179 & 294 & 0 & 1166 & **2597** & 4055 & 6989\
NGC 6218 (M12) & 0 & 19 & **37** & 58 & 103 & 0 & 412 & **840** & 1313 & 2193\
NGC 6254 (M10) & 0 & 10 & **26** & 42 & 59 & 0 & 180 & **544** & 939 & 1389\
NGC 6304 & 0 & 9 & **35** & 49 & 68 & 0 & 118 & **361** & 599 & 839\
NGC 6341 (M92) & 0 & 3 & **21** & 52 & 90 & 0 & 66 & **490** & 1247 & 2240\
NGC 6352 & 0 & 4 & **10** & 18 & 28 & 0 & 68 & **210** & 373 & 612\
NGC 6366 & 0 & 1 & **3** & 6 & 11 & 0 & 13 & **64** & 133 & 265\
NGC 6397 & 0 & 0 & **1** & 3 & 6 & 0 & 0 & **26** & 78 & 175\
NGC 6535 & 0 & 1 & **2** & 3 & 4 & 0 & 10 & **36** & 65 & 97\
NGC 6541 & 0 & 4 & **27** & 68 & 118 & 0 & 92 & **653** & 1664 & 2965\
NGC 6584 & 0 & 4 & **17** & 36 & 58 & 0 & 67 & **355** & 830 & 1408\
NGC 6624 & 2 & 66 & **78** & 91 & 105 & 123 & 326 & **515** & 703 & 887\
NGC 6637 (M69) & 1 & 21 & **46** & 67 & 89 & 0 & 400 & **920** & 1375 & 1854\
NGC 6652 & 0 & 1 & **4** & 11 & 18 & 0 & 13 & **91** & 230 & 403\
NGC 6656 (M22) & 0 & 9 & **61** & 125 & 352 & 0 & 125 & **1393** & 2903 & 9047\
NGC 6681 (M70) & 0 & 3 & **12** & 24 & 39 & 0 & 57 & **261** & 564 & 932\
NGC 6715 (M54) & 0 & 0 & **48** & 133 & 281 & 0 & 0 & **1193** & 3326 & 7056\
NGC 6717 (Pal9) & 0 & 0 & **1** & 4 & 7 & 0 & 2 & **33** & 87 & 166\
NGC 6723 & 8 & 44 & **92** & 136 & 276 & 79 & 886 & **1921** & 3014 & 6619\
NGC 6752 & 0 & 0 & **8** & 23 & 45 & 0 & 2 & **211** & 584 & 1156\
NGC 6779 (M56) & 2 & 15 & **31** & 46 & 61 & 8 & 276 & **645** & 1014 & 1358\
NGC 6809 (M55) & 7 & 31 & **68** & 158 & 269 & 29 & 517 & **1523** & 3744 & 6891\
NGC 6838 (M71) & 6 & 10 & **14** & 27 & 42 & 110 & 199 & **297** & 611 & 995\
NGC 6934 & 0 & 4 & **17** & 33 & 51 & 0 & 88 & **349** & 706 & 1110\
NGC 6981 (M72) & 13 & 29 & **50** & 70 & 102 & 263 & 613 & **1072** & 1555 & 2355\
NGC 7078 (M15) & 0 & 0 & **30** & 81 & 162 & 0 & 0 & **746** & 2044 & 4112\
NGC 7089 (M2) & 0 & 0 & **27** & 74 & 148 & 0 & 0 & **623** & 1736 & 3514\
NGC 7099 (M30) & 0 & 0 & **5** & 15 & 31 & 0 & 0 & **137** & 383 & 784\
\[T:bh\_dA\_Harris\]
|
---
author:
- 'John Loftin, Tengren Zhang'
title: 'Coordinates on the Augmented Moduli Space of Convex ${\mathbb{RP}}^2$ Structures'
---
Introduction
============
Let $S$ denote a smooth, connected, oriented, finite-type surface with negative Euler characteristic. A convex ${\mathbb{RP}}^2$ structure $\mu$ on $S$ is determined by a pair $(\phi,\rho)$, where $\rho:\pi_1(S)\to{\mathrm{PGL}}(3,{\mathbb{R}})$ is a representation and $\phi:\widetilde{S}\to\Omega$ is a $\rho$-equivariant diffeomorphism onto a properly convex domain $\Omega\subset{\mathbb{R}}{\mathbb{P}}^2$. The pair $(\phi,\rho)$ is usually known as a *developing pair* for $\mu$, while $\rho$ and $\phi$ are called a *holonomy representation* and a *developing map* of $\mu$ respectively. We will denote the deformation space of convex ${\mathbb{R}}{\mathbb{P}}^2$ structures on $S$ by ${\mathcal{C}}(S)$. (See Section \[sec:convex\] for more precise definitions.) Hyperbolic structures are then examples of convex ${\mathbb{RP}}^2$ structures via the Klein model of hyperbolic space, where $\Omega$ is a round disk. In this article, we give local coordinates that are adapted to describe degenerations of convex ${\mathbb{RP}}^2$ structures on $S$ that converge on the complement of a multi-curve ${\mathcal{D}}$. Our coordinates generalize natural orbifold coordinates, based on Fenchel-Nielsen coordinates, near the boundary of the Deligne-Mumford compactification $\overline{\mathcal M(S)}$ of the moduli space of finite-area hyperbolic structures on $S$.
In order to motivate this choice of limiting structure to study, we recall the case of hyperbolic structures on $S$ when $S$ is a closed oriented surface of genus at least $2$. The deformation space of all (marked) hyperbolic structures is the Teichmüller space $\mathcal T(S)$, which is homeomorphic to ${\mathbb{R}}^{6g-6}$. There are two major and essentially different ways to analyze degenerating families of hyperbolic structures on $S$. First, Thurston gives a natural compactification $\overline{{\mathcal{T}}(S)}$ of $\mathcal T(S)$ which may be seen as the set of limits in the projective space of hyperbolic lengths of all closed geodesics on $S$. Second, for (unmarked) hyperbolic structures on $S$, the moduli space $\mathcal M(S)$, which is the quotient of $\mathcal T(S)$ by the mapping class group $\mathrm{MCG}(S)$, has the structure of a quasi-projective algebraic variety with orbifold singularities. Its most natural compactification, the Deligne-Mumford compactification $\overline{\mathcal M(S)}$, is then formed by considering all complete, finite-area, hyperbolic structures on $S\setminus \mathcal D$ for all free homotopy classes of multi-curves $\mathcal D$ in $S$. One can also define an augmentation ${\mathcal{T}}(S)^{\mathrm{aug}}$ of ${\mathcal{T}}(S)$ to be ${\mathcal{T}}(S)$ together with all the possible limits of degenerating families of hyperbolic structures on $S$ so that the family converges on the complement of a multicurve. It is then well-known [@Abi; @HubKoch14] that $\overline{\mathcal M(S)}={\mathcal{T}}(S)^{\mathrm{aug}}/\mathrm{MCG}(S)$.
These two spaces $\overline{{\mathcal{T}}(S)}$ and ${\mathcal{T}}(S)^{\mathrm{aug}}$ are fundamentally different; most rays in Teichmüller space that converge to points in $\overline{{\mathcal{T}}(S)}$ do not project under the quotient map ${\mathcal{T}}(S)\to{\mathcal{M}}(S)$ to convergent rays in $\overline{\mathcal M(S)}$. The reason for this is the following: if a family of hyperbolic structures on $S$ is pinched along a multi-curve $\mathcal D$, its accumulation set in $\overline{\mathcal M(S)}$ depends on the limiting hyperbolic structures on the complement of ${\mathcal{D}}$. However, if we choose a lift of this sequence to Teichmüller space, then its limit in $\overline{\mathcal T(S)}$ only records the relative hyperbolic lengths of closed geodesics whose lengths are growing the fastest along this sequence. In particular, the hyperbolic structure on $S\setminus{\mathcal{D}}$ is forgotten in $\overline{\mathcal T(S)}$. The present work addresses, for the case of convex ${\mathbb{RP}}^2$ structures, analogs of the geometry of $\overline{\mathcal M(S)}$. In [@Loftin], the first author introduced *regular convex ${\mathbb{RP}}^2$ structures* which serve to augment the deformation space $\mathcal C(S)$. These are convex ${\mathbb{R}}{\mathbb{P}}^2$ structures on $S$, together with all the possible limits of degenerating families of convex ${\mathbb{R}}{\mathbb{P}}^2$ structures, with the property that the family converges on the complement of a multi-curve. The *augmented deformation space $\mathcal C(S)^{\rm aug}$* is then the set of all regular convex ${\mathbb{RP}}^2$ structures on $S$. One should think of ${\mathcal{C}}(S)^{\mathrm{aug}}$ as a generalization of ${\mathcal{T}}(S)^{\mathrm{aug}}$ to the setting of convex ${\mathbb{R}}{\mathbb{P}}^2$ structures on $S$. The first author also defined a natural topology on $\mathcal C(S)^{\rm aug}$. With this topology, $\mathcal C(S)^{\rm aug}$ has a stratification, where each stratum $\mathcal C(S,{\mathcal{D}})^{\rm adm}\subset{\mathcal{C}}(S)^{\mathrm{aug}}$ is determined by a multi-curve ${\mathcal{D}}$ on $S$. (See Section \[sec: aug top\] for more details.)
Despite its naturality, the topology on $\mathcal C(S)^{\rm aug}$ is rather abstract in terms of the geometry of the limiting surfaces. The purpose of this paper is elucidate the geometric properties of families of regular convex ${\mathbb{RP}}^2$ structures by using (global) Fenchel-Nielsen type coordinates on the space of holonomies of the convex ${\mathbb{R}}{\mathbb{P}}^2$ structures on $S$ to construct (local) coordinates on appropriate quotients of $\mathcal C(S)^{\rm aug}$ by certain subgroups of the mapping class group. We show that these coordinates induce the topology on the *augmented moduli space* $\mathcal C(S)^{\rm aug} / \mathrm{MCG}(S)$. More precisely, we have the following main theorem (also see Theorem \[thm: main\]).
\[thm: main intro\] Let $\mu\in{\mathcal{C}}(S)^{\mathrm{aug}}$, and let ${\mathcal{D}}$ be the multi-curve on $S$ so that $\mu\in{\mathcal{C}}(S,{\mathcal{D}})^{\mathrm{adm}}$. Also, let $G_{\mathcal{D}}\subset\mathrm{MCG}(S)$ be the subgroup generated by Dehn twists about the simple closed curves in ${\mathcal{D}}$. Then $${\mathcal{C}}(S)^{\mathrm{aug},{\mathcal{D}}}:=\bigcup_{{\mathcal{D}}'\subset{\mathcal{D}}}{\mathcal{C}}(S,{\mathcal{D}}')^{\rm adm}$$ is an open set of ${\mathcal{C}}(S)^{\mathrm{aug}}$ containing $\mu$ that is invariant under $G_{\mathcal{D}}$, and there is a homeomorphism $$\Psi_{\mathcal{D}}:{\mathcal{C}}(S)^{\mathrm{aug},{\mathcal{D}}}/G_{\mathcal{D}}\to{\mathbb{R}}^{10g-10+6n+2m}\times({\mathbb{R}}_+)^{6g-6+2n-2m},$$ where $m$ is the number of curves in ${\mathcal{D}}$, $n$ is the number of punctures of $S$, and $g$ is the genus of the compactification of $S$ in which each puncture is filled in. In particular, ${\mathcal{C}}(S)^{\mathrm{aug},{\mathcal{D}}}/G_{\mathcal{D}}$ is a cell of dimension $16g-16+8n$.
Furthermore, the coordinate functions of $\Psi_{\mathcal{D}}$ are explicitly described. See Section \[sec: coord\] for the description of $\Psi_{\mathcal{D}}$.
It turns out that if ${\mathcal{D}}$ is non-empty, then any open set of ${\mathcal{C}}(S)^{\mathrm{aug}}$ containing $\mu$ in the above theorem does not have compact closure. On the other hand, any open set of the augmented moduli space ${\mathcal{C}}(S)^{\mathrm{aug}}/\mathrm{MCG}(S)$ containing $[\mu]$ might have a complicated singular locus. (See Section \[sec: aug top\].) Thus, it is necessary to quotient ${\mathcal{C}}(S)^{\mathrm{aug},{\mathcal{D}}}$ by the appropriate subgroup $G_{{\mathcal{D}}}$ of $\mathrm{MCG}(S)$ for it to have a nice set of coordinates.
Theorem \[thm: main intro\] is a generalization of a standard result in Teichmüller theory describing the behavior of Fenchel-Nielsen coordinates at the boundary of $\overline{\mathcal M(S)}$, the Deligne-Mumford compactfication of the the moduli space of finite-area hyperbolic structures on $S$. The following theorem is well-known (see e.g. [@HubKoch14])
Let ${\mathcal{D}}$ be a multi-curve on $S$, and let $\mathcal T(S)^{{\rm aug},\mathcal D}$ be Teichmüller space augmented along curves in ${\mathcal{D}}$ only. Choose a pants decomposition ${\mathcal{P}}\supset{\mathcal{D}}$ on $S$, and let $\ell_i,\theta_i$ denote the length and twist coordinates on curves in ${\mathcal{P}}$, where the $\theta_i$ are normalized so that Dehn twists are represented by $\theta_i\mapsto\theta_i+2\pi$. Then ${\mathcal{T}}(S)^{{\rm aug},{\mathcal{D}}}/G_{\mathcal{D}}$ can be described by coordinates $$\left( \prod_{j=1}^m (\ell_j\cos(\theta_j),\ell_j\sin(\theta_j)) \right) \times \left(\prod_{j=m+1}^{3g-3+n}(\ell_j,\theta_j) \right) \in{\mathbb{R}}^{2m}\times ({\mathbb{R}}_+ \times {\mathbb{R}})^{3g-3+n-m}$$ to be homeomorphic to a cell of dimension $6g-6+2n$.
Note in this theorem that about each curve $c$ in ${\mathcal{D}}$, the length parameter is allowed to become 0 and the twist parameter is considered modulo $2\pi$. In our Theorem \[thm: main intro\], similar but more complicated constructions on generalized length and twist parameters about $c$ are needed to define $\Psi_{\mathcal{D}}$.
Theorem \[thm: main intro\] (together with Theorem \[thm: hol\]) gives us a complete description of the behavior of the boundary of the image of the developing map in families of convex ${\mathbb{RP}}^2$ structures degenerating to a regular convex ${\mathbb{RP}}^2$ structure, including new behavior which do not occur in the study of hyperbolic structures in $\overline{\mathcal M}_g$. The new phenomenon is that the limit set of the holonomy representation (restricted to a component of $S\setminus{\mathcal{D}}$) in the boundary of the image of the developing map might be a proper subset of the boundary. However, we can still describe the behavior of boundary in this case.
One can generalize the holonomy $\rho\!:\pi_1(S) \to {\mathrm{PGL}}(3,{\mathbb{R}})$ of a convex ${\mathbb{R}}{\mathbb{P}}^2$ structure to representations from $\pi_1(S)$ to other split real Lie groups (in particular for Hitchin representations into ${\mathrm{PGL}}(n+1,{\mathbb{R}})$, [@Hitchin92; @Labourie06; @Guichard05]). We hope the detailed model of the degeneration of the convex boundary curves given in Theorem \[thm: main intro\] will be of help to analyze families of Hitchin representations which degenerate along a multi-curve.
Via the uniformization theorem, one may also view ${\mathcal{M}}(S)$ as the moduli space of Riemann surfaces homeomorphic to $S$, which is itself naturally a complex orbifold. When $S$ is closed, there is a natural holomorphic vector bundle $\mathcal K(S)$ over ${\mathcal{M}}(S)$ whose fiber above every point $X\in{\mathcal{M}}(S)$ is the vector space of holomorphic cubic differentials on $X$. Labourie [@Labourie07] and the first author [@Loftin2001] independently constructed a natural homeomorphism $$\label{eqn}{\mathcal{C}}(S)/\mathrm{MCG}(S)\simeq\mathcal{K}(S).$$
Later, the first author [@Loftin] defined the notion of a regular convex ${\mathbb{RP}}^2$ structure on a (not necessarily compact) Riemann surface $S$. Also, in part by using [@Loftin2004; @BenHul13; @Nie15], he extended (\[eqn\]) by proving that there is a natural homeomorphism $\mathcal C(S)^{\rm aug} / \mathrm{MCG}(S)\simeq\mathcal K(S)^{\mathrm{reg}}$, where $\mathcal K(S)^{\mathrm{reg}}$ is the orbifold vector bundle of regular cubic differentials over the Deligne-Mumford compactification of the moduli space of Riemann surfaces $\overline{{\mathcal{M}}(S)}$. (See Section \[sec: conv cubic\] below.) In particular, the main theorem of [@Loftin] shows that there are (local) holomorphic coordinates (up to local finite group actions) on the augmented moduli space $\mathcal C(S)^{\rm aug}/\mathrm{MCG}(S)$.
As a consequence of Theorem \[thm: main intro\], we significantly simplify the proof of half of the main theorem in [@Loftin] by applying Brouwer’s Invariance of Domain Theorem to the real coordinates we construct here and the holomorphic coordinates induced by the regular cubic differentials. More precisely, we arrive at the following corollary.
\[cor:Loftin\] There is a natural homeomorphism ${\mathcal{C}}(S)^{\mathrm{aug}}/\mathrm{MCG}(S)\simeq\mathcal{K}(S)^{\mathrm{reg}}$. When $S$ is closed, this extends the homeomorphism (\[eqn\]).
The main tool used to prove Theorem \[thm: main intro\] are Fenchel-Nielsen type coordinates on the set of holonomies of convex ${\mathbb{R}}{\mathbb{P}}^2$ structures on $S$. In the case when $S$ is closed, these kinds of coordinates were first constructed by Goldman [@Go90]. Choi-Goldman [@ChoiGo1] also showed that these holonomies form the Hitchin component of the space of representations from $\pi_1(S)$ into ${\mathrm{PGL}}(3,{\mathbb{R}})$. In the more general setting of Hitchin representations into ${\mathrm{PGL}}(n,{\mathbb{R}})$, Fock-Goncharov, Bonahon-Dreyer, and the second author further developed analogs of Goldman’s coordinates [@FocGon; @BonDre; @Zhang16] which are more amenable to our construction than Goldman’s original coordinates. Marquis [@Mar10] extended Goldman’s coordinates to the case of finite area convex ${\mathbb{R}}{\mathbb{P}}^2$ structures on a (possibly) punctured surface $S$. In this paper, we study convex ${\mathbb{RP}}^2$ structures on punctured surfaces by extending the coordinates of [@FocGon; @BonDre; @Zhang16] instead. Other works studying noncompact convex ${\mathbb{RP}}^2$ surfaces from various points of view include [@BenHul13; @BenHul14; @Choi94II; @DumWolf15; @Loftin2004; @Mar12; @Nie15].
Here is a brief description of the structure of this paper. In Section \[admissible-section\], we follow [@Loftin] to define ${\mathcal{C}}(S)^{\mathrm{aug}}$ and its topology. Then in Section \[sec: hol\], we describe the (global) coordinates on the image ${\rm hol}(\mathcal C(S))$ of the holonomy map from ${\mathcal{C}}(S)$ to ${\mathcal{X}}(\pi_1(S),{\mathrm{PGL}}(3,{\mathbb{R}}))/{\mathrm{PGL}}(3,{\mathbb{R}})$, and use them to construct local coordinates on appropriate quotients of ${\mathcal{C}}(S)^{\mathrm{aug}}$. For this part, a key point is that unlike the case of compact $S$, the holonomy of a regular convex ${\mathbb{R}}{\mathbb{P}}^2$ structure on a non-compact surface $S$ of negative Euler characteristic does not always determine the projective structure at the ends of $S$. We then proceed to prove Theorem \[thm: main intro\] in Section \[sec: coord\] by showing that the coordinates constructed in Section \[sec: hol\] describe the topology on ${\mathcal{C}}(S)^{\mathrm{aug}}/\mathrm{MCG}(S)$ described in Section \[admissible-section\]. Section \[sec: conv cubic\] relates our description of the topology on ${\mathcal{C}}(S)^{\mathrm{aug}}/\mathrm{MCG}(S)$ to the regular cubic differentials studied in [@Loftin], which allows us to recover Corollary \[cor:Loftin\]. Finally, in the Appendix, we present the proof of Theorem \[thm: hol\], which describes how the image of the developing map of two regular convex ${\mathbb{R}}{\mathbb{P}}^2$ structures on $S$ with the same holonomy can differ. In a result which may be of independent interest, we also give in the Appendix a description of the limit set of any convex ${\mathbb{RP}}^2$ structure on $S$.
The first author would like to thank Bill Goldman for many inspiring discussions about ${\mathbb{RP}}^2$ structures. The authors acknowledge support from U.S. National Science Foundation grants DMS 1107452, 1107263, 1107367 “RNMS: GEometric structures And Representation varieties" (the GEAR Network). The second author was also partially supported by the National Science Foundation grant DMS 1566585, and by the NUS-MOE grant R-146-000-270-133.
Admissible convex real projective structures {#admissible-section}
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In this section, we define admissible convex ${\mathbb{R}}{\mathbb{P}}^2$ structures on finite type surfaces, as well as some terminology describing the holonomy of these structures about the ends of the surface.
Convex real projective structures {#sec:convex}
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We begin by recalling some standard definitions and properties of ${\mathbb{R}}{\mathbb{P}}^2$ structures on surfaces.
1. An *${\mathbb{R}}{\mathbb{P}}^2$ surface* $\Sigma$ is a smooth, connected, closed surface with finitely many punctures, that is equipped with a maximal collection of smooth maps $\{\psi_\alpha:U_\alpha\to{\mathbb{R}}{\mathbb{P}}^2\}_\alpha$ so that the following holds.
- Each $U_\alpha\subset\Sigma$ is a connected and simply connected open subset,
- For any pair of smooth maps $\psi_\alpha$ and $\psi_\beta$, $\psi_\alpha\circ\psi_\beta^{-1}:\psi_\beta(U_\alpha\cap U_\beta)\to\psi_\alpha(U_\alpha\cap U_\beta)$ is a restriction of a projective transformation on ${\mathbb{R}}{\mathbb{P}}^2$ to each connected component of $U_\alpha\cap U_\beta$.
The smooth maps $\psi_\alpha$ are called *charts* of $\Sigma$.
2. Let $\Sigma$ and $\Sigma'$ be two ${\mathbb{R}}{\mathbb{P}}^2$ surfaces with with maximal atlases $\{\psi_\alpha\}_\alpha$ and $\{\psi_\alpha'\}_\alpha$ respectively. A diffeomorphism $f:\Sigma\to\Sigma'$ is a *projective isomorphism* if for any charts $\psi_\alpha:U_\alpha\to{\mathbb{R}}{\mathbb{P}}^2$ of $\Sigma$ and $\psi'_\beta:U_\beta'\to{\mathbb{R}}{\mathbb{P}}^2$ of $\Sigma'$ so that $f(U_\alpha)\cap U_\beta'$ is non-empty, the composition $$\psi'_\beta\circ f\circ\psi_\alpha^{-1}:\psi_\alpha(U_\alpha\cap f^{-1}(U_\beta'))\to\psi'_\beta(f(U_\alpha)\cap U_\beta')$$ is the restriction of a projective transformation on ${\mathbb{R}}{\mathbb{P}}^2$ to each connected component of $\psi_\alpha(U_\alpha\cap f^{-1}(U_\beta'))$.
Let $\widetilde{\Sigma}$ be the universal cover of $\Sigma$. Then $\widetilde{\Sigma}$ is naturally an ${\mathbb{R}}{\mathbb{P}}^2$ surface. For any choice of chart $\widetilde{\psi}_\alpha:\widetilde{U}_\alpha\to{\mathbb{R}}{\mathbb{P}}^2$ of $\widetilde{\Sigma}$, one can construct via analytic continuation, a unique local diffeomorphism $\phi_\alpha:\widetilde{\Sigma}\to{\mathbb{R}}{\mathbb{P}}^2$ so that
- $\phi_\alpha|_{\widetilde{U}_\alpha}=\widetilde{\psi}_\alpha$,
- for any point $p\in\widetilde{\Sigma}$, there is an open set $\widetilde{U}\subset\widetilde{\Sigma}$ so that $p\in \widetilde{U}$ and $\phi_\alpha|_{\widetilde{U}}:\widetilde{U}\to{\mathbb{R}}{\mathbb{P}}^2$ is a chart for $\widetilde{\Sigma}$.
The local diffeomorphism $\phi_\alpha$ is usually known as a *developing map* for $\Sigma$. It induces a group homomorphism $\rho_\alpha:\pi_1(\Sigma)\to{\mathrm{PGL}}(3,{\mathbb{R}})$ with the defining property that $\phi_\alpha$ is $\rho_\alpha$-equivariant. This homomorphism is usually called a *holonomy representation* of $\Sigma$, and the pair $(\phi_\alpha,\rho_\alpha)$ is a *developing pair* for $\Sigma$.
Observe that for any pair of charts $\widetilde{\psi}_\alpha:\widetilde{U}_\alpha\to{\mathbb{R}}{\mathbb{P}}^2$ and $\widetilde{\psi}_\beta:\widetilde{U}_\beta\to{\mathbb{R}}{\mathbb{P}}^2$ of $\widetilde{\Sigma}$, there is some $g\in{\mathrm{PGL}}(3,{\mathbb{R}})$ so that $$(\phi_\alpha,\rho_\alpha)=(g\circ\phi_\beta,c_g\circ\rho_\beta),$$ where $c_g:{\mathrm{PGL}}(3,{\mathbb{R}})\to {\mathrm{PGL}}(3,{\mathbb{R}})$ is conjugation by $g$. In particular, the holonomy representation of $\Sigma$ is unique up to conjugation.
1. A domain $\Omega\subset{\mathbb{R}}{\mathbb{P}}^2$ is *properly convex* if its closure in ${\mathbb{R}}{\mathbb{P}}^2$ does not contain any projective lines, and for any pair of distinct points $p,q\in\Omega$, there is a projective line segment with endpoints $p,q$ that lies entirely in $\Omega$.
2. A connected ${\mathbb{R}}{\mathbb{P}}^2$ surface $\Sigma$ is *convex* if any (equivalently, some) developing map of $\Sigma$ is a diffeomorphism onto a properly convex domain in ${\mathbb{R}}{\mathbb{P}}^2$.
3. The *deformation space of convex ${\mathbb{R}}{\mathbb{P}}^2$ structures on $S$* is $${\mathcal{C}}(S):=\left\{(f,\Sigma):\begin{array}{l}
\Sigma\text{ is a convex }{\mathbb{R}}{\mathbb{P}}^2\text{ surface}\\
f:S\to\Sigma\text{ is a diffeomorphism}\end{array}\right\}\Bigg/\sim,$$ where $(f,\Sigma)\sim(f',\Sigma')$ if $f'\circ f^{-1}:\Sigma\to\Sigma'$ is homotopic to a projective isomorphism from $\Sigma$ to $\Sigma'$. An equivalence class $[f,\Sigma]\in{\mathcal{C}}(S)$ is a *(marked) convex ${\mathbb{R}}{\mathbb{P}}^2$ structure* on $S$.
Let $\mu\in{\mathcal{C}}(S)$, let $(f,\Sigma)$ be a representative of $\mu$, and let $(\phi_\alpha,\rho_\alpha)$ be any developing pair of $\Sigma$. The diffeomorphism $f:S\to\Sigma$ induces an isomorphism $f_*:\pi_1(S)\to\pi_1(\Sigma)$, and also lifts to a map $\widetilde{f}:\widetilde{S}\to\widetilde{\Sigma}$. Define the pair $$(\phi,\rho):=(\widetilde{f}\circ\phi_\alpha,f_*\circ\rho_\alpha).$$ Note that $\rho:\pi_1(S)\to{\mathrm{PGL}}(3,{\mathbb{R}})$ is injective, and that $\phi:\widetilde{S}\to{\mathbb{R}}{\mathbb{P}}^2$ is a $\rho$-equivariant diffeomorphism onto a properly convex domain. Furthermore, since $\rho(\pi_1(S))$ acts properly discontinuously on $\Omega$, it is a discrete subgroup of ${\mathrm{PGL}}(3,{\mathbb{R}})$. Thus $(\phi,\rho)$ is a developing pair for $(f,\Sigma)$.
The following well-known theorem (see e.g. Goldman [@Go90] Section 2.2) states that any developing pair for any representative $(f,\Sigma)$ of $\mu\in{\mathcal{C}}(S)$ determines $\mu$.
\[thm:dev\] Let $\rho:\pi_1(S)\to{\mathrm{PGL}}(3,{\mathbb{R}})$ be an injective representation, and $\phi:\widetilde{S}\to{\mathbb{R}}{\mathbb{P}}^2$ be a $\rho$-equivariant diffeomorphism onto a properly convex domain $\Omega\subset{\mathbb{R}}{\mathbb{P}}^2$. Then there is a unique $\mu\in{\mathcal{C}}(S)$ so that $(\phi,\rho)$ is the developing pair for $\mu$. Furthermore, $(\phi,\rho)$ and $(\phi',\rho')$ are developing pairs for $\mu\in{\mathcal{C}}(S)$ if and only if there is some $g\in{\mathrm{PGL}}(3,{\mathbb{R}})$ so that $\rho'=c_g\circ\rho$ and $\phi'$ is homotopic to $g\circ\phi$ via a $\rho'$-equivariant homotopy.
The domain $\Omega$ in the above proposition will be referred to as the $\rho$-*equivariant developing image* of $\mu$.
Given a properly convex domain $\Omega\subset{\mathbb{R}}{\mathbb{P}}^2$, we can define the *Hilbert metric* on $\Omega$ as follows.
Let $\Omega\subset{\mathbb{R}}{\mathbb{P}}^2$ be a properly convex domain and let $x,y\in\Omega$. Let $\ell$ be a projective line in ${\mathbb{R}}{\mathbb{P}}^2$ through $x$ and $y$, and let $a,b\in\ell$ so that $\{a,b\}=\partial\Omega\cap\ell$ and $a,x,y,b$ lie in $\ell$ in this order. Define $$d_\Omega(x,y):=\frac{1}{2}\log[a,x,y,b],$$ where $[a,x,y,b]$ is the cross ratio of four points $a,x,y,b$ on the projective line $\ell$.
One can verify that $d_\Omega$ defines a metric on $\Omega$, and $(\Omega,d_\Omega)$ is a complete, proper, path metric space. Furthermore, since the cross ratio is a projective invariant, $d_\Omega$ is invariant under the projective transformations that preserve $\Omega$. Thus, if $\Omega$ is the $\rho$-equivariant developing image of some $\mu\in{\mathcal{C}}(S)$, $d_\Omega$ descends to a metric on $\Omega/\rho(\pi_1(S))$. Projective line segments are geodesics of $d_\Omega$.
Using this, we show that any holonomy representation of $\mu$, together with the $\rho$-equivariant developing image, determines $\mu$.
\[prop:devimage\] Let $\mu_0,\mu_1\in{\mathcal{C}}(S)$ with developing pairs $(\phi_0,\rho_0)$ and $(\phi_1,\rho_1)$. If $\rho_0=\rho_1:\pi_1(S)\to{\mathrm{PGL}}(3,{\mathbb{R}})$ and $\phi_0(\widetilde{S})=\phi_1(\widetilde{S})\subset{\mathbb{R}}{\mathbb{P}}^2$, then $\mu_0=\mu_1$.
By Theorem \[thm:dev\], it is sufficient to show that $\phi$ is homotopic to $\phi'$ via a $\rho$-equivariant homotopy. Define $F:[0,1]\times\Omega\to\Omega$ by declaring $F(t,p)$ to be the unique point in the projective line segment in $\Omega$ between $\phi_0(p)$ and $\phi_1(p)$, so that $d_\Omega(\phi_0(p),F(t,p))=td_\Omega(\phi_0(p),\phi_1(p))$. It is clear that $F$ is a homotopy between $\phi_0$ and $\phi_1$, and it is easy to verify from the projective invariance of $d_\Omega$ that $F(t,\gamma\cdot p)=\rho(\gamma)\cdot F(t,p)$ for all $(t,p)\in[0,1]\times\Omega$ and all $\gamma\in\pi_1(S)$.
As a consequence of Theorem \[thm:dev\], we may define the *holonomy map* $$\mathrm{hol}:{\mathcal{C}}(S)\to{\mathcal{X}}_3(S):=\mathrm{Hom}(\pi_1(S),{\mathrm{PGL}}(3,{\mathbb{R}}))/{\mathrm{PGL}}(3,{\mathbb{R}}),$$ where ${\mathrm{PGL}}(3,{\mathbb{R}})$ acts on $\mathrm{Hom}(\pi_1(S),{\mathrm{PGL}}(3,{\mathbb{R}}))$ by conjugation. We will also denote $\mathrm{hol}_\mu:=\mathrm{hol}(\mu)$ when convenient.
We now describe natural topologies on ${\mathcal{C}}(S)$ and ${\mathcal{X}}_3(S)$. Let us start with the topology on ${\mathcal{X}}_3(S)$. Given a finite generating set ${\mathcal{S}}$ of $\pi_1(S)$, $\mathrm{Hom}(\pi_1(S),{\mathrm{PGL}}(3,{\mathbb{R}}))$ is realized as a subset of ${\mathrm{PGL}}(3,{\mathbb{R}})^{|{\mathcal{S}}|}$ by evaluating every $\rho\in\mathrm{Hom}(\pi_1(S),{\mathrm{PGL}}(3,{\mathbb{R}}))$ on ${\mathcal{S}}$. The standard topology on ${\mathrm{PGL}}(3,{\mathbb{R}})$ thus induces a subspace topology on $\mathrm{Hom}(\pi_1(S),{\mathrm{PGL}}(3,{\mathbb{R}}))\subset{\mathrm{PGL}}(3,{\mathbb{R}})^{|{\mathcal{S}}|}$, which can be verified to be independent of the choice of ${\mathcal{S}}$. This in turn induces the quotient topology on ${\mathcal{X}}_3(S)$.
Next, we define a topology in ${\mathcal{C}}(S)$. Choose a Riemannian metric on ${\mathbb{R}}{\mathbb{P}}^2$. For any properly convex domain $\Omega\subset{\mathbb{R}}{\mathbb{P}}^2$, let $U_{\Omega,\epsilon}$ denote the set of properly convex domains in ${\mathbb{R}}{\mathbb{P}}^2$, and whose Hausdorff distance from $\Omega$ is less than $\epsilon$. The set ${\mathcal{P}}{\mathcal{C}}:=\{\text{properly convex domains in }{\mathbb{R}}{\mathbb{P}}^2\}$ can then be equipped with the topology generated by $$\left\{U_{\Omega,\epsilon}:
\Omega\subset{\mathbb{R}}{\mathbb{P}}^2\text{ is a properly convex domain and }\epsilon>0.\right\}$$ This topology does not depend on the choice of Riemannian metric on ${\mathbb{R}}{\mathbb{P}}^2$.
Using this, we can topologize ${\mathcal{C}}(S)$ in the following way. Recall that any chart $\widetilde{\psi}_\alpha:\widetilde{U}_\alpha\to{\mathbb{R}}{\mathbb{P}}^2$ of $\widetilde{\Sigma}$ determines a developing pair $(\phi_\alpha,\rho_\alpha)$ for $\Sigma$. Equip $$\widetilde{{\mathcal{C}}}(S):=\left\{(f,\Sigma,\widetilde{\psi}_\alpha):\begin{array}{l}
\Sigma\text{ is a convex }{\mathbb{R}}{\mathbb{P}}^2\text{ surface}\\
f:S\to\Sigma\text{ is a diffeomorphism}\\
\widetilde{\psi}_\alpha:\widetilde{U}_\alpha\to{\mathbb{R}}{\mathbb{P}}^2\text{ is a chart of }\widetilde{\Sigma}\end{array}\right\}.$$ with the topology generated by $$\left\{U_{(f,\Sigma,\widetilde{\psi}_\alpha),U,V}:\begin{array}{l}(f,\Sigma,\widetilde{\psi}_\alpha)\in\widetilde{{\mathcal{C}}}(S),\,\, U\subset{\mathcal{P}}{\mathcal{C}}\text{ is an open set containing }\phi_\alpha(\widetilde\Sigma)\\
V\subset\mathrm{Hom}(\pi_1(S),{\mathrm{PGL}}(3,{\mathbb{R}}))\text{ is an open set containing }f_*\circ\rho_\alpha,
\end{array}\right\},$$ where $$U_{(f,\Sigma,\widetilde{\psi}_\alpha),U,V}:=\{(f',\Sigma',\widetilde{\psi}_\beta')\in\widetilde{{\mathcal{C}}}(S):\phi'_\beta(\widetilde{\Sigma}')\in U, \,\,f'_*\circ \rho'_\beta\in V\}.$$ Since ${\mathcal{C}}(S)$ can be realized as a quotient of $\widetilde{{\mathcal{C}}}(S)$, the topology on $\widetilde{{\mathcal{C}}}(S)$ induces a quotient topology on ${\mathcal{C}}(S)$. It is clear from the definition of this topology that the holonomy map $\mathrm{hol}:{\mathcal{C}}(S)\to{\mathcal{X}}_3(S)$ is continuous (the topology on ${\mathcal{X}}_3(S)$ is the one induced by the real topology on ${\mathrm{PGL}}(3,{\mathbb{R}})$).
The admissibility condition
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To build an augmentation of the deformation space of convex ${\mathbb{R}}{\mathbb{P}}^2$ structures, we need to consider a particular subset ${\mathcal{C}}(S)^{\mathrm{adm}}\subset{\mathcal{C}}(S)$ that satisfy an admissibility condition (see Definition \[def:admissible\]). The admissibility condition is a condition on the ends of convex ${\mathbb{R}}{\mathbb{P}}^2$ surfaces, which arises naturally in the course of studying degenerations of convex ${\mathbb{R}}{\mathbb{P}}^2$ structures. In the case when $S$ is closed, we have the following theorem. The first statement is due to Choi-Goldman [@ChoiGo1] while the second is due to Kuiper [@Kuiper54].
\[thm: determined by holonomy\] Suppose that $S$ is a closed oriented surface. Then
1. $\mathrm{hol}:{\mathcal{C}}(S)\to{\mathcal{X}}_3(S)$ is a homeomorphism onto a connected component of ${\mathcal{X}}_3(S)$, usually known as the ${\mathrm{PGL}}(3,{\mathbb{R}})$-Hitchin component.
2. for every $\gamma\in\pi_1(S)\setminus\{{\mathrm{id}}\}$ and for every $\mu\in{\mathcal{C}}(S)$, the conjugacy class $\mathrm{hol}_\mu(\gamma)$ contains a diagonal representative with pairwise distinct eigenvalues of the same sign.
Both statements in this theorem fail if $S$ has punctures. However, we still have the following results of Marquis [@Mar10].
\[thm: holonomy1\] Let $\mu\in{\mathcal{C}}(S)$.
1. If $\gamma\in\pi_1(S)\setminus\{{\mathrm{id}}\}$ is a non-peripheral element, then the conjugacy class $\mathrm{hol}_\mu(\gamma)$ contains a diagonal representative with pairwise distinct and positive eigenvalues.
2. If $\gamma\in\pi_1(S)$ is a peripheral element, then the conjugacy class $\mathrm{hol}_\mu(\gamma)$ has to contain $$\left[\begin{array}{ccc}
1&1&0\\
0&1&1\\
0&0&1
\end{array}\right],\,\,\,\left[\begin{array}{ccc}
\lambda_1&0&0\\
0&\lambda_2&1\\
0&0&\lambda_2
\end{array}\right]\,\,\,\text{or}\,\,\,\left[\begin{array}{ccc}
\lambda_1&0&0\\
0&\lambda_2&0\\
0&0&\lambda_3
\end{array}\right]$$ for some pairwise distinct and positive $\lambda_1,\lambda_2,\lambda_3$ .
The three group elements in ${\mathrm{PGL}}(3,{\mathbb{R}})$ listed in (2) of Theorem \[thm: holonomy1\] are known as the *standard parabolic element*, the *standard quasi-hyperbolic element* and the *standard hyperbolic element* respectively. Any group element in ${\mathrm{PGL}}(3,{\mathbb{R}})$ is *parabolic*, *quasi-hyperbolic* or *hyperbolic* if it is conjugate to the standard parabolic, quasi-hyperbolic or hyperbolic element respectively.
Motivated by the previous theorem, we will now restrict ourselves to convex ${\mathbb{R}}{\mathbb{P}}^2$ structures on $S$ whose ends are either parabolic type, quasi-hyperbolic type, or bulge $\pm\infty$ type, as described below.
### Parabolic type
Let $g\in{\mathrm{PGL}}(3,{\mathbb{R}})$ be the standard parabolic element. Note that $g$ has a unique fixed point $q=[1,0,0]^T\in{\mathbb{R}}{\mathbb{P}}^2$ and stabilizes a projective line $l=[0,0,1]$ through $q$. Choose any point $p\in{\mathbb{R}}{\mathbb{P}}^2\setminus l$ and let $l'$ be the projective line segment with endpoints $p$ and $g\cdot p$ that does not intersect $l$. Then the closed curve $$q\cup\left(\bigcup_{i=-\infty}^\infty g^i\cdot l'\right)$$ bounds a properly convex domain $\Omega_p\subset{\mathbb{R}}{\mathbb{P}}^2$ that is invariant under the $\langle g\rangle$-action, where $\langle g\rangle$ denotes the cyclic group generated by $g$ (see Figure \[parabolic-figure\]). Let $\Sigma_p:=\langle g\rangle\backslash\Omega_p$, and observe that $\Sigma_p$ is a convex ${\mathbb{R}}{\mathbb{P}}^2$ surface. Using $\Sigma_p$, we describe the first type of end that we allow the convex ${\mathbb{R}}{\mathbb{P}}^2$ surfaces we consider to have.
A puncture on a convex ${\mathbb{R}}{\mathbb{P}}^2$ surface is of *parabolic type* if there is some $p\in{\mathbb{R}}{\mathbb{P}}^2\setminus l$ so that $\Sigma_p$ is projectively isomorphic to a neighborhood of the puncture.
One can verify that if $\Sigma$ is a convex ${\mathbb{R}}{\mathbb{P}}^2$ surface with a puncture and $\gamma\in\pi_1(\Sigma)$ is a peripheral element corresponding to this puncture, then this puncture is of parabolic type if and only if some (equiv. any) holonomy representation of $\Sigma$ evaluated at $\gamma$ is parabolic.
![Parabolic[]{data-label="parabolic-figure"}](parabolic.eps)
### Quasi-hyperbolic type
Let $g\in{\mathrm{PGL}}(3,{\mathbb{R}})$ be a standard quasi-hyperbolic element. Observe that $g$ stabilizes two projective lines $l_1=l_1(g)$ and $l_2=l_2(g)$, and has two fixed points $q_0,q_1\in{\mathbb{R}}{\mathbb{P}}^2$, where $q_1\in l_1\cap l_2$ and $q_0\in l_2\setminus l_1$.
For any $p\in{\mathbb{R}}{\mathbb{P}}^2\setminus (l_1\cup l_2)$, let $l_3'$ be the projective line segment between $p$ and $g\cdot p$ that does not intersect $l_1$, and observe that there is a projective line segment $l_2'$ with endpoints $q_0$ and $q_1$ so that the closed curve $$\left(\bigcup_{i=-\infty}^\infty g^i\cdot l_3'\right)\cup l_2'$$ bounds a properly convex domain $\Omega_{p}\subset{\mathbb{R}}{\mathbb{P}}^2$ that is invariant under the $\langle g\rangle$-action (see Figure \[quasi-hyp-figure\]). This then defines a convex ${\mathbb{R}}{\mathbb{P}}^2$ surface $\Sigma_{g,p}:=\langle g\rangle\backslash\Omega_{p}$.
A puncture on a convex ${\mathbb{R}}{\mathbb{P}}^2$ surface is of *quasi-hyperbolic type* if there is some standard quasi-hyperbolic element $g\in{\mathrm{PGL}}(3,{\mathbb{R}})$ and some point $p\in{\mathbb{R}}{\mathbb{P}}^2\setminus (l_1(g)\cup l_2(g))$ so that $\Sigma_{g,p}$ is projectively isomorphic to a neighborhood of the puncture.
One can verify that if $\Sigma$ is a convex ${\mathbb{R}}{\mathbb{P}}^2$ surface with a puncture and $\gamma\in\pi_1(\Sigma)$ is a peripheral element corresponding to this puncture, then this puncture is of quasi-hyperbolic type if and only if some (equiv. any) holonomy representation of $\Sigma$ evaluated at $\gamma$ is quasi-hyperbolic.
![Quasi-Hyperbolic[]{data-label="quasi-hyp-figure"}](quasi-hyp.eps)
### Bulge $\pm\infty$ type
Let $g\in{\mathrm{PGL}}(3,{\mathbb{R}})$ be a standard hyperbolic element. Observe that $g$ has an attracting fixed point $q_1\in{\mathbb{R}}{\mathbb{P}}^2$, a repelling fixed point $q_3\in{\mathbb{R}}{\mathbb{P}}^2$, and a third fixed point $q_2\in{\mathbb{R}}{\mathbb{P}}^2$, which we refer to as the *saddle fixed point*. Let $l_1=l_1(g)$, $l_2=l_2(g)$, $l_3=l_3(g)$ be the projective lines through $q_1$ and $q_2$, $q_2$ and $q_3$, $q_3$ and $q_1$ respectively.
Choose any point $p\in{\mathbb{R}}{\mathbb{P}}^2\setminus(l_1\cup l_2\cup l_3)$, and let $l_4'$ be the projective line segment with endpoints $p,g\cdot p$ that does not intersect $l_1$. Then $l_4'$ lies in an open triangle $\Delta_p$ which is a connected component of ${\mathbb{R}}{\mathbb{P}}^2\setminus(l_1\cup l_2\cup l_3)$. For $i=1,2,3$, let $l_i':=\partial\Delta_p\cap l_i$, and let $l_i''$ be the closure of $l_i\setminus l_i'$. Note that the closed curves $$l_3'\cup\bigcup_{i=-\infty}^\infty g^i\cdot l_4'\,\,\,\,\text{and}\,\,\,\,l_1''\cup l_2''\cup\bigcup_{i=-\infty}^\infty g^i\cdot l_4'$$ both bound properly convex domains $\Omega_{p}'$ (see Figure \[bulge-minus-figure\]) and $\Omega_{p}''$ (see Figure \[bulge-plus-figure\]) respectively, which are invariant under the $\langle g\rangle$-action. Let $\Sigma_{g,p}':=\langle g\rangle\backslash\Omega_{p}'$ and $\Sigma_{g,p}'':=\langle g\rangle\backslash\Omega_{p}''$.
A puncture on a convex ${\mathbb{R}}{\mathbb{P}}^2$ surface $\Sigma$ is of *bulge $-\infty$ type* (resp. *bulge $+\infty$ type*) if there some standard hyperbolic element $g\in{\mathrm{PGL}}(3,{\mathbb{R}})$ and some $p\in{\mathbb{R}}{\mathbb{P}}^2\setminus (l_1(g)\cup l_2(g)\cup l_3(g))$ so that $\Sigma_{g,p}'$ (resp. $\Sigma_{g,p}''$) is projective isomorphic to a neighborhood of the puncture.
![Bulge $-\infty$[]{data-label="bulge-minus-figure"}](bulge-minus.eps)
![Bulge $+\infty$[]{data-label="bulge-plus-figure"}](bulge-plus.eps)
If $\Sigma$ is a convex ${\mathbb{R}}{\mathbb{P}}^2$ surface with a puncture and $\gamma\in\pi_1(\Sigma)$ is a peripheral element corresponding to this puncture, then the holonomy representation evaluated at $\gamma$ does not determine the type of the puncture. This is unlike the case when the puncture is of parabolic type or quasi-hyperbolic type.
Let $[f,\Sigma]\in{\mathcal{C}}(S)$. For every puncture $p$ of $S$, the diffeomorphism $f$ sends a neighborhood of $p$ to a neighborhood of some puncture $q$ of $\Sigma$. We will abuse notation and denote $q=f(p)$.
\[def:admissible\]
1. A convex ${\mathbb{R}}{\mathbb{P}}^2$ surface $\Sigma$ is *admissible* if every puncture of $\Sigma$ is either of parabolic type, quasi-hyperbolic type, bulge $\infty$ type, or bulge $-\infty$ type. Similarly, $[f,\Sigma]\in{\mathcal{C}}(S)$ is *admissible* if $\Sigma$ is admissible.
2. Let ${\mathcal{C}}(S)^{\mathrm{adm}}:=\{\mu\in{\mathcal{C}}(S):\mu\text{ is admissible}\}$. For any $\mu=[f,\Sigma]\in{\mathcal{C}}(S)^\mathrm{adm}$ and any puncture $p$ of $S$, the *$\mu$-type of $p$* is the type of $f(p)$.
We equip ${\mathcal{C}}(S)^\mathrm{adm}\subset{\mathcal{C}}(S)$ with the subspace topology.
The augmented deformation space. {#sec: aug}
--------------------------------
Next, we describe the *augmented deformation space of admissible convex ${\mathbb{R}}{\mathbb{P}}^2$ structures*, which was previously introduced and studied by the first author [@Loftin].
In [@Loftin], this augmented deformation space was constructed for closed $S$. However, the construction extends easily to general $S$ as long as we only consider admissible convex ${\mathbb{R}}{\mathbb{P}}^2$ structures.
We begin by describing multi-curves ${\mathcal{D}}$ in $S$.
A *multi-curve* is a collection of simple closed curves in $S$ that are pairwise non-intersecting, pairwise non-homotopic, non-contractible and non-peripheral. We allow $\emptyset$ as a multi-curve.
A multi-curve in $S$ is a collection of non-peripheral closed curves corresponding to the vertices of a simplex in the curve complex equipped with its usual simplicial structure.
Let ${\mathcal{D}}$ be a multi-curve in $S$, and let $S_1,\dots,S_k$ be the connected components of $S\setminus{\mathcal{D}}$. Consider a closed curve $c\in{\mathcal{D}}$, and choose an orientation on $c$. Let $S_i,S_j$ be the connected components of $S\setminus{\mathcal{D}}$ that that lie on the left and right of $c$ respectively (it is possible that $S_i=S_j$). The orientation on $c$ determines two conjugacy classes of group elements $[\gamma_L]\in[\pi_1(S_i)]$ and $[\gamma_R]\in[\pi_1(S_j)]$ on the left and right of $c$ respectively. Let $p_L$ and $p_R$ be the punctures of $S_i$ and $S_j$ respectively that correspond to $c$.
A tuple $(\mu_1,\dots,\mu_k)\in\prod_{j=1}^k{\mathcal{C}}(S_j)^{\mathrm{adm}}$ is *compatible across $c$* if $\mathrm{hol}_{\mu_i}(\gamma_L)$ is conjugate to $\mathrm{hol}_{\mu_j}(\gamma_R)$, and the $\mu_i$-type of $p_L$ is bulge $\pm\infty$ if and only if the $\mu_j$-type of $p_R$ is bulge $\mp\infty$.
In particular, if $(\mu_1,\dots,\mu_k)$ is compatible across $c$, then the $\mu_i$-type of $p_L$ is parabolic (resp. quasi-hyperbolic) if and only if the $\mu_j$-type of $p_R$ is parabolic (resp. quasi-hyperbolic).
Denote $${\mathcal{C}}(S,{\mathcal{D}})^\mathrm{adm}:=\left\{\mu\in\prod_{j=1}^k{\mathcal{C}}(S_j)^{\mathrm{adm}}:\mu\text{ is compatible across }c \text{ for all }c\in{\mathcal{D}}\right\}$$ and $${\mathcal{C}}(S)^\mathrm{aug}:=\bigcup_{{\mathcal{D}}\text{ a multi-curve}}{\mathcal{C}}(S,{\mathcal{D}})^\mathrm{adm}.$$ We will refer to each ${\mathcal{C}}(S,{\mathcal{D}})^\mathrm{adm}\subset{\mathcal{C}}(S)^\mathrm{aug}$ as the ${\mathcal{D}}$-*stratum* of ${\mathcal{C}}(S)^\mathrm{aug}$.
A *regular* marked convex ${\mathbb{RP}}^2$ structure on $S$ is a point in ${\mathcal{C}}(S)^{\mathrm{aug}}$.
Intuitively, one can think of ${\mathcal{C}}(S)^\mathrm{aug}$ as an “augmentation" of ${\mathcal{C}}(S)^\mathrm{adm}$ where we include all possible degenerations of the convex ${\mathbb{R}}{\mathbb{P}}^2$ structure on $S$ that converge on the complement of a multi-curve (see Theorem 2.5.1 in [@Loftin]). This is an analog of the augmented Teichmüller space, denoted ${\mathcal{T}}(S)^\mathrm{aug}$, for the deformation space of convex real projective structures. In fact, this construction of ${\mathcal{C}}(S)^\mathrm{aug}$, when restricted only to regular structures $(\mu_1,\dots,\mu_k)$ where the developing image of each $\mu_i$ is the Klein model of hyperbolic plane, gives the usual construction of ${\mathcal{T}}(S)^\mathrm{aug}$. From this, it is clear that ${\mathcal{T}}(S)^\mathrm{aug}$ naturally embeds in ${\mathcal{C}}(S)^\mathrm{aug}$. The case of ${\mathcal{T}}(S)^\mathrm{aug}$ is much simpler though, since the type of each puncture of $\mu_i$ is parabolic.
The topology on the augmented deformation space. {#sec: aug top}
------------------------------------------------
We will now define a topology on ${\mathcal{C}}(S)^\mathrm{aug}$. To do so, it is convenient to introduce the following terminology.
Let $\mu\in{\mathcal{C}}(S,{\mathcal{D}})^\mathrm{adm}$ and choose orientations on the closed curves $c\in{\mathcal{D}}$. Let $S'$ be the connected component of $S\setminus{\mathcal{D}}$ that lies to the left of $c$, and let $p$ be the puncture of $S'$ corresponding to $c$. Then the $\mu$-*type* of $c$ is the $\mu$-type of $p$.
Observe that the $\mu$-type of the oriented simple closed curve $c$ is parabolic, quasi-hyperbolic or bulge $\pm\infty$ if and only if the $\mu$-type of $c^{-1}$ is parabolic, quasi-hyperbolic or bulge $\mp\infty$ respectively.
The key to defining the topology on ${\mathcal{C}}(S)^\mathrm{aug}$ are the *pulling maps* that were introduced in [@Loftin]. As before, let ${\mathcal{D}}$ be any multi-curve in $S$ and let $S_1,\dots,S_k$ be the connected components of $S\setminus{\mathcal{D}}$. By making appropriate choice of base points, the inclusion $S_i\subset S$ identifies $\pi_1(S_i)$ as a subgroup of $\pi_1(S)$. Observe that there is a homeomorphism $g_i:S_i\to\widetilde{S}/\pi_1(S_i)$ so that the induced map on fundamental groups $(g_i)_*:\pi_1(S_i)\to\pi_1(\widetilde{S}/\pi_1(S_i))$ is an isomorphism of groups. The homeomorphism $g_i$ is unique up to homotopy. Let $\widetilde{g}_i:\widetilde{S}_i\to\widetilde{S}$ denote the lift of $g_i$. For any $\nu_i\in{\mathcal{C}}(S_i)$, if $(\phi,\rho)$ is a developing pair for $\nu_i$, then $\phi\circ\widetilde{g}_i:\widetilde{S}_i\to\mathbb{RP}^2$ is $\rho\circ (g_i)_*$-equivariant. Then Theorem \[thm:dev\] implies that $(\phi\circ\widetilde{g}_i,\rho\circ (g_i)_*)$ is the developing pair for some $\mu_i\in{\mathcal{C}}(S_i)$. Since $g_i$ is unique up to homotopy, $\mu_i$ does not depend on the choice of $g_i$. Hence, this allows us to define the ${\mathcal{D}}$-*pulling map* $$\mathrm{Pull}_{\mathcal{D}}:{\mathcal{C}}(S)\to\prod_{i=1}^k{\mathcal{C}}(S_i)$$ by $\mathrm{Pull}_{\mathcal{D}}(\mu)=(\mu_1,\dots,\mu_k)$. It is important to emphasize that each $\mu_i$ here has a representative $\rho_i\in\mathrm{hol}(\mu_i)$ so that the $\rho_i$-equivariant developing image of $\mu_i$ agree for all $i$. Note that if ${\mathcal{D}}$ is non-empty, then every $\mu_i$ is not admissible even if $\mu$ is admissible.
Next, if ${\mathcal{D}}'\subset{\mathcal{D}}$ are multi-curves in $S$, let $S_1,\dots,S_k$ be the connected components of $S\setminus{\mathcal{D}}$ and $S_1',\dots,S_{k'}'$ be the connected components of $S\setminus{\mathcal{D}}'$. Note that for all $i=1,\dots,k$, there is some $j=1,\dots,k'$ so that $S_i\subset S_j'$. Let ${\mathcal{D}}(S_i')$ denote the curves in ${\mathcal{D}}$ that lie in $S_i'$ but are non-peripheral in $S_i'$. This allows us to define the $({\mathcal{D}}',{\mathcal{D}})$-*pulling map* $$\mathrm{Pull}_{{\mathcal{D}}',{\mathcal{D}}}:\prod_{i=1}^{k'}{\mathcal{C}}(S_i')\to\prod_{i=1}^k{\mathcal{C}}(S_i)$$ by $\mathrm{Pull}_{{\mathcal{D}}',{\mathcal{D}}}(\mu_1,\dots,\mu_{k'}):=\Big(\textrm{Pull}_{{\mathcal{D}}(S_1')}(\mu_1),\dots,\mathrm{Pull}_{{\mathcal{D}}(S_{k'}')}(\mu_{k'})\Big)$.
Using the pulling maps, we can now define a basis for the topology on ${\mathcal{C}}(S)^\mathrm{aug}$. For any open $U\subset\prod_{i=1}^k{\mathcal{C}}(S_i)$ that intersects ${\mathcal{C}}(S,{\mathcal{D}})^\mathrm{adm}\subset\prod_{i=1}^k{\mathcal{C}}(S_i)$, let $${\mathcal{U}}(U,{\mathcal{D}}):=\bigcup_{{\mathcal{D}}'\subset{\mathcal{D}}}\Bigg(\mathrm{Pull}_{{\mathcal{D}}',{\mathcal{D}}}^{-1}(U)\cap{\mathcal{C}}(S,{\mathcal{D}}')^\mathrm{adm}\Bigg).$$ Note that ${\mathcal{U}}(U,{\mathcal{D}})\subset{\mathcal{C}}(S)^\mathrm{aug}$, and define $${\mathcal{A}}(S):=\left\{
{\mathcal{U}}(U,{\mathcal{D}}):\begin{array}{l}{\mathcal{D}}\text{ is a multi-curve in }S,\\
S_1,\dots,S_k\text{ are the connected components of }S\setminus{\mathcal{D}},\\
U\subset\prod_{i=1}^k{\mathcal{C}}(S_i)$ is an open set that intersects ${\mathcal{C}}(S,{\mathcal{D}})^\mathrm{adm}
\end{array}\right\}.$$ Then ${\mathcal{A}}(S)$ gives a basis for the topology on ${\mathcal{C}}(S)^{\rm aug}$. (Recall we allow the choice of ${\mathcal{D}}= \emptyset$.)
The topology on ${\mathcal{C}}(S)^\mathrm{aug}$ as defined is rather abstract. The philosophical purpose of this paper of this paper is to understand this topology in a concrete way. We begin by observing that this topology has several important features that we will record as the following preliminary remarks.
\[rem: top\]
1. For any $\mu\in{\mathcal{C}}(S)^\mathrm{aug}$, let ${\mathcal{D}}$ be the multi-curve so that $\mu\in{\mathcal{C}}(S,{\mathcal{D}})^\mathrm{adm}$, and let $S_1,\dots,S_k$ be the connected components of $S\setminus{\mathcal{D}}$. Then $${\mathcal{C}}(S)^{\mathrm{aug},{\mathcal{D}}}=V_{\mathcal{D}}:={\mathcal{U}}\left(\prod_{i=1}^k{\mathcal{C}}(S_i),{\mathcal{D}}\right)=\bigcup_{{\mathcal{D}}'\subset{\mathcal{D}}}{\mathcal{C}}(S,{\mathcal{D}}')^\mathrm{adm}$$ is an open set in ${\mathcal{C}}(S)^\mathrm{aug}$ that contains $\mu$. In particular, if a sequence $\{\mu^j\}_{j=1}^\infty\subset{\mathcal{C}}(S)^\mathrm{aug}$ converges to $\mu$, then by removing finitely many points from this sequence, we may assume that $\{\mu^j\}_{j=1}^\infty\subset V_{\mathcal{D}}$. Hence, there is some ${\mathcal{D}}^j\subset{\mathcal{D}}$ so that $\mu^j\in{\mathcal{C}}(S,{\mathcal{D}}^j)$, so $\mu$ and $\mathrm{Pull}_{{\mathcal{D}}^j,{\mathcal{D}}}(\mu^j)$ are of the form $$\mu=(\mu_1,\dots,\mu_k)\,\,\text{ and }\,\,\mathrm{Pull}_{{\mathcal{D}}^j,{\mathcal{D}}}(\mu^j)=(\mu^j_1,\dots,\mu^j_k)$$ for some $\mu_i,\mu^j_i\in{\mathcal{C}}(S_i)$. From the definition of the topology on ${\mathcal{C}}(S)^\mathrm{aug}$, one observes that $\lim_{j\to\infty}\mu^j=\mu$ in ${\mathcal{C}}(S)^\mathrm{aug}$ if and only if $\lim_{j\to\infty}\mu^j_i=\mu_i$ in ${\mathcal{C}}(S_i)$ for all $i=1,\dots,k$.
2. Let ${\mathcal{D}}$ be a multi-curve on $S$ and $S_1,\dots,S_k$ be the connected components of $S\setminus{\mathcal{D}}$. The holonomy map $\mathrm{hol}:{\mathcal{C}}(S_j)\to{\mathcal{X}}_3(S_j)$ extends to the map $$\begin{aligned}
\mathrm{hol}:\prod_{j=1}^k{\mathcal{C}}(S_j)&\to&\prod_{i=1}^k{\mathcal{X}}_3(S_j)\\
(\mu_1,\dots,\mu_k)&\mapsto&(\mathrm{hol}(\mu_1),\dots,\mathrm{hol}(\mu_k)).\end{aligned}$$ Restricting this to ${\mathcal{C}}(S,{\mathcal{D}})^\mathrm{adm}$ defines a continuous map $$\mathrm{hol}:{\mathcal{C}}(S,{\mathcal{D}})^\mathrm{adm}\to\prod_{i=1}^k{\mathcal{X}}_3(S_j).$$
3. This topology on ${\mathcal{C}}(S)^\mathrm{aug}$ is first countable. This was verified by the first author [@Loftin].
4. From the definition of the usual topology on the augmented Teichmüller space ${\mathcal{T}}(S)^\mathrm{aug}$ (see [@Abi]), it is easy to see that the natural inclusion of ${\mathcal{T}}(S)^\mathrm{aug}$ into ${\mathcal{C}}(S)^\mathrm{aug}$ as described above is a homeomorphism onto its image.
5. Let ${\mathcal{D}}$ be any non-empty multicurve in $S$, let $S_1,\dots,S_k$ be the connected components of $S\setminus{\mathcal{D}}$, and let $\mu=(\mu_1,\dots,\mu_k)\in{\mathcal{C}}(S,{\mathcal{D}})^\mathrm{adm}$. By the definition of the topology on ${\mathcal{C}}(S)^\mathrm{aug}$, for every open set $V\subset{\mathcal{C}}(S)^\mathrm{aug}$ containing $\mu$, there is some nonempty open $U\subset\prod_{i=1}^k{\mathcal{C}}(S_i)$ so that ${\mathcal{U}}(U,{\mathcal{D}})$ is non-empty and lies in $V$. Observe that the intersection of any fiber of the map $\mathrm{Pull}_{{\mathcal{D}}',{\mathcal{D}}}$ with ${\mathcal{C}}(S,{\mathcal{D}}')^\mathrm{adm}$ is either empty or non-compact. This implies that the closure of ${\mathcal{U}}(U,{\mathcal{D}})$ in ${\mathcal{C}}(S)^{\mathrm{aug}}$ is not compact, so the same holds for the closure of $V$ in ${\mathcal{C}}(S)^{\mathrm{aug}}$. Hence, ${\mathcal{C}}(S)^\mathrm{aug}$ is not locally compact.
6. The mapping class group $$\mathrm{MCG}(S):=\mathrm{Diffeo(S)^+}/\mathrm{Diffeo(S)^+_0}$$ acts naturally on the set of multi-curves on $S$. For any multi-curve ${\mathcal{D}}$ on $S$ and any $[g]\in{\mathrm{MCG}}(S)$, we may define ${\mathcal{D}}':=[g]\cdot{\mathcal{D}}$. Let $S_1,\dots,S_k$ be the connected components of $S\setminus{\mathcal{D}}$, and let $S_1',\dots,S_k'$ be the connected components of $S\setminus{\mathcal{D}}'$. One can verify that the map $$[g]:\prod_{i=1}^k{\mathcal{C}}(S_i)\to\prod_{i=1}^k{\mathcal{C}}(S_i')$$ defined by $\big([f_1,\Sigma_1],\dots,[f_k,\Sigma_k]\big)\mapsto\big([f_1\circ g,\Sigma_1],\dots,[f_k\circ g,\Sigma_k]\big)$ is a homeomorphism, so it restricts to a homeomorphism $[g]:{\mathcal{C}}(S,{\mathcal{D}})^\mathrm{adm}\to{\mathcal{C}}(S,{\mathcal{D}}')^\mathrm{adm}$. This in turn defines a homeomorphism $[g]:{\mathcal{C}}(S)^\mathrm{aug}\to{\mathcal{C}}(S)^\mathrm{aug}$. Thus, ${\mathrm{MCG}}(S)$ acts on ${\mathcal{C}}(S)^\mathrm{aug}$ by homeomorphisms. The first author [@Loftin] showed that the quotient $\overline{{\mathcal{M}}(S)}:={\mathcal{C}}(S)^\mathrm{aug}/\mathrm{MCG}(S)$ is topologically an orbifold, albeit with a complicated singular locus.
Describing $\mathrm{hol}({\mathcal{C}}(S))$ {#sec: hol}
===========================================
There is a well-known coordinate system on $\mathrm{hol}({\mathcal{C}}(S))$ that was originally due to Goldman [@Go90], and later modified by the second author [@Zhang16] using the work of Fock-Goncharov [@FocGon] and Bonahon-Dreyer [@BonDre]. These coordinates play a key role in our description of the topology on ${\mathcal{C}}(S)^\mathrm{aug}$, so we will devote this section to carefully constructing them.
Projective invariants
---------------------
We can think of ${\mathbb{R}}{\mathbb{P}}^2$ as the set of projective classes of vectors in ${\mathbb{R}}^3$. Similarly, $({\mathbb{R}}{\mathbb{P}}^2)^*$ can be thought of either as the set of projective classes of linear functionals in $({\mathbb{R}}^3)^*$, the set of projective lines in ${\mathbb{R}}{\mathbb{P}}^2$, or the set of projectivized planes through the origin in ${\mathbb{R}}^3$. Throughout the rest of this paper, we will assume these identifications without further comment.
Given any $L_1,L_2\in({\mathbb{R}}{\mathbb{P}}^2)^*$ and $p_1,p_2\in{\mathbb{R}}{\mathbb{P}}^2$ so that $p_i\notin L_j$ for all $i,j=1,2$, we can define the *cross ratio* $$C(L_1,p_1,p_2,L_2):=\frac{L_1(p_2)L_2(p_1)}{L_1(p_1)L_2(p_2)}.$$ Here, we choose a linear functional representative for each $L_i$ and a vector representative for each $p_j$ to evaluate $L_i(p_j)$. One can verify from the definition of the cross ratio that the choice of representatives is irrelevant. Furthermore, the cross ratio is a projective invariant, and satisfies the symmetries $$C(L_1,p_1,p_2,L_2)=C(L_2,p_2,p_1,L_1)=\frac{1}{C(L_1,p_2,p_1,L_2)}.$$
\[rem:C+\] Note that if $L_1=L_2$ or $p_1=p_2$ (or both), then $C(L_1,p_1,p_2,L_2)=1$. On the other hand, if $L_1\neq L_2$, then ${\mathbb{R}}{\mathbb{P}}^2\setminus(L_1\cup L_2)$ has two connected components, and $C(L_1,p_1,p_2,L_2)$ is positive (resp. negative) if and only if $p_1$ and $p_2$ lie in the same (resp. different) connected component of ${\mathbb{R}}{\mathbb{P}}^2\setminus(L_1\cup L_2)$.
Similarly, given any triple $L_1,L_2,L_3\in({\mathbb{R}}{\mathbb{P}}^2)^*$ and $p_1,p_2,p_3\in{\mathbb{R}}{\mathbb{P}}^2$ so that $p_i\notin L_{i-1}\cup L_{i+1}$ for all $i=1,2,3$ (arithmetic in the subscripts is done modulo 3), we can define the triple ratio $$T(L_1,L_2,L_3,p_1,p_2,p_3):=\frac{L_1(p_2)L_2(p_3)L_3(p_1)}{L_1(p_3)L_3(p_2)L_2(p_1)}.$$ As before, we choose linear functional representatives of the $L_i$ to and vector representatives of reach $p_j$ to evaluate $L_i(p_j)$, and one can verify that the independence of the triple ratio from these choices. The triple ratio is also a projective invariant satisfying the symmetries $$T(L_1,L_2,L_3,p_1,p_2,p_3)=T(L_2,L_3,L_1,p_2,p_3,p_1)=\frac{1}{T(L_2,L_1,L_3,p_2,p_1,p_3)}.$$
\[rem:T+\] Note that if $L_1$, $L_2$ and $L_3$ do not intersect at a common point, then ${\mathbb{R}}{\mathbb{P}}^2\setminus(L_1\cup L_2\cup L_3)$ has four connected components, each of which is a triangle. Further suppose that $p_i\in L_i$ for $i=1,2,3$, i.e. $(p_i,L_i)$ is a flag (see Section \[sec: limit maps\]). In this situation, it is straightforward to check that $T(L_1,L_2,L_3,p_1,p_2,p_3)$ is positive if and only if one of these four connected components contains all of $p_1$, $p_2$, $p_3$ in its boundary.
Ideal triangulations and pants decompositions {#sec: triangulation}
---------------------------------------------
Next, we will describe a particular ideal triangulation on $S$ that one can associate to any pants decomposition of $S$. We begin by precisely defining the notion of an ideal triangulation on the topological surface $S$.
Since $S$ has negative Euler characteristic, it is well-known that $\pi_1(S)$ is a hyperbolic group, and its Gromov boundary $\partial\pi_1(S)$ has a natural cyclic order induced by the orientation on $S$. More concretely, if we choose a convex cocompact hyperbolic metric on $S$, then the universal cover $\widetilde{S}$ of $S$ can be identified with the Poincaré disc ${\mathbb{D}}$ as oriented Riemannian metric spaces. For any $p\in\widetilde{S}={\mathbb{D}}$, the orbit map $\pi_1(S)\to{\mathbb{D}}$ defined by $\gamma\mapsto\gamma\cdot p$ is a quasi-isometric embedding, so it extends to an embedding of $\partial\pi_1(S)$ into $\partial{\mathbb{D}}^2$. The orientation on ${\mathbb{D}}\simeq\widetilde{S}$ then induces a counter-clockwise cyclic ordering on $\partial{\mathbb{D}}^2$, which restricts to a cyclic ordering on $\partial\pi_1(S)$. One can then verify that this cyclic ordering on $\partial\pi_1(S)$ does not depend on any of the choices made.
A *geodesic* on $\widetilde{S}$ is an (unordered) pair of distinct points $\{x,y\}\subset\partial\pi_1(S)$, so that $\{x,y\}$ is not the set of fixed points of some peripheral $\gamma\in\pi_1(S)$.
Denote the space of geodesics on $\widetilde{S}$ by ${\mathcal{G}}(\widetilde{S})$, and note that the natural $\pi_1(S)$ action on $\partial\pi_1(S)$ induces a $\pi_1(S)$ action on ${\mathcal{G}}(\widetilde{S})$. Also, we say that two geodesics $\{x,y\}$ and $\{x',y'\}$ *intersect transversely* if $x<x'<y<y'<x$ or $x<y'<y<x'<x$ in $\partial\pi_1(S)$.
An *ideal triangulation* $\widetilde{{\mathcal{T}}}\subset\widetilde{{\mathcal{G}}}(S)$ on $\widetilde{S}$ is then a maximal, $\pi_1(S)$-invariant collection of geodesics that pairwise do not intersect transversely, with the property that every $\{x,y\}\in\widetilde{{\mathcal{T}}}$ satisfies one of the following:
- $\{x,y\}$ is the set of fixed points of some $\gamma\in\pi_1(S)$
- there are points $z,w\in\partial\pi_1(S)$ so that $\{z,x\},\{z,y\},\{w,x\},\{w,y\}\in\widetilde{{\mathcal{T}}}$.
If the former holds, then $\{x,y\}$ is a *closed edge* of $\widetilde{{\mathcal{T}}}$. On the other hand, if the latter holds, then $\{x,y\}$ is a *isolated edge* of $\widetilde{{\mathcal{T}}}$. Every $\{x,y\}\in\widetilde{{\mathcal{T}}}$ is an *edge*, and $x,y$ are the *vertices* of the edge $\{x,y\}$. We can then define an *ideal triangulation* of $S$ to be the quotient ${\mathcal{T}}:=\widetilde{{\mathcal{T}}}/\pi_1(S)$ of some ideal triangulation of $\widetilde{{\mathcal{T}}}$ of $\widetilde{S}$, and let $[x,y]$ denote the element in ${\mathcal{T}}$ with $\{x,y\}$ as a representative.
1. For every $\gamma\in\pi_1(S)$, let $\gamma^-$, $\gamma^+\in\partial\pi_1(S)$ denote the attracting and repelling fixed points of $\gamma$ respectively. It is important to emphasize that by our definition of geodesics, if $\gamma\in\pi_1(S)$ is a peripheral group element, then $\{\gamma^-,\gamma^+\}$ is not a geodesic, and hence not an edge in ${\mathcal{T}}$. We use this convention as it will be convenient for our purposes later.
2. If one chooses a convex cocompact hyperbolic metric $\Sigma$ on $S$, then every ideal triangulation on $S$ is realized geometrically as an ideal triangulation of the convex core of the hyperbolic surface $\Sigma$ in the classical sense.
An *ideal triangle* of the ideal triangulation $\widetilde{{\mathcal{T}}}$ is a triple $\{x,y,z\}\subset\partial\pi_1(S)$ so that $\{x,y\},\{x,z\},\{y,z\}\in\widetilde{{\mathcal{T}}}$. We will refer to $x,y,z$ as the *vertices* of the ideal triangle $\{x,y,z\}$. Denote the set of ideal triangles of $\widetilde{{\mathcal{T}}}$ by $\widetilde{\Theta}_{\widetilde{{\mathcal{T}}}}=\widetilde{\Theta}$, and let $\widetilde{{\mathcal{V}}}=\widetilde{{\mathcal{V}}}_{\widetilde{{\mathcal{T}}}}\subset\partial\pi_1(S)$ denote the set of vertices of the ideal triangles in $\widetilde{\Theta}$.
Denote $\Theta:=\widetilde{\Theta}/\pi_1(S)$ and ${\mathcal{V}}:=\widetilde{{\mathcal{V}}}/\pi_1(S)$.
Next, we specialize to particular ideal triangulations coming from pants decompositions of $S$, i.e. maximal multi-curves. For every pants decomposition ${\mathcal{M}}$ of $S$, observe that each connected component of $S\setminus{\mathcal{M}}$ is a pair of pants. Denote this collection of pairs of pants by ${\mathbb{P}}={\mathbb{P}}_{\mathcal{M}}$. For each $P\in{\mathbb{P}}$, let $\alpha_P,\beta_P,\gamma_P\in\pi_1(S)$ be the group elements satisfying the following:
- $\alpha_P\beta_P\gamma_P={\mathrm{id}}$.
- $[\alpha_P]$, $[\beta_P]$, $[\gamma_P]$ correspond to the three boundary components of $P$, oriented so that $P$ lies to the left of the boundary component.
Then let $${\mathcal{Q}}:=\{[\alpha_P^-,\beta_P^-]:P\in{\mathbb{P}}\}\cup\{[\beta_P^-,\gamma_P^-]:P\in{\mathbb{P}}\}\cup\{[\gamma_P^-,\alpha_P^-]:P\in{\mathbb{P}}\},$$ $${\mathcal{P}}:=\{[\gamma^-,\gamma^+]:\gamma\in\pi_1(S)\text{ corresponds to some closed curve in }{\mathcal{M}}\},$$ and define ${\mathcal{T}}_{\mathcal{M}}:={\mathcal{Q}}\cup{\mathcal{P}}$. One may verify that ${\mathcal{T}}_{\mathcal{M}}$ is an ideal triangulation, and that ${\mathcal{P}}$ and ${\mathcal{Q}}$ are respectively the set of closed edges and isolated edges of ${\mathcal{T}}_{\mathcal{M}}$. Also, there is a natural identification between ${\mathcal{M}}$ and ${\mathcal{P}}$, so we will treat them as equal. Note that from the way we defined ${\mathcal{T}}_{\mathcal{P}}$, the vertices of every closed edge in $\widetilde{{\mathcal{T}}}_{\mathcal{P}}$ lie in $\widetilde{{\mathcal{V}}}_{\widetilde{{\mathcal{T}}}_{\mathcal{P}}}$. See Figure \[ideal-triangulation-figure\].
![Ideal Triangulation: $[x,y]\in{\mathcal{P}},\,\,[x,z_i],[y,w_j]\in{\mathcal{Q}}$[]{data-label="ideal-triangulation-figure"}](ideal-triangulation.eps)
We will refer to the elements in ${\mathcal{T}}$, ${\mathcal{P}}$, ${\mathcal{Q}}$, $\Theta$, ${\mathcal{V}}$ respectively as edges, closed edges, isolated edges, ideal triangles, and vertices of the ideal triangulation ${\mathcal{T}}$ of $S$. Also, the cyclic ordering on $\partial\pi_1(S)$ induces a cyclic orientation on $\widetilde{{\mathcal{V}}}$.
Flag maps {#sec: limit maps}
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Let $\mathcal F$ be the space of flags in ${\mathbb{RP}}^2$, i.e. $\mathcal F = \{(v,\ell)\in {\mathbb{RP}}^2 \times ({\mathbb{RP}}^2)^* : v\in\ell\}$. Also, let $\mu\in{\mathcal{C}}(S)$ and let $\rho:\pi_1(S)\to{\mathrm{PGL}}(3,{\mathbb{R}})$ be a representative of the conjugacy class $\mathrm{hol}(\mu)$. Observe that every point $x\in\widetilde{{\mathcal{V}}}$ is a repelling fixed point $\gamma^-$ of some $\gamma\in\pi_1(S)\setminus\{{\mathrm{id}}\}$. This allows us to construct a *flag map* $\xi_\rho:\widetilde{{\mathcal{V}}}\to \mathcal F$ in the following way. By (1) of Theorem \[thm: holonomy1\], we see that one of the following holds for $\rho(\gamma)$.
1. $\rho(\gamma)$ has exactly three fixed points in ${\mathbb{R}}{\mathbb{P}}^2$, one of which is attracting and another is repelling. The third fixed point that is neither attracting nor repelling is called the *saddle fixed point*.
2. $\rho(\gamma)$ has exactly two fixed points in ${\mathbb{R}}{\mathbb{P}}^2$, one of which is repelling. We will refer to the fixed point that is not repelling as the *quasi-attracting fixed point*. $\rho(\gamma)$ also stabilizes a unique line that contains the quasi-attracting fixed point, but not the repelling fixed point.
3. $\rho(\gamma)$ has exactly two fixed points in ${\mathbb{R}}{\mathbb{P}}^2$, one of which is attracting. We will refer to the fixed point that is not attracting as the *quasi-repelling fixed point*. $\rho(\gamma)$ also stabilizes a unique line that contains the quasi-repelling fixed point, but not the attracting fixed point.
4. $\rho(\gamma)$ has a unique fixed point in ${\mathbb{R}}{\mathbb{P}}^2$, and stabilizes a unique line through that fixed point.
\(1) holds when $\rho(\gamma)$ is hyperbolic, (2) or (3) holds when $\rho(\gamma)$ is quasi-hyperbolic, and (4) holds when $\rho(\gamma)$ is parabolic. Theorem \[thm: holonomy1\] implies that (2), (3) and (4) can happen only when $\gamma$ is a peripheral element.
Using this, define $\xi_\rho(x):=\left(\xi_\rho^{(1)}(x),\xi_\rho^{(2)}(x)\right)\in\mathcal F\subset {\mathbb{R}}{\mathbb{P}}^2\times({\mathbb{R}}{\mathbb{P}}^2)^*$ as follows.
- If (1) holds, define $\xi_\rho^{(1)}(x)$ to be the repelling fixed point of $\rho(\gamma)$, and $\xi_\rho^{(2)}(x)$ to be the projective line containing the repelling and saddle fixed points of $\rho(\gamma)$. See Figure \[hyperbolic-flag-figure\].
- If (2) holds, define $\xi_\rho^{(1)}(x)$ to be the repelling fixed point of $\rho(\gamma)$, and $\xi_\rho^{(2)}(x)$ to be the projective line containing both fixed points of $\rho(\gamma)$. See Figure \[quasi-hyp-flag1-figure\].
- If (3) holds, define $\xi_\rho^{(1)}(x)$ to be the quasi-repelling fixed point of $\rho(\gamma)$, and $\xi_\rho^{(2)}(x)$ to be projective line stabilized by $\rho(\gamma)$ that contains its quasi-repelling fixed point but not its attracting fixed point. See Figure \[quasi-hyp-flag2-figure\].
- If (4) holds, define $\xi_\rho^{(1)}(x)$ to be the unique fixed point of $\rho(\gamma)$, and define $\xi_\rho^{(2)}(x)$ to be the unique projective line stabilized by $\rho(\gamma)$. See Figure \[parabolic-flag-figure\].
![Flag for hyperbolic case[]{data-label="hyperbolic-flag-figure"}](hyperbolic-flag.eps)
![Flag for quasi-attracting case[]{data-label="quasi-hyp-flag1-figure"}](quasi-hyp-flag1.eps)
![Flag for quasi-repelling case[]{data-label="quasi-hyp-flag2-figure"}](quasi-hyp-flag2.eps)
![Flag for parabolic case[]{data-label="parabolic-flag-figure"}](parabolic-flag.eps)
Observe that the flag map can actually be defined on all fixed points of all non-identity elements in $\partial\pi_1(S)$. However, we will only consider the map restricted to $\widetilde{{\mathcal{V}}}$.
It is easy to verify in each of these cases that the flag map $\xi_\rho$ is $\rho$-equivariant. Also, for any $x\in\widetilde{{\mathcal{V}}}$, $\xi^{(1)}_\rho(x)$ is either a repelling fixed point, a quasi-repelling fixed point, or the unique fixed point of $\rho(\gamma)$ for some $\gamma\in\pi_1(S)$. It follows that $\xi^{(1)}_\rho(\widetilde{{\mathcal{V}}})$ lies in $\partial\Omega$, where $\Omega$ is the $\rho$-equivariant developing image of $\mu$. Furthermore, if $(\phi,\rho)$ is a developing pair for $\mu$, then the orientation on $S$ induces an orientation on $\widetilde{S}$, which induces an orientation on $\Omega$ via $\phi$. This orientation on $\Omega$ does not depend on the choice $(\phi,\rho)$, so it defines a counter-clockwise cyclic ordering on $\partial\Omega$.
\[prop:op\] Let $(a,b,c)$ be a pairwise distinct triple of points in $\widetilde{{\mathcal{V}}}$ so that $a<b<c<a$ in the cyclic ordering on $\widetilde{{\mathcal{V}}}$. Let $\mu\in{\mathcal{C}}(S)$, let $\rho\in\mathrm{hol}(\mu)$, and let $\xi_\rho$ be the $\rho$-equivariant flag map. Then $\xi^{(1)}_\rho(a)\leq\xi^{(1)}_\rho(b)\leq\xi^{(1)}_\rho(c)\leq\xi^{(1)}_\rho(a)$ in that cyclic order along $\partial\Omega$.
Observe that if $\xi^{(1)}_\rho(a)=\xi^{(1)}_\rho(b)$ or $\xi^{(1)}_\rho(b)=\xi^{(1)}_\rho(c)$ or $\xi^{(1)}_\rho(c)=\xi^{(1)}_\rho(a)$, then the proposition holds trivially. Hence, we only need to consider the case when $\xi^{(1)}_\rho(a)$, $\xi^{(1)}_\rho(b)$, and $\xi^{(1)}_\rho(c)$ are pairwise distinct.
Choose a convex cocompact hyperbolic metric on $S$, then $\widetilde{S}\simeq{\mathbb{D}}$ as oriented Riemannian metric spaces and $\widetilde{{\mathcal{V}}}$ is a subset of $\partial{\mathbb{D}}$. Let $\gamma_a$, $\gamma_b$ and $\gamma_c\in\pi_1(S)$ be group elements whose attracting fixed points are $a,b,c$ respectively. For any $p\in\widetilde{S}$ and any $\gamma\in\pi_1(S)$, let $l_{p,\gamma\cdot p}$ be the closed line segment between $p$ and $\gamma\cdot p$. Then define $$L_{a,p}:=\bigcup_{i=1}^\infty l_{\gamma_a^{i-1}\cdot p,\gamma_a^i\cdot p},\,\,\,\,L_{b,p}:=\bigcup_{i=1}^\infty l_{\gamma_b^{i-1}\cdot p,\gamma_b^i\cdot p},\,\,\,\,L_{c,p}:=\bigcup_{i=1}^\infty l_{\gamma_c^{i-1}\cdot p,\gamma_c^i\cdot p}.$$ Observe that $L_{a,p}$, $L_{b,p}$ and $L_{c,p}$ are simple curves starting at $p$ and going towards $a,b,c$ respectively. By choosing $p$ appropriately, we can further ensure that $L_{a,p}$, $L_{b,p}$ and $L_{c,p}$ are pairwise non-intersecting. Since $a<b<c<a$ in $\widetilde{{\mathcal{V}}}$, this ensures that if we take a small disc centered at $p$, then the boundary of this disc, when oriented counter-clockwise, intersects $L_{a,p}$, $L_{b,p}$ and $L_{c,p}$ in that order.
Suppose for contradiction that the proposition is false. Since $\xi^{(1)}_\rho(a)$, $\xi^{(1)}_\rho(b)$, and $\xi^{(1)}_\rho(c)$ are pairwise distinct, this implies that $\xi^{(1)}_\rho(a)<\xi^{(1)}_\rho(c)<\xi^{(1)}_\rho(b)<\xi^{(1)}_\rho(a)$ in this cyclic order around $\partial\Omega$. Let $\phi$ be the $\rho$-equivariant developing map for $\mu$, and recall that the orientation on $\Omega$ was chosen so that $\phi$ is orientation preserving. The $\rho$-equivariance of $\phi$ ensures that then $\phi(L_{a,p})$, $\phi(L_{b,p})$ and $\phi(L_{c,p})$ are three pairwise non-intersecting curves starting at $\phi(p)$ at going towards $\xi^{(1)}_\rho(a)$, $\xi^{(1)}_\rho(b)$ and $\xi^{(1)}_\rho(c)$ respectively. But this means that if we take a small disc centered at $\phi(p)$, then the boundary of this disc, when oriented counter-clockwise, intersects $\phi(L_{a,p})$, $\phi(L_{c,p})$ and $\phi(L_{b,p})$ in that order. This contradicts the assumption that $\phi$ is orientation preserving.
Pick an affine chart containing $\overline{\Omega}$. For any non-identity $\gamma\in\pi_1(S)$, the dynamics of the $\rho(\gamma)$ acting on $\Omega$ described above implies that $\Omega$ must lie on one side of $\xi_\rho^{(2)}(\gamma^+)$ in this affine chart. Thus one easily deduces the following proposition.
\[prop:transverse1\] Let $\Omega$ be the $\rho$-equivariant developing image of $\mu$. Then for all $x\in\widetilde{{\mathcal{V}}}$, $\xi_\rho^{(2)}(x)$ does not intersect $\Omega$.
Using this, we can prove the following proposition.
\[prop:transverse\] Suppose that $x,y\in\widetilde{{\mathcal{V}}}$ are distinct. Then
1. $\xi^{(1)}_\rho(x)\neq\xi^{(1)}_\rho(y)$. In particular, the weak inequalities in the statement of Proposition \[prop:op\] are in fact strict.
2. $\xi^{(1)}_\rho(x)$ does not lie in $\xi^{(2)}_\rho(y)$. In particular, $\xi^{(2)}_\rho(x)\neq \xi^{(2)}_\rho(y)$.
Proposition \[prop:transverse\] is false if we replace $\widetilde{{\mathcal{V}}}$ with $\partial\pi_1(S)$. This is the main reason why we excluded peripheral elements in our definition of geodesics.
By the definition of $\widetilde{{\mathcal{V}}}$, there are some $z,w\in\widetilde{{\mathcal{V}}}$ so that $x<z<y<w<x$ in this order along $\partial\pi_1(S)$. Let $\gamma\in\pi_1(S)$ be a non-peripheral element, and let $\gamma_z,\gamma_w\in\pi_1(S)$ be elements so that $\gamma_z^-=z$ and $\gamma_w^-=w$. Then for sufficiently large $n$, $$x<\gamma_z^{-n}\cdot \gamma^\pm<y<\gamma_w^{-n}\cdot \gamma^\pm<x.$$ Also, let $\Omega$ be the $\rho$-equivariant developing image of $\mu$. By Proposition \[prop:op\], we see that $$\xi^{(1)}_\rho(x)\leq \rho(\gamma_z)^{-n}\cdot \xi^{(1)}_\rho(\gamma^\pm)\leq \xi^{(1)}_\rho(y)\leq\rho(\gamma_w)^{-n}\cdot \xi^{(1)}_\rho(\gamma^\pm)\leq \xi^{(1)}_\rho(x).$$
*(1)* Suppose for contradiction that $\xi_\rho^{(1)}(x)=\xi_\rho^{(1)}(y)$. By (1) of Theorem \[thm: holonomy1\], we see that $\rho(\gamma)$ is hyperbolic, so it has distinct attracting and repelling fixed points $\rho(\gamma)^+$ and $\rho(\gamma)^-$ respectively. Also, since $\xi_\rho^{(1)}(\widetilde{{\mathcal{V}}})\subset\partial\Omega$, the convexity of $\Omega$ implies that either $\xi_\rho^{(1)}(x)=\rho(\gamma_w)^{-n}\cdot\rho(\gamma)^\pm=\xi_\rho^{(1)}(y)$ or $\xi_\rho^{(1)}(x)=\rho(\gamma_z)^{-n}\cdot\rho(\gamma)^\pm=\xi_\rho^{(1)}(y)$. Assume without loss of generality that the former holds. In particular, $\rho(\gamma_w)^{-n}\cdot\rho(\gamma)^+=\rho(\gamma_w)^{-n}\cdot\rho(\gamma)^-$. But $\rho(\gamma_w)^{-n}\cdot\rho(\gamma)^\pm$ are the attracting and repelling fixed points of $\rho(\gamma_w^{-n}\cdot\gamma\cdot\gamma_w^n)$, so they cannot be equal. This is clearly a contradiction.
*(2)* Suppose for contradiction that $\xi_\rho^{(1)}(x)\in\xi_\rho^{(2)}(y)$ for some distinct $x,y\in\widetilde{{\mathcal{V}}}$. By Proposition \[prop:transverse1\] and the convexity of $\Omega$, one deduces that there is an open line segment $L$ in ${\mathbb{R}}{\mathbb{P}}^2$ with endpoints $\xi_\rho^{(1)}(x)$ and $\xi_\rho^{(1)}(y)$, so that $L\subset\partial\Omega$ and either $\rho(\gamma_z)^{-n}\cdot\rho(\gamma)^\pm\in L$ or $\rho(\gamma_w)^{-n}\cdot\rho(\gamma)^\pm\in L$. Assume without loss of generality that the former holds. Then observe that $\lim_{k\to\infty}\rho(\gamma_z^{-n}\cdot\gamma\cdot\gamma_z^n)^k\cdot L$ is the entire projective line in ${\mathbb{R}}{\mathbb{P}}^2$ containing $L$. Since $\partial\Omega$ contains $L$ and is invariant under $\rho(\gamma_z^{-n}\cdot\gamma\cdot\gamma_z^n)$, we deduce that $\partial\Omega$ is a projective line, but this contradicts the properness of $\Omega$.
If we choose a different representative $\rho'=g\cdot\rho\cdot g^{-1}\in\mathrm{hol}(\mu)$, then $\xi_{\rho'}=g\cdot \xi_\rho$. Furthermore, if $\mathrm{hol}(\mu)=\mathrm{hol}(\mu')$, then the ${\mathrm{PGL}}(3,{\mathbb{R}})$-orbit of flag maps associated to $\mu$ and $\mu'$ agree. Hence, the map $\rho\mapsto\xi_\rho$ associates to the conjugacy class $\mathrm{hol}(\mu)$ a ${\mathrm{PGL}}(3,{\mathbb{R}})$-orbit of maps from $\widetilde{{\mathcal{V}}}\to{\mathbb{R}}{\mathbb{P}}^2\times({\mathbb{R}}{\mathbb{P}}^2)^*$, which we denote by $\xi_{\mathrm{hol}(\mu)}:=[\xi_\rho]$. The next proposition tells us that $\xi_{\mathrm{hol}(\mu)}$ varies continuously with $\mathrm{hol}(\mu)$.
\[prop:conv-flag\] Let $F:(-\epsilon,\epsilon)\to{\mathcal{X}}_3(S)$ given by $F:t\mapsto[\rho_t]$ be a map whose image lies in $\mathrm{hol}({\mathcal{C}}(S))$. Then $F$ is continuous if and only if $t\mapsto\xi_{\rho_t}(x)$ is a continuous path in ${\mathcal{F}}$ for every $x\in\widetilde{{\mathcal{V}}}$.
This proposition is a consequence of the following elementary fact.
\[conv-flag\] Let $L_i\to L$ be a convergent sequence of endomorphisms of ${\mathbb{R}}^n$, and let $\lambda^j_i,\lambda^j$ be the (generalized) eigenvalues of $L_i$ and $L$ respectively. Assume $\lambda^j_i,\lambda^j$ are real for all $j=1,\dots,n$. If there is some $m\in\{1,\dots,n\}$ so that for all $i$, $$\dim \ker \prod_{j=1}^m (L_i-\lambda_i^jI) = m= \dim \ker \prod_{j=1}^m (L-\lambda^jI),$$ where $I$ is the identity endomorphism on ${\mathbb{R}}^n$, then $$\ker \prod_{j=1}^m (L_i-\lambda_i^jI) \to \ker \prod_{j=1}^m (L-\lambda^jI)$$ in the Grassmannian ${\mathrm{Gr}}_m({\mathbb{R}}^n)$.
Let $x_i^1,\dots, x_i^m$ be an orthonormal basis of $\ker \prod_{j=1}^m (L_i-\lambda_i^jI)$. By taking subsequences, we may assume that $x_i^j$ converges to $x^j$. Since $L_i\to L$, we see that $\lambda^j_i\to\lambda^j$, so $x^1,\dots, x^m$ all lie in $\ker \prod_{j=1}^m (L-\lambda^jI)$. By the dimension hypothesis, $x^1,\dots, x^m$ is an orthonormal basis of $\ker \prod_{j=1}^m (L-\lambda^jI)$. Hence, up to taking subsequences, $\ker \prod_{j=1}^m (L_i-\lambda_i^jI)$ converges to $\ker \prod_{j=1}^m (L-\lambda^jI)$. Repeating this argument for all subsequences of $\{L_i\}_{i=1}^\infty$ proves the lemma.
For any $x\in\widetilde{{\mathcal{V}}}$, let $\gamma\in\pi_1(S)$ be the unique primitive group element so that $x$ is the repelling fixed point of $\gamma$. For any $\mu\in{\mathcal{C}}(S)$ and any representative $\rho\in\mathrm{hol}(\mu)$, let $L\in{\mathrm{SL}}(3,{\mathbb{R}})$ be a representative of $\rho(\gamma)$. Then let $\lambda_1\ge \lambda_2\ge \lambda_3$ be the generalized eigenvalues of $L$. Observe that as defined, $\xi_\rho^{(1)}(x)=\ker(L-\lambda_3)$ and $\xi_\rho^{(2)}(x)=\ker\big((L-\lambda_3)(L-\lambda_2)\big)$. To prove the forward direction, we simply apply Lemma \[conv-flag\].
For the backward direction, pick any $\gamma\in\pi_1(S)$, any triple of pairwise distinct points $x,y,z\in\widetilde{{\mathcal{V}}}$, and let $(x',y',z'):=\gamma\cdot (x,y,z)$. Also, let $$(a,b,c,d):=(\xi_\rho^{(1)}(x),\xi_\rho^{(1)}(y),\xi_\rho^{(1)}(z),\xi_\rho^{(2)}(x)\cap\xi_\rho^{(2)}(y)),$$ and let $$(a',b',c',d'):=(\xi_\rho^{(1)}(x'),\xi_\rho^{(1)}(y'),\xi_\rho^{(1)}(z'),\xi_\rho^{(2)}(x')\cap\xi_\rho^{(2)}(y')).$$ The $\rho$-equivariance of $\xi$ implies that $\rho(\gamma)\cdot (a,b,c,d)=(a',b',c',d')$.
We will now argue that the quadruple of points $a,b,c,d$ are in general position, i.e. no three of them lie in a line in ${\mathbb{R}}{\mathbb{P}}^2$. By (2) of Proposition \[prop:transverse\], we see that the triple $a,b,d$ do not lie in a line in ${\mathbb{R}}{\mathbb{P}}^2$. For the same reason, the same is true for the triples $a,c,d$ and $b,c,d$. On the other hand, suppose for contradition that $a,b,c$ lie in a line. By the same argument as the first part of the proof of Proposition \[prop:transverse\], there is some $\eta\in\pi_1(S)$ so that $x<\eta^-<\eta^+<y<z$. The convexity of $\Omega$ then implies that there is a projective (open) line segment $L$ with endpoints $a,b$ so that $L\subset\partial\Omega$ and $\rho(\eta)^-,\rho(\eta)^+\in L$. Since $\partial\Omega$ is $\rho$-equivariant, this means that $$\bigcup_{i=-\infty}^\infty \rho(\eta)^i\cdot L\subset\partial\Omega,$$ and is an entire projective line in ${\mathbb{R}}{\mathbb{P}}^2$. This violates the properness of $\Omega$, so $a,b,c$ cannot lie in a line. We have thus proven that $a,b,c,d$ are in general position, so $(a',b',c',d')=\rho(\gamma)\cdot (a,b,c,d)$ is also in general position.
If we normalize $a=[1:0:0]^T$, $b=[0:1:0]^T$, $c=[0:0:1]^T$ and $d=[1:1:1]^T$, then it is a straightforward exercise to explicitly write down a matrix representative for $\rho(\gamma)$ in terms of the coordinates of $a',b',c',d'$. From this, it is clear that $\rho(\gamma)$ varies continuously with $a',b',c',d'$.
The holonomy map.
-----------------
It will be important later that we understand the image of the map $\mathrm{hol}:{\mathcal{C}}(S)^\mathrm{adm}\to{\mathcal{X}}_3(S)$. To do so, we set up the following notation. For every $\mu\in{\mathcal{C}}(S)^\mathrm{adm}$, let ${\mathcal{A}}_{\mathrm{hol}(\mu)}$ denote the set of punctures of $S$ whose $\mu$-type is bulge $\pm\infty$, and let $\Gamma_{\mathrm{hol}(\mu)}\subset\pi_1(S)$ be the set of peripheral group elements corresponding to ${\mathcal{A}}_{\mathrm{hol}(\mu)}$.
For any representative $\rho\in\mathrm{hol}(\mu)$, let $\Omega$ denote the $\rho$-equivariant developing image of $\mu$. Let $\gamma\in\Gamma_{\mathrm{hol}(\mu)}$ and let $\rho(\gamma)^+=\xi^{(1)}(\gamma^+)$, $\rho(\gamma)^0$, $\rho(\gamma)^-=\xi^{(1)}(\gamma^-)$ be the attracting, saddle and repelling points of $\rho(\gamma)$. Note that $\rho(\gamma)^+$ and $\rho(\gamma)^-$ necessarily lie on the boundary of $\Omega$. Also, by Proposition \[prop:transverse1\] and (1) of Proposition \[prop:transverse\], we see that $\xi^{(1)}(\widetilde{{\mathcal{V}}}\setminus\{\gamma^+,\gamma^-\})$ lies entirely in one of the two connected components of ${\mathbb{R}}{\mathbb{P}}^2\setminus(\xi^{(2)}_\rho(\gamma^+)\cup\xi^{(2)}_\rho(\gamma^-))$, call it $A$. The projective line through $\rho(\gamma)^+$ and $\rho(\gamma)^-$ cuts $A$ into two open triangles, and the fact that $\gamma$ is a peripheral group element ensures that $\xi^{(1)}(\widetilde{{\mathcal{V}}}\setminus\{\gamma^+,\gamma^-\})$ lies entirely in one of these two open triangles, call it $\Delta'$.
Let $\rho\in\mathrm{hol}(\mu)$ and $\gamma\in\Gamma_{\mathrm{hol}(\mu)}$. The *principal triangle* of $\rho(\gamma)$ is the open triangle $\Delta=\Delta_\rho$ that is the connected component of $A\setminus{\mathrm{Span}}(\rho(\gamma)^+,\rho(\gamma)^-)$ that is not $\Delta'$.
It is clear that $\rho(\gamma)$ has a unique principal triangle, which depends only on $\rho$. Let ${\mathcal{G}}_\rho$ denote the set of principal triangles of the group elements in $\rho(\Gamma_{\mathrm{hol}(\mu)})$. Observe that there is a natural $\pi_1(S)$-action on ${\mathcal{G}}_\rho$ induced by $\rho$. The next theorem tells us to what extent different points in ${\mathcal{C}}(S,{\mathcal{D}})^{\mathrm{adm}}$ can have the same holonomy.
\[thm: hol\] Let ${\mathcal{D}}$ be any multi-curve in $S$, let $S_1,\dots,S_k$ be the connected components of $S\setminus{\mathcal{D}}$, and let $(\mu_1,\dots,\mu_k),(\mu_1',\dots,\mu_k')\in{\mathcal{C}}(S,{\mathcal{D}})^\mathrm{adm}$ so that $$\mathrm{hol}(\mu_1,\dots,\mu_k)=\mathrm{hol}(\mu_1',\dots,\mu_k').$$ For all $i=1,\dots,k$, let $\rho_i$ be representatives of $\mathrm{hol}(\mu_i)$, and let $\Omega_i, \Omega_i'\subset{\mathbb{R}}{\mathbb{P}}^2$ be the $\rho_i$-equivariant developing images of $\mu_i,\mu_i'\in{\mathcal{C}}(S_i)^{\mathrm{adm}}$ respectively. Then the interior of the symmetric difference $\Omega_i\, \triangle\, \Omega_i'$ is the union of a $\pi_1(S_i)$-invariant subset of triangles in ${\mathcal{G}}_{\rho_i}$.
The proof of Theorem \[thm: hol\] is in the Appendix. As an immediate consequence of Theorem \[thm: hol\], Proposition \[prop:devimage\], and the compatibility of $\mu$ across the closed curves in ${\mathcal{D}}$, we have the following corollary.
\[cor: hol\] Let $\mathrm{hol}:{\mathcal{C}}(S,{\mathcal{D}})^\mathrm{adm}\to\prod_{i=1}^k{\mathcal{X}}_3(S_j)$, and choose an orientation for each $c\in{\mathcal{D}}$. For any $\mu\in{\mathcal{C}}(S,{\mathcal{D}})^{\mathrm{adm}}$, let $${\mathcal{D}}_{\mathrm{hol}(\mu)}:=\{c\in{\mathcal{D}}:\mu\text{-type of c is bulge }\pm\infty\}.$$ Then $|\mathrm{hol}^{-1}(\mathrm{hol}(\mu))|=2^{|{\mathcal{D}}_{\mathrm{hol}(\mu)}\cup{\mathcal{A}}_{\mathrm{hol}(\mu)}|}$. Furthermore, each element in $\mathrm{hol}^{-1}(\mathrm{hol}(\mu))$ corresponds to the choice of whether the $\mu$-type of $c$ is bulge $+\infty$ or bulge $-\infty$ for each $c \in{\mathcal{D}}_{\mathrm{hol}(\mu)}$, and whether the $\mu$-type of $p$ is bulge $+\infty$ or bulge $-\infty$ for each $p \in{\mathcal{A}}_{\mathrm{hol}(\mu)}$.
Edge and triangle invariants {#sec:edgetriangle}
----------------------------
When $S$ is a closed surface, Goldman gave a parameterization for $\mathrm{hol}({\mathcal{C}}(S))$ that generalizes the Fenchel-Nielsen coordinates on ${\mathcal{T}}(S)$ [@Go90]. Briefly, he did this by first parameterizing $\mathrm{hol}({\mathcal{C}}(P))$, where $P$ is the oriented thrice punctured sphere. Then, he generalized this parameterization to all surfaces of negative Euler characteristic by parameterizing the space of ways to assemble convex ${\mathbb{R}}{\mathbb{P}}^2$ structures on pairs of pants together. This parameterization was later extended by Marquis to the setting of where $S$ is not closed [@Mar10].
By modifying previous parameterizations of $\mathrm{hol}({\mathcal{C}}(S))$ given by Bonahon-Dreyer [@BonDre] and Fock-Goncharov [@FocGon], the second author [@Zhang16] also gave a continuous (in fact real-analytic) coordinate system of $\mathrm{hol}({\mathcal{C}}(S))$ that is similar in flavor to the one given by Goldman. However, this coordinate system has the additional advantage that the parameters are naturally projective invariants, and thus have easier geometric interpretations. We will give a brief description of this coordinate system. To do, one first needs to define the edge and triangle invariants.
Choose a pants decomposition ${\mathcal{P}}$ of $S$ and let ${\mathcal{T}}={\mathcal{T}}_{\mathcal{P}}$ be the associated ideal triangulation described in Section \[sec: triangulation\]. For any closed edge $c=[x,y]\in{\mathcal{P}}$, choose a lift $\{x,y\}\in\widetilde{{\mathcal{P}}}$ of $c$, and let $\widetilde{T}_x,\widetilde{T}_y$ be triangles in $\widetilde{\Theta}$ with $x,y$ as a vertex respectively. We refer to the orbit $\pi_1(S)\cdot \{\widetilde{T}_x,\widetilde{T}_y\}$ as a *bridge* across $c$. See Figure \[bridge-figure\].
![Bridge[]{data-label="bridge-figure"}](bridge.eps)
For each edge $[x,y]\in{\mathcal{T}}$, choose a representative $\{x,y\}\in\widetilde{{\mathcal{T}}}$ of $[x,y]$. If $\{x,y\}\in\widetilde{{\mathcal{Q}}}$, let $z_1,z_2\in\widetilde{{\mathcal{V}}}$ so that $\{x,z_1\}, \{y,z_1\}, \{x,z_2\}, \{y,z_2\}\in\widetilde{{\mathcal{T}}}$ and $x<z_2<y<z_1<x$. On the other hand, if $\{x,y\}\in\widetilde{{\mathcal{P}}}$, let $\{\widetilde{T}_x,\widetilde{T}_y\}$ be an element in the bridge across $c$ so that $\widetilde{T}_{x}=\{x,z_1,w_1\}$ where $x<z_1<w_1<x$, and $\widetilde{T}_{y}=\{y,z_2,w_2\}$ where $y<z_2<w_2<y$. See Figures \[edge-inv-nonbridge-figure\] and \[edge-inv-bridge-figure\], noting that for $i=1,2$, $z_i=\xi^{(1)}(z_i)$ in these figures. Define $$\begin{array}{rrcl}
&s_{x,y}:\mathrm{hol}({\mathcal{C}}(S))&\to&{\mathbb{R}}\\
&[\rho]&\mapsto& C\left(\xi_\rho^{(2)}(x),\xi_\rho^{(1)}(z_2),\xi_\rho^{(1)}(z_1),\xi_\rho^{(1)}(x)+\xi_\rho^{(1)}(y)\right).
\end{array}$$ This is well-defined because the projective invariance of the cross ratio implies that all the choices we made are irrelevant (except for the choice of a bridge across each closed curve in ${\mathcal{P}}$). Hence, for each edge $[x,y]\in{\mathcal{T}}$, we have defined two invariants, $s_{x,y}$ and $s_{y,x}$. Observe using Remark \[rem:C+\] that $s_{x,y}<0$, so one can define $\sigma_{x,y}:=\log (-s_{x,y})$. These are called the *edge invariants* along $[x,y]$.
![Edge invariant for $\{x,y\}\in\mathcal Q$[]{data-label="edge-inv-nonbridge-figure"}](edge-inv-nonbridge.eps)
![Edge invariant for $\{x,y\}\in\mathcal P$[]{data-label="edge-inv-bridge-figure"}](edge-inv-bridge.eps)
Similarly, for every ideal triangle $[x,y,z]\in\Theta$, choose a representative $\{x,y,z\}\in\widetilde{\Theta}$ so that $x<y<z<x$ in $\partial\pi_1(S)$. Define $$\begin{array}{rrcl}
&t_{x,y,z}:\mathrm{hol}({\mathcal{C}}(S))&\to&{\mathbb{R}}\\
&[\rho]&\mapsto& T\left(\xi_\rho^{(2)}(x),\xi_\rho^{(2)}(y), \xi_\rho^{(2)}(z),\xi_\rho^{(1)}(x),\xi_\rho^{(1)}(y),\xi_\rho^{(1)}(z)\right).
\end{array}$$ See Figure \[triangle-inv-figure\]. The projective invariance of the triple ratio again guarantees that the $t_{x,y,z}$ do not depend on any of the choices made. Furthermore, the symmetry of the triple ratio implies that $t_{x,y,z}=t_{y,z,x}=t_{z,x,y}$, so we only have one such function for each $[x,y,z]\in\Theta$. Again, observe using Remark \[rem:T+\] that $t_{x,y,z}>0$, so one can define the *triangle invariants* for $[x,y,z]$ to be $\tau_{x,y,z}:=\log (t_{x,y,z})$.
![Triangle invariant[]{data-label="triangle-inv-figure"}](triangle-inv.eps)
Coordinates on $\mathrm{hol}({\mathcal{C}}(P))$
-----------------------------------------------
For now, we specialize to the case when $S=P$, the oriented thrice punctured sphere. Let $\alpha,\beta,\gamma\in\pi_1(P)$ be three group elements corresponding to oriented peripheral curves in $P$, so that $\gamma\beta\alpha={\mathrm{id}}$, and $P$ lies to the left of each oriented peripheral curve. Then observe that $$\widetilde{{\mathcal{T}}}:=\big\{\eta\cdot \{\alpha^-,\beta^-\}:\eta\in\pi_1(S)\big\}\cup\big\{\eta\cdot \{\beta^-,\gamma^-\}:\eta\in\pi_1(S)\big\}\cup\big\{\eta\cdot \{\gamma^-,\alpha^-\}:\eta\in\pi_1(S)\big\},$$ is an ideal triangulation of $\widetilde{P}$. With this ideal triangulation, $\widetilde{{\mathcal{P}}}=\emptyset$, $\widetilde{{\mathcal{Q}}}=\widetilde{{\mathcal{T}}}$, $$\widetilde{\Theta}:=\big\{\eta\cdot \{\alpha^-,\beta^-,\gamma^-\}:\eta\in\pi_1(S)\big\}\cup\big\{\eta\cdot \{\alpha^-,\alpha\cdot \beta^-,\gamma^-\}:\eta\in\pi_1(S)\big\},$$ and $$\widetilde{{\mathcal{V}}}:= \{\eta\cdot\alpha^-:\eta\in\pi_1(S)\}\cup\{\eta\cdot\beta^-:\eta\in\pi_1(S)\}\cup\{\eta\cdot\gamma^-:\eta\in\pi_1(S)\}.$$ In particular, ${\mathcal{T}}$ consists of three edges, $\Theta$ consists of two triangles, and ${\mathcal{V}}$ consists of three vertices. See Figure \[pants-figure\].
![Ideal triangulation of pair of pants[]{data-label="pants-figure"}](Q_e.pdf){width="90.00000%"}
Bonahon-Dreyer [@BonDre] computed an expression for the eigenvalues of $\rho(\alpha)$, $\rho(\beta)$ and $\rho(\gamma)$ in terms of the edge and triangle invariants associated to ${\mathcal{T}}$. Explicitly, if we denote the (generalized) eigenvalues of $\rho(\eta)$ by $\lambda_{1,\eta}(\rho)\geq\lambda_{2,\eta}(\rho)\geq\lambda_{3,\eta}(\rho)$ for any $\eta\in\pi_1(P)\setminus\{{\mathrm{id}}\}$, then for $i=1,2$, define $\ell_{i,[\eta]}:\mathrm{hol}({\mathcal{C}}(P))\to{\mathbb{R}}$ by $$\ell_{i,[\eta]}(\rho):=\log\left(\frac{\lambda_{i,\eta}(\rho)}{\lambda_{i+1,\eta}(\rho)}\right).$$ Note that the $\ell_{i,[\eta]}$ depends only on the conjugacy class $[\eta]$, and not on the choice of $\eta\in\pi_1(P)$. Then Bonahon-Dreyer showed that $$\begin{aligned}
\ell_{1,[\alpha]}&=&\sigma_{\alpha^-,\beta^-}+\sigma_{\alpha^-,\gamma^-}\\
\ell_{2,[\alpha]}&=&\sigma_{\beta^-,\alpha^-}+\sigma_{\gamma^-,\alpha^-}+\tau_{\alpha^-,\gamma^-,\beta^-}+\tau_{\alpha^-,\alpha\cdot\gamma^-,\beta^-}\\
\ell_{1,[\beta]}&=&\sigma_{\beta^-,\gamma^-}+\sigma_{\beta^-,\alpha^-}\\
\ell_{2,[\beta]}&=&\sigma_{\gamma^-,\beta^-}+\sigma_{\alpha^-,\beta^-}+\tau_{\alpha^-,\gamma^-,\beta^-}+\tau_{\alpha^-,\alpha\cdot\gamma^-,\beta^-}\\
\ell_{1,[\gamma]}&=&\sigma_{\gamma^-,\alpha^-}+\sigma_{\gamma^-\beta^-}\\
\ell_{2,[\gamma]}&=&\sigma_{\alpha^-,\gamma^-}+\sigma_{\beta^-\gamma^-}+\tau_{\alpha^-,\gamma^-,\beta^-}+\tau_{\alpha^-,\alpha\cdot\gamma^-,\beta^-}\end{aligned}$$
In particular, the expressions on the right have to be at least $0$. These six inequalities are known as the (weak) *closed leaf inequalities*. Bonahon-Dreyer then showed that these are the only relations satisfied by these parameters.
\[Bonahon-Dreyer\]\[thm: Bonahon-Dreyer\] The map $\Phi:\mathrm{hol}({\mathcal{C}}(P))\to{\mathbb{R}}^8$ given by $$\begin{array}{l}
\Phi:[\rho]\mapsto \left(\sigma_{\alpha^-,\beta^-}(\rho),\sigma_{\beta^-,\alpha^-}(\rho),\sigma_{\alpha^-,\gamma^-}(\rho),\sigma_{\gamma^-,\alpha^-}(\rho),\right.\\
\hspace{4.5cm}\left.\sigma_{\beta^-,\gamma^-}(\rho),\sigma_{\gamma^-,\beta^-}(\rho),\tau_{\alpha^-,\gamma^-,\beta^-}(\rho),\tau_{\alpha^-,\alpha\cdot\gamma^-,\beta^-}(\rho)\right)
\end{array}$$ is a homeomorphism onto the closed convex polytope in ${\mathbb{R}}^8$ cut out by the closed leaf inequalities.
Solving the six linear equations above then proves the following.
\[cor:coordinate\] The map $\Phi:\mathrm{hol}({\mathcal{C}}(P))\to({\mathbb{R}}_{\geq 0})^6\times{\mathbb{R}}^2$ given by $$\Phi:[\rho]\mapsto \left(\ell_{1,[\alpha]}(\rho),\ell_{2,[\alpha]}(\rho),\ell_{1,[\beta]}(\rho),\ell_{2,[\beta]}(\rho),\ell_{1,[\gamma]}(\rho),\ell_{2,[\gamma]}(\rho),\sigma_{\alpha^-,\beta^-}(\rho),\tau_{\alpha^-,\gamma^-,\beta^-}(\rho)\right)$$ is a homeomorphism.
Bonahon-Dreyer [@BonDre] and the second author [@Zhang16] were working in the more general setting of Hitchin representations, so they only stated their results for representations where the holonomy about each boundary component was required to be hyperbolic. However, in the case of convex ${\mathbb{R}}{\mathbb{P}}^2$ structures, Proposition \[prop:conv-flag\] extends their arguments verbatim to the cases where the holonomy about the boundary component is quasi-hyperbolic or parabolic.
In the coordinate system given in Corollary \[cor:coordinate\], the invariants $\sigma_{\alpha^-,\beta^-}$ and $\tau_{\alpha^-,\gamma^-,\beta^-}$ are called the *internal parameters* of $P$, and the six invariants $\ell_{1,[\alpha]}$, $\ell_{2,[\alpha]}$, $\ell_{1,[\beta]}$, $\ell_{2,[\beta]}$, $\ell_{1,[\gamma]}$, $\ell_{2,[\gamma]}$ are called the *length parameters*. We will simplify notation and denote $\sigma_{\alpha^-,\beta^-}$ and $\tau_{\alpha^-,\gamma^-,\beta^-}$ by $i_{1,P}$ and $i_{2,P}$ respectively.
Coordinates on $\mathrm{hol}({\mathcal{C}}(S))$ {#sec: coordhol}
-----------------------------------------------
Now, we will use the parameterization of $\mathrm{hol}({\mathcal{C}}(P))$ to parameterize $\mathrm{hol}({\mathcal{C}}(S))$. To do so, choose once and for all
- a pants decomposition ${\mathcal{P}}$ on $S$,
- a bridge across each closed curve in ${\mathcal{P}}$,
- an orientation for every closed curve in ${\mathcal{P}}$
- an orientation about each puncture of $S$.
For every $c\in{\mathcal{P}}$, let $p_1,p_2$ be the punctures of $S\setminus c$ corresponding to $c$. If $c$ is non-separating, then $c$ (equipped with its chosen orientation) determines two conjugacy classes $[\gamma_1],[\gamma_2]\in[\pi_1(S\setminus c)]$, so that $[\gamma_i]$ corresponds to the puncture $p_i$. On the other hand, if $c$ is separating, let $S_1$ and $S_2$ be the two connected components of $S\setminus c$, so that $p_i$ is a puncture of $S_i$. Then $c$ determines a conjugacy class $[\gamma_i]\in[\pi_1(S_i)]$ corresponding to $p_i$.
Goldman [@Go90] proved that if $c$ is non-separating, then for any $\mu'\in{\mathcal{C}}(S\setminus c)^\mathrm{adm}$, there is some $\mu\in{\mathcal{C}}(S)^\mathrm{adm}$ so that $\mu|_{S\setminus c}=\mu'$ if and only if the $\mu'$-type of $p_i$ is bulge $-\infty$ for both $i=1,2$, and $\mathrm{hol}_{\mu'}(\gamma_1)=\mathrm{hol}_{\mu'}(\gamma_2)$. Similarly, if $c$ is separating, then for any $(\mu_1,\mu_2)\in{\mathcal{C}}(S_1)^\mathrm{adm}\times{\mathcal{C}}(S_2)^\mathrm{adm}$, there is some $\mu\in{\mathcal{C}}(S)^\mathrm{adm}$ so that $\mu|_{S_i}=\mu_i$ for $i=1,2$ if and only if the $\mu_i$-type of $p_i$ is bulge $-\infty$ for both $i=1,2$, and $\mathrm{hol}_{\mu_1}(\gamma_1)=\mathrm{hol}_{\mu_2}(\gamma_2)$. Furthermore, regardless of whether $c$ is separating or not, Goldman [@Go90] also showed that the set of all such $\mu\in{\mathcal{C}}(S)^\mathrm{adm}$ is parameterized by two parameters, called *bulge* and *twist parameters* $b_c,t_c:{\mathcal{C}}(S)^\mathrm{adm}\to{\mathbb{R}}$. These are defined by $$b_c(\mu):=\sigma_{\gamma_c^-,\gamma_c^+}(\mathrm{hol}_\mu)-\sigma_{\gamma_c^+,\gamma_c^-}(\mathrm{hol}_\mu)\,\,\,\text{ and }\,\,\,t_c(\mu):= \sigma_{\gamma_c^-,\gamma_c^+}(\mathrm{hol}_\mu)+\sigma_{\gamma_c^+,\gamma_c^-}(\mathrm{hol}_\mu) .$$ In particular, $b_c$ and $t_c$ depend only on $\mathrm{hol}_\mu$, so we also denote $b_c(\mathrm{hol}_\mu):=b_c(\mu)$ and $t_c(\mathrm{hol}_\mu):=t_c(\mu)$. See Figure \[twist-bulge-figure\].
![Twist and bulge[]{data-label="twist-bulge-figure"}](twist-bulge.eps)
Given a simple closed curve $c$ in $S$, Goldman defined an ${\mathbb{R}}^2$ action on ${\mathcal{C}}(S)$ by *bulge and shearing deformations* along $c$. The bulge and twist parameters $b_c$ and $t_c$ were designed to precisely capture these deformations; performing a bulging deformation changes the bulge parameter while keeping the twist parameter fixed, while performing a twist deformation changes the twist parameter while keeping the bulge parameter fixed.
Goldman [@Go90] stated his results in the case when the $\mu$-type of all the punctures of $S$ are bulge $-\infty$, since he was mainly interested in the closed surface case. However, his arguments work in this more general setting as well.
Combining this together with Corollary \[cor:coordinate\] proves the following theorem.
\[thm: par\] Let $S$ be a connected, orientable surface with negative Euler characteristic, genus $g$ with $n$ punctures. Make the choices that we did at the start of this section, and let ${\mathcal{P}}=\{c_1,\dots,c_{3g-3+n}\}$, let $\{d_1,\dots,d_n\}$ be the punctures of $S$, and let ${\mathbb{P}}=\{P_1,\dots,P_{2g-2+n}\}$. Then $$\mathring{\Phi}:\mathrm{hol}({\mathcal{C}}(S))\to({\mathbb{R}}^2)^{3g-3+n}\times({\mathbb{R}}_+^2)^{3g-3+n}\times({\mathbb{R}}_{\geq 0}^2)^n\times({\mathbb{R}}^2)^{2g-2+n}$$ is a homeomorphism, where $$\mathring{\Phi}:=\prod_{i=1}^{3g-3+n}(b_{c_i},t_{c_i})\prod_{i=1}^{3g-3+n}(\ell_{1,c_i},\ell_{2,c_i})\prod_{i=1}^{n}(\ell_{1,d_i},\ell_{2,d_i})\prod_{j=1}^{2g-2+n}(i_{1,P_j},i_{2,P_j}).$$
Again, the second author [@Zhang16] proved Theorem \[thm: par\] for convex ${\mathbb{R}}{\mathbb{P}}^2$ structures where the holonomy about each boundary component was required to be hyperbolic. Proposition \[prop:conv-flag\] extends his proof verbatim to the cases where the holonomy about the boundary component is quasi-hyperbolic or parabolic.
The homeomorphism $\mathring{\Phi}$ is not ideal for our purposes because it does not behave well under Dehn twists about the curves in ${\mathcal{P}}$. We will thus further modify $\mathring{\Phi}$ to get a new homeomorphism that has that property. For each $c\in{\mathcal{P}}$, let $$r_c:=\frac{\ell_{1,c}\cdot\sigma_{\gamma_c^-,\gamma_c^+}-\ell_{2,c}\cdot\sigma_{\gamma_c^+,\gamma_c^-}}{3}$$ be the *reparameterized bulge parameters*. Observe that if we replace the parameters $b_{c_i}$ with $r_{c_i}$ for all $i$ in the homeomorphism $\mathring{\Phi}$, then this defines a new homeomorphism $$\Phi:\mathrm{hol}({\mathcal{C}}(S))\to({\mathbb{R}}^2)^{3g-3+n}\times({\mathbb{R}}_+^2)^{3g-3+n}\times({\mathbb{R}}_{\geq 0}^2)^n\times({\mathbb{R}}^2)^{2g-2+n}$$ given by $$\Phi:=\prod_{i=1}^{3g-3+n}(r_{c_i},t_{c_i})\prod_{i=1}^{3g-3+n}(\ell_{1,c_i},\ell_{2,c_i})\prod_{i=1}^{n}(\ell_{1,d_i},\ell_{2,d_i})\prod_{j=1}^{2g-2+n}(i_{1,P_j},i_{2,P_j}).$$ The next proposition describes how $\Phi$ behaves under Dehn twists about the curves in ${\mathcal{P}}$.
\[prop:Dehntwistcoords\] Let $D_c$ be the Dehn twist along the (oriented) closed curve $c\in{\mathcal{P}}$. Then all the coordinate functions of $\Phi$ agree at $\mathrm{hol}_\mu$ and $D_c\cdot \mathrm{hol}_\mu$, except for $t_c$ which satisfies $t_c(D_c\cdot\mathrm{hol}_\mu)=t_c(\mathrm{hol}_\mu)+\ell_{1,c}(\mathrm{hol}_\mu)+\ell_{2,c}(\mathrm{hol}_\mu)$.
It is clear from the projective invariance of the coordinate functions that the only possible coordinate functions of $\Phi$ that might differ at $\mathrm{hol}_\mu$ and $D_c\cdot\mathrm{hol}_\mu$ are $t_c$ and $r_c$. Let $\pi_1(S)\cdot\{\widetilde{T}_{\gamma_c^-},\widetilde{T}_{\gamma_c^+}\}$ be the bridge across $c$ that we chose to define $\Phi$. Observe that by choosing a basepoint in $S$, $D_c$ induces a group homomorphism, $D_c:\pi_1(S)\to\pi_1(S)$, which sends the bridge $\pi_1(S)\cdot\{\widetilde{T}_{\gamma_c^-},\widetilde{T}_{\gamma_c^+}\}$ across $c$ to the bridge $\pi_1(S)\cdot\{\widetilde{T}_{\gamma_c^-},\gamma_c\cdot\widetilde{T}_{\gamma_c^+}\}$ across $c$. See Figure \[dehn-twist-figure\].
![Dehn twist[]{data-label="dehn-twist-figure"}](dehn-twist.eps)
Choose representatives $\rho^1\in\mathrm{hol}_\mu$ and $\rho^2\in D_c\cdot \mathrm{hol}_\mu$ so that $$\xi_{\rho^j}^{(1)}(\gamma_c^+)=[0:0:1]^T,\,\,\, \xi_{\rho^j}^{(1)}(\gamma_c^-)=[1:0:0]^T,\,\,\, \xi_{\rho^j}^{(1)}(z_1)=[1:1:1]^T,$$ $$\xi_{\rho^j}^{(2)}(\gamma_c^+)=[1:0:0],\,\,\, \text{ and }\,\,\,\xi_{\rho^j}^{(2)}(\gamma_c^-)=[0:0:1]$$ for $j=1,2$. Then it follows that $$\xi^{(1)}_{\rho^2}(z_2)=\rho(\gamma_c)\cdot\xi^{(1)}_{\rho^1}(z_2)=
\left[\begin{array}{ccc}
\exp\left(\ell_{1,c}(\mathrm{hol}_\mu)\right)&0&0\\
0&1&0\\
0&0&\exp\left(-\ell_{2,c}(\mathrm{hol}_\mu)\right)
\end{array}\right]
\cdot\xi^{(1)}_{\rho^1}(z_2).$$
Let $a,d\in{\mathbb{R}}$ so that $\xi^{(1)}_{\rho^1}(z_1)=[a:1:d]^T$. Observe that $a,d<0$, and $$\xi^{(1)}_{\rho^2}(z_2)=[\exp\left(\ell_{1,c}(\mathrm{hol}_\mu)\right) a:1:\exp\left(-\ell_{2,c}(\mathrm{hol}_\mu)\right)d]^T.$$ One can then compute that $$\begin{aligned}
r_c(D_c\cdot\mathrm{hol}_\mu)&=&\frac{\ell_{1,c}(D_c\cdot\mathrm{hol}_\mu)\sigma_{\gamma_c^-,\gamma_c^+}(D_c\cdot\mathrm{hol}_\mu)-\ell_{2,c}(D_c\cdot\mathrm{hol}_\mu)\sigma_{\gamma_c^+,\gamma_c^-}(D_c\cdot\mathrm{hol}_\mu)}{3}\\
&=&\frac{\ell_{1,c}(\mathrm{hol}_\mu)(-\log(-d)+\ell_{2,c}(\mathrm{hol}_\mu))-\ell_{2,c}(\mathrm{hol}_\mu)(\log(-a)+\ell_{1,c}(\mathrm{hol}_\mu))}{3}\\
&=&\frac{-\ell_{1,c}(\mathrm{hol}_\mu)\log(-d)-\ell_{2,c}(\mathrm{hol}_\mu)\log(-a)}{3}\\
&=&\frac{\ell_{1,c}(\mathrm{hol}_\mu)\sigma_{\gamma_c^-,\gamma_c^+}(\mathrm{hol}_\mu)-\ell_{2,c}(\mathrm{hol}_\mu)\sigma_{\gamma_c^+,\gamma_c^-}(\mathrm{hol}_\mu)}{3}\\
&=&r_c(\mathrm{hol}_\mu).\end{aligned}$$ On the other hand, $$\begin{aligned}
t_c(D_c\cdot\mathrm{hol}_\mu)&=&\sigma_{\gamma_c^-,\gamma_c^+}(D_c\cdot\mathrm{hol}_\mu)+\sigma_{\gamma_c^+,\gamma_c^-}(D_c\cdot\mathrm{hol}_\mu)\\
&=&-\log(-d)+\ell_{2,c}(\mathrm{hol}_\mu)+\log(-a)+\ell_{1,c}(\mathrm{hol}_\mu)\\
&=&\sigma_{\gamma_c^-,\gamma_c^+}(\mathrm{hol}_\mu)+\sigma_{\gamma_c^+,\gamma_c^-}(\mathrm{hol}_\mu)+\ell_{1,c}(\mathrm{hol}_\mu)+\ell_{2,c}(\mathrm{hol}_\mu)\\
&=&t_c(\mathrm{hol}_\mu)+\ell_{1,c}(\mathrm{hol}_\mu)+\ell_{2,c}(\mathrm{hol}_\mu).\end{aligned}$$
Furthermore, the reparametrized bulge parameter going to $\infty$ or $-\infty$ has the following interpretation.
\[lem:bulge\] Let $z_1,z_2\in\widetilde{{\mathcal{V}}}$ so that $\gamma_c^-<z_2<\gamma_c^+<z_1<\gamma_c^-$ be the two points used to define $b_c$ (they correspond to a bridge across $c$). For all $j\in{\mathbb{Z}}^+$, let $\mu^j\in{\mathcal{C}}(S)$ and choose $\rho^j\in\mathrm{hol}(\mu^j)$ so that for all $i,j$, $$\xi_{\rho^j}(\gamma_c^+)=\xi_{\rho^i}(\gamma_c^+),\,\,\,\xi_{\rho^j}(\gamma_c^-)=\xi_{\rho^i}(\gamma_c^-),\,\,\,\text{ and }\,\,\,\xi_{\rho^j}^{(1)}(z_1)=\xi_{\rho^i}^{(1)}(z_1).$$ Also, let $\Omega^j$ be the $\rho^j$-equivariant developing image of $\mu^j$, and let $\Delta$ be the (open) triangle containing $\xi_{\rho^j}^{(1)}(z_2)$ whose vertices are $\rho_j^{(1)}(\gamma_c^+)$, $\rho_j^{(1)}(\gamma_c^-)$, and $\rho_j^{(2)}(\gamma_c^-)\cap \rho_j^{(2)}(\gamma_c^+)$. (Note that $\Delta$ does not depend on $j$.) If there is some $C>0$ so that $$-C<t_c(\mathrm{hol}_{\mu^j})<C\,\,\text{ and }\,\,\frac{1}{C}<\ell_{1,c}(\mathrm{hol}_{\mu^j}),\,\,\ell_{2,c}(\mathrm{hol}_{\mu^j})<C$$ for all $j$, then
1. $\lim_{j\to\infty}r_c(\mu^j)=\infty$ if and only if $\lim_{j\to\infty}\Omega^j\cap\Delta=\Delta$.
2. $\lim_{j\to\infty}r_c(\mu^j)=-\infty$ if and only if $\lim_{j\to\infty}\Omega^j\cap\Delta$ is empty.
We will only prove (1); the proof of (2) is similar. By transforming everything by a projective transformation, we may assume that for all $j$, $$\xi_{\rho^j}^{(1)}(\gamma_c^+)=[0:0:1]^T,\,\,\, \xi_{\rho^j}^{(1)}(\gamma_c^-)=[1:0:0]^T,\,\,\, \xi_{\rho^j}^{(1)}(z_1)=[1:1:1]^T,$$ $$\xi_{\rho^j}^{(2)}(\gamma_c^+)=[1:0:0],\,\,\, \text{ and }\,\,\,\xi_{\rho^j}^{(2)}(\gamma_c^-)=[0:0:1].$$ Let $\xi^{(1)}_{\rho^j}(z_2)=[a_j:1:d_j]^T$. Then $a_j,d_j<0$, and a straightforward computation shows that $$r_c(\mathrm{hol}_{\mu_j})=\frac{-\ell_{1,c}(\mathrm{hol}_{\mu_j})\log(-d_j)-\ell_{2,c}(\mathrm{hol}_{\mu_j})\log(-a_j)}{3}$$ and $$t_c(\mathrm{hol}_{\mu_j})=\log(-a_j)-\log(-d_j).$$ Hence, $e^{-C}<\frac{a_j}{d_j}<e^C$ for all $j$. Since $\frac{1}{C}<\ell_{1,c}(\mathrm{hol}_{\mu^j}),\,\,\ell_{2,c}(\mathrm{hol}_{\mu^j})<C$ for all $j$, $\lim_{j\to\infty}r_c(\mu_j)=\infty$ if and only if $\lim_{j\to\infty}a_j=0=\lim_{j\to\infty}d_j$. This implies that $\lim_{j\to\infty}r_c(\mu_j)=\infty$ if and only if $$\lim_{j\to\infty}[a_j:1:d_j]^T=[0:1:0]^T,$$ which happens if and only if $\lim_{j\to\infty}\Omega^j\cap\Delta=\Delta$ by the convexity of $\Omega_j$.
From our construction, it is clear that for any multi-curve ${\mathcal{D}}\subset{\mathcal{P}}$ and any connected component $S'$ of $S\setminus{\mathcal{D}}$, the oriented pants decomposition ${\mathcal{P}}$ on $S$ restricts to an oriented pants decomposition ${\mathcal{P}}'$ on $S'$. Also, the ideal triangulation ${\mathcal{T}}_{{\mathcal{P}}'}$ on $S'$ is naturally a subset of the ideal triangulation ${\mathcal{T}}_{\mathcal{P}}$ on $S$, so the choice of bridge across each closed edge in ${\mathcal{T}}_{\mathcal{P}}$ induces a choice of bridge across each closed edge of ${\mathcal{T}}_{{\mathcal{P}}'}$. Thus, ${\mathcal{P}}$ together with the choice of a bridge across each closed edge in ${\mathcal{T}}_{\mathcal{P}}$ determines a coordinate system on $\mathrm{hol}({\mathcal{C}}(S'))$. Furthermore,
1. The length parameters on $\mathrm{hol}({\mathcal{C}}(S'))$ are exactly the length parameters on $\mathrm{hol}({\mathcal{C}}(S))$ associated to the closed curves in ${\mathcal{P}}'\subset{\mathcal{P}}$ and the boundary curves of $S'$.
2. The internal parameters on $\mathrm{hol}({\mathcal{C}}(S'))$ are exactly the internal parameters on $\mathrm{hol}({\mathcal{C}}(S))$ associated to the pairs of pants in ${\mathbb{P}}_{\mathcal{P}}$ that lie in $S'$.
3. The twist, bulge, and reparameterized bulge parameters on $\mathrm{hol}({\mathcal{C}}(S'))$ are exactly the twist, bulge, and reparameterized bulge parameters on $\mathrm{hol}({\mathcal{C}}(S))$ associated to the closed curves in ${\mathcal{P}}'\subset{\mathcal{P}}$.
This immediately implies the following remark.
\[rem: coordinates\] For all ${\mathcal{D}}\subset{\mathcal{P}}$, the coordinate system on $\mathrm{hol}({\mathcal{C}}(S))$ determines a coordinate system on $\mathrm{hol}({\mathcal{C}}(S,{\mathcal{D}})^\mathrm{adm})\subset\prod_{i=1}^k\mathrm{hol}({\mathcal{C}}(S_i))$. More precisely, if $\{d_1,\dots,d_n\}$ are the punctures of $S$, ${\mathcal{P}}\setminus{\mathcal{D}}=\{c_1,\dots,c_{3g-3+n-k}\}$, ${\mathcal{D}}=\{e_1,\dots,e_k\}$, and ${\mathbb{P}}=\{P_1,\dots,P_{2g-2+n}\}$, then this coordinate system is given explicitly by the homeomorphism $$\Phi_{\mathcal{D}}:\mathrm{hol}({\mathcal{C}}(S,{\mathcal{D}})^\mathrm{adm})\to({\mathbb{R}}^2)^{3g-3+n-k}\times({\mathbb{R}}_+^2)^{3g-3+n-k}\times({\mathbb{R}}_{\geq 0}^2)^{k+n}\times({\mathbb{R}}^2)^{2g-2+n},$$ where $$\begin{aligned}
\Phi_{\mathcal{D}}&:=&\prod_{i=1}^{3g-3+n-k}(r_{c_i},t_{c_i})\prod_{i=1}^{3g-3+n-k}(\ell_{1,c_i},\ell_{2,c_i})\\
&&\hspace{2cm}\prod_{i=1}^{k}(\ell_{1,e_i},\ell_{2,e_i})\prod_{i=1}^{n}(\ell_{1,d_i},\ell_{2,d_i})\prod_{j=1}^{2g-2+n}(i_{1,P_j},i_{2,P_j}).\end{aligned}$$
Coordinate description of the topology of ${\mathcal{C}}(S)^\mathrm{aug}$ {#sec: coord}
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As we observed in (4) and (5) of Remark \[rem: top\], ${\mathcal{C}}(S)^\mathrm{aug}$ is not locally compact and ${\mathcal{C}}(S)^\mathrm{aug}/{\mathrm{MCG}}(S)$ is an orbifold with a complicated singular locus. As such, it is not easy to give local coordinates for either ${\mathcal{C}}(S)^\mathrm{aug}$ or ${\mathcal{C}}(S)^\mathrm{aug}/{\mathrm{MCG}}(S)$.
This however, tells us that taking the quotient of ${\mathcal{C}}(S)^\mathrm{aug}$ by the trivial group is “too big", while taking the quotient of ${\mathcal{C}}(S)^\mathrm{aug}$ by all of ${\mathrm{MCG}}(S)$ is “too small". The naive dream is then to find a subgroup $G\subset{\mathrm{MCG}}(S)$ so that the topology on ${\mathcal{C}}(S)^\mathrm{aug}/G$ admits a nice local description about every point $p$. It turns out that this too is impossible. However, the next theorem tells us that if we allow $G$ to change depending on the stratum of ${\mathcal{C}}(S)^\mathrm{aug}$ where $p$ lies, then there is an explicit description of the topology on a neighborhood of $p$ in ${\mathcal{C}}(S)^\mathrm{aug}/G$.
More precisely, let ${\mathcal{D}}$ be any multi-curve and let $G_{\mathcal{D}}\subset{\mathrm{MCG}}(S)$ be the subgroup generated by Dehn twists about ${\mathcal{D}}$. Recall that we previously defined $$V_{\mathcal{D}}:=\bigcup_{{\mathcal{D}}'\subset{\mathcal{D}}}{\mathcal{C}}(S,{\mathcal{D}}')^\mathrm{adm}.$$ Observe that there is a natural $G_{\mathcal{D}}$-action on $X_{{\mathcal{D}}'}:={\mathcal{C}}(S,{\mathcal{D}}')^\mathrm{adm}$ for all ${\mathcal{D}}'\subset{\mathcal{D}}$, so we can define $W_{{\mathcal{D}}',{\mathcal{D}}}:=X_{{\mathcal{D}}'}/G_{\mathcal{D}}$ and $U_{\mathcal{D}}:=V_{\mathcal{D}}/G_{\mathcal{D}}$. Equip $W_{{\mathcal{D}}',{\mathcal{D}}}$ and $U_{\mathcal{D}}$ with the quotient topology. By (1) of Remark \[rem: top\], $V_{\mathcal{D}}\subset{\mathcal{C}}(S)^\mathrm{aug}$ is an open set about any point in $X_{\mathcal{D}}\subset{\mathcal{C}}(S)^\mathrm{aug}$.
The goal of this section is to prove the following theorem.
\[thm: main\] Let ${\mathcal{D}}$ be any multi-curve, let ${\mathcal{P}}\supset{\mathcal{D}}$ be an oriented pants decomposition on $S$ and choose orientations about the punctures of $S$. Let ${\mathcal{T}}_{\mathcal{P}}$ be the induced ideal triangulation as described in Section \[sec: triangulation\], and choose a bridge across every closed edge of ${\mathcal{T}}_{\mathcal{P}}$. Then there is an explicit homeomorphism $$\Psi_{\mathcal{D}}:U_{\mathcal{D}}\to({\mathbb{R}}^2)^{3g-3+n-k}\times({\mathbb{R}}_+^2)^{3g-3+n-k}\times({\mathbb{R}}^4)^k\times ({\mathbb{R}}^2)^n\times({\mathbb{R}}^2)^{2g-2+n}.$$ In particular, $U_{\mathcal{D}}$ is homeomorphic to a cell.
The homeomorphism $\Theta_{{\mathcal{D}}',{\mathcal{D}}}$
---------------------------------------------------------
As a preliminary step to define $\Psi_{\mathcal{D}}$, we first define a continuous parameterization $\Theta_{{\mathcal{D}}',{\mathcal{D}}}$ of $\mathrm{hol}(W_{{\mathcal{D}}',{\mathcal{D}}}):=\mathrm{hol}(X_{{\mathcal{D}}'})/G_{\mathcal{D}}$ for any ${\mathcal{D}}'\subset{\mathcal{D}}$. Per the hypothesis of Theorem \[thm: main\], choose an oriented pants decomposition ${\mathcal{P}}\supset{\mathcal{D}}$, an orientation on every boundary component of $S$, and a bridge across every closed edge in ${\mathcal{T}}_{\mathcal{P}}$. Remark \[rem: coordinates\] tells us that for any ${\mathcal{D}}'\subset{\mathcal{D}}$, these choices determine a coordinate system on $\mathrm{hol}(X_{{\mathcal{D}}'})$.
Let $\{S_1',\dots,S_k'\}$ be the connected components of $S\setminus \mathcal{D}'$. Observe that if $c\in{\mathcal{D}}'$, then the Dehn twist $D_c\in G_{\mathcal{D}}$ about $c$ acts as the identity on $\prod_{j=1}^k{\mathcal{C}}(S_j')$. Also, if $c\in{\mathcal{D}}\setminus{\mathcal{D}}'$, then $c$ lies in the interior of $S_j'$ for some $j=1,\dots,k$. In that case, $D_c$ acts as the identity on ${\mathcal{C}}(S_i')$ for all $i\neq j$, and its action on ${\mathcal{C}}(S_j')$ induces an action on $\mathrm{hol}({\mathcal{C}}(S_j'))$, which we have described explicitly in terms of the coordinates on $\mathrm{hol}({\mathcal{C}}(S_j'))$ in Proposition \[prop:Dehntwistcoords\].
Proposition \[prop:Dehntwistcoords\] implies that aside from the twist coordinates in $\{t_c:c\in{\mathcal{D}}\setminus{\mathcal{D}}'\}$, all the other coordinate functions on $\mathrm{hol}(X_{{\mathcal{D}}'})$ descend to well-defined functions on $\mathrm{hol}(W_{{\mathcal{D}}',{\mathcal{D}}})$, which in turn give well-defined functions on $W_{{\mathcal{D}}',{\mathcal{D}}}$ by precomposing with the holonomy map $\mathrm{hol}:W_{{\mathcal{D}}',{\mathcal{D}}}\to\mathrm{hol}(W_{{\mathcal{D}}',{\mathcal{D}}})$. Although $t_c$ does not descend to a well-defined function on $\mathrm{hol}(W_{{\mathcal{D}}',{\mathcal{D}}})$ for all $c\in{\mathcal{D}}\setminus{\mathcal{D}}'$, we may replace $t_c$ with the map $\theta_c:\mathrm{hol}(X_{{\mathcal{D}}'})\to{\mathbb{S}}:= {\mathbb{R}}/(2\pi\cdot{\mathbb{Z}})$ defined by $$\label{eqn:theta}
\theta_c:\mathrm{hol}(\mu)=(\mathrm{hol}(\mu_1),\dots,\mathrm{hol}(\mu_k))\mapsto \frac{2\pi t_c(\mathrm{hol}(\mu_j))}{\ell_{1,c}(\mathrm{hol}(\mu_j))+\ell_{2,c}(\mathrm{hol}(\mu_j))}.$$ By Proposition \[prop:Dehntwistcoords\], this map descends to a well-defined map $\theta_c:\mathrm{hol}(W_{{\mathcal{D}}',{\mathcal{D}}})\to {\mathbb{S}}$, so we may think of it as a map from $W_{{\mathcal{D}}',{\mathcal{D}}}$ to ${\mathbb{S}}$ by pre-composing with $\mathrm{hol}:W_{{\mathcal{D}}',{\mathcal{D}}}\to\mathrm{hol}(W_{{\mathcal{D}}',{\mathcal{D}}})$.
If $c\in {\mathcal{D}}\setminus{\mathcal{D}}'$, we may define the functions $g_{1,c},\dots,g_{4,c}:\mathrm{hol}(W_{{\mathcal{D}}',{\mathcal{D}}})\to{\mathbb{R}}$ by $$\begin{aligned}
g_{1,c}(\cdot)&:=&\ell_{1,c}(\cdot) \ell_{2,c}(\cdot) \cos\left(\frac{\pi}{2}f(r_c(\cdot))\right)\cos(\theta_c(\cdot)),\\
g_{2,c}(\cdot)&:=&\ell_{1,c}(\cdot) \ell_{2,c}(\cdot) \cos\left(\frac{\pi}{2}f(r_c(\cdot))\right)\sin(\theta_c(\cdot)),\\
g_{3,c}(\cdot)&:=&\ell_{1,c}(\cdot)\ell_{2,c}(\cdot) \sin\left(\frac{\pi}{2}f(r_c(\cdot))\right),\\
g_{4,c}(\cdot)&:=&\ell_{1,c}(\cdot)^2-\ell_{2,c}(\cdot)^2.\end{aligned}$$ In the above formulas, $f:{\mathbb{R}}\to{\mathbb{R}}$ is the smooth function given by $f(s)=\frac{e^s-1}{e^s+1}$. Again, for $i=1,\dots,4$ and $c\in {\mathcal{D}}\setminus{\mathcal{D}}'$, $g_{i,c}$ can also be viewed as functions on $W_{{\mathcal{D}}',{\mathcal{D}}}$ by pre-composing with $\mathrm{hol}:W_{{\mathcal{D}}',{\mathcal{D}}}\to\mathrm{hol}(W_{{\mathcal{D}}',{\mathcal{D}}})$.
On the other hand, if $c\in{\mathcal{D}}'$ or $c$ is a puncture of $S$, define $g_{1,c},\dots,g_{4,c}:W_{{\mathcal{D}}',{\mathcal{D}}}\to{\mathbb{R}}$ by $$\begin{aligned}
g_{1,c}(\cdot)&:=&0\\
g_{2,c}(\cdot)&:=&0\\
g_{3,c}(\cdot)&:=&\left\{\begin{array}{ll}
0&\text{if }\mu\text{-type of } c\text{ is parabolic or quasi-hyperbolic}\\
\ell_{1,c}(\cdot)\ell_{2,c}(\cdot)&\text{if }\mu\text{-type of } c\text{ is bulge }+\infty\\
-\ell_{1,c}(\cdot)\ell_{2,c}(\cdot)&\text{if }\mu\text{-type of } c\text{ is bulge }-\infty
\end{array}\right.,\\
g_{4,c}(\cdot)&:=&\ell_{1,c}(\cdot)^2-\ell_{2,c}(\cdot)^2.\end{aligned}$$ Note that if $c\in{\mathcal{D}}'$ or $c$ is a puncture of $S$, $g_{i,c}$ is a function on $\mathrm{hol}(W_{{\mathcal{D}}',{\mathcal{D}}})$ for $i=1,2,4$. This is not so for $g_{3,c}$, but its absolute value $|g_{3,c}(\cdot)|=\ell_{1,c}(\cdot)\ell_{2,c}(\cdot)$ is a function on $\mathrm{hol}(W_{{\mathcal{D}}',{\mathcal{D}}})$. Moreover, in the cases in which the reparametrized bulge parameter $r_c(\cdot) = \pm\infty$, we have simply extended the formulas above by identifying $f(\infty)=1$ and $f(-\infty)=-1$.
\[not:standard\] Let ${\mathcal{P}}\setminus{\mathcal{D}}=:\{c_1,\dots,c_{3g-3+n-m}\}$, ${\mathcal{D}}\setminus{\mathcal{D}}'=:\{e_{m'+1},\dots,e_m\}$, ${\mathcal{D}}'=:\{e_1,\dots,e_{m'}\}$, and $\{d_1,\dots,d_n\}$ be the punctures of $S$. Note that $m:=|{\mathcal{D}}|$, $m':=|{\mathcal{D}}'|\leq m$.
With this notation, define $$\Theta_{{\mathcal{D}}',{\mathcal{D}}}:\mathrm{hol}(W_{{\mathcal{D}}',{\mathcal{D}}})\to({\mathbb{R}}^2)^{3g-3+n-m}\times({\mathbb{R}}_+^2)^{3g-3+n-m}\times({\mathbb{R}}^4)^m\times({\mathbb{R}}^2)^n\times({\mathbb{R}}^2)^{2g-2+n}$$ by $$\begin{aligned}
\Theta_{{\mathcal{D}}',{\mathcal{D}}}&\:=&\prod_{i=1}^{3g-3+n-m}(r_{c_i},t_{c_i})\prod_{i=1}^{3g-3+n-m}(\ell_{1,c_i},\ell_{2,c_i})\prod_{i=m'+1}^{m}(g_{1,e_i},g_{2,e_i},g_{3,e_i},g_{4,e_i})\\
&&\hspace{1cm}\prod_{i=1}^{m'}(g_{1,e_i},g_{2,e_i},|g_{3,e_i}|,g_{4,e_i})\prod_{i=1}^{n}(|g_{3,d_i}|,g_{4,d_i})\prod_{j=1}^{2g-2+n}(i_{1,P_j},i_{2,P_j}).\end{aligned}$$
\[lem:besthol\] Let $E_1:=({\mathbb{R}}^2\setminus(0,0))\times {\mathbb{R}}^2\subset{\mathbb{R}}^4$ and $E_2:=(0,0)\times{\mathbb{R}}_{\geq 0}\times{\mathbb{R}}\subset{\mathbb{R}}^4$. The map $\Theta_{{\mathcal{D}}',{\mathcal{D}}}$ is a homeomorphism onto its image, which is $$({\mathbb{R}}^2)^{3g-3+n-m}\times({\mathbb{R}}_+^2)^{3g-3+n-m}\times E_1^{m-m'}\times E_2^{m'}\times({\mathbb{R}}_{\geq 0}\times{\mathbb{R}})^n\times({\mathbb{R}}^2)^{2g-2+n}.$$
Recall from Remark \[rem: coordinates\] that the map $$\Phi_{{\mathcal{D}}'}:\mathrm{hol}(X_{{\mathcal{D}}'})\to({\mathbb{R}}^2)^{3g-3+n-m'}\times({\mathbb{R}}_+^2)^{3g-3+n-m'}\times({\mathbb{R}}_{\geq 0}^2)^{m'+n}\times({\mathbb{R}}^2)^{2g-2+n}$$ given by $$\begin{aligned}
\Phi_{{\mathcal{D}}'}&:=&\left(\prod_{i=1}^{3g-3+n-m}(r_{c_i},t_{c_i})\prod_{i=m'+1}^{m}(r_{e_i},t_{e_i})\right)\\
&&\hspace{1.5cm}\left(\prod_{i=1}^{3g-3+n-m}(\ell_{1,c_i},\ell_{2,c_i})\prod_{i=m'+1}^{m}(\ell_{1,e_i},\ell_{2,e_i})\right)\\
&&\hspace{3cm}\prod_{i=1}^{m'}(\ell_{1,e_i},\ell_{2,e_i})\prod_{i=1}^{n}(\ell_{1,d_i},\ell_{2,d_i})\prod_{j=1}^{2g-2+n}(i_{1,P_j},i_{2,P_j}).\end{aligned}$$ is a homeomorphism. From this and the definition of the $G_{\mathcal{D}}$-action on $\mathrm{hol}(X_{{\mathcal{D}}'})$ defined above, we see that $$\begin{aligned}
\overline{\Phi}_{{\mathcal{D}}',{\mathcal{D}}}:\mathrm{hol}(W_{{\mathcal{D}}',{\mathcal{D}}})&\to&({\mathbb{R}}^2)^{3g-3+n-m}\times({\mathbb{R}}_+^2)^{3g-3+n-m}\times({\mathbb{R}}\times{\mathbb{S}}\times{\mathbb{R}}_+^2)^{m-m'}\times\\
&&\hspace{4cm}({\mathbb{R}}_{\geq 0}^2)^{m'}\times({\mathbb{R}}_{\geq 0}^2)^n\times({\mathbb{R}}^2)^{2g-2+n}\end{aligned}$$ given by $$\begin{aligned}
\overline{\Phi}_{{\mathcal{D}}',{\mathcal{D}}}&:=&\prod_{i=1}^{3g-3+n-m}(r_{c_i},t_{c_i})\prod_{i=1}^{3g-3+n-m}(\ell_{1,c_i},\ell_{2,c_i})\prod_{i=m'+1}^{m}(r_{e_i},\theta_{e_i},\ell_{1,e_i},\ell_{2,e_i})\\
&&\hspace{3cm}\prod_{i=1}^{m'}(\ell_{1,e_i},\ell_{2,e_i})\prod_{i=1}^{n}(\ell_{1,d_i},\ell_{2,d_i})\prod_{j=1}^{2g-2+n}(i_{1,P_j},i_{2,P_j}).\end{aligned}$$ is also a homeomorphism.
From the definition of $(g_{1,e_i},\dots,g_{4,e_i})$, one sees that to finish the proof, it is sufficient to prove that the maps $F_1:{\mathbb{R}}\times{\mathbb{S}}\times{\mathbb{R}}^2_+\to E_1$ and $F_2:({\mathbb{R}}_{\geq 0})^2\to E_2$ defined by $$\begin{aligned}
F_1(a_1,a_2,a_3,a_4)&:=&\left(a_3a_4\cos\left(\frac{\pi}{2}f(a_1)\right)\cos(a_2),\right.\\
&&a_3a_4\cos\left(\frac{\pi}{2}f(a_1)\right)\sin(a_2),\left.a_3a_4\sin\left(\frac{\pi}{2}f(a_1)\right),a_3^2-a_4^2\right),\\
F_2(a_3,a_4)&:=&(0,0,a_3a_4,a_3^2-a_4^2)\end{aligned}$$ are homeomorphisms. But this can be verified easily by writing down explicit continuous formulas for the inverse maps for both $F_1$ and $F_2$.
$\Psi_{\mathcal{D}}$ is a bijection
-----------------------------------
Next, we will explicitly describe the map $\Psi_{\mathcal{D}}$ in Theorem \[thm: main\] and show that it is a bijection. Define $$\Psi_{{\mathcal{D}}',{\mathcal{D}}}:W_{{\mathcal{D}}',{\mathcal{D}}}\to({\mathbb{R}}^2)^{3g-3+n-m}\times({\mathbb{R}}_+^2)^{3g-3+n-m}\times({\mathbb{R}}^4)^m\times ({\mathbb{R}}^2)^n\times({\mathbb{R}}^2)^{2g-2+n},$$ by $$\begin{aligned}
\Psi_{{\mathcal{D}}',{\mathcal{D}}}&:=&\prod_{i=1}^{3g-3+n-m}(r_{c_i},t_{c_i})\prod_{i=1}^{3g-3+n-m}(\ell_{1,c_i},\ell_{2,c_i})\prod_{i=1}^m(g_{1,e_i},g_{2,e_i},g_{3,e_i},g_{4,e_i})\\
&&\hspace{5cm}\prod_{i=1}^n(g_{3,d_i},g_{4,d_i})\prod_{j=1}^{2g-2+n}(i_{1,P_j},i_{2,P_j}).\end{aligned}$$
\[lem:bijection on strata\] Let $E_1:=({\mathbb{R}}^2\setminus(0,0))\times {\mathbb{R}}^2\subset{\mathbb{R}}^4$ and $E_3:={\mathbb{R}}^4\setminus E_1=(0,0)\times{\mathbb{R}}^2\subset{\mathbb{R}}^4$. The map $\Psi_{{\mathcal{D}}',{\mathcal{D}}}$ is a bijection onto its image, which is $$({\mathbb{R}}^2)^{3g-3+n-m}\times({\mathbb{R}}_+^2)^{3g-3+n-m}\times E_1^{m-m'}\times E_3^{m'}\times({\mathbb{R}}^2)^n\times({\mathbb{R}}^2)^{2g-2+n}.$$
For any $\mu\in X_{{\mathcal{D}}'}$, let ${\mathcal{A}}_{\mathrm{hol}(\mu)}\subset\{d_1,\dots,d_n\}$ be the punctures of $S$ whose $\mu$-type is bulge $\pm\infty$, and let ${\mathcal{D}}'_{\mathrm{hol}(\mu)}:=\{c\in{\mathcal{D}}':\mu\text{-type of c is bulge }\pm\infty\}$. By Corollary \[cor: hol\], we see that $\mathrm{hol}^{-1}(\mathrm{hol}(\mu))\subset X_{{\mathcal{D}}'}$ has $2^{|{\mathcal{D}}'_{\mathrm{hol}(\mu)}\cup{\mathcal{A}}_{\mathrm{hol}(\mu)}|}$ elements, each of which corresponds to the choice of whether the $\mu$-type of $c$ is bulge $+\infty$ or bulge $-\infty$ for each $c\in{\mathcal{D}}'_{\mathrm{hol}(\mu)}\cup{\mathcal{A}}_{\mathrm{hol}(\mu)}$. The same is true for $\mathrm{hol}^{-1}(\mathrm{hol}[\mu])\subset W_{{\mathcal{D}}',{\mathcal{D}}}$ as well, because the only element in $G_{\mathcal{D}}$ that sends $\mathrm{hol}^{-1}(\mathrm{hol}(X_{{\mathcal{D}}'}))$ to itself is the identity.
Note that by replacing the coordinate functions $|g_{3,c}|$ of $\Theta_{{\mathcal{D}}',{\mathcal{D}}}$ with $g_{3,c}$ for $c\in\{e_1,\dots,e_{m'}\}\cup\{d_1,\dots,d_n\}$ allows us to distinguish whether the $\mu$-type of $c$ is bulge $+\infty$ or bulge $-\infty$. The lemma follows immediately from this observation.
As defined, the target of $\Psi_{{\mathcal{D}}',{\mathcal{D}}}$ does not depend on $m'$, but only on $m$. Since $$U_{\mathcal{D}}=\bigcup_{{\mathcal{D}}'\subset{\mathcal{D}}}W_{{\mathcal{D}}',{\mathcal{D}}},$$ is a disjoint union, we may define $$\Psi_{\mathcal{D}}:U_{\mathcal{D}}\to({\mathbb{R}}^2)^{3g-3+n-m}\times({\mathbb{R}}_+^2)^{3g-3+n-m}\times({\mathbb{R}}^4)^m\times ({\mathbb{R}}^2)^n\times({\mathbb{R}}^2)^{2g-2+n}$$ by $\Psi_{\mathcal{D}}[\mu]:=\Psi_{{\mathcal{D}}',{\mathcal{D}}}[\mu]$ if $[\mu]\in W_{{\mathcal{D}}',{\mathcal{D}}}$. As a consequence of Lemma \[lem:bijection on strata\], we have the following proposition.
The map $\Psi_{\mathcal{D}}$ is a bijection.
Simply note that $E_1\cup E_2={\mathbb{R}}^4$ is a disjoint union, where $E_1$ and $E_2$ are as defined in the statement of Lemma \[lem:bijection on strata\].
$\Psi_{\mathcal{D}}$ is a homeomorphism
---------------------------------------
To finish the proof of Theorem \[thm: main\], we need to show that the bijection $\Psi_{\mathcal{D}}$ is a homeomorphism. Recall that $\mathcal C(S)^{\rm aug}$ is first countable (see (3) of Remark \[rem: top\]). Hence, it is sufficient to show that if $\mu=[\mu]\in X_{\mathcal{D}}=W_{{\mathcal{D}},{\mathcal{D}}}$ and $\{[\mu^j]\}_{j=1}^\infty$ is a sequence in $U_{\mathcal{D}}$, then $\lim_{j\to\infty}[\mu^j]=[\mu]$ in $U_{\mathcal{D}}$ if and only if $\lim_{j\to\infty}\Psi_{\mathcal{D}}[\mu^j]=\Psi_{\mathcal{D}}[\mu]$.
Let $S_1,\dots,S_k$ be the connected components of $S\setminus{\mathcal{D}}$. Since ${\mathcal{D}}$ is a finite set, by considering the subsequences of $\{[\mu^j]\}_{j=1}^\infty$ that lie in different strata separately, we may further assume that $\{[\mu^j]\}_{j=1}^\infty\subset W_{{\mathcal{D}}',{\mathcal{D}}}$ for some fixed ${\mathcal{D}}'\subset{\mathcal{D}}$. Also, note that the map $\mathrm{Pull}_{{\mathcal{D}}',{\mathcal{D}}}:X_{{\mathcal{D}}'}\to X_{\mathcal{D}}$ descends to $\mathrm{Pull}_{{\mathcal{D}}',{\mathcal{D}}}:W_{{\mathcal{D}}',{\mathcal{D}}}\to W_{{\mathcal{D}},{\mathcal{D}}}$. Then (1) of Remark \[rem: top\] implies that it is sufficient to prove the following theorem.
\[thm: special case\] Let ${\mathcal{D}}$ be an oriented multi-curve on $S$, let $S_1,\dots,S_k$ be the connected components of $S\setminus{\mathcal{D}}$, and let $\mu=(\mu_1,\dots,\mu_k)\in X_{\mathcal{D}}$. Also, let ${\mathcal{D}}'\subset{\mathcal{D}}$, let $\{[\mu^j]\}_{j=1}^\infty\subset W_{{\mathcal{D}}',{\mathcal{D}}}$, and let $\mu^j_i\in{\mathcal{C}}(S_i)$ so that $\mathrm{Pull}_{{\mathcal{D}}',{\mathcal{D}}}[\mu^j]=(\mu^j_1,\dots,\mu^j_k)$.
1. If $\lim_{j\to\infty}\mu^j_i=\mu_i$ for all $i=1,\dots,k$, then $\lim_{j\to\infty}\Psi_{{\mathcal{D}}',{\mathcal{D}}}[\mu^j]=\Psi_{{\mathcal{D}},{\mathcal{D}}}(\mu)$.
2. If $\lim_{j\to\infty}\Psi_{{\mathcal{D}}',{\mathcal{D}}}[\mu^j]=\Psi_{{\mathcal{D}},{\mathcal{D}}}(\mu)$, then $\lim_{j\to\infty}\mu^j_i=\mu_i$ for all $i=1,\dots,k$.
In the above theorem, we again choose an oriented pants decomposition ${\mathcal{P}}\supset{\mathcal{D}}$, a bridge across every closed edge of ${\mathcal{T}}_{\mathcal{P}}$, and an orientation about every puncture of $S$ to define $\Psi_{\mathcal{D}}$.
We first prove (1) of Theorem \[thm: special case\].
First, observe that if $\lim_{j\to\infty}\mu^j_i=\mu_i$ for all $i=1,\dots,k$, then $\lim_{j\to\infty}\mathrm{hol}(\mu^j_i)=\mathrm{hol}(\mu_i)$ for all $i=1,\dots,k$. So Lemma \[lem:besthol\] implies that
- $\displaystyle\lim_{j\to\infty}(i_{1,P_i}[\mu^j],i_{2,P_i}[\mu^j])=(i_{1,P_i}(\mu),i_{2,P_i}(\mu))$ for all $i=1,\dots,2g-2+n$,
- $\displaystyle\lim_{j\to\infty}(r_{c_i}[\mu^j],t_{c_i}[\mu^j])=(r_{c_i}(\mu),t_{c_i}(\mu))$ for all $i=1,\dots,3g-3+n-m$,
- $\displaystyle\lim_{j\to\infty}(\ell_{1,c_i}[\mu^j],\ell_{2,c_i}[\mu^j])=(\ell_{1,c_i}(\mu),\ell_{2,c_i}(\mu))$ for all $i=1,\dots,3g-3+n-m$,
- $\displaystyle\lim_{j\to\infty}(g_{1,e_i}[\mu^j],g_{2,e_i}[\mu^j],g_{3,e_i}[\mu^j],g_{4,e_i}[\mu^j])=(g_{1,e_i}(\mu),g_{2,e_i}(\mu),g_{3,e_i}(\mu),g_{4,e_i}(\mu))$ for all $i=m'+1,\dots,m$
- $\displaystyle\lim_{j\to\infty}(g_{1,e_i}[\mu^j],g_{2,e_i}[\mu^j],|g_{3,e_i}[\mu^j]|,g_{4,e_i}[\mu^j])=(g_{1,e_i}(\mu),g_{2,e_i}(\mu),|g_{3,e_i}(\mu)|,g_{4,e_i}(\mu))$ for all $i=1,\dots,m'$
- $\displaystyle\lim_{j\to\infty}(|g_{3,d_i}[\mu^j]|,g_{4,d_i}[\mu^j])=(|g_{3,d_i}(\mu)|,g_{4,d_i}(\mu))$ for all $i=1,\dots,n$
Thus, to prove (1) of Theorem \[thm: special case\], it is sufficient to prove that $$\label{eqn:g3}\lim_{j\to\infty}g_{3,e_i}[\mu^j]=g_{3,e_i}(\mu)\text{ for all }i=1,\dots,m'$$ and $$\label{eqn:g3'}\lim_{j\to\infty}g_{3,d_i}[\mu^j]=g_{3,d_i}(\mu)\text{ for all }i=1,\dots,n.$$ We will only give the proof of (\[eqn:g3\]); the same argument also works for (\[eqn:g3’\]).
Let $a\in\{1,\dots,k\}$ so that $S_a$ is the connected component of $S\setminus {\mathcal{D}}$ that lies on the left of $e_i$. Also, let $p$ be the puncture of $S_a$ that corresponds to $e_i$. If the $\mu_a$-type of $p$ is quasi-hyperbolic or parabolic, then observe that $g_{3,e_i}(\mu)=0$. Hence, $\lim_{j\to\infty}|g_{3,e_i}[\mu^j]|=|g_{3,e_i}(\mu)|=0$, so $$\lim_{j\to\infty}g_{3,e_i}[\mu^j]=0=g_{3,e_i}(\mu).$$
On the other hand, if the $\mu_a$-type of $p$ is bulge $\pm\infty$, then $\ell_{1,e_i}(\mu),\ell_{2,e_i}(\mu)>0$, which implies that there is some $C>1$ so that $$\frac{1}{2C}<\frac{1}{C}<\ell_{1,e_i}[\mu^j],\ell_{2,e_i}[\mu^j]<C<2C$$ for all $j$. Since we can choose representatives $\mu^j\in[\mu^j]$ so that $0\leq t_{e_i}(\mu^j)<\ell_{1,e_i}(\mu^j)+\ell_{2,e_i}(\mu^j)$, this implies that $$-2C< t_{e_i}(\mu^j)< 2C.$$ Lemma \[lem:bulge\] then implies that $\lim_{j\to\infty}r_{e_i}(\mu^j)=\pm\infty$ if the $\mu_a$-type of $p_L$ is bulge $\pm\infty$, so $r_{e_i}[\mu^j]=r_{e_i}(\mu^j)$ is positive (resp. negative) for sufficiently large $j$ if the $\mu_a$-type of $p$ is $+\infty$ (resp. $-\infty$). A straightforward calculation then proves that for sufficiently large $j$, $g_{3,e_i}[\mu^j]$ is positive (resp. negative) if the $\mu_a$-type of $p$ is $+\infty$ (resp. $-\infty$). By definition, $g_{3,e_i}(\mu)$ is positive (resp. negative) if the $\mu_a$-type of $p$ is $+\infty$ (resp. $-\infty$). Since $\lim_{j\to\infty}|g_{3,e_i}[\mu^j]|=|g_{3,e_i}(\mu)|$, this immediately proves (\[eqn:g3\]).
For any convex real projective structure $\mu\in{\mathcal{C}}(S)$, let $\rho\in\mathrm{hol}(\mu)$, and let $\Omega$ be the $\rho$-equivariant developing image of $\mu$. To prove (2) of Theorem \[thm: special case\], we need to define two properly convex domains $\widehat{\Omega}$ and $\widecheck{\Omega}$ so that
- $\rho(\pi_1(S))$ acts properly discontinuously on both $\widehat{\Omega}$ and $\widecheck{\Omega}$,
- $\widehat{\Omega}$ and $\widecheck{\Omega}$ depend only on $\rho$, and
- $\widecheck{\Omega}\subset\Omega\subset\widehat{\Omega}$.
First, we define $\widehat{\Omega}$. Proposition \[prop:transverse1\] states that $\xi_{\rho}^{(2)}(x)$ and $\Omega$ do not intersect for any $x\in\widetilde{{\mathcal{V}}}$. Also, Proposition \[prop:transverse\] implies that $\xi_{\rho}^{(1)}(y)$ does not lie in $\xi_{\rho}^{(2)}(x)$ for all distinct $x,y\in\widetilde{{\mathcal{V}}}$, which in particular implies that $\xi_{\rho}^{(2)}(x)\neq\xi_{\rho}^{(2)}(y)$. Since $\xi^{(1)}_{\rho}(\widetilde{{\mathcal{V}}})\subset\partial\Omega$, this means that one of the two connected components of ${\mathbb{R}}{\mathbb{P}}^2\setminus(\xi_{\rho}^{(2)}(x)\cup\xi_{\rho}^{(2)}(y))$, denoted $H(x,y)$, contains $\xi_\rho^{(1)}(\widetilde{{\mathcal{V}}}\setminus\{x,y\})$. With this, we can define $\widehat{\Omega}$ to be the interior of $$\bigcap_{\mathrm{distinct }\,x,y\in\widetilde{{\mathcal{V}}}}H(x,y).$$ It is clear that $\widehat{\Omega}$ is open, $\Omega\subset\widehat{\Omega}$, and $\rho(\pi_1(S))$ acts on $\widehat{\Omega}$. In particular, $\xi_\rho^{(1)}(\widetilde{{\mathcal{V}}})\subset\partial\widehat{\Omega}$.
1. The open set $\widehat{\Omega}\subset{\mathbb{R}}{\mathbb{P}}^2$ is properly convex.
2. The action of $\rho(\pi_1(S))$ on $\widehat{\Omega}$ is properly discontinuous.
\(1) Let $x,y,z\in\widetilde{{\mathcal{V}}}$ be a pairwise distinct triple of points. It is straightforward to check that since $t_{x,y,z}>0$, ${\mathbb{R}}{\mathbb{P}}^2\setminus(\xi_\rho^{(2)}(x)\cup\xi_\rho^{(2)}(y)\cup\xi_\rho^{(2)}(z))$ is a union of four properly convex (open) triangles. By definition, $\widehat{\Omega}$ has to lie in one of these four triangles, so $\overline{\widehat{\Omega}}$ does not contain an entire projective line in ${\mathbb{R}}{\mathbb{P}}^2$. With this, it is clear from the definition of $\widehat{\Omega}$ that $\widehat{\Omega}$ is properly convex.
\(2) Since $\widehat{\Omega}$ is properly convex, we can define the Hilbert metric $d_{\widehat{\Omega}}$ on $\widehat{\Omega}$ (see proof of Proposition \[prop:devimage\]), which is invariant under the $\rho(\pi_1(S))$ action on $\widehat{\Omega}$. Recall that $(\widehat{\Omega},d_{\widehat{\Omega}})$ is a proper path metric space. Hence, $\rho(\pi_1(S))$ acts properly discontinuously on $\widehat{\Omega}$ because $\rho(\pi_1(S))$ is a discrete subgroup of the isometry group of the Hilbert metric.
Next, we define $\widecheck{\Omega}$ to be the interior of the convex hull of $\xi_{\rho}^{(1)}(\widetilde{{\mathcal{V}}})$ in $\widehat{\Omega}$. Since $\Omega$ is properly convex and $\xi_{\rho}^{(1)}(\widetilde{{\mathcal{V}}})\subset\overline{\Omega}$, we see that $\xi_{\rho}^{(1)}(\widetilde{{\mathcal{V}}})$ is not contained in a projective line in ${\mathbb{R}}{\mathbb{P}}^2$. Thus, $\widecheck{\Omega}$ is non-empty. Furthermore, since $\xi_{\rho}^{(1)}(\widetilde{{\mathcal{V}}})\subset\partial\Omega$, the convexity of $\Omega$ implies that $\widecheck{\Omega}\subset \Omega$. In particular, $\widecheck{\Omega}$ is a non-empty, properly convex subset of ${\mathbb{R}}{\mathbb{P}}^2$, on which $\rho(\pi_1(S))$ acts properly discontinuously.
\[rem:cont\] By construction, we see that $\widehat{\Omega}$ and $\widecheck{\Omega}$ depend only on $\rho$, and vary continuously with $\rho$.
With this, we can prove (2) of Theorem \[thm: special case\]
Since $\lim_{j\to\infty}\Psi_{{\mathcal{D}}',{\mathcal{D}}}[\mu^j]=\Psi_{{\mathcal{D}},{\mathcal{D}}}(\mu)$, it is clear that $\lim_{j\to\infty}\Theta_{{\mathcal{D}}',{\mathcal{D}}}(\mathrm{hol}[\mu^j])=\Theta_{{\mathcal{D}},{\mathcal{D}}}(\mathrm{hol}(\mu))$. It then follows from Remark \[rem: coordinates\] that $\lim_{j\to\infty}\mathrm{hol}(\mu^j_i)=\mathrm{hol}(\mu_i)$ for all $i=1,\dots,k$. This means that we can find a representative $\rho^j_i\in\mathrm{hol}(\mu^j_i)$ and a representative $\rho_i\in\mathrm{hol}(\mu_i)$ so that $\lim_{j\to\infty}\rho^j_i=\rho_i$, and $0\leq t_{e}(\mu_i^j)<\ell_{1,e}[\mu^j]+\ell_{2,e}[\mu^j]$ for all $e\in{\mathcal{D}}$. In particular, there is some $C>0$ so that $-2C\leq t_e(\mu^j_i)\leq 2C$ for all $j,i$.
Since $\lim_{j\to\infty}\rho^j_i=\rho_i$, we see that $\lim_{j\to\infty}\xi_{\rho^j_i}=\xi_{\rho_i}$ uniformly. Also, by Remark \[rem:cont\], $\lim_{j\to\infty}\widehat{\Omega}^j_i=\widehat{\Omega}_i$ and $\lim_{j\to\infty}\widecheck{\Omega}^j_i=\widecheck{\Omega}_i$. Since $\widecheck{\Omega}^j_i\subset\Omega^j_i\subset\widehat{\Omega}^j_i$ and $\widecheck{\Omega}_i\subset\Omega_i\subset\widehat{\Omega}_i$, it follows from Theorem \[thm: hol\] and Lemma \[lem:bulge\] that $\lim_{j\to\infty}\Omega_i^j=\Omega_i$. (2) of Theorem \[thm: special case\] follows.
Convex real projective structures via cubic differentials {#sec: conv cubic}
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For a closed oriented surface $S$ of genus at least 2, Labourie [@Labourie07] and the first author [@Loftin2001] independently showed that a convex ${\mathbb{RP}}^2$ structure on $S$ is equivalent to a pair $(X,U)$, where $X$ is a complex structure on $S$ and $U$ is a holomorphic cubic differential on $X$. This correspondence was later extended to regular convex ${\mathbb{RP}}^2$ structures on the one hand and pairs $(X,U)$, where $X$ is a noded, connected Riemann surface and $U$ is a regular cubic differential over $X$ [@BenHul13; @Nie15; @Loftin2004; @Loftin]. The notion of a regular $k$-differential is due to Bers [@Bers74], while the geometric and analytic foundation of the relationship between cubic differentials and convex ${\mathbb{RP}}^2$ structures follows largely from deep work on hyperbolic affine spheres of Cheng-Yau [@ChengYau77; @ChengYau86].
To formally state this result, we recall some standard terminology from the theory of Riemann surfaces. Let $\bar X$ be a compact, noded Riemann surface. A neighborhood of each node of $\bar X$ is biholomorphic to a neighborhood of the origin in $\{(z,w)\in{\mathbb{C}}^2: zw=0\}$. We refer to $z$ and $w$ here as *local coordinates* near the node. Let $P$ be a (possibly empty) finite collection of points in $\bar X$ that are not nodes, and let $X=\bar X \setminus P$. We refer to the points in $P$ as *punctures* of $X$. Also let $\mathring{X}$ denote the complement of the nodes in $X$. The *normalization of $X$* is a smooth (possibly disconnected) Riemann surface equipped with a projection map to $X$ which is a biholomorphism restricted to the preimage of $\mathring X$ and is two-to-one over each node.
A *regular cubic differential* on $X$ is a meromorphic section $\sigma$ of the third tensor power of the holomorphic cotangent bundle over the normalization of $\bar X$ with the following properties:
- $\sigma$ is holomorphic on $\mathring{X}$,
- $\sigma$ has poles of order at most $3$ at each node and puncture of $\bar{X}$.
- the [*residues*]{} of $\sigma$ sum to 0 at each node, i.e. in terms $z,w$ local coordinates near the nodes, the third-order terms of the cubic differential are $R\,dz^3/z^3$ and $-R \, dw^3/w^3$, for a complex constant $R$.
The set of regular cubic differentials on $X$ is naturally a finite dimensional complex vector space.
Via the unifomization theorem, the Teichmüller space ${\mathcal{T}}(S)$ can be thought of as the deformation space of complex structures on $S$. From this point of view, the augmented Teichmüller space $\mathcal T(S)^{\rm aug}$ is a stratified space with strata $\mathcal T(S,\mathcal D)^{\rm aug}$ enumerated by the multi-curves $\mathcal D$ on $S$. Each stratum $\mathcal T(S,\mathcal D)^{\rm aug}$ is the deformation space of marked noded, compact Riemann surfaces with punctures $P$, with the property that the marking $f:S\setminus{\mathcal{D}}\to\mathring{X}$ identifies
- neighborhoods of the punctures of $\mathring{X}$ with neighborhoods of the punctures of $S$,
- neighborhoods of the nodes of $\mathring{X}$ to neighborhoods in $S$ of the curves in ${\mathcal{D}}$.
Now for a fixed multi-curve $\mathcal D$, define $$\mathcal T(S)^{\rm aug,\mathcal D} = \bigcup_{\mathcal D'\subset \mathcal D} \mathcal T(S,\mathcal D')^{\rm aug},$$ with the subspace topology induced from that on $\mathcal T(S)^{\rm aug}$. Recall $G_{\mathcal D}$ is the subgroup of the mapping class group generated by Dehn twists around loops in $\mathcal D$, and define $\mathcal Q_{\mathcal D}:=\mathcal T^{\rm aug,\mathcal D}(S)/ G_{\mathcal D}$. Let ${\mathcal{X}}_{\mathcal D}$ be the proper flat family of noded Riemann surfaces parametrized by $\mathcal Q_{\mathcal D}$, and let $\mathcal K^{\rm reg}_{\mathcal{D}}$ be the complex vector bundle of regular cubic differentials over ${\mathcal{Q}}_{\mathcal{D}}$. In other words, the fiber of ${\mathcal{X}}_{{\mathcal{D}}}$ above the point $X\in{\mathcal{Q}}_{\mathcal{D}}$ is $X$ (see e.g. [@HubKoch14] for a full discussion), and the fiber of $\mathcal K^{\rm reg}_{\mathcal{D}}$ above the point $X\in{\mathcal{Q}}_{\mathcal{D}}$ is the vector space of regular cubic differentials on $X$. With this notation, we can state the following theorem.
\[thm: cx-manifold\] $\mathcal Q_{\mathcal D}$ carries the natural structure of a complex manifold, and $\mathcal K^{\rm reg}_{\mathcal{D}}$ is a holomorphic vector bundle over ${\mathcal{Q}}_{\mathcal{D}}$. In particular, the total space of $\mathcal K^{\rm reg}_{\mathcal{D}}$ has the structure of a complex manifold.
Hubbard-Koch [@HubKoch14] construct the complex structure on $\mathcal Q_{\mathcal D}$. See [@Loftin] and [@HubKoch14] for a proof that $\mathcal K^{\rm reg}_{\mathcal{D}}$ is a holomorphic vector bundle.
In the setting when $S$ is a closed surface, the first author [@Loftin2001] and Labourie [@Labourie07] independently established the following theorem.
\[LL-thm\] Let $S$ be a closed connected oriented surface of genus at least two. Then there is a canonical bijective correspondence between ${\mathcal{C}}(S)$ and the total space of the vector bundle over $\mathcal T(S)$ whose fibers over a point $X\in{\mathcal{T}}(S)$ is the space of cubic differentials on $X$. In particular, this defines a canonical complex structure on ${\mathcal{C}}(S)$.
The first author later extended this theorem to ${\mathcal{C}}(S)^{\rm aug}$. More precisely, he proved the following (see Theorem 4.3.1 and Section 5.1 of [@Loftin]).
\[thm: Loftin\] There is a canonical continuous bijection $\Xi$ from the total space of $\mathcal K^{\rm reg}_{{\mathcal{D}}}$ to the quotient space $\mathcal C(S)^{\rm aug,\mathcal D} / G_{\mathcal D}$.
See (1) of Remark \[rem: top\] for the definition of $\mathcal C(S)^{\rm aug,\mathcal D} $.
The first author worked in the setting when $S$ is a closed surface, but his arguments show that Theorem \[thm: Loftin\] holds for finite type surfaces with negative Euler characteristic as well.
Thus Theorem \[thm: main\], together with the above theorem shows that $\Xi$ is a continuous one-to-one correspondence between manifolds of the same dimension. Then Brouwer’s Invariance of Domain Theorem shows $\Xi$ is a homeomorphism. Corollary \[cor:Loftin\] follows immediately.
The Proof of Theorem \[thm: hol\]
=================================
In this appendix, we give a proof of Theorem \[thm: hol\]. We define an *interval* in $\partial\Omega$ to be a subset of $\partial \Omega$ homeomorphic to an interval in $\mathbb R$. Note intervals need not be straight line segments. The key step in this proof is summarized in the following proposition.
\[rule-out-gaps\] Let $\mu\in{\mathcal{C}}(S)$, let $\rho\in\mathrm{hol}_\mu$, and let $\Omega$ be the $\rho$-equivariant developing image of $\mu$. Also, let $$\begin{aligned}
\mathscr{E}(\rho)=\mathscr{E}&:=&\{\text{saddle fixed points of hyperbolic elements in }\rho(\pi_1(S))\},\\
\mathscr{F}(\rho)=\mathscr{F}&:=&\{\text{fixed points of non-identity elements in }\rho(\pi_1(S))\}\setminus{\mathcal{E}},\\
\Gamma(\rho)=\Gamma&:=&\{\gamma\in\pi_1(S):\gamma\text{ is peripheral and }\rho(\gamma)\text{ is hyperbolic or quasi-hyperbolic}\},\\
\mathscr{J}(\rho)=\mathscr{J}&:=&\{\langle\rho(\gamma)\rangle:\gamma\in\Gamma\}.\end{aligned}$$ For each $H=\langle\rho(\gamma)\rangle\in\mathscr{J}$, let $I_H\subset\partial\Omega$ be the unique interval that does not contain any points in $\mathscr F$, and whose endpoints are:
- the attracting and repelling fixed points of $\rho(\gamma)$ if $\rho(\gamma)$ is hyperbolic,
- the two fixed points of $\rho(\gamma)$ if $\rho(\gamma)$ is quasi-hyperbolic.
Then $$\partial \Omega \setminus\overline{\mathscr{F}}=\bigcup_{H\in\mathscr{J}}I_H.$$
The interval $I_H$ in Proposition \[rule-out-gaps\] exists because the every element in $H$ is peripheral. Assuming Proposition \[rule-out-gaps\], we will now prove Theorem \[thm: hol\].
Let $\gamma\in\pi_1(S)\setminus\{{\mathrm{id}}\}$. It is easy to observe from the dynamics of the $\langle\rho_i(\gamma)\rangle$ action on ${\mathbb{R}}{\mathbb{P}}^2$ that if $\rho_i(\gamma)$ has a fixed point $p$ that does not lie in $\partial\Omega_i\cap\partial\Omega_i'$, then $\rho_i(\gamma)$ is necessarily hyperbolic and $p$ is the saddle fixed point of $\rho_i(\gamma)$. Since $\partial\Omega_i\cap\partial\Omega_i'\subset{\mathbb{R}}{\mathbb{P}}^2$ is closed, it follows that $\overline{{\mathcal{F}}}\subset\partial\Omega_i\cap\partial\Omega_i'$. Proposition \[rule-out-gaps\] then implies that $$\partial \Omega_i\setminus\overline{\mathscr{F}(\rho_i)}=\bigcup_{H\in\mathscr{J}(\rho_i)}I_H\,\,\,\text{ and }\,\,\,\partial \Omega_i'\setminus\overline{\mathscr{F}(\rho_i)}=\bigcup_{H\in\mathscr{J}(\rho_i)}I_H',$$ where $I_H\subset\partial\Omega_i$ and $I_H'\subset\partial\Omega_i'$ are the unique open intervals that do not contain any points in $\mathscr{F}(\rho_i)$, and whose endpoints are
- the attracting and repelling fixed points of $\rho_i(\gamma)$ if $\rho_i(\gamma)$ is hyperbolic,
- the two fixed points of $\rho_i(\gamma)$ if $\rho_i(\gamma)$ is quasi-hyperbolic.
If $\rho_i(\gamma)$ is quasi-hyperbolic, then observe from the dynamics of the $H:=\langle\rho_i(\gamma)\rangle$ action on ${\mathbb{R}}{\mathbb{P}}^2$ that $I_H=I_H'$ is the straight line segment between the two fixed points of $\rho_i(\gamma)$, or else the convexity of $\Omega$ is violated. (See for example [@Loftin2004]).
On the other hand, if $\rho_i(\gamma)$ is hyperbolic, and $H:=\langle\rho_i(\gamma)\rangle$, then the admissibility of $\mu_i$ implies that $I_H$ is either the (open) edge of the principal triangle of $\rho_i(\gamma)$ whose endpoints are $\{\rho_i(\gamma)^+,\rho_i(\gamma)^-\}$ (bulge $-\infty$), or the union of $\rho_i(\gamma)^0$ with the two edges of the principal triangle of $\rho_i(\gamma)$ that have $\rho_i(\gamma)^0$ as a common vertex (bulge $+\infty$). In either case, the endpoints of $I_H$ are $\{\rho_i(\gamma)^+,\rho_i(\gamma)^-\}$. The admissibility of $\mu_i'$ implies the same for $I_H'$. Thus, if $I_H\neq I_H'$, then $I_H\cup I_H'$ is the union of $\rho_i(\gamma)^0$ with the three edges of the principal triangle of $\rho_i(\gamma)$. See Figure \[admissible-hyperbolic-figure\].
![Principal triangle[]{data-label="admissible-hyperbolic-figure"}](admissible-hyperbolic.eps)
It follows immediately from this that the interior of the symmetric difference $\Omega_i\, \triangle\, \Omega_i'$ is the union of a $\pi_1(S_i)$-invariant subset of triangles in ${\mathcal{G}}_{\rho_i}$.
It is thus sufficient to prove Proposition \[rule-out-gaps\]. To do so, we need the notion of a limit set. First of all, recall that $\widecheck{\Omega}\subset\Omega$ is the interior of the convex hull in $\Omega$ of $\overline{\mathscr{F}}$, and that $\rho(\pi_1(S))$ acts on $\widecheck{\Omega}$ freely properly discontinuously.
For $p\in \widecheck{\Omega}$, define the *limit set* $\Lambda_p=\Lambda_p(\rho)$ of the $\rho(\pi_1(S))$-action on $\widecheck{\Omega}$ to be the set of accumulation points in $\partial\widecheck{\Omega}$ of $\rho(\pi_1(S))\cdot p$.
Next, we want to prove that $\Lambda_p=\overline{\mathscr{F}}$ for all $p\in\widecheck{\Omega}$.
\[limit-set-lemma\] For every $p\in\widecheck{\Omega}$, $\Lambda_p=\overline{\mathscr{F}}$. In particular, $\Lambda_{p_1}=\Lambda_{p_2}$ for all $p_1,p_2\in\widecheck{\Omega}$.
The proof of this lemma in large part follows Kuiper [@Kuiper54], p. 208. We recall the argument for the reader’s convenience.
It is clear that $\overline{\mathscr{F}}\subset\Lambda_p$. Let $m$ be any maximal open subinterval of $\partial\widecheck{\Omega}\setminus\overline{\mathscr{F}}$. By our definition of $\widecheck{\Omega}$, $m$ is a projective line segment. We will now prove that no points in $m$ can lie in $\Lambda_p$. This immediately implies that $\Lambda_p\subset\overline{\mathscr{F}}$.
Suppose for contradiction that there is some $q\in m$ so that $q\in\Lambda_p$. This means that there is a sequence $\{\gamma_i\}_{i=1}^\infty\subset\pi_1(S)$ so that $\rho(\gamma_i)\cdot p\to q$ as $i\to\infty$. The sequence $\{\rho(\gamma_i)\}_{i=1}^\infty$ lies in ${\mathrm{PGL}}(3,{\mathbb{R}})\subset{\mathbb{P}}(\mathrm{End}({\mathbb{R}}^3))$, and ${\mathbb{P}}(\mathrm{End}({\mathbb{R}}^3))$ is compact,. Thus (possibly passing to a subsequence), we may assume this sequence $\rho(\gamma_i)$ has a limit $g_\infty\in{\mathbb{P}}(\mathrm{End}({\mathbb{R}}^3))$, which is the projectivization of a linear endomorphism $L_\infty:{\mathbb{R}}^3\to{\mathbb{R}}^3$ of rank $1$ or $2$. In other words, $g_\infty$ is a projective map $g_\infty:{\mathbb{R}}{\mathbb{P}}^2\setminus{\mathbb{P}}(\ker L_\infty)\to{\mathbb{R}}{\mathbb{P}}^2$ whose image is ${\mathbb{P}}({\mathrm{im}}\,L_\infty)$. Moreover, $\rho(\gamma_i)\to g_\infty$ uniformly on compact subsets of ${\mathbb{RP}}^2 \setminus {\mathbb{P}}(\ker L_\infty)$.
Consider a geodesic ball $B$ of radius $\epsilon>0$ centered at $p$ with respect to the Hilbert metric on $\widecheck{\Omega}$. The uniform convergence $\rho(\gamma_i)\to g_\infty$ on compact sets imply that $$g_\infty(B) =\lim_{i\to\infty} \rho(\gamma_i)\cdot B,$$ so the proper discontinuity of $\rho$ implies that $g_\infty(B) \subset \partial\widecheck{\Omega}$. By the definition of the Hilbert metric and the fact that $q\in\partial\widecheck{\Omega}$ is in the interior of a line segment in the boundary, it is straightforward to see that there is a neighborhood $H_\epsilon$ of $q \in {\mathbb{RP}}^2$ so that $H_\epsilon\cap\widecheck{\Omega}$ does not contain any balls of radius $\epsilon$ with respect to the Hilbert metric. (To see this, consider the Hilbert distance along a sequence of line segments in $\widecheck\Omega$ with endpoints in $\partial\widecheck\Omega\setminus\overline m$ which converges to $m$.) Since each $\rho(\gamma_i)$ is an isometry, this implies that $g_\infty(B)$ is not a point. The fact that $L_\infty$ has rank either $1$ or $2$ then gives us that $g_\infty(B)$ has to be an open subsegment of $m$. In particular, $L_\infty$ has rank 2, so ${\mathbb{P}}(\ker L_\infty)$ is a single point $t$, and ${\mathbb{P}}({\mathrm{im}}\, L_\infty)=\ell$. The maximality of $m$ then implies that $\overline{m}=\partial\widecheck\Omega\cap\ell$. Also, $t$ also has to be a point in $\partial{\widecheck{\Omega}}$, because for any $x\in\widecheck\Omega$, $\rho(\gamma_i^{-1})\cdot x$ converges to $t$.
The remainder of the proof proceeds in two cases, depending on whether $t\in\ell$ or not. In either case, we will show that $\widecheck{\Omega}$ is an open triangle. This will be a contradiction because it is easy to see that the projective automorphism group of such a triangle is virtually Abelian, so there is no injective representation $\pi_1(S)\to {\rm Aut}(\widecheck{\Omega})$ since $S$ has negative Euler characteristic.
Observe that the preimage under $g_\infty$ of any point in $\ell$ is a projective line in ${\mathbb{RP}}^2$ containing $t$, minus the point $t$ itself. Moreover, $g_\infty(\widecheck\Omega) \subset m$, since for every $x\in \widecheck\Omega$, $g_\infty(x) = \lim \rho(\gamma_i)\cdot x \in \partial \widecheck\Omega$, and $m$ is the largest open subset in $\ell\cap \partial \widecheck\Omega$.
First we consider the case when $t\not\in\ell$. Let $T$ and $T'$ be the two open triangles in ${\mathbb{R}}{\mathbb{P}}^2$ with $t$ as a vertex and $m$ as the opposite edge. Since $m,\{t\} \subset\partial \widecheck\Omega$, the convexity of $\widecheck\Omega$ shows that either $T$ or $T'$ is contained in $\widecheck\Omega$. Thus we may assume without loss of generality that $T\subset\widecheck\Omega$ and thus $g_\infty(T)\subset m$. Since $g_\infty$ is projective linear, $g_\infty(T')=g_\infty(T)\subset m$ as well. Moreover, $g_\infty$ maps ${\mathbb{RP}}^2 \setminus (\overline T \cup \overline T')$ to $\ell \setminus \overline m$. To prove a contradiction, assume there is an $x\not\in \overline T \cup \overline T'$ so that $g_\infty(x) \in \overline m$. Let $\xi$ be the point of intersection of $\ell$ and the line $\overline{tx}$. Note $\xi\not\in\overline m$. If $g_\infty(x) \in m$, then $g_\infty(\xi) = g_\infty(x)\in m$. For any neighborhood of $\xi$, the image under $\rho(\gamma_i)$ of this neighborhood intersects $m$ and also $\widecheck\Omega$ for large $i$, by the convergence of $\rho(\gamma_i)$ to $g_\infty$. Since $\partial\widecheck\Omega$ is invariant under the action of $\rho(\gamma_i)^{-1}$, we see then that $\xi\in \partial\widecheck\Omega$. But this is a contradiction since $\xi \in \ell\setminus \overline m$ and $\overline m = \partial\widecheck \Omega \cap \ell$.
To recap, $g_\infty$ maps both $T$ and $T'$ to $m$, and also maps ${\mathbb{RP}}^2 \setminus (\overline T\cup \overline T')$ to $\ell\setminus \overline m$. Since $\widecheck\Omega$ is properly convex and contains $m$ and $t$ in its boundary, and since we have assumed that $\widecheck\Omega$ contains $T$, $\widecheck\Omega$ remains disjoint from $T'$. To prove that $\widecheck\Omega=T$, it is now sufficient to show that $\widecheck\Omega$ does not intersect the complement of $\overline{T}\cup\overline{T}'$. If $\widecheck\Omega$ contains a point $x\in{\mathbb{RP}}^2 \setminus (\overline T\cup \overline T')$, then $g_\infty(x)\in \ell\setminus \overline{m}$ is a boundary point of $\widecheck{\Omega}$, which is impossible by the convexity of $\widecheck\Omega$ and the maximality of $m$. Next, consider the case when $t\in\ell$. In this case, the preimage under $g_\infty$ of each point in $\ell$ is a projective line in ${\mathbb{RP}}^2$ containing $t$, minus the point $t$ itself. This case then further splits into two subcases: when $t\in \overline{m}$ and when $t\in\ell\setminus\overline{m}$. If $t\in\ell\setminus \overline m$, then the maximality of $m$ and the proper convexity of $\widecheck{\Omega}$ implies that $t\not\in\partial{\widecheck{\Omega}}$, which is a contradiction.
On the other hand, if $t\in \overline m$, let $$X = \{ x\in {\mathbb{RP}}^2\setminus\{t\} : \overline{tx} \cap \widecheck{\Omega}\neq \emptyset\},$$ where $\overline{tx}$ is the projective line in ${\mathbb{RP}}^2$ passing through $t$ and $x$. Also let $ Y = {\mathbb{RP}}^2 \setminus \overline X$. Since $\rho(\gamma_i)$ leaves $\widecheck\Omega$ invariant for all $i$ and $\overline m=\ell\cap\partial\widecheck\Omega$, we see that $g_\infty(X) = m$ and $g_\infty(Y) = \ell \setminus \overline m$ \[the inclusion $g_\infty(Y)\subset \ell\setminus \overline m$ follows as in the argument above to show $g_\infty({\mathbb{RP}}^2\setminus(\overline T \cup \overline T'))\subset\ell\setminus\overline m$\]. Furthermore, since $\widecheck{\Omega}$ is a properly convex subset of ${\mathbb{R}}{\mathbb{P}}^2$, $\overline{m}$ is a proper connected subset of $\ell$, which implies that $Y$ is non-empty and $t\in\partial m$. Thus, $X$ is an open, connected region in ${\mathbb{R}}{\mathbb{P}}^2$ whose boundary consists of two distinct projective lines, $\ell_1,\ell_2$, which meet at $t$, and $g_\infty(\partial X\setminus\{t\}) = \partial m$.
Since $t\in\partial\widecheck\Omega$, the proper convexity of $\widecheck\Omega$ implies that $\partial\widecheck\Omega$ is a union of at most three paths in $\overline{X}$. There are two cases. In the first case, there are two line segment paths, call them $a_1$ and $a_2$ in $\ell_1$ and $\ell_2$ respectively, both of which have $t$ as an endpoint. In the second case, there is a single line segment $a_1\subset\ell_1$ which has $t$ as an endpoint. Note that we may assume $\ell_1=\ell$ and $a_1=m$ in both cases. In the first case, let $x_2$ be the endpoint of $\ell_2$ which is not $t$. In the second case, let $x_2=t$. In both cases then, the remaining path $a_3$ in $\partial\widecheck\Omega$ has $x_1$ and $x_2$ as endpoints, and its interior lies in $X$. It is now sufficient to show that $a_3$ is a projective line segment, since in the first case $\widecheck\Omega$ is an open triangle and in the second case, $\partial\widecheck\Omega$ consists of only two line segments, and so $\widecheck\Omega$ is not properly convex.
For any $y$ in the interior of $a_3$, $g_\infty(y) \in m$ because $y\in X$. Since $m$ is open in $\partial\widecheck{\Omega}$ and $\rho(\gamma_i)$ leaves $\partial \widecheck{\Omega}$ invariant for all $i$, we see that $\rho(\gamma_i)\cdot y \in m$ for large $i$. Thus, for any connected, compact neighborhood $I_y$ of $y$ in $m$, $\rho(\gamma_i)\cdot I_y\subset m$ for sufficiently large $i$. This means that $I_y$ is a projective line segment for all $y$ in the interior of $a_3$, so $a_3$ is a projective line segment.
In the previous lemma, we can weaken the restriction that $p\in\widecheck{\Omega}$. In fact, $\Lambda_p = \overline{ \mathcal F}$ for any $p\in\Omega$, by Proposition \[rule-out-gaps\] and basic facts about the dynamics of hyperbolic elements on the principal triangle.
With this, we can now finish the proof of Proposition \[rule-out-gaps\].
It is clear from the definitions that $$\partial \Omega \setminus\overline{\mathscr F}\supset\bigcup_{H\in\mathscr J}I_H,$$ so it is sufficient to prove the other inclusion, i.e. every maximal open interval in $\partial\Omega\setminus\overline{\mathscr F}$ is of the form $I_H$ for some $H\in\mathscr J$. Let $\widecheck{\Omega}\subset \Omega$ be the convex in $\Omega$ of $\overline{\mathscr F}$. Since $\partial\Omega$ and $\partial\widecheck\Omega$ are both oriented topological circles and the cyclic ordering on $\overline{\mathscr F}$ induced by both $\partial\Omega$ and $\partial\widecheck\Omega$ agree, we see that there is a canonical bijection between connected components of $\partial\Omega\setminus\overline{\mathscr{F}}$ and $\partial\widecheck\Omega\setminus\overline{\mathscr{F}}$. It is thus sufficient to prove the proposition for $\widecheck\Omega$ in place of $\Omega$.
Then $\partial\widecheck{\Omega} \setminus \overline{\mathscr F}$ is a disjoint union of line segments. Let $\Sigma := \widecheck{\Omega}/\rho(\pi_1(S))$ and let $\pi:\widecheck{\Omega}\to\Sigma$ be the projection map. Choose any $p\in\widecheck{\Omega}$, and let $B_r(p)\subset\widecheck{\Omega}$ denote the open ball (with respect to the Hilbert metric) centered at $p$ with radius $r$. Consider $r$ large enough so that $\Sigma\setminus \pi(\overline{B_r(p)})$ is a disjoint collection of open cylinders, one for each end of $S$. Then observe that $$D := \bigcup_{\gamma\in\pi_1(S)} \rho(\gamma) \cdot\overline{B_r(p)},$$ is connected. Furthermore, our choice of $r$ ensures that each connected component of $\widecheck{\Omega}\setminus D$ is the image of the developing map of a connected component of $\Sigma\setminus \pi(\overline{B_r(p)})$, which is a cylinder.
Let $m$ be a maximal open line segment in $\partial \widecheck{\Omega}\setminus\overline{{\mathcal{F}}}$. By Lemma \[limit-set-lemma\], every $q\in m$ is not a limit point of the $\rho(\pi_1(S))$ action on $\widecheck{\Omega}$. In other words, each such $q$ has a neighborhood in $\overline{\widecheck{\Omega}}$ which does not contain any $\pi_1(S)$-translate of $p$. By considering the union of such neighborhoods for all points in $m$, we produce a neighborhood $N\subset \overline{\widecheck{\Omega}}$ of $m$ so that $\rho(\gamma)\cdot p\not\in N$ for all $\gamma\in\pi_1(S)$.
Let $bN :=\partial N\cap \widecheck{\Omega}$ and let $\widehat N := \{x\in N\cap\widecheck{\Omega} : d_{\widecheck{\Omega}}(x,bN)>r\}$. Since $\rho(\gamma)$ is an isometry with respect to the $d_{\widecheck{\Omega}}$ for every $\gamma\in\Gamma$, we find that $\widehat N \cap D = \emptyset.$ Moreover, the completeness of $d_{\widecheck{\Omega}}$ implies that $\widehat N \cup m$ is also a neighborhood of $m$ in $\overline{\widecheck{\Omega}}$. Let $\widecheck N$ denote the connected component of $\overline{\widecheck{\Omega}} \setminus \overline{D}$ which contains $\widehat N$. Since $D$ is connected and both endpoints of $m$ lie in $\overline{D}$, we see that $\widecheck N\cap\partial\widecheck{\Omega}=m$. See Figure \[classify-ends-figure\].
![A gap in the limit set[]{data-label="classify-ends-figure"}](classify-ends.eps)
Since $\widecheck N$ is the image of the developing map of a convex ${\mathbb{R}}{\mathbb{P}}^2$ cylinder, there is a infinite cyclic subgroup $H=\langle \rho(\gamma)\rangle\subset\rho(\pi_1(S))$ that preserves $\widecheck N$. The subgroup $H$ also preserves $\partial\widecheck{\Omega}$, so $H$ preserves $\overline{m}$, which implies that it fixes both endpoints of $m$ and preserves $m$. This means that $\rho(\gamma)$ is either hyperbolic or quasi-hyperbolic. The fact that $m$ does not contain any points in $\overline{\mathscr F}$ immediately implies that $\gamma\in\pi_1(S)$ is peripheral, and so $m=I_H$ for some $H\in\mathscr J$.
This completes the proof of Proposition \[rule-out-gaps\].
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---
abstract: 'Exceptional sequences are certain ordered sequences of quiver representations. We introduce a class of objects called strand diagrams and use this model to classify exceptional sequences of representations of a quiver whose underlying graph is a type $\mathbb{A}_n$ Dynkin diagram. We also use variations of this model to classify **c**-matrices of such quivers, to interpret exceptional sequences as linear extensions of posets, and to give a simple bijection between exceptional sequences and certain chains in the lattice of noncrossing partitions. This work is an extension of the classification of exceptional sequences for the linearly-ordered quiver by the first and third authors.'
address:
- 'Laboratoire de Combinatoire et d’Informatique Mathématique, Université du Québec à Montréal, Montréal, QC H3C 3P8, Canada'
- 'Department of Mathematics, Brandeis University, Waltham, MA 02454, USA'
- 'Department of Mathematics and Statistics, University of Massachusetts Amherst, Amherst, MA 01003, USA'
- 'Department of Mathematics, University of Washington, Seattle, WA 98195, USA'
author:
- Alexander Garver
- Kiyoshi Igusa
- 'Jacob P. Matherne'
- Jonah Ostroff
bibliography:
- 'refs.bib'
nocite: '[@*]'
title: Combinatorics of exceptional sequences in type A
---
Introduction
============
Exceptional sequences are certain sequences of quiver representations with strong homological properties. They were introduced in [@gr87] to study exceptional vector bundles on $\mathbb{P}^2$, and more recently, Crawley-Boevey showed that the braid group acts transitively on the set of complete exceptional sequences (exceptional sequences of maximal length) [@c93]. This result was generalized to hereditary Artin algebras by Ringel [@r94]. Since that time, Meltzer has also studied exceptional sequences for weighted projective lines [@m04], and Araya for Cohen-Macaulay modules over onedimensional graded Gorenstein rings with a simple singularity [@a99]. Exceptional sequences have been shown to be related to many other areas of mathematics since their invention:
- chains in the lattice of noncrossing partitions [@b03; @hk13; @it09],
- $\textbf{c}$-matrices and cluster algebras [@st13],
- factorizations of Coxeter elements [@is10], and
- $t$-structures and derived categories [@bez03; @bk89; @r90].
Despite their ubiquity, very little work has been done to concretely describe exceptional sequences, even for path algebras of Dynkin quivers [@a13; @gm1]. In this paper, we give a concrete description of exceptional sequences for type $\operatorname{\mathbb{A}}_n$ quivers of any orientation. This work extends and elaborates on a classification of exceptional sequences for the linearly-ordered quiver obtained in [@gm1] by the first and third authors.
Exceptional sequences are composed of indecomposable representations which have a particularly nice description. For a quiver $Q$ of type $\mathbb{A}_n$, the indecomposable representations are completely determined by their dimension vectors, which are of the form $(0,\ldots,0,1,\ldots,1,0,\ldots,0) \ \in \mathbb{Z}_{\ge 0}^n$. Let us denote such a representation by $X_{i,j}^{\epsilon}$, where $\epsilon$ is a vector that keeps track of the orientation of the quiver, and $i+1$ and $j$ are the positions where the string of $1$’s begins and ends, respectively.
This simple description allows us to view exceptional sequences as combinatorial objects. Define a map $\Phi_\epsilon$ which associates to each indecomposable representation $X_{i,j}^{\epsilon}$ a curve $\Phi_\epsilon(X_{i,j}^{\epsilon})$ connecting two of $n+1$ points in $\mathbb{R}^2$. We will refer to such curves as **strands**.[^1]
![An example of the indecomposable representation $X_{0,1}^\epsilon$ on a type $\mathbb{A}_2$ quiver and the corresponding strand $\Phi_\epsilon(X_{0,1}^{\epsilon})$[]{data-label="strex"}](strex.pdf)
As exceptional sequences are collections of representations, the map $\Phi_\epsilon$ allows one to regard them as collections of strands. The following lemma is the foundation for all of our results in this paper (it characterizes the homological data encoded by a pair of strands and thus by a pair of representations). An exceptional sequence is completely determined by its **exceptional pairs** (i.e., its subsequences of length 2). Thus, Lemma \[maintechlemma\], which we now state, allows us to completely classify exceptional sequences using strand diagrams.
Let $Q_\epsilon$ be a type $\mathbb{A}$ Dynkin quiver. Fix two distinct indecomposable representations $U, V \in \text{ind}(\text{rep}_\Bbbk(Q_\epsilon))$.
$\begin{array}{rll}
a) & \text{The strands $\Phi_{\epsilon}(U)$ and $\Phi_{\epsilon}(V)$ intersect nontrivially if and only if neither $(U,V)$ nor $(V,U)$ are}\\
& \text{exceptional pairs.}\\
b) & \text{The strand $\Phi_{\epsilon}(U)$ is clockwise from $\Phi_{\epsilon}(V)$ if and only if $(U,V)$ is an exceptional pair and $(V,U)$}\\
& \text{is not an exceptional pair.}\\
c) & \text{The strands $\Phi_{\epsilon}(U)$ and $\Phi_{\epsilon}(V)$ do not intersect at any of their endpoints and they do not intersect}\\
& \text{nontrivially if and only if $(U, V)$ and $(V, U)$ are both exceptional pairs.}
\end{array}$
The paper is organized in the following way. In Section \[sec:prelim\], we give the preliminaries on quivers and their representations which are needed for the rest of the paper.
In Section \[sec:strands\], we introduce strand diagrams. Later, we decorate our strand diagrams with strand-labelings and oriented edges so that they can keep track of both the ordering of the representations in a complete exceptional sequence as well as the signs of the rows in the $\operatorname{\textbf{c}}$-matrix it came from. While unlabeled diagrams classify complete exceptional collections (Theorem \[ECbij\]), we show that the new decorated diagrams classify exceptional sequences (Theorem \[ESbij\]). Although Lemma \[maintechlemma\] is the main tool that allows us to obtain these results, we delay its proof until Section \[firstproofs\].
The work of Speyer and Thomas (see [@st13]) allows one to obtain complete exceptional sequences from $\operatorname{\textbf{c}}$-matrices. In [@onawfr13], the number of complete exceptional sequences in type $\operatorname{\mathbb{A}}_n$ is given, and there are more of these than there are $\operatorname{\textbf{c}}$-matrices. Thus, it is natural to ask exactly which $\operatorname{\textbf{c}}$-matrices appear as strand diagrams. By establishing a bijection between the mixed cobinary trees of Igusa and Ostroff [@io13] and a certain subcollection of strand diagrams, we give an answer to this question in Section \[sec:mct\].
In Section \[sec:pos\], we ask how many complete exceptional sequences can be formed using the representations in a complete exceptional collection. It turns out that two complete exceptional sequences can be formed in this way if they have the same underlying strand diagram without strand labels. We interpret this number as the number of linear extensions of a poset determined by the strand diagram of the complete exceptional collection. This also gives an interpretation of complete exceptional sequences as linear extensions.
In Section \[sec:app\], we give several applications of the theory in type $\operatorname{\mathbb{A}}$, including combinatorial proofs that two reddening sequences produce isomorphic ice quivers (see [@k12] for a general proof in all types using deep category-theoretic techniques) and that there is a bijection between exceptional sequences and saturated chains in the lattice of noncrossing partitions.
[**Acknowledgements. **]{} A. Garver and J. Matherne gained helpful insight through conversations with E. Barnard, J. Geiger, M. Kulkarni, G. Muller, G. Musiker, D. Rupel, D. Speyer, and G. Todorov. A. Garver and J. Matherne also thank the 2014 AMS Mathematics Research Communities program for giving us a stimulating place to work. The authors thank an anonymous referee for comments that helped to improve the exposition.
Preliminaries {#sec:prelim}
=============
We will be interested in the connection between exceptional sequences and the **c**-matrices of an acyclic quiver $Q$ so we begin by defining these. After that we review the basic terminology of quiver representations and exceptional sequences, which serve as the starting point in our study of exceptional sequences. We conclude this section by explaining the notation we will use to discuss exceptional representations of quivers that are orientations of a type $\mathbb{A}_n$ Dynkin diagram.
Quiver mutation {#subsec:quivers}
---------------
A **quiver** $Q$ is a directed graph without loops or 2-cycles. In other words, $Q$ is a 4-tuple $(Q_0,Q_1,s,t)$, where $Q_0 = [m] := \{1,2, \ldots, m\}$ is a set of **vertices**, $Q_1$ is a set of **arrows**, and two functions $s, t:Q_1 \to Q_0$ defined so that for every $a \in Q_1$, we have $s(a) \xrightarrow{a} t(a)$. An **ice quiver** is a pair $(Q,F)$ with $Q$ a quiver and $F \subset Q_0$ **frozen vertices** with the additional restriction that any $i,j \in F$ have no arrows of $Q$ connecting them. We refer to the elements of $Q_0\backslash F$ as **mutable vertices**. By convention, we assume $Q_0\backslash F = [n]$ and $F = [n+1,m] := \{n+1, n+2, \ldots, m\}.$ Any quiver $Q$ can be regarded as an ice quiver by setting $Q = (Q, \emptyset)$.
The [**mutation**]{} of an ice quiver $(Q,F)$ at mutable vertex $k$, denoted $\mu_k$, produces a new ice quiver $(\mu_kQ,F)$ by the three step process:
\(1) For every $2$-path $i \to k \to j$ in $Q$, adjoin a new arrow $i \to j$.
\(2) Reverse the direction of all arrows incident to $k$ in $Q$.
\(3) Remove all 2-cycles in the resulting quiver as well as all of the arrows between frozen vertices. We show an example of mutation below, depicting the mutable (resp. frozen) vertices in black (resp. blue). $$\begin{array}{c c c c c c c c c}
\raisebox{.35in}{$(Q,F)$} & \raisebox{.35in}{=} & {\includegraphics[scale = .7]{Q2.pdf}} & \raisebox{.35in}{$\stackrel{\mu_2}{\longmapsto}$} & {\includegraphics[scale = .7]{M2.pdf}} & \raisebox{.35in}{=} & \raisebox{.35in}{$(\mu_2Q,F)$}
\end{array}$$
The information of an ice quiver can be equivalently described by its (skew-symmetric) **exchange matrix**. Given $(Q,F),$ we define $B = B_{(Q,F)} = (b_{ij}) \in \mathbb{Z}^{n\times m} := \{n \times m \text{ integer matrices}\}$ by $b_{ij} := \#\{i \stackrel{a}{\to} j \in Q_1\} - \#\{j \stackrel{a}{\to} i \in Q_1\}.$ Furthermore, ice quiver mutation can equivalently be defined as **matrix mutation** of the corresponding exchange matrix. Given an exchange matrix $B \in \mathbb{Z}^{n\times m}$, the **mutation** of $B$ at $k \in [n]$, also denoted $\mu_k$, produces a new exchange matrix $\mu_k(B) = (b^\prime_{ij})$ with entries $$b^\prime_{ij} := \left\{\begin{array}{ccl}
-b_{ij} & : & \text{if $i=k$ or $j=k$} \\
b_{ij} + \frac{|b_{ik}|b_{kj}+ b_{ik}|b_{kj}|}{2} & : & \text{otherwise.}
\end{array}\right.$$
For example, the mutation of the ice quiver above (here $m=4$ and $n=3$) translates into the following matrix mutation. Note that mutation of matrices [(and of ice quivers)]{} is an involution (i.e. $\mu_k\mu_k(B) = B$).
$$\begin{array}{c c c c c c c c c c}
B_{(Q,F)} & = & \left[\begin{array}{c c c | r}
0 & 2 & 0 & 0 \\
-2 & 0 & 1 & 0\\
0 & -1 & 0 & -1\\
\end{array}\right]
& \stackrel{\mu_2}{\longmapsto} &
\left[\begin{array}{c c c | r}
0 & -2 & 2 & 0 \\
2 & 0 & -1 & 0\\
-2 & 1 & 0 & -1\\
\end{array}\right]
& = & B_{(\mu_2Q,F)}.
\end{array}$$
[Given a quiver $Q$, we define its **framed** (resp. **coframed**) quiver to be the ice quiver $\widehat{Q}$ (resp. $\widecheck{Q}$) where $\widehat{Q}_0\ (= \widecheck{Q}_0) := Q_0 \sqcup [n+1, 2n]$, $F = [n+1, 2n]$, and $\widehat{Q}_1 := Q_1 \sqcup \{i \to n+i: i \in [n]\}$ (resp. $\widecheck{Q}_1 := Q_1 \sqcup \{n+i \to i: i \in [n]\}$).]{} Now given $\widehat{Q}$ we define the **exchange tree** of $\widehat{Q}$, denoted $ET(\widehat{Q})$, to be the (a priori infinite) graph whose vertices are quivers obtained from $\widehat{Q}$ by a finite sequence of mutations and with two vertices connected by an edge if and only if the corresponding quivers are obtained from each other by a single mutation. Similarly, define the **exchange graph** of $\widehat{Q}$, denoted $EG(\widehat{Q})$, to be the quotient of $ET(\widehat{Q})$ where two vertices are identified if and only if there is a **frozen isomorphism** of the corresponding quivers (i.e. an isomorphism that fixes the frozen vertices). Such an isomorphism is equivalent to a simultaneous permutation of the rows and columns of the corresponding exchange matrices.
Given $\widehat{Q}$, we define the **c**-**matrix** $C(n) = C_R(n)$ (resp. $C = C_R$) of $R \in ET(\widehat{Q})$ (resp. $R \in EG(\widehat{Q})$) to be the submatrix of $B_R$ where $C(n) := (b_{ij})_{i \in [n], j \in [n+1, 2n]}$ (resp. $C := (b_{ij})_{i \in [n], j \in [n+1,2n]}$). We let **c**-mat($Q$) $:= \{C_R: R \in EG(\widehat{Q})\}$. By definition, $B_R$ (resp. $C$) is only defined up to simultaneous permutations of its rows and first $n$ columns (resp. up to permutations of its rows) for any $R \in EG(\widehat{Q})$.
A row vector of a **c**-matrix, $\overrightarrow{c}$, is known as a **c**-**vector**. The celebrated theorem of Derksen, Weyman, and Zelevinsky [@dwz10 Theorem 1.7], known as [sign-coherence]{} of $\textbf{c}$-vectors, states that for any $R \in ET(\widehat{Q})$ and $i \in [n]$ the **c**-vector $\overrightarrow{c_i}$ is a nonzero element of $\mathbb{Z}_{\ge 0}^n$ or $\mathbb{Z}_{\le0}^n$. In the former case, we say a **c**-vector is **positive**, and in the latter case, we say a **c**-vector is **negative**.
Representations of quivers
--------------------------
A **representation** $V = ((V_i)_{i \in Q_0}, (\varphi_a)_{a \in Q_1})$ of a quiver $Q$ is an assignment of a finite dimensional $\Bbbk$-vector space $V_i$ to each vertex $i$ and a $\Bbbk$-linear map $\varphi_a: V_{s(a)} \rightarrow V_{t(a)}$ to each arrow $a$ where $\Bbbk$ is a field. The **dimension vector** of $V$ is the vector $\underline{\dim}(V):=(\dim V_i)_{i\in Q_0}$. The **support** of $V$ is the set $\text{supp}(V) := \{i\in Q_0 : V_i \neq 0\}$. Here is an example of a representation, with $\underline{\dim}(V) = (3,3,2)$, of the **mutable part** of the quiver depicted in Section \[subsec:quivers\]. $${\includegraphics[scale = .8]{R2prime.pdf}}$$ Let $V = ((V_i)_{i \in Q_0}, (\varphi_a)_{a \in Q_1})$ and $W = ((W_i)_{i \in Q_0}, (\varrho_a)_{a \in Q_1})$ be two representations of a quiver $Q$. A **morphism** $\theta : V \rightarrow W$ consists of a collection of linear maps $\theta_i : V_i \rightarrow W_i$ that are compatible with each of the linear maps in $V$ and $W$. That is, for each arrow $a \in Q_1$, we have $\theta_{t(a)} \circ \varphi_a = \varrho_a \circ \theta_{s(a)}$. An **isomorphism** of quiver representations is a morphism $\theta: V \to W$ where $\theta_i$ is a $\Bbbk$-vector space isomorphism for all $i \in Q_0$. We define $V \oplus W := ((V_i\oplus W_i)_{i \in Q_0}, (\varphi_a \oplus \varrho_a)_{a \in Q_1})$ to be the **direct sum** of $V$ and $W$. We say that a nonzero representation $V$ is **indecomposable** if it is not isomorphic to a direct sum of two nonzero representations. Note that the representations of a quiver $Q$ along with morphisms between them form an abelian category, denoted by $\text{rep}_{\Bbbk}(Q)$, with the indecomposable representations forming a full subcategory, denoted by $\text{ind}(\text{rep}_\Bbbk(Q))$.
We remark that representations of $Q$ can equivalently be regarded as modules over the **path algebra** $\Bbbk Q$. As such, one can define $\operatorname{\mathrm{Ext}}_{\Bbbk Q}^s(V,W)$ for $s \ge 1$ and $\operatorname{\mathrm{Hom}}_{\Bbbk Q}(V,W)$ for any representations $V$ and $W$, and $\operatorname{\mathrm{Hom}}_{\Bbbk Q}(V,W)$ is isomorphic to the vector space of all morphisms $\theta:V\to W$. We refer the reader to [@ass06] for more details on representations of quivers.
An **exceptional sequence** $\xi = (V_1,\ldots, V_k)$ ($k \le n:= \#Q_0$) is a sequence of **exceptional representations**[^2] $V_j$ of $Q$ (i.e. $V_j$ is indecomposable and $\operatorname{\mathrm{Ext}}_{\Bbbk Q}^s(V_j,V_j) = 0$ for all $s\ge 1$) satisfying $\operatorname{\mathrm{Hom}}_{\Bbbk Q}(V_j,V_i) = 0$ and $\operatorname{\mathrm{Ext}}_{\Bbbk Q}^s(V_j,V_i) = 0$ if $i < j$ for all $s \ge 1$. We use the term **exceptional pair** to mean an exceptional sequence consisting of exactly two exceptional representations. We define an **exceptional collection** $\overline{\xi} = \{V_1,\ldots, V_k\}$ to be a *set* of exceptional representations $V_j$ of $Q$ that can be ordered in such a way that they define an exceptional sequence. When $k = n$, we say $\xi$ (resp. $\overline{\xi}$) is a **complete exceptional sequence** (CES) (resp. **complete exceptional collection** (CEC)).
The following result of Speyer and Thomas gives a beautiful connection between $\operatorname{\textbf{c}}$-matrices of an acyclic quiver $Q$ and CESs. It serves as motivation for our work. Before stating it we remark that for any $R \in ET(\widehat{Q})$ and any $i \in [n]$ where $Q$ is an acyclic quiver, the $\operatorname{\textbf{c}}$-vector $\overrightarrow{c_i} = \overrightarrow{c_i}(R) = \pm \underline{\dim}(V_i)$ for some exceptional representation of $Q$ (see [@c12] or [@st13]).
Let $\overrightarrow{c}$ be a **c**-vector of an acyclic quiver $Q$. Define $$\begin{array}{rcl}
|\overrightarrow{c}| & := & \left\{\begin{array}{rcl} \overrightarrow{c} & : & \text{if $\overrightarrow{c}$ is positive}\\ -\overrightarrow{c} & : & \text{if $\overrightarrow{c}$ is negative.} \end{array}\right.
\end{array}$$
Let $C \in \textbf{c}$-mat$(Q)$, let $\{\overrightarrow{c_i}\}_{i \in [n]}$ denote the **c**-vectors of $C$, and let $|\overrightarrow{c_i}| = \underline{\dim}(V_i)$ for some exceptional representation of $Q$. There exists a permutation $\sigma \in \mathfrak{S}_n$ such that $(V_{\sigma(1)},...,V_{\sigma(n)})$ is a CES with the property that if there exist positive **c**-vectors in $C$, then there exists $k \in [n]$ such that $\overrightarrow{c_{\sigma(i)}}$ is positive if and only if $i \in [k,n]$, and $\operatorname{\mathrm{Hom}}_{\Bbbk Q}(V_j,V_{j^\prime})=0$ for any $\overrightarrow{c_j}, \overrightarrow{c_{j^\prime}}$ that have the same sign. Conversely, any set of $n$ vectors $\overrightarrow{c}_1,\ldots, \overrightarrow{c}_n$ having these properties defines a $\operatorname{\textbf{c}}$-matrix whose row vectors are $\{\overrightarrow{c_i}\}_{i\in[n]}$.
Quivers of type $\mathbb{A}_n$
------------------------------
For the purposes of this paper, we will only be concerned with quivers of type $\operatorname{\mathbb{A}}_n$. We say a quiver $Q$ is of **type** $\mathbb{A}_n$ if the underlying graph of $Q$ is a Dynkin diagram of type $\mathbb{A}_n$. By convention, two vertices $i$ and $j$ with $i < j$ in a type $\mathbb{A}_n$ quiver $Q$ are connected by an arrow if and only if $j = i+1$ and $i \in [n-1]$.
It will be convenient to denote a given type $\mathbb{A}_n$ quiver $Q$ using the notation $Q_\epsilon$, which we now define. Let $\epsilon = (\epsilon_0,\epsilon_1,\ldots, \epsilon_{n}) \in \{+,-\}^{n+1}$ and for $i \in [n-1]$ define $a_{i}^{\epsilon_i} \in Q_1$ by $$\begin{array}{cccccccccccc}
a_{i}^{\epsilon_i} & := & \left\{\begin{array}{rcl} i\leftarrow i+1 &: & \epsilon_i = -\\ i\rightarrow i+1 & : & \epsilon_i = +. \end{array}\right.
\end{array}$$ Then $Q_\epsilon := ((Q_\epsilon)_0 := [n], (Q_\epsilon)_1 := \{a_i^{\epsilon_i}\}_{i \in [n-1]}) = Q.$ One observes that the values of $\epsilon_0$ and $\epsilon_{n}$ do not affect $Q_\epsilon.$
Let $n = 5$ and $\epsilon = (-,+, -, +,-,+)$ so that $Q_{\epsilon} = 1 \stackrel{a_1^+}{\longrightarrow} 2 \stackrel{a_2^-}{\longleftarrow} 3 \stackrel{a_3^+}{\longrightarrow} 4 \stackrel{a_4^-}{\longleftarrow} 5.$ Below we show its framed quiver $\widehat{Q}_\epsilon$. $$\widehat{Q}_\epsilon = \begin{xy} 0;<1pt,0pt>:<0pt,-1pt>::
(0,10) *+{1} ="0",
(30,10) *+{2} ="1",
(60,10) *+{3} ="2",
(90,10) *+{4} ="3",
(120,10) *+{5} ="4",
(0,-20) *+{\textcolor{blue}{\text{$6$}}} ="5",
(30,-20) *+{\textcolor{blue}{\text{$7$}}} ="6",
(60, -20) *+{\textcolor{blue}{\text{$8$}}} = "9",
(90,-20) *+{\textcolor{blue}{\text{$9$}}} ="7",
(120,-20) *+{\textcolor{blue}{\text{$10$}}} ="8",
"0", {\ar"1"},
"0", {\ar"5"},
"2", {\ar"1"},
"1", {\ar"6"},
"2", {\ar"3"},
"4", {\ar"3"},
"3", {\ar"7"},
"4", {\ar"8"},
"2", {\ar"9"},
\end{xy}$$
Let $Q_\epsilon$ be given where $\epsilon = (\epsilon_0,\epsilon_1,\ldots, \epsilon_{n}) \in \{+,-\}^{n+1}.$ Let $i, j \in [0,n] := \{0,1,\ldots, n\}$ where $i < j$ and let $X^\epsilon_{i,j} = ((V_\ell)_{\ell \in (Q_\epsilon)_0}, (\varphi^{i,j}_a)_{a \in (Q_\epsilon)_1}) \in \text{rep}_{\Bbbk}(Q_\epsilon)$ be the indecomposable representation defined by
$$\begin{array}{cccccccccccc}
V_\ell & := & \left\{\begin{array}{rcl} \Bbbk &: & i+1 \le \ell \le j\\ 0 & : & \text{otherwise} \end{array}\right. & & & & \varphi^{i,j}_{a} & := & \left\{\begin{array}{rcl} 1 & : & a = a^{\epsilon_k}_k \text{ where } i+1 \le k \le j-1\\ 0 & : & \text{otherwise.} \end{array}\right.
\end{array}$$
The objects of $\text{ind}(\text{rep}_{\Bbbk}(Q_\epsilon))$ are those of the form $X^\epsilon_{i,j}$ where $0 \le i < j \le n$, up to isomorphism.
\[rem:hom0\] If $X^\epsilon_{i,j}$ and $X^\epsilon_{k,\ell}$ are distinct indecomposables of $\text{rep}_\Bbbk(Q_\epsilon)$, then $\text{Hom}_{\Bbbk Q_\epsilon}(X^\epsilon_{i,j}, X^\epsilon_{k,\ell}) = 0$ or $\text{Hom}_{\Bbbk Q_\epsilon}(X^\epsilon_{k,\ell}, X^\epsilon_{i,j}) = 0.$ This follows from the well-known fact that the Auslander–Reiten quiver of $\Bbbk Q_\epsilon$ is acyclic.
Strand diagrams
===============
In this section, we define three different types of combinatorial objects: strand diagrams, labeled strand diagrams, and oriented strand diagrams. We will use these objects to classify exceptional collections, exceptional sequences, and **c**-matrices of a given type $\mathbb{A}_n$ quiver $Q_\epsilon$, so we fix such a quiver $Q_\epsilon$.
Exceptional sequences and strand diagrams {#sec:strands}
-----------------------------------------
Let $\mathcal{S}_{n,\epsilon} \subset \mathbb{R}^2$ be a collection of $n+1$ points arranged in a horizontal line. We identify these points with $\epsilon_0, \epsilon_1, \ldots, \epsilon_{n}$ where $\epsilon_j$ appears to the right of $\epsilon_i$ for any $i,j \in [0,n] := \{0,1,2, \ldots, n\}$ where $i < j$. Using this identification, we can write $\epsilon_i = (x_i,y_i)\in \mathbb{R}^2$.
[Let $i,j \in [0,n]$ where $i\neq j$. A **strand** $c(i,j)$ on $\mathcal{S}_{n,\epsilon}$ is an isotopy class of simple curves in $\mathbb{R}^2$ where any $\gamma \in c(i,j)$ satisfies:]{}
- the endpoints of [$\gamma$]{} are $\epsilon_i$ and $\epsilon_j$,
- as a subset of $\mathbb{R}^2$, ${\gamma} \subset \{(x,y) \in \mathbb{R}^2: x_i \le x \le x_j \}\backslash\{\epsilon_{i+1}, \epsilon_{i+2}, \ldots, \epsilon_{j-1}\}$,
- if $k \in [0,n]$ satisfies $i \le k \le j$ and $\epsilon_k = +$ (resp. $\epsilon_k = -$), then [$\gamma$]{} is locally below (resp. locally above) $\epsilon_k$. By locally below (resp. locally above) $\epsilon_k$, we mean that for a given parameterization of $\gamma = (\gamma^{(1)}, \gamma^{(2)}) :[0,1]\to \mathbb{R}^2$ there exists $\delta > 0$ such that $\gamma$ satisfies $\gamma^{(2)}(t) \le y_k$ (resp. $\gamma^{(2)}(t) \ge y_k$) for all $t \in [0,1]$ where $\gamma(t) \in \{p \in \mathbb{R}^2: \ \text{dist}(p, \epsilon_{k}) < \delta\}$.
There is a natural bijection $\Phi_{\epsilon}$ from the objects of $\text{ind}(\text{rep}_\Bbbk(Q_\epsilon))$ to the set of strands on $\mathcal{S}_{n,\epsilon}$ given by $\Phi_{\epsilon}(X^\epsilon_{i,j}) := c(i,j).$
It is clear that any strand $c(i,j)$ can be represented by a **monotone curve** $\gamma \in c(i, j).$ That is, there exists a curve $\gamma \in c(i,j)$ with a parameterization $\gamma = (\gamma^{(1)}, \gamma^{(2)}): [0,1] \to \mathbb{R}^2$ such that if $t, s \in [0,1]$ and $t < s$, then $\gamma^{(1)}(t) < \gamma^{(1)}(s)$.
We say that two strands $c(i_1, j_1)$ and $c(i_2,j_2)$ **intersect nontrivially** if any two curves $\gamma_\ell \in c(i_\ell, j_\ell)$ with $\ell \in \{1,2\}$ have at least one transversal crossing. Otherwise, we say that $c(i_1, j_1)$ and $c(i_2,j_2)$ **do not intersect nontrivially**. For example, $c(1,3),c(2,4)$ intersect nontrivially if and only if $\epsilon_2=\epsilon_3$. Additionally, we say that $c(i_2,j_2)$ is **clockwise** from $c(i_1,j_1)$ (or, equivalently, $c(i_1,j_1)$ is **counterclockwise** from $c(i_2,j_2)$) if and only if some ${\gamma_1} \in c(i_1,j_1)$ and ${\gamma_2} \in c(i_2,j_2)$ share an endpoint $\epsilon_k$ and locally appear in one of the following six configurations up to isotopy. It follows from Lemma \[nocommonendpt\] below that if a strand is clockwise or counterclockwise from another strand, then the two do not intersect nontrivially. $$\includegraphics[scale=1.5]{clockwise2.pdf} \ \ \ \ \ \ \ \ \ \includegraphics[scale=1.5]{clockwise5.pdf} \ \ \ \ \ \ \ \ \ \includegraphics[scale=1.5]{clockwise4.pdf} \ \ \ \ \ \ \ \ \ \includegraphics[scale=1.5]{clockwise1.pdf} \ \ \ \ \ \ \ \ \ \includegraphics[scale=1.5]{clockwise6.pdf} \ \ \ \ \ \ \ \ \ \includegraphics[scale=1.5]{clockwise3.pdf}.$$
A given collection of strands $d = \{c(i_\ell, j_\ell)\}_{\ell \in [k]}$ with $k \le n$, naturally defines a graph with vertex set $\{\epsilon_0, \ldots, \epsilon_n\}$ and edge set $\{\{\epsilon_s, \epsilon_t\}: \ c(s, t) \in d\}$. We refer to this graph as the **graph determined by** $d$.
A **strand diagram** $d = \{c(i_\ell, j_\ell)\}_{\ell \in [k]}$ ($k \le n$) on $\mathcal{S}_{n,\epsilon}$ is a collection of strands on $\mathcal{S}_{n,\epsilon}$ that satisfies the following conditions:
$\begin{array}{rll}
a) & \text{distinct strands do not intersect nontrivially, and} \\
b) & \text{the graph determined by $d$ is a \textbf{forest} (i.e. a disjoint union of trees)} \\
\end{array}$
Let $\mathcal{D}_{k,\epsilon}$ denote the set of strand diagrams on $\mathcal{S}_{n,\epsilon}$ with $k$ strands and let $\mathcal{D}_\epsilon$ denote the set of strand diagrams with any positive number of strands. Then $$\mathcal{D}_\epsilon = \bigsqcup_{k \in [n]} \mathcal{D}_{k,\epsilon}.$$
\[firstexample\] Let $n = 4$ and $\epsilon = (-,+, -, +,+)$ so that $Q_{\epsilon} = 1 \stackrel{a_1^+}{\longrightarrow} 2 \stackrel{a_2^-}{\longleftarrow} 3 \stackrel{a_3^+}{\longrightarrow} 4.$ Then we have that $d_1 = \{c(0,1), c(0,2), c(2,3), c(2,4)\} \in \mathcal{D}_{4,\epsilon}$ and $d_2 = \{c(0,4), c(1,3), c(2,4)\}\in \mathcal{D}_{3,\epsilon}$. We draw these strand diagrams below. $$\begin{array}{ccccccc}
\includegraphics[scale=1]{firstexampleAprime1.pdf} & & & & \includegraphics[scale=1]{firstexampleBprime.pdf}
\end{array}$$
The following technical lemma classifies when two distinct indecomposable representations of $Q_\epsilon$ define 0, 1, or 2 exceptional pairs. Its proof appears in Section \[firstproofs\].
\[maintechlemma\] Let $Q_\epsilon$ be given. Fix two distinct indecomposable representations $U, V \in \text{ind}(\text{rep}_\Bbbk(Q_\epsilon))$.
$\begin{array}{rll}
a) & \text{The strands $\Phi_{\epsilon}(U)$ and $\Phi_{\epsilon}(V)$ intersect nontrivially if and only if neither $(U,V)$ nor $(V,U)$ are}\\
& \text{exceptional pairs.}\\
b) & \text{The strand $\Phi_{\epsilon}(U)$ is clockwise from $\Phi_{\epsilon}(V)$ if and only if $(U,V)$ is an exceptional pair and $(V,U)$}\\
& \text{is not an exceptional pair.}\\
c) & \text{The strands $\Phi_{\epsilon}(U)$ and $\Phi_{\epsilon}(V)$ do not intersect at any of their endpoints and they do not intersect}\\
& \text{nontrivially if and only if $(U, V)$ and $(V, U)$ are both exceptional pairs.}
\end{array}$
Using Lemma \[maintechlemma\] we obtain our first main result. The following theorem says that the data of an exceptional collection is completely encoded in the strand diagram it defines.
\[ECbij\] Let $\overline{\mathcal{E}}_\epsilon := \{\text{exceptional collections of } Q_\epsilon\}$. There is a bijection $\overline{\mathcal{E}}_\epsilon \to \mathcal{D}_{\epsilon}$ defined by $$\overline{\xi}_\epsilon = \{X^\epsilon_{i_\ell,j_\ell}\}_{\ell \in [k]} \mapsto \{\Phi_{\epsilon}(X^\epsilon_{i_\ell,j_\ell})\}_{\ell \in [k]}.$$
Let $\overline{\xi}_\epsilon = \{X^\epsilon_{i_\ell,j_\ell}\}_{\ell \in [k]}$ be an exceptional collection of $Q_\epsilon$. Let $\xi_\epsilon$ be an exceptional sequence gotten from $\overline{\xi}_\epsilon$ by reordering its representations. Without loss of generality, assume $\xi_\epsilon = (X^\epsilon_{i_\ell,j_\ell})_{\ell \in [k]}$ is an exceptional sequence. Thus, $(X^\epsilon_{i_\ell,j_\ell},X^\epsilon_{i_p,j_p})$ is an exceptional pair for all $\ell < p$. Lemma \[maintechlemma\] a) implies that distinct strands of $\{\Phi_{\epsilon}(X^\epsilon_{i_\ell,j_\ell})\}_{\ell \in [k]}$ do not intersect nontrivially.
Now we will show that $\{\Phi_{\epsilon}(X^\epsilon_{i_\ell,j_\ell})\}_{\ell \in [k]}$ has no cycles. Suppose that $\Phi_{\epsilon}(X^\epsilon_{i_{\ell_1},j_{\ell_1}}),\ldots,\Phi_{\epsilon}(X^\epsilon_{i_{\ell_p},j_{\ell_p}})$ is a cycle of length $p \leq k$ in $\Phi_{\epsilon}(\xi_\epsilon)$. Then, there exist $\ell_a, \ell_b \in [k]$ with $\ell_b > \ell_a$ such that $\Phi_{\epsilon}(X^\epsilon_{i_{\ell_b},j_{\ell_b}})$ is clockwise from $\Phi_{\epsilon}(X^\epsilon_{i_{\ell_a},j_{\ell_a}})$. Thus, by Lemma \[maintechlemma\] b), $(X^\epsilon_{i_{\ell_a},j_{\ell_a}},X^\epsilon_{i_{\ell_b},j_{\ell_b}})$ is not an exceptional pair. This contradicts the fact that $(X^\epsilon_{i_{\ell_1},j_{\ell_1}},\ldots,X^\epsilon_{i_{\ell_p},j_{\ell_p}})$ is an exceptional sequence. Hence, the graph determined by $\{\Phi_{\epsilon}(X^\epsilon_{i_\ell,j_\ell})\}_{\ell \in [k]}$ is a tree. We have shown that $\Phi_{\epsilon}(\overline{\xi}_\epsilon) \in \mathcal{D}_{k,\epsilon}$.
Now let $d = \{c(i_\ell, j_\ell)\}_{\ell \in [k]} \in \mathcal{D}_{k,\epsilon}$. Since $c(i_\ell,j_\ell)$ and $c(i_m,j_m)$ do not intersect nontrivially, it follows that $(\Phi_{\epsilon}^{-1}(c(i_\ell,j_\ell)),\Phi_{\epsilon}^{-1}(c(i_m,j_m)))$ or $(\Phi_{\epsilon}^{-1}(c(i_m,j_m)),\Phi_{\epsilon}^{-1}(c(i_\ell,j_\ell)))$ is an exceptional pair for every $\ell \neq m$. Notice that there exists $c(i_{\ell_1},j_{\ell_1}) \in d$ such that $(\Phi_{\epsilon}^{-1}(c(i_{\ell_1},j_{\ell_1})),\Phi_{\epsilon}^{-1}(c(i_\ell,j_\ell)))$ is an exceptional pair for every $c(i_\ell,j_\ell) \in d \setminus \{c(i_{\ell_1},j_{\ell_1})\}$. This is true because if such $c(i_{\ell_1},j_{\ell_1})$ did not exist, then $d$ must have a cycle. Set $E_1 = \Phi_{\epsilon}^{-1}(c(i_{\ell_1},j_{\ell_1}))$. Now, choose $c(i_{\ell_p},j_{\ell_p})$ such that $(\Phi_{\epsilon}^{-1}(c(i_{\ell_p},j_{\ell_p})),\Phi_{\epsilon}^{-1}(c(i_\ell,j_\ell)))$ is an exceptional pair for every $c(i_\ell,j_\ell) \in d \setminus \{c(i_{\ell_1},j_{\ell_1}),\ldots,c(i_{\ell_p},j_{\ell_p})\}$ inductively and put $E_p = \Phi_{\epsilon}^{-1}(c(i_{\ell_p},j_{\ell_p}))$. By construction, $(E_1,\ldots,E_k)$ is a complete exceptional sequence, as desired.
Our next step is to add distinct integer labels to each strand in a given strand diagram $d$. When these labels have what we call a **good** labeling, these labels will describe exactly the order in which to put the representations corresponding to strands of $d$ so that the resulting sequence of representations is an exceptional sequence.
\[Def:labeled\_diag\] A **labeled diagram** $d(k) = \{(c(i_\ell, j_\ell), s_\ell)\}_{\ell \in [k]}$ on $\mathcal{S}_{n,\epsilon}$ is a set of pairs $(c(i_\ell, j_\ell), s_\ell)$ where $c(i_\ell, j_\ell)$ is a strand on $\mathcal{S}_{n,\epsilon}$ and $s_\ell \in [k]$ such that $d := \{c(i_\ell, j_\ell)\}_{\ell \in [k]}$ is a strand diagram on $\mathcal{S}_{n,\epsilon}$ and $s_\ell \neq s_{\ell^\prime}$ for any distinct $\ell, \ell^\prime \in [k]$. We refer to the pairs $(c(i_\ell, j_\ell), s_\ell)$ as **labeled strands** and to $d$ as the **underlying diagram** of $d(k)$. We define the **endpoints** of a labeled strand $(c(i_\ell, j_\ell), s_\ell)$ to be the endpoints of $c(i_\ell, j_\ell)$. Let $\epsilon_i \in \mathcal{S}_{n,\epsilon}$ and let $((c(i,j_1),s_1),\ldots, (c(i,j_r),s_r))$ be the sequence of all labeled strands of $d(k)$ that have $\epsilon_i$ as an endpoint, and assume that this sequence is ordered so that strand $c(i,j_k)$ is clockwise from $c(i,j_{k^\prime})$ if $k^\prime < k$. We say the strand labeling of $d(k)$ is **good** if for each point $\epsilon_i \in \mathcal{S}_{n,\epsilon}$ one has $s_1 < \cdots < s_r.$ Let $\mathcal{D}_{k,\epsilon}(k)$ denote the set of labeled strand diagrams on $\mathcal{S}_{n,\epsilon}$ with $k$ strands and with good strand labelings.
\[secondexample\] Let $n = 4$ and $\epsilon = (-,+, -, +,+)$ so that $Q_{\epsilon} = 1 \stackrel{a_1^+}{\longrightarrow} 2 \stackrel{a_2^-}{\longleftarrow} 3 \stackrel{a_3^+}{\longrightarrow} 4.$ Below we show the labeled diagrams $d_1(4) = \{(c(0,1),1), (c(0,2), 2), (c(2,3), 3), (c(2,4), 4)\}$ and $d_2(3) = \{(c(0,4), 1), (c(2,4), 2), (c(1,3), 3)\}.$
$$\begin{array}{ccccccc}
\includegraphics[scale=1]{secondexampleAprime.pdf} & & & & \includegraphics[scale=1]{secondexampleBprime.pdf}
\end{array}$$
We have that $d_1(4) \in \mathcal{D}_{4,\epsilon}(4)$, but $d_2(3) \not \in \mathcal{D}_{3,\epsilon}(3).$
\[ESbij\] Let $k \in [n]$ and let $\mathcal{E}_\epsilon(k) := \{\text{exceptional sequences of } Q_\epsilon\text{ of length }k\}.$ There is a bijection $\widetilde{\Phi}_\epsilon: \mathcal{E}_\epsilon(k) \to \mathcal{D}_{k,\epsilon}(k)$ defined by $$\xi_\epsilon = (X^\epsilon_{i_\ell,j_\ell})_{\ell \in [k]} \longmapsto \{(c(i_\ell,j_\ell), k+1-\ell)\}_{\ell \in [k]}.$$
Let $\xi_\epsilon \in \mathcal{E}_\epsilon(k)$. By Lemma \[maintechlemma\] a), $\widetilde{\Phi}_{\epsilon}(\xi_\epsilon)$ has no strands that intersect nontrivially. Let $(V_1,V_2)$ be an exceptional pair appearing in $\xi_\epsilon$ with $V_i$ corresponding to strand $c_i$ in $\widetilde{\Phi}_{\epsilon}(\xi_\epsilon)$ for $i=1,2$, where $c_1$ and $c_2$ intersect only at one of their endpoints. Note that by the definition of $\widetilde{\Phi}_{\epsilon}$, the strand label of $c_1$ is larger than that of $c_2$. From Lemma \[maintechlemma\] b), strand $c_1$ is clockwise from $c_2$ in $\widetilde{\Phi}_{\epsilon}(\xi_\epsilon)$. Thus the strand labeling of $\widetilde{\Phi}_{\epsilon}(\xi_\epsilon)$ is good, so $\widetilde{\Phi}_{\epsilon}(\xi_\epsilon) \in \operatorname{\mathcal{D}}_{k,\epsilon}(k)$ for any $\xi_\epsilon \in \mathcal{E}_\epsilon(k)$.
Let $\widetilde{\Psi}_{\epsilon}: \operatorname{\mathcal{D}}_{k,\epsilon}(k) \rightarrow \mathcal{E}_\epsilon(k)$ be defined by $\{(c(i_\ell,j_\ell),\ell)\}_{\ell \in[k]} \mapsto (X_{i_k,j_k}^{\epsilon},X_{i_{k-1},j_{k-1}}^\epsilon,\ldots,X_{i_1,j_1}^\epsilon)$. We will show that $\widetilde{\Psi}_\epsilon(d(k)) \in \mathcal{E}_\epsilon(k)$ for any $d(k) \in \mathcal{D}_{k,\epsilon}(k)$ and that $\widetilde{\Psi}_\epsilon = \widetilde{\Phi}_{\epsilon}^{-1}.$ Let $$\widetilde{\Psi}_\epsilon(\{(c(i_\ell, j_\ell), \ell)\}_{\ell \in [k]}) = (X_{i_k,j_k}^\epsilon, X_{i_{k-1}, j_{k-1}}^\epsilon,\ldots, X_{i_1, j_1}^\epsilon).$$ Consider the pair $(X_{i_s,j_s}^\epsilon, X_{i_{s^\prime}, j_{s^\prime}}^\epsilon)$ with $s > s^\prime.$ We will show that $(X_{i_s,j_s}^\epsilon, X_{i_{s^\prime}, j_{s^\prime}}^\epsilon)$ is an exceptional pair and thus conclude that $\widetilde{\Psi}_\epsilon(\{(c(i_\ell, j_\ell), \ell)\}_{\ell \in [k]}) \in \mathcal{E}_\epsilon(k)$ for any $d(k) \in \mathcal{D}_{k,\epsilon}(k)$. Clearly, $c(i_s,j_s)$ and $c(i_{s^\prime}, j_{s^\prime})$ do not intersect nontrivially. If $c(i_s, j_s)$ and $c(i_{s^\prime}, j_{s^\prime})$ do not intersect at one of their endpoints, then by Lemma \[maintechlemma\] c) $(X_{i_s,j_s}^\epsilon, X_{i_{s^\prime}, j_{s^\prime}}^\epsilon)$ is exceptional. Now suppose $c(i_s, j_s)$ and $c(i_{s^\prime}, j_{s^\prime})$ intersect at one of their endpoints. Because the strand-labeling of $\{(c(i_\ell, j_\ell), \ell)\}_{\ell \in [k]}$ is good, $c(i_s, j_s)$ is clockwise from $c(i_{s^\prime}, j_{s^\prime})$. By Lemma \[maintechlemma\] b), we have that $(X_{i_s,j_s}^\epsilon, X_{i_{s^\prime}, j_{s^\prime}}^\epsilon)$ is exceptional.
To see that $\widetilde{\Psi}_\epsilon = \widetilde{\Phi}_\epsilon^{-1}$, observe that $$\begin{array}{rcl}
\widetilde{\Phi}_\epsilon \left(\widetilde{\Psi}_\epsilon(\{(c(i_\ell, j_\ell), \ell)\}_{\ell \in [k]})\right) & = & \widetilde{\Phi}_\epsilon \left((X_{i_k,j_k}^\epsilon, X_{i_{k-1}, j_{k-1}}^\epsilon,\ldots, X_{i_1,j_1}^\epsilon)\right)\\
& = & \{(c(i_\ell, j_\ell), k+1 - (k+1 - \ell))\}_{\ell \in [k]} \\
& = & \{(c(i_\ell, j_\ell), \ell)\}_{\ell \in [k]}.
\end{array}$$ Thus $\widetilde{\Phi}_\epsilon \circ \widetilde{\Psi}_\epsilon = 1_{\mathcal{D}_{n,\epsilon}(k)}.$ Similarly, one shows that $\widetilde{\Psi}_\epsilon \circ \widetilde{\Phi}_\epsilon = 1_{\mathcal{E}_\epsilon(k)}.$ Thus $\widetilde{\Phi}_\epsilon$ is a bijection.
The last combinatorial objects we discuss in this section are called **oriented diagrams**. These are strand diagrams whose strands have a direction.
Let $\overrightarrow{c}(i_\ell, j_\ell)$ denote the data of the strand $c(i_\ell,j_\ell)$ on $\mathcal{S}_{n,\epsilon}$ and an orientation of each curve in $c(i_\ell, j_\ell)$ from $\epsilon_{i_\ell}$ to $\epsilon_{j_\ell}$. We refer to $\overrightarrow{c}(i_\ell, j_\ell)$ as an **oriented strand** on $\mathcal{S}_{n,\epsilon}$ and we define the **endpoints** of $\overrightarrow{c}(i_\ell, j_\ell)$ to be the endpoints of $c(i_\ell, j_\ell)$. An **oriented diagram** $\overrightarrow{d}=\{\overrightarrow{c}(i_\ell, j_\ell)\}_{\ell \in [k]}$ on $\mathcal{S}_{n,\epsilon}$ is a collection of oriented strands on $\mathcal{S}_{n,\epsilon}$ where $d = \{c(i_\ell, j_\ell)\}_{\ell \in [k]}$ is a strand diagram on $\mathcal{S}_{n,\epsilon}.$ We refer to $d$ as the **underlying diagram** of $\overrightarrow{d}$.
When it is clear from the context what the values of $n$ and $\epsilon$ are, we will often refer to a strand diagram on $\mathcal{S}_{n,\epsilon}$ simply as a **diagram**. Similarly, we will often refer to labeled diagrams (resp. oriented diagrams) on $\mathcal{S}_{n,\epsilon}$ as **labeled diagrams** (resp. **oriented diagrams**). Additionally, if we have two diagrams $d_1$ and $d_2$ (both on $\mathcal{S}_{n, \epsilon}$) where $d_1 \subset d_2$, we say that $d_1$ is a **subdiagram** of $d_2$. One analogously defines **labeled subdiagrams** (resp. **oriented subdiagrams**) of a labeled diagram (resp. oriented diagram).
We now define a special subset of the oriented diagrams on $\mathcal{S}_{n,\epsilon}$. As we will see, each element in this subset of oriented diagrams, denoted $\overrightarrow{\mathcal{D}}_{n,\epsilon}$, will correspond to a unique $\textbf{c}$-matrix $C \in \textbf{c}\text{-mat}(Q_\epsilon)$ and vice versa. Thus we obtain a diagrammatic classification of **c**-matrices (see Theorem \[c-matClassif\]).
\[def:cmatdiag\] Let $\overrightarrow{\mathcal{D}}_{n,\epsilon}$ denote the set of oriented diagrams $\overrightarrow{d} = \{\overrightarrow{c}(i_\ell,j_\ell)\}_{\ell \in [n]}$ on $\mathcal{S}_{n,\epsilon}$ with the property that for each $k \in [0,n]$ there exist integers $i_1, i_2, k, j \in [0,n]$ where $i_1< k< i_2$ and $j \in [0,n]\backslash\{i_1, k, i_2\}$ such that the oriented subdiagram $\overrightarrow{d}_1$ of $\overrightarrow{d}$ consisting of the oriented strands of $\overrightarrow{d}$ with $\epsilon_k$ as an endpoint is an oriented subdiagram of one of the following two oriented diagrams on $\mathcal{S}_{n,\epsilon}$:
- $\overrightarrow{d}_+ = \{\overrightarrow{c}(k,i_1),\overrightarrow{c}(k,i_2), \overrightarrow{c}(j,k)\}$ where $\epsilon_k = +$ (shown in Figure \[local1\] (left)) or
- $\overrightarrow{d}_- = \{\overrightarrow{c}(i_1,k),\overrightarrow{c}(i_2,k), \overrightarrow{c}(k,j)\}$ where $\epsilon_k = -$ (shown in Figure \[local1\] (right)).
![[]{data-label="local1"}](oriented2.pdf "fig:") ![[]{data-label="local1"}](oriented1.pdf "fig:")
\[cmatdiag\] Let $\{\overrightarrow{c_i}\}_{i \in [k]}$ be a collection of **c**-vectors of $Q_\epsilon$ where $k \le n$. Let $\overrightarrow{c_i} = \pm\underline{\dim}(X^\epsilon_{i_1,i_2})$ where the sign is determined by $\overrightarrow{c_i}$. There is an injective map $$\begin{array}{rcl}
\{ \text{\textbf{noncrossing collections} $\{\overrightarrow{c_i}\}_{i\in[k]}$ of $Q_\epsilon$}\} & \longrightarrow & \{\text{oriented diagrams } \overrightarrow{d} = \{\overrightarrow{c}(i_\ell, j_\ell)\}_{\ell \in [k]}\}\end{array}$$ defined by $$\begin{array}{rcl}
\overrightarrow{c_i} &\longmapsto & \left\{\begin{array}{r c l c c c c c} \overrightarrow{c}(i_1,i_2) & : & \text{\overrightarrow{c_i} is positive}\\ \overrightarrow{c}(i_2,i_1) & : & \text{\overrightarrow{c_i} is negative,}
\end{array}\right.
\end{array}$$
where $\{\overrightarrow{c_i}\}_{i \in [k]}$ is a **noncrossing collection** of **c**-vectors if $\Phi_{\epsilon}(X^\epsilon_{i_1,i_2})$ and $\Phi_{\epsilon}(X^\epsilon_{i^\prime_1,i^\prime_2})$ do not intersect nontrivially for any $i,i^\prime \in [k]$. In particular, each **c**-matrix $C_\epsilon \in \textbf{c}\text{-mat}(Q_\epsilon)$ determines a unique oriented diagram denoted $\overrightarrow{d}_{C_\epsilon}$ with $n$ oriented strands.
\[thirdexample\] Let $n = 4$ and $\epsilon = (+,+, -, +,-)$ so that $Q_{\epsilon}= 1 \stackrel{a_1^+}{\longrightarrow} 2 \stackrel{a_2^-}{\longleftarrow} 3 \stackrel{a_3^+}{\longrightarrow} 4.$ After performing the mutation sequence $\mu_3 \circ \mu_2$ to the corresponding framed quiver, we have the $\operatorname{\textbf{c}}$-matrix with its oriented diagram. $$\left[
\begin{array}{c c c c}
1 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & -1 & -1 & 0 \\
0 & 0 & 0 & 1
\end{array}\right] \ \ \ \ \ \ \ \ \ \
\raisebox{-.1in}{\includegraphics[scale=1.5]{thirdexample.pdf}}$$
The following theorem shows oriented diagrams belonging to $\overrightarrow{\mathcal{D}}_{n,\epsilon}$ are in bijection with **c**-matrices of $Q_\epsilon.$ We delay its proof until Section \[sec:mct\] because it makes use of the concept of a mixed cobinary tree.
\[c-matClassif\] The map $\textbf{c}\text{-mat}(Q_\epsilon)\to \overrightarrow{\mathcal{D}}_{n,\epsilon}$ induced by the map defined in Lemma \[cmatdiag\] is a bijection.
Proof of Lemma \[maintechlemma\] {#firstproofs}
--------------------------------
The proof of Lemma \[maintechlemma\] requires some notions from representation theory of finite dimensional algebras, which we now briefly review. For a more comprehensive treatment of the following notions, we refer the reader to [@ass06].
Given a quiver $Q$ with $\#Q_0=n$, the **Euler characteristic** (of $Q$) is the $\operatorname{\mathbb{Z}}$-bilinear (nonsymmetric) form $\operatorname{\mathbb{Z}}^n \times \operatorname{\mathbb{Z}}^n \rightarrow \operatorname{\mathbb{Z}}$ defined by $$\langle \underline{\dim}(V),\underline{\dim}(W)\rangle = \sum_{i\ge 0}(-1)^i\dim \operatorname{\mathrm{Ext}}_{\Bbbk Q}^i(V,W)$$ for every $V,W \in \operatorname{\mathrm{rep}}_\Bbbk(Q)$.
For hereditary algebras $A$ (e.g. path algebras of acyclic quivers), $\operatorname{\mathrm{Ext}}_{A}^i(V,W)=0$ for $i \geq 2$ and the formula reduces to
$$\langle \underline{\dim}(V),\underline{\dim}(W)\rangle = \dim \operatorname{\mathrm{Hom}}_{\Bbbk Q}(V,W) - \dim \operatorname{\mathrm{Ext}}_{\Bbbk Q}^1(V,W)$$
The following result gives a simple formula for the Euler characteristic. We note that this formula is independent of the orientation of the arrows of $Q$. The following lemma can be found in [@ass06].
Given an acyclic quiver $Q$ with $\#Q_0=n$ and integral vectors $x=(x_1,x_2,\ldots,x_n),y=(y_1,y_2,\ldots,y_n) \in \operatorname{\mathbb{Z}}^n$, the Euler characteristic of $Q$ has the form $$\langle x,y\rangle = \sum_{i \in Q_0}x_iy_i - \sum_{\alpha \in Q_1}x_{s(\alpha)}y_{t(\alpha)}$$
Next, we give a slight simplification of the previous formula. Recall that the support of $V \in \operatorname{\mathrm{rep}}_\Bbbk(Q)$ is the set $\text{supp}(V) := \{i\in Q_0 : V_i \neq 0\}$. Thus for quivers of the form $Q_\epsilon$, any indecomposable representation $X^\epsilon_{i,j} \in \text{ind}(\text{rep}_\Bbbk(Q_\epsilon))$ has $\text{supp}(X^\epsilon_{i,j}) = [i+1,j].$
\[eulerform\] Let $X^\epsilon_{k,\ell}, X^\epsilon_{i,j} \in \text{ind}(\text{rep}_\Bbbk(Q_\epsilon))$ and $A := \{a \in (Q_\epsilon)_1: s(a), t(a) \in \text{supp}(X^\epsilon_{k,\ell})\cap \text{supp}(X^\epsilon_{i,j})\}$. Then $$\langle \underline{\dim}(X^\epsilon_{k,\ell}), \underline{\dim}(X^\epsilon_{i,j})\rangle = \chi_{\text{supp}(X^\epsilon_{k,\ell}) \cap \text{supp}(X^\epsilon_{i,j})} - \#\left(\{a \in (Q_\epsilon)_1: s(a) \in \text{supp}(X^\epsilon_{k,\ell}), \ t(a) \in \text{supp}(X^\epsilon_{i,j})\}\backslash A\right)$$ where $\chi_{\text{supp}(X^\epsilon_{k,\ell}) \cap \text{supp}(X^\epsilon_{i,j})} = 1$ if $\text{supp}(X^\epsilon_{k,\ell})\cap \text{supp}(X^\epsilon_{i,j}) \neq \emptyset$ and 0 otherwise.
We have that $$\begin{array}{rcl}
\langle \underline{\dim}(X^\epsilon_{k,\ell}), \underline{\dim}(X^\epsilon_{i,j})\rangle & = & \displaystyle \sum_{m \in ({Q_\epsilon})_0} \underline{\dim}(X^\epsilon_{k,\ell})_m\underline{\dim}(X^\epsilon_{i,j})_m
- \sum_{a \in ({Q_\epsilon})_1} \underline{\dim}(X^\epsilon_{k,\ell})_{s(a)}\underline{\dim}(X^\epsilon_{i,j})_{t(a)} \\
& = & \#\left(\text{supp}(X^\epsilon_{k,\ell})\cap \text{supp}(X^\epsilon_{i,j})\right)\\
& &- \#\{\alpha\in (Q_\epsilon)_1: s(a) \in \text{supp}(X^\epsilon_{k,\ell}), \ t(a) \in \text{supp}(X^\epsilon_{i,j})\}\\
& = & \#\left(\text{supp}(X^\epsilon_{k,\ell})\cap \text{supp}(X^\epsilon_{i,j})\right) - \#A\\
& &- \#\left(\{a \in (Q_\epsilon)_1: s(a) \in \text{supp}(X^\epsilon_{k,\ell}), \ t(a) \in \text{supp}(X^\epsilon_{i,j})\}\backslash A\right).
\end{array}$$ Observe that if $\text{supp}(X^\epsilon_{k,\ell})\cap \text{supp}(X^\epsilon_{i,j}) \neq \emptyset$, then $\#A = \#(\text{supp}(X^\epsilon_{k,\ell})\cap \text{supp}(X^\epsilon_{i,j})) - 1.$ Otherwise $\#A = 0.$ Thus $$\langle \underline{\dim}(X^\epsilon_{k,\ell}), \underline{\dim}(X^\epsilon_{i,j})\rangle = \chi_{\text{supp}(X^\epsilon_{k,\ell}) \cap \text{supp}(X^\epsilon_{i,j})} - \#\left(\{a \in (Q_\epsilon)_1: s(a) \in \text{supp}(X^\epsilon_{k,\ell}), \ t(a) \in \text{supp}(X^\epsilon_{i,j})\}\backslash A\right) .$$
We now present the lemmas that we will use in the proof of Lemma \[maintechlemma\]. The proofs of the next four lemmas use very similar techniques so we only prove Lemma \[interlaced\]. The following four lemmas characterize when $\operatorname{\mathrm{Hom}}_{\Bbbk Q_\epsilon}(-,-)$ and $\operatorname{\mathrm{Ext}}^1_{\Bbbk Q_\epsilon}(-,-)$ vanish for a given type $\mathbb{A}_n$ quiver $Q_\epsilon.$ The conditions describing when $\operatorname{\mathrm{Hom}}_{\Bbbk Q_\epsilon}(-,-)$ and $\operatorname{\mathrm{Ext}}^1_{\Bbbk Q_\epsilon}(-,-)$ vanish are given in terms of inequalities satisfied by the indices that describe a pair of indecomposable representations of $Q_\epsilon$ and the entries of $\epsilon.$
\[interlaced\] Let $X^\epsilon_{k,\ell}, X^\epsilon_{i,j} \in \text{ind}(\text{rep}_\Bbbk(Q_\epsilon))$. Assume $0 \le i < k < j < \ell \le n$.
$\begin{array}{rll}
i) & \text{$\operatorname{\mathrm{Hom}}_{\Bbbk Q_\epsilon}(X^\epsilon_{i,j}, X^\epsilon_{k,\ell}) \neq 0$ if and only if $\epsilon_k = -$ and $\epsilon_j = -$.}\\
ii) & \text{$\operatorname{\mathrm{Hom}}_{\Bbbk Q_\epsilon}(X^\epsilon_{k,\ell}, X^\epsilon_{i,j}) \neq 0$ if and only if $\epsilon_k = +$ and $\epsilon_j = +$.}\\
iii) & \text{$\operatorname{\mathrm{Ext}}^1_{\Bbbk Q_\epsilon}(X^\epsilon_{i,j}, X^\epsilon_{k,\ell}) \neq 0$ if and only if $\epsilon_k = +$ and $\epsilon_j = +$.}\\
iv) & \text{$\operatorname{\mathrm{Ext}}^1_{\Bbbk Q_\epsilon}(X^\epsilon_{k,\ell}, X^\epsilon_{i,j}) \neq 0$ if and only if $\epsilon_k = -$ and $\epsilon_j = -$.}
\end{array}$
We only prove $i)$ and $iv)$ as the proof of $ii)$ is very similar to that of $i)$, and the proof of $iii)$ is very similar to that of $iv)$. To prove $i)$, first assume there is a nonzero morphism $\theta: X^\epsilon_{i,j} \to X^\epsilon_{k,\ell}.$ Clearly, $\theta_s = 0$ if $s \not \in [k+1,j]$. If $\theta_s \neq 0$ for some $s \in [n]$, then $\theta_s = \lambda\cdot \text{id}_{\Bbbk}$ for some nonzero $\lambda \in \Bbbk$ (i.e. $\theta_s$ is a nonzero scalar transformation). As $\theta$ is a morphism of representations, for any $a \in (Q_\epsilon)_1$ the equality $\theta_{t(a)}\varphi^{i,j}_a = \varphi^{k,\ell}_a \theta_{s(a)}$ holds. Thus for any $a \in \{a_{k+1}^{\epsilon_{k+1}}, \ldots, a_{j - 1}^{\epsilon_{j - 1}}\}$, we have $\theta_{t(a)} = \theta_{s(a)}.$ As $\theta$ is nonzero, this implies that $\theta_s = \lambda\cdot \text{id}_{\Bbbk}$ for any $s \in [k+1,j]$. If $a = a_{k}^{\epsilon_k}$, then we have $$\begin{array}{rcl}
\theta_{t(a)}\varphi^{i,j}_a & = & \varphi^{k,\ell}_a\theta_{s(a)} \\
\theta_{t(a)} & = & 0.
\end{array}$$ Thus $\epsilon_k = -$. Similarly, $\epsilon_j = -.$
Conversely, it is easy to see that if $\epsilon_k = \epsilon_j = -,$ then $\theta: X^\epsilon_{i,j} \to X^\epsilon_{k,\ell}$ defined by $\theta_s = 0$ if $s \not \in [k+1,j]$ and $\theta_s = 1$ otherwise is a nonzero morphism.
Next, we prove $iv)$. Observe that by Lemma \[eulerform\] we have $$\begin{array}{rcl}
\dim\text{Ext}^1_{\Bbbk Q_\epsilon}(X^\epsilon_{k,\ell}, X^{\epsilon}_{i,j}) & = & \dim\text{Hom}_{\Bbbk Q_\epsilon}(X^\epsilon_{k,\ell},X^\epsilon_{i,j}) -\langle \underline{\dim}(X^\epsilon_{k,\ell}), \underline{\dim}(X^\epsilon_{i,j})\rangle \\
& = & \dim\text{Hom}_{\Bbbk Q_\epsilon}(X^\epsilon_{k,\ell},X^\epsilon_{i,j}) - 1\\
& & + \#\left(\{b \in (Q_\epsilon)_1: s(b) \in \text{supp}(X^\epsilon_{k,\ell}), \ t(b) \in \text{supp}(X^\epsilon_{i,j})\}\backslash A\right).
\end{array}$$ Note that $\#\left(\{b \in (Q_\epsilon)_1: s(b) \in \text{supp}(X^\epsilon_{k,\ell}), \ t(b) \in \text{supp}(X^\epsilon_{i,j})\}\backslash A\right) \le 2$ with equality if and only if $\epsilon_k = \epsilon_j = -.$ Suppose $\epsilon_k = \epsilon_j = -$. Then by $i)$, we have that $\text{Hom}_{\Bbbk Q_\epsilon}(X_{i,j}^\epsilon, X_{k,\ell}^\epsilon) \neq 0$ so $\text{Hom}_{\Bbbk Q_\epsilon}(X_{k,\ell}^\epsilon, X_{i,j}^\epsilon) = 0$. This means $$\begin{array}{rcl}
\dim\text{Ext}^1_{\Bbbk Q_\epsilon}(X_{k,\ell}^\epsilon,X_{i,j}^\epsilon) & = & \#\left(\{b \in (Q_\epsilon)_1: s(b) \in \text{supp}(X^\epsilon_{k,\ell}), \ t(b) \in \text{supp}(X^\epsilon_{i,j})\}\backslash A\right) - 1\\
& = & 1.
\end{array}$$
Conversely, suppose $\text{Ext}^1_{\Bbbk Q_\epsilon}(X_{k,\ell}^\epsilon, X_{i,j}^\epsilon) \neq 0$. Thus, one checks that there is a nonsplit extension $$0 \longrightarrow X^\epsilon_{i,j} \stackrel{f}{\longrightarrow} X^\epsilon_{i,\ell} \oplus X^\epsilon_{k,j} \stackrel{g}{\longrightarrow} X^\epsilon_{k,\ell} \longrightarrow 0.$$ This implies that $\text{Hom}_{\Bbbk Q_\epsilon}(X_{k,\ell}^\epsilon, X_{i,j}^\epsilon) = 0$, since the composition $h: X^\epsilon_{i,j} \stackrel{f_1}{\to} X^\epsilon_{i,\ell} \stackrel{g_1}{\to} X_{k,\ell}^\epsilon$ is nonzero. Using again that $\text{dim}\text{Ext}^1_{\Bbbk Q_\epsilon}(X_{k,\ell}^\epsilon, X_{i,j}^\epsilon) \neq 0$, the formula above for this dimension tells us that $\epsilon_k = \epsilon_j = -$.
\[nested\] Let $X^\epsilon_{k,\ell}, X^\epsilon_{i,j} \in \text{ind}(\text{rep}_\Bbbk(Q_\epsilon))$. Assume $0 \le i < k < \ell < j \le n$.
$\begin{array}{rll}
i) & \text{$\operatorname{\mathrm{Hom}}_{\Bbbk Q_\epsilon}(X^\epsilon_{i,j}, X^\epsilon_{k,\ell}) \neq 0$ if and only if $\epsilon_k = -$ and $\epsilon_\ell = +$.}\\
ii) & \text{$\operatorname{\mathrm{Hom}}_{\Bbbk Q_\epsilon}(X^\epsilon_{k,\ell}, X^\epsilon_{i,j}) \neq 0$ if and only if $\epsilon_k = +$ and $\epsilon_\ell = -$.}\\
iii) & \text{$\operatorname{\mathrm{Ext}}^1_{\Bbbk Q_\epsilon}(X^\epsilon_{i,j}, X^\epsilon_{k,\ell}) \neq 0$ if and only if $\epsilon_k = +$ and $\epsilon_\ell = -$.}\\
iv) & \text{$\operatorname{\mathrm{Ext}}^1_{\Bbbk Q_\epsilon}(X^\epsilon_{k,\ell}, X^\epsilon_{i,j}) \neq 0$ if and only if $\epsilon_k = -$ and $\epsilon_\ell = +$.}
\end{array}$
\[shareendpt\] Assume $0 \le i < k < j \le n$. Then
$\begin{array}{rll}
i) & \text{$\operatorname{\mathrm{Hom}}_{\Bbbk Q_\epsilon}(X^\epsilon_{i,k}, X^\epsilon_{k,j}) = 0$ and $\operatorname{\mathrm{Hom}}_{\Bbbk Q_\epsilon}(X^\epsilon_{k,j}, X^\epsilon_{i,k}) = 0$.}\\
ii) & \text{$\operatorname{\mathrm{Ext}}^1_{\Bbbk Q_\epsilon}(X^\epsilon_{i,k}, X^\epsilon_{k,j}) \neq 0$ if and only if $\epsilon_k = +$.}\\
iii) & \text{$\operatorname{\mathrm{Ext}}^1_{\Bbbk Q_\epsilon}(X^\epsilon_{k,j}, X^\epsilon_{i,k}) \neq 0$ if and only if $\epsilon_k = -$.}\\
iv) & \text{$\operatorname{\mathrm{Hom}}_{\Bbbk Q_\epsilon}(X^\epsilon_{i,k}, X^\epsilon_{i,j}) \neq 0$ if and only if $\epsilon_k = -$.}\\
v) & \text{$\operatorname{\mathrm{Hom}}_{\Bbbk Q_\epsilon}(X^\epsilon_{i,j}, X^\epsilon_{i,k}) \neq 0$ if and only if $\epsilon_k = +$.}\\
vi) & \operatorname{\mathrm{Ext}}^1_{\Bbbk Q_\epsilon}(X^\epsilon_{i,k}, X^\epsilon_{i,j}) = 0 \text{ and } \operatorname{\mathrm{Ext}}^1_{\Bbbk Q_\epsilon}(X^\epsilon_{i,j}, X^\epsilon_{i,k}) = 0.\\
vii) & \text{$\operatorname{\mathrm{Hom}}_{\Bbbk Q_\epsilon}(X^\epsilon_{k,j}, X^\epsilon_{i,j}) \neq 0$ if and only if $\epsilon_k = +$.}\\
viii) & \text{$\operatorname{\mathrm{Hom}}_{\Bbbk Q_\epsilon}(X^\epsilon_{i,j}, X^\epsilon_{k,j}) \neq 0$ if and only if $\epsilon_k = -$.}\\
ix) & \operatorname{\mathrm{Ext}}^1_{\Bbbk Q_\epsilon}(X^\epsilon_{k,j}, X^\epsilon_{i,j}) = 0 \text{ and } \operatorname{\mathrm{Ext}}^1_{\Bbbk Q_\epsilon}(X^\epsilon_{i,j},X^\epsilon_{k,j}) = 0.
\end{array}$
\[separate\] Let $X^\epsilon_{k,\ell}, X^\epsilon_{i,j} \in \text{ind}(\text{rep}_\Bbbk(Q_\epsilon))$. Assume $0 \le i < j < k < \ell \le n$. Then
$\begin{array}{rl}
i) & \text{$\operatorname{\mathrm{Hom}}_{\Bbbk Q_\epsilon}(X^\epsilon_{i,j},X^\epsilon_{k,\ell}) = 0,$ $\operatorname{\mathrm{Hom}}_{\Bbbk Q_\epsilon}(X^\epsilon_{k,\ell},X^\epsilon_{i,j}) = 0,$}\\
ii) & \text{$\operatorname{\mathrm{Ext}}^1_{\Bbbk Q_\epsilon}(X^\epsilon_{i,j},X^\epsilon_{k,\ell}) = 0,$ $\operatorname{\mathrm{Ext}}^1_{\Bbbk Q_\epsilon}(X^\epsilon_{k,\ell},X^\epsilon_{i,j}) = 0$.}
\end{array}$
Next, we present three geometric facts about pairs of distinct strands. These geometric facts will be crucial in our proof of Lemma \[maintechlemma\].
\[uniqcross\] If two distinct strands $c(i_1,j_1)$ and $c(i_2,j_2)$ on $\mathcal{S}_{n,\epsilon}$ intersect nontrivially, then $c(i_1,j_1)$ and $c(i_2,j_2)$ can be represented by a pair of [monotone]{} curves that have a unique transversal crossing.
[Suppose $c(i_1,j_1)$ and $c(i_2,j_2)$ intersect nontrivially. Without loss of generality, we assume $i_1 \le i_2$. Let $\gamma_k \in c(i_k,j_k)$ with $k \in [2]$ be monotone curves. There are two cases: ]{}
$\begin{array}{rcl}
a) & i_1 \le i_2 < j_1 \le j_2\\
b) & i_1 \le i_2 < j_2 \le j_1.
\end{array}$
Suppose that case $a)$ holds. Let $(x',y') \in \{(x,y) \in \mathbb{R}^2: x_{i_2} \le x \le x_{j_1}\}$ denote a point where $\gamma_1$ crosses $\gamma_2$ transversally. If $\epsilon_{i_2} = -$ (resp. $\epsilon_{i_2} = +$), isotope $\gamma_1$ relative to $\epsilon_{i_1}$ and $(x', y')$ in such a way that the monotonocity of $\gamma_1$ is preserved and so that $\gamma_1$ lies strictly above (resp. strictly below) $\gamma_2$ on $\{(x,y) \in \mathbb{R}^2: x_{i_2} \le x < x'\}.$
Next, if $\epsilon_{j_1} = -$ (resp. $\epsilon_{j_1} = +$), isotope $\gamma_2$ relative to $(x',y')$ and $\epsilon_{j_2}$ in such a way that the monotonicity of $\gamma_2$ is preserved and so that $\gamma_2$ lies strictly above (resp. strictly below) $\gamma_1$ on $\{(x,y) \in \mathbb{R}^2: x' < x \le x_{j_1}\}.$ This process produces two monotone curves $\gamma_1 \in c(i_1,j_1)$ and $\gamma_2 \in c(i_2,j_2)$ that have a unique transversal crossing. The proof in case $b)$ is very similar.
\[nocommonendpt\] Let $c(i_1,j_1)$ and $c(i_2,j_2)$ be distinct strands on $\mathcal{S}_{n,\epsilon}$ that intersect nontrivially. Then $c(i_1,j_1)$ and $c(i_2,j_2)$ do not share an endpoint.
Suppose $c(i_1,j_1)$ and $c(i_2,j_2)$ share an endpoint. Thus, there exist curves $\gamma_k \in c(i_k,j_k)$ with $k \in \{1,2\}$ such that $\gamma_1$ and $\gamma_2$ are isotopic relative to their endpoints to curves with no transversal crossing.
\[mononoint\] If $c(i_1, j_1)$ and $c(i_2, j_2)$ are two distinct strands on $\mathcal{S}_{n,\epsilon}$ that do not intersect nontrivially, then $c(i_1,j_1)$ and $c(i_2,j_2)$ can be represented by a pair of monotone curves $\gamma_\ell \in c(i_\ell, j_\ell)$ where $\ell \in [2]$ that are nonintersecting, except possibly at their endpoints.
We now arrive at the proof of Lemma \[maintechlemma\]. The proof is a case by case analysis where the cases are given in terms of the entries of $\epsilon$ and the inequalities satisfied by the indices that describe a pair of indecomposable representations of $Q_\epsilon$.
Let $X^\epsilon_{i,j}:= U$ and $X^\epsilon_{k,\ell}:= V$. Assume that the strands $\Phi_{\epsilon}(X^\epsilon_{i,j})$ and $\Phi_{\epsilon}(X^\epsilon_{k,\ell})$ intersect nontrivially. By Lemma \[nocommonendpt\], we can assume without loss of generality that either $0\le i < k < j < \ell \le n$ or $0 \le i < k < \ell < j \le n$. By Lemma \[uniqcross\], we can represent $\Phi_{\epsilon}(X^\epsilon_{i,j})$ and $\Phi_{\epsilon}(X^\epsilon_{k,\ell})$ by monotone curves $\gamma_{i,j}$ and $\gamma_{k,\ell}$ that have a unique transversal crossing. Furthermore, we can assume that this unique crossing occurs between $\epsilon_k$ and $\epsilon_{k+1}$. There are four possible cases:
$\begin{array}{rll}
i) & \epsilon_k = \epsilon_{k+1} = -,\\
ii) & \epsilon_k = - \text{ and } \epsilon_{k+1} = +, \\
iii) & \epsilon_k = \epsilon_{k+1} = +,\\
iv) & \epsilon_k = + \text{ and } \epsilon_{k+1} = -.
\end{array}$
We illustrate these cases up to isotopy in Figure \[crossings\]. We see that in cases $i)$ and $ii)$ (resp. $iii)$ and $iv)$) $\gamma_{k,\ell}$ lies
above (resp. below) $\gamma_{i,j}$ inside of $\{(x,y) \in \mathbb{R}^2: x_{k+1} \le x \le x_{\text{min}\{\ell, j\}}\}$.
Suppose $\gamma_{k,\ell}$ lies above $\gamma_{i,j}$ inside $\{(x,y) \in \mathbb{R}^2: x_{k+1} \le x \le x_{\text{min}\{\ell, j\}}\}$. Then
$$\begin{array}{rcl}
\epsilon_{\text{min}\{\ell,j\}} & = & \left\{\begin{array}{rll} + & : & \text{min}\{\ell, j\} = \ell\\
- & : & \text{min}\{\ell,j\} = j\end{array}\right.
\end{array}$$ otherwise $\gamma_{k,\ell}$ and $\gamma_{i,j}$ would have a nonunique transversal crossing. If $\text{min}\{\ell,j\} = \ell,$ we have $0\le i < k < \ell < j \le n$, $\epsilon_k = -$, and $\epsilon_{\ell} = +.$ Now by Lemma \[nested\], we have that $\operatorname{\mathrm{Hom}}_{\Bbbk Q_\epsilon}(X^\epsilon_{i,j},X^\epsilon_{k,\ell}) \neq 0$ and $\operatorname{\mathrm{Ext}}^1_{\Bbbk Q_\epsilon}(X^\epsilon_{k,\ell},X^\epsilon_{i,j}) \neq 0.$ If $\min\{\ell, j\} = j$, then $0 \le i < k < j < \ell \le n$, $\epsilon_k = -,$ and $\epsilon_j = -.$ Thus, by Lemma \[interlaced\], we have that $\operatorname{\mathrm{Hom}}_{\Bbbk Q_\epsilon}(X^\epsilon_{i,j},X^\epsilon_{k,\ell}) \neq 0$ and $\operatorname{\mathrm{Ext}}^1_{\Bbbk Q_\epsilon}(X^\epsilon_{k,\ell},X^\epsilon_{i,j}) \neq 0.$
Similarly, if $\gamma_{i,j}$ lies above $\gamma_{k,\ell}$ inside $\{(x,y) \in \mathbb{R}^2: x_{k+1} \le x \le x_{\text{min}\{\ell, j\}}\}$, it follows that
$$\begin{array}{rcl}
\epsilon_{\text{min}\{\ell,j\}} & = & \left\{\begin{array}{rll} - & : & \text{min}\{\ell, j\} = \ell\\
+ & : & \text{min}\{\ell,j\} = j.\end{array}\right.
\end{array}$$
If $\min\{\ell,j\} = \ell,$ then Lemma \[nested\] implies that $\operatorname{\mathrm{Hom}}_{\Bbbk Q_\epsilon}(X^\epsilon_{k,\ell},X^\epsilon_{i,j}) \neq 0$ and $\operatorname{\mathrm{Ext}}^1_{\Bbbk Q_\epsilon}(X^\epsilon_{i,j},X^\epsilon_{k,\ell}) \neq 0.$ If $\min\{\ell,j\} = j,$ then Lemma \[interlaced\] implies that $\operatorname{\mathrm{Hom}}_{\Bbbk Q_\epsilon}(X^\epsilon_{k,\ell},X^\epsilon_{i,j}) \neq 0$ and $\operatorname{\mathrm{Ext}}^1_{\Bbbk Q_\epsilon}(X^\epsilon_{i,j},X^\epsilon_{k,\ell}) \neq 0.$ Thus we conclude that neither $(X^\epsilon_{i,j},X^\epsilon_{k,\ell})$ nor $(X^\epsilon_{k,\ell},X^\epsilon_{i,j})$ are exceptional pairs.
Conversely, assume that neither $(U,V)$ nor $(V,U)$ are exceptional pairs where $X^\epsilon_{i,j} := U$ and $X^\epsilon_{k,\ell} := V$. Then at least one of the following is true:
$\begin{array}{rll}
a) & \text{Hom}_{\Bbbk Q_\epsilon}(X^\epsilon_{i,j}, X^\epsilon_{k,\ell}) \neq 0 \text{ and } \text{Hom}_{\Bbbk Q_\epsilon}(X^\epsilon_{k,\ell}, X^\epsilon_{i,j}) \neq 0,\\
b) & \text{Hom}_{\Bbbk Q_\epsilon}(X^\epsilon_{i,j}, X^\epsilon_{k,\ell}) \neq 0 \text{ and } \text{Ext}^1_{\Bbbk Q_\epsilon}(X^\epsilon_{k,\ell},X^\epsilon_{i,j})\neq 0, \\
c) & \text{Ext}^1_{\Bbbk Q_\epsilon}(X^\epsilon_{i,j}, X^\epsilon_{k,\ell}) \neq 0 \text{ and } \text{Hom}_{\Bbbk Q_\epsilon}(X^\epsilon_{k,\ell}, X^\epsilon_{i,j}) \neq 0,\\
d) & \text{Ext}^1_{\Bbbk Q_\epsilon}(X^\epsilon_{i,j}, X^\epsilon_{k,\ell}) \neq 0 \text{ and } \text{Ext}^1_{\Bbbk Q_\epsilon}(X^\epsilon_{k,\ell}, X^\epsilon_{i,j})\neq 0.
\end{array}$
As $X^\epsilon_{i,j}$ and $X^\epsilon_{k,\ell}$ are indecomposable and distinct, we have that $\text{Hom}_{\Bbbk Q_\epsilon}(X^\epsilon_{i,j}, X^\epsilon_{k,\ell}) = 0$ or $\text{Hom}_{\Bbbk Q_\epsilon}(X^\epsilon_{k,\ell},X^\epsilon_{i,j}) = 0$ by Remark \[rem:hom0\]. Without loss of generality, assume that $\text{Hom}_{\Bbbk Q_\epsilon}(X^\epsilon_{k,\ell},X^\epsilon_{i,j}) = 0.$ Thus $b)$ or $d)$ hold so $\text{Ext}^1_{\Bbbk Q_\epsilon}(X^\epsilon_{k,\ell}, X^\epsilon_{i,j})\neq 0$. Then Lemma \[shareendpt\] and Lemma \[separate\] imply that $0 \le i < k < j < \ell \le n$ or $0 \le i < k < \ell < j \le n.$
If $0 \le i < k < j < \ell < n$, we have $\epsilon_k = \epsilon_j = -$ by Lemma \[interlaced\] as $\operatorname{\mathrm{Hom}}_{\Bbbk Q_\epsilon}(X^\epsilon_{i,j},X_{k,\ell}^\epsilon) \neq 0$ and $\operatorname{\mathrm{Ext}}^1_{\Bbbk Q_\epsilon}(X^\epsilon_{k,\ell},X^\epsilon_{i,j}) \neq 0.$ Let $\gamma_{i,j} \in \Phi_{\epsilon}(X^\epsilon_{i,j})$ and $\gamma_{k,\ell} \in \Phi_{\epsilon}(X^\epsilon_{k,\ell})$. We can assume that there exists $\delta(k) > 0$ such that $\gamma_{i,j}$ and $\gamma_{k,\ell}$ have no transversal crossing inside $\{(x,y) \in \mathbb{R}^2: x_k \le x \le x_k + \delta(k)\}.$ This implies that $\gamma_{i,j}$ lies above $\gamma_{k,\ell}$ inside $\{(x,y) \in \mathbb{R}^2: x_k\le x\le x_k + \delta(k)\}$. Similarly, we can assume there exists $\delta(j) > 0 $ such that $\gamma_{i,j}$ and $\gamma_{k,\ell}$ have no transversal crossing inside $\{(x,y) \in \mathbb{R}^2: x_j - \delta(j) \le x \le x_j\}$. Thus $\gamma_{i,j}$ lies below $\gamma_{k,\ell}$ inside $\{(x,y) \in \mathbb{R}^2: x_j - \delta(j) \le x \le x_j\}.$ This means $\gamma_{i,j}$ and $\gamma_{k,\ell}$ must have at least one transversal crossing. Thus $\Phi_{\epsilon}(X^\epsilon_{i,j})$ and $\Phi_{\epsilon}(X^\epsilon_{k,\ell})$ intersect nontrivially. An analogous argument shows that if $0 \le i < k < \ell < j \le n$, then $\Phi_{\epsilon}(X^\epsilon_{i,j})$ and $\Phi_{\epsilon}(X^\epsilon_{k,\ell})$ intersect nontrivially.
Assume that $\Phi_{\epsilon}(U)$ is clockwise from $\Phi_{\epsilon}(V)$. Then we have that one of the following holds:
$\begin{array}{rll}
a) & X^\epsilon_{k,j}= U \text{ and } X^\epsilon_{i,k}= V \text{ for some } 0 \le i < k < j \le n,\\
b) & X^\epsilon_{i,k}= U \text{ and } X^\epsilon_{k,j}= V \text{ for some } 0 \le i < k < j \le n,\\
c) & X^\epsilon_{i,j}= U \text{ and } X^\epsilon_{i,k} = V \text{ for some } 0 \le i < j \le n \text{ and } 0 \le i < k \le n,\\
d) & X^\epsilon_{i,j}= U \text{ and } X^\epsilon_{k,j}= V \text{ for some } 0 \le i < j \le n \text{ and } 0 \le k < j \le n.
\end{array}$
In Case $a)$, we have that $\epsilon_k = -$ since $\Phi_{\epsilon}(X^\epsilon_{k,j})$ is clockwise from $\Phi_{\epsilon}(X^\epsilon_{i,k})$. By Lemma \[shareendpt\] $i)$ and $ii)$, we have that $\operatorname{\mathrm{Hom}}_{\Bbbk Q_\epsilon}(X^\epsilon_{i,k},X^\epsilon_{k,j}) = 0$ and $\operatorname{\mathrm{Ext}}^1_{\Bbbk Q_\epsilon}(X^\epsilon_{i,k},X^\epsilon_{k,j}) = 0.$ Thus $(X^\epsilon_{k,j}, X^\epsilon_{i,k})$ is an exceptional pair. By Lemma \[shareendpt\] $iii)$, we have that $\operatorname{\mathrm{Ext}}^1_{\Bbbk Q_\epsilon}(X^\epsilon_{k,j}, X^\epsilon_{i,k}) \neq 0$. Thus $(X^\epsilon_{i,k},X^\epsilon_{k,j})$ is not an exceptional pair.
In Case $b)$, we have that $\epsilon_k = +$ since $\Phi_{\epsilon}(X^\epsilon_{i,k})$ is clockwise from $\Phi_{\epsilon}(X^\epsilon_{k,j})$. By Lemma \[shareendpt\] $i)$ and $iii)$, we have that $\operatorname{\mathrm{Hom}}_{\Bbbk Q_\epsilon}(X^\epsilon_{k,j},X^\epsilon_{i,k}) = 0$ and $\operatorname{\mathrm{Ext}}^1_{\Bbbk Q_\epsilon}(X^\epsilon_{k,j},X^\epsilon_{i,k}) = 0.$ Thus $(X^\epsilon_{i,k}, X^\epsilon_{k,j})$ is an exceptional pair. By Lemma \[shareendpt\] $ii)$, we have that $\operatorname{\mathrm{Ext}}^1_{\Bbbk Q_\epsilon}(X^\epsilon_{i,k}, X^\epsilon_{k,j}) \neq 0$. Thus $(X^\epsilon_{k,j},X^\epsilon_{i,k})$ is not an exceptional pair.
In Case $c)$, if $j < k$, it follows that $\epsilon_j = -$. Indeed, since $\Phi_{\epsilon}(X^\epsilon_{i,j})$ and $\Phi_{\epsilon}(X^\epsilon_{i,k})$ share an endpoint, the two do not intersect nontrivially by Lemma \[nocommonendpt\]. As $\Phi_{\epsilon}(X^\epsilon_{i,j})$ is clockwise from $\Phi_{\epsilon}(X^\epsilon_{i,k})$, Remark \[mononoint\] asserts that we can choose monotone curves $\gamma_{i,k} \in \Phi_{\epsilon}(X^\epsilon_{i,k})$ and $\gamma_{i,j} \in \Phi_{\epsilon}(X^\epsilon_{i,j})$ such that $\gamma_{i,k}$ lies strictly above $\gamma_{i,j}$ on $\{(x,y) \in \mathbb{R}^2: x_i < x \le x_j\}$. Thus $\epsilon_j = -.$ By Lemma \[shareendpt\] $v)$ and $vi)$, we have that $\operatorname{\mathrm{Hom}}_{\Bbbk Q_\epsilon}(X^\epsilon_{i,k}, X^\epsilon_{i,j}) = 0$ and $\operatorname{\mathrm{Ext}}^1_{\Bbbk Q_\epsilon}(X^\epsilon_{i,k}, X^\epsilon_{i,j}) = 0$ so that $(X^\epsilon_{i,j},X^\epsilon_{i,k})$ is an exceptional pair. By Lemma \[shareendpt\] $iv)$, we have that $\operatorname{\mathrm{Hom}}_{\Bbbk Q_\epsilon}(X^\epsilon_{i,j},X^\epsilon_{i,k}) \neq 0$. Thus $(X^\epsilon_{i,k}, X^\epsilon_{i,j})$ is not an exceptional pair.
Similarly, one shows that if $k < j$, then $\epsilon_k = +$. By Lemma \[shareendpt\] $iv)$ and $vi)$, we have that $\operatorname{\mathrm{Hom}}_{\Bbbk Q_\epsilon}(X^\epsilon_{i,k}, X^\epsilon_{i,j}) = 0$ and $\operatorname{\mathrm{Ext}}^1_{\Bbbk Q_\epsilon}(X^\epsilon_{i,k}, X^\epsilon_{i,j}) = 0$ so that $(X^\epsilon_{i,j},X^\epsilon_{i,k})$ is an exceptional pair. By Lemma \[shareendpt\] $v)$, we have that $\operatorname{\mathrm{Hom}}_{\Bbbk Q_\epsilon}(X^\epsilon_{i,j},X^\epsilon_{i,k}) \neq 0$. Thus $(X^\epsilon_{i,k}, X^\epsilon_{i,j})$ is not an exceptional pair. The proof in Case $d)$ is completely analogous to the proof in Case $c)$ so we omit it.
Conversely, let $U = X^\epsilon_{i,j}$ and $V = X^\epsilon_{k,\ell}$ and assume that $(X^\epsilon_{i,j}, X^\epsilon_{k,\ell})$ is an exceptional pair and $(X^\epsilon_{k,\ell}, X^\epsilon_{i,j})$ is not an exceptional pair. This implies that at least one of the following holds:
$\begin{array}{rll}
1) & \text{Hom}_{\Bbbk Q_\epsilon}(X^\epsilon_{k,\ell}, X^\epsilon_{i,j}) = 0, \text{Ext}^1_{\Bbbk Q_\epsilon}(X^\epsilon_{k,\ell}, X^\epsilon_{i,j}) = 0, \text{ and } \operatorname{\mathrm{Hom}}_{\Bbbk Q_\epsilon}(X^\epsilon_{i,j},X^\epsilon_{k,\ell}) \neq 0,\\
2) & \text{Hom}_{\Bbbk Q_\epsilon}(X^\epsilon_{k,\ell}, X^\epsilon_{i,j}) = 0, \text{Ext}^1_{\Bbbk Q_\epsilon}(X^\epsilon_{k,\ell}, X^\epsilon_{i,j}) = 0, \text{ and } \operatorname{\mathrm{Ext}}^1_{\Bbbk Q_\epsilon}(X^\epsilon_{i,j},X^\epsilon_{k,\ell}) \neq 0.
\end{array}$
By Lemma \[separate\], we know that $[i,j]\cap[k,\ell] \neq \emptyset.$ This implies that either
$\begin{array}{rll}
i) & \Phi_{\epsilon}(X^\epsilon_{i,j}) \text{ and } \Phi_{\epsilon}(X^\epsilon_{k,\ell}) \text{ share an endpoint},\\
ii) & 0 \le i < k < j < \ell \le n,\\
iii) & 0 \le i < k < \ell < j \le n, \\
iv) & 0 \le k < i < \ell < j \le n,\\
v) & 0 \le k < i < j < \ell \le n.
\end{array}$
We will show that $\Phi_{\epsilon}(X^\epsilon_{i,j})$ and $\Phi_{\epsilon}(X^\epsilon_{k,\ell})$ share an endpoint.
Suppose $0 \le i < k < j < \ell \le n.$ Then since $\text{Hom}_{\Bbbk Q_\epsilon}(X^\epsilon_{k,\ell}, X^\epsilon_{i,j}) = 0, \text{Ext}^1_{\Bbbk Q_\epsilon}(X^\epsilon_{k,\ell}, X^\epsilon_{i,j}) = 0$, we have by Lemma \[interlaced\] $ii)$ and $iv)$ that either $\epsilon_k = -$ and $\epsilon_j = +$ or $\epsilon_{k} = +$ and $\epsilon_j = -$. However, as $\operatorname{\mathrm{Hom}}_{\Bbbk Q_\epsilon}(X^\epsilon_{i,j},X^\epsilon_{k,\ell}) \neq 0$ or $\operatorname{\mathrm{Ext}}^1_{\Bbbk Q_\epsilon}(X^\epsilon_{i,j},X^\epsilon_{k,\ell}) \neq 0$, Lemma \[interlaced\] $i)$ and $iii)$ assert that $\epsilon_k = \epsilon_j = -$ or $\epsilon_k = \epsilon_j = +.$ This is a contradiction. Thus, $i, j, k, \ell$ do not satisfy $0 \leq i < k < j < \ell \leq n$, and by a similar argument, they also do not satisfy $0 \le k < i < \ell < j \le n$.
Suppose $0 \le i < k < \ell < j \le n.$ Then since $\text{Hom}_{\Bbbk Q_\epsilon}(X^\epsilon_{k,\ell}, X^\epsilon_{i,j}) = 0, \text{Ext}^1_{\Bbbk Q_\epsilon}(X^\epsilon_{k,\ell}, X^\epsilon_{i,j}) = 0$, we have by Lemma \[nested\] $ii)$ and $iv)$ that either $\epsilon_k =\epsilon_\ell = +$ or $\epsilon_k = \epsilon_\ell = -$. However, as $\operatorname{\mathrm{Hom}}_{\Bbbk Q_\epsilon}(X^\epsilon_{i,j},X^\epsilon_{k,\ell}) \neq 0$ or $\operatorname{\mathrm{Ext}}^1_{\Bbbk Q_\epsilon}(X^\epsilon_{i,j},X^\epsilon_{k,\ell}) \neq 0$, Lemma \[nested\] $i)$ and $iii)$ we have that $\epsilon_k = -$ and $\epsilon_\ell = +$ or $\epsilon_k = +$ and $\epsilon_\ell = -.$ This is a contradiction. Thus, $i,j,k,\ell$ do not satisfy $0 \le i < k < \ell < j \le n,$ and by an analogous argument, they also do not satisfy $0 \le k < i < j < \ell \le n.$
We conclude that $\Phi_{\epsilon}(U)$ and $\Phi_{\epsilon}(V)$ share an endpoint. Thus we have that one of the following holds where we forget the previous roles played by $i,j,k$:
$\begin{array}{rll}
a) & X^\epsilon_{k,j}= U \text{ and } X^\epsilon_{i,k}= V \text{ for some } 0 \le i < k < j \le n,\\
b) & X^\epsilon_{i,k}= U \text{ and } X^\epsilon_{k,j}= V \text{ for some } 0 \le i < k < j \le n,\\
c) & X^\epsilon_{i,j}= U \text{ and } X^\epsilon_{i,k} = V \text{ for some } 0 \le i < j \le n \text{ and } 0 \le i < k \le n,\\
d) & X^\epsilon_{i,j}= U \text{ and } X^\epsilon_{k,j}= V \text{ for some } 0 \le i < j \le n \text{ and } 0 \le k < j \le n.
\end{array}$
Suppose Case $a)$ holds. Then since $(U,V)$ is an exceptional pair, we have $\operatorname{\mathrm{Ext}}^1_{\Bbbk Q_\epsilon}(X^\epsilon_{i,k}, X^\epsilon_{k,j}) = 0.$ By Lemma \[shareendpt\] $ii)$, we have that $\epsilon_k = -$. Thus $\Phi_{\epsilon}(U)$ is clockwise from $\Phi_{\epsilon}(V)$.
Suppose Case $b)$ holds. Then since $(U,V)$ is an exceptional pair, we have $\operatorname{\mathrm{Ext}}^1_{\Bbbk Q_\epsilon}(X^\epsilon_{k,j}, X^\epsilon_{i,k}) = 0.$ By Lemma \[shareendpt\] $iii)$, we have that $\epsilon_k = +$. Thus $\Phi_{\epsilon}(U)$ is clockwise from $\Phi_{\epsilon}(V)$.
Suppose Case $c)$ holds. Assume $k < j$. Then Lemma \[shareendpt\] $iv)$ and the fact that $\operatorname{\mathrm{Hom}}_{\Bbbk Q_\epsilon}(X^\epsilon_{i,k},X^\epsilon_{i,j}) = 0$ imply that $\epsilon_k = +.$ Thus we have that $\Phi_{\epsilon}(U) = \Phi_{\epsilon}(X^\epsilon_{i,j})$ is clockwise from $\Phi_{\epsilon}(V) = \Phi_{\epsilon}(X^\epsilon_{i,k})$. Now suppose $j < k$. Then Lemma \[shareendpt\] $v)$ and $\operatorname{\mathrm{Hom}}_{\Bbbk Q_\epsilon}(X^\epsilon_{i,k}, X^\epsilon_{i,j}) = 0$ imply that $\epsilon_j = -.$ Thus we have that $\Phi_{\epsilon}(U) = \Phi_{\epsilon}(X^\epsilon_{i,j})$ is clockwise from $\Phi_{\epsilon}(V) = \Phi_{\epsilon}(X^\epsilon_{i,k})$. The proof in Case $d)$ is very similar so we omit it.
Observe that two strands $c(i_1,j_1)$ and $c(i_2, j_2)$ share an endpoint if and only if one of the two strands is clockwise from the other. Thus Lemma \[maintechlemma\] $a)$ and $b)$ implies that $\Phi_{\epsilon}(U)$ and $\Phi_{\epsilon}(V)$ do not intersect at any of their endpoints and they do not intersect nontrivially if and only if both $(U,V)$ and $(V,U)$ are exceptional pairs.
Mixed cobinary trees {#sec:mct}
====================
We recall the definition of an $\epsilon$-mixed cobinary tree and construct a bijection between the set of (isomorphism classes of) such trees and the set of maximal oriented strand diagrams on $\mathcal{S}_{n,\epsilon}.$
\[def: MCT\] Given a sign function $\epsilon:[0,n]\to\{+,-\}$, an **$\epsilon$-mixed cobinary tree** (MCT) is a tree $T$ embedded in $\mathbb R^2$ with vertex set $\{(i,y_i): i\in[0,n]\}$ and edges straight line segments and satisfying the following conditions.
$\begin{array}{rl}
a) & \text{None of the edges is horizontal.}\\
b) & \text{If $\epsilon_i=+$ then $y_i\ge z$ for any $(i,z)\in T$. So, the tree goes under $(i,y_i)$.}\\
c) & \text{If $\epsilon_i=-$ then $y_i\le z$ for any $(i,z)\in T$. So, the tree goes over $(i,y_i)$.}\\
d) & \text{If $\epsilon_i=+$ then there is at most one edge descending from $(i,y_i)$ and}\\
&\text{at most two edges ascending from $(i,y_i)$ and not on the same side.}\\
e) & \text{If $\epsilon_i=-$ then there is at most one edge ascending from $(i,y_i)$ and}\\
&\text{at most two edges descending from $(i,y_i)$ and not on the same side.}\\
\end{array}$
Two MCTs $T,T'$ are **isomorphic** as MCTs if there is a graph isomorphism $T\cong T'$ which sends $(i,y_i)$ to $ (i,y_i')$ and so that corresponding edges have the same sign of their slopes.
Given a MCT $T$, there is a partial ordering on $[0,n]$ given by $i<_Tj$ if the unique path from $(i,y_i)$ to $(j,y_j)$ in $T$ is monotonically increasing. Isomorphic MCTs give the same partial ordering by definition. Conversely, the partial ordering $<_T$ determines $T$ uniquely up to isomorphism since $T$ is the Hasse diagram of the partial ordering $<_T$. We sometimes decorate MCTs with **leaves** at vertices so that the result is **trivalent**, i.e., with three edges incident to each vertex. See, e.g., Figure \[figMCTa\]. The ends of these leaves are not considered to be vertices. In that case, each vertex with $\epsilon=+$ forms a “Y” and this pattern is vertically inverted for $\epsilon=-$. The position of the leaves is uniquely determined.
\[fig:MCT example\]
(-1.8,1) node[ ]{}; (0,0)–(1,1)–(3,0)–(4,1)(2,-1)–(3,0); (4,1) circle\[radius=2pt\] node\[right\][$(4,1)$]{} (1,1) circle\[radius=2pt\]node\[above\][$(1,1)$]{} (0,0) circle\[radius=2pt\]node\[left\][$(0,0)$]{} (3,0)circle\[radius=2pt\](3,-.2)node\[right\][$(3,0)$]{} (2,-1) circle\[radius=2pt\]node\[left\][$(2,-1)$]{};
(-2.3,1) node[ ]{}; (0,0)–(1,1)–(2,2)–(3,3); in [0,2,3]{}(,)–+(.4,-.4); (3,3)–+(.4,.4) (0,0)–(-.4,-.4) (1,1)–+(-.4,.4); in [0,...,3]{}(,) ellipse\[x radius=2pt,y radius=3pt\];
In Figure \[figMCTa\], the four vertices have coordinates $(0,y_0),(1,y_1),(2,y_2),(3,y_3)$ where the $y_i$ can be any real numbers such that $y_0<y_1<y_2<y_3$. This inequality defines an open subset of $\mathbb R^4$ which is called the [region]{} of this tree $T$. More generally, for any MCT $T$, the **region** of $T$, denoted $\mathcal R_\epsilon(T)$, is the set of all points $y\in\mathbb R^{n+1}$ with the property that there exists a mixed cobinary tree $T'$ which is isomorphic to $T$ so that the vertex set of $T'$ is $\{(i,y_i): i\in[0,n]\}$.
Let $n \in \operatorname{\mathbb{Z}}_{\geq 0}$ and $\epsilon:[0,n]\to \{+,-\}$ be fixed. Then, for every MCT $T$, the region $\mathcal R_\epsilon(T)$ is convex and nonempty. Furthermore, every point $y=(y_0,\ldots,y_n)$ in $\mathbb R^{n+1}$ with distinct coordinates lies in $\mathcal R_\epsilon(T)$ for a unique $T$ (up to isomorphism). In particular these regions are disjoint and their union is dense in $\mathbb R^{n+1}$.
For a fixed $n$ and $\epsilon:[0,n]\to \{+,-\}$ we will construct a bijection between the set $\mathcal T_\epsilon$ of isomorphism classes of mixed cobinary trees with sign function $\epsilon$ and the set $\overrightarrow{\mathcal{D}}_{n,\epsilon}$ defined in Definition \[def:cmatdiag\].
\[lem: orientation of paths in OSDs\] Let $\overrightarrow{d} = \{\overrightarrow{c}(i_\ell, j_\ell)\}_{\ell \in [n]} \in \overrightarrow{\mathcal{D}}_{n,\epsilon}$ and let $\overrightarrow{\gamma}_\ell \in \overrightarrow{c}(i_\ell, j_\ell)$ with $\ell \in [n]$ be monotone paths from $i_\ell$ to $j_\ell$ that are pairwise non-intersecting except possibly at their endpoints. If $p, q$ are two points on the union of these paths where $q$ lies directly above $p$, then the paths $\overrightarrow{\gamma}_{i_1}, \ldots, \overrightarrow{\gamma}_{i_k}$ appearing in the unique curve $\overrightarrow{\gamma}$ connecting $p$ to $q$ are oriented in the same direction as $\overrightarrow{\gamma}$.
The proof will be by induction on the number $m$ of internal vertices of the path $\gamma$. If $m=1$ with internal vertex $\epsilon_i$ then the path $\gamma$ has only two edges of $\overrightarrow d$: one going from $p$ to $\epsilon_i$, say to the left, and the other going from $\epsilon_i$ back to $q$. Since $\overrightarrow{d} \in \overrightarrow{\mathcal{D}}_{n,\epsilon}$, the edge coming into $\epsilon_i$ from its right is below the edge going out from $\epsilon_i$ to $q$. Therefore the orientation of these two edges in $\overrightarrow{d}$ matches that of $\gamma$.
Now suppose that $m\ge2$ and the lemma holds for smaller $m$. There are two cases. Case 1: The path $\gamma$ lies entirely on one side of $p$ and $q$ (as in the case $m=1$). Case 2: $\gamma$ has internal vertices on both sides of $p,q$.
: Suppose by symmetry that $\gamma$ lies entirely on the left side of $p$ and $q$. Let $j$ be maximal so that $\epsilon_j$ is an internal vertex of $\gamma$. We claim that $\epsilon_j$ cannot be a local maximum of the ($x$-coordinate on the) curve $\gamma$ since, if it were, it would need to be either both below or both above the paths from $\epsilon_j$ to $p$ and to $q$. But, in that case, we can apply induction on $m$ to conclude that the edges in $\gamma$ adjacent to $\epsilon_j$ are oriented in the same direction contradicting the definition of $\overrightarrow{\mathcal{D}}_{n,\epsilon}$. Thus, $\gamma$ contains an edge connecting $\epsilon_j$ to either $p$ or $q$, say $p$. And the edge of $\gamma$ ending in $q$ contains a unique point $r$ which lies directly above $\epsilon_j$. This forces the sign to be $\epsilon_j=-$. By induction on $m$, we can conclude that each edge of $\overrightarrow d$ is oriented in the same direction as the path from $\epsilon_j$ to $r$. So, it must be oriented outward from $\epsilon_j$. Since $\epsilon_j=-$, any other edge at $\epsilon_j$ is oriented inward. So, the edge between $p$ and $\epsilon_j$ is oriented from $p$ to $\epsilon_j$ as required. The edge coming into $r$ from the left is oriented to the right (by induction). So, this same edge continues to be oriented to the right as it goes from $r$ to $q$. The other subcases (when $\epsilon_j$ is connected to $q$ instead of $p$ and when $\gamma$ lies to the right of $p$ and $q$) are analogous.
: Suppose that $\gamma$ is on both sides of $p$ and $q$. Then $\gamma$ passes through a third point, say $r$, on the same vertical line containing $p$ and $q$. Let $\gamma_0$ and $\gamma_1$ denote the parts of $\gamma$ going from $p$ to $r$ and from $r$ to $q$ respectively. Then $\gamma_0,\gamma_1$ each have at least one internal vertex. So, the lemma holds for each of them separately. There are three subcases: either (a) $r$ lies below $p$, (b) $r$ lies above $q$ or (c) $r$ lies between $p$ and $q$. In subcase (a), we have, by induction on $m$, that $\gamma_0,\gamma_1$ are both oriented away from $r$. So, $r=\epsilon_k=+$ which contradicts the assumption that $q$ lies above $r$. Similarly, subcase (b) is not possible. In subcase (c), we have by induction on $m$ that the orientations of the edges of $\overrightarrow d$ are compatible with the orientations of $\gamma_0$ and $\gamma_1$. So, the lemma holds in subcase (c), which is the only subcase of Case 2 which is possible. Therefore, the lemma holds in all cases.
\[thm: bijection between MCTs and OSDs\] For each $\overrightarrow{d} = \{\overrightarrow{c}(i_\ell,j_\ell)\}_{\ell \in [n]} \in \overrightarrow{\mathcal{D}}_{n,\epsilon}$, let $\mathcal R(\overrightarrow{d})$ denote the set of all $y\in \mathbb R^{n+1}$ so that $y_i<y_j$ for any $\overrightarrow c(i,j)$ in $\overrightarrow{d}$. Then $\mathcal R(\overrightarrow{d})=\mathcal R_\epsilon(T)$ for a uniquely determined mixed cobinary tree $T\in \mathcal T_\epsilon$. Furthermore, this gives a bijection $$\overrightarrow{\mathcal{D}}_{n,\epsilon}\cong \mathcal T_\epsilon.$$
We first verify the existence of a mixed cobinary tree $T$ for every choice of $y\in\mathcal R(\overrightarrow d)$. Since the strand diagram is a tree, the vector $y$ is uniquely determined by $y_0\in\mathbb R$ and $y_{j_\ell}-y_{i_\ell}>0$, $\ell\in [n]$, which are arbitrary. Given such a $y$, we claim that the $n$ line segments $L_\ell$ in $\mathbb R^2$ connecting the pairs of points $(i_\ell,y_{i_\ell}), (j_\ell,y_{j_\ell})$ meet only [at]{} their endpoints. If not then two of these line segments, say $L_s,L_t$, meet at some point $(a,b)\in \mathbb R^2$. This leads to a contradiction of Lemma \[lem: orientation of paths in OSDs\] as follows. Choose representatives $\overrightarrow{\gamma}_\ell \in \overrightarrow{c}(i_\ell, j_\ell)$ as in the lemma. Let $p\in \overrightarrow{\gamma}_s$ and $q\in \overrightarrow{\gamma}_t$ be the points on those curves with $x$-coordinate $a$. By symmetry assume $p$ is below $q$. Let $\overrightarrow{\gamma}_{w_1}, \ldots, \overrightarrow{\gamma}_{w_k}$ be the monotone curves in the unique path $\overrightarrow\gamma$ connecting $p$ and $q$ so that $w_1=s$ and $w_k=t$. By Lemma \[lem: orientation of paths in OSDs\] these curves are oriented in the same direction as $\overrightarrow\gamma$. By definition of the vector $y\in \mathcal R(\overrightarrow{d})$ we have $y_{i_\ell}<y_{j_\ell}$ for each $\ell=w_1,\ldots,w_k$. Then $b<y_{j_s}<y_{i_t}<b$ is a contradiction. So, $T$ is a linearly embedded tree. The lemma also implies that the tree $T$ lies above all negative vertices and below all positive vertices. The other parts of Definition \[def: MCT\] follow from the definition of an oriented strand diagram. Therefore $T\in \mathcal T_\epsilon$. Since this argument works for every $y\in\mathcal R(\overrightarrow d)$, we see that $\mathcal R(\overrightarrow d)=\mathcal R_\epsilon(T)$ as claimed.
A description of the inverse mapping $\mathcal T_\epsilon\to \overrightarrow{\mathcal{D}}_{n,\epsilon}$ is given as follows. Take any $\epsilon$-MCT $T$ and deform the tree by moving all vertices vertically to the subset $[0,n]\times 0$ on the $x$-axis and deforming the edges in such a way that they are always embedded in the plane with no vertical tangents and so that their interiors do not meet. The result is an oriented strand diagram $\overrightarrow d$ with $\mathcal R(\overrightarrow d)=\mathcal R_\epsilon(T)$.
It is clear that these are inverse mappings giving the desired bijection $\overrightarrow{\mathcal{D}}_{n,\epsilon}\cong \mathcal T_\epsilon$.
The MCTs in Figures \[figMCTb\] and \[figMCTa\] above give the oriented strand diagrams:
(0,0) .. controls (0.2,1) and (.8,-1)..(.97,-.07); (1,0) .. controls (1.2,-1) and (1.8,1)..(1.97,.07); (2,0) .. controls (2.2,.5) and (2.8,.5)..(2.97,.07); in [0,1,2,3]{}(,0) circle\[radius=2pt\];
(0,0) .. controls (0.2,.5) and (.8,.5)..(.97,.07); (2,0) .. controls (2.2,1) and (2.8,-1)..(3,-.07); (3,0) .. controls (3.2,-.5) and (3.8,-.5)..(3.97,-.07); (3,0) .. controls (2.5,-1) and (3,1.5)..(1.03,.07); in [0,...,4]{}(,0) circle\[radius=2pt\];
and the oriented strand diagram in Example \[thirdexample\] gives the MCT:
(0,0) .. controls (1.6,-1.3) and (1.7,.7)..(1.97,.07); (2,0) .. controls (2.2,1) and (2.8,-1)..(3,-.07); (3,0) .. controls (3.2,-.5) and (3.8,-.5)..(3.97,-.07); (3,0) .. controls (2.7,-.7) and (3,.5)..(2,.5) ..controls (1,.5) and (1.3,-.7)..(1.03,-.07); in [0,...,4]{}(,0) circle\[radius=2pt\];
(-1.5,0) node[$\iff$]{}; (0,-.5)–(1.95,-.03); (2,0)–(2.95,.48); (3,.5)–(3.95,.98); (3,.5)–(1.05,.98); (4,1) circle\[radius=2pt\] (1,1) circle\[radius=2pt\] (0,-.5) circle\[radius=2pt\] (3,.5)circle\[radius=2pt\](3,-.2) (2,0) circle\[radius=2pt\];
We now arrive at the proof of Theorem \[c-matClassif\]. This theorem follows from the fact that oriented diagrams belonging to $\overrightarrow{\mathcal{D}}_{n,\epsilon}$ can be regarded as mixed cobinary trees by Theorem \[thm: bijection between MCTs and OSDs\].
Let $f$ be the map $\textbf{c}\text{-mat}(Q_\epsilon)\to \overrightarrow{\mathcal{D}}_{n,\epsilon}$ induced by the map defined in Lemma \[cmatdiag\], and let $g$ be the bijective map $\mathcal T_\epsilon\to \overrightarrow{\mathcal{D}}_{n,\epsilon}$ defined in Theorem \[thm: bijection between MCTs and OSDs\]. We will assert the existence of a map $h:\textbf{c}\text{-mat}(Q_\epsilon) \to \mathcal T_\epsilon$ which fits into the diagram
**c**(Q\_) \[swap\][f]{} & & T\_\[swap\][\~]{}\
& \_[n,]{} &
The theorem will follow after verifying that $h$ is a bijection and that $f = g \circ h$.
We will define two new notions of $\textbf{c}\text{-matrix}$, one for MCTs and one for oriented strand diagrams. Let $T \in \mathcal T_\epsilon$ with internal edges $\ell_i$ having endpoints $(i_1,y_{i_1})$ and $(i_2,y_{i_2})$. For each $\ell_i$, define the ‘$\textbf{c}\text{-vector}$’ of $\ell_i$ to be $c_i(T) := \sum_{i_1< j \le i_2} \text{sgn}(\ell_i)e_j$, where $\text{sgn}(\ell_i)$ is the sign of the slope of $\ell_i$. Define $c(T)$ to be the ‘$\textbf{c}\text{-matrix}$’ of $T$ whose rows are the $\textbf{c}\text{-vectors } c_i(T)$. Now, let $\overrightarrow{d} = \{\overrightarrow{c}(i_\ell,j_\ell)\}_{\ell \in [n]} \in \overrightarrow{\mathcal{D}}_{n,\epsilon}$. For each oriented strand $\overrightarrow{c}(i_\ell,j_\ell)$, define the ‘$\textbf{c}\text{-vector}$’ of $\overrightarrow{c}(i_\ell,j_\ell)$ to be $$\begin{array}{rcl}
c_\ell(\overrightarrow{d}) & := & \left\{\begin{array}{rll} \sum_{i_\ell< k \le j_\ell} \text{sgn}(\overrightarrow{c}(i_\ell,j_\ell))e_k & : & i_\ell < j_\ell\\
\sum_{j_\ell< k \le i_\ell} \text{sgn}(\overrightarrow{c}(i_\ell,j_\ell))e_k & : & i_\ell > j_\ell\end{array}\right.
\end{array}$$ where $\text{sgn}(\overrightarrow{c}(i_\ell,j_\ell))$ is positive if $i_\ell < j_\ell$ and negative if $i_\ell > j_\ell$. Define $c(\overrightarrow{d})$ to be the ‘$\textbf{c}\text{-matrix}$’ of $\overrightarrow{d}$ whose rows are the $\textbf{c}\text{-vectors } c_\ell(\overrightarrow{d})$.
It is known that the notion of $\textbf{c}\text{-matrix}$ for MCTs coincides with the original notion of $\textbf{c}\text{-matrix}$ defined in Section \[subsec:quivers\], and that there is a bijection between $\textbf{c}\text{-mat}(Q_\epsilon)$ and $\mathcal T_\epsilon$ which preserves $\textbf{c}\text{-matrices}$ (see [@io13 Remarks 2 and 4] for details). Thus, we have a bijective map $h: \textbf{c}\text{-mat}(Q_\epsilon)\to \mathcal T_\epsilon$. On the other hand, the bijection $g : \mathcal T_\epsilon\to \overrightarrow{\mathcal{D}}_{n,\epsilon}$ defined in Theorem \[thm: bijection between MCTs and OSDs\] also preserves $\textbf{c}\text{-matrices}$. The map $f: \textbf{c}\text{-mat}(Q_\epsilon) \to \mathcal{T}_\epsilon$ preserves $\textbf{c}\text{-matrices}$ by definition. Hence, we have $f = g \circ h$ and $f$ is a bijection, as desired.
For linearly-ordered quivers (those with $\epsilon = (+,\ldots,+)$ or $\epsilon = (-,\ldots,-))$, this bijection was established by the first and third authors in [@gm1] using a different approach. The bijection was given by hand without going through MCTs. This was more tedious, and the authors feel that some aspects (such as mutation) are better phrased in terms of MCTs.
Exceptional sequences and linear extensions {#sec:pos}
===========================================
In this section, we consider the problem of counting the number of CESs arising from a given CEC. We show that this problem can be restated as the problem of counting the number of linear extensions of certain posets.
We define the **poset** $\mathcal{P}_d = (\{c(i_\ell,j_\ell)\}_{\ell \in [n]}, \le)$ **associated to $d$** as the partially ordered set whose elements are the strands of $d$ and where $c(k,\ell)$ covers $c(i,j)$, denoted by $c(i,j) \lessdot c(k,\ell)$, if and only if the strand $c(k,\ell)$ is clockwise from $c(i,j)$ and there does not exist another strand $c(i^\prime,j^\prime)$ distinct from $c(i,j)$ and $c(k,\ell)$ such that $c(i^\prime,j^\prime)$ is clockwise from $c(i,j)$ and counterclockwise from $c(k,\ell)$.
The construction defines a poset because any oriented cycle in the Hasse diagram of $\mathcal{P}_d$ arises from a cycle in the graph determined by $d$. Since the graph determined by $d$ is a tree, it has no cycles. In Figure \[fig:PosetExample\], we show a diagram $d \in \mathcal{D}_{4,\epsilon}$ where $\epsilon :=(-,+,-,+,+)$ and its poset $\mathcal{P}_d.$
![Two diagrams with the same poset.[]{data-label="fig:SamePosets"}](SamePosetsprime.pdf){width="\textwidth"}
In general, the map $\mathcal{D}_{n,\epsilon} \rightarrow \mathscr{P}(\mathcal{D}_{n,\epsilon}):=\{\mathcal{P}_d: d \in \mathcal{D}_{n,\epsilon}\}$ is not injective. For instance, each of the two diagrams in Figure \[fig:SamePosets\] have $\mathcal{P}_d = \textbf{4}$ where $\textbf{4}$ denotes the linearly-ordered poset with 4 elements. It is thus natural to ask which posets are obtained from strand diagrams.
Our next result describes the posets arising from diagrams in $\mathcal{D}_{n,\epsilon}$ where $\epsilon = (-,\ldots,-)$ or $\epsilon = (+,\ldots,+)$. Before we state it, we remark that diagrams in $\mathcal{D}_{n,\epsilon}$ where $\epsilon = (-,\ldots,-)$ or $\epsilon = (+,\ldots,+)$ can be regarded as **chord diagrams**.[^3] Figure \[ident\_chord\_strand\] gives an example of this identification. Under this identification, the term strand is synonymous with **chord**.
$$\includegraphics[scale = .75]{todisksprime.pdf}$$
Let $d \in \mathcal{D}_{n,\epsilon}$ where $\epsilon = (-,\ldots,-)$ or $\epsilon = (+,\ldots,+)$. Let $c(i,j)$ be a strand of $d$. There is an obvious action of $\mathbb{Z}/(n+1)\mathbb{Z}$ on chord diagrams. Let $\tau \in \mathbb{Z}/(n+1)\mathbb{Z}$ denote a generator and define $\tau c(i,j) := c(i-1, j-1)$ and $\tau^{-1} c(i,j) := c(i+1, j+1)$ where we consider $i\pm 1$ and $j\pm 1$ mod $n+1$. We also define $\tau d := \{\tau c(i_\ell,j_\ell)\}_{\ell \in [n]}$ and $\tau^{-1}d := \{\tau^{-1} c(i_\ell,j_\ell)\}_{\ell \in [n]}$. The next lemma, which is easily verified, shows that the order-theoretic properties of CECs are invariant under the action of $\tau^{\pm 1}$.
\[rotationinv\] Let $d \in \mathcal{D}_{n,\epsilon}$ where $\epsilon = (-,\ldots,-)$ or $\epsilon = (+,\ldots,+)$. Then we have the following isomorphisms of posets $\mathcal{P}_d \cong \mathcal{P}_{\tau d}$ and $\mathcal{P}_d \cong \mathcal{P}_{\tau^{-1}d}.$
\[posetclassif\] Let $\epsilon = (-,\ldots,-)$ or let $\epsilon = (+,\ldots, +).$ Then a poset $\mathcal{P} \in \mathscr{P}(\mathcal{D}_{n,\epsilon})$ if and only if
$\begin{array}{rllc}
i) & \text{each $x \in \mathcal{P}$ has at most two covers and covers at most two elements,}\\
ii) & \text{the underlying graph of the Hasse diagram of $\mathcal{P}$ has no cycles,}\\
iii) & \text{the Hasse diagram of $\mathcal{P}$ is connected}.\\
\end{array}$
Let $\mathcal{P}_d \in \mathscr{P}(\mathcal{D}_{n,\epsilon})$. By definition, $\mathcal{P}_d$ satisfies $i)$. It is also clear that the Hasse diagram of $\mathcal{P}_d$ is connected since the graph determined by $d$ is connected. To see that $\mathcal{P}_d$ satisfies $ii)$, [suppose that $C$ is a full subposet of $\mathcal{P}_d$ whose Hasse diagram is a **minimal cycle** (i.e. the underlying graph of $C$ is a cycle, but does not contain a proper subgraph that is a cycle).]{} Thus there exists $c_0 \in C$ that is covered by two distinct elements $c_1, c_\ell \in C$ in $\mathcal{P}_d$ where $\ell \le n$. Observe that $C$ can be regarded as a sequence of distinct chords $\{c_i\}_{i = 0}^\ell$ of $d$ where for all $i \in [0,\ell],$ $c_i$ and $c_{i+1}$ (we consider the indices modulo $\ell + 1$) share an endpoint $j$, and no chord adjacent to $j$ appears between $c_i$ and $c_{i+1}$. Observe that the graph determined by the diagram $d \backslash c_0$ has two connected components. We conclude that such a sequence $\{c_i\}_{i = 0}^\ell$ cannot exist in $d$. Thus the Hasse diagram of $\mathcal{P}_d$ has no cycles.
To prove the converse, we proceed by induction on the number of elements of $\mathcal{P}$ where $\mathcal{P}$ is a poset satisfying conditions $i), ii), iii)$. If $\#\mathcal{P} = 1$, then $\mathcal{P}$ is the unique poset with one element and $\mathcal{P} = \mathcal{P}_d$ where $d$ is the unique chord diagram with a single chord in a disk with exactly two boundary vertices. Assume that for any poset $\mathcal{P}$ satisfying conditions $i), ii), iii)$ with $\#\mathcal{P} = r$ for any positive integer $r < n+1$ there exists a chord diagram $d$ such that $\mathcal{P} = \mathcal{P}_d$. Let $\mathcal{Q}$ be a poset satisfying the above conditions where $\#\mathcal{Q} = n+1,$ and let $x \in \mathcal{Q}$ be a maximal element. We know $x$ covers either one or two elements of $\mathcal{Q}$.
Assume $x$ covers two elements $y,z \in \mathcal{Q}$. Since the Hasse diagram of $\mathcal{Q}$ has no cycles, we have that $\mathcal{Q} - \{x\} = \mathcal{Q}_1 + \mathcal{Q}_2$ where $y \in \mathcal{Q}_1$, $z \in \mathcal{Q}_2$, and $\mathcal{Q}_i$ satisfies $i), ii), iii)$ for $i \in [2]$. By induction, there exists positive integers $k_1, k_2$ satisfying $k_1 + k_2 = n$ and chord diagrams $d_i \in \mathcal{D}_{k_i,\epsilon^{(i)}}$ where $\mathcal{Q}_i = \mathcal{P}_{d_i}$ for $i \in [2]$ and where $\epsilon^{(i)} \in \{+,-\}^{k_i + 1}$ has all of its entries equal to the entries of $\epsilon$. By Lemma \[rotationinv\], we can further assume that the chord corresponding to $y \in \mathcal{Q}_1$ (resp. $z \in \mathcal{Q}_2$) is $c_1(i(y),k_1) \in d_1$ for some $i(y) \in [0,k_1-1]$ (resp. [$c_2(j(z),k_2) \in d_2$]{} [ for some $j(z) \in [1,k_2]$).]{} Define $d_1\sqcup d_2 := \{c^\prime(i^\prime_\ell,j^\prime_\ell)\}_{\ell \in [n]}$ to be the chord diagram in the disk with $n+2$ boundary vertices as follows (see Figure \[disjunion\]): [$$\begin{array}{rclcc}
c^\prime(i^\prime_\ell, j^\prime_\ell) & := & \left\{\begin{array}{lcl} c_1(i_\ell, j_\ell) & : & \text{if $\ell \in [k_1]$}\\ \tau^{-(k_1+1)}c_2(i_{\ell - k_1},j_{\ell - k_1}) & : & \text{if $\ell \in [k_1 + 1, n]$}. \end{array}\right.
\end{array}$$]{}
![An example with $k_1 = 3$ and $k_2 = 2$ so that $n = k_1 + k_2 = 5.$[]{data-label="disjunion"}](d1.pdf "fig:") ![An example with $k_1 = 3$ and $k_2 = 2$ so that $n = k_1 + k_2 = 5.$[]{data-label="disjunion"}](d2.pdf "fig:") ![An example with $k_1 = 3$ and $k_2 = 2$ so that $n = k_1 + k_2 = 5.$[]{data-label="disjunion"}](d1d2.pdf "fig:")
[Define $c^\prime(i^\prime_{n+1}, j^\prime_{n+1}): = c(k_1, n+1)$ ]{}and then $d:= \{c^\prime(i^\prime_\ell, j^\prime_\ell)\}_{\ell \in [n+1]}$ satisfies $i), ii), iii),$ and $\mathcal{Q} = \mathcal{P}_d.$
Now assume $x$ covers only the element $y \in \mathcal{Q}$. In this case, the Hasse diagram of $\mathcal{Q}-\{x\}$ is connected. Now by induction the poset $\mathcal{Q} - \{x\} = \mathcal{P}_d$ for some diagram $d = \{c(i_\ell, j_\ell)\}_{\ell \in [n]} \in \mathcal{D}_{n,\epsilon}$ where we assume $i_\ell < j_\ell$. Let $y = c(i(y),j(y)) \in \mathcal{Q} - \{x\}$ with $i(y) < j(y)$ denote the unique element that is covered by $x$ in $\mathcal{Q}$. This means that there are no chords in $d$ that are clockwise from $c(i(y), j(y))$. Without loss of generality, we assume that there are no chords in $d$ that are clockwise from $c(i(y), j(y))$ about $i(y)$.
We regard $d$ as an element of $\mathcal{D}_{n+1,\epsilon^\prime}$ by replacing it with $\widetilde{d}:= \{c^\prime(i^\prime_\ell, j^\prime_\ell)\}_{\ell \in [n]} \in \mathcal{D}_{n+1,\epsilon^\prime}$ as follows (see Figure \[addpoint\]):
[$$\begin{array}{rclcc}
c^\prime(i^\prime_\ell, j^\prime_\ell) & := & \left\{\begin{array}{lcl} \rho^{-1}c(i_\ell, j_\ell) & : & \text{if $i_\ell \le i(y)$ and $j(y) \le j_\ell$,}\\ \tau^{-1}c(i_{\ell},j_{\ell}) & : & \text{if $j(y) \le i_\ell$,}\\
c(i_\ell, j_\ell) & : & \text{otherwise}. \end{array}\right.
\end{array}$$]{}
$$\begin{array}{ccccccccc}
\raisebox{.35in}{$d$} & \raisebox{.35in}{$=$} & \includegraphics[scale=.85]{d.pdf} \raisebox{.35in}{ \text{ $\Large \longrightarrow$ } } & \raisebox{.35in}{$\widetilde{d}$} & \raisebox{.35in}{$=$} & \includegraphics[scale=.85]{widetilded.pdf}
\end{array}$$
Define $c^\prime(i^\prime_{n+1}, j^\prime_{n+1}) := c(i(y), i(y)+1)$ and put $d^\prime := \{c^\prime(i^\prime_\ell, j^\prime_\ell)\}_{\ell \in [n+1]}$. As $\mathcal{Q}-\{x\}$ satisfies $i), ii),$ and $iii)$, it is clear that the resulting chord diagram $d^\prime$ also satisfies $i), ii),$ and $iii)$, and that $\mathcal{P} = \mathcal{P}_{d^\prime}$.
Let $\mathcal{P}$ be a finite poset with $m = \#\mathcal{P}$. Let $f: \mathcal{P} \to \textbf{m}$ be an injective, order-preserving map (i.e. $x \le y$ implies $f(x) \le f(y)$ for all $x,y \in \mathcal{P}$) where $\textbf{m}$ is the linearly-ordered poset with $m$ elements. We call $f$ a **linear extension** of $\mathcal{P}$. We denote the set of linear extensions of $\mathcal{P}$ by $\mathscr{L}(\mathcal{P})$. Note that since $f$ is an injective map between sets of the same cardinality, $f$ is a bijective map between those sets.
\[CESsandlinexts\] Let $d = \{c(i_\ell, j_\ell)\}_{\ell\in [n]} \in \mathcal{D}_{n, \epsilon}$ and let $\overline{\xi}_{\epsilon}$ denote the corresponding complete exceptional collection. Let $\text{CES}(\overline{\xi}_{\epsilon})$ denote the set of CESs that can be formed using only the representations appearing in $\overline{\xi}_{\epsilon}$. Then the map $\chi: \text{CES}(\overline{\xi}_{\epsilon}) \to \mathscr{L}(\mathcal{P}_d)$ defined by $(X_{i_1,j_1}^{\epsilon},\ldots, X_{i_n,j_n}^{\epsilon}) \stackrel{\chi_2}{\longmapsto} \{(c(i_\ell,j_\ell), n+1-\ell)\}_{\ell \in [n]} \stackrel{\chi_1}{\longmapsto} (f(c(i_\ell,j_\ell)) := n+1-\ell)$ is a bijection.
The map $\chi_2 = \Phi : \text{CES}(\overline{\xi}_{\epsilon}) \to \mathcal{D}_{n,\epsilon}(n)$ is a bijection by Theorem \[ESbij\]. Thus it is enough to prove that $\chi_1: \mathcal{D}_{n,\epsilon}(n) \to \mathscr{L}(\mathcal{P}_d)$ is a bijection.
First, we show that $\chi_1(d(n)) \in \mathscr{L}(\mathcal{P}_d)$ for any $d(n) \in \mathcal{D}_{n, \epsilon}(n)$. Let $d(n) \in \mathcal{D}_{n, \epsilon}(n)$ and let $f := \chi_1(d(n))$. Since the strand-labeling of $d(n)$ is good, if $(c_1, \ell_1)$ and $(c_2, \ell_2)$ are two labeled strands of $d(n)$ satisfying $c_1 \le c_2$, then $f(c_1) = \ell_1 \le \ell_2 = f(c_2).$ Thus $f$ is order-preserving. As the strands of $d(n)$ are bijectively labeled by $[n]$, we have that $f$ is bijective so $f \in \mathscr{L}(\mathcal{P}_d)$.
Next, define a map $$\begin{array}{rcl}
\mathscr{L}(\mathcal{P}_d) & \stackrel{\varphi}{\longrightarrow} & \mathcal{D}_{n,\epsilon}(n)\\
f & \longmapsto & \{(c(i_\ell, j_\ell), f(c(i_\ell, j_\ell)))\}_{\ell \in [n]}.
\end{array}$$ To see that $\varphi(f) \in \mathcal{D}_{n,\epsilon}(n)$ for any $f \in \mathscr{L}(\mathcal{P}_d)$, consider two labeled strands $(c_1, f(c_1))$ and $(c_2, f(c_2))$ belonging to $\varphi(f)$ where $c_1 \le c_2$. Since $f$ is order-preserving, $f(c_1) \le f(c_2).$ Thus the strand-labeling of $\varphi(f)$ is good so $\varphi(f) \in \mathscr{L}(\mathcal{P}_d)$.
Lastly, we have that $$\chi_1(\varphi(f)) = \chi_1(\{(c(i_\ell, j_\ell), f(c(i_\ell, j_\ell)))\}_{\ell \in [n]}) = f$$ and $$\varphi(\chi_1(\{(c(i_\ell, j_\ell), \ell)\}_{\ell \in [n]})) = \varphi(f(c(i_\ell, j_\ell)):= \ell) = \{(c(i_\ell, j_\ell), \ell)\}_{\ell \in [n]}$$ so $\varphi = \chi^{-1}_1$. Thus $\chi_1$ is a bijection.
Applications {#sec:app}
============
Here we showcase some interesting results that follow easily from our main theorems.
Labeled trees
-------------
In [@sw86 p. 67], Stanton and White gave a nonpositive formula for the number of vertex-labeled trees with a fixed number of leaves. By connecting our work with that of Goulden and Yong [@gy02], we obtain a positive expression for this number. Here we consider diagrams in $\mathcal{D}_{n,\epsilon}$ where $\epsilon = (-,\ldots,-)$ or $\epsilon = (+,\ldots,+)$. We regard these as chord diagrams to make clear the connection between our work and that of [@gy02].
\[trees\]Let $T_{n+1}(r) := \{\text{trees on } [n+1] \text{ with } r \text{ leaves}\}$ and $\mathcal{D}_{n,\epsilon} := \{\text{diagrams } d = \{c(i_\ell,j_\ell)\}_{\ell \in [n]}\}$. Then $$\begin{array}{rcl}
\#T_{n+1}(r) & = &\displaystyle \sum_{\begin{array}{c}\small\text{$d \in \mathcal{D}_{n,\epsilon}:$ \ $d \text{ has $r$ chords }c(i_j,i_j+1)$}\end{array}}\#\mathscr{L}(\mathcal{P}_d).
\end{array}$$
Observe that $$\begin{array}{rcl}
\displaystyle \sum_{\begin{array}{c}\small\text{$d \in \mathcal{D}_{n,\epsilon}$ : $d$ \text{ has $r$}}\\ \small \text{chords $c(i_j,i_j+1)$}\end{array}}\#\mathscr{L}(\mathcal{P}_d) & = & \displaystyle \sum_{\begin{array}{c}\small\text{$d \in \mathcal{D}_{n,\epsilon}: d \text{ has $r$}$}\\ \small \text{$\text{chords $c(i_j, i_j + 1)$}$}\end{array}}\#\{\text{good labelings of $d$}\} \\
& = & \#\left\{d(n) \in \mathcal{D}_{n,\epsilon}(n): \begin{array}{l}d(n) \text{ has $r$ chords $c(i_j,i_j + 1)$}\\ \text{for some $i_1,\ldots, i_r \in [0,n]$}\end{array}\right\}
\end{array}$$ where we consider $i_j + 1$ mod $n+1$. By [@gy02 Theorem 1.1], we have a bijection between diagrams $d \in \mathcal{D}_{n,\epsilon}$ with $r$ chords of the form $c(i_j, i_j + 1)$ for some $i_1, \ldots, i_r \in [0,n]$ with good labelings and elements of $T_{n+1}(r)$.
We have $(n+1)^{n-1} = \sum_{d \in \mathcal{D}_{n,\epsilon}} \#\mathscr{L}(\mathcal{P}_d)$.
Let $T_{n+1} := \{\text{trees on [n+1]}\}.$ One has that $$\begin{array}{cclccc}
(n+1)^{n-1} & = & \#T_{n+1} \\
& = & \displaystyle\sum_{r \ge 0} \#T_{n+1}(r) \\
& = & \displaystyle\sum_{r \ge 0} \sum_{\begin{array}{c}\small\text{$d \in \mathcal{D}_{n,\epsilon}$ : $d$ \text{ has $r$}}\\ \small \text{chords $c(i_j,i_j+1)$}\end{array}}\#\mathscr{L}(\mathcal{P}_d) & \text{(by Theorem~\ref{trees})} \\
& = & \displaystyle \sum_{d \in \mathcal{D}_{n,\epsilon}} \#\mathscr{L}(\mathcal{P}_d).
\end{array}$$
Reddening sequences
-------------------
In [@k12], Keller proves that for any quiver $Q$, any two reddening mutation sequences applied to $\widehat{Q}$ produce isomorphic ice quivers. As mentioned in [@kel13], his proof is highly dependent on representation theory and geometry, but the statement is purely combinatorial—we give a combinatorial proof of this result for type $\mathbb{A}_n$ quivers $Q_\epsilon$.
Let $R \in EG(\widehat{Q})$. A mutable vertex [$i \in R_0$]{} is called **green** if there are no arrows $j \to i$ in $R$ with $j \in [n+1,m]$. Otherwise, $i$ is called **red**. A sequence of mutations $\mu_{i_r}\circ \cdots \circ \mu_{i_1}$ is **reddening** if all mutable vertices of the quiver $\mu_{i_r}\circ \cdots \circ \mu_{i_1}(\widehat{Q})$ are red. Recall that an isomorphism of quivers that fixes the frozen vertices is called a **frozen isomorphism**. We now state the theorem.
If $\mu_{i_r}\circ \cdots \circ \mu_{i_1}$ and $\mu_{j_s}\circ \cdots \circ \mu_{j_1}$ are two reddening sequences of $\widehat{Q}_\epsilon$ for some $\epsilon \in \{+,-\}^{n+1}$, then there is a frozen isomorphism $\mu_{i_r}\circ \cdots \circ \mu_{i_1}(\widehat{Q}_{\epsilon}) \cong \mu_{j_s}\circ \cdots \circ \mu_{j_1}(\widehat{Q}_{\epsilon})$.
Let $\mu_{i_r}\circ \cdots \circ \mu_{i_1}$ be any reddening sequence. Denote by $C$ the $\operatorname{\textbf{c}}$-matrix of $\mu_{i_r}\circ \cdots \circ \mu_{i_1}(\widehat{Q}_{\epsilon})$. By Theorem \[c-matClassif\], $C$ corresponds to an oriented strand diagram $\overrightarrow{d}_C \in \overrightarrow{\mathcal{D}}_{n,\epsilon}$ with all strands of the form $\overrightarrow{c}(j,i)$ for some $i$ and $j$ satisfying $i < j$. As $\overrightarrow{d}_C$ avoids the configurations described in Definition \[def:cmatdiag\], we conclude that $\overrightarrow{d}_C = \{\overrightarrow{c}(i, i-1)\}_{i\in[n]}$ and $C = -I_n$. Since **c**-matrices are in bijection with ice quivers in $EG(\widehat{Q}_\epsilon)$ (see [@nz12 Thm 1.2]) and since $\widecheck{Q}_\epsilon$ is an ice quiver in $EG(\widehat{Q}_\epsilon)$ whose **c**-matrix is $-I_n$, we obtain the desired result.
Noncrossing partitions and exceptional sequences
------------------------------------------------
In this section, we give a combinatorial proof of Ingalls’ and Thomas’ result that complete exceptional sequences are in bijection with maximal chains in the lattice of noncrossing partitions [@it09]. We remark that their result is more general than that which we present here. Throughout this section, we assume that $Q_\epsilon$ has $\epsilon= (-,\ldots,-)$ and we regard the strand diagrams of $Q_\epsilon$ as chord diagrams.
A **partition** of $[n]$ is a collection $\pi = \{B_\alpha\}_{\alpha \in I} \in 2^{[n]}$ of subsets of $[n]$ called **blocks** that are nonempty, pairwise disjoint, and whose union is $[n].$ We denote the lattice of set partitions of $[n]$, ordered by refinement, by $\Pi_n$. A set partition $\pi = \{B_{\alpha}\}_{\alpha \in I} \in \Pi_n$ is called **noncrossing** if for any $i < j < k < \ell$ where $i, k \in B_{\alpha_1}$ and $j, \ell \in B_{\alpha_2}$, one has $B_{\alpha_1} = B_{\alpha_2}.$ We denote the lattice of noncrossing partitions of $[n]$ by $NC^{\mathbb{A}}(n)$.
Label the vertices of a convex $n$-gon $\mathcal{S}$ with elements of $[n]$ so that reading the vertices of $\mathcal{S}$ counterclockwise determines an increasing sequence mod $n$. We can thus regard $\pi = \{B_\alpha\}_{\alpha \in I} \in NC^\mathbb{A}(n)$ as a collection of convex hulls $B_\alpha$ of vertices of $\mathcal{S}$ where $B_\alpha$ has empty intersection with any other block $B_{\alpha^\prime}$.
Let $n = 5$. The following partitions all belong to $\Pi_5$, but only $\pi_1, \pi_2, \pi_3 \in NC^\mathbb{A}(5).$ $$\pi_1 = \{\{1\}, \{2,4,5\}, \{3\}\}, \pi_2 = \{\{1,4\}, \{2,3\}, \{5\}\}, \pi_3 = \{\{1,2,3\}, \{4,5\}\}, \pi_4 = \{\{1,3,4\}, \{2,5\}\}$$ Below we represent the partitions $\pi_1, \ldots, \pi_4$ as convex hulls of sets of vertices of a convex pentagon. We see from this representation that $\pi_4 \not \in NC^\mathbb{A}(5).$
\[chains\] Let $k \in [n].$ There is a bijection between $\mathcal{D}_{k,\epsilon}(k)$ and the following chains in $NC^{\operatorname{\mathbb{A}}}(n+1)$ $$\left\{(\pi_1 = \{\{i\}\}_{i \in [n+1]}, \pi_2, \ldots, \pi_{k+1}) \in (NC^\mathbb{A}(n+1))^{k+1}: \begin{array}{c}\pi_j = (\pi_{j-1} \backslash\{B_{\alpha}, B_\beta\})\sqcup \{B_\alpha \sqcup B_\beta\}\\ \text{for some $B_\alpha \neq B_\beta$ in $\pi_{j-1}$}\end{array}\right\}.$$ In particular, when $k = n$, there is a bijection between $\mathcal{D}_{n,\epsilon}(n)$ and maximal chains in $NC^{\mathbb{A}}(n+1)$. We remark that each chain described above is **saturated** (i.e. each inequality appearing in $\{\{i\}\}_{i \in [n+1]} < \pi_1 < \cdots < \pi_{k}$ is a covering relation).
Let $d(k) = \{(c(i_\ell,j_\ell),\ell)\}_{\ell \in [k]} \in \mathcal{D}_{k,\epsilon}(k)$. Define $\pi_{d(k),1} := \{\{i\}\}_{i \in [n+1]} \in \Pi_{n+1}.$ Next, define $\pi_{d(k),2} := \left(\pi_{d(k),1}\backslash \{\{i_1 + 1\}, \{j_1+1\}\}\right) \sqcup \{\{i_1+1,j_1+1\}\}$. Now assume that $\pi_{d(k),s}$ has been defined for some $s \in [k]$. Define $\pi_{d(k),s+1}$ to be the partition obtained by merging the blocks of $\pi_{d(k),s}$ containing $i_{s}+1$ and $j_{s}+1$. Now define $f(d(k)) := (\pi_{d(k),1}, \ldots, \pi_{d(k), k+1}).$
It is clear that $f(d(k))$ is a chain in $\Pi_{n+1}$ with the desired property as $\pi_1 \lessdot \pi_2$ in $\Pi_{n+1}$ if and only if $\pi_2$ is obtained from $\pi_1$ by merging exactly two distinct blocks of $\pi_1.$ To see that each $\pi_{d(k),s} \in NC^\mathbb{A}(n+1)$, suppose a crossing of two blocks occurs in a partition appearing in $f(d(k))$. Let $\pi_{d(k),s}$ be the smallest partition of $f(d(k))$ (with respect to the partial order on set partitions) with two blocks crossing blocks $B_1$ and $B_2$. Without loss of generality, we assume that $B_2 \in \pi_{d(k),s}$ is obtained by merging the blocks $B_{\alpha_1}, B_{\alpha_2} \in \pi_{d(k),s-1}$ containing $i_{s-1} + 1$ and $j_{s-1}+1$, respectively. This means that $c(i_{s-1},j_{s-1}) \in d(k)$ and $c(i_{s-1}, j_{s-1})$ crosses at least one other chord of $d(k)$. This contradicts that $d(k) \in \mathcal{D}_{k,\epsilon}(k)$. Thus $f(d(k))$ is a chain in $NC^\mathbb{A}(n+1)$ with the desired property.
Next, we define a map $g$ that is the inverse of $f$. Let $C = (\pi_1 = \{\{i\}\}_{i \in [n+1]}, \pi_2, \cdots, \pi_{k+1}) \in (NC^\mathbb{A}(n+1))^{k+1}$ be a chain where each partition in $C$ satisfies $\pi_j = (\pi_{j-1}\backslash\{B_\alpha, B_\beta\}) \sqcup \{B_\alpha \sqcup B_\beta\}$ for some $B_\alpha \neq B_\beta$ in $\pi_{j-1}$. As $\pi_2 = \left(\pi_1 \backslash \{\{s_1\}, \{t_1\}\}\right)\sqcup \{\{s_1, t_1\}\}$, define $c(i_1,j_1) := c(s_1-1, t_1-1)$ where we consider $s_1 - 1$ and $t_1-1$ mod $n+1.$ Now for $r \ge 2$ let $B_1, B_2 \in \pi_{r-1}$ be the blocks that one merges to obtain $\pi_r$. Define $s_1 \in B_1$ (resp., $t_1 \in B_1$) to be the last element of $B_1$ (resp., $B_2$) that one encounters before any element of $B_2$ (resp., $B_1$) while reading counterclockwise through the integers $1, \ldots, n$. Let $c(i_{r-1},j_{r-1}) := c(s_1-1,t_2-1)$. Finally, put $g(C) := \{(c(i_\ell,j_\ell),\ell): \ell \in [k]\}$.
We claim that $g(C)$ has no crossing chords. Suppose $(c(s_i,t_i), i)$ and $(c(s_j,t_j),j)$ are crossing chords in $g(C)$ with $i < j$ and $i,j \in [k]$. We further assume that $$j = \min\{j^\prime \in [i+1,k]: \text{$(c(s_{j^\prime},t_{j^\prime}), j^\prime)$ crosses $(c(s_i,t_i),i)$ in $g(C)$}\}.$$ We observe that $s_i + 1, t_i + 1 \in B_1$ for some block $B_1 \in \pi_{j}$ and that $s_j+1, t_j+1 \in B_2$ for some block $B_2 \in \pi_{j+1}$. We further observe that $s_j+1, t_j+1 \not \in B_1$ otherwise, by the definition of the map $g$, the chords $(c(s_i,t_i),i)$ and $(c(s_j,t_j),j)$ would be noncrossing. Thus $B_1, B_2 \in \pi_{j+1}$ are distinct blocks that cross so $\pi_{j+1} \not \in NC^\mathbb{A}(n+1).$ We conclude that $g(C)$ has no crossing chords so $g(C) \in \mathcal{D}_{k,\epsilon}(k)$.
To complete the proof, we show that $g\circ f = 1_{\mathcal{D}_{k,\epsilon}(k)}$. The proof that $f \circ g$ is the identity map is similar. Let $d(k) \in \mathcal{D}_{k,\epsilon}(k)$. Then $f(d(k)) = (\pi_1 = \{\{i\}\}_{i \in [n+1]}, \pi_2, \ldots, \pi_{k+1})$ where for any $s \in [k]$ we have $$\pi_s = \left(\pi_{s-1}\backslash\{B_\alpha, B_\beta\}\right) \sqcup \{B_\alpha, B_\beta\}$$ where $i_{s-1} + 1 \in B_\alpha$ and $j_{s-1} + 1 \in B_\beta.$ Then we have $g(f(d(k))) = \{c((i_\ell +1) - 1, (j_\ell +1) - 1), \ell)\}_{\ell \in [k]} = \{(c(i_\ell, j_\ell), \ell)\}_{\ell \in [k]}.$
If $\epsilon = (-,\ldots, -)$ of $\epsilon = (+, \ldots, +)$, then the exceptional sequences of $Q_\epsilon$ are in bijection with saturated chains in $NC^\mathbb{A}(n+1)$ of the form $$\left\{(\pi_1\{\{i\}\}_{i \in [n+1]}, \pi_2, \ldots, \pi_{k+1}) \in (NC^\mathbb{A}(n+1))^{k+1}: \begin{array}{c}\pi_j = (\pi_{j-1} \backslash\{B_{\alpha}, B_\beta\})\sqcup \{B_\alpha \sqcup B_\beta\}\\ \text{for some $B_\alpha \neq B_\beta$ in $\pi_{j-1}$}\end{array}\right\}.$$
$$\begin{array}{ccc}
\includegraphics[scale=.7]{maxchaindiags1prime1.pdf} & \raisebox{.3in}{$\longmapsto$} & \raisebox{.1in}{\includegraphics[scale=.7]{maxchains1prime1.pdf}}\\
\includegraphics[scale=.7]{maxchaindiags1prime2.pdf} & \raisebox{.3in}{$\longmapsto$} & \raisebox{.1in}{\includegraphics[scale=.7]{maxchains1prime2.pdf}}
\end{array}$$
In Figure \[fig\_max\_chain\_bij\], we give two examples of the bijection from Theorem \[chains\] with $k = 4$.
\[sec:biblio\]
[^1]: The curves $\Phi_\epsilon(X_{i,j}^{\epsilon})$ will have some additional topological conditions that we omit here.
[^2]: [In general, not all indecomposable representations are exceptional. For Dynkin quivers, it is well-known that a representation is exceptional if and only if it is indecomposable.]{}
[^3]: These noncrossing trees embedded in a disk with vertices lying on the boundary have been studied by Araya in [@a13], Goulden and Yong in [@gy02], and the first and third authors in [@gm1].
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---
abstract: 'We study diffusion with a bias towards a target node in networks. This problem is relevant to efficient routing strategies in emerging communication networks like optical networks. Bias is represented by a probability $p$ of the packet/particle to travel at every hop towards a site which is along the shortest path to the target node. We investigate the scaling of the mean first passage time (MFPT) with the size of the network. We find by using theoretical analysis and computer simulations that for Random Regular (RR) and Erdős-Rényi (ER) networks, there exists a threshold probability, $p_{th}$, such that for $p<p_{th}$ the MFPT scales anomalously as $N^\alpha$, where $N$ is the number of nodes, and $\alpha$ depends on $p$. For $p>p_{th}$ the MFPT scales logarithmically with $N$. The threshold value $p_{th}$ of the bias parameter for which the regime transition occurs is found to depend only on the mean degree of the nodes. An exact solution for every value of $p$ is given for the scaling of the MFPT in RR networks. The regime transition is also observed for the second moment of the probability distribution function, the standard deviation.'
author:
- Loukas Skarpalezos
- Aristotelis Kittas
- Panos Argyrakis
- Reuven Cohen
- Shlomo Havlin
bibliography:
- 'netbias3.bib'
title: Anomalous biased diffusion in networks
---
Recently there has been growing interest in investigating the properties of complex networks [@barabasi_sci286; @albert_rev74; @newman_siam45; @dorogovtsev_adv51; @caldarelli2012networks]. These include systems from markedly different disciplines, as communication networks, the Internet itself, social networks, networks of collaboration between scientists, transport networks, gene regulatory networks, and many other examples in biology, sociology, economics and even linguistics, with new systems being added continuously to the list [@Pastor-Satorras:2004:ESI:1076357; @DorogovtsevBook; @Cohen_Havlin:2010; @newman_book; @Barrat:2008:DPC:1521587; @Bashan_natcom].
Sending messages through a network in the form of packets in an efficient way is one of the most challenging problems in today’s communication technologies. It is obvious that a fully biased walk (with probability to stay on the shortest path equal to 1) would be the most efficient way to send a message, if the exact structure of the network is known. But, quite often, as in the case of wireless sensor networks [@avin_query], ad-hoc networks [@yossef_mobi] and peer-to-peer networks [@gkantsidis_peer], due to the continuously dynamically changing infrastructure, the application of routing tables is not possible, and the so called hot potato/random walk routing protocol is preferable, because it can naturally cope with failures or disconnections of nodes. The problem with such a procedure, in which data packets traverse the network in a random fashion, is a significant increase of the hitting time. For this reason, new protocols have been proposed recently [@beraldi1], that are based on the idea of biased random walks and which can significantly reduce the hitting time in such networks. For example, the Lukewarm Potato Protocol [@beraldi2], is totally tunable (with the value of just one threshold parameter) and can operate anywhere in the continuum from the hot potato/random walk forwarding protocol to a deterministic shortest path forwarding protocol. But also, in a more general manner, we can consider that every routing protocol, which uses deflection (hot potato) routing in certain circumstances (e.g. insufficient storage space of the node or a disconnected node), can be represented by a biased random walk process since it uses the shortest path only if it is possible. This problem is also very relevant in optical networks where optical switches pay a large price for packet storing (with the conversion of light to elecronic signals). The result is a limited storing capacity of optical switches that must route packages in a random direction in the case the destination path is overloaded or they have reached the storage limit. Therefore, the probabilty to stay on the shortest path may, in certain cases, have a small value (optical switches with unsufficient storage capacity), and in other cases, a large one (few disfunctioning nodes). It is consequently of great interest, and it is the subject of this work, to understand how the diffusion process is affected when a tunable bias along the shortest path is used and to theoretically study the scaling properties of such biased diffusion processes.
Random Regular (RR) networks are networks where all nodes have exactly the same number of edges (connections). They constitute a well studied mathematical model which is suitable for exact analysis of its properties. The Erdős-Rényi (ER) model [@erdos_publ6; @erdos_1960; @Bollobas_RandGraph] is a well known simple model, which generates random graphs by setting an edge between each pair of nodes with a probability $q$, independently of the other edges. This yields (in the limit $N\rightarrow\infty$) a Poisson distribution (for $q<1$) of the node degree $k$: $P(k)=\frac{\av{k}^k}{k!}e^{-\av{k}}$ with $\av{k}=q(N-1)$, with $q=1$ giving the completely connected graph. An important property of networks is the average path length $D$ between two nodes. For the case of RR and ER networks it has been shown that $D$ scales as $\ln N $. This dependence is the origin of the well known small world phenomena in networks.
Random walks have interesting properties which may depend on the dimension and the structure of the medium in which they are confined [@avraham_diffusion; @weiss_random; @redner_mfpt; @havlin_adv36; @gallos_prl], e.g. lattices or complex networks. Diffusion is a very natural mode of transport, where hopping from one node to the next is unaffected by the history of the walk [@weiss_random; @bollt_njphys7]. A measure of diffusion which has been extensively studied (see e.g. [@redner_mfpt; @condamin_nat450; @condamin_prl95; @baronchelli_pre73; @sood_prl99; @sood_jphys38; @argy_physa363]), is the first-passage time (FPT), which is the time required for a random walker to reach a given target point for the first time. The importance of FPT originates from the crucial role played by first encounter properties in various real situations, including transport in disordered media, neuron firing dynamics, buying/selling on the stock market, spreading of diseases or target search processes [@redner_mfpt; @condamin_nat450].
The properties of the first-passage time have been investigated in a variety of networks. Baronchelli and Loreto [@baronchelli_pre73], using the concept of rings, have shown that the FPT probability distribution in ER networks is an exponential decay and FPT vs the degree of the target node is a power law for various networks, such as ER networks, the Barabási-Albert model (BA) [@barabasi99emergence], as well as the Internet. An analytical formula has also been derived for the mean first-passage time (MFPT) of a random walker from one node to another, namely $\av{T_{ij}}$ (mean transit time), on networks [@noh_prl92]. Note that in this case a random walk motion from node $i$ to $j$ is not symmetric with the motion in the opposite direction. The size scaling of $\av{T_{ij}}$ has been studied in a variety of systems and geometries [@bollt_njphys7]. The trapping problem on networks which is closely related to MFPT was studied by Kittas et al [@kittas08]. Biased random walks on networks have also been studied [@sood_prl99], including local navigation rules (see e.g. [@fronczak_pre80; @adamic_pre64; @tadic_physa332; @wang_pre73]).
We use Monte Carlo computer simulations implemented by the following algorithm: Initially, a source and a target node are selected at random. The particle travels from the source to the target node either randomly, or with a bias (for a schematic see Fig. \[fig1\]). The bias is expressed by a parameter $p$, which is the probability that the particle at each time step travels towards the target node using the shortest path to it. To calculate the shortest path we use the Breadth-First-Search (BFS) algorithm as described in [@cormen_algorithms]. Given a graph $G=(V, E)$ and a specific source vertex $s$, BFS systematically explores the edges of $G$ to record every vertex that is reachable from $s$. It computes the distance from $s$ to each reachable vertex, which is the smallest number of edges. We use the target node as the BFS “source” $s$ and identify the geodesic distances from the target to every node in the network, i.e. the number of links in the shortest path from the target to any arbitrary node. Thus, each node is assigned a number, which indicates its distance from the target. When the particle moves, it jumps to one of its adjacent nodes, which belong to the shortest path with probability $p$, or to a random node (including the ones in the shortest path) with probability $1-p$. Consequently, for $p=1$ the particle always travels on the shortest path, while for $p=0$ it performs a stochastic random walk. We consider the process only on the largest cluster of the network (also discovered with the BFS algorithm). We perform $10^5$ total runs (1000 networks, considering 100 pairs of random source-target nodes for each network realization).
Firstly, we investigate the scaling of the MFPT with system size $N$ (number of nodes of the network) for RR networks. We find that the value of $p$ has a large effect on the scaling of MFPT, with one range of large $p$ having a logarithmic scaling and another of small $p$ having a power law function of $N$ (see Fig. \[fig2\]). As $p$ increases the system size becomes less relevant and the MFPT scales logarithmically with the system size, similar to the diameter of the network.
For the analytical approach of the case of RR networks, we consider a walk on a finite tree of depth $D$ with reflecting boundary conditions at the leaves (ends). We go towards the root with probability $p$ and hop to a random neighbor with probability $1-p$. Since there are $k$ neighbors to each node, there is a probability $(1-p)/k$ that we may choose the link going towards the root. Eventually, this can be mapped to a random walk on a finite segment $\{0,1,\ldots,D\}$. Since the number of nodes at a distance $d$ from the source is approximately $n_d=k(k-1)^{d-1}$, and the total number of nodes is $$N=1+\sum_{i=1}^D k(k-1)^{d-1}=1+k\frac{(k-1)^D-1}{k-2}\;,$$ it follows that $$D=\frac{\log\left(1+(k-2)(N-1)/k\right)}{\log (k-1)}\approx\frac{\log\left((k-2)N/k\right)}{\log (k-1)}$$ is the average distance and the probability of going towards the target is $p'=p+(1-p)/k$. Denote by $T_i$ the average time it takes the walker to reach the destination when it is at distance $i$ from it. The recurrence equations are $$\label{eq:recur}
T_i=1+p'T_{i-1}+(1-p')T_{i+1}\;,$$ for $0<i<D$ and $$\begin{aligned}
\label{eq:recur_t0}
T_0&=&0\;,\\
\label{eq:recur_tD}
T_D&=&1+p'T_{D-1}+(1-p')T_D\;.\end{aligned}$$ The solution of Eq. (\[eq:recur\]) is $$T_i=\frac{i}{2p'-1}+c_1+c_2\left(\frac{p'}{1-p'}\right)^i\;.$$ Substituting in Eq. (\[eq:recur\_t0\]) and (\[eq:recur\_tD\]) one obtains $$c_1=-c_2=\frac{1-p'}{(2p'-1)^2}\left(\frac{1-p'}{p'}\right)^D$$ Thus, $$\label{eq:sol}
T_D=\frac{D}{2p'-1}+\frac{1-p'}{(2p'-1)^2}\left[\left(\frac{1-p'}{p'}\right)^D-1\right]\;.$$ A better approximation is obtained when taking into consideration the probability of selecting a pair of nodes at distance $i$ from each other. The probability of choosing such a pair is approximately $$P(i)=\frac{k(k-1)^{i-1}}{\sum_{i=1}^D k(k-1)^{i-1}}\;.$$ Thus, the expected time is $$\begin{aligned}
E[T]=\sum_{i=1}^D P(i)T_i=\frac{D(k-1)^{D+1}-(D+1)(k-1)^D+1}{(k-2)((k-1)^D-1)}+c_1-\nonumber\\
c_1\frac{(k-1)\frac{p'}{1-p'}\left(\left((k-1)\frac{p'}{1-p'}\right)^D-1\right)}{\left((k-1)\frac{p'}{1-p'}-1\right)
\left((k-1)^D-1)\right)}\;.\end{aligned}$$
Therefore, if $p'>1/2$ (i.e. $p>(k-2)/(2k-2)$), the first term of Eq.(\[eq:sol\]) dominates and we have that the first passage time is approximately $$\label{eq:log}
T_D\approx\frac{D}{2p'-1}\approx \frac{\log\left((k-2)N/k\right)}{(2p+2(1-p)/k-1)\log(k-1)}\;.$$ Whereas, if $p'<1/2$ the second term dominates and we have $$\label{eq:pow}
T_D\approx\frac{1-p'}{(2p'-1)^2}\left(\frac{1-p'}{p'}\right)^D
\approx \frac{1-p'}{(2p'-1)^2}\left(\frac{1-p'}{p'}\right)^{\log\left((k-2)N/k\right)/\log(k-1)}\propto N^\alpha\;,$$ where $$\label{eq:slope}
\alpha= \frac{\log\frac{1-p'}{p'}}{\log(k-1)}\;.$$ The minimum value for $p'$ is $p'=1/k$ (obtained for $p=0$). In this case $1-p'=(k-1)/k$ and $\alpha=1$, i.e., on average the walk moves randomly with no preferred direction and reaches a large fraction of the nodes in the network before reaching the target, as expected.
For the case $p'=1/2$ the solution for the equations becomes $$T_i=(2D+1)i-i^2\;,$$ and therefore, $$T_D=D(D+1)=\frac{\log\left((k-2)N/k\right)}{\log(k-1)}\left(\frac{\log\left((k-2)N/k\right)}{\log(k-1)}+1\right)\;.$$ Thus, it behaves like normal diffusion, where the time needed to reach distance $D$ is of the order $D^2$ [@weiss_random; @redner_mfpt; @avraham_diffusion].
From the above analysis we clearly see that for RR networks there exists an abrupt change from a power law behavior to logarithmic dependence on $N$ for the MFPT. The limit between these two radically different scaling behaviors corresponds to the threshold value of the bias parameter $p_{th}=(k-2)/(2k-2)$. In Fig. \[fig2\] we compare the analytical solution (\[eq:sol\]) with the results of the Monte Carlo simulations for $k=3$ and $k=10$. The measured slopes of the power law regime and the prefactors of the logarithmic regime are in excellent agreement with the values given by (\[eq:slope\]) and (\[eq:log\]), respectively.
In Fig. \[fig3\] we investigate the behavior of the standard deviation $\sigma$, which is the second moment of the probability distribution function of the FPT for RR networks. In Fig. \[fig3:a\] and \[fig3:b\] we see that the scaling of the standard deviation with the size of the network largely resembles that of the MFPT, with two different regimes separated by the threshold value $p_{th}$. This resemblance and the existance of the regime transition for the same threshold value $p_{th}$ is made more clear in \[fig3:c\] where the scaling of the ratio $\sigma$/MFPT is represented. We see that for $p<p_{th}$, the scaling of the two quantities is the same, while for $p>p_{th}$, the standard deviation scales slower with $N$ than the MFPT. In Fig. \[fig3:d\] we see the dependence of the standard deviation on the value of $p$ for a fixed network size. We see that the standard deviation decreases to reach a very small value for a fully biased diffusion. This is expected since for a fully biased walk the probability distribution function corresponds to a delta function.
We now investigate the case of ER networks. A notable result is the fact that the MFPT in ER networks behaves in the same way as in RR networks i.e. the previous analytical relations are also applicable for ER networks by simply substituing $k$ by $\av{k}$. In fact, in Fig. \[fig4:a\] and \[fig4:b\] we see that for ER networks there is also a very good agreement between theoretical and computed results, and the threshold value of the bias parameter is given now by $p_{th}=(\av{k}-2)/(2\av{k}-2)$. This is an important result since ER networks constitute a more general ensemble than RR networks. In Fig. \[fig4:c\] and \[fig4:d\] we see the two regimes of the scaling of the standard deviation $\sigma$ of the FPT for ER networks.
In summary, a model was developed to study the efficiency of biased random walks in networks. The bias is expressed by the parameter $p$, which is the probability that the particle remains in the shortest path to a target node, in the range of extreme values 0 (unbiased case) and 1 (fully biased case). In both RR and ER networks, the MFPT scaling with the size of the system shows a sudden transition from power law to logarithmic behavior and this transition occurs for the value of the bias parameter $p_{th}=(\av{k}-2)/(2\av{k}-2)$. This was shown by means of Monte Carlo simulations, but also demonstrated analytically with an exact solution for the case of RR networks. Also, a similar transition between two regimes is observed for the standard deviation.
Aknowledgements: This research has been co-financed by the European Union (European Social Fund - ESF) and Greek national funds through the Operational Program “Education and Lifelong Learning” of the National Strategic Reference Framework (NSRF) - Research Funding Program: Heracleitus II (to LS). SH wishes to thank the LINC and the Epiwork EU projects, the DFG and the Israel Science Foundation for support.
![ Illustration of the biased diffusion process. The arrows represent moves. Full arrows represents movement along the shortest path, while dashed arrows represent random steps. The destination node is represented by a square.[]{data-label="fig1"}](illustration.ps){width="6.in"}
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---
abstract: 'Based on the proposed earlier by the Author approach to macroscopic description of scalar interaction, this paper develops the macroscopic model of relativistic plasma with a fantom scalar interaction of elementary particles. In the given model as opposed to previous models a restriction on nonnegativeness of the particles’ effective mass is removed.'
---
[**Statistical system with a fantom scalar interaction in the Gravitation Theory. I. The Microscopic Dynamics.**]{}\
Yu.G. Ignatyev and D.Yu. Ignatyev\
Kazan Federal University,\
Kremlyovskaya str., 35, Kazan 420008, Russia
[**keywords**]{}: Early Universe, fantom scalar interaction, relativistic kinetics, cosmological acceleration, numerical simulation.\
[**PACS**]{}: 04.20.Cv, 98.80.Cq, 96.50.S 52.27.Ny
Introduction
============
In the recent years there emerged a large number of articles devoted to scalar fields, which are necessary for an explanation of a secondary acceleration of the Universe. It is reasonable to take models with minimal interaction, in which interaction of a scalar field with an <<ordinary matter>> is implemented by means of a gravitational interaction. Also it is reasonable to take models with non-minimal interaction, in which interaction of a scalar field with an <<ordinary matter>> is implemented by means of some connection. This connection could be realized through the factor depending on the scalar field potential in the Lagrange matter function (potential connection) or on the factor which depends on the scalar field’s derivatives (kinetic connection). These models are intended to describe to some extent at least qualitative behavior of the Universe at large and small times. However, being essentially highly phenomenological, these models are not able to satisfy aesthetical requirements of theorists and unlikely are long-living. From the other hand, there exist theoretical models based on the gravitation theory’s modifications, which are able to describe a kinematics of the cosmological extension. These are, primarily, $f(R)$ gravitation models of A.Starobinsky [@Starobinsky], and also other modifications of the gravitation theory such as Poincare calibration theories of gravitation[^1]. By no means negating a possibility of modification of the Einstein theory we nonetheless consider a cosmological model constructed on the fundamental scalar interaction since the scalar field being highly symmetrical one, from one hand can be the foundation of the elementary particle physics and, from the other hand, can realize a concept of physical vacuum. In the article we consider fantom fields, i.e. fields possessing a <<kinetic>> energy.
In contrast to specified nonminimal models of scalar interaction we consider statistical systems of scalar charged particles in which some sort of particles can directly interact with a scalar field by means of fundamental *scalar charge*. From the other hand, statistical system possessing non-zero scalar charge and being itself a source of a scalar field, can efficiently influence on scalar field managing its behavior. Such a scalar interaction was introduced in the general relativistic theory in 1983 by the Author [@Ignatev1; @Ignatev2; @Ignatev3; @Ignatev4] and a bit later - by G.G.Ivanov [@Ivanov]. In particular, in articles [@Ignatev2; @Ignatev3] based on the kinetic theory the self-consistent set of equations describing a statistical system of particles with scalar interaction was obtained. In [@kuza] there were investigated group properties of equilibrium statistical configurations with scalar interaction and in [@Ignat_Popov] it was established a tight connection between the variable mass particle dynamics and the dynamics of scalar charge particle. In recent Author’s articles [@YuNewScalar1; @YuNewScalar2; @YuNewScalar3][^2] the macroscopic theory of statistical systems with scalar interaction was significantly improved and extended to the case of fantom scalar fields[^3].
As opposed to the cited above nonminimal models of scalar interaction, the considered strict dynamic model of interaction of a scalar field with elementary particles is based on the Hamilton microscopic equations of motion and following statistical averaging procedure. It turns out that an interaction of a scalar field with particles at that could be introduced by a single way; consequently the macroscopic equations of matter and scalar field are obtained in a single way by means of standard averaging procedures. Thereby a tight connection between micro and macro levels of a scalar field description is established. Naturally enough that obtained in that way model of scalar interaction should have more complex construction than similar phenomenological models at the same time it should reveal more rich possibilities of behavior[^4].
The Canonical Equations Of Motion
=================================
The canonical equations of relativistic particle motion relative to a pair of canonically conjugated dynamic variables $x^{i} $ (coordinates) and $P_{i} $ (generalized momentum) have a form (see e.g. [@Ignatev2])[^5]:
$$\label{EQ1}
\frac{dx^{i} }{ds} =\frac{\partial H}{\partial P_{i} } ;\quad \quad \frac{dP_{i} }{ds} =-\frac{\partial H}{\partial x^{i} } ,$$
where $H(x,P)$ is a relativistically invariant Hamilton function. Calculating the full derivative of the dynamic variables $\psi (x^{i} ,P_{k} )$, with an account of we obtain:
$$\label{EQ2}
\frac{d\psi }{ds} =[H,\psi ],$$
where the invariant Poisson brackets are introduced: $$\label{EQ3}
[H,\psi ]=\frac{\partial H}{\partial P_{i} } \frac{\partial \psi }{\partial x^{i} } -\frac{\partial H}{\partial x^{i} } \frac{\partial \psi }{\partial P_{i} } \; .$$ As a result of the Hamilton function is an integral of particle motion: $$\label{EQ4}
\frac{dH}{ds} =[H,H]=0,\Rightarrow H= {\rm Const}.$$ The relation could be named the normalization ratio. The invariant Hamilton function is determined ambiguously. Indeed, in consequence of if $H(x,P)$ is a Hamilton function, then any continuously differentiable function $f(H)$ is also a Hamilton function. The sole possibility of introduction of the invariant Hamilton function quadratic by generalized particle momentum at presence of only gravitational and scalar fields is: $$\label{EQ7}
H(x,P)=\frac{1}{2} \left[\psi(x)(P,P)-\varphi(x) \right],$$ where $(a,b)$ are here and then a scalar product of 4-dimensional vectors $a$ and $b$, and $\psi(x)$ and $\varphi(x)$ are certain scalar functions. Choosing a non-zero normalization of the Hamilton function [@Ignat_Popov; @YuNewScalar1], $$\label{EQ7a}
H(x,P)=\frac{1}{2} \left[\psi(x)(P,P)-\varphi(x) \right]=0.$$ we obtain: $$\label{EQ7b}
(P,P)=\frac{\varphi}{\psi},$$ and from the first group of the canonical equations of motion(\[EQ1\]) we obtain the relation between the generalized momentum and a particle velocity vector: $$\label{EQ10a}
u^{i} \equiv \frac{dx^{i} }{ds} =\psi P^{i} \Rightarrow P^{i} =\psi^{-1} u^{i} ,$$ Substituting the last relation to the normalization ratio (\[EQ7b\]) and requiring the unit normalization of particle velocity vector $$\label{EQ11}
(u,u)=1.$$ we obtain: $$\psi\varphi=1 \Rightarrow \psi=\varphi^{-1},$$ thereby particle’s invariant Hamilton function could be defined by only one scalar function $\varphi(x)$. Taking into account the last relation, let us write down the Hamilton function in the final form: $$\label{EQ7 }
H(x,P)=\frac{1}{2} \left[\varphi^{-1}(x)(P,P)-\varphi(x) \right]=0,$$ and from the canonical equations we obtain the relation between the generalized momentum and particle velocity vector: $$\label{EQ10} P^i=\varphi \frac{dx^i}{ds}.$$ From the definition it follows that the generalized momentum vector is timelike: $$\label{EQ8} (P,P)=\varphi^2.$$ Let us notice useful for future reasoning a relation which is a consequence of , and :
$$\label{EQ9}
[H,P^{k} ]=\nabla ^{k} \varphi \equiv g^{ik} \partial _{i} \varphi.$$
The Lagrange Equations Of Motion
--------------------------------
From the second group of the canonical equations we obtain the equations of motion in Lagrange forulation [@Yubook1]:
$$\label{EQ12}
\frac{d^{2} x^{i} }{ds^{2} } +\Gamma _{jk}^{i} \frac{dx^{j} }{ds} \frac{dx^{k} }{ds} =\partial _{,k} \ln |\varphi|{\rm {\mathcal P}}^{ik} ,$$
where:
$$\label{EQ13}
{\rm {\mathcal P}}^{ik} ={\rm {\mathcal P}}^{ki} =g^{ik} -u^{i} u^{k}$$
is a tensor of an orthogonal projection to the direction $u$, such that:
$$\label{EQ14}
{\rm {\mathcal P}}^{ik} u_{k} \equiv 0;\quad {\rm {\mathcal P}}^{ik} g_{ik} \equiv 3.$$
From these relations and the Lagrange equations it follows the strict consequence of velocity and acceleration vectors orthogonality:
$$\label{EQ15}
g_{ik} u^{i} \frac{du^{k} }{ds} \equiv 0.$$
Let us note that the Lagrange equations of motion (\[EQ12\]) are invariant relative to a sign of the scalar function $\varphi(x)$: $$\label{EQ16a}
\varphi(x)\rightarrow -\varphi(x).$$ The Hamilton function (\[EQ7 \]) at its zero normalization $H\rightarrow -H$ is also invariant relative to the transformation (\[EQ16a\]). Therefore from the relations , , and the Lagrange equations it follows that the quadrate of $\varphi $ scalar has a meaning of a quadrate of *the efficient inert mass of particle, $m_{*} $, in a scalar field*:
$$\label{EQ16}
\varphi^2 =m_{*}^2 .$$
Let us note that a following action function formally coinciding with the Lagrange function of a relativistic particle with a rest mass $\varphi$ in a gravitational field corresponds to the cited choice of the Hamilton function: $$\label{EQ17}
S=\int m_* ds,$$ where an effective mass $m_*$ due to invariance of the Lagrange equations relative to the transformation (\[EQ16a\]) can be defined as $m_*=\varphi$, or $m_*=|\varphi|$. In this article we resolve the relation (\[EQ16\]) by next form: $$\label{phi=m}
m_*=\varphi,$$ as opposed to how it wa proposed in the previous articles: $m_*=|\varphi|$. Thus at (\[phi=m\]) an effective mass of a particle in general could be negative.
Integrals Of Motion
-------------------
Let us now find conditions of existence of canonical equations’ of state line integral which is connected to particle’s total energy and momentum. For that let us calculate the scalar product’s $(\xi ,P)$ full derivative by the canonical parameter. Using the canonical equations of motion , the normalization ratio , and the relation of the generalized momentum to the kinematic one , we find:
$$\label{EQ18}
\frac{d(\xi ,P)}{ds} =\frac{1}{m_{*} } P^{i} P^{k} \mathop{L}\limits_{\xi } g_{ik} +\mathop{L}\limits_{\xi } m_{*} ,$$
where $\mathop{L}\limits_{\xi } $ is a Lie derivative by the direction $\xi $ [^6]. Assuming further
$$\label{EQ19}
\frac{d(\xi ,P)}{ds} =0\Leftrightarrow (\xi ,P)={\kern 1pt} {\rm Const}{\kern 1pt} ,$$
with an account of arbitrariness of the generalized momentum’s vector and its normalization ratio we obtain the conditions of the equation fulfillment:
$$\label{EQ20}
\mathop{L}\limits_{\xi } g_{ik} =\sigma g_{ik} \Rightarrow \sigma =-\mathop{L}\limits_{\xi } \ln |m_{*} |,$$
where $\sigma$ is a arbitrary scalar. Substituting this result back to the relation , we find the necessary and sufficient conditions of existence of the canonical equations’ line integral (see e.g. [@Yubook1]):
$$\label{EQ21}
\mathop{L}\limits_{\xi } m_{*} g_{ik} =0.$$
Thus *to exist the line integral, it is enough and sufficient that conformal corresponding space with metrics $m_{*} g_{ik} $ allows a group of motions with the Killing vector $\xi $*. Let us note that line integrals have the meaning of total momentum (at spacelike vector $\xi $) or total energy (at timelike vector $\xi $).
The Choice Of mass Function
---------------------------
There emerges the question of choice of function $m_{*} (\Phi )$. Let us highlight one important circumstance yet not concretizing this function. Let us consider statistical fields $g_{ik} $ and $\Phi $, allowing timelike Killing vector $\xi ^{i} =\delta _{4}^{i} $, when particle’s total energy $P_{4} $ is conserved. Further let us consider the reference frame in which $g_{\alpha 4} =0$, so that coordinate $x^{4} $ coincides with a world time $t$. Then from the relations of connection between the kinematic velocity $u^{i} $ and vector of particle’s total momentum $P_{i} $ if follows:
$$\label{EQ22}
P_{4} ds=m_{*} dt,$$
where $P_{4} =E_{0} ={\kern 1pt} {\rm Const}{\kern 1pt} >0$ is a full energy of a charged particle. Therefore if we want to conserve the same orientation of world and proper time (i.e. $dt/ds>0$), it is required to choose such a mass function which always stay nonnegative. In the article we nonetheless remove the requirement of positiveness of an effective mass of particle.[^7]
Further, from one hand, in absence of a scalar field or, more precisely, in a constant scalar field, mass function should change to particle rest mass, $m\ge 0$. From the other hand, the Lagrange equations in case of a weak scalar field and small velocities should change to classical equations of motion in a scalar field: $$\label{weak_F}
m\frac{d^2x^\alpha}{dt^2}=-q\partial_\alpha\Phi.$$ Hence following from the correspondence principle we should have:
$$\label{EQ25}
\varphi(0)=m;\quad \lim\limits_{\Phi=0}\left( \frac{\varphi_{,k}}{\varphi}\right)=q\Phi _{,k} ,$$
where $\varphi=\varphi(\Phi)$. Supposing $q\Phi=f$, let us rewrite in a equivalent form: $$\label{m_0}
\varphi(0)=m;\quad \lim\limits_{f=0}\frac{d\varphi}{df}=1,$$ whence mean that at small values of the scalar potential $\Phi $ function $m_{*} (\Phi )$ should have an expansion of form:
$$\label{EQ26}
m_{*} (\Phi ){\rm \simeq } m+q\Phi+\ldots.$$
To this condition corresponds the linear function $$\label{m_*}
%
m_{*} (\Phi )=m+q\Phi,$$ as well as function used in the cited papers: $$\label{|m|}
m_{*} (\Phi )=|m+q\Phi|.$$ Further we will choose particle’s effective mass function in form (\[m\_\*\]). Then according to(\[EQ10\]) we obtain the relation between the particle’s generalized momentum and the kinematic velocity: $$\label{Pu}
P^i=m_*u^i=(m+q\Phi)u^i;$$ the normalization ratio for the generalized momentum vector takes form $$\label{norm_m}
(P,P)=m_*^2=(m+q\Phi)^2$$ and has an identical form for mass function (\[m\_\*\]), as well as for mass function (\[|m|\]).
To refine the meaning of introduced kinematic and dynamic values let us consider a problem of scalar charged particle’s movement in the Minkovsky space in a statistical scalar field, which potential depends only on one coordinate, $x^{1} =x$, and let for the simplicity $m_{*} =m+q\Phi =x$. This we have 3 Killing vectors - one of which is timelike and two are spacelike: \#1\#2 $$\stackunder{1}{\xi}^i=\delta _{4}^{i};\; \stackunder{2}{\xi}^i=\delta _{2}^{i} ;\; \stackunder{3}{\xi}^i =\delta _{3}^{i} .$$ Correspondingly to these vectors there exist three linear integrals of motion:
$$\begin{aligned}
\label{EQ30}
P_{2} =P_{2}^{0} = {\rm Const} ;\; P_{3} =P_{3}^{0} = {\rm Const};\nonumber\\
P_{4} =P_{4}^{0} = {\rm Const}.\end{aligned}$$
Let for the simplicity $P_{2} =P_{3} =0$. Then with an account of the normalization ratio we obtain from the canonical equations of motions the nontrivial one: $$\begin{aligned}
\label{EQ31}
\frac{P^1}{m_*}=u^1\equiv\frac{dx}{ds} =\mp \frac{\sqrt{(P^0_{4})^{2} -x^{2}}}{x }.\\
\label{EQ31a}
u^4\equiv\frac{dt}{ds}\equiv \frac{P_4}{m_*}=\frac{P^0_4}{x}\Rightarrow\end{aligned}$$ $$\label{Eqx}
\frac{dx}{dt}=\mp \frac{1}{P^0_4}\sqrt{(P^0_4)^2-x^2},$$ where signs $\mp$ correspond to a movement to the left or to the right. Integrating (\[Eqx\]) with the initial condition $x(0)=0$, we find the solution: $$x=\mp P^0_4\sin(t/P^0_4).$$ Thus a particle performs harmonic oscillations by clocks of the world time with an amplitude $P^0_4$ and frequency $P^0_4$. However at that, following (\[EQ31a\]) kinematic velocity’s components $u^i$ in moments of time $t=\pi k$ undergo discontinuities of the second kind and change the sign. However components of the generalized momentum, $P_i$, as well as components of the kinematic momentum, $p^i$, being the observed variables defined by the standard procedure as $$\label{p_k}
p^i=m_*\frac{dx^i}{ds}\equiv P^i,$$ remain continuous functions and kinematic momentum’s component $p^4$ does not change the sign, remaining constant. Nevertheless, the sign of microscopic proper time’s relation to the world time is changed. Let us notice, that this outcome is not dependent on the massive function’s choice method; the same outcome has been obtained in [@YuNewScalar1], where positively defined mass function had been being used. Therefore reasonings cited in more recent Author’s papers in favor of nonnegativeness of the mass function and formula (\[|m|\]) on the basis of the Lagrange equations’ of motion analysis, are not strict. Apparently, cited example of particle interaction with a scalar field presents the realization of A.Vlasov [@Vlasov] concept about the possibility of discontinuity of the relation of microscopic proper time of the particle with the world macroscopic time.
Quantum equations
-----------------
Let us note that with a use of the standard procedure of quantum equations’ obtainment from the classical Hamilton function it is required to make a change in the function:[^8] $$\label{quant_trans}
P_i\rightarrow i\hbar \frac{\partial}{\partial x^i},$$ As a result of covariant generalization, Hamilton operator can be obtained from the Hamilton function: $$\label{Hamiltonian}
\hat{{\rm H}}=-m_*^{-1}(\hbar^2 g^{ik}\nabla_i\nabla_k + m^2_*).$$ Thus for a free massive scalar field we could have obtained the wave equations in form of standard Klein-Gordon equations with the only difference that bosons rest mass should be changed by the effective mass: $$\label{free_bozon}
(\square+m_*^2/\hbar^2)\Psi=0,$$ and for free fermions we could have obtained corresponding Dirac equations: $$\label{Dirac_eq}
(\hbar\gamma^i\nabla_i+m_*)\Psi=0,$$ where $\gamma$ are spinors.
Let us note that from (\[free\_bozon\]) having substituted $\Psi=\Phi$ and chosen simplest mass function $m_*=|q\Phi|$ it right away follows an equation of the free scalar field with a cubic nonlinearity: $$\label{Phi3}
\square\Phi+q^2/\hbar^2\Phi^3=0.$$ Thus constant of self-action in a scalar field’s cubic equation takes quite defined meaning: $$\lambda=\frac{q^2}{\hbar^2}.$$
[15]{} A.A. Starobinsky, Phys. Lett. B [**91 (1)**]{}, 99 (1980). A.V. Minkevich, Gravitation & Cosmology, [**12**]{}, 11 (2006). Yu.G. Ignat’ev, Russian Physics Journal, [**25**]{}, No 4, 92 (1982). Yu.G. Ignat’ev, Russian Physics Journal, [**26**]{}, No 8, 15 (1983). Yu.G. Ignat’ev, Russian Physics Journal, [**26**]{}, No 8, 19 (1983). Yu.G. Ignat’ev, Russian Physics Journal, [**26**]{}, No 12, 9 (1983). G.G. Ivanov, Russian Physics Journal, [**26**]{}, No 1, 32 (1983). Yu.G. Ignat’ev, R.R. Kuzeev, Ukr. Fiz. J. [**29**]{}, 1021 (1984). Yu.G. Ignatev and A.A. Popov, Actrophysics and Space Science, [**163**]{}, 153 (1990); arXiv:1101.4303v1 \[gr-qc\]. Yu. Ignat’ev, R. Miftakhov, Gravitation & Cosmology, [**12**]{}, 179 (2006); arXiv:1011.5774\[gr-qc\]. Yu. G. Ignat’ev, Russian Physics Journal, [**55**]{}, No 2, 166 (2012); DOI: 10.1007/s11182-012-9790-9. Yu. G. Ignat’ev, Russian Physics Journal, [**55**]{}, No 5, 550 (2012); DOI: 10.1007/s11182-012-9847-9. Yu. G. Ignat’ev, Russian Physics Journal, [**55**]{}, No 11, 1345 (2013); arXiv:1307.2509 \[gr-qc\]. Yurii G. Ignatyev, Relativistic Kinetic Theory of Nonequilibrium Processes in Gravitational Fields. Kazan, Foliant-Press, – 2010; http://rgs.vniims.ru/books/const.pdf. Yurii G. Ignatyev, The Nonequilibrium Universe: The Kinetics Models of the Cosmological Evolution, Kazan: Kazan University Press, 2013; http://www.stfi.ru/archive\_rus/2013\_2\_ Ignatiev.pdf Yu,G. Ignat’ev, Spase, Time and Foudamental Interections, No 1, 47 (2014) (In Russian). A.Z.Petrov, New Methods in General Theory of Gravitation. Moskow, Nauka, 1966. A.A. Vlasov, Statistical Distribution Functions. Moskow, Nauka, 1966.
[^1]: see e.g., A.V.Minkevich [@Minkevich].
[^2]: see also monographes [@Yubook1; @Yubook2].
[^3]: see also review [@YuSTFI]
[^4]: see e.g. [@YuMif]
[^5]: Here and then the generic unit system is accepted $G=c=\hbar =1$
[^6]: see e.g. [@Petrov].
[^7]: It should be noted that as early as A.A. Vlasov in his fundamental work [@Vlasov] supposed a possibility of discontinuity of a standard connection between these times.
[^8]: In this place we temporarily depart from universal system of units, in which $\hbar=1$.
|
---
author:
- |
B. Baller$^a$, C. Bromberg$^b$, N. Buchanan$^c$, F. Cavanna$^d$, H. Chen$^e$, E. Church$^d$, V. Gehman$^f$, H. Greenlee$^a$, E. Guardincerri$^g$, B. Jones $^h$, T. Junk$^a$, T. Katori$^h$, M. Kirby$^a$, K. Lang$^i$, B. Loer$^a$, A. Marchionni$^a$, T. Maruyama$^j$, C. Mauger$^g$, A. Menegolli$^k$, D. Montanari$^a$, S. Mufson$^l$, B. Norris$^a$, S. Pordes$^a$, J. Raaf$^a$, B. Rebel$^a$, R. Sanders$^a$, M. Soderberg$^{a,m}$, J. St. John$^n$, T. Strauss$^o$, A. Szelc$^d$, C. Touramanis$^p$, C. Thorn$^e$, J. Urheim$^l$, R. Van de Water$^g$, H. Wang$^q$, B. Yu$^e$, M. Zuckerbrot$^a$\
Fermi National Accelerator Laboratory, Batavia, IL 60510, USA\
Michigan State University, East Lansing, MI 48824, USA\
Colorado State University, Fort Collins, CO 80523, USA\
Yale University, New Haven, CT 06520, USA\
Brookhaven National Laboratory, Upton, NY 11973, USA\
Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA\
Los Alamos National Laboratory, Los Alamos, NM 87545, USA\
Massachusetts Institute of Technology, Cambridge, MA 02139, USA\
University of Texas at Austin, TX 78712, USA\
High Energy Accelerator Research Organization, KEK, Tsukuba, Ibaraki 305-0801, Japan\
Istituto Nazionale di Fisica Nucleare, Pavia 6-27100, Italy\
Indiana University, Bloomington, IN 47405, USA\
Syracuse University, NY 13210, USA\
University of Cincinnati, Cincinnati, OH 45220, USA\
University of Bern, 3012 Bern, Switzerland\
University of Liverpool, Liverpool, Merseyside L69 3BX, United Kingdom\
University of California Los Angeles, Los Angeles, CA 90095, USA
bibliography:
- 'sumbib.bib'
title: Liquid Argon Time Projection Chamber Research and Development in the United States
---
Introduction {#sec:Introduction}
============
Physics Requirements for Neutrino and Dark Matter Experiments {#sec:PhysicsRequirements}
=============================================================
Argon Purity {#sec:Purity}
============
Cryogenic Systems {#sec:Cryogenics}
=================
TPC and High Voltage {#sec:TPCHV}
====================
Electronics, Data Acquisition, and Triggering {#sec:Electronics}
=============================================
Scintillation Light Detection {#sec:Photons}
=============================
Calibration and Test Beams {#sec:TestBeams}
==========================
Software {#sec:Software}
========
World Wide R&D Efforts {#sec:World}
======================
Summary {#sec:Summary}
=======
Acknowledgments
===============
The speakers at this workshop represented many colleagues from both collaborations and local university groups. The work they presented was supported by a variety of funding agencies. The following statements of acknowledgement were supplied by the speakers and have not been edited.
We thank the ICARUS collaboration for sharing their experiences and lessons learned in building and operating the first large LArTPC. The High Energy Astrophysics Group at Indiana University is supported by the U.S. Department of Energy Office of Science with grant DE-FG02-91ER40661 to Indiana University and LBNE project funding from Brookhaven National Laboratory with grant BNL 240296 to Indiana University. The LArIAT collaboration is supported by the U.S. Department of Energy Office of Science and the National Science Foundation. The CAPTAIN detector has been designed and is being built by the Physics and the Theory divisions of Los Alamos National Laboratory under the auspices of the LDRD program. T. Strauss spoke on behalf of the Albert Einstein Center, Laboratory of High Energy Physics of the University of Bern. The MicroBooNE and LBNE collaborations have participated in the development of cold electronics as supported by the U.S. Department of Energy Office of Science.
|
---
abstract: 'Gaussian random fields (GRF) are a fundamental stochastic model for spatiotemporal data analysis. An essential ingredient of GRF is the covariance function that characterizes the joint Gaussian distribution of the field. Commonly used covariance functions give rise to fully dense and unstructured covariance matrices, for which required calculations are notoriously expensive to carry out for large data. In this work, we propose a construction of covariance functions that result in matrices with a hierarchical structure. Empowered by matrix algorithms that scale linearly with the matrix dimension, the hierarchical structure is proved to be efficient for a variety of random field computations, including sampling, kriging, and likelihood evaluation. Specifically, with $n$ scattered sites, sampling and likelihood evaluation has an $O(n)$ cost and kriging has an $O(\log n)$ cost after preprocessing, particularly favorable for the kriging of an extremely large number of sites (e.g., predicting on more sites than observed). We demonstrate comprehensive numerical experiments to show the use of the constructed covariance functions and their appealing computation time. Numerical examples on a laptop include simulated data of size up to one million, as well as a climate data product with over two million observations.'
author:
- 'Jie Chen[^1]'
- 'Michael L. Stein[^2]'
bibliography:
- 'reference.bib'
---
Keywords {#keywords .unnumbered}
========
Gaussian sampling; Kriging; Maximum likelihood estimation; Hierarchical matrix; Climate data
Introduction
============
A Gaussian random field (GRF) $Z(\bm{x}):\real^d\to\real$ is a random field where all of its finite-dimensional distributions are Gaussian. Often termed as *Gaussian processes*, GRFs are widely adopted as a practical model in areas ranging from spatial statistics [@Stein1999], geology [@Chiles2012], computer experiments [@Koehler1996], uncertainty quantification [@Smith2013], to machine learning [@Rasmussen2006]. Among the many reasons for its popularity, a computational advantage is that the Gaussian assumption enables many computations to be done with basic numerical linear algebra.
Although numerical linear algebra [@Golub1996] is a mature discipline and decades of research efforts result in highly efficient and reliable software libraries (e.g., BLAS [@Goto2008] and LAPACK [@Anderson1999])[^3], the computation of GRF models cannot overcome a fundamental scalability barrier. For a collection of $n$ scattered sites $\bm{x}_1$, $\bm{x}_2$, …, $\bm{x}_n$, the computation typically requires $O(n^2)$ storage and $O(n^2)$ to $O(n^3)$ arithmetic operations, which easily hit the capacity of modern computers when $n$ is large. In what follows, we review the basic notation and a few computational components that underlie this challenge.
Denote by $\mu(\bm{x}):\real^d\to\real$ the mean function and $k(\bm{x},\bm{x}'):\real^d\times\real^d\to\real$ the covariance function, which is (strictly) positive definite. Let $X=\{\bm{x}_i\}_{i=1}^n$ be a set of sampling sites and let $\bm{z}=[Z(\bm{x}_1),\ldots,Z(\bm{x}_n)]^T$ (column vector) be a realization of the random field at $X$. Additionally, denote by $\bm{\mu}$ the mean vector with elements $\mu_i=\mu(\bm{x}_i)$ and by $K$ the covariance matrix with elements $K_{ij}=k(\bm{x}_i,\bm{x}_j)$.
Sampling
: Realizing a GRF amounts to sampling the multivariate normal distribution $\mathcal{N}(\bm{\mu},K)$. To this end, one performs a matrix factorization $K=GG^T$ (e.g., Cholesky), samples a vector $\bm{y}$ from the standard normal, and computes $$\label{eqn:sampling}
\bm{z}=\bm{\mu}+G\bm{y}.$$
Kriging
: Kriging is the estimation of $Z(\bm{x}_0)$ at a new site $\bm{x}_0$. Other terminology includes *interpolation*, *regression*[^4], and *prediction*. The random variable $Z(\bm{x}_0)$ admits a conditional distribution $\mathcal{N}(\mu_0,\sigma_0^2)$ with $$\label{eqn:kriging}
\mu_0=\mu(\bm{x}_0)+\bm{k}_0^TK^{-1}(\bm{z}-\bm{\mu})
\quad\text{and}\quad
\sigma_0^2=k(\bm{x}_0,\bm{x}_0)-\bm{k}_0^TK^{-1}\bm{k}_0,$$ where $\bm{k}_0$ is the column vector $[k(\bm{x}_1,\bm{x}_0),k(\bm{x}_2,\bm{x}_0),\ldots,k(\bm{x}_n,\bm{x}_0)]^T$.
Log-likelihood
: The log-likelihood function of a Gaussian distribution $\mathcal{N}(\bm{\mu},K)$ is $$\label{eqn:loglik}
\mathcal{L}=-\frac{1}{2}(\bm{z}-\bm{\mu})^TK^{-1}(\bm{z}-\bm{\mu})-\frac{1}{2}\log\det K-\frac{n}{2}\log2\pi.$$ The log-likelihood $\mathcal{L}$ is a function of $\bm{\theta}\in\real^p$ that parameterizes the mean function $\mu$ and the covariance function $k$. The evaluation of $\mathcal{L}$ is an essential ingredient in maximum likelihood estimation and Bayesian inference.
A common characteristic of these examples is the expensive numerical linear algebra computations: Cholesky-like factorization in , linear system solutions in and , and determinant computation in . In general, the covariance matrix $K$ is dense and thus these computations have $O(n^2)$ memory cost and $O(n^3)$ arithmetic cost. Moreover, a subtlety occurs in the kriging of more than a few sites. In dense linear algebra, a preferred approach for solving linear systems is not to form the matrix inverse explicitly; rather, one factorizes the matrix as a product of two triangular matrices with $O(n^3)$ cost, followed by triangular solves whose costs are only $O(n^2)$. Then, if one wants to krige $m=O(n)$ sites, the formulas in , particularly the variance calculation, have a total cost of $O(n^2m)=O(n^3)$. This cost indicates that speeding up matrix factorization alone is insufficient for kriging, because $m$ vectors $\bm{k}_0$ create another computational bottleneck.
Existing Approaches
-------------------
Scaling up the computations for GRF models has been a topic of great interest in the statistics community for many years and has recently attracted the attention of the numerical linear algebra community. Whereas it is not the focus of this work to extensively survey the literature, we discuss a few representative approaches and their pros and cons.
A general idea for reducing the computations is to restrict oneself to covariance matrices $K$ that have an exploitable structure, e.g., sparse, low-rank, or block-diagonal. Covariance tapering [@Furrer2006; @Kaufman2008; @Wang2011; @Stein2013a] approximates a covariance function $k$ by multiplying it with another one $k_{\text{t}}$ that has a compact support. The resulting compactly supported function $kk_{\text{t}}$ potentially introduces sparsity to the matrix. However, often the appropriate support for statistical purposes is not narrow, which undermines the use of sparse linear algebra to speed up computation. Low-rank approximations [@Cressie2008; @Eidsvik2012] generally approximate $K$ by using a low-rank matrix plus a diagonal matrix. In many applications, such an approximation is quite limited, especially when the diagonal component of $K$ does not dominate the small-scale variation of the random field [@Stein2008; @Stein2014]. In machine learning under the context of kernel methods, a number of randomized low-rank approximation techniques were proposed (e.g., Nyström approximation [@Drineas2005] and random Fourier features [@Rahimi2007]). In these methods, often the rank may need to be fairly large relative to $n$ for a good approximation, particularly in high dimensions [@Huang2014]. Moreover, not every low-rank approximation can krige $m=O(n)$ sites efficiently. The block-diagonal approximation casts an artificial independence assumption across blocks, which is unappealing, although this simple approach can outperform covariance tapering and low-rank methods in many circumstances [@Stein2008; @Stein2014].
There also exists a rich literature focusing on only the parameter estimation of $\bm{\theta}$. Among them, spectral methods [@Whittle1954; @Guyon1982; @Dahlhaus1987] deal with the data in the Fourier domain. These methods work less well for high dimensions [@Stein1995] or when the data are ungridded [@Fuentes2007]. Several methods focus on the approximation of the likelihood, wherein the log-determinant term may be approximated by using Taylor expansions [@Zhang2006] or Hutchinson approximations [@Aune2014; @Han2017; @Dong2017; @Ubaru2017]. The composite-likelihood approach [@Vecchia1988; @Stein2004a; @Caragea2007; @Varin2011] partitions $X$ into subsets and expands the likelihood by using the law of successive conditioning. Then, the conditional likelihoods in the product chain are approximated by dropping the conditional dependence on faraway subsets. This approach is often competitive. Yet another approach is to solve unbiased estimating equations [@Anitescu2012; @Stein2013; @Anitescu2017] instead of maximizing the log-likelihood $\mathcal{L}$. This approach rids the computation of the determinant term, but its effectiveness relies on fast matrix-vector multiplications [@Chen2014b] and effective preconditioning of the covariance matrix [@Stein2012a; @Chen2013].
Recently, a multi-resolution approach [@Katzfuss2017] based on successive conditioning was proposed, wherein the covariance structure is approximated in a hierarchical manner. The remainder of the approximation at the coarse level is filled by the finer level. This approach shares quite a few characteristics with our approach, which falls under the umbrella of “hierarchical matrices” in numerical linear algebra. Detailed connections and distinctions are drawn in Section \[sec:compare.hierarchical.matrix\].
Proposed Approach
-----------------
In this work, we take a holistic view and propose an approach applicable to the various computational components of GRF. The idea is to construct covariance functions that render a linear storage and arithmetic cost for (at least) the computations occurring in to . Specifically, for any (strictly) positive definite function $k(\cdot,\cdot)$, which we call the “base function,” we propose a recipe to construct (strictly) positive definite functions $k_{\text{h}}(\cdot,\cdot)$ as alternatives. The base function $k$ is not necessarily stationary. The subscript “h” standards for “hierarchical,” because the first step of the construction is a hierarchical partitioning of the computation domain. With the subscript “h”, the storage of the corresponding covariance matrix $K_{\text{h}}$, as well as the additional storage requirement incurred in matrix computations, is $O(n)$. Additionally,
1. the arithmetic costs of matrix construction $K_{\text{h}}$, factorization $K_{\text{h}}=G_{\text{h}}G_{\text{h}}^T$, explicit inversion $K_{\text{h}}^{-1}$, and determinant calculation $\det(K_{\text{h}})$ are $O(n)$;
2. for any dense vector $\bm{y}$ of matching dimension, the arithmetic costs of matrix-vector multiplications $G_{\text{h}}\bm{y}$ and $K_{\text{h}}^{-1}\bm{y}$ are $O(n)$; and
3. for any dense vector $\bm{w}$ of matching dimension, the arithmetic costs of the inner product $\bm{k}_{\text{h},0}^T\bm{w}$ and the quadratic form $\bm{k}_{\text{h},0}^TK_{\text{h}}^{-1}\bm{k}_{\text{h},0}$ are $O(\log n)$, provided that an $O(n)$ preprocessing is done independently of the new site $\bm{x}_0$.
The last property indicates that the overall cost of kriging $m=O(n)$ sites and estimating the uncertainties is $O(n\log n)$, which dominates the preprocessing $O(n)$.
The essence of this computationally attractive approach is a special covariance structure that we coin “recursively low-rank.” Informally speaking, a matrix $A$ is recursively low-rank if it is a block-diagonal matrix plus a low-rank matrix, with such a structure recursive in the main diagonal blocks. The “recursive” part mandates that the low-rank factors share bases across levels. The matrix $K_{\text{h}}$ resulting from the proposed covariance function $k_{\text{h}}$ is a symmetric positive definite version of recursively low-rank matrices. Interesting properties of the recursively low-rank structure of $A$ include that $A^{-1}$ admits exactly the same structure, and that if $A$ is symmetric positive definite, it may be factorized as $GG^T$ where $G$ also admits the same structure, albeit not being symmetric. These are the essential properties that allow for the development of $O(n)$ algorithms throughout. Moreover, the recursively low-rank structure is carried out to the out-of-sample vector $\bm{k}_{\text{h},0}$, which makes it possible to compute inner products $\bm{k}_{\text{h},0}^T\bm{w}$ and quadratic forms $\bm{k}_{\text{h},0}^TK_{\text{h}}^{-1}\bm{k}_{\text{h},0}$ in an $O(\log n)$ cost, asymptotically lower than $O(n)$.
This matrix structure is closely connected to the rich literature of fast kernel approximation methods in scientific computing, reflected through a similar hierarchical framework but fine distinctions in design choices. A holistic design that aims at fitting the many computational components of GRF simultaneously however narrows down the possible choices and rationalizes the one that we take. After the presentation of the technical details, we will discuss in depth the subtle distinctions with many related hierarchical matrix approaches in Section \[sec:compare.hierarchical.matrix\].
Recursively Low-Rank Covariance Function
========================================
Let $k:S\times S\to\real$ be positive definite for some domain $S$; that is, for any set of points $\bm{x}_1,\ldots,\bm{x}_n\in S$ and any set of coefficients $\alpha_1,\ldots,\alpha_n\in\real$, the quadratic form $\sum_{ij}\alpha_i\alpha_jk(\bm{x}_i,\bm{x}_j)\ge0$. We say that $k$ is *strictly* positive definite if the quadratic form is strictly greater than $0$ whenever the $\bm{x}$’s are distinct and not all of the $\alpha_i$’s are 0. Given any $k$ and $S$, in this section we propose a recipe for constructing functions $k_{\text{h}}$ that are (strictly) positive definite if $k$ is so. We note the often confusing terminology that a strictly positive definite function always yields a positive definite covariance matrix for $n$ distinct observations, whereas, for a positive definite function, this matrix is only required to be positive semi-definite.
Some notations are necessary. Let $X$ be an ordered list of points in $S$. We will use $k(X,X)$ to denote the matrix with elements $k(\bm{x},\bm{x}')$ for all pairs $\bm{x},\bm{x}'\in X$. Similarly, we use $k(X,\bm{x})$ and $k(\bm{x},X)$ to denote a column and a row vector, respectively, when one of the arguments passed to $k$ contains a singleton $\{\bm{x}\}$. The construction of $k_{\text{h}}$ is based on a hierarchical partitioning of $S$. For simplicity, let us first consider a partitioning with only one level. Let $S$ be partitioned into disjoint subdomains $S_1,\ldots,S_t$ such that $S=S_1\cup\cdots\cup S_t$. Let $\ud{X}$ be a set of $r$ distinct points in $S$. If $k(\ud{X},\ud{X})$ is invertible, define $$\label{eqn:k.one.level}
k_{\text{h}}(\bm{x},\bm{x}')=
\begin{cases}
k(\bm{x},\bm{x}'), & \text{if $\bm{x},\bm{x}'\in S_j$ for some $j$},\\
k(\bm{x},\ud{X})k(\ud{X},\ud{X})^{-1}k(\ud{X},\bm{x}'), & \text{otherwise}.
\end{cases}$$ In words, states that the covariance for a pair of sites $\bm{x},\bm{x}'$ is equal to $k(\bm{x},\bm{x}')$ if they are located in the same subdomain; otherwise, it is replaced by the Nyström approximation $k(\bm{x},\ud{X})k(\ud{X},\ud{X})^{-1}k(\ud{X},\bm{x}')$. The Nyström approximation is always no greater than $k(\bm{x},\bm{x}')$ and when $k$ is strictly positive definite, it attains $k(\bm{x},\bm{x}')$ only when either $\bm{x}$ or $\bm{x}'$ belongs to $\ud{X}$. Following convention, we call the $r$ points in $\ud{X}$ *landmark points*. Throughout this work, we will reserve underscores to indicate a list of landmark points. The term “low-rank” comes from the fact that a matrix generated from Nyström approximation generically has rank $r$ (when $n\ge r$), regardless of how large $n$ is.
The positive definiteness of $k_{\text{h}}$ follows a simple Schur-complement split. Furthermore, we have a stronger result when $k$ is assumed to be strictly positive definite; in this case, $k_{\text{h}}$ carries over the strictness. We summarize this property in the following theorem, whose proof is given in the appendix.
\[thm:k.one.level\] The function $k_{\text{h}}$ defined in is positive definite if $k$ is positive definite and $k(\ud{X},\ud{X})$ is invertible. Moreover, $k_{\text{h}}$ is strictly positive definite if $k$ is so.
We now proceed to hierarchical partitioning. Such a partitioning of the domain $S$ may be represented by a partitioning tree $T$. We name the tree nodes by using lower case letters such as $j$ and let the subdomain it corresponds to be $S_j$. The root is always $j=1$ and hence $S\equiv S_1$. We write $\text{Ch}(j)$ to denote the set of all child nodes of $j$. Equivalently, this means that a (sub)domain $S_j$ is partitioned into disjoint subdomains $S_l$ for all $l\in\text{Ch}(j)$. An example is illustrated in Figure \[fig:tree\], where $S_1=S_2\cup S_3\cup S_4$, $S_2=S_5\cup S_6\cup S_7$, and $S_4=S_8\cup S_9$.
![Domain $S$ and partitioning tree $T$.[]{data-label="fig:tree"}](partition "fig:"){width=".48\linewidth"} ![Domain $S$ and partitioning tree $T$.[]{data-label="fig:tree"}](tree1 "fig:"){width=".48\linewidth"}
We now define a covariance function $k_{\text{h}}$ based on hierarchical partitioning. For each nonleaf node $i$, let $\ud{X}_i$ be a set of $r$ landmark points in $S_i$ and assume that $k(\ud{X}_i,\ud{X}_i)$ is invertible. The main idea is to cascade the definition of covariance to those of the child subdomains. Thus, we recursively define a function $k_{\text{h}}^{(i)}:S_i\times S_i\to\real$ such that if $\bm{x}$ and $\bm{x}'$ belong to the same child subdomain $S_j$ of $S_i$, then $k_{\text{h}}^{(i)}(\bm{x},\bm{x}')=k_{\text{h}}^{(j)}(\bm{x},\bm{x}')$; otherwise, $k_{\text{h}}^{(i)}(\bm{x},\bm{x}')$ resembles a Nyström approximation. Formally, our covariance function $$\label{eqn:k.multilevel.part1}
k_{\text{h}}\equiv k_{\text{h}}^{(1)},$$ where for any tree node $i$, $$\label{eqn:k.multilevel.part2}
k_{\text{h}}^{(i)}(\bm{x},\bm{x}')=
\begin{cases}
k(\bm{x},\bm{x}'), & \text{if $i$ is leaf},\\
k_{\text{h}}^{(j)}(\bm{x},\bm{x}'), & \text{if $\bm{x},\bm{x}'\in S_j$ for some $j\in\text{Ch}(i)$},\\
\psi^{(i)}(\bm{x},\ud{X}_i)k(\ud{X}_i,\ud{X}_i)^{-1}\psi^{(i)}(\ud{X}_i,\bm{x}'), & \text{otherwise}.
\end{cases}$$ The auxiliary function $\psi^{(i)}(\bm{x},\ud{X}_i)$ cannot be the same as $k(\bm{x},\ud{X}_i)$, because positive definiteness will be lost. Instead, we make the following recursive definition when $\bm{x}\in S_i$: $$\label{eqn:k.multilevel.part3}
\psi^{(i)}(\bm{x},\ud{X}_i)=
\begin{cases}
k(\bm{x},\ud{X}_i), & \text{if $\bm{x}\in S_j$ for some $j\in\text{Ch}(i)$ and $j$ is leaf},\\
\psi^{(j)}(\bm{x},\ud{X}_j)k(\ud{X}_j,\ud{X}_j)^{-1}k(\ud{X}_j,\ud{X}_i), & \text{otherwise}.
\end{cases}$$
To understand the definition, we expand the recursive formulas – for a pair of points $\bm{x}\in S_j$ and $\bm{x}'\in S_l$, where $j$ and $l$ are two leaf nodes. If $j=l$, it is trivial that $k_{\text{h}}(\bm{x},\bm{x}')=k(\bm{x},\bm{x}')$. Otherwise, they have a unique least common ancestor $p$. Then, $$\begin{gathered}
k_{\text{h}}(\bm{x},\bm{x}')=k_{\text{h}}^{(p)}(\bm{x},\bm{x}')\\
=\underbrace{k(\bm{x},\ud{X}_{j_1})k(\ud{X}_{j_1},\ud{X}_{j_1})^{-1}k(\ud{X}_{j_1},\ud{X}_{j_2})\cdots k(\ud{X}_{j_s},\ud{X}_{j_s})^{-1}k(\ud{X}_{j_s},\ud{X}_{p})}_{\psi^{(p)}(\bm{x},\ud{X}_p)}k(\ud{X}_{p},\ud{X}_{p})^{-1}\\
\cdot\underbrace{k(\ud{X}_{p},\ud{X}_{l_t})k(\ud{X}_{l_t},\ud{X}_{l_t})^{-1}\cdots k(\ud{X}_{l_2},\ud{X}_{l_1})k(\ud{X}_{l_1},\ud{X}_{l_1})^{-1}k(\ud{X}_{l_1},\bm{x}')}_{\psi^{(p)}(\ud{X}_p,\bm{x}')},\end{gathered}$$ where $(j,j_1,j_2,\ldots,j_s,p)$ is the path in the tree connecting $j$ and $p$ and similarly $(l,l_1,l_2,\ldots,l_t,p)$ is the path connecting $l$ and $p$. The vectors $\psi^{(p)}(\bm{x},\ud{X}_p)$ and $\psi^{(p)}(\ud{X}_p,\bm{x}')$ on the two sides of $k(\ud{X}_{p},\ud{X}_{p})^{-1}$ come from recursively applying .
Similar to Theorem \[thm:k.one.level\], the positive definiteness of $k$ follows from recursive Schur-complement splits across the hierarchy tree. Furthermore, we have that $k_{\text{h}}$ is strictly positive definite if $k$ is so. We summarize the overall result in the following theorem, whose proof is given in the appendix.
\[thm:k.multilevel\] The function $k_{\text{h}}$ defined in – is positive definite if $k$ is positive definite and $k(\ud{X}_i,\ud{X}_i)$ is invertible for all nonleaf nodes $i$. Moreover, $k_{\text{h}}$ is strictly positive definite if $k$ is so.
Recursively Low-Rank Matrix $A$
===============================
An advantage of the proposed covariance function $k_{\text{h}}$ is that when the number of landmark points in each subdomain is considered fixed, the covariance matrix $K_{\text{h}}\equiv k_{\text{h}}(X,X)$ for a set $X$ of $n$ points admits computational costs only linear in $n$. Such a desirable scaling comes from the fact that $K_{\text{h}}$ is a special case of *recursively low-rank matrices* whose computational costs are linear in the matrix dimension. In this section, we discuss these matrices and their operations (such as factorization and inversion). Then, in the section that follows, we will show the specialization of $K_{\text{h}}$ and discuss additional vector operations tied to $k_{\text{h}}$.
Let us first introduce some notation. Let $I=\{1,\ldots,n\}$. The index set $I$ may be recursively (permuted and) partitioned, resulting in a hierarchical formation that resembles the second panel of Figure \[fig:tree\]. Then, corresponding to a node $i$ is a subset $I_i\subset I$. Moreover, we have $I_i=\cup_{j\in\text{Ch}(i)} I_j$ where the $I_j$’s under union are disjoint. For an $n\times n$ real matrix $A$, we use $A(I_j,I_l)$ to denote a submatrix whose rows correspond to the index set $I_j$ and columns to $I_l$. We also follow the Matlab convention and use $:$ to mean all rows/columns when extracting submatrices. Further, we use $|I|$ to denote the cardinality of an index set $I$. We now define a recursively low-rank matrix.
\[def:matrix\] A matrix $A\in\real^{n\times n}$ is said to be *recursively low-rank* with a partitioning tree $T$ and a positive integer $r$ if
1. for every pair of sibling nodes $i$ and $j$ with parent $p$, the block $A(I_i,I_j)$ admits a factorization $$A(I_i,I_j)=U_i\Sigma_pV_j^T$$ for some $U_i\in\real^{|I_i|\times r}$, $\Sigma_p\in\real^{r\times r}$, and $V_j\in\real^{|I_j|\times r}$; and
2. for every pair of child node $i$ and parent node $p$ not being the root, the factors $$U_p(I_i,:)=U_iW_p \quad\text{and}\quad V_p(I_i,:)=V_iZ_p$$ for some $W_p,Z_p\in\real^{r\times r}$.
In Definition \[def:matrix\], the first item states that each off-diagonal block of $A$ is a rank-$r$ matrix. The middle factor $\Sigma_p$ is shared by all children of the same parent $p$, whereas the left factor $U_i$ and the right factor $V_j$ may be obtained through a change of basis from the corresponding factors in the child level, as detailed by the second item of the definition. As a consequence, if $\text{Ch}(i)=\{i_1,\ldots,i_s\}$ and $\text{Ch}(j)=\{j_1,\ldots,j_t\}$, then $$A(I_i,I_j)=
\underbrace{\begin{bmatrix} U_{i_1} \\ \vdots \\ U_{i_s} \end{bmatrix}W_i}_{U_i}
\Sigma_p
\underbrace{Z_j^T\begin{bmatrix} V_{j_1}^T & \cdots & V_{j_t}^T \end{bmatrix}}_{V_j^T}.$$
From now on, we use the shorthand notation $A_{ii}$ to denote a diagonal block $A(I_i,I_i)$ and $A_{ij}$ to denote an off-diagonal block $A(I_i,I_j)$. A pictorial illustration of $A$, which corresponds to the tree in Figure \[fig:tree\], is given in Figure \[fig:matrix\]. Then, $A$ is completely represented by the factors $$\label{eqn:A.factors}
\{A_{ii}, U_i, V_i, \Sigma_p, W_q, Z_q \mid i \text{ is leaf, } p \text{ is nonleaf, } q \text{ is neither leaf nor root}\}.$$ In computer implementation, we store these factors in the corresponding nodes of the tree. See Figure \[fig:tree2\] for an extended example of Figure \[fig:tree\]. Clearly, $A$ is symmetric when $A_{ii}$ and $\Sigma_p$ are symmetric, $U_i=V_i$, and $W_q=Z_q$ for all appropriate nodes $i$, $p$, and $q$. In this case, the computer storage can be reduced by approximately a factor of $1/3$ through omitting the $V_i$’s and $Z_q$’s; meanwhile, matrix operations with $A$ often have a reduced cost, too.
![The matrix $A$ corresponding to the partitioning tree in Figure \[fig:tree\].[]{data-label="fig:matrix"}](matrix){width=".45\linewidth"}
![Data structure for storing $A$. The partitioning tree is the same as that in Figure \[fig:tree\].[]{data-label="fig:tree2"}](tree2_tight-eps-converted-to)
It is useful to note that not all matrix computations concerned in this paper are done with a symmetric matrix, although the covariance matrix is always so. One instance with unsymmetric matrices is sampling, where the matrix is a Cholesky-like factor of the covariance matrix. Hence, in this section, general algorithms are derived whenever $A$ may be unsymmetric, but we note the simplification for the symmetric case as appropriate.
The four matrix operations under consideration are:
1. matrix-vector multiplication $\bm{y}=A\bm{b}$;
2. matrix inversion $\widetilde{A}=A^{-1}$;
3. determinant $\det(A)$; and
4. Cholesky-like factorization $A=GG^T$ (when $A$ is symmetric positive definite).
The detailed algorithms are presented in the appendix. Suffice it to mention here that interestingly, all algorithms are in the form of tree walks (e.g., preorder or postorder traversals) that heavily use the tree data structure illustrated in Figure \[fig:tree2\]. The inversion and Cholesky-like factorization rely on existence results summarized in the following. The proofs of these theorems are constructive, which simultaneously produce the algorithms. Hence, one may find the proofs inside the algorithms given in the appendix.
\[thm:invA\] Let $A$ be recursively low-rank with a partitioning tree $T$ and a positive integer $r$. If $A$ is invertible and additionally, $A_{ii}-U_i\Sigma_pV_i^T$ is also invertible for all pairs of nonroot node $i$ and parent $p$, then there exists a recursively low-rank matrix $\widetilde{A}$ with the same partitioning tree $T$ and integer $r$, such that $\widetilde{A}=A^{-1}$. Following , we denote the corresponding factors of $\widetilde{A}$ to be $$\{\widetilde{A}_{ii}, \widetilde{U}_i, \widetilde{V}_i, \widetilde{\Sigma}_p, \widetilde{W}_q, \widetilde{Z}_q \mid i \text{ is leaf, } p \text{ is nonleaf, } q \text{ is neither leaf nor root}\}.$$
\[thm:cholA\] Let $A$ be recursively low-rank with a partitioning tree $T$ and a positive integer $r$. If $A$ is symmetric, by convention let $A$ be represented by the factors $$\{A_{ii}, U_i, U_i, \Sigma_p, W_q, W_q \mid i \text{ is leaf, } p \text{ is nonleaf, } q \text{ is neither leaf nor root}\}.$$ Furthermore, if $A$ is positive definite and additionally, $A_{ii}-U_i\Sigma_pU_i^T$ is also positive definite for all pairs of nonroot node $i$ and parent $p$, then there exists a recursively low-rank matrix $G$ with the same partitioning tree $T$ and integer $r$, and with factors $$\{G_{ii}, U_i, V_i, \Omega_p, W_q, Z_q \mid i \text{ is leaf, } p \text{ is nonleaf, } q \text{ is neither leaf nor root}\},$$ such that $A=GG^T$.
Covariance Matrix $K_{\text{h}}$ as a Special Case of $A$ and Out-Of-Sample Extension {#sec:out.of.sample}
=====================================================================================
As noted at the beginning of the preceding section, the covariance matrix $K_{\text{h}}=k_{\text{h}}(X,X)$ is a special case of recursively low-rank matrices. This fact may be easily verified through populating the factors of $A$ defined in Definition \[def:matrix\]. Specifically, let $X$ be a set of $n$ distinct points in $S$ and let $X_j=X\cap S_j$ for all (sub)domains $S_j$. To avoid degeneracy assume $X_j\ne\emptyset$ for all $j$. Assign a recursively low-rank matrix $A$ in the following manner:
1. for every leaf node $i$, let $A_{ii}=k(X_i,X_i)$;
2. for every nonleaf node $p$, let $\Sigma_p=k(\ud{X}_p,\ud{X}_p)$;
3. for every leaf node $i$, let $U_i=V_i=k(X_i,\ud{X}_p)k(\ud{X}_p,\ud{X}_p)^{-1}$ where $p$ is the parent of $i$; and
4. for every nonleaf node $p$ not being the root, let $W_p=Z_p=k(\ud{X}_p,\ud{X}_q)k(\ud{X}_q,\ud{X}_q)^{-1}$ where $q$ is the parent of $p$.
Then, one sees that $A=K_{\text{h}}$. Clearly, $A$ is symmetric. Moreover, such a construction ensures that the preconditions of Theorems \[thm:invA\] and \[thm:cholA\] be satisfied.
In this section, we consider two operations with the vector $\bm{v}=k_{\text{h}}(X,\bm{x})$, where $\bm{x}\notin X$ is an out-of-sample (i.e., unobserved site). The quantities of interest are
1. the inner product $\bm{w}^T\bm{v}$ for a general length-$n$ vector $\bm{w}$; and
2. the quadratic form $\bm{v}^T\widetilde{A}\bm{v}$, where $\widetilde{A}$ is a *symmetric* recursively low-rank matrix with the same partitioning tree $T$ and integer $r$ as that used for constructing $k_{\text{h}}$.
For the quadratic form, in practical use $\widetilde{A}=K_{\text{h}}^{-1}$, but the algorithm we develop here applies to a general symmetric $\widetilde{A}$. The inner product is used to compute prediction (first equation of ) whereas the quadratic form is used to estimate standard error (second equation of ).
The detailed algorithms are presented in the appendix. Similar to those in the preceding section, they are organized as tree algorithms. The difference is that both algorithms in this section are split into a preprocessing computation independent of $\bm{x}$ and a separate $\bm{x}$-dependent computation. The preprocessing still consists of tree traversals that visit all nodes of the hierarchy tree, but the $\bm{x}$-dependent computation visits only one path that connects the root and the leaf node that $\bm{x}$ lies in. In all cases, one needs not explicitly construct the vector $\bm{v}$, which otherwise costs $O(n)$ storage.
Cost Analysis {#sec:cost.analysis}
=============
All the recipes and algorithms developed in this work apply to a general partitioning of the domain $S$. As is usual, if the tree is arbitrary, cost analysis of many tree-based algorithms is unnecessarily complex. To convey informative results, here we assume that the partitioning tree $T$ is binary and perfect and the associated partitioning of the point set $X$ is balanced. That is, with some positive integer $n_0$, $|X_i|=n_0$ for all leaf nodes $i$. Then, with a partitioning tree of height $h$, the number of points is $|X|=n=n_02^h$. We assume that the number of landmark points, $r$, is equal to $n_0$ for simplicity.
Since the factors $A_{ii}$, $U_i$ and $V_i$ are stored in the leaf nodes $i$ and $\Sigma_p$, $W_p$, and $Z_p$ are stored in the nonleaf nodes $p$ (in fact, at the root there is no $W_p$ or $Z_p$), the storage is clearly $$\underbrace{(2^h)(n_0^2)}_{\text{for } A_{ii}}
+\underbrace{2(2^h)(n_0r)}_{\text{for } U_i \text{ and } V_i}
+\underbrace{(2^h-1)(r^2)}_{\text{for } \Sigma_p}
+\underbrace{2(2^h-2)(r^2)}_{\text{for } W_p \text{ and } Z_p}
=O(nr).$$ An alternative way to conclude this result is that the tree has $O(n/r)$ nodes, each of which contains an $O(1)$ number of matrices of size $r\times r$. Therefore, the storage is $O(n/r\times r^2)=O(nr)$. This viewpoint also applies to the additional storage needed when executing all the matrix algorithms, wherein temporary vectors and matrices are allocated. This additional storage is $O(r)$ or $O(r^2)$ per node, hence it does not affect the overall assessment $O(nr)$.
The analysis of the arithmetic cost of each matrix operation is presented in the appendix. In brief summary, matrix construction is $O(n\log n+nr^2)$, matrix-vector multiplication is $O(nr)$, matrix inversion and Cholesky-like factorization are $O(nr^2)$, determinant computation is $O(n/r)$, inner product is $O(r^2\log_2(n/r))$ with $O(nr)$ preprocessing, and quadratic form is $O(r^2\log_2(n/r))$ with $O(nr^2)$ preprocessing.
Connections and Distinctions to Hierarchical Matrices {#sec:compare.hierarchical.matrix}
=====================================================
The proposed recursively low-rank matrix structure builds on a number of previous efforts. For decades, researchers in scientific computing have been keenly developing fast methods for multiplying a dense matrix with a vector, $K\bm{y}$, where the matrix $K$ is defined based on a kernel function (e.g., Green’s function) that resembles a covariance function. Notable methods include the tree code [@Barnes1986], the fast multipole method (FMM) [@Greengard1987; @Sun2001], hierarchical matrices [@Hackbusch1999; @Hackbusch2002; @Boerm2003], and various extensions [@Gimbutas2002; @Ying2004; @Chandrasekaran2006; @Martinsson2007; @Fong2009; @Ho2013; @Ambikasaran2014; @March2015]. These methods were either co-designed, or later generalized, for solving linear systems $K^{-1}\bm{y}$. They are all based on a hierarchical partitioning of the computation domain, or equivalently, a hierarchical block partitioning of the matrix. The diagonal blocks at the bottom level remain unchanged but (some of) the off-diagonal blocks are low-rank approximated. The differences, however, lie in the fine details, including whether all off-diagonal blocks are low-rank approximated or the ones immediately next to the diagonal blocks should remain unchanged; whether the low-rank factors across levels share bases; and how the low-rank approximations are computed. The aim of this work is an approach applicable to as many computational components as possible of GRF. Hence, the aforementioned design details necessarily differ from those for other applications. Moreover, certain compromises may need to be made for a broad coverage; for example, a structure optimal for kriging is out of the question if not generalizable to likelihood calculation. The rationale of our design choice is best conveyed through comparing with related methods. Our work distinguishes from them in the following aspects.
1. We explicitly define the covariance function on $\real^d\times\real^d$, which is shown to be (strictly) positive definite. Whereas the related methods are all understood as matrix approximations, to the best of our knowledge, none of these works considers the underlying kernel function that corresponds to the approximate matrix. The knowledge of the underlying function is important for out-of-sample extensions, because, for example in kriging , one should approximate also the vector $\bm{k}_0$ in addition to the matrix $K$.
One may argue that if $K$ is well approximated (e.g., accurate to many digits), then it suffices to use the nonapproximate $\bm{k}_0$ for computation. It is important to note, however, that the matrix approximations are elementwise, which does not guarantee good spectral approximations. As a consequence, numerical error may be disastrously amplified through inversion, especially when there is no or a small nugget effect. Moreover, using the nonapproximate $\bm{k}_0$ for computation will incur a computational bottleneck if one needs to krige a large number of sites, because constructing the vector $\bm{k}_0$ alone incurs an $O(n)$ cost.
On the other hand, we start from the covariance function and hence one needs not interpret the proposed approach as an approximation. *All the linear algebra computations are exact in infinite precision, including inversion and factorization*. Additionally, positive definiteness is proved. Few methods under comparison hold such a guarantee.
2. A substantial flexibility in the design of methods under comparison is the low-rank approximation of the off-diagonal blocks. If the approximation is algebraic, the common objective is to minimize the approximation error balanced with computational efficiency (otherwise the standard truncated singular value decomposition suffices). Unfortunately, rarely does such a method maintain the positive definiteness of the matrix, which poses difficulty for Cholesky-like factorization and log-determinant computation. A common workaround is some form of compensation, either to the original blocks of the matrix [@Bebendorf2007] or to the Schur complements [@Xia2010a]. Our approach requires no compensation because of the guaranteed positive definiteness.
3. The fine distinctions in matrix structures lead to substantially different algorithms for matrix operations, if even possible. Our structure is almost the same as that of HSS matrices [@Chandrasekaran2006; @Xia2010] and of H$^2$ matrices with weak admissiblity [@Hackbusch2002], but distant from that of tree code [@Barnes1986], FMM [@Greengard1987], H matrices [@Hackbusch1999], and HODLR matrices [@Ambikasaran2014]. Whereas fast matrix-vector multiplications are a common capability of different matrix structures, the picture starts to diverge for solving linear systems: some structures (e.g., HSS) are amenable for direct factorizations [@Chandrasekaran2006a; @Xia2010a; @Li2012; @Wang2013], while the others must apply preconditioned iterative methods. An additional complication is that direct factorizations may only be approximate, and thus if the approximation is not sufficiently accurate, it can serve only as a preconditioner but cannot be used in a direct method [@Iske2017]. Then, it will be nearly impossible for these matrix structures to perform Cholesky-like factorizations accurately.
In this regard, our matrix structure is the most clean. Thanks to the property that the matrix inverse and the Cholesky-like factor admit the same structure as that of the original matrix, all the matrix operations considered in this work are exact. Moreover, the explicit covariance function also allows for the development of $O(\log n)$ algorithms for computing inner products and quadratic forms, which, to the best of our knowledge, has not been discussed in the literature for other matrix structures.
4. In the proposed approach, the factors are defined by exploiting the base covariance function, as opposed to HSS and H$^2$ approaches where the factors are generally computed through algebraic factorization and approximation. The delicate definition of the factors ensures positive definiteness, which is lacked by the algebraic methods and even by the methods that exploit the base kernel (e.g., @Fong2009). The guarantee of positive definiteness necessitates certain sacrifice in approximation accuracy. Thus, the proposed approach is well suited for GRF but for other applications, such as solving partial differential equations, more specialized methods such as HSS and H$^2$ are preferred.
5. Although most of the methods under this category enjoy an $O(n)$ or $O(n\log^p n)$ (for some small $p$) arithmetic cost, not every one does so. For example, the cost of skeletonization [@Ho2013; @Minden2016] is dimension dependent; in two dimensions it is approximately $O(n^{3/2})$ and in higher dimensions it will be even higher. In general, all these methods are considered matrix approximation methods, and hence there exists a likely tradeoff between approximation accuracy and computational cost. What confounds the approximation is that the low-rank phenomenon exhibited in the off-diagonal blocks fades as the dimension increases [@Ambikasaran2016]. In this regard, it is beneficial to shift the focus from covariance matrices to covariance functions where approximation holds in a more meaningful sense. We conduct experiments to show that predictions and likelihoods are well preserved with the proposed approach.
Practical Considerations {#sec:practical}
========================
So far, we have presented a hierarchical framework for constructing valid covariance functions and revealed their appealing computational consequences. The framework is general but there remain instantiations for specific use. In this section, we discuss details tailored to GRF, a low dimensional use case as opposed to the more general (often high-dimensional) case of reproducing kernel Hilbert space.
Partitioning of Domain
----------------------
For GRF, the sampling sites often reside on a regular grid or a structured (e.g., triangular) mesh. Large spatial datasets with irregular locations commonly occur in remote sensing, although even in this setting, there is usually substantial regularity in the locations due to, for example, the periodicity in a polar-orbiting satellite. When the sites are on a regular grid, a natural choice of the partitioning is axis aligned and balanced. We recommend the following bounding box approach: Begin with the bounding box of the grid, select the longest dimension, cut it into equal halves, and repeat. If the number of grid points along the partitioning dimension in each partitioning is even, the procedure results in a perfect binary tree, whose leaf nodes have exactly the same bounding box volume and the same number of sites. If the number of grid points is odd in some occasion, one shifts the cutting point by half the grid spacing, so that the sampling sites in the middle are not cut.
This bounding box approach straightforwardly generalizes to the mesh or random configuration: Each time the longest dimension of the bounding box is selected and the box is cut into two halves, each of which contains approximately the same number of sampling sites. For random points without exploitable structures, the resulting partitioning tree is known as the k-d tree [@Bentley1975].
Landmark Points
---------------
Assume that the partitioning tree is balanced. As explained in the cost analysis, we consolidate the two parameters, leaf size $n_0$ and the number of landmark points, $r$, into one for convenience. To achieve so, we set the tree height $h$ to be some integer such that the leaf size $n_0=n/2^h$ is greater than or equal to $r$ but less than $2r$. Even if the partitioning is not balanced, the same effect can still be achieved: the recursive partitioning is terminated when each leaf size is $\ge r$ but $<2r$.
The appropriate $r$ is case dependent. There exists a tradeoff between approximation accuracy and computational cost. The larger $r$, the closer $k_{\text{h}}$ is to $k$ but the more expensive is the computation (the cost of matrix-vector multiplication is linear in $r$, whereas those for inversion, Cholesky, inner product, and quadratic forms are all quadratic in $r$). Although there exists analysis (see, e.g., @Drineas2005) on the approximation error of the covariance matrix under Nyström approximation (which is part of our one-level construction), extending it to the error analysis of kriging or likelihood is challenging, let alone to the analysis under the multilevel setting. For empirical evidence, we show later a computational example of the kriging error and the log-likelihood, as $r$ varies. We suggest that in practice, one sets $r$ through balancing the tolerable error (which may be estimated, for example, by using a hold out set) and the computational resources at hand.
The configuration of the landmark points is flexible. Because of the low dimension, a regular grid is feasible. One may set the number of grid points along each dimension to be approximately proportional to the size of the bounding box. An advantage of using regular grids is that the results are deterministic. An alternative is randomization. The landmark points may either be uniformly random within the bounding box, or uniformly sampled from the sampling sites. A later experiment indicates that the random choice yields a worse approximation on average, but the variance is nonnegligible such that sometimes a better approximation is obtained compared with the regular-grid choice.
Numerical Experiments
=====================
In this section, we show a comprehensive set of experiments to demonstrate the practical use of the proposed covariance function $k_{\text{h}}$ for various GRF computations. These computations are centered around simulated data and data from test functions, based on a simple stationary covariance model $k$. In the next section we will demonstrate an application with real-life data and a more realistic nonstationary covariance model.
The base covariance function $k$ in this section is the Matérn model $$\label{eqn:matern}
k(\bm{x},\bm{x}')=
\frac{10^{\alpha}}{2^{\nu-1}\Gamma(\nu)}
\left(\frac{\sqrt{2\nu}\|\bm{r}\|}{\ell}\right)^{\nu}
\bessel_{\nu}\left(\frac{\sqrt{2\nu}\|\bm{r}\|}{\ell}\right)
+10^{\tau}\cdot\bm{1}(\bm{r}=\bm{0})
\quad\text{with}\quad
\bm{r}=\bm{x}-\bm{x}',$$ where $10^{\alpha}$ is the sill, $\ell$ is the range, $\nu$ is the smoothness, and $10^{\tau}$ is the nugget. In each experiment, the vector $\bm{\theta}$ of parameters include some of them depending on appropriate setting. We have reparameterized the sill and the nugget through a power of ten, because often the plausible search range is rather wide or narrow. Note that for the extremely smooth case (i.e., $\nu=\infty$), becomes equivalently the squared-exponential model $$\label{eqn:sq.exp}
k(\bm{x},\bm{x}')=
10^{\alpha}\exp\left(-\frac{\|\bm{r}\|^2}{2\ell^2}\right)
+10^{\tau}\cdot\bm{1}(\bm{r}=\bm{0}).$$ We will use this covariance function in one of the experiments. Throughout we assume zero mean for simplicity.
Small-Scale Example {#sec:exp.closed.loop}
-------------------
We first conduct a closed-loop experiment whereby data are simulated on a two-dimensional grid from some prescribed parameter vector $\bm{\theta}$. We discard (uniformly randomly) half the data and perform maximum likelihood estimation to verify that the estimated $\widehat{\bm{\theta}}$ is indeed close to $\bm{\theta}$. Afterward, we perform kriging by using the estimated $\widehat{\bm{\theta}}$ to recover the discarded data. Because it is a closed-loop setting and there is no model misspecification, the kriging errors should align well with the square root of the variance of the conditional distribution (see ). We do not use a large $n$, since we will compare the results of the proposed method with those from the standard method that requires $O(n^3)$ expensive linear algebra computations.
The prescribed parameter vector $\bm{\theta}$ consists of three elements: $\alpha$, $\ell$, and $\nu$. We choose to use a zero nugget because in some real-life settings, measurements can be quite precise and it is unclear one always needs a nugget effect. This experiment covers such a scenario. Further, note that numerically accurate codes for evaluating the derivatives with respect to $\nu$ are unknown. Such a limitation poses constraints when choosing optimization methods.
Further details are as follows. We simulate data on a grid of size $40\times50$ occupying a physical domain $[-0.8,0.8]\times[-1,1]$, by using prescribed parameters $\alpha=0$, $\ell=0.2$, and $\nu=2.5$. Half of the data are discarded, which results in $n=1000$ sites for estimation and $m=1000$ sites for kriging.
For the proposed method, we build the partitioning tree by using the bounding box approach elaborated in Section \[sec:practical\]. We specify the number of landmark points, $r$, to be $125$, and make the height of the partitioning tree $h=\lfloor\log_2(n/r)\rfloor$ such that the number of points in each leaf node is approximately $r$. The landmark points for each subdomain in the hierarchy are placed on a regular grid.
Figure \[fig:exp.closed.loop.1\](a) illustrates the random field simulated by using $k$. With this data, maximum likelihood estimation is performed, by using separately $k$ and $k_{\text{h}}$. The parameter estimates and their standard errors are given in Table \[tab:exp.closed.loop\]. The numbers between the two methods are both quite close to the truth. With the estimated parameters, kriging is performed, with the results shown in Figure \[fig:exp.closed.loop.1\](b) and (c). The kriging errors are sorted in the increasing order of the prediction variance. The red curves in the plots are three times the square root of the variance; not surprisingly almost all the errors are below this curve.
------------------------------- ---------- ----------- --------- ----------- -------- ----------
Truth $0.000$ $0.200$ $2.50$
Estimated with $k$ $-0.172$ $(0.076)$ $0.182$ $(0.012)$ $2.56$ $(0.11)$
Estimated with $k_{\text{h}}$ $-0.150$ $(0.075)$ $0.186$ $(0.012)$ $2.53$ $(0.11)$
------------------------------- ---------- ----------- --------- ----------- -------- ----------
: True parameters and estimates.[]{data-label="tab:exp.closed.loop"}
Comparison of Log-Likelihoods and Estimates {#sec:exp.loglik}
-------------------------------------------
One should note that the base covariance function $k$ and the proposed $k_{\text{h}}$ are not particularly close, because the number $r$ of landmarks for defining $k_{\text{h}}$ is only $125$ (compare this number with the number of observed sites, $n=1000$). Hence, if one compares the covariance matrix $K$ with $K_{\text{h}}$, they agree in only a limited number of digits. However, the reason why $k_{\text{h}}$ is a good alternative of $k$ is that the shapes of the likelihoods are similar, as well as the locations of the optimum.
\
We graph in Figure \[fig:exp.closed.loop.3\] the cross sections of the log-likelihood centered at the truth $\bm{\theta}$. The top row corresponds to $k$ and the bottom row to $k_{\text{h}}$. One sees that in both cases, the center (truth $\bm{\theta}$) is located within a reasonably concave neighborhood, whose contours are similar to each other.
-------------------------------------------------- ---------------------------------------------- -------------------------------------------- ------------------------------------------------------------------------------------------
$|\widehat{\alpha}-\widehat{\alpha}_{\text{h}}|$ $|\widehat{\ell}-\widehat{\ell}_{\text{h}}|$ $|\widehat{\nu}-\widehat{\nu}_{\text{h}}|$ $|\mathcal{L}_k(\widehat{\bm{\theta}})-\mathcal{L}_k(\widehat{\bm{\theta}}_{\text{h}})|$
$0.0120$ $(0.0098)$ $0.0018$ $(0.0018)$ $0.0240$ $(0.0211)$ $0.1151$ $(0.0880)$
$\text{stderr}(\widehat{\alpha})$ $\text{stderr}(\widehat{\ell})$ $\text{stderr}(\widehat{\nu})$
$0.0841$ $(0.0050)$ $0.0137$ $(0.0016)$ $0.1002$ $(0.0074)$
-------------------------------------------------- ---------------------------------------------- -------------------------------------------- ------------------------------------------------------------------------------------------
: Difference of estimates and log-likelihoods under $k$ and $k_{\text{h}}$. The unparenthesized number is the mean and the number with parenthesis is the standard deviation. For reference, the uncertainties (denoted as stderr) of the estimates are listed in the second part of the table.[]{data-label="tab:exp.loglik"}
In fact, the maxima of the log-likelihoods are rather close. We repeat the simulation ten times and report the statistics in Table \[tab:exp.loglik\]. The quantities with a subscript “h” correspond to the proposed covariance function $k_{\text{h}}$. One sees that for each parameter, the differences of the estimates are generally about $20\%$ of the standard errors of the estimates. Furthermore, the difference of the true log-likelihoods at the two estimates is always substantially less than one unit. These results indicate that the proposed $k_{\text{h}}$ produces highly comparable parameter estimates with the base covariance function $k$.
Landmark Points {#sec:exp.landmark}
---------------
In the preceding two subsections, we fixed the number of landmark points, $r$, to be $125$ and placed them on a regular grid within each subdomain. Here, we study the effect of $r$ and the locations.
In Figure \[fig:exp.landmark\], we show two plots on the kriging error and the log-likelihood, one obtained by using the ground truth parameters $[\alpha,\ell,\nu]=[0,0.2,2.5]$ and the other by using $[\alpha,\ell,\nu]=[0.2,0.24,2.7]$, which results in a noticeably different covariance function as judged from the likelihood surface exhibited in Figure \[fig:exp.closed.loop.3\]. The experimented values of $r$ are $7$, $15$, $31$, $62$, $125$, $250$, and $500$, geometrically progressing toward the number of observed sites, $n=1000$. The solid curve corresponds to a regular grid of landmark points, whereas the dashed curve corresponds to the randomized choice, with one times standard deviation shown as a shaded region. “RMSE” denotes root mean squared error.
One sees that the error decreases monotonically as $r$ increases. There thus forms a tradeoff between error and time, since the computational cost is quadratic in $r$. In this particular case, it appears that $125$ yields a significant decrease in RMSE while being reasonably small. The likelihood shows a similar trend of change as $r$ varies (except that it increases rather than decreases). Moreover, the randomized choice of landmark points is inferior to the regular-grid choice, considering the mean and standard deviation. However, one should note that if three times standard deviation is considered instead, the shaded region will cover the solid curve for large $r$, indicating that the advantage of regular grid diminishes as $r$ increases. Finally, an interesting observation is that the kriging error remains highly comparable when one uses less accurate covariance parameters, although in this case the reduction of likelihood is substantial.
Comparison with Nyström and Block-Diagonal Approximation {#sec:exp.nystrom}
--------------------------------------------------------
In this subsection, we compare with two methods: Nyström and block-diagonal approximation. The former is a part of our one-level construction, whereas the latter performs kriging in each fine-level subdomain independently (equivalent to applying a block-diagonal approximation of the covariance matrix $K$). The experiment setting is the same as that of the preceding subsections.
Figure \[fig:exp.nystrom\](a) shows the kriging error of Nyström normalized by that of the proposed method. First, all error ratios are greater than one, indicating that the hierarchical approach clearly strengthens the approximation with only one level as in Nyström. Moreover, this observation is consistent regardless of what covariance parameters are used. Interestingly, the ratio is slightly smaller when the used parameters are less accurate, suggesting that one-level approximation appears to suffer less when the parameters are not close to the ground truth. Finally, as $r$ increases, the error ratio generally decreases, which is expected since the number of levels that strengthen the approximation becomes fewer. Nyström performs disastrously in light of the fact that the error ratio is greater than 2 when $r<500$.
Similarly, Figure \[fig:exp.nystrom\](b) shows the kriging error of block-diagonal approximation, normalized. This method performs much better than Nyström, with the normalized errors only slightly greater than 1. Interestingly, contrary to Nyström, this method suffers more when the parameters are not close to the ground truth. Since the method performs essentially local kriging by ignoring the long-range correlation, this phenomenon is expected.
Scaling {#sec:exp.scaling}
-------
In this subsection, we verify that the linear algebra costs for the proposed method indeed agree with the theoretical analysis. Namely, random field simulation and log-likelihood evaluation are both $O(n)$, and the kriging of $m=n$ sites is $O(n\log n)$. Note that all these computations require the construction of the covariance matrix, which is $O(n\log n)$. The experiment setting is the same as that of the preceding subsections, except that we restrict the number of log-likelihood evaluations to $125$ to avoid excessive computation. We vary the grid size from $40\times50$ to $640\times800$ to observe the scaling. The random removal of sites has a minimal effect on the partitioning and hence on the overall time. The computation is carried out on a laptop with eight Intel cores (CPU frequency 2.8GHz) and 32GB memory. Six parallel threads are used.
Figure \[fig:exp.scaling\] plots the computation times, which indeed well agree with the theoretical scaling. As expected, log-likelihood evaluations are the most expensive, particularly when many evaluations are needed for optimization. The simulation of a random field follows, with kriging being the least expensive, even when a large number of sites are kriged.
Large-Scale Example Using Test Function {#sec:exp.large.scale}
---------------------------------------
The above scaling results confirm that handling a large $n$ is feasible on a laptop. In this subsection, we perform an experiment with up to one million data sites. Different from the closed-loop setting that uses a known covariance model, here we generate data by using a test function. We estimate the covariance parameters and krige with the estimated model.
The test function is $$Z(\bm{x}) = \exp(1.4x_1)\cos(3.5\pi x_1)[\sin(2\pi x_2)+0.2\sin(8\pi x_2)]$$ on $[0,1]^2$. This function is rather smooth (see Figure \[fig:exp.closed.loop\](a) for an illustration). Hence, we use the squared-exponential model for estimation. The high smoothness results in a too ill-conditioned matrix; therefore, a nugget is necessary. The vector of parameters is $\bm{\theta}=[\alpha, \ell, \tau]^T$. We inject independent Gaussian noise $\mathcal{N}(0,\,0.1^2)$ to the data so that the nugget will not vanish. As before, we randomly select half of the sites for parameter estimation and the other half for kriging. The number of landmark points, $r$, remains $125$.
Our strategy for large-scale estimation is to first perform a small-scale experiment with the base covariance function $k$ that quickly locates the optimum. The result serves as a reference for later use of the proposed $k_{\text{h}}$ in the larger-scale setting. The results are shown in Figure \[fig:exp.closed.loop\] (for the largest grid) and Table \[tab:exp.large.scale\].
Grid Est. w/
------------------ ---------------- --------- ----------- ---------- ------------ ----------- ------------
$50\times50$ $k$ $0.313$ $(0.098)$ $0.1199$ $(0.0035)$ $-2.0109$ $(0.0186)$
$100\times100$ $k_{\text{h}}$ $0.389$ $(0.095)$ $0.1238$ $(0.0029)$ $-1.9923$ $(0.0089)$
$1000\times1000$ $k_{\text{h}}$ $0.919$ $(0.134)$ $0.1395$ $(0.0031)$ $-2.0011$ $(0.0009)$
: Estimated parameters.[]{data-label="tab:exp.large.scale"}
Each of the cross sections of the log-likelihood on the bottom row of Figure \[fig:exp.closed.loop\] is plotted by setting the unseen parameter at the estimated value. For example, the $\alpha$-$\ell$ plane is located at $\widehat{\tau}=-2.0011$. From these contour plots, we see that the estimated parameters are located at a local maximum with nicely concave contours in a neighborhood of this maximum. The estimated nugget ($\approx -2$) well agrees with $\log_{10}$ of the actual noise variance. The kriged field (plot (b)) is visually as smooth as the test function. The kriging errors for predicting the test function $Z(\cdot)$, again sorted by their estimated standard errors, are plotted in (c). As one would expect, nearly all of the errors are less than three times their estimated standard errors. Note that the kriging errors are counted without the perturbed noise; they are substantially lower than the noise level.
Analysis of Climate Data {#sec:exp.noaa}
========================
In this section, we apply the proposed method to analyze a climate data product developed by the National Centers for Environmental Prediction (NCEP) of the National Oceanic and Atmospheric Administration (NOAA).[^5] The Climate Forecast System Reanalysis (CFSR) data product [@Saha2010] offers hourly time series as well as monthly means data with a resolution down to one-half of a degree (approximately 56 km) around the Earth, over a period of 32 years from 1979 to 2011. For illustration purpose, we extract the temperature variable at 500 mb height from the monthly means data and show a snapshot on the top of Figure \[fig:noaa\]. Temperatures at this pressure (generally around a height of 5 km) provide a good summary of large-scale weather patterns and should be more nearly stationary than surface temperatures. We will estimate a covariance model for every July over the 32-year period.
![Snapshot of CFSR global temperature at 500 mb and the resulting data after subtraction of pixelwise mean for the same month over 32 years.[]{data-label="fig:noaa"}](pgbhnl_gdas_197907.pdf){width="0.8\linewidth"}
Through preliminary investigations, we find that the data appears fairly Gaussian after a subtraction of pixelwise mean across time. An illustration of the demeaned data for the same snapshot is given at the bottom of Figure \[fig:noaa\]. Moreover, the correlation between the different snapshots are so weak that we shall treat them as independent anomalies. Although temperatures have warmed during this period, the warming is modest compared to the interannual variation in temperatures at this spatial resolution, so we assume the anomalies have mean 0. We use $\bm{z}_i$ to denote the anomaly at time $i$. Then, the log-likelihood with $N=32$ zero-mean independent anomalies $\bm{z}_i$ is $$\mathcal{L}=-\sum_{i=1}^N\frac{1}{2}\bm{z}_i^TK^{-1}\bm{z}_i-\frac{N}{2}\log\det K-\frac{Nn}{2}\log2\pi.$$
For random fields on a sphere, a reasonable covariance function for a pair of sites $\bm{x}$ and $\bm{x}'$ may be based on their great-circle distance, or equivalently the chordal distance, because of their monotone relationship. Specifically, let a site $\bm{x}$ be represented by latitude $\phi$ and longitude $\psi$. Then, the chordal distance between two sites $\bm{x}$ and $\bm{x}'$ is $$\label{eqn:chordal}
r=2\left[\sin^2\left(\frac{\phi-\phi'}{2}\right)+\cos\phi\cos\phi'\sin^2\left(\frac{\psi-\psi'}{2}\right)\right]^{1/2}.$$ Here, we assume that the radius of the sphere is $1$ for simplicity, because it can always be absorbed into a range parameter later. We still use the Matérn model $$\label{eqn:matern2}
k(\bm{x},\bm{x}')=
\frac{10^{\alpha}}{2^{\nu-1}\Gamma(\nu)}
\left(\frac{\sqrt{2\nu}r}{\ell}\right)^{\nu}
\bessel_{\nu}\left(\frac{\sqrt{2\nu}r}{\ell}\right)
+10^{\tau}\cdot\bm{1}(r=0)$$ to define the covariance function, where $r$ is the chordal distance , so that the model is isotropic on the sphere. More sophisticated models based on the same Matérn function and the chordal distance $r$ are proposed in [@Jun2008]. Note that this model depends on the longitudes for $\bm{x}$ and $\bm{x}'$ only through their differences modulo $2\pi$. Such a model is called *axially symmetric* [@Jones1963].
A computational benefit of an axially symmetric model and gridded observations is that one may afford computations with $k$ even when the latitude-longitude grid is dense. The reason is that for any two fixed latitudes, the cross-covariance matrix between the observations is circulant and diagonalizing it requires only one discrete Fourier transform (DFT), which is efficient. Thus, diagonalizing the whole covariance matrix amounts to diagonalizing only the blocks with respect to each longitude, apart from the DFT’s for each latitude. Hence, we will perform computations with both the base covariance function $k$ and the proposed function $k_{\text{h}}$ and compare the results. We subsample the grid with every other latitude and longitude for parameter estimation. We also remove the two grid lines 90N and 90S due to their degeneracy at the pole. Because of the half-degree resolution, this results in a coarse grid of size $180\times360$ for parameter estimation, for a total of $180\times 360\times 32 = 2{,}073{,}600$ observations. The rest of the grid points are used for kriging. As before, we set the number $r$ of landmark points to be $125$.
$\nu$ Log-likelihood
------- --------------------- ---------- ---------------- --------------------- ---------- ----------- --------------------
$0.5$ ($-0.285$ $ 0.156$ $-4.935$) ($-0.794$ $ 1.446$ $-7.165$) $ 3.938\times10^6$
($-0.794$ $ 1.446$ $-7.165$)
$1.0$ ($-0.285$ $ 0.156$ $-4.935$) ($-0.279$ $ 0.411$ $-5.133$) $ 4.696\times10^6$
($-0.279$ $ 0.411$ $-5.133$) ($\phantom{-}0.838$ $ 1.494$ $-5.125$) $ 4.700\times10^6$
$1.5$ ($\phantom{-}0.124$ $ 0.215$ $-4.933$) ($-0.285$ $ 0.156$ $-4.935$) $ 4.757\times10^6$
($-0.285$ $ 0.156$ $-4.935$) ($-0.285$ $ 0.156$ $-4.935$) $ 4.757\times10^6$
$2.0$ ($-0.285$ $ 0.156$ $-4.935$) ($-0.279$ $ 0.094$ $-4.933$) $ 4.643\times10^6$
($-0.279$ $ 0.094$ $-4.933$) ($-0.545$ $ 0.081$ $-4.821$) $ 4.653\times10^6$
: Optimization results for different $\nu$’s using the base covariance function $k$.[]{data-label="tab:exp.noaa0"}
We set the parameter vector $\bm{\theta}=[\alpha,\ell,\tau]^T$, considering only several values for the smoothness parameter $\nu$ because of the difficulties of numerical optimization of the loglikelihood over $\nu$. To our experience, blackbox optimization solvers do not always find accurate optima. We show in Table \[tab:exp.noaa0\] several results of the Matlab solver `fminunc` when one varies $\nu$. For each $\nu$, we start the solver at some initial guess $\bm{\theta}_0$ until it claims a local optimum $\widehat{\bm{\theta}}$. Then, we use this optimum as the initial guess to run the solver again. Ideally, the solver should terminate at $\widehat{\bm{\theta}}$ if it indeed is an optimum. However, reading Table \[tab:exp.noaa0\], one finds that this is not always the case.
When $\nu=0.5$, the second search diverges from the initial $\widehat{\bm{\theta}}$. The cross-section plots of the log-likelihood (not shown) indicate that $\widehat{\bm{\theta}}$ is far from the center of the contours. The solver terminates merely because the gradient is smaller than a threshold and the Hessian is positive-definite (recall that we *minimize* the negative log-likelihood). The diverging search starting from $\widehat{\bm{\theta}}$ (with $\alpha$ and $\ell$ continuously increasing) implies that the infimum of the negative log-likelihood may occur at infinity, as can sometimes happen in our experience.
When $\nu=1.0$, although the search starting at $\widehat{\bm{\theta}}$ does not diverge, it terminates at a location quite different from $\widehat{\bm{\theta}}$, with the log-likelihood increased by about 4000, which is arguably a small amount given the number of observations. Such a phenomenon is often caused by the fact that the peak of the log-likelihood is flat (at least along some directions); hence, the exact optimizer is hard to locate. This phenomenon similarly occurs in the case $\nu=2.0$. Only when $\nu=1.5$ does restarting the optimization yield $\widehat{\bm{\theta}}$ that is essentially the same as the initial estimate. Incidentally, the log-likelihood in this case is also the largest. Hence, all subsequent results are produced for only $\nu=1.5$.
Est. w/
---------------- ----------- ------------ ----------- ------------- ----------- ------------
$k$ $-0.2875$ $(0.0047)$ $0.15620$ $(0.00058)$ $-4.9360$ $(0.0014)$
$k_{\text{h}}$ $-0.2275$ $(0.0044)$ $0.16640$ $(0.00058)$ $-4.9300$ $(0.0015)$
: Estimation results ($\nu=1.5$).[]{data-label="tab:exp.noaa"}
at $\widehat{\bm{\theta}}$ at $\widehat{\bm{\theta}}_{\text{h}}$
---------------------- ---------------------------- ---------------------------------------
Using $k$ $4757982$ $4756981$
Using $k_{\text{h}}$ $4557568$ $4558731$
: Log-likelihood (left) and root mean squared prediction error (right).[]{data-label="tab:exp.noaa2"}
at $\widehat{\bm{\theta}}$ at $\widehat{\bm{\theta}}_{\text{h}}$
---------------------- ---------------------------- ---------------------------------------
Using $k$ $0.01394$ $0.01394$
Using $k_{\text{h}}$ $0.01556$ $0.01556$
: Log-likelihood (left) and root mean squared prediction error (right).[]{data-label="tab:exp.noaa2"}
Near $\widehat{\bm{\theta}}$, we further perform a local grid search and obtain finer estimates, as shown in Table \[tab:exp.noaa\]. One sees that the estimated parameters produced by $k$ and $k_{\text{h}}$ are qualitatively similar, although their differences exceed the tiny standard errors. To distinguish the two estimates, we use $\widehat{\bm{\theta}}$ to denote the one resulting from $k$ and $\widehat{\bm{\theta}}_{\text{h}}$ from $h_{\text{h}}$. In Table \[tab:exp.noaa2\], we list the log-likelihood values and the kriging errors when the covariance function is evaluated at both locations. One sees that the estimate $\widehat{\bm{\theta}}_{\text{h}}$ is quite close to $\widehat{\bm{\theta}}$ in two important regards: first, the root mean squared prediction errors using $k$ are the same to four significant figures, and the log-likelihood under $k$ differs by 1000, which we would argue is a very small difference for more than 2 million observations. On the other hand, $k_{\text{h}}$ does not provide a great approximation to the loglikelihood itself and the predictions using $k_{\text{h}}$ are slightly inferior to those using $k$ no matter which estimate is used. Figure \[fig:exp.noaa\] plots the log-likelihoods centered around the respectively optimal estimates. The shapes are visually identical, which supports the use of $k_{\text{h}}$ for parameter estimation. We see that the statistical efficacy of the proposed covariance function depends on the purpose to which it is put.
\
Conclusions
===========
We have presented a computationally friendly approach that addresses the challenge of formidably expensive computations of Gaussian random fields in the large scale. Unlike many methods that focus on the approximation of the covariance matrix or of the likelihood, the proposed approach operates on the covariance function such that positive definiteness is maintained. The hierarchical structure and the nested bases in the proposed construction allow for organizing various computations in a tree format, achieving costs proportional to the tree size and hence to the data size $n$. These computations range from the simulation of random fields to kriging and likelihood evaluations. More desirably, kriging has an amortized cost of $O(\log n)$ and hence one may perform predictions for as many as $O(n)$ sites easily. Moreover, the efficient evaluation of the log-likelihoods paves the way for maximum likelihood estimation as well as Markov Chain Monte Carlo. Numerical experiments show that the proposed construction yields comparable prediction results and likelihood surfaces with those of the base covariance function, while being scalable to data of ever increasing size.
Proof of Theorem \[thm:k.one.level\]
====================================
For a proof of positive definiteness, we write $k_{\text{h}}$ as a sum of two functions $\xi_1$ and $\xi_2$, where $$\xi_1(\bm{x},\bm{x}')=k(\bm{x},\ud{X})k(\ud{X},\ud{X})^{-1}k(\ud{X},\bm{x}')$$ is the Nyström approximation in the whole domain $S$ and hence positive definite, and $$\xi_2(\bm{x},\bm{x}')=
\begin{cases}
k(\bm{x},\bm{x}')-k(\bm{x},\ud{X})k(\ud{X},\ud{X})^{-1}k(\ud{X},\bm{x}'), & \text{if $\bm{x},\bm{x}'\in S_j$ for some $j$},\\
0, & \text{otherwise},
\end{cases}$$ is a Schur complement in each subdomain $S_j$ and hence also positive definite. Then, the constructed $k_{\text{h}}$ is positive definite.
To prove the strict positive definiteness, we need the following lemma.
\[lem:strict\] Let $k$ be strictly positive definite. For any set of points $X=\{\bm{x}_1,\ldots,\bm{x}_n\}$ such that $X\cap\ud{X}=\emptyset$ and for any set of coefficients $\alpha_1,\ldots,\alpha_n\in\real$ that are not all zero, we have $$\sum_{i,j=1}^n\alpha_i\alpha_j\left[k(\bm{x}_i,\bm{x}_j)-k(\bm{x}_i,\ud{X})k(\ud{X},\ud{X})^{-1}k(\ud{X},\bm{x}_j)\right]>0.$$
The result is equivalent to saying that the matrix $k(X,X)-k(X,\ud{X})k(\ud{X},\ud{X})^{-1}k(\ud{X},X)$ is positive definite. To see so, consider $$k(X\cup\ud{X},X\cup\ud{X})=
\begin{bmatrix}
k(X,X) & k(X,\ud{X}) \\
k(\ud{X},X) & k(\ud{X},\ud{X})
\end{bmatrix}.$$ Because of the strict positive definiteness of the function $k$, the matrix $k(X\cup\ud{X},X\cup\ud{X})$ is positive definite. Then, the Schur complement matrix $k(X,X)-k(X,\ud{X})k(\ud{X},\ud{X})^{-1}k(\ud{X},X)$ is also positive definite.
We now continue the proof of Theorem \[thm:k.one.level\]. For a set of coefficients $\alpha_1,\ldots,\alpha_n\in\real$, $$\label{eqn:sum}
\sum_{i,j=1}^n\alpha_i\alpha_jk_{\text{h}}(\bm{x}_i,\bm{x}_j)
=\underbrace{\sum_{i,j=1}^n\alpha_i\alpha_j \xi_1(\bm{x}_i,\bm{x}_j)}_{B_1}+
\underbrace{\sum_{i,j=1}^n\alpha_i\alpha_j \xi_2(\bm{x}_i,\bm{x}_j)}_{B_2}.$$ If we want the left-hand side to be zero, $B_1$ and $B_2$ must be simultaneously zero. Because $\xi_2(\bm{x},\bm{x}')$ is zero whenever $\bm{x}$ or $\bm{x}'$ belongs to $\ud{X}$, based on Lemma \[lem:strict\], $B_2=0$ implies that $\alpha_i=0$ for all $\bm{x}_i\notin\ud{X}$. In such a case, $B_1$ is simplified to $$B_1=\sum_{\bm{x}_i,\bm{x}_j\in\ud{X}}\alpha_i\alpha_j \xi_1(\bm{x}_i,\bm{x}_j)
=\ud{\bm{\alpha}}^Tk(\ud{X},\ud{X})\ud{\bm{\alpha}},$$ where $\ud{\bm{\alpha}}$ is the column vector of $\alpha_i$’s for all $\bm{x}_i\in\ud{X}$. Then, because of the strict positive definiteness of $k$, $B_1=0$ implies that $\alpha_i=0$ for all $\bm{x}_i\in\ud{X}$. Thus, all coefficients $\alpha_i$ must be zero for the left-hand side of to be zero. This concludes that $k_{\text{h}}$ is strictly positive definite.
Proof of Theorem \[thm:k.multilevel\]
=====================================
To avoid notational cluttering, for the covariance function $k$ we write $$k_{\bm{x},\bm{x}'}\equiv k(\bm{x},\bm{x}'),\quad
k_{\bm{x},\ud{i}}\equiv k(\bm{x},\ud{X}_i),\quad
k_{\ud{i},\ud{j}}\equiv k(\ud{X}_i,\ud{X}_j),$$ and similarly for the auxiliary function $\psi^{(i)}$.
To prove positive definiteness, we will define a function $\xi^{(i)}:S\times S\to\real$ for each nonroot node $i$, which has a support on only $S_i\times S_i$ and which acts as a Schur complement. Additionally, for the root $i=1$, define $\xi^{(1)}$ to be Nyström-like. Specifically, for any node $i$ with parent $p$ (if any), let $$\label{eqn:xi}
\begin{split}
\xi^{(i)}(\bm{x},\bm{x}') &= 0 \text{ if either $\bm{x}$ or $\bm{x}'\notin S_i$; otherwise: } \\
\xi^{(i)}(\bm{x},\bm{x}') &=
\begin{cases}
k_{\bm{x},\bm{x}'}-k_{\bm{x},\ud{p}}k_{\ud{p},\ud{p}}^{-1}k_{\ud{p},\bm{x}'}, & \text{if $i$ is leaf},\\
\psi^{(i)}_{\bm{x},\ud{i}}k_{\ud{i},\ud{i}}^{-1}
\Big(k_{\ud{i},\ud{i}}-k_{\ud{i},\ud{p}}k_{\ud{p},\ud{p}}^{-1}k_{\ud{p},\ud{i}}\Big)
k_{\ud{i},\ud{i}}^{-1}\psi^{(i)}_{\ud{i},\bm{x}'}, & \text{if $i$ is neither leaf nor root},\\
\psi^{(i)}_{\bm{x},\ud{i}}k_{\ud{i},\ud{i}}^{-1}\psi^{(i)}_{\ud{i},\bm{x}'}, & \text{if $i$ is root}.
\end{cases}
\end{split}$$ Then, through telescoping, one sees that the proposed covariance function $k_{\text{h}}$ is the sum of $\xi^{(i)}$ for all nodes $i$ in the partitioning tree; that is, $$k_{\text{h}}(\bm{x},\bm{x}')=\sum_i\xi^{(i)}(\bm{x},\bm{x}').$$ Inspecting each case of , we clearly see that $\xi^{(i)}$ is positive definite. Thus, the sum $k_{\text{h}}$ is positive definite.
To prove the strict positive definiteness, we need the following lemma.
\[lem:psik\] Let $l$ be a leaf descendant of some nonleaf node $i$ and let $(l,l_1,l_2,\ldots,l_s,i)$ be the path connecting $l$ and $i$. Then, $$\psi^{(i)}_{\bm{x},\ud{i}}=k_{\bm{x},\ud{i}}$$ if $\bm{x}\in \ud{X}_{l_1}\cap\ud{X}_{l_2}\cap\cdots\cap\ud{X}_{l_s}$.
The result is a straightforward verification. For an array of distinct points which contains some point $\bm{x}$ at the $j$-th location, we use the notation $\bm{e}_{\bm{x}}$ to denote a column vector whose $j$-th element is $1$ and otherwise $0$. Then, for $\bm{x}\in S_l$ and also $\in \ud{X}_{l_1}$, $$\psi^{(i)}_{\bm{x},\ud{i}}
=k_{\bm{x},\ud{l_1}}k_{\ud{l_1},\ud{l_1}}^{-1}k_{\ud{l_1},\ud{l_2}}k_{\ud{l_2},\ud{l_2}}^{-1}\cdots k_{\ud{l_s},\ud{l_s}}^{-1}k_{\ud{l_s},\ud{i}}
=\bm{e}_{\bm{x}}^Tk_{\ud{l_1},\ud{l_2}}k_{\ud{l_2},\ud{l_2}}^{-1}\cdots k_{\ud{l_s},\ud{l_s}}^{-1}k_{\ud{l_s},\ud{i}}
=k_{\bm{x},\ud{l_2}}k_{\ud{l_2},\ud{l_2}}^{-1}\cdots k_{\ud{l_s},\ud{l_s}}^{-1}k_{\ud{l_s},\ud{i}}.$$ Iteratively simplifying by noting that $\bm{x}$ also belongs to $\ud{X}_{l_1},\ldots,\ud{X}_{l_s}$, we eventually reach $$\psi^{(i)}_{\bm{x},\ud{i}}
=k_{\bm{x},\ud{l_s}}k_{\ud{l_s},\ud{l_s}}^{-1}k_{\ud{l_s},\ud{i}}
=\bm{e}_{\bm{x}}^Tk_{\ud{l_s},\ud{i}}
=k_{\bm{x},\ud{i}}.$$
We now continue the proof of Theorem \[thm:k.multilevel\]. The strategy resembles induction. For a set of coefficients $\alpha_1,\ldots,\alpha_n\in\real$, write $$\label{eqn:sum2}
\sum_{j,l=1}^n\alpha_j\alpha_lk_{\text{h}}(\bm{x}_j,\bm{x}_l)
=\sum_i\underbrace{\sum_{j,l=1}^n\alpha_j\alpha_l\xi^{(i)}(\bm{x}_j,\bm{x}_l)}_{B_i}.$$ If we want the left-hand side to be zero, all the $B_i$’s on the right must be simultaneously zero. When $i$ is a leaf node, $\xi^{(i)}(\bm{x},\bm{x}')$ is zero whenever $\bm{x}$ or $\bm{x}'$ belongs to $\ud{X}_p$, where $p$ is the parent of $i$. Then, $B_i=0$ implies that $\alpha_j=0$ for all $\bm{x}_j\in S_i\backslash\ud{X}_p$.
For any nonleaf node $p$, we use $Q_p$ to denote the union of the intersections of landmark points: $$Q_p\equiv\bigcup_{l \text{ is leaf descendant of } p}
\{ \ud{X}_{l_1}\cap\cdots\cap\ud{X}_{l_s}\cap\ud{X}_p \mid
\text{$(l,l_1,\ldots,l_s,p)$ is a path connecting $l$ and $p$} \}.$$ Clearly, $Q_p\subset S_p$. As a special case, if all the children of $p$ are leaf nodes, $Q_p=\ud{X}_p$. We now have an induction hypothesis: for a nonroot node $i$ with parent $p$, there holds $\alpha_j=0$ for all $\bm{x}_j\in S_i\backslash (Q_i\cap\ud{X}_p)$. Assume that the hypothesis is true for all child nodes of some node $p$, who has a parent $q$. Then, summarizing the results for all these child nodes, we have $\alpha_j=0$ for all $\bm{x}_j\in S_p\backslash Q_p$. Furthermore, based on Lemma \[lem:psik\], $\xi^{(p)}(\bm{x},\bm{x}')$ is zero whenever $\bm{x}$ or $\bm{x}'$ belongs to $Q_p\cap\ud{X}_q$. Then, $B_p=0$ implies that $\alpha_j=0$ for all $\bm{x}_j\in (S_p\backslash Q_p)\cup(Q_p\backslash\ud{X}_q)=S_p\backslash (Q_p\cap\ud{X}_q)$. This finishes the induction step.
At the end of the induction, we reach the root node $p$. Summarizing the results for all the child nodes of the root, we have $\alpha_j=0$ for all $\bm{x}_j\in S_p\backslash Q_p$. Invoking Lemma \[lem:psik\] again, we have $\xi^{(p)}(\bm{x},\bm{x}')=k_{\bm{x},\bm{x}'}$ whenever $\bm{x}$ or $\bm{x}'$ belongs to $Q_p$. Then, by the strict positive definiteness of $k$, $B_p=0$ implies that $\alpha_j=0$ for all $\bm{x}_j\in Q_p$. Thus, all coefficients $\alpha_i$ must be zero for the left-hand side of to be zero. This concludes that $k_{\text{h}}$ is strictly positive definite.
Algorithm for Matrix-vector Multiplication {#sec:matvec}
==========================================
The objective is to compute $\bm{y}=A\bm{b}$. We will use a shorthand notation $\bm{b}_i$ to denote a subvector of $\bm{b}$ that corresponds to the index set $I_i$; and similarly for the vector $\bm{y}$. In computer implementation, only the subvectors corresponding to leaf nodes are stored therein. On the other hand, we need auxiliary vectors $\bm{c}_j$ and $\bm{d}_j$, all of length $r$, to be stored in each nonroot node $j$. These auxiliary vectors are defined in the following context.
The vector $\bm{y}$ is the sum of two parts: the first part comes from $A_{ll}\bm{b}_l$ for every leaf node $l$ and the second part comes from $A_{ij}\bm{b}_j$ for every pair of sibling nodes $i$ and $j$. The first part is straightforward to calculate. The second part, however, needs an expansion through change of basis according to Definition \[def:matrix\]. In particular, let $l$ be a leaf descendant of $i$. Then, the subvector of $A_{ij}\bm{b}_j$ corresponding to the index set $I_l$ is $$U_lW_{l_1}W_{l_2}\cdots W_{l_s}W_i\Sigma_pZ_j^T\left(
\sum_{\substack{q \text{ is leaf} \\ (q,q_1,q_2,\ldots,q_t,j) \text{ is path}}}
Z_{q_t}^T\cdots Z_{q_2}^TZ_{q_1}^TV_q^T\bm{b}_q\right),$$ where $p$ is the parent of $i$ and $j$, $(l,l_1,l_2,\ldots,l_s,i)$ is the path connecting $l$ and $i$, and the bracketed expression to the right of $Z_j^T$ sums over all the contributions from any descendant leaf $q$ of $j$.
Many computations in the above summation are duplicated. For example, the term $V_q^T\bm{b}_q$ at a leaf node $q$ appears in all $A_{ij}\bm{b}_j$ whenever $q$ is a leaf descendant of $j$. Hence, we define two sets of auxiliary vectors $$\bm{c}_i=\begin{dcases}
V_i^T\bm{b}_i, & \text{if $i$ is leaf},\\
Z_i^T\sum_{j\in \text{Ch}(i)}\bm{c}_j, & \text{otherwise},\\
\end{dcases}$$ and $$\bm{d}_j=W_i\bm{d}_i+\sum_{j'\in \text{Ch}(i)\backslash\{j\}}\Sigma_i\bm{c}_{j'},
\quad\text{for $j$ being a child of $i$; \quad $W_i\bm{d}_i=0$ if $i$ is root},$$ as temporary storage to avoid duplicate computation. It is not hard to see that for any leaf node $l$, the final output subvector is $\bm{y}_l=A_{ll}\bm{b}_l+U_l\bm{d}_l$.
By definition, the set of auxiliary vectors $\bm{c}_i$ may be recursively computed from children to parent, whereas the other set $\{\bm{d}_j\}$ may be computed in a reverse order, from parent to children. Then, the overall computation consists of two tree walks, one upward and the other downward. This computation is summarized in Algorithm \[algo:Ab\]. The blue texts highlight the modification of the algorithm when $A$ is symmetric. All subsequent algorithms similarly use blue texts to indicate modifications for symmetry.
Initialize $\bm{d}_i\gets\bm{0}$ for each nonroot node $i$ of the tree
$\bm{c}_i\gets V_i^T\bm{b}_i$; $\bm{y}_i\gets A_{ii}\bm{b}_i$ **for all** children $j$ of $i$ **do** **end for** $\bm{c}_i\gets Z_i^T\left(\sum_{j\in \text{Ch}(i)}\bm{c}_j\right)$ **if** $i$ is not root **for all** siblings $l$ of $i$ **do** $\bm{d}_l\gets \bm{d}_l+\Sigma_p \bm{c}_i$ **end for**
**if** $i$ is leaf **then** $\bm{y}_i\gets \bm{y}_i+U_i\bm{d}_i$ and return **end if** $\bm{d}_j\gets \bm{d}_j+W_i\bm{d}_i$, **if** $i$ is not root
Algorithm for Matrix Inversion
==============================
The objective is to compute $A^{-1}$. We first note that $A^{-1}$ has exactly the same structure as that of $A$. We repeat this observation mentioned in the main paper:
Let $A$ be recursively low-rank with a partitioning tree $T$ and a positive integer $r$. If $A$ is invertible and additionally, $A_{ii}-U_i\Sigma_pV_i^T$ is also invertible for all pairs of nonroot node $i$ and parent $p$, then there exists a recursively low-rank matrix $\widetilde{A}$ with the same partitioning tree $T$ and integer $r$, such that $\widetilde{A}=A^{-1}$. We denote the corresponding factors of $\widetilde{A}$ to be $$\label{eqn:factors.invA}
\{\widetilde{A}_{ii}, \widetilde{U}_i, \widetilde{V}_i, \widetilde{\Sigma}_p, \widetilde{W}_q, \widetilde{Z}_q \mid i \text{ is leaf, } p \text{ is nonleaf, } q \text{ is neither leaf nor root}\}.$$
This theorem may be proved by construction, which simultaneously gives all the factors in . Consider a pair of child node $p$ and parent $q$ and let $p$ have children such as $i$ and $j$. By noting that a diagonal block of $A_{pp}$ is $A_{ii}$ and an off-diagonal block is $A_{ij}=U_i\Sigma_pV_j^T$, we may write $A_{pp}-U_p\Sigma_qV_p^T$ as a block diagonal matrix (with diagonal blocks equal to $A_{ii}-U_i\Sigma_pV_i^T$) plus a rank-$r$ term: $$\label{eqn:AA}
A_{pp}-U_p\Sigma_qV_p^T=
\diag\Big[A_{ii}-U_i\Sigma_pV_i^T\Big]_{i\in\text{Ch}(p)}
+\begin{bmatrix}\vdots \\ U_i \\ \vdots\end{bmatrix}
(\Sigma_p-W_p\Sigma_qZ_p^T)
\begin{bmatrix}\cdots & V_i^T & \cdots\end{bmatrix}.$$ In fact, this equation also applies to $p=$ root, in which case one treats $\Sigma_q,W_p,Z_p=0$. Then, the Sherman–Morrison–Woodbury formula gives the inverse $$\label{eqn:iA}
(A_{pp}-U_p\Sigma_qV_p^T)^{-1}=
\diag\Big[(A_{ii}-U_i\Sigma_pV_i^T)^{-1}\Big]_{i\in\text{Ch}(p)}
+\begin{bmatrix}\vdots \\ \widetilde{U}_i \\ \vdots\end{bmatrix}
\widetilde{\Pi}_p
\begin{bmatrix}\cdots & \widetilde{V}_i^T & \cdots\end{bmatrix},$$ where the tilded factors are related to the non-tilded factors through $$\label{eqn:UV}
\widetilde{U}_i=(A_{ii}-U_i\Sigma_pV_i^T)^{-1}U_i, \qquad
\widetilde{V}_i=(A_{ii}-U_i\Sigma_pV_i^T)^{-T}V_i,$$ and $$\label{eqn:Pi}
\widetilde{\Pi}_p=
-(I+\widetilde{\Lambda}_p\widetilde{\Xi}_p)^{-1}\widetilde{\Lambda}_p
\quad\text{with}\quad
\widetilde{\Lambda}_p=\Sigma_p-W_p\Sigma_qZ_p^T
\quad\text{and}\quad
\widetilde{\Xi}_p=\sum_{i\in \text{Ch}(p)}V_i^T\widetilde{U}_i.$$ Equation immediately gives the $\widetilde{U}_i$ and $\widetilde{V}_i$ factors of $\widetilde{A}$ for all leaf nodes $i$. Further, right-multiplying $U_p$ to both sides of and similarly left-multiplying $V_p^T$ to both sides, we obtain $$\widetilde{W}_p=(I+\widetilde{\Pi}_p\widetilde{\Xi}_p)W_p \quad\text{and}\quad
\widetilde{Z}_p=(I+\widetilde{\Pi}_p^T\widetilde{\Xi}_p^T)Z_p,$$ which give the $\widetilde{W}_p$ and $\widetilde{Z}_p$ factors of $\widetilde{A}$ for all nonleaf and nonroot nodes $p$.
Additionally, may be interpreted as relating the inverse of $A_{pp}-U_p\Sigma_rV_p^T$ at some parent level $p$, to that of $A_{ii}-U_i\Sigma_pV_i^T$ at the child level $i$ with a rank-$r$ correction. Then, let $i$ be a leaf node and $(i,i_1,i_2,\ldots,i_s,1)$ be the path connecting $i$ and the root $=1$. We expand the chain of corrections and obtain $$\label{eqn:Aii}
\widetilde{A}(I_i,I_i)=(A_{ii}-U_i\Sigma_{i_1}V_i^T)^{-1}
+\widetilde{U}_i\widetilde{\Pi}_{i_1}\widetilde{V}_i^T
+\widetilde{U}_i\widetilde{W}_{i_1}\widetilde{\Pi}_{i_2}\widetilde{Z}_{i_1}^T\widetilde{V}_i^T
+\cdots
+(\widetilde{U}_i\widetilde{W}_{i_1}\cdots\widetilde{W}_{i_s}\widetilde{\Pi}_{1}\widetilde{Z}_{i_s}^T\cdots\widetilde{Z}_{i_1}^T\widetilde{V}_i^T).$$ Meanwhile, for any nonleaf node $p$, the factor $\widetilde{\Sigma}_p$ admits a similar chain of corrections: $$\label{eqn:Sigmap}
\widetilde{\Sigma}_p=\widetilde{\Pi}_p
+\widetilde{W}_p\widetilde{\Pi}_{p_1}\widetilde{Z}_p^T
+\widetilde{W}_p\widetilde{W}_{p_1}\widetilde{\Pi}_{p_2}\widetilde{Z}_{p_1}^T\widetilde{Z}_p^T
+\cdots
+(\widetilde{W}_p\widetilde{W}_{p_1}\cdots\widetilde{W}_{p_t}\widetilde{\Pi}_{1}\widetilde{Z}_{p_t}^T\cdots\widetilde{Z}_{p_1}^T\widetilde{Z}_p^T),$$ where $(p,p_1,p_2,\ldots,p_t,1)$ is the path connecting $p$ and the root $=1$. The above two formulas give the $\widetilde{A}_{ii}$ and $\widetilde{\Sigma}_p$ factors of $\widetilde{A}$ for all leaf nodes $i$ and nonleaf nodes $p$.
Hence, the computation of $\widetilde{A}$ consists of two tree walks, one upward and the other downward. In the upward phase, $\widetilde{U}_i$, $\widetilde{V}_i$, $\widetilde{W}_p$, and $\widetilde{Z}_p$ are computed. This phase also computes $(A_{ii}-U_i\Sigma_{i_1}V_i^T)^{-1}$ and $\widetilde{\Pi}_p$ as the starting point of corrections. Then, in the downward phase, a chain of corrections as detailed by and is performed from parent to children, which eventually yields the correct $\widetilde{A}_{ii}$ and $\widetilde{\Sigma}_p$. The overall computation is summarized in Algorithm \[algo:invA\]. The algorithm also includes straightforward modifications for the case of symmetric $A$.
\[algo.ln:patch1\] $\widetilde{A}_{ii}\gets (A_{ii}-U_i\Sigma_pV_i^T)^{-1}$ $\widetilde{U}_i\gets \widetilde{A}_{ii}U_i$ $\widetilde{V}_i\gets \widetilde{A}_{ii}^TV_i$ $\widetilde{\Theta}_i\gets V_i^T\widetilde{U}_i$ return $\widetilde{W}_j\gets (I+\widetilde{\Sigma}_j\widetilde{\Xi}_j)W_j$ **if** $j$ is not leaf $\widetilde{Z}_j\gets (I+\widetilde{\Sigma}_j^T\widetilde{\Xi}_j^T)Z_j$ **if** $j$ is not leaf $\widetilde{\Theta}_j\gets Z_j^T\widetilde{\Xi}_j\widetilde{W}_j$ **if** $j$ is not leaf $\widetilde{\Xi}_i\gets\sum_{j\in \text{Ch}(i)}\widetilde{\Theta}_j$ **if** $i$ is not root **then** $\widetilde{\Lambda}_i\gets \Sigma_i-W_i\Sigma_pZ_i^T$ **else** $\widetilde{\Lambda}_i\gets \Sigma_i$ **end if** \[algo.ln:patch2\] $\widetilde{\Sigma}_i\gets -(I+\widetilde{\Lambda}_i\widetilde{\Xi}_i)^{-1}\widetilde{\Lambda}_i$ $\widetilde{E}_j\gets\widetilde{W}_j\widetilde{\Sigma}_i\widetilde{Z}_j^T$ **if** $j$ is not leaf $\widetilde{E}_i\gets0$ **if** $i$ is root $\widetilde{A}_{ii}\gets\widetilde{A}_{ii}+\widetilde{U}_i\widetilde{\Sigma}_p\widetilde{V}_i^T$ **if** $i$ is not root $\widetilde{E}_i\gets \widetilde{E}_i+\widetilde{W}_i\widetilde{E}_p\widetilde{Z}_i^T$ **if** $i$ is not root $\widetilde{\Sigma}_i\gets\widetilde{\Sigma}_i+\widetilde{E}_i$ **for all** children $j$ of $i$ **do** **end for**
Algorithm for Determinant Computation
=====================================
The computation of the determinant $\delta=\det(A)$ is rather simple if done simultaneously with the inversion of $A$. The key idea is that one may apply Sylvester’s determinant theorem on to obtain $$\label{eqn:det}
\det(A_{pp}-U_p\Sigma_qV_p^T)=\det(I+\widetilde{\Lambda}_p\widetilde{\Xi}_p)
\prod_{i\in \text{Ch}(p)}\det(A_{ii}-U_i\Sigma_pV_i^T),$$ where $\widetilde{\Lambda}_p$ and $\widetilde{\Xi}_p$ are given in . In fact, $I+\widetilde{\Lambda}_p\widetilde{\Xi}_p$ must have been factorized in order to compute $\widetilde{\Pi}_p$ in ; hence its determinant is trivial to obtain. Then, the determinant of $A_{pp}-U_p\Sigma_qV_p^T$ at the parent $p$ is the product of those at the children $i$, multiplied by $\det(I+\widetilde{\Lambda}_p\widetilde{\Xi}_p)$. A simple recursion suffices for obtaining the determinant at the root. The procedure is summarized as Algorithm \[algo:detA\]. It is organized as an upward tree walk.
Note that the determinant easily overflows or underflows in finite precision arithmetics. A common treatment is to compute the log-determinant instead, in which case the multiplications in becomes summation. However, the log-determinant may be complex if $\det(A)$ is negative. Hence, if one wants to avoid complex arithmetic, as we do in Algorithm \[algo:detA\], one may use two quantities, the log-absolute-determinant $\log|\delta|$ and the sign $\sgn(\delta)$, to uniquely represent $\delta$.
Patch Algorithm \[algo:invA\]: Line \[algo.ln:patch1\]: Store $\log|\delta_i|$ and $\sgn(\delta_i)$, where $\delta_i=\det(A_{ii}-U_i\Sigma_pV_i^T)$ Line \[algo.ln:patch2\]: Store $\log|\delta_i|$ and $\sgn(\delta_i)$, where $\delta_i=\det(I+\widetilde{\Lambda}_i\widetilde{\Xi}_i)$
$\log|\delta|\gets\log|\delta_i|$; $\sgn(\delta)\gets\sgn(\delta_i)$ $\log|\delta|\gets\log|\delta|+\log|\delta_j|$; $\sgn(\delta)\gets\sgn(\delta)\cdot\sgn(\delta_j)$ $\log|\delta|$ and $\sgn(\delta)$
Algorithm for Cholesky-like Factorization
=========================================
The objective is to compute a factorization $A=GG^T$ when $A$ is symmetric positive definite. This factorization is not Cholesky in the traditional sense, because $G$ is not triangular. Rather, we would like to compute a $G$ that has the same structure as $A$, so that we can reuse the matrix-vector multiplication developed in Section \[sec:matvec\] on $G$. We repeat the existence theorem of $G$ mentioned in the main paper:
Let $A$ be recursively low-rank with a partitioning tree $T$ and a positive integer $r$. If $A$ is symmetric, by convention let $A$ be represented by the factors $$\{A_{ii}, U_i, U_i, \Sigma_p, W_q, W_q \mid i \text{ is leaf, } p \text{ is nonleaf, } q \text{ is neither leaf nor root}\}.$$ Furthermore, if $A$ is positive definite and additionally, $A_{ii}-U_i\Sigma_pU_i^T$ is also positive definite for all pairs of nonroot node $i$ and parent $p$, then there exists a recursively low-rank matrix $G$ with the same partitioning tree $T$ and integer $r$, and with factors $$\{G_{ii}, U_i, V_i, \Omega_p, W_q, Z_q \mid i \text{ is leaf, } p \text{ is nonleaf, } q \text{ is neither leaf nor root}\},$$ such that $A=GG^T$.
Note that in the theorem, $G$ and $A$ share factors $U_i$ and $W_q$. In other words, only the factors $G_{ii}$, $V_i$, $\Omega_p$, and $Z_q$ are to be determined. Similar to matrix inversion, we will prove this theorem through constructing these factors. Consider a pair of child node $p$ and parent $q$ and let $p$ have children such as $i$ and $j$. We repeat for the symmetric case in the following $$\label{eqn:AA2}
\underbrace{A_{pp}-U_p\Sigma_qU_p^T}_{B_{pp}}=
\diag\Big[\underbrace{A_{ii}-U_i\Sigma_pU_i^T}_{B_{ii}}\Big]_{i\in\text{Ch}(p)}
+\begin{bmatrix}\vdots \\ U_i \\ \vdots\end{bmatrix}
(\underbrace{\Sigma_p-W_p\Sigma_rW_p^T}_{\Lambda_p})
\begin{bmatrix}\cdots & U_i^T & \cdots\end{bmatrix},$$ and also write $$\label{eqn:GG2}
\underbrace{G_{pp}-U_p\Omega_qV_p^T}_{C_{pp}}=
\diag\Big[\underbrace{G_{ii}-U_i\Omega_pV_i^T}_{C_{ii}}\Big]_{i\in\text{Ch}(p)}
+\begin{bmatrix}\vdots \\ U_i \\ \vdots\end{bmatrix}
D_p
\begin{bmatrix}\cdots & V_i^T & \cdots\end{bmatrix}$$ for some $D_p$. Suppose we have computed $B_{ii}=C_{ii}C_{ii}^T$ for all $i\in\text{Ch}(p)$, then equating $B_{pp}=C_{pp}C_{pp}^T$ we obtain $$\label{eqn:V}
C_{ii}V_i=U_i$$ and $$\label{eqn:D}
\Lambda_p=D_p^T+D_p+D_p\Xi_pD_p^T
\quad\text{where}\quad
\Xi_p=\sum_{i\in \text{Ch}(p)}V_i^TV_i.$$ When $i$ is a leaf node, we let $C_{ii}$ be the Cholesky factor of $B_{ii}=A_{ii}-U_i\Sigma_pU_i^T$. Then, gives the factors $V_i$ of $G$ for all leaf nodes $i$: $V_i=C_{ii}^{-1}U_i$. Further, right-multiplying $V_p$ to both sides of and substituting , we have $W_p=(I+D_p\Xi_p)Z_p$, which gives the factors $Z_p$ of $G$ for all nonleaf and nonroot nodes $p$, provided that $D_p$ and $\Xi_p$ are known. The term $\Xi_p$ enjoys a simple recurrence relation that we omit here to avoid tediousness. On the other hand, the term $D_p$ is solved from . Equation is a continuous-time algebraic Riccati equation and it admits a symmetric solution $D_p$ when all the eigenvalues of $I+\Xi_p\Lambda_p$ are positive. It is not hard to see that the eigenvalues of $I+\Xi_p\Lambda_p$ are positive if and only if $B_{pp}$ is symmetric positive definite, which is satisfied based on the assumptions of the theorem. The solution $D_p$ may be computed by using the well-known Schur method [@Laub1979; @Arnold1984].
Additionally, may be interpreted as relating the Cholesky-like factor of $B_{pp}$ at some parent level $p$, to that of $B_{ii}$ at the child level $i$ with a rank-$r$ correction. Then, let $i$ be a leaf node and $(i,i_1,i_2,\ldots,i_s,1)$ be the path connecting $i$ and the root $=1$. We expand the chain of corrections and obtain $$\label{eqn:Gii}
G_{ii}=C_{ii}
+U_iD_{i_1}V_i^T
+U_iW_{i_1}D_{i_2}Z_{i_1}^TV_i^T
+\cdots
+(U_iW_{i_1}\cdots W_{i_s}D_{1}Z_{i_s}^T\cdots Z_{i_1}^TV_i^T).$$ Meanwhile, for any nonleaf node $p$, the factor $\Omega_p$ admits a similar chain of corrections: $$\label{eqn:Omegap}
\Omega_p=D_p
+W_pD_{p_1}Z_p^T
+W_pW_{p_1}D_{p_2}Z_{p_1}^TZ_p^T
+\cdots
+(W_pW_{p_1}\cdots W_{p_t}D_{1}Z_{p_t}^T\cdots Z_{p_1}^TZ_p^T),$$ where $(p,p_1,p_2,\ldots,p_t,1)$ is the path connecting $p$ and the root $=1$. The above two formulas give the $G_{ii}$ and $\Omega_p$ factors of $G$ for all leaf nodes $i$ and nonleaf nodes $p$.
Hence, the computation of $G$ consists of two tree walks, one upward and the other downward. In the upward phase, $V_i$ and $Z_p$ are computed. This phase also computes $C_{ii}$ and $D_p$ as the starting point of corrections. Then, in the downward phase, a chain of corrections as detailed by and are performed from parent to children, which eventually yields the correct $G_{ii}$ and $\Omega_p$. The overall computation is summarized in Algorithm \[algo:cholA\].
Copy all factors $U_i$ and $W_i$ from $A$ to $G$
Factorize $G_{ii}G_{ii}^T\gets A_{ii}-U_i\Sigma_pU_i^T$;$V_i\gets G_{ii}^{-1}U_i$;$\Theta_i\gets V_i^TV_i$ return $Z_j\gets (I+\Omega_j\Xi_j)^{-1}W_j$ **if** $j$ is not leaf $\Theta_j\gets Z_j^T\Xi_jZ_j$ **if** $j$ is not leaf $\Xi_i\gets\sum_{j\in \text{Ch}(i)}\Theta_j$ **if** $i$ is not root **then** $\Lambda_i\gets \Sigma_i-W_i\Sigma_pW_i^T$ **else** $\Lambda_i\gets \Sigma_i$ **end if** Solve $\Lambda_i=\Omega_i^T+\Omega_i+\Omega_i\Xi_i \Omega_i^T$ for $\Omega_i$ $E_j\gets W_j\Omega_iZ_j^T$ **if** $j$ is not leaf $E_i\gets0$ **if** $i$ is root $G_{ii}\gets G_{ii}+U_i\Omega_pV_i^T$ **if** $i$ is not root $E_i\gets E_i+W_iE_pZ_i^T$ **if** $i$ is not root $\Omega_i\gets\Omega_i+E_i$ **for all** children $j$ of $i$ **do** **end for**
Algorithm for Constructing $K_{\text{h}}$
=========================================
The computation is summarized in Algorithm \[algo:constructing.A\]. See Section \[sec:out.of.sample\] of the main paper.
Construct a partitioning tree and for every nonleaf node $i$, find landmark points $\ud{X}_i$
$A_{ii}\gets k(X_i,X_i)$;$U_i\gets k(X_i,\ud{X}_p)k(\ud{X}_p,\ud{X}_p)^{-1}$ $V_i\gets\text{empty matrix}$ return $\Sigma_i\gets k(\ud{X}_i,\ud{X}_i)$; $W_i\gets k(\ud{X}_i,\ud{X}_p)k(\ud{X}_p,\ud{X}_p)^{-1}$ **if** $i$ is not root $Z_i\gets\text{empty matrix}$ **for all** children $j$ of $i$ **do** **end for**
Algorithm for Computing $\bm{w}^T\bm{v}$ with $\bm{v}=k_{\text{h}}(X,\bm{x})$
=============================================================================
To begin with, note that $\bm{x}$ must lie in one of the subdomains $S_j$ for some leaf node $j$. We will abuse language and say that “$\bm{x}$ lies in the leaf node $j$” for simplicity. In such a case, the subvector $\bm{v}_j=k(X_j,\bm{x})$ and for any leaf node $l\ne j$, the subvector $$\bm{v}_l=U_lW_{l_1}W_{l_2}\cdots W_{l_s}\Sigma_pW_{j_t}^T\cdots W_{j_2}^TW_{j_1}^Tk(\ud{X}_{j_1},\ud{X}_{j_1})^{-1}k(\ud{X}_{j_1},\bm{x}),$$ where $p$ is the least common ancestor of $j$ and $l$, $(l,l_1,l_2,\ldots,l_s,p)$ is the path connecting $l$ and $p$, and $(j,j_1,j_2,\ldots,j_t,p)$ is the path connecting $j$ and $p$. Then, the inner product $$\bm{w}^T\bm{v}=\bm{w}_j^Tk(X_j,\bm{x})+\sum_{l\ne j,\,\, l \text{ is leaf}}
\bm{w}_l^TU_lW_{l_1}W_{l_2}\cdots W_{l_s}\Sigma_pW_{j_t}^T\cdots W_{j_2}^TW_{j_1}^Tk(\ud{X}_{j_1},\ud{X}_{j_1})^{-1}k(\ud{X}_{j_1},\bm{x}).$$
Similar to matrix-vector multiplications, we may define a few sets of auxiliary vectors to avoid duplicate computations. Specifically, define $\bm{x}$-independent vectors $$\bm{e}_i=\begin{dcases}
U_i^T\bm{w}_i, & \text{if $i$ is leaf},\\
W_i^T\sum_{j\in \text{Ch}(i)}\bm{e}_j, & \text{otherwise},\\
\end{dcases}$$ and $$\bm{c}_l=\Sigma_p^T\bm{e}_i\quad\text{for $i$ and $l$ being siblings with parent $p$},$$ and $\bm{x}$-dependent vectors $$\bm{d}_p=W_p^T\bm{d}_i\quad\text{for $p$ being the parent of $i$}; \qquad
\bm{d}_j=k(\ud{X}_{j_1},\ud{X}_{j_1})^{-1}k(\ud{X}_{j_1},\bm{x})\quad\text{for $\bm{x}$ lying in $j$}.$$ Then, the inner product is simplified as $$\bm{w}^T\bm{v}=\bm{w}_j^Tk(X_j,\bm{x})+\sum_{j_t\,\in\,\text{path connecting } j \text{ and root}}\bm{c}_{j_t}^T\bm{d}_{j_t}.$$
Hence, the computation of $\bm{w}^T\bm{v}$ consists of a full tree walk and a partial one, both upward. The first upward phase computes $\bm{e}_i$ from children to parent and simultaneously $\bm{c}_l$ by crossing sibling nodes from $i$ to $l$. This computation is independent of $\bm{x}$ and hence is considered preprocessing. The second upward phase computes $\bm{d}_{j_t}$ for all $j_t$ along the path connecting $j$ and the root. This phase visits only one path but not the whole tree, which is the reason why it costs less than $O(n)$. We summarize the detailed procedure in Algorithm \[algo:inprod\].
$\triangleright$ The above step is independent of $\bm{x}$ and is treated as preprocessing. In computer implementation, the intermediate results $\bm{c}_i$ are carried over to the next step , whereas the contents in $\bm{d}_i$ are discarded and the allocated memory is reused. $\bm{d}_i\gets U_i^T\bm{w}_i$ **for all** children $j$ of $i$ **do** **end for** $\bm{d}_i\gets W_i^T\left(\sum_{j\in \text{Ch}(i)}\bm{d}_j\right)$ **if** $i$ is not root **for all** siblings $l$ of $i$ **do** $\bm{c}_l\gets\Sigma_p^T \bm{d}_i$ **end for** $\bm{d}_i\gets k(\ud{X}_p,\ud{X}_p)^{-1}k(\ud{X}_p,\bm{x})$ $z\gets \bm{w}_i^Tk(X_i,\bm{x})$ Find the child $j$ (among all children of $i$) where $\bm{x}$ lies in $\bm{d}_i\gets W_i^T\bm{d}_j$ **if** $i$ is not root $z\gets z+\bm{c}_i^T\bm{d}_i$ **if** $i$ is not root
Algorithm for Computing $\bm{v}^T\widetilde{A}\bm{v}$ with $\bm{v}=k_{\text{h}}(X,\bm{x})$ for Symmetric $\widetilde{A}$
========================================================================================================================
We consider the general case where $\widetilde{A}$ is not necessarily related to the covariance function $k_{\text{h}}$; what is assumed is only symmetry. We recall that $\widetilde{A}$ is represented by the factors $$\{\widetilde{A}_{ii}, \widetilde{U}_i, \widetilde{U}_i, \widetilde{\Sigma}_p, \widetilde{W}_q, \widetilde{W}_q \mid i \text{ is leaf, } p \text{ is nonleaf, } q \text{ is neither leaf nor root}\}.$$
The derivation of the algorithm is more involved than that of the previous ones; hence, we need to introduce further notations. Let $\text{p}(i)$ denote the parent of a node $i$ and similarly $\text{p}(i,j)$ denote the common parent of $i$ and $j$. Let $(l,l_1,l_2,\ldots,l_t,p)$ be a path connecting nodes $l$ and $p$, where $l$ is a descendant of $p$. Denote this path as $\text{path}(l,p)$ for short. We will use subscripts $l\to p$ and $p\gets l$ to simplify the notation of the product chain of the $W$ factors: $$W_{l\to p}\equiv W_{l_1}W_{l_2}\cdots W_{l_t}
\quad\text{and}\quad
W_{p\gets l}^T\equiv W_{l_t}^T\cdots W_{l_2}^TW_{l_1}^T.$$ Note that the two ends of the path (i.e., $l$ and $p$) are not included in the product chain. If $l$ is a leaf and $p$ is the root, then every node $i\in\text{path}(l,p)$, except the root, has the parent also in the path, but its siblings are not. We collect all these sibling nodes to form a set $\text{B}(l)$. It is not hard to see that $\text{B}(l)\cup\{l\}$ is a disjoint partitioning of whole index set. Moreover, any two nodes from the set $\text{B}(l)\cup\{l\}$ must have a least common ancestor belonging to $\text{path}(l,\text{root})$; and this ancestor is the parent of (at least) one of the two nodes. If $\bm{x}$ lies in a leaf node $l$, $i$ is some node $\in\text{path}(l,\text{root})$, and $j\in \text{B}(l)$ is a sibling of $i$, then by reusing the $\bm{d}$ vectors defined in the preceding subsection, we have $$\label{eqn:vlj}
\bm{v}_l=k(X_l,\bm{x}) \quad\text{and}\quad
\bm{v}_j=U_j\Sigma_{\text{p}(j)}W_{\text{p}(j)\gets l}^Tk(\ud{X}_{\text{p}(l)},\ud{X}_{\text{p}(l)})^{-1}k(\ud{X}_{\text{p}(l)},\bm{x})
=U_j\Sigma_{\text{p}(j,i)}\bm{d}_i.$$
Because $\text{B}(l)\cup\{l\}$ forms a disjoint partitioning of whole index set, the quadratic form $\bm{v}^T\widetilde{A}\bm{v}$ consists of three parts: $$\bm{v}^T\widetilde{A}\bm{v}
=\bm{v}_l^T\widetilde{A}_{ll}\bm{v}_l
+\sum_{i\in \text{B}(l)} \bm{v}_i^T\widetilde{A}_{ii}\bm{v}_i
+\sum_{\substack{i,j\in \text{B}(l)\\i\ne j}} \bm{v}_i^T\widetilde{A}_{ij}\bm{v}_j.$$ The first part involving the leaf node $l$ is straightforward. For the second part, we expand $\bm{v}_i$ by using and define two quantities therein: $$\bm{v}_i^T\widetilde{A}_{ii}\bm{v}_i
=\Big( \bm{d}_t^T\underbrace{\Sigma_{\text{p}(i,t)}^T\overbrace{U_i^T \Big) \widetilde{A}_{ii} \Big( U_i}^{\Xi_i}\Sigma_{\text{p}(i,t)}}_{\widetilde{\Xi}_i}\bm{d}_t \Big),$$ where $t$ as a sibling of $i$ belongs to $\text{path}(l,\text{root})$. For the third part, we similarly expand each individual term and define additionally two quantities: $$\bm{v}_i^T\widetilde{A}_{ij}\bm{v}_j
=\Big( \bm{d}_s^T
\underbrace{\Sigma_{\text{p}(i,s)}^T\overbrace{U_i^T \Big) \Big( \widetilde{U}_i}^{\Theta_i^T}}_{\widetilde{\Theta}_i^T}
\widetilde{W}_{i\rightarrow q}\widetilde{\Sigma}_q\widetilde{W}_{q\leftarrow j}^T
\underbrace{\overbrace{\widetilde{U}_j^T \Big) \Big( U_j}^{\Theta_j}\Sigma_{\text{p}(j,t)}}_{\widetilde{\Theta}_j}
\bm{d}_t \Big),$$ where $s$ as a sibling of $i$ belongs to $\text{path}(l,\text{root})$, $t$ as a sibling of $j$ belongs to the same path, and $q$ is the least common ancestor of $i$ and $j$. The four newly introduced quantities $\Xi_i$, $\widetilde{\Xi}_i$, $\Theta_i$, and $\widetilde{\Theta}_i$ are independent of $\bm{x}$ and may be computed in preprocessing, in a recursive manner from children to parent. We omit the simple recurrence relation here to avoid tediousness. Then, the quadratic form $\bm{v}^T\widetilde{A}\bm{v}$ admits the following expression: $$\bm{v}^T\widetilde{A}\bm{v}
=\bm{v}_l^T\widetilde{A}_{ll}\bm{v}_l
+\sum_{i\in \text{B}(l)} \bm{d}_t^T\widetilde{\Xi}_i\bm{d}_t
+\sum_{\substack{i,j\in \text{B}(l)\\i\ne j}} \bm{d}_s^T\widetilde{\Theta}_i^T\widetilde{W}_{i\rightarrow q}\widetilde{\Sigma}_q\widetilde{W}_{q\leftarrow j}^T\widetilde{\Theta}_j\bm{d}_t.$$
We may further simplify the summation in the last term of this equation to avoid duplicate computation. As mentioned, any two nodes in $\text{B}(l)$ have a least common ancestor that happens to be the parent of one of them. Assume that this node is $i$. Then, we write $$\sum_{\substack{i,j\in \text{B}(l)\\i\ne j}} \bm{d}_s^T\widetilde{\Theta}_i^T\widetilde{W}_{i\rightarrow q}\widetilde{\Sigma}_q\widetilde{W}_{q\leftarrow j}^T\widetilde{\Theta}_j\bm{d}_t
=2\sum_{i\in \text{B}(l)} \bm{d}_s^T\widetilde{\Theta}_i^T\widetilde{\Sigma}_{\text{p}(i)}\sum_{\substack{j\in\text{B}(l),\,\,j\ne i \\ j \text{ is descendant of p}(i)}}\widetilde{W}_{\text{p}(i)\leftarrow j}^T\widetilde{\Theta}_j\bm{d}_t.$$ Note the inner summation on the right-hand side of this equality. This quantity iteratively accumulates as $i$ moves up the tree. Therefore, we define $$\bm{c}_i=\begin{dcases}
\widetilde{\Theta}_i\bm{d}_s, & \text{if $i\in \text{B}(l)$},\\
\widetilde{W}_i^T\sum_{j\in \text{Ch}(i)}\bm{c}_j, & \text{if $i\in \text{path}(l,\text{root})$},
\end{dcases}$$ where recall that $s$ as a sibling of $i$ belongs to $\text{path}(l,\text{root})$. Then, the inner summation becomes $\bm{c}_s$. In other words, $$\sum_{\substack{i,j\in \text{B}(l)\\i\ne j}} \bm{d}_s^T\widetilde{\Theta}_i^T\widetilde{W}_{i\rightarrow q}\widetilde{\Sigma}_q\widetilde{W}_{q\leftarrow j}^T\widetilde{\Theta}_j\bm{d}_t
=2\sum_{i\in \text{B}(l)}\bm{c}_i^T\widetilde{\Sigma}_{\text{p}(i,s)}\bm{c}_s.$$
To summarize, the computation of $\bm{v}^T\widetilde{A}\bm{v}$ consists of a full tree walk and a partial one, both upward. The first upward phase computes $\Xi_i$, $\widetilde{\Xi}_i$, $\Theta_i$, and $\widetilde{\Theta}_i$ recursively from children to parent. This computation is independent of $\bm{x}$ and hence is considered preprocessing. The second upward phase computes $\bm{d}_s$ and $\bm{c}_s$ for all $s$ along the path connecting $l$ and the root (assuming $\bm{x}\in S_l$), as well as all $\bm{c}_i$ for $i$ being sibling nodes of $s$. This phase visits only one path but not the whole tree, which is the reason why it costs less than $O(n)$. The detailed procedure is given in Algorithm \[algo:quadratic.form\].
$\triangleright$ The above step is independent of $\bm{x}$ and is treated as preprocessing. $\Theta_i\gets\widetilde{U}_i^TU_i$;$\widetilde{\Theta}_i\gets\Theta_i\Sigma_p$ $\Xi_i\gets U_i^T\widetilde{A}_iU_i$;$\widetilde{\Xi}_i\gets\Sigma_p^T\Xi_i\Sigma_p$ return **for all** children $j$ of $i$ **do** **end for** $\Theta_i\gets\widetilde{W}_i^T\left(\sum_{j\in \text{Ch}(i)}\Theta_j\right)W_i$;$\widetilde{\Theta}_i\gets\Theta_i\Sigma_p$ $\Xi_i\gets W_i^T\left(\sum_{j\in \text{Ch}(i)}\Xi_j+\sum_{\substack{j,k\in \text{Ch}(i)\\j\ne k}}\Theta_j^T\widetilde{\Sigma}_i\Theta_k\right)W_i$;$\widetilde{\Xi}_i\gets\Sigma_p^T\Xi_i\Sigma_p$ $\bm{d}_i\gets k(\ud{X}_p,\ud{X}_p)^{-1}k(\ud{X}_p,\bm{x})$ $\bm{c}_i\gets\widetilde{U}_i^Tk(X_i,\bm{x})$ $z\gets k(\bm{x},X_i)\widetilde{A}_ik(X_i,\bm{x})$ Find the child $j$ (among all children of $i$) where $\bm{x}$ lies in $\bm{d}_i\gets W_i^T\bm{d}_j$ **if** $i$ is not root $\bm{c}_l\gets\widetilde{\Theta}_l\bm{d}_i$ $z\gets z+\bm{d}_i^T\widetilde{\Xi}_l\bm{d}_i+2\bm{c}_l^T\widetilde{\Sigma}_p\bm{c}_i$ $\bm{c}_p\gets \widetilde{W}_p^T\left(\sum_{j\in \text{Ch}(p)}\bm{c}_j\right)$ **if** $p$ is not root
Cost Analysis {#cost-analysis}
=============
The storage cost has been analyzed in the main paper. In what follows is the analysis of arithmetic costs.
Arithmetic Cost of Matrix-Vector Multiplication (Algorithm \[algo:Ab\])
-----------------------------------------------------------------------
The algorithm consists of two tree walks, each of which visits all the $O(n/r)$ nodes. Inside each tree node, the computation is dominated by $O(1)$ matrix-vector multiplications with $r\times r$ matrices; hence the per-node cost is $O(r^2)$. Then, the overall cost is $O(n/r\times r^2)=O(nr)$.
Arithmetic Cost of Matrix Inversion (Algorithm \[algo:invA\])
-------------------------------------------------------------
The algorithm consists of two tree walks, each of which visits all the $O(n/r)$ nodes. Inside each tree node, the computation is dominated by $O(1)$ matrix operations (matrix-matrix multiplications and inversions) with $r\times r$ matrices; hence the per-node cost is $O(r^3)$. Then, the overall cost is $O(n/r\times r^3)=O(nr^2)$.
Arithmetic Cost of Determinant Computation (Algorithm \[algo:detA\])
--------------------------------------------------------------------
The algorithm requires patching Algorithm \[algo:invA\] with additional computations that do not affect the $O(nr^2)$ cost of Algorithm \[algo:invA\]. Omitting the patching, Algorithm \[algo:detA\] visits every tree node once and the computation per node is $O(1)$. Hence, the cost of this algorithm is only $O(n/r)$.
In practice, we indeed implement the patching inside Algorithm \[algo:invA\].
Arithmetic Cost of Cholesky-like Factorization (Algorithm \[algo:cholA\])
-------------------------------------------------------------------------
The cost analysis of this algorithm is almost the same as that of Algorithm \[algo:invA\], except that the dominating per-node computation also includes Cholesky factorization of $r\times r$ matrices and the solving of continuous-time algebraic Riccati equation of size $r\times r$. Both costs are $O(r^3)$, the same as that of matrix-matrix multiplications and inversions. Hence, the overall cost of this algorithm is $O(nr^2)$.
Arithmetic Cost of Constructing $K_{\text{h}}$ (Algorithm \[algo:constructing.A\])
----------------------------------------------------------------------------------
The algorithm consists of three parts: (i) hierarchical partitioning of the domain; (ii) finding landmark points; and (iii) instantiating the factors of a symmetric recursively low-rank matrix.
For part (i), much flexibility exists. In practice, partitioning is data driven, which ensures that the number of points is balanced in all leaf nodes. If we assume that the cost of partitioning a set of $n$ points is $O(n)$, then the overall partitioning cost counting recursion is $O(n\log n)$.
Similarly, part (ii) depends on the specific method used for choosing the landmark points. In general, we may assume that choosing $r$ landmark points costs $O(r)$. Then, because each of the $O(n/r)$ nonleaf nodes has a set of landmark points, the cost is $O(n/r\times r)=O(n)$.
Part (iii) is a tree walk that visits each of the $O(n/r)$ nodes once. The per-node computation is dominated by constructing one or a few $r\times r$ covariance matrices and performing matrix-matrix multiplications and inversions. We assume that constructing an $r\times r$ covariance matrix costs $O(r^2)$, which is less expensive than the $O(r^3)$ cost of matrix-matrix multiplications and inversions. Then, the overall cost for instantiating the overall matrix is $O(n/r\times r^3)=O(nr^2)$.
Arithmetic Cost of Computing $\bm{w}^T\bm{v}$ (Algorithm \[algo:inprod\])
-------------------------------------------------------------------------
The algorithm consists of two tree walks (one full and one partial): the first one is $\bm{x}$-independent preprocessing and the second one is $\bm{x}$-dependent.
For preprocessing, the tree walk visits all the $O(n/r)$ nodes. Inside each tree node, the computation is dominated by $O(1)$ matrix-vector multiplications with $r\times r$ matrices; hence the per-node cost is $O(r^2)$. Then, the overall preprocessing cost is $O(n/r\times r^2)=O(nr)$.
For the $\bm{x}$-dependent computation, only $O(h)=O(\log_2(n/r))$ tree nodes are visited. Inside each visited node, the computation is dominated by $O(1)$ matrix-vector multiplications with $r\times r$ matrices; hence the per-node cost is $O(r^2)$. Here, we assume that finding the child node where $\bm{x}$ lies in has $O(1)$ cost. Note also that although the computation of the $\bm{d}$ vectors requires a matrix inverse, the matrix in fact has been prefactorized when constructing $K_{\text{h}}$ (that is, inside Algorithm \[algo:constructing.A\]). Hence, the per-node cost is not $O(r^3)$. To conclude, the $\bm{x}$-dependent cost is $O(r^2\log_2(n/r))$.
Arithmetic Cost of Computing $\bm{v}^T\widetilde{A}\bm{v}$ (Algorithm \[algo:quadratic.form\])
----------------------------------------------------------------------------------------------
The cost analysis of this algorithm is almost the same as that of Algorithm \[algo:inprod\], except that in the preprocessing phase, the dominant per-node computation is $O(1)$ matrix-matrix multiplications with $r\times r$ matrices. Hence, the preprocessing cost is $O(n/r\times r^3)=O(nr^2)$ whereas the $\bm{x}$-dependent cost is still $O(r^2\log_2(n/r))$.
[^1]: MIT-IBM Watson AI Lab, IBM Research. Email: `chenjie@us.ibm.com`
[^2]: University of Chicago. Emails: `stein@galton.uchicago.edu`
[^3]: These libraries are the elementary components of commonly used software such as R, Matlab, and python.
[^4]: Regression often assumes a noise term that we omit here for simplicity. An alternative way to view the noise term is that the covariance function has a nugget.
[^5]: <https://www.ncdc.noaa.gov/data-access/model-data/model-datasets/climate-forecast-system-version2-cfsv2>
|
---
abstract: 'To obtain the eigenfrequencies of a protoneutron star (PNS) in the postbounce phase of core-collapse supernovae (CCSNe), we perform a linear perturbation analysis of the angle-averaged PNS profiles using results from a general relativistic CCSN simulation of a $15 M_{\odot}$ star. In this work, we investigate how the choice of the outer boundary condition could affect the PNS oscillation modes in the linear analysis. By changing the density at the outer boundary of the PNS surface in a parametric manner, we show that the eigenfrequencies strongly depend on the surface density. By comparing with the gravitational wave (GW) signatures obtained in the hydrodynamics simulation, the so-called surface $g$-mode of the PNS can be well ascribed to the fundamental oscillations of the PNS. The frequency of the fundamental oscillations can be fitted by a function of the mass and radius of the PNS similar to the case of cold neutron stars. In the case that the position of the outer boundary is chosen to cover not only the PNS but also the surrounding postshock region, we obtain the eigenfrequencies close to the modulation frequencies of the standing accretion-shock instability (SASI). However, we point out that these oscillation modes are unlikely to have the same physical origin of the SASI modes seen in the hydrodynamics simulation. We discuss possible limitations of applying the angle-averaged, linear perturbation analysis to extract the full ingredients of the CCSN GW signatures.'
author:
- Hajime Sotani
- Takami Kuroda
- Tomoya Takiwaki
- Kei Kotake
title: ' Dependence of outer boundary condition on protoneutron star asteroseismology with gravitational-wave signatures '
---
Introduction {#sec:I}
============
Success of direct observations of gravitational waves (GWs) from the compact binary mergers ushered in a new era of GW astronomy. Up to now, GWs from five binary black hole (BH) mergers, i.e., GW150914 [@GW1], GW151226 [@GW2], GW170104 [@GW3], GW170608 [@GW4], and GW170814 [@GW5], and one binary neutron star (NS) merger, i.e., GW170817 [@GW6], have been detected by LIGO (Laser Interferometer Gravitational-wave Observatory) Scientific Collaboration and Virgo Collaboration. In the event of GW170817 [@EM], the electromagnetic-wave counterpart has been detected, which opens yet another new era of multi-messenger astronomy. In addition to the advanced LIGO and advanced Virgo, KAGRA will be operational in the coming years [@aso13]. Furthermore, the third-generation detectors have been proposed such as Einstein Telescope and Cosmic Explorer [@punturo; @CE]. At such high level of precision, these detectors are sensitive enough to a wide variety of compact objects. Next to the primary targets of the compact binary coalescence, other intriguing sources include core-collapse supernovae (CCSNe) [@KotakeGWreview], which mark the catastrophic end of massive stars and produce all these compact objects.
In order to study the GW signatures from CCSNe, numerical simulations have been done extensively (e.g., [@MJM2013; @CDAF2013; @KKT2016; @Ott13; @Andresen16; @Murphy09; @Yakunin15; @OC2018]). The most distinct GW emission process commonly seen in recent self-consistent three-dimensional (3D) models is associated with the excitation of core/protoneutron star (PNS) oscillatory modes [@KKT2016; @radice2018; @Andresen18]. This is supported by the evidence that the Brunt-Väisälä frequency estimated at the PNS surface is in good accordance with that of the strongest GW component. The typical GW frequency of the surface $g$-mode is approximately expressed by $GM_{\rm PNS}/R^2_{\rm PNS}$ [@MJM2013; @CDAF2013; @KKT2016; @Murphy09] with $G$ the gravitational constant, $M_{\rm PNS}$ and $R_{\rm PNS}$ the mass and radius of the PNS, respectively. In the postbounce phase, the PNS mass increases with time due to the mass accretion and the PNS radius decreases with time due to the mass accretion onto the PNS and neutrino cooling. Accordingly, the typical GW frequency of the surface $g$-mode increases with time after bounce [@MJM2013; @Murphy09], which is roughly in the range of $\sim 500-1000$ Hz. These oscillations are excited because the PNS surface is chimed by the mass motions. Recent studies indicate that the dominant excitation process may be sensitive to the spacial dimension in the hydrodynamics simulations. In axisymmetric two-dimensional (2D) models, the mass accretion from above the PNS, where the mass accretion activity to the PNS is influenced by the growth of neutrino-driven convection and the standing accretion-shock instability (SASI) [@Blondin03; @Foglizzo06], are the main excitation process of the PNS surface oscillations [@Murphy09; @MJM2013]. While Ref. [@Andresen16] showed in the 3D models that the PNS convection could also significantly contribute to the postbounce GW emission. In addition to the PNS oscillations, recent 3D CCSN models have shown another remarkable GW signatures whose frequencies are close to the modulation frequency of the SASI motion, i.e., $\sim 100$ Hz [@KKT2016; @Andresen16; @OC2018; @radice2018]. Thus, the detection of the GWs with $\sim 100$ Hz separately from those with $\sim 500-1000$ Hz may provide a probe into the SASI activity in the pre-explosion supernova core [@KKT2016; @Andresen16].
The hydrodynamics modeling is really powerful to clarify the inner-workings of the forming compact objects, while linear perturbation approaches are also valuable to understand the physics behind the numerical results obtained by simulations. Given angle-averaged profiles obtained in hydrodynamics models, oscillation spectra are determined by a linear analysis. Then, if one could find a correlation between the properties of the background model and the resultant oscillation spectra, one can extract the information about the background model through observations of the spectra. This technique is known as asteroseismology, which has been extensively investigated in the context of cold NSs. With this technique, it has been suggested that the properties of the NSs such as the mass ($M$), radius ($R$), and EOS, would be constrained with GW asteroseismology, where one would get an information about the source object with the GW spectra (e.g., [@AK1996; @AK1998; @KS1999; @STM2001; @SH2003; @SYMT2011; @PA2012; @DGKK2013]).
Compared to a lot of studies with the linear perturbation analysis on cold NSs, similar studies on PNSs are very few [@FMP2003; @FKAO2015; @ST2016; @Camelio17; @SKTK2017; @TCPF2018; @MRBV2018; @TCPOF2019]. The paucity of the perturbative studies on the PNSs may come from the difficulty for preparing for the background model of the PNSs. That is, unlike the case of nearly hydrostatic cold NSs, one also needs the time dependent radial distributions of the electron fraction and, e.g., the entropy per baryon for constructing the finite temperature PNS models. However, these time dependent spatial profiles are determined only via the self-consistent CCSN simulations which are computationally expensive. Among the recent studies to tackle with this problem [@FMP2003; @FKAO2015; @ST2016; @Camelio17; @SKTK2017; @TCPF2018; @MRBV2018; @TCPOF2019], we have found that the frequencies of the fundamental $(f)$ and the space-time $(w)$ modes [@ST2016; @SKTK2017] can be respectively expressed as a function of the average density and compactness of the PNSs almost independently of the EOS of PNSs, in a similar way to the case of cold NSs [@AK1996; @AK1998]. In this context, a universal relation of the CCSN GW spectra is recently reported in Ref. [@TCOMF2019a].
Up to now, two representative ways have been proposed for constructing the background PNS models for determining the eigenfrequencies in the linear perturbation analysis. They differ in the definition of the PNS surface. One is the PNS model, in which the surface density is fixed as a specific value, for example, of $\sim10^{10}$ g cm$^{-3}$ [@ST2016; @SKTK2017; @MRBV2018]. In this case, one can impose the boundary condition similarly as taken in the stellar oscillation analysis and can classify the stellar oscillations. However, unlike the usual cold NS case, the low density matter still hovers and the accretion shock also exists outside this density region, whose influences might not be negligible. Therefore the PNS model covering up to the shock radius is also proposed [@TCPF2018; @TCPOF2019]. With the boundary condition imposed at the shock, one can investigate the global oscillations inside the whole postshock region, although the eigenvalue problem to solve is significantly different from that with the PNS model with the fixed surface density.
In this study, we calculate the eigenfrequencies in the PNS models by the linear perturbation analysis with the two different boundary conditions, i.e., either at the PNS surface with a fixed specific density or at the shock radius, with an attempt to identify the excitation mechanism of the GW signatures seen in the numerical simulation. This paper is organized as follows. Section \[PNSmodel\] starts with a brief summary of the PNS models employed in this work. In Section \[sec:III\], we describe the linear perturbation analysis to solve the eigenvalue problem. Section \[sec:IV\] presents our results and the comparison with the GW signal computed in the numerical simulations. We summarize our results and discuss their implications in Section \[sec:V\]. Unless otherwise mentioned, we adopt geometric units in the following, $c=G=1$, where $c$ denotes the speed of light, and the metric signature is $(-,+,+,+)$. The time is measured after bounce ($T_{\rm pb} = 0)$
PNS Models {#PNSmodel}
==========
The line element is expressed as $$ds^2=-\alpha^2dt^2+\gamma_{ij}(dx^i+\beta^idt)(dx^j+\beta^jdt), \label{eq:ds_BSSN}$$ where $\alpha$, $\beta^i$, and $\gamma_{ij}$ are the lapse, shift vector, and three metric, respectively. To prepare the background of PNS models, the metric functions $\alpha$, $\beta^i$, and $\gamma_{ij}$ from hydrodynamics simulations, which are not spherically symmetric, are transformed into the spherically symmetric properties, assuming that the hydrodynamic background at each time step is also static and spherically symmetric. In this procedure, all variables defined on the Cartesian coordinates in numerical relativity simulations are transformed into those in polar coordinates by spatially linear interpolation at each time step. Then, the space-time in the isotropic coordinates can be rewritten as $$\begin{aligned}
ds^2 =&-\alpha^2 dt^2+ \gamma_{\hat{r}\hat{r}}
(d\hat{r}^2+\hat{r}^2d\theta^2+\hat{r}^2\rm{sin}^2\theta d\phi^2),
\label{eq:ds_isotropic}\end{aligned}$$ where $\hat{r}$ denotes the isotropic radius $\hat{r}=\sqrt{x^2+y^2+z^2}$.
In the calculations of stellar oscillations, we adopt the following spherically symmetric space-time $$ds^2 =-e^{2\Phi} dt^2 + e^{2\Lambda} dr^2 + r^2\left(d\theta^2 + \sin^2\theta d\phi^2\right), \label{eq:ds_spherical}$$ as a background space-time, where $\Phi$ and $\Lambda$ are functions of only $r$. We remark that the metric expressed by Eq. (\[eq:ds\_spherical\]) is similar to the Schwarzschild metric and is given by the coordinate transformation from the isotropic coordinates, i.e. Eqs. (\[eq:ds\_BSSN\]) or (\[eq:ds\_isotropic\]). Additionally, the metric function $\Lambda$ is associated with the mass function $m$ in such a way that $e^{-2\Lambda}=1-2m/r$. Then, the background four-velocity of the fluid element is given by $u^\mu=(e^{-\Phi},0,0,0)$. Comparing Eqs.(\[eq:ds\_isotropic\]) and (\[eq:ds\_spherical\]), the conversion relation is expressed as followings
& &&e\^[2]{} =\^2,&& \[eq:conv\_1\]\
& &&r\^2=\_\^2,&& \[eq:conv\_2\]\
&&&&&\
& &&e\^[2]{}dr\^2=\_d\^2.&& \[eq:conv\_3\]
From these, one can deduce the following relations
dr=&(\_+)d,\[eq:dr\]\
m=&. \[eq:m\]
In this study, instead of using Eq. (\[eq:m\]), we evaluate the enclosed gravitational mass $m$ within $\hat r$ and use a simple conversion relation $r=\hat r(1+m/2\hat r)^2$, from isotropic to Schwarzschild coordinates. Although this simple conversion relation can originally be applied to the exterior of the object, we employ it as it can suppress the high frequency structural noise that appears when using Eq. (\[eq:m\]) without some appropriate smoothing. Since we use the spatial derivative of $\Lambda$ that is a function of $m$ in the following seismology analysis, spurious noise should be suppressed. We consider that the difference between the correct, i.e. Eq. (\[eq:m\]), and simple evaluations is not so significant. The highest values of $\exp{(2\Lambda)}=(1-2m/r)^{-1}$ appear at $\hat r\sim 1.3\times10^6$ cm and they differ approximately 1 % between both evaluations.
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In the present study, we especially focus on the numerical results constructed with SFHx EOS [@SHF2013]. The initial hydrodynamic profile is taken from a $15M_\odot$ progenitor model [@WW95] in the simulation [@KKT2016]. In Fig. \[fig:back\], we show the radial profiles of the rest mass density $\rho$, entropy per baryon $s$, and electron fraction $Y_e$ at several time snapshots after bounce. From this figure, one can observe that the profiles at 248 ms is almost the same as that at 348 ms. On these background properties, we consider the specific oscillations in PNS at each time step. As PNS models, we consider two different approaches, i.e., 1) as in Ref. [@MRBV2018], the position, where the rest mass density is equivalent to be $\rho_s=5\times 10^9$, $10^{10}$, and $10^{11}$ g cm$^{-3}$, is considered as the stellar surface of a background PNS, or 2) the domain inside the shock radius is adopted for calculating the frequencies of stellar oscillations as in Refs. [@TCPF2018; @TCPOF2019]. Here, we define the position of the shock radius, where the entropy per baryon becomes $s=7$ $k_{\rm B}$ baryon$^{-1}$ at the outermost radial position with excluding obviously infalling unshocked stellar mantles. In Fig. \[fig:MtRt\], we show the time evolution of the PNS gravitational mass and radius, which are determined with different definitions of the PNS surface, as a function of the postbounce time $T_{\rm pb}$. One can observe that the gravitational masses after $\sim 150$ ms are almost independent from the definition of PNS surface, while the PNS radius still depends on the surface density. In the right panel of Fig. \[fig:MtRt\], we also show the shock radius, which does not change monotonically with time due to the vigorous SASI motion [@KKT2016].
Perturbation equations in the Cowling approximation {#sec:III}
===================================================
In this paper, we simply assume the relativistic Cowling approximation [@Finn1988], i.e., the metric perturbations are neglected during the stellar oscillations, where the oscillation frequencies can be discussed qualitatively but the damping of oscillations (or the imaginary part of complex frequencies) due to the GW emission can not be calculated. We remark that our perturbation formalism is basically the same as in Ref. [@MRBV2018] with $\delta\hat{\alpha}=0$, noting that this should be improved in our future work as in [@MRBV2018]. The Lagrangian displacement vector of fluid element $\xi^i$ for the polar type oscillations is given by $$\xi^i(t,r,\theta,\phi) = \left(e^{-\Lambda}W, -V\partial_\theta, -\frac{V}{\sin^2\theta}\partial_\phi\right)\frac{1}{r^2}Y_{\ell k}(\theta,\phi),$$ where $W$ and $V$ are a function of $t$ and $r$, while $Y_{\ell k}(\theta,\phi)$ denotes the spherical harmonics with the azimuthal quantum number $\ell$ and the magnetic quantum number $k$. With $\xi^i$, one can obtain the perturbed four-velocity $\delta u^\mu$ as $$\delta u^\mu = \left(0, e^{-\Lambda}\dot{W}, -\dot{V}\partial_\theta, -\frac{\dot V}{\sin^2\theta}\partial_\phi\right)
\frac{1}{r^2}e^{-\Phi}Y_{\ell k}, \label{eq:du}$$ where the dot denotes the partial derivative with respect to $t$. In addition, one should add the perturbations of the baryon number density $n_{\rm b}$, the pressure $p$, and the energy density $\varepsilon$.
From the baryon number conservation with the Cowling approximation, one can obtain the relation as $$\frac{\Delta n_{\rm b}}{n_{\rm b}} = -\left[e^{-\Lambda}W' + \ell(\ell+1)V\right]\frac{1}{r^2}Y_{\ell k}, \label{eq:dn}$$ where $\Delta n_{\rm b}$ is the Lagrangian perturbation of the baryon number density and the prime denotes the partial derivative with respect to $r$. Assuming the adiabatic perturbations, the Lagrangian perturbations of the pressure ($\Delta p$) and $\Delta n_{\rm b}$ are related to the adiabatic index $\Gamma_1$ via $$\Gamma_1 \equiv \left(\frac{\partial \ln p}{\partial \ln n_{\rm b}}\right)_s = \frac{n_{\rm b}}{p}\frac{\Delta p}{\Delta n_{\rm b}},
\label{eq:Gamma1}$$ while one can get the additional equation from the energy conservation law (or the first law of thermodynamics), i.e., $$\Delta\varepsilon = (\varepsilon + p)\frac{\Delta n_{\rm b}}{n_{\rm b}}, \label{eq:e_con}$$ where $\Delta \varepsilon$ denotes the Lagrangian perturbation of the energy density. Since the Lagrangian perturbation of a property $x$, i.e., $\Delta x$, is associated with the Eulerian perturbation ($\delta x$) in the linear analysis, such as $\Delta x = \delta x + \xi^i\partial_i x$, by combining Eqs. (\[eq:Gamma1\]) and (\[eq:e\_con\]), one can obtain that $$\delta p = c_s^2 \delta\varepsilon + p\Gamma_1 {\cal A}\xi^r, \label{eq:dp0}$$ where $c_s$ is the sound velocity and ${\cal A}$ is the relativistic Schwarzschild discriminant given by $$\begin{gathered}
c_s^2 \equiv \left(\frac{\partial p}{\partial \varepsilon}\right)_s
= \frac{\Delta p}{\Delta \varepsilon}
= \frac{p\Gamma_1}{\varepsilon + p}, \\
{\cal A}(r) \equiv \frac{\varepsilon'}{\varepsilon + p} - \frac{p'}{p\Gamma_1}
= \frac{1}{\varepsilon + p}\left(\varepsilon' - \frac{p'}{c_s^2}\right).\end{gathered}$$ We remark that ${\cal A}$ is a little different from that introduced in Ref. [@Finn1988], where the factor $e^{-\Lambda}$ is also included in the discriminant. We also remark that $c_s$ is determined from the adopted EOS independently of the stellar structure, while ${\cal A}$ is determined only with the stellar structure. With this discriminant ${\cal A}$, the relativistic Brunt-Väisälä frequency, $f_{\rm BV}$, is given by $$f_{\rm BV} = {\rm sgn}({\cal N}^2)\sqrt{|{\cal N}^2|} / 2\pi,$$ where ${\cal N}^2$ is given by [@Finn1988] $${\cal N}(r)^2 = -\Phi'e^{2\Phi-2\Lambda}{\cal A}(r).$$ We remark that the region with ${\cal A}>0$ (${\cal A}<0$), which corresponds to ${\cal N}^2<0$ (${\cal N}^2>0$), is stable (unstable) with respect to the convection. The radial profiles of ${\cal A}$ and $f_{\rm BV}$ at $T_{\rm pb}=48$, 148, 248, and 348 ms are shown in Fig. \[fig:BV\]. From this figure, one can see that most regions of PNS seem to be stable with respect to the convection, although this may be an original feature in the PNS model obtained in Ref. [@KKT2016] in which the 3D hydrodynamic motion rapidly washes out the negative entropy gradient (see the middle panel in Fig. \[fig:back\]). One can also see that the absolute value of $f_{\rm BV}$ becomes larger in the earlier phases after core bounce. Now, $\delta p$ and $\delta \varepsilon$ are generally expressed as $\delta p(t,r)Y_{\ell k}$ and $\delta\varepsilon(t,r)Y_{\ell k}$, respectively. Thus, one obtains the following equations for any $\ell$-th perturbations, $$\begin{gathered}
\delta\varepsilon = -\frac{\varepsilon + p}{r^2}\left[e^{-\Lambda}W' + \ell(\ell+1)V\right] - \frac{\varepsilon'}{r^2}e^{-\Lambda}W,
\label{eq:de} \\
\delta p = c_s^2\delta\varepsilon + \frac{p\Gamma_1 {\cal A}}{r^2}e^{-\Lambda}W, \label{eq:dp}\end{gathered}$$ where Eq. (\[eq:de\]) comes from Eqs. (\[eq:dn\]) and (\[eq:e\_con\]), while Eq. (\[eq:dp\]) comes from Eq. (\[eq:dp0\]).
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In addition, the perturbed energy-momentum conservation law, i.e., $\nabla_\nu \delta T^{\mu\nu}=0$, gives us the following equations, $$\begin{gathered}
\frac{\varepsilon + p}{r^2}e^{-2\Phi}\ddot{W} + e^{-\Lambda}\delta p' + \Phi' e^{-\Lambda}\left(\delta\varepsilon + \delta p\right) = 0,
\label{eq:mur} \\
\delta p = (\varepsilon + p)e^{-2\Phi}\ddot{V}, \label{eq:muth}\end{gathered}$$ which correspond to the $r$- and $\theta$-components of the perturbed energy-momentum conservation law. We remark that the $t$-component of the perturbed energy-momentum conservation law is exactly the same as Eq. (\[eq:de\]). By combining Eqs. (\[eq:de\]) – (\[eq:muth\]) and assuming that $W(t,r)=e^{i\omega t}W(r)$ and $V(t,r)=e^{i\omega t}V(r)$, one can get the perturbation equations for $W$ and $V$ as $$\begin{gathered}
W' = \frac{1}{c_s^2}\left(\Phi'W + \omega^2 r^2e^{-2\Phi+\Lambda}V\right) - \ell(\ell+1)e^{\Lambda}V, \\
V' = -\frac{1}{r^2}e^{\Lambda}W + 2\Phi' V - {\cal A}\left(\frac{1}{\omega^2r^2}\Phi'e^{2\Phi-\Lambda}W + V\right).\end{gathered}$$ In order to solve this equation system, one has to impose appropriate boundary conditions. The regularity condition should be imposed at the stellar center, i.e., $$\begin{gathered}
W = W_0 r^{\ell+1} \ \ {\rm and}\ \ V=-\frac{W_0}{\ell}r^{\ell},\end{gathered}$$ where $W_0$ is constant. The boundary condition is that the Lagrangian perturbation of pressure should be zero at the surface of PNS, i.e., $$\Phi'e^{-\Lambda}W + \omega^2 r^2 e^{-2\Phi}V = 0,$$ for the case that the PNS surface is determined by the critical density, while it is that the radial displacement should be zero at the shock radius, i.e., $W=0$, for the case that the oscillations are considered in the domain inside the shock radius. At last, the problem to solve becomes the eigenvalue problem with respect to $\omega$. Once the eigenfrequency $\omega$ is determined, it is connected to the oscillation frequency, $f$, via $f=\omega/2\pi$.
Asteroseismology of PNS {#sec:IV}
=======================
Recently, the sophisticated time-frequency analysis [@Kawahara2018] showed that the various GW signatures with wide frequency ranges can be extracted from the GW spectrogram for the 3D-GR model (SFHx) employed in this work [@KKT2016]. For instance, as shown in Fig. \[fig:Kawahara\], they found the sequences of A, B, C, C\#, and D. The sequence A has been observed in the several previous studies, which is considered as “the surface $g$-mode" [@MJM2013; @CDAF2013; @KKT2016] of the PNS. The sequences B and D could come from the mass accretion influenced by SASI [@KKT2016]. It is noteworthy that the low frequency component B has been also reported in other recent 3D studies [@OC2018; @Andresen18]. The excitation mechanism of the sequence C (and C\#) is still unclear. In this paper, we attempt to compare the eigenfrequencies derived from the perturbation analysis with the GW frequencies obtained from the hydrodynamics simulations as in Fig. \[fig:Kawahara\]. As a baseline, we mainly focus on the sequence A in this work.
![The characteristic GW frequencies extracted by the time-frequency analysis [@Kawahara2018] for the 3D model SFHx in [@KKT2016]. []{data-label="fig:Kawahara"}](Kawahara)
![Eigenfrequencies in the PNS model with $\rho_s=10^{11}$ g cm$^{-3}$. In particular, the $f$ and $p_i$ for $i=1-6$ are explicitly shown with the open-squares together with the dotted lines, where the double squares denote the cases that the node number in the eigenfunction is different from the standard definition (see Fig. \[fig:wf-1e11-128\]). For reference, the various excited GW frequencies derived from the simulation data are also shown with the red lines. []{data-label="fig:mode-1e11"}](gmode-1e11c)
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![The eigenfunctions of $W$ for the $f$-modes are shown in the left panel, where the amplitude of $W$ is normalized by the amplitude at the stellar surface and is shifted a little for easily distinguishing the lines. The radial-dependent pulsation energy density $E$ given by Eq. (\[eq:energy\]) is shown in the right panel, where the amplitudes are normalized appropriately. []{data-label="fig:wE-1e11"}](wf-1e11b "fig:") ![The eigenfunctions of $W$ for the $f$-modes are shown in the left panel, where the amplitude of $W$ is normalized by the amplitude at the stellar surface and is shifted a little for easily distinguishing the lines. The radial-dependent pulsation energy density $E$ given by Eq. (\[eq:energy\]) is shown in the right panel, where the amplitudes are normalized appropriately. []{data-label="fig:wE-1e11"}](Ef-1e11b "fig:")
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PNS surface determined by the fixed density {#sec:IV-A}
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First, we consider the PNS model, whose surface density ($\rho_s$) is changed in a parametric manner. In Fig. \[fig:mode-1e11\], we show the eigenfrequencies determined in the PNS model with $\rho_s=10^{11}$ g cm$^{-3}$. Among many eigenfrequencies, we identify the $f$ and $p_i$-modes for $i=1-6$, which are shown with open squares (and double squares) connected by thin-dotted lines. In addition, for reference the various excited GW frequencies derived from the simulation data are shown in Fig. \[fig:Kawahara\]. The eigenfunctions of $W$ for the $f$-modes with various time steps are shown in the left panel of Fig. \[fig:wE-1e11\], where the amplitude of $W$ is normalized by the value at the PNS surface and is shifted upward in order to distinguish each line easily. From this figure, one can observe that the eigenfunctions of $W$ at any time step become as the standard definition of $f$-mode, i.e., the eigenfunction monotonically increases outward without any nodes. However, in fact, the eigenfunctions at 108 and 128 ms are different from the standard definition, where the node number is more than one. Even so, since the both eigenfunctions are very similar to the other $f$-mode eigenfunctions, we identify these modes as $f$-modes. For example, the eigenfunction of $|W|$ at 128 ms is shown in Fig. \[fig:wf-1e11-128\]. In the similar way, by checking the shape of the eigenfunctions and by counting the node number, we identify the $p_1$- and $p_2$-modes as shown in Fig. \[fig:mode-1e11\], where the eigenfunctions of $W$ for the $p_1$-modes at each time step are shown in Fig. \[fig:wp1-1e11\]. Again, the double squares denote the eigenfrequencies that are different from the standard definition by the node number.
![The details of the eigenfunction of $|W|$ for the $f$-modes at 128ms for the PNS model with $\rho_s=10^{11}$ g/cm$^3$ is shown, which is an example of the specific case shown with the double squares in Fig. \[fig:mode-1e11\]. This sample has one node in the eigenfunction shown by the arrow, but the shape of eigenfunction is almost the same as that of the other $f$-mode shown in the left panel of Fig. \[fig:wE-1e11\]. []{data-label="fig:wf-1e11-128"}](wf-1e11-128ms)
![The eigenfunctions of $W$ for the $p_1$-modes are shown for the PNS model with $\rho_s=10^{11}$ g/cm$^3$, where the amplitude is normalized by the surface amplitude and is shifted a little. []{data-label="fig:wp1-1e11"}](wp1-1e11a)
![For the PNS model with $\rho_s=10^{11}$ g/m$^3$, the $f$-mode frequencies are shown as a function of the square root of the PNS average density. The solid line denotes the fitting formula given by Eq. (\[eq:fit\]), while the dashed line is the analytical formula of the $f$-mode frequency for the star with uniform incompressible fluid given by Eq. (\[eq:ana-f\]). []{data-label="fig:fave-1e11"}](f-ave-1e11)
In the right panel of Fig. \[fig:wE-1e11\], we also show the radial dependent pulsation energy density in the $f$-mode oscillation at each time step, where in the same way as in Refs. [@TCPF2018; @MRBV2018], the Newtonian pulsation energy density at each radial position can be estimated with our variables as $$E(r) \sim \frac{\omega^2\varepsilon}{r^4} \left[W^2 + \ell(\ell+1)r^2V^2\right]. \label{eq:energy}$$ One can observe that the amplitude of the $f$-mode eigenfunction becomes maximum at the stellar surface. The pulsation energy density, however, becomes maximum at around $80-90 \%$ of the PNS radius. From Fig. \[fig:mode-1e11\] we can see a good agreement of the sequence A (which is referred as the surface $g$-mode [@MJM2013; @CDAF2013; @KKT2016]) with the $f$-mode oscillations in the PNS, when we take the specific surface density of $10^{11}$ g cm$^{-3}$. In Fig. \[fig:mode-1e11\], the filled squares correspond to the eigenfrequencies, which we can not unambiguously identify as either $f$-, $p_1$-, or $p_2$-modes (e.g., Fig. \[fig:w-1e10-108\]). These modes are left unidentified in this work.
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![The $f$-mode frequencies from the PNS models with different definition of the surface density are shown as a function of the corresponding PNS average density, where the squares, diamonds, and circles correspond to the results with $\rho_s=10^{11}$, $10^{10}$, and $5\times 10^{9}$ g cm$^{-3}$, respectively. []{data-label="fig:fave"}](f-ave)
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![Same as in the left panel of Fig. \[fig:wE-1e11\], but for the PNS models with $\rho_s=5\times 10^9$ g/cm$^3$. []{data-label="fig:wf"}](wf-5e09b "fig:") ![Same as in the left panel of Fig. \[fig:wE-1e11\], but for the PNS models with $\rho_s=5\times 10^9$ g/cm$^3$. []{data-label="fig:wf"}](Ef-5e09b "fig:")
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![An example of the eigenfunction $W$, of which eigenmode is left unidentified (from the radial node number of $W$). This corresponds to the frequency shown by the asterisk in the right panel of Fig. \[fig:modes\], which is close to the sequence A. []{data-label="fig:w-1e10-108"}](w-1e10-108)
The identification of the sequence A with the $f$-mode oscillation indicates that one could extract the PNS properties by observing the $f$-mode originated GWs. In practice, it is well-known that, since the $f$-mode is associated with the acoustic waves, its frequency can be characterized with the stellar average density almost independently of the adopted EOS (see [@AK1996; @AK1998] for cold NSs and also [@ST2016; @SKTK2017] even for the PNS models). In Fig. \[fig:fave-1e11\], we show the $f$-mode frequency from the PNS model for each time step as a function of the corresponding square root of the PNS average density. From this figure, as in the previous studies, one can observe that the $f$-mode frequencies can be expressed as a linear function of the square root of the PNS average density. Additionally, with this data, we obtain the fitting formula expressing the $f$-mode frequency, i.e., $$f_f\ {\rm (Hz)} = -87.34 + 4080.78 \left(\frac{M_{\rm PNS}}{1.4M_\odot}\right)^{1/2}\left(\frac{R_{\rm PNS}}{10\ {\rm km}}\right)^{-3/2},
\label{eq:fit}$$ where $M_{\rm PNS}$ and $R_{\rm PNS}$ denotes the PNS (gravitational) mass and radius, respectively. The resultant fitting formula is also shown in Fig. \[fig:fave-1e11\] with the thick-solid line. With this fitting formula, one may know the time evolution of the PNS average density from the observation of the gravitational waves. We remark that the $\ell$-th $f$-mode frequency for the star with uniform incompressible fluid has been derived analytically as $$f_f^{(\rm a)} = \frac{1}{2\pi}\sqrt{\frac{2M_{\rm PNS}}{R_{\rm PNS}^3}\frac{2\ell(\ell-1)}{2\ell+1}}, \label{eq:ana-f}$$ which is known as a Kelvin $f$-mode. The expected $\ell=2$ frequency is also shown in Fig. \[fig:fave-1e11\] with dashed line, but it seems that this formula assuming the incompressible fluid is not suitable for expressing the $f$-mode frequencies for the PNS models.
In the similar way, we determine the eigenfunctions in the PNS models with $\rho_s=5 \times 10^9$ and $10^{10}$ g/cm$^3$, which are shown in Fig. \[fig:modes\]. From this figure together with Fig. \[fig:mode-1e11\], we find that the eigenfrequencies depend on the selection of the surface density of the PNS model. In fact, the frequencies of $f$ and $p_i$-modes decrease, as $\rho_s$ decreases. This tendency may be understood as a result of the decrease of the average density of the PNS, as $\rho_s$ decreases, because the $f$ and $p_i$-modes are a kind of acoustic waves, whose frequencies can be characterized by the average density of the PNS. In fact, as shown in Fig. \[fig:fave\], the $f$-mode frequencies can be expressed well as a function of the PNS average density, even though the definition of the surface density is different.
The dependence of the eigenfrequencies on the surface density seems to be consistent with Ref. [@MRBV2018] as least in the early postbounce phase. On the other hand, in the phase later than $\sim 500$ ms after bounce, Morozova et al. [@MRBV2018] showed that the eigenfrequencies are almost independent from the selection of the surface densities of the PNS models. This could be because the density gradient in the vicinity of the background PNS models becomes steeper in the later phase, making the average density less sensitive to the choice of the surface densities. Thus, although the GW signal (the sequence A) obtained in the 3D numerical simulation [@KKT2016] is well ascribed to the $f$-mode oscillations in the PNS model with $\rho_s=10^{11}$ g/cm$^3$, this result may not be universal at least in the early postbounce phase, i.e., one may have to select a specific surface density to identify the GW signal. In such a case, it could be more difficult to extract the PNS information from direct GW observations.
Additionally, for the PNS model with $\rho_s=5 \times 10^9$ g/cm$^3$, the eigenfunction of $W$ and the radial dependent pulsation energy density at each time step are shown in Fig. \[fig:wf\]. We remark that eigenfunction of $W$ and the radial dependent pulsation energy density for the PNS model with $\rho_s=10^{10}$ g/cm$^3$ are more or less similar to those for the PNS model with $\rho_s=5 \times 10^9$ g/cm$^3$. The eigenfunctions of $W$ look similar to those shown in Fig. \[fig:mode-1e11\], but one can see the difference in the radial dependent pulsation energy density. From this figure, it seems that the oscillations around the stellar surface become more important in the PNS model with lower $\rho_s$. Furthermore, as an example of the eigenmode that could not be identified as a specific mode, we show the eigenfunction of $W$ for the PNS model with $\rho_s=10^{10}$ g/cm$^3$ at 108 ms, which is shown with the asterisk in the right panel of Fig. \[fig:modes\]. Obviously, this eigenfunction is satisfied the boundary condition but the shape of eigenfunction is apparently different from the other $f$- or $p_1$ mode. We note that the lower frequencies, such as sequences B or D in Fig. \[fig:Kawahara\], are not excited in the PNS models with the specific surface density irrespective of its value. In addition, some of the eigenfrequencies lower than the $f$-mode in Figs. \[fig:mode-1e11\] and \[fig:modes\] could be considered as $g$-mode oscillations. However these modes are left unidentified because of the lack of the clear node structure in the eigenfunctions as mentioned above.
Finally, the GW signal (the sequence A) is compared with the $f$-mode frequencies calculated in this study with different surface density and the surface $g$-mode with the formula proposed in Ref. [@MJM2013], i.e., $$f_{\rm peak} = \frac{1}{2\pi}\frac{M_{\rm PNS}}{R_{\rm PNS}^2}\sqrt{\frac{1.1 m_n}{\langle E_{{\bar{\nu}}_e}\rangle}}\left(1-\frac{M_{\rm PNS}}{R_{\rm PNS}}\right)^2, \label{eq:fpeak}$$ where $\langle E_{{\bar{\nu}}_e}\rangle$ denotes the mean energy of electron antineutrinos and $m_n$ is the neutron mass. We remark that the Brunt-Väisälä frequencies estimated at the PNS surface is original “surface $g$-mode", with which Eq. (\[eq:fpeak\]) is approximately derived. Those frequencies are shown in Fig. \[fig:comparison\], where the left and right panels correspond to the results of $f$-mode frequencies obtained in the linear analysis and surface $g$-mode frequencies calculated with Eq. (\[eq:fpeak\]), respectively, for the PNS models with $\rho_s=10^{11}$ (circles), $10^{10}$ (squares), and $5\times 10^{9}$ g/cm$^3$ (diamonds). From this figure, one can observe that the both frequencies strongly depend on the surface density, but agree well with the GW signal of the sequence of A for the PNS model with $\rho_s=10^{11}$ g/cm$^3$. Even so, since the surface $g$-mode (or the Brunt-Väisälä frequency at the PNS surface) is the local value while $f$-mode is the global oscillations of PNS, it may be more natural that the GW signal (sequence of A) is considered as a result of the $f$-mode oscillations.
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![The GW signal (the sequence of A) is compared with the $f$-mode GW of PNS in the left panel and with the surface $g$-mode calculated with Eq. (\[eq:fpeak\]) in the right panel, where the circles, squares, and diamonds denote the PNS models constructed with $\rho_s=10^{11}$, $10^{10}$, and $5\times 10^{9}$ g/cm$^3$, respectively. []{data-label="fig:comparison"}](comp1.eps "fig:") ![The GW signal (the sequence of A) is compared with the $f$-mode GW of PNS in the left panel and with the surface $g$-mode calculated with Eq. (\[eq:fpeak\]) in the right panel, where the circles, squares, and diamonds denote the PNS models constructed with $\rho_s=10^{11}$, $10^{10}$, and $5\times 10^{9}$ g/cm$^3$, respectively. []{data-label="fig:comparison"}](comp3.eps "fig:")
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PNS inside the shock radius {#sec:IV-B}
---------------------------
Next, we consider the oscillations inside the shock radius. In this case, as mentioned before, the boundary condition at the shock radius is that the radial component of the Lagrangian displacement should be zero. That is, the eigenfunction of $W$ is always zero at the shock radius, where the standard classification of the eigenmode may not be adopted. Intrinsically, the eigenvalue problem to solve with this PNS model is significantly different from that with the PNS model whose surface density is fixed. Anyway, as an advantage of this PNS model, the ambiguity for selecting the position of boundary disappears, while the spherical symmetric model may not be a good assumption in the region whose density is very low, because the matter motion is not neglected in such a region. Furthermore the excitation of GWs in the numerical simulation may come from such an oscillation inside the whole shocked region, although there are currently a few studies [@TCPF2018; @TCPOF2019] examining this effect.
We try to determine the eigenfrequencies with the PNS model inside the shock radius. The resultant eigenfrequencies are shown in Fig. \[fig:shock\], where the same modes are connected with dotted lines. From this figure, one can observe that even lower frequencies can be excited with the PNS model inside the shock radius, which is different feature compared to the results with the PNS model whose surface density is fixed. In fact, in some time interval, it seems that the eigenfrequencies are excited close to the sequences B and D.
![The eigenfrequencies calculated in this study are shown with marks, while the excited GW frequencies in the numerical simulation are again shown with various red lines. The double circles are the lowest and the second lowest eigenfrequencies at 268 ms, which are focused in Fig. \[fig:268ms\]. The open circles are the examples, of which eigenfunctions will be discussed in Fig. \[fig:wshock\]. The modes, whose eigenfunctions are the similar to each other, are connected with the dotted lines. []{data-label="fig:shock"}](gmode-shock-wb)
------------------------ -------------------------
 
------------------------ -------------------------
In order to see the oscillation behavior for such eigenfrequencies, we especially focus on the lowest and the second lowest eigenfrequencies at 268 ms after core bounce, which are denoted with the double circles in Fig. \[fig:shock\]. The corresponding eigenfunction of $W$ and the radial dependent pulsation energy density are shown in Fig. \[fig:268ms\], where the solid and dashed lines correspond to the results with the lowest and the second lowest eigenfrequencies, respectively. One can see that the eigenfunctions are very similar to the standard classification of stellar oscillation except for the behavior close to the stellar surface, i.e., the lowest and the second lowest eigenfrequencies may correspond to the $f$- and $p_1$-modes. In addition, in the similar fashion to the results with the PNS model whose surface density is fixed, the amplitude of eigenfunction $W$ and the radial dependent pulsation energy density become significant on the outer part of the oscillation region. However, it has been reported that the excitation of the GW signal according to the sequence B (or maybe also D) effectively comes from the inner part of the PNS, such as $\sim 20$ km [@KKT2016; @Andresen16]. Thus, although the lower eigenfrequencies obtained via the eigenvalue problem inside the shock radius appear close to the sequences B and D, these frequencies may not physically correspond to the excitation of gravitational wave signal in the sequences B and D. We remark that in our model we found only $f$- and $p_i$-mode like frequencies, while not only $f$- and $p_i$-mode like frequencies but also $g_i$-mode like frequencies are found in the previous similar analysis [@TCPF2018; @TCPOF2019]. This discrepancy may come from the different PNS models obtained by the numerical simulations. We need further studies changing the PNS models to draw a robust conclusion on this choice of the boundary condition.
![Examples of the eigenfunctions of $|W|$ with the frequencies shown by the open circles in Fig. \[fig:shock\]. []{data-label="fig:wshock"}](W-shock)
There are a few possible reasons to explain the discrepancy between the the sequence B (and D) and the eigenmodes obtained by the linear analysis. The first and perhaps the main reason is that, in our perturbation analysis using the static background model, the restoring force against the perturbations is assumed to be the acoustic mode. On the contrary, the SASI that is considered to be the emission mechanism of the component B [@KKT2016; @Andresen16] is sustained by the cycle of the fluid advection and the acoustic mode [@Foglizzo06]. It may thus not be suitable to use the static background model that completely omits the fluid advection. As the second reason, the background model is actually far from spherical symmetry particularly in the non-linear SASI phase ($T_{\rm pb}\gtrsim150$ ms, [@KKT2016]). These facts would make it even harder to extract the proper eigenmodes for the corresponding GW components. We also remark that one can not clearly find a specific correspondence between the eigenfrequencies and the sequence A on Fig. \[fig:shock\]. In practice, we show examples of the eigenfunction of $|W|$ in Fig. \[fig:wshock\] with the frequencies shown by the open circles in Fig. \[fig:shock\], which are close to the sequence A, but one can not straightforwardly identify these modes as the same eigenmode by checking the shape of the eigenfunction (or the radial node numbers). Anyway, in this study we have made a linear analysis with only one result obtained by the numerical simulation in Ref. [@KKT2016], i.e., our result may not be always acceptable for any PNS models. In order to make a robust statement for the CCSN GW signals, we have to make more systematical analyses somewhere by adopting various results of different numerical simulations.
Conclusion and Discussions {#sec:V}
==========================
In an attempt to obtain the eigenfrequencies of a PNS in the postbounce phase of CCSNe, we have performed a linear perturbation analysis of the angle-averaged PNS profiles using results from a general relativistic CCSN simulation of a $15 M_{\odot}$ star. Particularly, we paid attention to how the choice of the outer boundary condition could affect the PNS oscillation modes in the linear analysis. By changing the density at the outer boundary of the PNS surface in a parametric manner, we showed that the eigenfrequencies strongly depend on the surface density. By comparing with the GW signals from the hydrodynamics model, it was shown that the so-called surface $g$-mode of the PNS can be well ascribed to the fundamental oscillations of the PNS. The best match was obtained when the PNS surface is chosen at $10^{11}$ g/cm$^3$. We pointed out that the frequency of the fundamental oscillations can be fitted by a function of the mass and radius of the PNS similar to the case of cold NSs. In the case that the position of the outer boundary is chosen to cover not only the PNS but also the surrounding postshock region, we obtained the eigenfrequencies close to the modulation frequencies of the SASI. On the other hand, our results suggested that these oscillation modes are unlikely to have the same physical origin of the SASI modes obtained in the hydrodynamics simulation. We have discussed possible limitations of applying the angle-averaged, linear perturbation analysis to extract the full facets of the CCSN GW signatures. In order to identify the GW signatures in the spectrograms more in a systematic manner, one may need to conduct a more detailed linear analysis as in Ref. [@TCOMF2019a].
In this study we adopted the relativistic Cowling approximation, which could be applicable to the early postbounce phase because the stellar compactness is not so large and the relativistic effect may not be so significant. To apply the similar analysis to the late postbounce phase or to the very massive progenitor stars leading to a BH formation as reported in Ref. [@Kuroda2018], we need to perform the linear analysis taking into account the metric perturbation, which we shall leave for the future work. Towards the observation of the most remarkable spectral GW signature (i.e., the ramp-up $f$-mode) in the laser interferometers, dedicated data analysis schemes (e.g., [@astone2018]) need to be further developed.
This study was supported in part by the Grants-in-Aid for the Scientific Research of Japan Society for the Promotion of Science (JSPS, Nos. JP17K05458, JP26707013, JP17H01130, JP17K14306, JP18H01212 ), the Ministry of Education, Science and Culture of Japan (MEXT, Nos. JP15H00789, JP15H01039, JP15KK0173, JP17H05206 JP17H06357, JP17H06364) by the Central Research Institute of Fukuoka University (Nos.171042,177103) and the Research Institute of Explosive Stellar Phenomena (REISEP), and JICFuS as a priority issue to be tackled by using Post ‘K’ Computer. TK was supported by the European Research Council (ERC; FP7) under ERC Starting Grant EUROPIUM-677912.
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abstract: 'It is shown that electromagnetic potentials convey physical information beyond that supplied by electric and magnetic fields alone, and are thus more fundamental. Observable physical properties can impose conditions on the selection of electromagnetic gauge (i.e. sets of potentials) that are explicit and restrictive. This is true both classically and quantum mechanically. The implication that the choice of gauge carries physical information is confirmed by exhibiting a set of potentials that describes fields correctly, but that violates physical constraints. The basic conclusions are that physical requirements place limits on acceptable gauges; and that potentials are more fundamental than fields in both classical and quantum physics, representing a major generalization of the quantum-only Aharonov-Bohm effect. These important properties are obscured if the dipole approximation is employed. The properties demonstrated here relate directly to conditions that exist in strong-field laser applications.'
author:
- 'H. R. Reiss'
title: Physical restrictions on the choice of electromagnetic gauge and their practical consequences
---
Introduction
============
Electric and magnetic fields of electrodynamics can be represented by scalar and vector potentials. Particular derivatives of these potentials will generate the fields. These potentials are not unique, with alternative sets of potentials connected by a mathematical procedure called a *gauge transformation*. The conventional point of view is that physical processes depend only on the fields, and the potentials can be regarded as nothing more than useful auxiliary quantities [@jackson]. An important exception to this rule is the Aharonov-Bohm effect [@siday; @ab], in which a charged particle passing near a solenoid containing a magnetic field can be deflected by the potential that exists outside the solenoid in a field-free region. The effect is particular to quantum mechanics, and provides only a cautionary limitation to the general notion that fields are more basic than potentials.
It is shown here that such a ubiquitous phenomenon as a laser field imposes strong limitations on possible gauge transformations when the demand is made that the propagation property of the laser field must be sustained. This limitation applies both classically and quantum mechanically. It is further shown that if the laser field (a transverse field) impinges on a charged particle that is simultaneously subjected to a Coulomb binding potential (a longitudinal field), then the unique allowable gauge is the radiation gauge (also known as the Coulomb gauge) if all physical constraints are to be satisfied. This combination of transverse and longitudinal fields pervades Atomic, Molecular, and Optical (AMO) physics. The common AMO practice of using the dipole approximation in the description of laser-induced effects amounts to replacing the transverse field of a laser by the fundamentally different longitudinal field, thus altering basic physical constraints.
A brief review of the basic features of gauge transformations is given in the next Section. The standard requirement is that the generating function for the gauge transformation can be any scalar function that satisfies the homogeneous wave equation [@jackson]. Section III considers the important case of a propagating field. This application applies to all transverse fields, including laser fields. It is shown that the only possible departure from the familiar radiation gauge must be such that the 4-vector potential describing the field can have added to it only a contribution that depends on the light-cone coordinates appropriate to that field. This is an important supplement to the standard gauge requirements of electrodynamics.
Section IV examines the more restrictive case where a charged particle is simultaneously subjected to a transverse field (like a laser field) and a longitudinal field (like a Coulomb binding potential). Since this combination of fields describes most AMO situations, the basic electrodynamic principles established here are directly applicable to strong-field laser experiments. The operative limitation in this case comes from the properties of the relativistic quantum equations of motion and the inferences that persist in the nonrelativistic limit. These considerations are not important for laser fields in the perturbative domain, but they are applicable under the conditions that exist when fields are strong enough to require nonperturbative methods. The reason is that laser fields propagate at the speed of light, thereby introducing relativistic considerations even into nominally nonrelativistic problems [@hr90]. Relativistic conditions exist and must be properly accounted for in strong-field applications, since they signal the importance of the magnetic field in ways that are invisible within the dipole approximation.
Section V exhibits the general conclusion that potentials are more fundamental than the fields that are derived from them, by exhibiting two sets of potentials that describe exactly the same electric and magnetic fields; but one set satisfies all physical requirements, and the other set gives incorrect predictions for such basic matters as the propagation property, Lorentz symmetries, and the ponderomotive potential of a charged particle in the field. This marks a major generalization of the important Aharonov-Bohm effect, presently the sole practical example of the dominance of potentials over fields.
The final Section is an overview of the essential results, including an appraisal of the practical consequences of the results arrived at here. A simple summary is that physical intuition or physical interpretation is dependent on the choice of electromagnetic gauge. The use of the dipole approximation severely limits those benefits and can lead to the adoption of physical pictures that do not match laboratory reality. A leading example is the tunneling limit that envisions a very low frequency laser field as a nearly static electric field, in contrast to the actuality of a laser field that propagates at the velocity of light for all frequencies, and cannot possibly have a static limit. It is emphasized that some seriously misdirected criteria have been adopted, unchallenged, in strong-field physics.
Basic Gauge Transformation
==========================
For notational simplicity only vacuum conditions are considered, and Gaussian units are employed. The electric field $\mathbf{E}$ and the magnetic field $\mathbf{B}$ can be represented by the scalar potential $\phi$ and the vector potential $\mathbf{A}$ as$$\begin{aligned}
\mathbf{E} & =-\mathbf{\nabla}\phi-\frac{1}{c}\partial_{t}\mathbf{A},\label{a}\\
\mathbf{B} & =\mathbf{\nabla}\times\mathbf{A}. \label{b}$$ A gauge transformation of the $\phi,$ $\mathbf{A}$ potentials to a new set $\widetilde{\phi},$ $\widetilde{\mathbf{A}}$ can be achieved with the scalar generating function $\Lambda$ with the connection that$$\begin{aligned}
\widetilde{\phi} & =\phi+\frac{1}{c}\partial_{t}\Lambda,\label{c}\\
\widetilde{\mathbf{A}} & =\mathbf{A}-\mathbf{\nabla}\Lambda. \label{d}$$ The substitution of Eqs. (\[c\]) and (\[d\]) into (\[a\]) and (\[b\]) leaves the field expressions unchanged. The only constraint on $\Lambda$ is that it should satisfy the homogeneous wave equation. This is to enable a decoupling of the equations for the scalar and vector potentials. Relativistic notation is useful. The 4-vector potential that encompasses both the scalar and 3-vector potentials is$$A^{\mu}:\left( \phi,\mathbf{A}\right) , \label{e}$$ and the basic spacetime 4-vector is$$x^{\mu}:\left( ct,\mathbf{r}\right) . \label{f}$$ The expressions (\[c\]) and (\[d\]) are then subsumed into the single expression$$\widetilde{A}^{\mu}=A^{\mu}+\partial^{\mu}\Lambda\label{g}$$ subject to the constraint on $\Lambda$ that it must satisfy$$\partial^{\mu}\partial_{\mu}\Lambda=0. \label{h}$$
Gauge Limitation for Transverse Fields
======================================
Now the important example is considered wherein $A^{\mu}$ represents a *transverse field*, with the equivalent terminologies that it is a *propagating field* or a *plane-wave field*. Such a field propagates in vacuum with the speed of light $c$, with the additional proviso of special relativity that this speed of propagation must be the same in all inertial frames of reference. This is equivalent to the statement (see, for example, Refs. [@schwinger] and [@ss]) that any occurrence of $x^{\nu
}$ in the potential $A^{\mu}$ can only be in the form of the scalar product with the propagation 4-vector $k^{\nu}$:$$\varphi\equiv k^{\mu}x_{\mu}, \label{i}$$ where$$\begin{aligned}
k^{\mu} & :\left( \frac{\omega}{c},\mathbf{k}\right) ,\label{j}\\
\left\vert \mathbf{k}\right\vert & =\omega/c. \label{k}$$ In other words, $k^{\mu}$ is a lightlike 4-vector with the 3-vector component $\mathbf{k}$ in the propagation direction of the transverse field. The 4-vector potential $A^{\mu}$ can depend on the spacetime 4-vector $x^{\mu}$ only as $A^{\mu}\left( \varphi\right) $, and the same condition must apply to $\widetilde{A}^{\mu}.$ Therefore, Eq. (\[g\]) requires that $\partial^{\mu}\Lambda$ must also be a function of $\varphi$ alone. This requirement can be satisfied by the condition that $\Lambda$ depends on $x^{\mu}$ only as $\Lambda=\Lambda\left( \varphi\right) $, since then$$\partial^{\mu}\Lambda\left( \varphi\right) =\partial^{\mu}\left(
\varphi\right) \frac{d}{d\varphi}\Lambda\left( \varphi\right) =k^{\mu
}\Lambda^{\prime}\left( \varphi\right) , \label{l}$$ where $\Lambda^{\prime}$ is the total derivative of $\Lambda$ with respect to $\varphi$. The gauge-transformed 4-vector potential $\widetilde{A}^{\mu}$ can therefore differ from the original $A^{\mu}$ only by a quantity that lies on the light cone, since the gauge-transformation condition (\[g\]) must be of the form$$\widetilde{A}^{\mu}=A^{\mu}+k^{\mu}\Lambda^{\prime}. \label{m}$$
The condition (\[m\]) is very restrictive. One important consequence is that the squared 4-vector potential is gauge invariant [@hrmod; @hrpond], as follows from the light-cone condition$$k^{\mu}k_{\mu}=0, \label{n}$$ and the transversality condition$$k^{\mu}A_{\mu}=0, \label{o}$$ so that Eq. (\[m\]) leads to$$\widetilde{A}^{\mu}\widetilde{A}_{\mu}=A^{\mu}A_{\mu}. \label{p}$$ The ponderomotive potential of a charged particle in a transverse field is proportional to $A^{\mu}A_{\mu}$, meaning that this fundamentally important quantity [@hrmod; @hrpond] is gauge-invariant.
An important *caveat* is that the use of the dipole approximation, a standard procedure in AMO physics, has the effect of losing altogether the propagation property of a laser field. The dipole approximation amounts to the replacement of the propagating, transverse field by a simple oscillatory electric field, with the significance that the basic condition of Eq. (\[m\]) is discarded. That is, Eq. (\[g\]) no longer leads to Eq. (\[m\]) when the dipole approximation is employed.
It is well-known from long experience in nuclear and high energy physics that calculations of the effects of plane wave fields on charged particles can be successfully applied in the context of the radiation gauge. A convenient way to describe the radiation gauge is that it is the gauge within which a pure transverse field is described by the 3-vector potential $\mathbf{A}$ alone, and a pure longitudinal field is described by a scalar potential $\phi$ (or $A^{0}$) alone.
Combined Transverse and Longitudinal Fields
===========================================
The ionization of an atom by a laser field typifies AMO processes. An atomic electron in a laser field experiences both the transverse field of the laser and the longitudinal field of the binding potential. The dipole approximation has been useful in AMO physics since it offers the simplifying property that the laser field is replaced by an oscillatory electric field, thus substituting another longitudinal field for the transverse field. That is, traditional AMO physics replaces a combination of a transverse laser field and a longitudinal binding potential by two longitudinal fields. This is true whether the so-called length gauge is used, where the interaction Hamiltonian is of the form $\mathbf{r\cdot
E}\left( t\right) $ or by the gauge-equivalent [@gm] velocity gauge where the interaction Hamiltonian contains $\mathbf{A}\left( t\right) \mathbf{\cdot p}$.
When laser fields are very strong, the fact that the field propagates with the velocity of light becomes an important feature [@hr90]. Replacement of the propagating field by the dipole-equivalent oscillatory electric field is no longer sufficient. Even when magnetic forces remain small, the complete neglect of the magnetic field removes all possibility of propagation. This has major importance in practical applications. For example, the laboratory detection of Above-Threshold Ionization (ATI) in 1979 [@ati], where the perturbation-theory dominance of the lowest-order process gives way to the participation of higher orders of interaction with the applied field, caused a major sensation in the AMO community and triggered theoretical efforts lasting more than a decade (see the introductory remarks in Ref. [@cbc1]) to achieve some understanding of how this could happen in a dipole-approximation context. By contrast, a theory based on the nonrelativistic limit of a relativistically formulated theory [@hr90] provided an anticipatory prediction of all of the ATI features [@hr80] in a theory paper prepared in advance of the observation of ATI. It is important to note that a nonrelativistic limit of a relativistic theory leads to analytical forms that resemble theories based on *a priori* employment of the dipole approximation, but the seemingly slight differences are nevertheless critical.
A laser-induced phenomenon that is unquestionably relativistic is the production of electron-positron pairs. It was predicted in 1971 [@hrpairs] and confirmed in 1997 [@burke] that this was possible with a laser wavelength of the order of $1\mu m$ at a focused field intensity of about $10^{18}W/cm^{2}$. Many laboratories can now produce such intensities; it is simply a confirmation of the need to recognize the relativistic foundations of laser effects.
Reduction of a relativistic treatment to a nonrelativistic limit of the effects of combined transverse and longitudinal fields introduces a feature that had not been anticipated. Following the usual practice of neglecting effects of the spin of the electron, the universally employed relativistic description of the electron is the Klein-Gordon (KG) equation [@pauliweiss; @schweber]$$\left[ \left( i\hbar\partial_{\mu}-\frac{q}{c}A_{\mu}\right) \left(
i\hbar\partial^{\mu}-\frac{q}{c}A^{\mu}\right) -m^{2}c^{2}\right] \psi=0.
\label{p1}$$ A separation of time and space parts gives the form$$\left[ \left( \frac{i\hbar}{c}\partial_{t}-\frac{q}{c}A^{0}\right)
^{2}-\left( -i\hbar\mathbf{\nabla}-\frac{q}{c}\mathbf{A}\right) ^{2}-m^{2}c^{2}\right] \psi=0. \label{p2}$$ It is the first parenthesis in the square bracket of Eq.(\[p2\]) that is of interest here, since $A^{0}$ can represent the time part of the laser-field 4-vector potential $A_{PW}^{0}$ (where the subscript $PW$ stands for plane wave) as well as a binding potential $V$ that may be present:$$A^{0}=A_{PW}^{0}+V. \label{p3}$$ When $V$ is a Coulomb binding potential,$$V\sim1/r, \label{p4}$$ this singularity causes problems [@schoene] in the reduction of the KG equation to the Schrödinger equation in the nonrelativistic limit [@schiff]. The act of squaring indicated for the first term in Eq.(\[p2\]) introduces a cross coupling $VA_{PW}^{0}$ that is also singular at the origin of spatial coordinates, but it is an unacceptable term because the magnitude of this singular term depends upon the properties of the laser field. This unphysical behavior will not occur if$$A_{PW}^{0}=0, \label{p5}$$ which corresponds to selection of the radiation gauge (also known as the Coulomb gauge) wherein longitudinal fields are represented by scalar potentials and transverse fields are represented by 3-vector potentials. If this gauge selection must be enforced in a relativistic problem, that gauge must also refer to the nonrelativistic limit.
The same considerations arising in the reduction of the KG equation to the Schrödinger equation in the nonrelativistic limit applies also to the reduction of the Dirac equation to the Pauli equation for a spin-${\frac12}$ particle. The is most easily seen from the second-order form of the Dirac equation [@feyngell; @schweber]:$$\begin{gathered}
\left[ \left( i\hbar\partial_{\mu}-\frac{e}{c}A_{\mu}\right) \left(
i\hbar\partial^{\mu}-\frac{e}{c}A^{\mu}\right) +\frac{1}{2}\frac{e\hbar}{c}\sigma^{\mu\nu}F_{\mu\nu}-m^{2}c^{2}\right] \psi=0,\label{q1}\\
\sigma^{\mu\nu}=\frac{1}{2i}\left( \gamma^{\mu}\gamma^{\nu}-\gamma^{\nu
}\gamma^{\mu}\right) , \label{q2}$$ where the $\gamma^{\mu}$ are the standard Dirac matrices and $F^{\mu\nu}$ is the electromagnetic field tensor. This equation for a spin-${\frac12}$ particle is the same as the KG equation (\[p1\]) for a spin-0 particle, but with the addition of a term representing the spin interaction. It presents the same dilemma in reduction to the Pauli equation as does the KG equation in reduction to the Schrödinger equation. The second-order Dirac equation has the dual advantages of its similarity to the KG equation, as well as lacking the *Zitterbewegung* problem [@schweber; @bd] of the first order Dirac equation in reduction to the nonrelativistic limit.
Elimination of the unphysical coupling of the laser field to a singular quantity means that $$A_{PW}^{0}=0, \label{r}$$ a feature of the radiation gauge, is a general requirement for attainment of the correct equation of motion. Explicitly, since Eq. (\[m\]) is a general requirement for a propagating field, and it has now been shown that the additional presence of a binding potential requires that any $\widetilde
{A}^{0}$ must also vanish, the condition (\[m\]) means that $$\Lambda^{\prime}=0,\quad\Lambda=\text{ constant} \label{s}$$ must hold true. That is, no departure from the radiation gauge for the transverse field is allowable if all field conditions are to be satisfied exactly.
None of the above reasoning arises if the dipole approximation is imposed from the outset. This does not mean that the dipole approximation is more convenient because of this; it means rather that the dipole approximation infers a measure of approximation beyond the usual interpretation. This is not consequential for fields that are perturbatively weak, but it is of fundamental importance when applied transverse fields are strong. A practical example of this has already been mentioned: the ATI phenomenon is perplexing within the dipole approximation [@cbc1], but it is obvious when propagating-field considerations are *a priori* present [@hr80; @hr90].
Potentials Are More Fundamental Than Fields
===========================================
The strong constraints that have been found to apply to potentials, but without reference to the fields associated with those potentials, has immediate significance. That is, potentials have introduced essential probes into physical phenomena such as the propagation phenomenon, preservation of the ponderomotive energy, proper reduction to the Schrödinger equation, and so on. These properties become evident from the potentials, but not from the fields.
A specific simple (but fundamental) example is now given where two sets of potentials can be written for description of the same fields, where one set of potentials is acceptable, but the other is nonphysical. A monochromatic plane-wave field of constant amplitude can be described by the 4-potential$$A^{\mu}\left( \varphi\right) =A_{c}^{\mu}\cos\varphi, \label{t}$$ where the phase $\varphi$ is given in Eq. (\[i\])$$\varphi=k^{\mu}x_{\mu}=\omega t-\mathbf{k\cdot r}, \label{u}$$ and $A_{c}^{\mu}$ is a constant 4-vector amplitude. It is noted here that this 4-potential satisfies the Lorenz condition$$\partial^{\mu}A_{\mu}=k^{\mu}A_{\mu}^{\prime}\left( \varphi\right) =0
\label{v}$$ because of the transversality condition of Eq. (\[o\]). (The Danish physicist L. V. Lorenz should not be confused with the Dutch physicist H. A. Lorentz.)
Now consider the gauge transformation generated by the function [@hr79]$$\Lambda=-A^{\mu}\left( \varphi\right) x_{\mu}. \label{w}$$ This gives the transformed 4-potential$$\widetilde{A}^{\mu}=-k^{\mu}\left( x^{\nu}A_{\nu}^{\prime}\right) .
\label{x}$$ It is readily verified that $$\partial^{\mu}\partial_{\mu}\Lambda=0, \label{y}$$ the sole condition normally required of the generating function of a gauge transformation [@jackson]. Because $\widetilde{A}^{\mu}$ was obtained from the $A^{\mu}$ of Eq. (\[t\]) by a gauge transformation, the electric and magnetic fields obtained from (\[x\]) are identical to those obtained from (\[t\]), as can be verified by direct computation. It is also true that $\widetilde{A}^{\mu}$ is a Lorenz gauge, and it is even true that $\widetilde{A}^{\mu}$ is transverse because of the light-cone condition (\[n\]).
However, the vector potential $\widetilde{A}^{\mu}$ given in Eq. (\[x\]) is not a physically acceptable gauge. It has the incorrect Lorentz transformation property of being lightlike rather than spacelike. It predicts that the all-important ponderomotive energy [@hrmod; @hrpond] vanishes, because$$\widetilde{A}^{\mu}\widetilde{A}_{\mu}=0, \label{z}$$ and it does not possess the basic property required by relativity that it depend on the spacetime 4-vector $x^{\mu}$ only in the combination $k^{\mu
}x_{\mu}$ as demanded by the condition (\[i\]). All of these failures occur for the simple reason that the gauge transformation (\[w\]) that produced $\widetilde{A}^{\mu}$ does not depend on $x^{\mu}$ solely in the form of the scalar product (\[i\]). Nevertheless, the unphysical nature of $\widetilde{A}^{\mu}$ is not evident from the normal rules for performing a gauge transformation. Judged by prediction of the correct electric and magnetic fields, one would be justified in employing the $\widetilde{A}^{\mu}$ of (\[x\]) as the gauge-equivalent version of (\[t\]). However, this seemingly safe conclusion based on the fields is incorrect because of the unphysical properties that are evident only by noting that the physical properties of the 4-potential (\[t\]) are different from those of the 4-potential (\[x\]).
It is not enough to know the fields; one must know the appropriate potentials.
Practical Consequences
======================
The focus of attention throughout this article is on the properties of propagating fields, with the specific case of laser fields as the most important practical example. It has been shown that when a laser field interacts with matter, so that bound charged particles are subjected simultaneously to both transverse and longitudinal fields, then the only formally acceptable electromagnetic gauge that can be employed is the radiation gauge (also called the Coulomb gauge). It has been remarked that this restriction is not of major importance when fields are perturbatively weak, but there is a great and growing interest in the effects of very strong laser fields. The practical models currently employed in strong field applications are based on the dipole approximation, which amounts to treating the laser field as an oscillatory electric field, with no propagation property at all, and the results here obtained do not apply in the dipole context. Since the dipole approximation gives the appearance of introducing important conceptual and practical simplifications, and many successes have been achieved in this manner, it is natural to inquire about the practical consequences of the results shown above. That is a fundamental question, and a comprehensive answer is proposed.
Since laser fields are, in actuality, transverse, propagating fields, it is to be expected that physical understanding of practical consequences of laser interactions with matter should be based on the properties of propagating fields. The dipole approximation reduces the laser field to an oscillatory electric field, which is a longitudinal field that differs fundamentally from an actual laser field. One important example has already been mentioned. The ATI phenomenon, so startling and unexpected within the AMO community, is actually an obvious and commonplace consequence of all strong-field phenomena. For example, in the context of pair production by strong laser fields, one finds the 1971 comment [@hrpairs]: ...an extremely high order process can be competitive with – and even dominate – the lowest order ... process. In the context of strong-field bound-bound transitions, it was shown in 1970 that [@prl25]: ...as the intensity gets very high, ... the lowest order process gets less probable ... \[and\] higher-order processes become increasingly important. The 1980 strong-field approximation (SFA) paper demonstrates the basic aspects of ATI, including some that were not observed in the laboratory until much later. For example, the character of spectra generated by strong, circularly polarized fields, exhibiting a multi-peaked spectrum with a near-Gaussian envelope with the most probable order being significantly higher than the lowest order, was observed with astonishment in a 1986 experiment [@bucks86], but this was already predicted in 1980, and the 1980 theory was accurate in exhibiting [@hr87] the explicit behavior found in the 1986 experiment. The reminder is important here that, although the 1980 paper superficially resembles dipole-approximation theories, it is actually the nonrelativistic limit of a relativistic theory of laser-induced ionization [@hr90; @hrrel]. The distinction is vital.
The above paragraph reveals that a propagating-field theory, since it models the actual laser field, can produce results that are more insightful and more successful than theories based on an oscillating-electric-field model. Furthermore, the 1980 SFA theory is actually easier to apply than the dipole-approximation versions of the SFA.
A recent example is instructive. In very precise spectrum measurements in an ionization experiment with circularly polarized light, it was found to be possible to detect the effects of radiation pressure on the photoelectrons [@smeenk]. Attempts to provide a theoretical explanation for the effect in a dipole-approximation context proved to be extremely difficult and inconclusive [@smeenk; @cbc1; @cbc2]. This is not surprising. Radiation pressure arises from photon momentum that does not exist in a dipole-approximation theory. In the context of a transverse-field description, the most probable kinetic energy of a photoelectron released by a strong, circularly polarized field is just the ponderomotive energy $U_{p}$. The number of photons above threshold needed to produce such a photoelectron is $n=U_{p}/\hbar\omega$. Each photon carries a momentum of $\hbar\omega/c$, with all photon momenta aligned in the direction of propagation of the laser field, so the field-induced momentum in the propagation direction is just $U_{p}/c$, and this is independent of the atom being ionized when the field is strong. This is precisely what the laboratory measurements reveal [@smeenk; @hrpress]. Transverse-field concepts produce insightful and quantitatively accurate results as shown in the span of three sentences given above, as contrasted with three journal articles [@smeenk; @cbc1; @cbc2].
The seeming simplicity of dipole-approximation methods is actually counter-productive in strong fields, as shown by the ATI and radiation pressure examples. The dipole approximation can lead to complication rather than simplicity.
Perhaps the most consequential of all misconceptions that arise from dependence on a dipole-approximation model of laser effects is the matter of low frequency behavior [@hr101; @hrtun]. The oscillatory electric fields that arise from a dipole-approximation theory approach a constant electric field as the frequency declines. This limit (sometimes called the *tunneling limit*) has been applied as a test of the accuracy of theoretical models. For example, a textbook on the subject of strong laser-field effects altogether rejects models based on transverse fields, since they do not approach the tunneling limit [@jkp]. Another example is a paper that assesses the accuracy of analytical approximations based on their behavior as the field frequency approaches zero [@jarek]. However, actual transverse fields in vacuum always propagate at the speed of light independently of frequency. There is no limit possible in which a real propagating field becomes a static field. The effect on strong-field theory of this zero-frequency misconception continues to the present. It is related to the equally problematic concept that the final arbiter of validity is to be found in the exact numerical solution of the Schrödinger equation, generally referred to as TDSE (Time-Dependent Schrödinger Equation). Since TDSE as customarily employed is based on the dipole approximation, it reinforces the critically misleading concept that there is a zero-frequency limit of laser effects equivalent to a constant electric field.
It is difficult to conceive of a notion more consequential for an entire field of inquiry than this reliance on the criterion that laser-induced effects have a zero-frequency limit equivalent to that of a constant electric field. When joined with the equally difficult concept, championed by K.-H. Yang [@yang] and others [@lamb], that the scalar potential known as the length gauge (accurate for a scalar field like a longitudinal field) can somehow be a privileged gauge for the description of a vector field like the transverse field of a laser beam, the discipline of strong-field physics is laboring under a burden of misdirected criteria. This has stood nearly unchallenged since the 1979 observation of ATI [@ati]. The scrutiny provided by an approach based on the radiation gauge creates the necessary challenge.
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---
abstract: 'We study a continuous dynamics for a class of Petri nets which allows the routing at non-free choice places to be determined by priorities rules. We show that this dynamics can be written in terms of policies which identify the bottleneck places. We characterize the stationary solutions, and show that they coincide with the stationary solutions of the discrete dynamics of this class of Petri nets. We provide numerical experiments on a case study of an emergency call center, indicating that pathologies of discrete models (oscillations around a limit different from the stationary limit) vanish by passing to continuous Petri nets.'
address:
- 'INRIA and CMAP, École polytechnique, CNRS, Université de Paris-Saclay'
- 'École des Ponts ParisTech INRIA and CMAP, École polytechnique, CNRS, Université de Paris-Saclay Brigade de sapeurs-pompiers de Paris'
author:
- Xavier Allamigeon
- Vianney Bœuf
- Stéphane Gaubert
title: Stationary solutions of discrete and continuous Petri nets with priorities
---
[^1]
Introduction
============
Context {#context .unnumbered}
-------
The study of continuous analogues of Petri nets dates back to the works of David and Alla [@david1987continuous] and Silva and Colom [@silva1987structural] in 1987. It has given rise to a large scope of research in the field of Petri nets. Whereas classical (discrete) Petri nets belong to the class of discrete event dynamic systems, the circulation of tokens in continuous Petri nets is a continuous phenomenon: tokens are assumed to be fluid, [*i.e.*]{}, a transition can fire an infinitesimal quantity of tokens. In this way, the continuous dynamics can be represented by a system of ordinary differential equations or differential inclusions.
Continuous Petri nets are usually introduced as a relaxed approximation of Petri nets, that helps understanding some of the properties of the underlying discrete model, allowing one to overcome the state space explosion that can occur in the latter. The continuous framework can also be seen as a scaling limit of a class of stochastic Petri nets (see [@darling2008diff]), where the marking $M_p$ of place $p$ in the fluid model is the finite limit of $M_p(N)/N$, with $N$ being a scaling ratio tending to infinity, and where the firing times of transitions follow a Poisson distribution.
An important effort has been devoted to the comparison between continuous nets and their discrete counterparts. For example, the relationship between reachability of continuous Petri nets and of discrete Petri nets is well understood (see [@recalde1999autonomous]). A recent introduction to continuous models can be found in [@vazquez2013introduction], while a more extensive reference is [@david2010discrete].
In order to evaluate the long-term performance of Petri nets, one has to characterize the stationary or steady states of the Petri nets dynamics. Cohen, Gaubert and Quadrat [@cohen1995asymptotic] introduced an approximation of a discrete Petri net by a fluid, piecewise affine dynamics with finite delays, and showed that the limit throughput does exist for a class of consistent and free choice Petri nets. In the more recent work of Gaujal and Giua [@gaujal2004optimal], the result is extended to larger classes of Petri nets, and the stationary throughputs are computed as the solutions of a linear program. The results obtained using this fluid approximation hardly apply to the discrete model, up to a remarkable exception identified by Bouillard, Gaujal and Mairesse [@bouillard2006extremal] (bounded Petri nets under total allocation). This reference illustrates the many difficulties that arise from the discrete setting (*e.g.*, some firing sequences may lead to a deadlock).
In the continuous dynamics setting, with time attached to transitions, Recalde and Silva [@recalde2000pn] showed that the steady states of free choice Petri nets as well as upper bounds of the throughputs in larger classes of Petri nets can be determined by linear programming. However, in general, the asymptotic throughputs are non-monotone with respect to the initial marking or the firing rates of the transitions [@mahulea2006performance]. An example of oscillations in infinite time around a steady state is also given in [@mahulea2008steady].
Contributions {#contributions .unnumbered}
-------------
We propose a continuous dynamics of Petri nets where time is attached to places and not to transitions. The main novelty is that it handles a class of Petri nets in which tokens can be routed according to priority rules (Section \[sec:semantics\]). We initially studied this class in [@allamigeon2015performance] in the discrete setting, motivated by an application to the performance analysis of an emergency call center.
We show that the continuous dynamics can equivalently be expressed in terms of [*policies*]{}. A policy is a map associating with every transition one of its upstream places. In this way, the dynamics of the Petri net can be written as an infimum of the dynamics of subnets induced by the different policies. The policies reaching the infimum indicate the places which are bottleneck in the Petri net. On any time interval in which a fixed policy reaches the infimum, the dynamics reduces to a linear dynamics (Section \[sec:policies\]).
We characterize the stationary solutions in terms of the policies of the Petri net. This allows us to set up a correspondence between the (ultimately affine) stationary solutions of the discrete dynamics that were described in [@allamigeon2015performance] and the stationary solutions of the continuous dynamics (Section \[sec:4\]). We also relate the continuous stationary solutions to the initial marking of the Petri net. This relies on restrictive assumptions, in particular the semi-simplicity of a 0 eigenvalue of a matrix associated with a policy.
We finally provide some numerical simulations of the continuous dynamics. We consider a model of emergency call center with two hierarchical levels for handling calls, originating from a real case study (17-18-112 call center in the Paris area) [@allamigeon2015performance]. On this Petri net, numerical experiments illustrate the convergence of the trajectory towards the stationary solution. This exhibits an advantage of the continuous setting in comparison to the discrete one, in which, for certain values of the parameters, the asymptotic throughputs computed by simulations differ from the stationary solutions (Section \[sec:numerical experiments\]).
Related work {#related-work .unnumbered}
------------
The motivation of this work stems from our previous study [@allamigeon2015performance], in which we addressed the same class of Petri nets with priorities in the discrete setting, and applied it to the performance analysis of an emergency call center. The discrete dynamics is shown there to be given by piecewise affine equations (tropical analogues of rational equations). The idea of modeling priority rules by piecewise affine dynamics originated from Farhi, Goursat and Quadrat [@farhi], who applied it to a special class of road traffic models. In the discrete setting, limit time-periodic behaviors can occur. They may lead to asymptotic throughputs different from the affine stationary solutions of the dynamics, a pathology which motivates our study of a continuous version of the dynamics.
The “continuization” of our dynamics draws inspiration from the original continuous model where time is attached to transitions. In particular, the situation in which the routing of a token at a given place is influenced by the firing times of the output transitions through a race policy has received much attention, see [@vazquez2013introduction]. Here, we address the situation in which the routing is specified by priority or preselection rules which are independent of the processing rates. To do so, it is convenient to attach times to places, instead of attaching firing rates to transitions. We point out in Remark \[rk:comparison\] that our model can be reduced to a variant of the standard continuous model [@vazquez2013introduction] in which we allow immediate transitions and require non-trivial routings to occur only at these transitions. A benefit of our presentation is to allow a more transparent comparison between the continuous model and the discrete time piecewise affine models studied in [@cohen1995asymptotic; @gaujal2004optimal; @allamigeon2015performance].
The use of the term “policy” refers to the theory of Markov decision processes, owing to the analogy between the discrete time dynamics and the value function of a semi-Markovian decision process. Note that in the context of continuous Petri nets, policies are also known as “configurations”, see [@mahulea2006performance] for an example.
Continuous dynamics of Petri nets {#sec:semantics}
=================================
General notation {#sec:notations}
----------------
A Petri net consists of a set ${\mathcal{P}}$ of places, a set ${\mathcal{Q}}$ of transitions and a set of arcs $\mathcal{E} \subset ({\mathcal{P}}\times {\mathcal{Q}}) \cup ({\mathcal{Q}}\times {\mathcal{P}})$. Every arc is given a valuation in ${\mathbb{N}}$. Each place $p \in {\mathcal{P}}$ is given an initial marking $M^0_p \in {\mathbb{N}}$, which represents the number of tokens initially occurring in the place.
We denote by $a^+_{qp}$ the valuation of the arc from transition $q$ to place $p$, with the convention that $a^+_{qp}=0$ if there is no such arc. Similarly, we denote by $a^-_{qp}$ the valuation of the arc from place $p$ to transition $q$, with the same convention. We set $a_{qp} := a^+_{qp} - a^-_{qp}$. The ${\mathcal{Q}}\times {\mathcal{P}}$ matrix $A = (a_{qp})_{q \in {\mathcal{Q}}, p \in {\mathcal{P}}}$ is referred to as the *incidence matrix* of the Petri net, and its transpose matrix $C := A{^\mathsf{T}}$ as its *token flow matrix*. We also denote by $C^+$ (resp. $C^-$) the ${\mathcal{P}}\times {\mathcal{Q}}$ matrix with entry $a_{qp}^+$ (resp. $a_{qp}^-$), so that $C = C^+ - C^-$. We limit our attention to *pure* Petri nets, [*i.e.*]{}, Petri nets with no self-loop: for every pair $(q,p)$, at least one of $a^+_{qp}$ and $a^-_{qp}$ is zero.
We denote by $q{^{\textrm{in}}}$ the set of upstream places of transition $q$ and by $q{^{\textrm{out}}}$ the set of downstream places of transition $q$. Similarly, we use the notation $p{^{\textrm{in}}}$ and $p{^{\textrm{out}}}$ to refer to the sets of input and output transitions of a place $p$.
Petri nets with free choice and priority routing
------------------------------------------------
In this paper, we consider a class of Petri nets in which places are either free choice or subject to priority. Recall that a place $p\in {\mathcal{P}}$ is said to be *free choice* if either all the output transitions $q \in p{^{\textrm{out}}}$ satisfy $q{^{\textrm{in}}}= \{p\}$ (*conflict*, see Figure \[fig:configuration\](a)), or $|p{^{\textrm{out}}}| = 1$ (*synchronization*, see Figure \[fig:configuration\](b)). A place is *subject to priority* if its tokens are routed to output transitions according to a priority rule. We refer to Figure \[fig:configuration\](c) for an illustration. For the sake of simplicity, we assume that each place subject to priority has exactly two output transitions, and that any transition has at most one upstream place subject to priority. Given a place $p$ subject to priority, we denote by $q^+(p)$ and $q^-(p)$ its two output transitions, with the convention that $q^+(p)$ has priority over $q^-(p)$. For the sake of readability, we use the notation $q^+$ and $q^-$ when the place $p$ is clear from context.
The set of transitions such that every upstream place $p$ satisfies $|p{^{\textrm{out}}}| = 1$ is referred to as ${\mathcal{Q}_\textsf{sync}}$ and the set of free choice places that have at least two output transitions is referred to as ${\mathcal{P}_\textsf{conflict}}$. We denote by ${\mathcal{P}_\textsf{priority}}$ the set of places subject to priority. The sets $({\mathcal{P}_\textsf{conflict}}){^{\textrm{out}}}$, ${\mathcal{Q}_\textsf{sync}}$ and $({\mathcal{P}_\textsf{priority}}){^{\textrm{out}}}$ form a partition of ${\mathcal{Q}}$. Figure \[fig:configuration\] hence summarizes the three possible place/transition patterns that can occur in this class of Petri nets.
![Conflict, synchronization and priority patterns[]{data-label="fig:configuration"}](main-FINAL-figure0.pdf)
Continuous dynamics and routing rules
-------------------------------------
We now equip the Petri net with a continuous semantics. Given a transition $q$, we associate a flow $f_q(t)$ which represents the instantaneous firing rate of transition $q$ at time $t$. We also associate with each place $p$ a [*marking*]{} $M_p(t)$, which is a continuous real valued function of the time $t$. In the case of [*discrete*]{} timed Petri nets, one typically requires that every token stays a minimum time in the place, — at this stage, the token may be considered as [*under processing*]{} — before becoming available for the firing of output transitions. To capture this property in the continuous setting, we assume that the marking $M_p(t)$ can be decomposed as $M_p(t)=m_p(t)+w_p(t)$, where $m_p(t)$ is the quantity of tokens under processing and $w_p(t)$ is the quantity of tokens waiting to contribute to the firing of an output transition.
We associate with each place $p$ a time constant $\tau_p>0$. Each token entering in a place is processed with the rate $1/\tau_p$. This leads to the following differential equation: $$\label{eq:mp}
\dot{m}_p(t) = \sum_{q \in p{^{\textrm{in}}}}a_{qp}^+f_q(t) - \dfrac{m_p(t)}{\tau_p} \, .$$ The evolution of the number of tokens waiting in place $p$ is described by the relation: $$\label{eq:wp}
\dot{w}_p(t) = \dfrac{m_p(t)}{\tau_p} - \sum_{q\in p{^{\textrm{out}}}} a_{qp}^- f_{q}(t) \, .$$ Moreover, for all transition $q$, we require that $$\min_{p\in q{^{\textrm{in}}}, \, w_p(t)=0}\Big(
\dfrac{m_p(t)}{\tau_p} - \sum_{q'\in p{^{\textrm{out}}}} a^-_{q'p}f_{q'}(t)\Big) =0
\,.
\label{eq:earliest}$$ In particular, this implies that at least one place $p \in q{^{\textrm{in}}}$ verifies $w_p(t) = 0$. In this case, means that each of the upstream places $p$ that has a zero quantity of waiting tokens ($w_p(t)=0$) must satisfy $\dot{w}_p(t)\geq 0$, and that at least one of these places satisfies $\dot{w}_p(t)=0$. In other words, there is at least one *bottleneck* upstream place $p$ of $q$, which has no waiting tokens and whose outgoing flow $\sum_{q'\in p{^{\textrm{out}}}} a^-_{q'p}f_{q'}(t)$ coincides with its processing flow ${m_p(t)}/{\tau_p}$.
The relation provided in can be simplified in the case of conflict and synchronization patterns. In more detail, if $q$ has a unique upstream place $p$, and this place is free choice ([conflict]{}), then reduces to: $$\label{eq:conflict}
\dfrac{m_p(t)}{\tau_p} - \sum_{q'\in p{^{\textrm{out}}}} a^-_{q'p}f_{q'}(t) =0 \,.$$ Now, if $q$ has several upstream places, which are all free choice ([synchronization]{}), then reads as: $$\label{eq:sync}
f_{q}(t) = \min_{p\in q{^{\textrm{in}}}, \, w_p(t)=0} \dfrac{m_p(t)}{a^-_{qp}\tau_p} \, .$$ This equation also holds if $|q{^{\textrm{in}}}| = 1$ and if the upstream place of $q$ has a single output transition.
We respectively denote by $m(t)$, $w(t)$ and $f(t)$ the vectors of entries $m_p(t)$, $w_p(t)$ and $f_q(t)$.
Albeit the dynamics that we presented so far is piecewise affine, a trajectory $t \mapsto (m(t), w(t), f(t))$ may be discontinuous. Indeed, in , the set of the places over which the minimum is taken may change over time. If at time $t$, there is a new place $p \in q{^{\textrm{in}}}$ such that $w_p(t)$ cancels, and if the quantity ${m_p(t)}/(a^-_{qp}\tau_p)$ is sufficiently small, then the minimum in (and subsequently the flow $f_q(t)$) discontinuously jumps to the latter value.
Initial conditions of the dynamics are specified by a pair $(m(t_i),w(t_i))$ such that the minimum in makes sense, [*i.e.*]{}, at least one $w_p(t_i)$ is equal to $0$ for each set of places $q{^{\textrm{in}}}$. One can easily show that if the set $\{ p \in q{^{\textrm{in}}}\colon w_p(t) = 0\} $ is nonempty for all transition $q \in {\mathcal{Q}}$ at time $t = t_i$, then it remains nonempty for all time $t \geq t_i$.
The dynamics – may admit different trajectories for a given initial condition. These correspond to different routings of tokens in places with several output transitions. However, each of these trajectories satisfies the conservation law: $$\label{eq:flow_conservation_eq}
\dot{m}(t) + \dot{w}(t) = C f(t) \, ,$$ where $C$ is the token flow matrix of the Petri net. Recall that a *P-invariant* of the Petri net refers to a nonnegative solution $y$ of the system $y{^\mathsf{T}}C = 0$. In the discrete setting, a P-invariant corresponds to a weighting of places that is constant for any reachable marking, meaning that the quantity $y {^\mathsf{T}}M$ is preserved under any firing of transition. An analogous statement holds in the continuous setting:
Given a P-invariant $y$ of the Petri net, the quantity $y{^\mathsf{T}}(m(t) + w(t))_{p \in {\mathcal{P}}}$ is independent of $t$.
In particular, if the entries of $y$ are all positive, then the Petri net is bounded, [*i.e.*]{}, each function $t \mapsto M_p(t)$ is bounded.
The proof consists in multiplying both sides of by the invariant $y$. It follows that the derivative of $y{^\mathsf{T}}(m(t) + w(t))_{p \in {\mathcal{P}}}$ is zero. If the entries of $y$ are all positive, then $\bigl(y{^\mathsf{T}}(m(0) + w(0))_{p \in {\mathcal{P}}}\bigr)/y_p$ is an upper bound to the marking of any place $p$.
The following proposition collects several homogeneity properties of the continuous dynamics:
\[thm:prop2\] Let $(m(t),w(t),f(t))$ be a trajectory solution of the dynamics –, with the initial markings $(m_p(0))_{p \in {\mathcal{P}}}$, and the holding times $(\tau_p)_{p \in {\mathcal{P}}}$ and let $\alpha \in {\mathbb{R}_{> 0}}$, then:
(i) $(\alpha m(t), \alpha w(t), \alpha f(t))$ is a trajectory solution of the dynamics, associated with the initial markings $(\alpha m_p(0))_{p \in {\mathcal{P}}}$.
(ii) $(m(t/\alpha),w(t/\alpha),(1/\alpha)f(t/\alpha))$ is a trajectory solution of the dynamics, associated with the holding times $(\alpha \tau_p)_{p \in {\mathcal{P}}}$ and the same initial conditions.
(iii) let $x$ be a vector of the kernel of $C$, and $D = \text{diag}(\tau)$ be the ${\mathcal{P}}\times {\mathcal{P}}$ diagonal matrix such that $D_{pp} = \tau_p$, then $(m(t) + \alpha D C^+ x, w(t), f(t) + \alpha x)$ is a trajectory solution of the dynamics, associated with the initial markings $(m(0) + \alpha D C^+ x)$.
The first two statements derive easily from the homogeneity properties of Equations –. For the third statement, one can note that adding $\alpha x_q$ to the $f_q(t)$ and adding $\alpha \sum_{q\in p{^{\textrm{in}}}} a_{qp}^+ x_q$ to the $m_p(t)/\tau_p$ in – does not change the right hand sides of and , or the expression within the minimum in . For and , this is due to the fact that $(C^+ - C^-) x = C x = 0$.
We now complete the description of the continuous dynamics by additional equations which arise from the specification of routing rules. Such rules occur in the following two situations:
Given $p \in {\mathcal{P}_\textsf{conflict}}$, we suppose that tokens are routed according to a stationary distribution specified by weights $\mu_{qp} > 0$ associated with each output transition $q$. Therefore, $$\label{eq:conflict_routing}
\forall p \in {\mathcal{P}_\textsf{conflict}}, \; \forall q \in p{^{\textrm{out}}}, \quad a^-_{qp}f_q(t) = \mu_{qp} \dfrac{m_p(t)}{\tau_p} \, .$$
Let $p \in {\mathcal{P}_\textsf{priority}}$, and $q_+$ and $q_-$ be the two output transitions, as illustrated in (c). In order to specify that the flow is routed in priority to transition $q_+$, we require that: $$\begin{aligned}
f_{q_+}(t) &= \min_{r \in q_+{^{\textrm{in}}}, \, w_r(t) = 0} \dfrac{m_r(t)}{a^-_{q_+{r}}\tau_r} \,,
\label{eq:p1}\\
f_{q_-}(t) &= \begin{cases}
\min_{r \in q_-{^{\textrm{in}}}\setminus \{{p}\}, w_r(t) = 0} \dfrac{m_r(t)}{a^-_{q_-r}\tau_r} & \text{if}\ w_{{p}}(t) \neq 0 \, ,\\
\min \Bigl(\dfrac{m_{{p}}(t)}{a^-_{q{p}}\tau_{{p}}} - \dfrac{a^-_{q_+{p}}}{a^-_{q{p}}}f_{q_+}(t) \, , \min_{r \in q_-{^{\textrm{in}}}\setminus \{{p}\}, w_r(t) = 0} \dfrac{m_r(t)}{a^-_{q_-r}\tau_r}
\Big)
& \text{if}\ w_{{p}}(t) = 0 \, .
\end{cases}
\label{eq:p2}\end{aligned}$$
The expression of $f_{q_-}(t)$ in , when $w_{p}=0$, indicates that only the outgoing flow from ${p}$ that is not already consumed by the priority transition $q_+$ is available to $q_-$.
The first two properties of homogeneity in Proposition \[thm:prop2\] are still satisfied by the dynamics extended by the routing rules –.
\[rk:comparison\] We already mentioned in the introduction that our model differs from the standard continuous Petri net model in which transitions are equipped with firing rates, in the sense that in the latter model, the flows of the output transitions of a given place are pairwise independent. To overcome this limitation, *immediate transitions* have been introduced [@recalde2006improving]. These transitions come with the specification of routing rules, for instance, in the case of conflict pattern. In this way, our model could be reduced to a classical continuous model enriched with immediate transitions. In this reduction, we require timed transitions to have exactly one upstream place and one downstream place, so that all the routing is determined by immediate transitions, which inherit the equations defined in our place-timed dynamics.
Simply put, our model is the continuous analogue of discrete Petri nets equipped with “holding durations”, in which tokens are frozen during processing, whereas the usual continuous Petri net model can be seen as the continuous analogues of Petri nets with “enabling durations”, in which transitions preempt tokens. We refer to [@bowden2000brief] for a discussion on the meaning of time in Petri nets.
Policies and bottleneck places {#sec:policies}
==============================
The analysis of the piecewise affine dynamical system – leads to introduce the notion of [*policy*]{}. Fixing a policy allows one to solve the dynamics on a region where it is linear. We shall see in that policies also arise in the characterization of stationary solutions.
Even if our continuous dynamics holds for more general classes of Petri nets, we focus in the following on strongly connected, autonomous Petri nets, so that each transition has at least one upstream place.
We observe that the dynamics of Petri nets with free choice and priority routing –, and – is linear on each region where the arguments of the minimum operators do not change. More precisely, at any time $t$, for any transition $q \in {\mathcal{Q}}$, there exists a place $p \in q{^{\textrm{in}}}$ such that $w_p(t)=0$ and $p$ is the unique upstream place of $q$ or $p$ realizes the minimum in the expression , or of $f_q(t)$. Place $p$ is then referred to as the *bottleneck place* of transition $q$ at time $t$.
We define a *policy* $\pi$ as a function from ${\mathcal{Q}}$ to ${\mathcal{P}}$, which maps any transition $q$ to one of its upstream places $p_{\pi}(q) \in q{^{\textrm{in}}}$. A policy is meant to indicate the bottleneck place of each transition $q$. We denote by $S_{\pi}$ the [*selection matrix*]{} associated with $\pi$, that is, the ${\mathcal{Q}}\times {\mathcal{P}}$ matrix such that $(S_{\pi})_{qp} = 1$ if $p = p_{\pi}(q)$, and $0$ otherwise. In particular, $(S_{\pi})_{qp} = 1$ implies that $a_{qp} < 0$.
Note that, if $p$ realizes the minimum in one of the equations , or for some transition, then $p$ also realizes the minimum in . The converse is not true if places are subject to priority. For $p$ denoting a priority place and $q_+$ its priority output transition, if $p$ realizes the minimum in for transition $q_+$, then, $p$ does not necessarily realize the minimum in . In other words, our definition of a bottleneck place is dependent on the routing rules of the net.
We point out that notions comparable to policies are used in [@mahulea2006performance] in the context of continuous Petri nets with time attached to transitions.
The dynamics of a Petri net can be expressed in terms of the different policies of the net: at any time $t$, there is a policy $\pi^*$ (we can note $\pi^*(t)$ if we want to emphasize the dependence on time) such that $$\forall q \in {\mathcal{Q}}\, , \quad w_{p_{\pi^*}(q)}(t) = 0 \,, $$ and $$\begin{aligned}
&\forall q \in {\mathcal{Q}}\text{ s.t.\ } p_{\pi^*}(q) \not\in ({\mathcal{P}_\textsf{conflict}}\cup {\mathcal{P}_\textsf{priority}}) \,, \; \dfrac{m_{p_{\pi^*}(q)}(t)}{\tau_{p_{\pi^*}(q)}} = a^-_{qp_{\pi^*}(q)} f_q(t) \,,\\
&\forall q \in {\mathcal{Q}}\text{ s.t.\ } p_{\pi^*}(q) \in {\mathcal{P}_\textsf{conflict}}\,, \quad \dfrac{m_{p_{\pi^*}(q)}(t)}{\tau_{p_{\pi^*}(q)}} = \dfrac{a^-_{qp_{\pi^*}(q)}}{\mu_{qp_{\pi^*}(q)}} f_q(t) \,,\\
&\forall q_+ \in {\mathcal{Q}}\text{ s.t.\ } p_{\pi^*}(q_+) \in {\mathcal{P}_\textsf{priority}}\,, \; \dfrac{m_{p_{\pi^*}(q_+)}(t)}{\tau_{p_{\pi^*}(q_+)}} = a^-_{q_+p_{\pi^*}(q_+)} f_{q_+}(t) \,,\\
&\forall q_- \in {\mathcal{Q}}\text{ s.t.\ } p_{\pi^*}(q_-) \in {\mathcal{P}_\textsf{priority}}\,, \; \dfrac{m_{p_{\pi^*}(q_-)}(t)}{\tau_{p_{\pi^*}(q_-)}}
= a^-_{q_-p_{\pi^*}(q_-)} f_{q_-}(t)
+ a^-_{q_+p_{\pi^*}(q_+)} f_{q_+}(t) \,.
\end{aligned}$$
Now, for any policy $\pi$, we denote by ${C^-_{\pi}}$ the ${\mathcal{Q}}\times {\mathcal{Q}}$ matrix such that the right-hand side of this system of equations reads ${C^-_{\pi}}f(t)$ , where $f(t)$ is the vector of the $(f_q(t))_{q \in {\mathcal{Q}}}$. In particular, the above system writes $$S_{\pi^*} \left(\dfrac{m(t)}{\tau}\right) = {C^-_{\pi^*}}f(t) \,,$$ where $(m(t)/\tau)$ is the vector of the $(m_p(t)/\tau_p)_{p \in {\mathcal{P}}}$. The diagonal entries of ${C^-_{\pi}}$ are positive. Moreover, if we order each transition of type $q_+$ before its associated transition $q_-$, the matrix ${C^-_{\pi}}$ becomes lower triangular.[^2] Hence, ${C^-_{\pi}}$ is invertible. Matrix ${C^-_{\pi}}$ can be seen as a specification of the downstream token flow matrix of the Petri net $C^-$ (introduced in Section \[sec:notations\]), associated with the policy $\pi$.
With this notation, the continuous dynamics of Petri nets with free choice and priority routing reads:
\[eq:matrix\_dyn\_system\] $$\begin{aligned}
f(t) &= \inf_{\pi \text{ s.t. } S_{\pi} w (t) = 0} ({C^-_{\pi}})^{-1} S_{\pi} \left( \dfrac{m(t)}{\tau} \right) \label{eq:matrix_dyn_f} \,,\\
\dot{m}(t) &= C^+ f(t) - \dfrac{m(t)}{\tau} \label{eq:matrix_dyn_dot_m} \,,\\
\dot{w}(t) &= \dfrac{m(t)}{\tau} - C^- f(t) \label{eq:matrix_dyn_dot_w} \,,
\end{aligned}$$
where the infimum must be understood for the partial order over ${\mathbb{R}}^{\mathcal{Q}}$ induced by $\leq$. Note that there is at least one policy $\pi^*$ (depending on $t$) attaining the infimum. It suffices to choose the policy $\pi^*$ introduced earlier ([*i.e.*]{}, to choose a policy that attains the minimum componentwise).
By choosing an upstream place for each transition, a policy actually defines a candidate “bottleneck net” of the Petri net, that is, a subnet with all the transitions of the original Petri net, and such that each transition has a unique upstream place. On each of these subnets, the dynamics is linear and yields a unique trajectory for a given initial condition. The trajectory is solved on a subnet of the original Petri net, but one can easily recover the solution over the whole Petri net. This applies to the original dynamics of the system, on any time interval over which the infimum is reached by a constant policy, as stated in the following proposition.
\[thm:fc\_prio\_unicity\] Suppose that there is a policy $\pi^*$ which reaches the infimum in for all time $t$ in the interval $[t_i, t_f]$. Then the dynamics of the Petri net with free choice and priority routing reduces to a linear system, which admits a unique solution, given the initial conditions $(m(t_i), w(t_i))$.
If $\pi^*$ reaches the infimum for any $t \in [t_i, t_f]$, then the continuous dynamics of the Petri net reads:
\[eq:const\_pi\_dyn\_system\] $$\begin{aligned}
{C^-_{\pi^*}}f(t) &= S_{\pi^*} \left(\dfrac{m(t)}{\tau} \right) \label{eq:const_pi_dyn_f} \,, \\
\dot{m}(t) &= C^+ f(t) - \dfrac{m(t)}{\tau} \label{eq:const_pi_dyn_dot_m} \,, \\
\dot{w}(t) &= \dfrac{m(t)}{\tau} - C^- f(t) \label{eq:const_pi_dyn_dot_w} \,, \\
S_{\pi^*} w (t) &= 0 \,, \label{eq:const_pi_dyn_w}
\end{aligned}$$
which is a linear system.
We multiply by $S_{\pi^*}$, and replace the term $S_{\pi^*} (m(t)/\tau)$ by its expression given in . This leads to: $$S_{\pi^*} \dot{m}(t) = S_{\pi^*}C^+ f(t) - {C^-_{\pi^*}}f(t) \,.$$
Let $D = \text{diag}(\tau)$ be the ${\mathcal{P}}\times {\mathcal{P}}$ diagonal matrix such that $D_{pp} = \tau_p$, then $D_{\pi} \mathrel{:=} S_{\pi} D {S_{\pi}}{^\mathsf{T}}$ is the ${\mathcal{Q}}\times {\mathcal{Q}}$ diagonal matrix such that $(D_{\pi})_{qq} = \tau_{p_{\pi}(q)}$. Equation then writes $S_{\pi^*} m(t) = D_{\pi^*} {C^-_{\pi^*}}f(t)$. This leads to: $$\dot{f}(t) = ({C^-_{\pi^*}})^{-1} D_{\pi^*}^{-1} \left( S_{\pi^*} C^+ - {C^-_{\pi^*}}\right) f(t) \, ,$$ which is an ordinary differential system. Moreover, the $f(t_i)$ can be obtained from the $m(t_i)$ by , so that this system admits a unique solution $f$ for all $t \in [t_i, t_f]$.
Given this solution $f$, one can successively solve the differential system in ${m}$ given by and the differential system in ${w}$ given by , whose initial conditions are known, so that the whole dynamics admits a unique trajectory.
Stationary solutions {#sec:4}
====================
In this section, we prove that the stationary solutions of the continuous and discrete dynamics of a timed Petri net with free-choice and priority routing are the same. To do so, we first recall in Section \[sec:stationary\_discrete\] the formulation of the discrete dynamics and the associated stationary solutions given in [@allamigeon2015performance].
Stationary solutions of the discrete dynamics {#sec:stationary_discrete}
---------------------------------------------
The discrete dynamics of Petri nets with free choice and priority is expressed in terms of *counter variables* associated with transitions and places. Given a transition $q$, the counter variable $z_q: {\mathbb{R}_{\geq 0}}\to {\mathbb{N}}$ denotes the number of firings of $q$ that occurred up to time $t$ included. Similarly, the counter variable of place $p$ is a function $x_p:{\mathbb{R}_{\geq 0}}\to {\mathbb{N}}$ which represents the number of tokens that have visited place $p$ up to time $t$ included (taking into account the initial marking). On top of being non-decreasing, the counter variables are *[[càdlàg]{}]{}* functions, which means that they are right continuous and have a left limit at any time.
In this setting, the parameter $\tau_p$ associated with the place $p$ represents a minimal holding time. It is shown in [@allamigeon2015performance] that, if tokens are supposed to be fired as early as possible, the counter variables satisfy the following equations (we generalize the equations to the case with valuations):
\[eq:discrete\] $$\begin{aligned}
\forall p \in {\mathcal{P}}\,&, \quad x_p(t) = M_p^0 + \sum_{q\in p{^{\textrm{in}}}} a^+_{qp} z_q(t) \, , \label{eq:pnpriority1} \\
\forall p \in {\mathcal{P}_\textsf{conflict}}\,&, \quad \sum_{q \in p{^{\textrm{out}}}} a^-_{qp} z_q(t) = x_p(t - \tau_p) \, ,\label{eq:pnpriority2} \\
\forall q \in {\mathcal{Q}_\textsf{sync}}\,&, \quad z_q(t) = \min_{p \in q{^{\textrm{in}}}} x_p(t - \tau_p) / a^-_{qp} \, ,\label{eq:pnpriority3}
\end{aligned}$$ $$\begin{aligned}
\forall & p \in {\mathcal{P}_\textsf{priority}}\,, \nonumber\\
&
z_{q_+}(t) = \min\biggl(
\Bigl(\dfrac{1}{a^-_{q_+p}}x_p(t-\tau_p) - \dfrac{a^-_{q_-p}}{a^-_{q_+p}} \lim_{s \uparrow t} z_{q_-}(s) \Bigr),\min_{{r \in q_+{^{\textrm{in}}}, r \neq p}} \dfrac{1}{a^-_{q_+r}}x_r(t - \tau_r) \biggr)\, ,
\label{eq:pnpriority4} \\
&
z_{q_-}(t) = \min \biggr(
\Bigl(\dfrac{1}{a^-_{q_-p}}x_p(t-\tau_p) - \dfrac{a^-_{q_+p}}{a^-_{q_-p}} z_{q_+}(t)\Bigr),
\min_{{r \in q_-{^{\textrm{in}}}, r \neq p}}\dfrac{1}{a^-_{q_-r}} x_r(t - \tau_r) \biggr) \, ,
\label{eq:pnpriority5}\end{aligned}$$
where $q_+$ ($q_-$) is the priority (non priority) output transition of $p \in {\mathcal{P}_\textsf{priority}}$.
Note that if all the holding times $\tau_p$ are integer multiples of a fixed time $\delta$, the left limit $\lim_{s \uparrow t}z_{q_-}(s)$ in can be replaced by $z_{q_-}(t-\delta)$. This is helpful in particular to simulate these equations.
In the setting of [@allamigeon2015performance], all conflicts are solved by a stationary distribution routing. The equivalent of the routing rule introduced to solve conflicts in the continuous setting is obtained here by allowing the tokens to be shared in fractions, so that the counter functions take real values. This corresponds to a *fluid approximation* of the discrete dynamics. In this setting, for each $p \in {\mathcal{P}_\textsf{conflict}}$ and $q\in p{^{\textrm{out}}}$, we fix $\mu_{qp}>0$, giving the proportion of the tokens routed from $p$ to $q$. We have: $$\forall p \in {\mathcal{P}_\textsf{conflict}}\, , \, \forall q \in p{^{\textrm{out}}}\,, \quad z_q(t) = \dfrac{\mu_{qp}}{a^-_{qp}} x_p(t - \tau_p) \ .
\label{eq:pnpriority6fluid}$$
The stationary solutions of the discrete dynamics are defined as functions $x_p$ and $z_q$ satisfying the relations – and which ultimately behave as affine functions, [*i.e.*]{}, $x_p(t) = u_p + t \rho_p$ and $z_q(t) = u_q + t \rho_q$ for all $t$ large enough. In this case, $\rho_p$ (resp. $\rho_q$) represents the asymptotic throughput of place $p$ (resp. transition $q$). We have shown in [@allamigeon2015performance Theorem 3] that these stationary solutions are precisely given by following system (we generalize the equations to the case with valuations):
\[eq:fixpoint\_germ\_rho\] $$\begin{aligned}
\forall p \in {\mathcal{P}}\, , \qquad \rho_p& = \sum_{q \in p{^{\textrm{in}}}} a^+_{qp} \rho_q \label{eq:fixpoint_germ1_rho} \,, \\
\forall p \in {\mathcal{P}_\textsf{conflict}}\, , \forall q \in p{^{\textrm{out}}}\, , \qquad \rho_q & = \mu_{qp} \rho_p / a^-_{qp}
\label{eq:fixpoint_germ2_rho} \,, \\
\forall q \in {\mathcal{Q}_\textsf{sync}}\, , \qquad \rho_q & = \min_{p \in q{^{\textrm{in}}}} \rho_p/a^-_{qp} \label{eq:fixpoint_germ3_rho}\,, \\
\forall p \in {\mathcal{P}_\textsf{priority}}\, , \qquad
\rho_{q_+} &= \min_{r \in q_+{^{\textrm{in}}}} \rho_r / a^-_{q_+r} \label{eq:fixpoint_germ4_rho}\,, \\
\forall p \in {\mathcal{P}_\textsf{priority}}\, , \quad
\rho_{q_-} = \min \Big(
&{\big(\rho_p - a^-_{q_+p}\rho_{q_+}\big)}/{a^-_{q_-p} }, \min_{r \in q_-{^{\textrm{in}}}\setminus \{ p \}} {\rho_r}/{a^-_{q_-r}} \Big) \,,
\label{eq:fixpoint_germ5_rho} \end{aligned}$$
\[eq:fixpoint\_germ\_u\] $$\begin{aligned}
\forall p \in {\mathcal{P}}\, &, \quad u_p = M_p^0 + \sum_{q \in p{^{\textrm{in}}}} a^+_{qp} u_q \label{eq:fixpoint_germ1_u} \,, \\
\forall p \in {\mathcal{P}_\textsf{conflict}}\, , \forall q \in p{^{\textrm{out}}}&, \; u_q = (\mu_{qp}/a^-_{qp}) (u_p - \rho_p \tau_p)
\label{eq:fixpoint_germ2_u} \,,\end{aligned}$$ $$\forall q \in {\mathcal{Q}_\textsf{sync}}\, , \quad u_q = \min_{p \in q{^{\textrm{in}}}, \rho_q = \rho_p} (u_p - \rho_p \tau_p)/a^-_{qp} \label{eq:fixpoint_germ3_u} \,,$$ $$\begin{aligned}
\forall p &\in {\mathcal{P}_\textsf{priority}}\, , \nonumber\\
u_{q_+} & = \begin{dcases}
\min
\begin{multlined}[t]
\Big( (u_p - \rho_p \tau_p - a^-_{q_-p}u_{q_-})/a^-_{q_+p}, \\
\min_{r \in q_+{^{\textrm{in}}}\setminus \{p\}, \, \rho_{q_+} = \rho_r} (u_r - \rho_r \tau_r)/a^-_{q_+r} \Big) \end{multlined}
\quad &\text{if } \rho_{q_-} = 0 \, ,
\\
\min_{r \in q_+{^{\textrm{in}}}\setminus \{p\}, \, \rho_{q_+} = \rho_r} (u_r - \rho_r \tau_r)/a^-_{q_+r} \quad &\text{otherwise,}
\end{dcases}\label{eq:fixpoint_germ4_u}
\\
u_{q_-} & = \begin{dcases}
\min
\begin{multlined}[t]
\Big( (u_p - \rho_p \tau_p - a^-_{q_+p} u_{q_+})/a^-_{q_-p}, \\
\min_{r \in q_-{^{\textrm{in}}}\setminus \{p\}, \, \rho_{q_-} = \rho_r} (u_r - \rho_r \tau_r)/a^-_{q_-r} \Big)
\end{multlined}
\quad &\!\!\! \text{if } \rho_{q_-} + \rho_{q_+} = \rho_p \, ,
\\
\min_{r \in q_-{^{\textrm{in}}}\setminus \{p\}, \, \rho_{q_-} = \rho_r} (u_r - \rho_r \tau_r)/a^-_{q_-r} \quad &\text{otherwise.}
\end{dcases}\label{eq:fixpoint_germ5_u}\end{aligned}$$
The above equations are expressed in a more compact form in [@allamigeon2015performance], using a semiring of germs of affine functions, which encodes lexicographic minimization operations.
Stationary solutions of the continuous time dynamics {#sec:stationary_continuous}
----------------------------------------------------
In the continuous setting, we define a *stationary solution* as a solution $(m,w,f)$ of the continuous dynamics such that for any place, ${m_p}$ is constant and ${w_p}$ is affine ($\dot{w_p}$ is constant). The following theorem provides a characterization of the stationary solutions.
\[th:1\] A triple $(m,w,f)$ of vectors of resp. $|{\mathcal{P}}|$, $|{\mathcal{P}}|$ and $|{\mathcal{Q}}|$ functions from ${\mathbb{R}_{\geq 0}}$ to ${\mathbb{R}_{\geq 0}}$, with all the $m_p$ constant and all the $w_p$ affine, is a stationary solution of the continuous dynamics if and only if the following conditions hold:
$$\begin{aligned}
\dfrac{m}{\tau} &= C^+ f
\, , \label{eq:mpcns} \\
\dot{w} &= \dfrac{m}{\tau} - C^- f
\, , \label{eq:wpcns} \\
C f &\geq 0 \, ,\label{eq:stat_nodes_law} \\
\shortintertext{and there exists a policy $\pi^*$, such that}
\label{eq:wp_annuler}
\forall t \,, \quad S_{\pi^*} w(t) &= 0 \, , \\
\label{eq:right_annuler}
\left( S_{\pi^*} C^+ - {C^-_{\pi^*}}\right) f &= 0 \, .\end{aligned}$$
Note that the existence of an $f \gneq 0$ that satisfies provides a simple algebraic necessary condition to the existence of a stationary flow in a Petri net. This corresponds to the net being [*partially repetitive*]{} (see [@murata1989petri] for a definition).
Equations and are derived from and , with $\dot{m} =0$ for a stationary solution.
In a stationary solution, for any place $p$, $\dot{w}_p$ is constant, so that one cannot have $\dot{w}_p < 0$, otherwise this would yield $\lim_{t \rightarrow \infty} w_p(t) = -\infty$. Therefore, by , $(m/\tau) \geq C^- f$, and by , we can replace $(m/\tau)$ by $C^+ f$, and get .
As the $\dot{w}$ are constant, if, for some place $p$ and at some time $t_0 > 0$, $w_p(t_0) = 0$, then $\dot{w}_p =0$, (otherwise it would contradict $w_p(t) \geq 0$ for $0 \leq t < t_0$ or for $t > t_0$). Hence, the set of places $p$ such that $w_p(t) = 0$ is independent of time for $t > 0$.
Moreover, the $m_p$ are constant, so that, if a policy $\pi$ attains the minimum in at some time, then it attains the minimum at any time. This means that, if $(m, w, f)$ is a solution of the continuous dynamics, then there exists a policy $\pi^*$ such that: $$\begin{aligned}
\forall t \,, \quad S_{\pi^*} w(t) &= 0 \, , \\
{C^-_{\pi^*}}f &= S_{\pi^*} \left( \dfrac{m}{\tau} \right) \,.\end{aligned}$$ Now, by again, we can replace $m/\tau$ by $C^+ f$ in the above equation, and we get Equations and .
Conversely, suppose that a triple of functions $(m,w,f)$ satisfies the conditions of the theorem, with policy $\pi^*$. We prove that the the relations given in describing the dynamics are satisfied. First, and are derived from and , with $\dot{m}_p = 0$. We also note that, in Equations and , replacing the term $C^+ f$ by $m/\tau$ (by ) leads to the following equations: $$\begin{aligned}
C^- f &\leq \dfrac{m}{\tau} \label{eq:stationary_ineq_mp} \,, \\
f &= (C_{\pi^*}^-)^{-1} S_{\pi^*} (\dfrac{m}{\tau})\,. \label{eq:right_annuler_mp}\end{aligned}$$
Equations and show that $\pi^*$ attains the equality in . Hence, in order to prove , it is sufficient to prove that, for any $\pi$, we have $${C^-_{\pi}}f \leq S_{\pi} \left(\dfrac{m}{\tau}\right)\,. \label{eq:to_prove}$$ We prove this inequality row by row. Let $q$ be a transition. We distinguish the following cases:
- if $q \in {\mathcal{Q}_\textsf{sync}}$, then $(C^- f)_{p_{\pi}(q)} = ({C^-_{\pi}}f)_q$ for any $\pi$ (for any choice of an upstream place of $q$) so that follows from .
- if $q$ has a unique upstream place $p$, with $p \in {\mathcal{P}_\textsf{conflict}}$, then for any $\pi$, $p_{\pi}(q) = p_{\pi^*}(q)$ so that follows from .
- assume now that $q_+$ is the priority transition of a place $p$ subject to priority. Then, by , $m_p/\tau_p \geq a^-_{q_+p} f_{q_+} + a^-_{q_-p} f_{q_-} \geq a^-_{q_+p} f_{q_+}$ and for $r \in q_+{^{\textrm{in}}}\setminus \{p\}$, ${m_r}/{\tau_r} \geq a^-_{{q_+}r} f_{q_+}$. Finally, for any $r \in q_+{^{\textrm{in}}}$, $a^-_{q_+p} f_{q_+} \leq {m_r}/{\tau_r}$. This proves .
- let $q_-$ be the non priority transition of a place $p$ subject to priority. Then $(C^- f)_{p_{\pi}(q_-)} = ({C^-_{\pi}}f)_{q_-}$ for any policy $\pi$, so that follows from .
As a consequence of , we obtain a correspondence between the stationary solutions of the continuous dynamics and the stationary solutions of the discrete dynamics. In order to highlight the parallel between the discrete and the continuous setting, we denote by $f_p$ the processing flow $m_p/\tau_p$ for every place $p$.
\[coro:corresp\]
(i) Suppose $(m,w,f)$ defines a stationary solution of the continuous dynamics. Then, for the initial marking $M_p^0= m_p$, setting $\rho:=f$, $u_p:=M_p^0$, and $u_q:=0$ yields a stationary solution of the discrete dynamics.
(ii) Conversely, suppose $(\rho,u)$ is a stationary solution of the discrete dynamics. Then, defining $f:=\rho$, setting $m_p:=\rho_p\tau_p$ for every place $p$, and defining $w$ according to and yields a stationary solution of the continuous dynamics.
Both statements are straightforward. We point out that reads $\rho_p = C^+ \rho_q$ and that – are equivalent to $\rho_q = \min_{\pi} ({C^-_{\pi}})^{-1} S_{\pi} \rho_p$. The same relationship between the $f_q$ and the $f_p$ was established in the proof of Theorem \[th:1\].
An important problem is to relate the stationary flow to the initial marking. On top of the relations given by the invariants of the Petri nets, most results in this direction are limited to nets without priorities, as they rely on monotonicity properties of the dynamics. The next theorem identifies, however, a somehow special situation in which such a relation persists even in the presence of priority. This applies in particular to the Petri net of the next section.
If a trajectory of the continuous Petri net converges towards a stationary solution $(m^{\infty},w^{\infty},f^{\infty})$, if for this trajectory, there exists a policy $\pi$ that reaches the infimum in at any time, and if $0$ is a semi-simple eigenvalue of $(S_{\pi} C^+ - {C^-_{\pi}})$ associated with this policy, then $f^{\infty}$ is uniquely determined by the initial marking.
(Recall that the eigenvalue $\lambda$ of a matrix $B$ is said to be [*semi-simple*]{} if the dimension of its eigenspace is equal to its algebraic multiplicity, that is, to the multiplicity of $\lambda$ as the root of the characteristic polynomial of $B$. In particular, if $0$ is a semi-simple eigenvalue of $B$, then the kernel of $B$ and its range space are complementary subspaces.)
Under the conditions of the theorem, there exists a policy $\pi$ such that, for any $t$, $$\begin{aligned}
S_{\pi} \dot{m}(t) &= (S_{\pi}C^+ - {C^-_{\pi}}) f(t) \,, \label{eq:eq1} \\
S_{\pi} m(t) &= D_{\pi} {C^-_{\pi}}f(t) \,,\end{aligned}$$ as shown in the proof of Proposition \[thm:fc\_prio\_unicity\].
Since $0$ is a semi-simple eigenvalue of $(S_{\pi}C^+ - {C^-_{\pi}})$, the same property holds for the matrix $(S_{\pi}C^+ - {C^-_{\pi}}) (D_\pi {C^-_{\pi}})^{-1} = (S_{\pi}C^+ ({C^-_{\pi}})^{-1} - I)D_{\pi}^{-1}$. Therefore, the kernel of this matrix and its range space are complementary subspaces. We denote by $Q$ the projection onto the former along the latter.
By , we obtain that $Q S_{\pi} \dot{m}(t) = Q (S_{\pi}C^+ - {C^-_{\pi}}) f(t) = 0$, so that $Q S_{\pi} {m}(t)$ is independent of time, and $$Q S_{\pi} {m}(0) = Q S_{\pi}m_{\infty} = Q D_{\pi} {C^-_{\pi}}f_{\infty} \,.$$ Moreover, as $(m^{\infty},w^{\infty},f^{\infty})$ is a stationary solution of the continuous dynamics, Equation holds and $D_{\pi} {C^-_{\pi}}f_{\infty}$ belongs to the kernel of $(S_{\pi}C^+ ({C^-_{\pi}})^{-1} - I)D_{\pi}^{-1}$. Therefore, $$f_{\infty} = ({C^-_{\pi}})^{-1} D_{\pi}^{-1} Q S_{\pi} {m}(0)\,. \qedhere$$
Experimental results {#sec:numerical experiments}
====================
In this section, we illustrate our results on the model of an emergency call center with two treatment levels, introduced in [@allamigeon2015performance]. In this simplified model of an emergency call center, emergency calls are handled by a first level of operators who dispatch them into three categories: extremely urgent, urgent and non urgent. Non-urgent calls (proportion $\mu_4$ of the calls) are entirely processed by level 1 operators. Extremely urgent ($\mu_2$) and urgent calls ($\mu_3$) are transfered to level 2 operators. Extremely urgent calls have priority over urgent calls (but cannot interrupt a talk between an operator of level 2 and an urgent call).
This emergency call center can be modeled by a Petri net with free choice and priority routing, as depicted in Figure \[fig:call\_center\]. Place $p_3$ is a conflict place with a fluid stationary routing, with proportions $\mu_2$, $\mu_3$, $\mu_4$, representing the dispatching of calls into the categories “extremely urgent”, “urgent” and “non urgent” respectively. Every arc has a valuation equal to one. The initial marking $M_1^0$ (resp. $M_2^0$) of place $p_1$ ($p_2$) denotes the available number of operators of level 1 (level 2) in the call center.
It was observed in [@allamigeon2015performance] that the discrete dynamics has a pathological feature: when certain arithmetic relations between the time delays are satisfied, the discrete time trajectory may not converge to a stationary solution, and its asymptotic throughput may differ from the throughput of the stationary solution. It follows from our correspondence result (Corollary \[coro:corresp\]) that the continuous dynamics has the same stationary solutions. We shall observe that, in this continuous setting, the trajectory converges towards a stationary solution, so that the former pathology vanishes.
To compute the (fluid approximation) of the discrete dynamics, simulations have been performed in exact (rational) arithmetics, using the GMP library [@gmplib]. The throughput of transitions $q_5$ and $q_6$ (see Figure \[fig:call\_center\]), for the discrete dynamics, are compared in to the throughputs of the stationary solutions, computed by Equations and .
![Petri net of a simplified emergency call center. Place $p_2$ is subject to priority routing. The initial markings of the places different from $p_1$ and $p_2$ are null. The holding times $(\tau_1,\tau_2,\tau_3,\tau_4,\tau_5,\tau_6,\tau_7,\tau_8,\tau_9,\tau_{10})$ are the following: $(0.01, 0.01, 0.01, 4, 3, 3, 1, 0.01, 6, 7)$.[]{data-label="fig:call_center"}](main-FINAL-figure1.pdf)
The dynamics expressed by –, and – belongs to the class of hybrid automata [@henzinger2000theory], which can handle piecewise linear but discontinuous dynamics like ours. We simulate our dynamics with the tool SpaceEx [@frehse2011spaceex], which is a verification platform for hybrid systems. The particularity of SpaceEx is that it computes a sound over-approximation of the trajectories.
At the scale of Figure \[fig:phasesDiagramm\], the lower and upper bounds to the values of the throughputs, computed by SpaceEx, coincide with the shape of the stationary throughputs curve. Table \[tab:1\] compares the numerical values of these lower and upper bounds to the stationary throughputs for a few values of $M_2^0/M_1^0$. We observe that the over-approximation computed by SpaceEx provides an accurate estimate of the stationary throughput computed via Equations –. This tends to show that the continuous dynamics converges towards the stationary throughputs, unlike the discrete dynamics.
Note that the experiments made with SpaceEx did not terminate for $M_2^0/M_1^0 = 0.6$: this seems to be related with the larger number of switches between the states of the automaton at this frontier between two different phases.
![Comparison of the throughputs of the discrete dynamics simulations with the theoretical throughputs (fluid model). []{data-label="fig:phasesDiagramm"}](main-FINAL-figure2.pdf)
$M_2^0/M_1^0$ 0.2 0.4 0.6 0.8 1.0 1.2
--------------------- ----------- ----------- ------- ------- ------- -------
$\rho_5$ 2.857 5.714 8.333 8.333 8.333 8.333
$f_5^{\text{up}}$ 2.865 5.716 — 8.338 8.339 8.340
$f_5^{\text{down}}$ 2.849 5.707 — 8.328 8.328 8.327
$\rho_6$ 0 0 0.238 3.095 5.952 8.333
$f_6^{\text{up}}$ $< 0.001$ $< 0.001$ — 3.107 5.968 8.340
$f_6^{\text{down}}$ 0 0 — 3.083 5.936 8.327
: Lower and upper bounds of the throughputs of the continuous dynamics computed by SpaceEx, and comparison to the stationary throughputs[]{data-label="tab:1"}
Conclusion
==========
We introduced a hybrid dynamical system model for continuous Petri nets having both free choice and priority places, and showed that there is a correspondence between the stationary solutions of the continuous dynamics and the discrete one. An advantage of the continuous setting is that some pathologies of the discrete model (failure of convergence to a stationary solution) may vanish. This is the case in particular on a case study (emergency call center). We leave it for further work to see under which generality the convergence to the stationary solution can be established.
Acknowledgments {#acknowledgments .unnumbered}
---------------
An abridged version of the present work appeared in the Proceedings of the conference Valuetools 2016. We thank the referees for their detailed comments and for pointing out relevant references.
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[^1]: The three authors were partially supported by the programme “Concepts, Syst[è]{}mes et Outils pour la Sécurité Globale” of the French National Agency of Research (ANR), project “DEMOCRITE”, number ANR-13-SECU-0007-01, and by the PGMO program of EDF and Fondation Mathématique Jacques Hadamard. The first and last authors were partially supported by the programme “Ingénierie Numérique & Sécurité” of ANR, project “MALTHY”, number ANR-13-INSE-0003.
[^2]: We recall that, in our class of Petri nets, we assume that each transition has at most one upstream place subject to priority, so that this re-ordering is valid.
|
[**[Mixture Modeling for Marked Poisson Processes]{}**]{}
[ Matthew A. Taddy]{}\
[taddy@chicagobooth.edu]{}\
[ *The University of Chicago Booth School of Business\
5807 South Woodlawn Ave, Chicago, IL 60637, USA* ]{}
.5cm [Athanasios Kottas]{}\
[thanos@ams.ucsc.edu]{}\
[ *Department of Applied Mathematics and Statistics\
University of California, Santa Cruz\
1156 High Street, Santa Cruz, CA 95064, USA*]{}
[ [Abstract:]{} We propose a general modeling framework for marked Poisson processes observed over time or space. The modeling approach exploits the connection of the nonhomogeneous Poisson process intensity with a density function. Nonparametric Dirichlet process mixtures for this density, combined with nonparametric or semiparametric modeling for the mark distribution, yield flexible prior models for the marked Poisson process. In particular, we focus on fully nonparametric model formulations that build the mark density and intensity function from a joint nonparametric mixture, and provide guidelines for straightforward application of these techniques. A key feature of such models is that they can yield flexible inference about the conditional distribution for multivariate marks without requiring specification of a complicated dependence scheme. We address issues relating to choice of the Dirichlet process mixture kernels, and develop methods for prior specification and posterior simulation for full inference about functionals of the marked Poisson process. Moreover, we discuss a method for model checking that can be used to assess and compare goodness of fit of different model specifications under the proposed framework. The methodology is illustrated with simulated and real data sets. ]{}
[ [Keywords:]{} Bayesian nonparametrics; Beta mixtures; Dirichlet process; Marked point process; Multivariate normal mixtures; Non-homogeneous Poisson process; Nonparametric regression.]{}
Introduction
============
Marked point process data, occurring on either spatial or temporal domains, is encountered in research for biology, ecology, economics, sociology, and numerous other disciplines. Whenever interest lies in the intensity of event occurrences as well as the spatial or temporal distribution of events, the data analysis problem will involve inference for a non-homogeneous point process. Moreover, many applications involve [*marks*]{} – a set of random variables associated with each point event – such that the data generating mechanism is characterized as a marked point process. In marketing, for example, interest may lie in both the location and intensity of purchasing behavior as well as consumer choices, and the data may be modeled as a spatial point process with purchase events and product choice marks. As another example, in forestry interest often lies in estimating the wood-volume characteristics of a plot of land by understanding the distribution and type of tree in a smaller subplot. Hence, the forest can be modeled as a spatial point process with tree events marked by trunk size and tree species.
Non-homogeneous Poisson processes (NHPPs) play a fundamental role in inference for data consisting of point event patterns [e.g., @Gutt1995; @MollWaag2004], and marked NHPPs provide the natural model extension when the point events are accompanied by random marks. One reason for the common usage of Poisson processes is their general tractability and the simplicity of the associated data likelihood. In particular, for a NHPP, ${\mathrm{PoP}}({\mathcal{R}},\lambda)$, defined on the observation window ${\mathcal{R}}$ with intensity $\lambda({\mathbf{x}})$ for ${\mathbf{x}} \in {\mathcal{R}}$, which is a non-negative and locally integrable function for all bounded ${\mathcal{B}} \subseteq {\mathcal{R}}$, the following hold true:
- For any such ${\mathcal{B}}$, the number of points in ${\mathcal{B}}$, $N({\mathcal{B}}) \sim$ ${\mathrm{Po}}(\Lambda({\mathcal{B}}))$, where $\Lambda({\mathcal{B}})=$ $\int_{\mathcal{B}} \lambda({\mathbf{x}}) d{\mathbf{x}}$ is the NHPP cumulative intensity function.
- Given $N({\mathcal{B}})$, the point locations within ${\mathcal{B}}$ are i.i.d. with density $\lambda({\mathbf{x}})/ \int_{\mathcal{B}}
\lambda({\mathbf{x}}) d{\mathbf{x}}$.
Here, ${\mathrm{Po}}(\mu)$ denotes the Poisson distribution with mean $\mu$. Although ${\mathcal{R}}$ can be of arbitrary dimension, we concentrate on the common settings of temporal NHPPs with ${\mathcal{R}} \subset {\mathds{R}}^+$, or spatial NHPPs where ${\mathcal{R}} \subset {\mathds{R}}^2$.
This paper develops Bayesian nonparametric mixtures to model the intensity function of NHPPs, and will provide a framework for combining this approach with flexible (nonparametric or semiparametric) modeling for the associated mark distribution. Since we propose fully nonparametric mixture modeling for the point process intensity, but within the context of Poisson distributions induced by the NHPP assumption, the nature of our modeling approach is [*semiparametric*]{}. We are able to take advantage of the above formulation of the NHPP and specify the sampling density $f({\mathbf{x}})=$ $\lambda({\mathbf{x}})/\Lambda_{{\mathcal{R}}}$ through a Dirichlet process (DP) mixture model, where $\Lambda_{{\mathcal{R}}} \equiv$ $\Lambda({\mathcal{R}})=$ $\int_{\mathcal{R}} \lambda({\mathbf{x}}) d{\mathbf{x}}$ is the total integrated intensity. Crucially, items $i$ and $ii$ above imply that the likelihood for a NHPP generated point pattern $\{{\mathbf{x}}_1,\ldots,{\mathbf{x}}_N\} \subset {\mathcal{R}}$ factorizes as $$\label{eqn:lhd}
{\mathrm{p}}\left(\{{\mathbf{x}}_i\}_{i=1}^N ; \lambda(\cdot) \right) \equiv
{\mathrm{p}}\left(\{{\mathbf{x}}_i\}_{i=1}^N ; \Lambda_{{\mathcal{R}}}, f(\cdot) \right) \propto
\Lambda_{{\mathcal{R}}}^{N} \exp(-\Lambda_{{\mathcal{R}}}) \prod_{i=1}^{N} f({\mathbf{x}}_{i}),$$ such that the NHPP density, $f(\cdot)$, and integrated intensity, $\Lambda_{{\mathcal{R}}}$, can be modeled separately. In particular, the DP mixture modeling framework for $f(\cdot)$ allows for data-driven inference about non-standard intensity shapes and quantification of the associated uncertainty.
This approach was originally developed by @KottSans2007 in the context of spatial NHPPs with emphasis on extreme value analysis problems, and has also been applied to analysis of immunological studies [@JiMerlKeplWest2009] and neuronal data analysis [@KottasBehseta2010]. Here, we generalize the mixture model to alternative kernel choices that provide for conditionally conjugate models and, in the context of temporal NHPPs, for monotonicity restrictions on the intensity function. However, in addition to providing a more general approach for intensity estimation, the main feature of this paper is an extension of the intensity mixture framework to modeling marked Poisson processes. Indeed, the advantage of a Bayesian nonparametric model-based approach will be most clear when it is combined with modeling for the conditional mark distribution, thus providing unified inference for point pattern data.
General theoretical background on Poisson processes can be found, for instance, in @Cres1993, @King1993, and @DaleVere2003. @Digg2003 reviews likelihood and classical nonparametric inference for spatial NHPPs, and @MollWaag2004 discusses work on simulation-based inference for spatial point processes.
A standard approach to (approximate) Bayesian inference for NHPPs is based upon log-Gaussian Cox process models, wherein the random intensity function is modeled on logarithmic scale as a Gaussian process . In particular, @LianCarlGelf2009 present a Bayesian hierarchical model for marked Poisson processes through an extension of the log-Gaussian Cox process to accommodate different types of covariate information. Early Bayesian nonparametric modeling focused on the cumulative intensity function, $\int_{0}^{t} \lambda(s) ds$, for temporal point processes, including models based on gamma, beta or general Lévy process priors [e.g., @Hjor1990; @Lo1992; @KuoGhos1997; @GutiNiet2003]. An alternative approach is found in @HeikArja1998 [@HeikArja1999], where piecewise constant functions, driven by Voronoi tessellations and Markov random field priors, are used to model spatial NHPP intensities.
The framework considered herein is more closely related to approaches that involve a mixture model for $\lambda(\cdot)$. In particular, @LoWeng1989 and @IshwJame2004 utilize a mixture representation for the intensity function based upon a convolution of non-negative kernels with a weighted gamma process. Moreover, @WolpIcks1998 include the gamma process as a special case of convolutions with a general Lévy random field, while @IcksWolp1999 and @BestIcksWolp2000 describe extensions of the gamma process convolution model to regression settings. @IcksWolp1999 also provide a connection to modeling for marked processes through an additive intensity formulation. Since these mixture models have the integrated intensity term linked to their nonparametric prior for $\lambda(\cdot)$, they can be cast as a generalization of our model of independent $\Lambda_{{\mathcal{R}}}$.
A distinguishing feature of the proposed approach is that it builds the modeling from the NHPP density. By casting the nonparametric modeling component as a density estimation problem, we can develop flexible classes of nonparametric mixture models that allow relatively easy prior specification and posterior simulation, and enable modeling for multivariate mark distributions comprising both categorical and continuous marks. Most importantly, in the context of marked NHPPs, the methodology proposed herein provides a unified inference framework for the joint location-mark process, the marginal point process, and the conditional mark distribution. In this way, our framework offers a nice simplification of some of the more general models discussed in the literature, providing an easily interpretable platform for applied inference about marked Poisson processes. The combination of model flexibility and relative simplicity of our approach stands in contrast to various extensions of Gaussian process frameworks: continuous marks lead to additional correlation function modeling or a separate mark distribution model; it is not trivial to incorporate categorical marks; and a spatially changing intensity surface requires complicated non-stationary spatial correlation.
The plan for the paper is as follows. Section \[NHPP\] presents our general framework of model specification for the intensity function of unmarked temporal or spatial NHPPs. Section \[NHPP-marked\] extends the modeling framework to general marked Poisson processes in both a semiparametric and fully nonparametric manner. Section \[implementation\] contains the necessary details for application of the models developed in Sections \[NHPP\] and \[NHPP-marked\], including posterior simulation and inference, prior specification, and model checking (with some of the technical details given in an Appendix). We note that Section \[inference\] discusses general methodology related to conditional inference under a DP mixture model framework, and is thus relevant beyond the application to NHPP modeling. Finally, Section \[data-examples\] illustrates the methodology through three data examples, and Section \[summary\] concludes with discussion.
Mixture specification for process intensity {#NHPP}
===========================================
This section outlines the various models for unmarked NHPPs which underlie our general framework. As described in the introduction, the ability to factor the likelihood as in (\[eqn:lhd\]) allows for modeling of $f({\mathbf{x}})=$ $\lambda({\mathbf{x}})/\Lambda_{{\mathcal{R}}}$, the process density, independent of $\Lambda_{{\mathcal{R}}}$, the integrated process intensity. The Poisson assumption implies that $N$ is sufficient for $\Lambda_{{\mathcal{R}}}$ in the posterior distribution and, in Section 4, we describe standard inference under both conjugate and reference priors for $\Lambda_{{\mathcal{R}}}$. Because the process density has domain restricted to the observation window ${\mathcal{R}}$, we seek flexible models for densities with bounded support that can provide inference for the NHPP intensity and its functionals without relying on specific parametric forms or asymptotic arguments.
We propose a general family of models for NHPP densities $f({\mathbf{x}})$ built through DP mixtures $f({\mathbf{x}};G)$ of arbitrary kernels, ${\mathrm{k}}^{{\mathbf{x}}}({\mathbf{x}}; \theta)$, with support on ${\mathcal{R}}$. Specifically, $$\label{eqn:dpm}
f({\mathbf{x}}; G) = \int {\mathrm{k}}^{{\mathbf{x}}}({\mathbf{x}}; \theta) dG(\theta),
\hspace{0.3cm} {\mathrm{with}}
\hspace{0.3cm} {\mathrm{k}}^{{\mathbf{x}}}({\mathbf{x}};\theta)=0~{\mathrm{for}}~{\mathbf{x}} \notin {\mathcal{R}},
\hspace{0.3cm} {\mathrm{and}} \hspace{0.3cm} G \sim \text{DP}(\alpha,G_{0}),$$ where $\theta$ is the (typically multi-dimensional) kernel parametrization. The kernel support restriction guarantees that $\int_{\mathcal{R}} f({\mathbf{x}};G) d{\mathbf{x}}=1$ and hence $\Lambda_{{\mathcal{R}}}=$ $\int_{\mathcal{R}} \lambda({\mathbf{x}}) d{\mathbf{x}}$. The random mixing distribution $G$ is assigned a DP prior [@Ferg1973; @Anto1974] with precision parameter $\alpha$ and base (centering) distribution $G_{0}(\cdot) \equiv$ $G_{0}(\cdot;\psi)$ which depends on hyperparameters $\psi$. For later reference, recall the DP constructive definition [@Sethuraman1994] according to which the DP generates (almost surely) discrete distributions with a countable number of atoms drawn i.i.d. from $G_0$. The corresponding weights are generated using a [*stick-breaking*]{} mechanism based on i.i.d. Beta$(1,\alpha)$ (a beta distribution with mean $(1 + \alpha)^{-1}$) draws, $\{ \zeta_{s}: s=1,2,... \}$ (drawn independently of the atoms); specifically, the first weight is equal to $\zeta_{1}$ and, for $l \geq 2$, the $l$-th weight is given by $\zeta_{l} \prod_{s=1}^{l-1} (1-\zeta_{s})$. The choice of a DP prior allows us to draw from the existing theory, and to utilize well-established techniques for simulation-based model fitting.
The remainder of this section describes options for specification of the kernel and base distribution for the model in (\[eqn:dpm\]): for temporal processes in Section 2.1 and for spatial processes in Section 2.2. In full generality, NHPPs may be defined over an unbounded space, so long as the intensity is locally integrable, but in most applications the observation window is bounded and this will be a characteristic of our modeling framework. Indeed, the specification of DP mixture models for densities with bounded support is a useful aspect of this work in its own right. Hence, temporal point processes can be rescaled to the unit interval, and we will thus assume that ${\mathcal{R}}=$ $(0,1)$. Furthermore, we assume that spatial processes are observed over rectangular support, such that the observation window can also be rescaled, in particular, ${\mathcal{R}}=(0,1)\times(0,1)$ in Section \[spatial-NHPP\] and elsewhere for spatial data.
Temporal Poisson processes {#temporal-NHPP}
--------------------------
Denote by $\{ t_{1},\ldots,t_N \}$ the temporal point pattern observed in interval ${\mathcal{R}} = (0,1)$, after the rescaling described above. Following our factorization of the intensity as $\lambda(t)=$ $\Lambda_{{\mathcal{R}}} f(t)$ and conditional on $N$, the observations are assumed to arise i.i.d. from $f(t;G)=$ $\int {\mathrm{k}}^t(t;\theta)dG(\theta)$ and $G$ is assigned a DP prior as in (\[eqn:dpm\]). We next consider specification for ${\mathrm{k}}^t(t;\theta)$.
Noting that mixtures of beta densities can approximate arbitrarily well any continuous density defined on a bounded interval [e.g., @DiacYlvi1985 Theorem 1], the beta emerges as a natural choice for the NHPP density kernel. Therefore, the DP mixture of beta densities model for the NHPP intensity is given by $$\label{beta-model}
\lambda(t;G) = \Lambda_{{\mathcal{R}}} \int b(t;\mu,\tau) dG(\mu,\tau),
\hspace{0.3cm} t \in (0,1);
\hspace{0.5cm} G \sim {\mathrm{DP}}(\alpha,G_{0}).$$ Here, $b(\cdot;\mu,\tau)$ denotes the density of the beta distribution parametrized in terms of its mean $\mu \in (0,1)$ and a scale parameter $\tau > 0$, i.e., $b(t;\mu,\tau) \propto$ $t^{\mu \tau - 1} (1-t)^{\tau (1 - \mu) - 1}$, $t \in (0,1)$. Regarding the DP centering distribution $G_{0} \equiv$ $G_{0}(\mu,\tau)$, we work with independent components, specifically, a uniform distribution on $(0,1)$ for $\mu$, and an inverse gamma distribution for $\tau$ with fixed shape parameter $c$ and mean $\beta/(c-1)$ (provided $c>1$). Hence, the density of $G_{0}$ is $g_{0}(\mu,\tau) \propto$ $\tau^{-(c+1)}\exp(-\beta \tau^{-1}){\mathds{1}}_{\mu \in (0,1)}$, where $\beta$ can be assigned an exponential hyperprior.
The beta kernel is appealing due to its flexibility and the fact that it is directly bounded to the unit interval. However, there are no commonly used conjugate priors for its parameters; there are conjugate priors for parameters of the exponential family representation of the beta density, such as the beta-conjugate distribution in @GrunRaftGutt1993a, but none of these are easy to work with or intuitive to specify. There are substantial benefits (refer to Section 4) to be gained from the Rao-Blackwellization of posterior inference for mixture models [see, e.g., @MacEClydLiu1999 for empirical demonstration of the improvement in estimators] that is only possible with conditional conjugacy – that is, in this context, when the base distribution is conjugate for the kernel parametrization. Moreover, the nonparametric mixture allows inference to be robust to a variety of reasonable kernels, such that the convenience of conjugacy will not usually detract from the quality of analysis.
We are thus motivated to provide a conditionally conjugate alternative to the beta model, and do so by first applying a logit transformation, ${\mathrm{logit}}(t)=$ $\log\left(t/(1-t)\right)$, $t \in (0,1)$, and then using a Gaussian density kernel. In detail, the logit-normal DP mixture model is then, $$\label{normal-model}
\lambda(t;G) = \Lambda_{{\mathcal{R}}}
\int {\mathrm{N}}\left({\mathrm{logit}}(t);\mu,\sigma^2\right) \frac{1}{t(1-t)}
dG(\mu,\sigma^{2}), \hspace{0.3cm} t \in (0,1);
\hspace{0.5cm} G \sim {\mathrm{DP}}(\alpha,G_{0}).$$ The base distribution is taken to be of the standard conjugate form [as in, e.g., @EscoWest1995], such that $g_{0}(\mu, \sigma^{2})=$ ${\mathrm{N}}(\mu; \delta, \sigma^2/\kappa){\mathrm{ga}}(\sigma^{-2} ;\nu,\omega)$, where ${\mathrm{ga}}(\cdot;\nu,\omega)$ denotes the gamma density with ${\mathds{E}}[\sigma^{-2}]=$ $\nu/\omega$. A gamma prior is placed on $\omega$ whereas $\kappa$, $\nu$ and $\delta$ are fixed (however, a normal prior for $\delta$ can be readily added).
The price paid for conditional conjugacy is that the logit-normal model is susceptible to boundary effects: the density specification in (\[normal-model\]) [*must*]{} be zero in the limit as $t$ approaches the boundaries of the observation window (such that ${\mathrm{logit}}(t)
\rightarrow \pm \infty$). In contrast, the beta model is not restricted to any single type of boundary behavior, and will thus be more appropriate whenever there is a need to model processes which maintain high intensity at the edge of the observation window. Section 5 offers empirical comparison of the two models.
The beta and logit-normal mixtures form the basis for our approach to modeling marked Poisson processes, and Section 2.2 will extend these models to spatial NHPPs. Both schemes are developed to be as flexible as possible, in accordance with our semiparametric strategy of having point event data restricted by the Poisson assumption but modeled with an unrestricted NHPP density. However, in some situations it may be of interest to constrain the model further by making structural assumptions about the NHPP density, including monotonicity assumptions for the intensity function as in, for example, software reliability applications [e.g., @KuoYang1996]. To model monotonic intensities for temporal NHPPs, we can employ the representation of non-increasing densities on ${\mathds{R}}^{+}$ as scale mixtures of uniform densities. In particular, for any non-increasing density $h(\cdot)$ on ${\mathds{R}}^{+}$ there exists a distribution function $G$, with support on ${\mathds{R}}^{+}$, such that $h(t) \equiv$ $h(t;G)=$ $\int \theta^{-1} {\mathds{1}}_{t \in (0,\theta)}
\text{d}G(\theta)$ [see, e.g., @BrunLo1989; @KottGelf2001]. In the context of NHPPs, a DP mixture formulation could be written $\lambda(t;G)=$ $\Lambda_{{\mathcal{R}}} \int \theta^{-1} {\mathds{1}}_{t\in(0,\theta)}dG(\theta)$, $t \in (0,1)$, with $G \sim {\mathrm{DP}}(\alpha,G_{0})$, where $G_{0}$ has support on $(0,1)$, e.g., it can be defined by a beta distribution. Then, $\lambda(t;G)$ defines a prior model for non-increasing intensities. Similarly, a prior model for non-decreasing NHPP intensities can be built from $f(t;G)=$ $\int \theta^{-1} {\mathds{1}}_{(t-1) \in (-\theta,0)} dG(\theta)$, $t \in (0,1)$, with $G \sim {\mathrm{DP}}(\alpha,G_{0})$, where again $G_0$ has support on $(0,1)$.
Spatial Poisson processes {#spatial-NHPP}
-------------------------
We now present modeling for spatial NHPPs as an extension of the framework in Section \[temporal-NHPP\]. As mentioned previously, we assume that the bounded event data has been rescaled such that point locations $\{{\mathbf{x}}_1,\ldots,{\mathbf{x}}_N\}$ all lie within the unit square, ${\mathcal{R}}=$ $(0,1) \times (0,1)$. The extra implicit assumption of a rectangular observation window is standard in the literature on spatial Poisson process modeling [see, e.g., @Digg2003].
The most simple extension of our models for temporal NHPPs is to build a bivariate kernel out of two independent densities. For example, a two-dimensional version of the beta mixture density in (\[beta-model\]) could be written $f({\mathbf{x}};G)=$ $\int b(x_1;\mu_1,\tau_1) b(x_2;\mu_2,\tau_2)dG({\boldsymbol{\mu}},{\boldsymbol{\tau}})$, where ${\boldsymbol{\mu}}=$ $(\mu_{1},\mu_{2})$ and ${\boldsymbol{\tau}}=$ $(\tau_{1},\tau_{2})$. However, although dependence between $x_1$ and $x_2$ will be induced by mixing, it will typically be more efficient to allow for explicit dependence in the kernel. A possible two-dimensional extension of (\[beta-model\]) is that of @KottSans2007, which employs a Sarmanov dependence factor to induce a bounded bivariate density with beta marginals. The corresponding model for the spatial NHPP intensity is given by $$\begin{aligned}
\label{bivarbeta}
\lambda({\mathbf{x}};G) = \Lambda_{{\mathcal{R}}} \int b(x_1; \mu_1, \tau_1) b(x_2; \mu_2,
\tau_2)\left(1+\rho(x_1-\mu_1)(x_2-\mu_2)\right)dG({\boldsymbol{\mu}},{\boldsymbol{\tau}}, \rho),\end{aligned}$$ where $G \sim {\mathrm{DP}}(\alpha,G_{0})$ and $G_0$ is built from independent centering distributions as in (\[beta-model\]) for each dimension, multiplied by a conditional uniform distribution for $\rho$ over the region such that $1 + \rho(x_1 - \mu_1) (x_2 - \mu_2) > 0$, for all ${\mathbf{x}} \in {\mathcal{R}}$. Thus, $g_{0}({\boldsymbol{\mu}}, {\boldsymbol{\tau}}, \rho)=$ $ {\mathds{1}}_{\rho \in ( C_{{\boldsymbol{\mu}}}, C^{{\boldsymbol{\mu}}})}
(C^{{\boldsymbol{\mu}}}-C_{{\boldsymbol{\mu}}})^{-1} \prod_{i=1}^{2}
{\mathrm{ga}}(\tau^{-1}_i ; \nu_i, \beta_i ) {\mathds{1}}_{\mu_i\in (0,1)}$, where $C_{{\boldsymbol{\mu}}}=$ $-\left({\mathrm{max}}\{\mu_1\mu_2,(1-\mu_1)(1-\mu_2)\}\right)^{-1}$ and $C^{{\boldsymbol{\mu}}} $ $=
-\left({\mathrm{min}}\{\mu_1(\mu_2-1),\mu_2(\mu_1-1)\}\right)^{-1}$. Gamma hyperpriors can be placed on $\beta_{1}$ and $\beta_{2}$.
Model (\[bivarbeta\]) has appealing flexibility, including resistance to edge effects, but a lack of conditional conjugacy requires the use of an augmented Metropolis-Hastings algorithm for posterior simulation (discussed in Appendix A.2). The inefficiency of this approach is only confounded in higher dimensions, and becomes especially problematic when we extend the models to incorporate process marks. Hence, we are again motivated to seek a conditionally conjugate alternative for spatial NHPPs, and this is achieved in a straightforward manner by applying individual logit transformations to each coordinate dimension and mixing over bivariate Gaussian density kernels. Specifically, the spatial NHPP logit-normal model is $$\label{bivarnormal}
\lambda({\mathbf{x}};G) = \Lambda_{{\mathcal{R}}} \int {\mathrm{N}}\left({\mathrm{logit}}({\mathbf{x}});
{\boldsymbol{\mu}},{\boldsymbol{\Sigma}} \right)\frac{1}{\prod_{i=1}^2 x_i (1-x_i)}
dG({\boldsymbol{\mu}},{\boldsymbol{\Sigma}}),
\hspace{0.5cm} G \sim {\mathrm{DP}}(\alpha,G_{0}),$$ where ${\mathrm{logit}}({\mathbf{x}})$ is shorthand for $\left[
{\mathrm{logit}}(x_1), {\mathrm{logit}}(x_2) \right]'$. The base distribution is again of the standard conjugate form, such that $g_{0}({\boldsymbol{\mu}},{\boldsymbol{\Sigma}})=$ ${\mathrm{N}}({\boldsymbol{\mu}}; {\boldsymbol{\delta}},
{\boldsymbol{\Sigma}}/\kappa){\mathrm{W}}({\boldsymbol{\Sigma}}^{-1} ; \nu, {\boldsymbol{\Omega}})$, with fixed $\kappa$, $\nu$, ${\boldsymbol{\delta}}$ and a Wishart hyperprior for ${\boldsymbol{\Omega}}$. Here, ${\mathrm{W}}(\cdot;\nu,{\boldsymbol{\Omega}})$ denotes a Wishart density such that ${\mathds{E}}[{\boldsymbol{\Sigma}}^{-1}]=$ $\nu {\boldsymbol{\Omega}}^{-1}$ and ${\mathds{E}}[{\boldsymbol{\Sigma}}]=$ $(\nu - \frac{3}{2})^{-1}{\boldsymbol{\Omega}}$.
Frameworks for modeling marked Poisson processes {#NHPP-marked}
================================================
The models for unmarked NHPPs, as introduced in Section 2, are essentially density estimators for distributions with bounded support. As mentioned in the Introduction, the Bayesian nonparametric approach is most powerful when embedded in a more complex model for marked point processes. Section 3.1 describes how the methodology of Section 2 can be coupled with general regression modeling for marks, whereas in Section 3.2, we develop a fully nonparametric Bayesian modeling framework for marked Poisson processes.
Semiparametric modeling for the mark distribution {#marks-SP}
-------------------------------------------------
In the standard marked point process setting, one is interested in inference for the process intensity over time or space and the associated conditional distribution for the marks.
Regarding the data structure, for each temporal or spatial point ${\mathbf{x}}_{i}$, $i=1,...,N$, in the observation window ${\mathcal{R}}$ there is an associated mark ${\mathbf{y}}_{i}$ taking values in the mark space ${\mathcal{M}}$, which may be multivariate and may comprise both categorical and continuous variables. Let $h({\mathbf{y}} \mid {\mathbf{x}})$ denote the conditional mark density at point ${\mathbf{x}}$. (Note that we use ${\mathbf{y}}$ and ${\mathbf{y}}_{i}$ as simplified notation for ${\mathbf{y}}({\mathbf{x}})$ and ${\mathbf{y}}({\mathbf{x}}_{i})$.) Under the semiparametric approach, we build the joint model for the marks and the point process intensity through $$\label{eqn:joint}
\phi({\mathbf{x}}, {\mathbf{y}}) = \lambda({\mathbf{x}}) h({\mathbf{y}} \mid {\mathbf{x}}) =
\Lambda_{{\mathcal{R}}} f({\mathbf{x}}) h({\mathbf{y}} \mid {\mathbf{x}}), \,\,\,\,\,
{\mathbf{x}} \in {\mathcal{R}}, \, {\mathbf{y}} \in {\mathcal{M}}.$$ Note that the conditioning in $h({\mathbf{y}} \mid {\mathbf{x}})$ does not involve any portion of the point process other than point ${\mathbf{x}}$; for instance, for temporal processes, the conditional mark density at time $t$ does not depend on earlier times $t^{\prime} < t$. Under this setting, the [*Marking theorem*]{} [e.g., proposition 3.9 in @MollWaag2004; @King1993 p. 55] yields that the marked point process $\{ ({\mathbf{x}},{\mathbf{y}}): {\mathbf{x}} \in {\mathcal{R}}, {\mathbf{y}} \in {\mathcal{M}} \}$ is a NHPP with intensity function given by (\[eqn:joint\]) for $({\mathbf{x}},{\mathbf{y}}) \in$ ${\mathcal{R}} \times {\mathcal{M}}$, and by its extension to ${\mathcal{B}} \times {\mathcal{M}}$ for any bounded ${\mathcal{B}} \supset {\mathcal{R}}$.
This intensity factorization, combined with the general NHPP likelihood factorization in (\[eqn:lhd\]), results in convenient semiparametric modeling formulations for the marked process through a DP mixture model for $f(\cdot)$ (as in Section \[NHPP\]) and a separate parametric or semiparametric regression specification for the conditional mark distribution. In particular, assuming that the marks $\{{\mathbf{y}}_i\}_{i=1}^N$ are mutually independent given $\{{\mathbf{x}}_i\}_{i=1}^N$, and combining (\[eqn:lhd\]) and (\[eqn:joint\]), we obtain $$\label{markslhd}
{\mathrm{p}}\left(\{{\mathbf{x}}_i,{\mathbf{y}}_i\}_{i=1}^{N} ;\Lambda_{{\mathcal{R}}},f(\cdot),h(\cdot) \right)
\propto \Lambda_{{\mathcal{R}}}^{N} \exp(-\Lambda_{{\mathcal{R}}})\prod_{i=1}^{N} f({\mathbf{x}}_i)
\prod_{i=1}^{N} h({\mathbf{y}}_i \mid {\mathbf{x}}_i),$$ such that the conditional mark density can be modeled independent of the process intensity.
The consequence of this factorization of integrated intensity, process density, and the conditional mark density, is that any regression model for $h$ can be added onto the modeling schemes of Section \[NHPP\] and provide an extension to marked processes. In some applications, it will be desirable to use flexible semiparametric specifications for $h$, such as a Gaussian process regression model, while in other settings it will be useful to fit $h$ parametrically, such as through the use of a generalized linear model. As an illustration, Section \[simulation\] explores a Gaussian process-based specification, however, the important point is that this aspect of the modeling does not require any further development of the underlying nonparametric model for the NHPP intensity. Moreover, despite the posterior independence of $f$ and $h$, combining them as in (\[eqn:joint\]) leads to a practical semiparametric inference framework for the joint mark-location Poisson process. The fully nonparametric approach developed in the following section provides an alternative for settings where further modeling flexibility is needed.
Fully nonparametric joint and implied conditional mark modeling {#marks-NP}
---------------------------------------------------------------
While the semiparametric approach of Section 3.1 provides a convenient extension of the NHPP models in Section 2, the connection between joint and marked processes provides the opportunity to build fully nonparametric models for marked point event data. Here, we introduce a general modeling approach, built through fully nonparametric models for joint mark-location Poisson processes, and describe how this provides a unified inference framework for the joint process, the conditional mark distribution, and the marginal point process.
Instead of specifying directly a model for the marked process, we begin by writing the joint Poisson process, PoP$({\mathcal{R}} \times {\mathcal{M}},\phi)$, defined over the joint location-mark observation window with intensity $\phi({\mathbf{x}},{\mathbf{y}})$. The inverse of the marking theorem used to obtain equation (\[eqn:joint\]) holds that, if the marginal intensity $\int_{\mathcal{M}} \phi({\mathbf{x}},{\mathbf{y}})d{\mathbf{y}}=$ $\lambda({\mathbf{x}})$ is locally integrable, then the joint process just defined is also the marked Poisson process of interest.
Analogously to the model development in Section 2, we define a process over the joint location-mark space with intensity function $$\begin{aligned}
\phi({\mathbf{x}},{\mathbf{y}};G) =
\Lambda_{{\mathcal{R}}} \int {\mathrm{k}}^{{\mathbf{x}}}({\mathbf{x}} ; \theta^{{\mathbf{x}}})
{\mathrm{k}}^{{\mathbf{y}}}({\mathbf{y}} ; \theta^{{\mathbf{y}}}) dG(\theta^{{\mathbf{x}}}, \theta^{{\mathbf{y}}}) =
\Lambda_{{\mathcal{R}}} f({\mathbf{x}},{\mathbf{y}};G),
\hspace{0.5cm} G \sim {\mathrm{DP}}(\alpha,G_{0}),
\label{lmproc}\end{aligned}$$ where the mark kernel ${\mathrm{k}}^{{\mathbf{y}}}({\mathbf{y}}; \theta^{{\mathbf{y}}})$ has support on ${\mathcal{M}}$ and the integrated intensity can be defined in terms of either the joint or marginal process, such that $\Lambda_{{\mathcal{R}}}=$ $\int_{\mathcal{R}} \lambda({\mathbf{x}}) d{\mathbf{x}}=$ $\int_{\mathcal{R}} \left[\int_{\mathcal{M}} \phi({\mathbf{x}},{\mathbf{y}}) d{\mathbf{y}} \right]d{\mathbf{x}}$. Note that the marginal intensity, and hence the marked point process, are properly defined with locally integrable intensity functions. Specifically, we can move integration over ${\mathcal{M}}$ inside the infinite sum and $$\begin{aligned}
\int_{\mathcal{M}} \phi({\mathbf{x}}, {\mathbf{y}}) d{\mathbf{y}} & = & \Lambda_{{\mathcal{R}}}
\int_{\theta_{\mathbf{x}}} {\mathrm{k}}^{{\mathbf{x}}}({\mathbf{x}} ; \theta^{{\mathbf{x}}}) \int_{\theta_{\mathbf{y}}}
\left[ \int_{\mathcal{M}} {\mathrm{k}}^{{\mathbf{y}}}({\mathbf{y}} ; \theta^{{\mathbf{y}}}) d{\mathbf{y}} \right]
dG(\theta^{{\mathbf{x}}}, \theta^{{\mathbf{y}}}) \label{eqn:locint} \\
& = & \Lambda_{{\mathcal{R}}}
\int {\mathrm{k}}^{{\mathbf{x}}}({\mathbf{x}};\theta^{{\mathbf{x}}})dG^{{\mathbf{x}}}(\theta^{{\mathbf{x}}}) =
\Lambda_{{\mathcal{R}}} f({{\mathbf{x}}}; G) = \lambda({\mathbf{x}}). \notag\end{aligned}$$ Here, $G^{{\mathbf{x}}}(\theta^{{\mathbf{x}}})$ is the marginal mixing distribution, which has an implied DP prior with base density $g_{0}^{{\mathbf{x}}}(\theta^{{\mathbf{x}}})=$ $\int g_{0}(\theta^{{\mathbf{x}}},\theta^{{\mathbf{y}}}) d\theta^{{\mathbf{y}}}$, and we have thus recovered the original DP mixture model of Section 2 for the marginal location NHPP $\mathrm{PoP}({\mathcal{R}},\lambda)$. As an aside we note that, through a similar argument and since $\phi({\mathbf{x}},{\mathbf{y}})=$ $\lambda({\mathbf{x}}) h({\mathbf{y}} \mid {\mathbf{x}})$, the joint location-mark process of (\[lmproc\]) satisfies the requirements of proposition 3.9 in [@MollWaag2004], and hence the marks alone are marginally distributed as a Poisson process defined on ${\mathcal{M}}$ with intensity $\int_{{\mathcal{R}}} \phi({\mathbf{x}},{\mathbf{y}}) d{\mathbf{x}}=$ $\Lambda_{{\mathcal{R}}}
\int {\mathrm{k}}^{{\mathbf{y}}}({\mathbf{y}};\theta^{{\mathbf{y}}})dG^{{\mathbf{y}}}(\theta^{{\mathbf{y}}})$.
In general, both the mixture kernel and base distributions will be built from independent components corresponding to marks and to locations, and the random mixing measure is relied upon to induce dependence between these random variables. This technique has been employed in regression settings by [@TaddKott2010], and provides a fairly automatic procedure for nonparametric model building in mixed data-type settings. For example, suppose that a spatial point process is accompanied by categorical marks, such that marks $\{y_1,\ldots,y_N\}$ are each a member of the set ${\mathcal{M}}=$ $\{1,2,\ldots,{\mathrm{M}}\}$. The joint intensity model can be specified as $$\label{catmodel}
\phi({\mathbf{x}},y;G) = \Lambda_{{\mathcal{R}}} \int {\mathrm{k}}^{\mathbf{x}}({\mathbf{x}};\theta^{{\mathbf{x}}})
q_{y} dG(\theta^{{\mathbf{x}}},{\mathbf{q}}),
\hspace{0.5cm}
G \sim {\mathrm{DP}}(\alpha,G_{0}^{{\mathbf{x}}}(\theta^{{\mathbf{x}}}){\mathrm{Dir}}({\mathbf{q}};{\mathbf{a}})),$$ where ${\mathbf{q}} = [q_1,\ldots,q_{{\mathrm{M}}}]$ is a probability vector with $q_y=$ ${\mathrm{Pr}}(Y=y \mid {\mathbf{q}})$, ${\mathrm{Dir}}({\mathbf{q}};{\mathbf{a}})$ is the Dirichlet distribution, with ${\mathbf{a}}=$ $(a_{1},...,a_{M})$, such that $\mathbb{E}(q_y \mid {\mathbf{a}})=$ $a_y/\sum_{s=1}^{{\mathrm{M}}} a_s$, and the location-specific kernel, ${\mathrm{k}}^{{\mathbf{x}}}$, and centering distribution, $G_{0}^{{\mathbf{x}}}$, are specified as in either (\[bivarbeta\]) or (\[bivarnormal\]) and thereafter. Additional marks can be incorporated in the same manner by including additional independent kernel and base distribution components.
Similarly, continuous marks can be modeled through an appropriate choice for the independent mark kernel. For example, in the case of real-valued continuous marks (i.e., ${\mathcal{M}}=\mathbb{R}$) for a temporal point process, the choice of a normal density kernel leads to the intensity model $$\begin{aligned}
\label{ctsmodel}
\phi(t,y;G) = \Lambda_{{\mathcal{R}}} \int {\mathrm{k}}^t(t;\theta^t) {\mathrm{N}}(y;\eta,\sigma^2)
dG(\theta^t,\eta,\sigma^{2}), \,\,\,
G \sim {\mathrm{DP}}\left(\alpha,G_{0}^{t}(\theta^{t})G_{0}^{y}(\eta,\sigma^{2}) \right).\end{aligned}$$ The location specific kernel, ${\mathrm{k}}^t$, and base measure, $G_{0}^{t}$, can be taken from Section \[temporal-NHPP\]; $G_{0}^{y}$ can be specified through the conjugate normal inverse-gamma form as in (\[normal-model\]). Other possible mark kernels are negative-binomial or Poisson for count data (as in Section \[coal-mining\]), a Weibull for failure time data, or a log-normal for positive continuous marks (as in Section \[pine-trees\]).
As an alternative to this generic independent kernel approach, the special case of a combination of real-valued continuous marks with the logit-normal kernel models in either (\[normal-model\]) or (\[bivarnormal\]) allows for joint multivariate-normal kernels. Thus, instead of the model in (\[ctsmodel\]), a temporal point process with continuous marks is specified via bivariate normal kernels as $$\label{mvnmarks}
\phi(t,y;G) = \Lambda_{{\mathcal{R}}} \int {\mathrm{N}}\left([{\mathrm{logit}}(t),y]';
{\boldsymbol{\mu}},{\boldsymbol{\Sigma}} \right)\frac{1}{t(1-t)} d
G({\boldsymbol{\mu}},{\boldsymbol{\Sigma}}),
\hspace{0.5cm} G \sim {\mathrm{DP}}(\alpha,G_{0}),$$ with base distribution of the standard conjugate form, exactly as described following (\[bivarnormal\]). Specification is easily adapted to spatial processes or multivariate continuous marks through the use of higher dimensional normal kernels (see Section \[pine-trees\] for an illustration).
A key feature of the joint mixture modeling framework for the location-mark process is that it can provide flexible specifications for multivariate mark distributions comprising both categorical and continuous marks. For any of the joint intensity models specified in this section, inference for the conditional mark density is available through $$\label{markcnd}
h({\mathbf{y}} \mid {\mathbf{x}} ; G) =
\frac{f({\mathbf{x}},{\mathbf{y}} ; G)}{f({\mathbf{x}} ; G)} =
\frac{\int {\mathrm{k}}^{{\mathbf{x}}}({\mathbf{x}} ; \theta^{{\mathbf{x}}})
{\mathrm{k}}^{{\mathbf{y}}}({\mathbf{y}} ; \theta^{{\mathbf{y}}}) dG(\theta^{{\mathbf{x}}}, \theta^{{\mathbf{y}}}) }
{\int {\mathrm{k}}^{{\mathbf{x}}}({\mathbf{x}} ; \theta^{{\mathbf{x}}}) dG^{{\mathbf{x}}}(\theta^{{\mathbf{x}}}) }.$$ Of course, other conditioning arguments are also possible if, for example, some subset of the marks is viewed as covariates for a specific mark of interest. In any case, the integrals in (\[markcnd\]) are actually infinite sums induced by discrete realizations from the posterior distribution for $G$. In Section \[inference\], we show that truncation approximations to the infinite sums allow for proper conditional inference and, hence, for fully nonparametric inference about any functional of the conditional mark distribution.
Implementation
==============
This section provides guidelines for application of the models proposed in Sections \[NHPP\] and \[NHPP-marked\], with prior specification and posterior simulation briefly discussed in Section \[MCMC\] (further details can be found in the Appendix), inference for marked NHPP functionals in Section \[inference\], and model checking in Section \[model-checking\].
Prior specification and posterior simulation {#MCMC}
--------------------------------------------
As with our approach to model building, we can specify the prior for integrated intensity independent of the prior for parameters of the DP mixture density model. The marginal likelihood for $\Lambda_{{\mathcal{R}}}$ corresponds to a Poisson density for $N$, such that the conjugate prior for $\Lambda_{{\mathcal{R}}}$ is a gamma distribution. As a default alternative, we make use of the (improper) reference prior for $\Lambda_{{\mathcal{R}}}$, which can be derived as $\pi(\Lambda_{{\mathcal{R}}})
\propto \Lambda_{{\mathcal{R}}}^{-1}$ for $\Lambda_{{\mathcal{R}}} > 0$. The posterior distribution for the integrated intensity is then available analytically as a gamma distribution, since the posterior distribution for the NHPP intensity factorizes as ${\mathrm{p}}(f(\cdot),\Lambda_{{\mathcal{R}}} \mid {\mathrm{data}})=$ ${\mathrm{p}}(f(\cdot)
\mid {\mathrm{data}}) {\mathrm{p}}(\Lambda_{{\mathcal{R}}} \mid N)$. In particular, ${\mathrm{p}}(\Lambda_{{\mathcal{R}}} \mid N)=$ ${\mathrm{ga}}(N,1)$ under our default reference prior. Similarly, under the semiparametric approach of Section \[marks-SP\], prior specification and posterior inference for any model applied to the conditional mark distribution can be dealt with separately from the intensity function model, and will generally draw on existing techniques for the regression model of interest.
What remains is to establish general prior specification and MCMC simulation algorithms for the DP mixture process density models of Sections \[NHPP\] and \[marks-NP\]. In a major benefit of our approach – one which should facilitate application of these models – we are able here to make use of standard results and methodology from the large literature on DP mixture models. Our practical implementation guidelines are detailed in the Appendix, with prior specification in A.1 and a posterior simulation framework in A.2.
Inference about NHPP functionals {#inference}
--------------------------------
Here, we describe the methods for posterior inference about joint or marginal intensity functions and for conditional density functions. We outline inference for a general NHPP with events $\{{\mathbf{z}}_i\}_{i=1}^N$, possibly consisting of both point location and marks, and leave specifics to the examples of Section \[data-examples\].
Due to the almost sure discreteness of the DP, a generic representation for the various mixture models for NHPP densities is given by $f({\mathbf{z}};G)=$ $\sum_{l=1}^{\infty} p_{l} {\mathrm{k}}({\mathbf{z}};
\vartheta_{l})$, where the $\vartheta_{l}$, given the base distribution hyperparameters $\psi$, are i.i.d. from $G_{0}$, and the weights $p_{l}$ are generated according to the stick-breaking process discussed in Section \[NHPP\]. Here, ${\mathbf{z}}$ may include only point locations (as in the models of Section \[NHPP\]) or both point locations and marks whence ${\mathrm{k}}({\mathbf{z}}; \vartheta)=$ ${\mathrm{k}}^{{\mathbf{x}}}({\mathbf{x}};\vartheta^{{\mathbf{x}}}) {\mathrm{k}}^{{\mathbf{y}}}({\mathbf{y}} ;
\vartheta^{{\mathbf{y}}})$ (as in Section \[marks-NP\]). Hence, the DP induces a clustering of observations: for $\text{data}=$ $\{{\mathbf{z}}_1,\ldots,{\mathbf{z}}_N\}$, if we introduce latent mixing parameters ${\boldsymbol{\theta}}=$ $\{\theta_1,\ldots,\theta_N\}$ such that ${\mathbf{z}}_i \mid \theta_{i} \stackrel{ind}{\sim}$ ${\mathrm{k}}({\mathbf{z}}_i;\theta_i)$, with $\theta_i \mid G
\stackrel{iid}{\sim} G$, for $i=1,\ldots,N$, and $G \mid \alpha,\psi
\sim$ DP$(\alpha,G_{0}(\cdot;\psi))$, then observations can be grouped according to the number, $m \le N$, of distinct mixing parameters in ${\boldsymbol{\theta}}$. This group of distinct parameter sets, ${\boldsymbol{\theta}}^{\star} =$ $\{\theta^\star_1,\ldots,\theta_m^\star\}$, maps back to data through the latent allocation vector, ${\mathbf{s}}=$ $[s_1,\ldots,s_N]$, such that $\theta_i=$ $\theta^{\star}_{s_i}$. The expanded parametrization is completed by the number of observations allocated to each unique component, ${\mathbf{n}}=$ $[n_1,\ldots,n_m]$, where $n_j=$ $\sum_{i=1}^{N} {\mathds{1}}_{s_i=j}$, and the associated groups of observations $\{{\mathbf{z}}_i: s_i=j\}$. If $G$ is marginalized over its DP prior, we obtain the Pólya urn expression for the DP prior predictive distribution, $$\label{polya}
{\mathrm{p}}(\theta_{0} \mid {\boldsymbol{\theta}}^\star,\alpha,\psi) =
d{\mathds{E}}\left[G(\theta_{0}) \mid {\boldsymbol{\theta}}^{\star},\alpha,\psi \right] \propto
\alpha g_{0}(\theta_{0};\psi) + \sum_{j=1}^{m} n_{j} \delta_{\theta^{\star}_j}(\theta_{0})$$ where $\delta_{a}$ denotes a point mass at $a$. Moreover, based on the DP Pólya urn structure, the prior for ${\boldsymbol{\theta}}^\star$, given $m$ and $\psi$, is such that $\theta^{\star}_{j} \mid \psi \stackrel{iid}{\sim} G_{0}(\cdot;\psi)$, for $j=1,\ldots,m$.
Within the DP mixture framework, estimation of linear functionals of the mixture is possible via posterior expectations conditional on only this finite dimensional representation (i.e., it is not necessary to draw $G$). In particular, with the NHPP density modeled as our generic DP mixture, the posterior expectation for the intensity function can be written as ${\mathds{E}}\left[\lambda({\mathbf{z}};G) \mid \text{data} \right]=$ ${\mathds{E}}(\Lambda_{{\mathcal{R}}} \mid N) {\mathrm{p}}({\mathbf{z}} \mid \text{data})$, where ${\mathrm{p}}({\mathbf{z}} \mid \text{data})=$ ${\mathds{E}}\left[ f({\mathbf{z}};G) \mid \text{data} \right]$ is the posterior predictive density given by $$\label{expect}
\int \frac{1}{\alpha+N}
\left(
\alpha \int {\mathrm{k}}({\mathbf{z}};\theta) dG_0(\theta;\psi)
+ \sum_{j=1}^{m} n_{j} {\mathrm{k}}({\mathbf{z}};\theta^{\star}_{j})
\right) {\mathrm{p}}({\boldsymbol{\theta}}^{\star},{\mathbf{s}},\alpha,\psi \mid \text{data})
d{\boldsymbol{\theta}}^{\star} d{\mathbf{s}} d\alpha d\psi.$$ Hence, a point estimate for the intensity function is readily available through ${\mathds{E}}\left[ f({\mathbf{z}};G) \mid \text{data} \right]$ estimated as the average, for each point in a grid in ${\mathbf{z}}$, over realizations of (\[expect\]) calculated for each MCMC posterior sample for ${\boldsymbol{\theta}}^{\star}$, ${\mathbf{s}}$, $\alpha$ and $\psi$.
However, care must be taken when moving to posterior inference about the conditional mark distribution in (\[markcnd\]). As a general point on conditioning in DP mixture models for joint distributions, Pólya urn-based posterior expectation calculations, such as (\[expect\]), are invalid for the estimation of non-linear functionals of $\lambda$ or $f$. For example, @MullErkaWest1996 develop a DP mixture curve fitting approach that, in the context of our model, would estimate the conditional mark density by $$\label{muller}
\widehat{h({\mathbf{y}} | {\mathbf{x}} )} =
\int \frac{ \int
{\mathrm{k}}^{{\mathbf{x}}}({\mathbf{x}} ; \theta^{{\mathbf{x}}})
{\mathrm{k}}^{{\mathbf{y}}}({\mathbf{y}} ; \theta^{{\mathbf{y}}})
d{\mathds{E}}\left[G(\theta ) \mid {\boldsymbol{\theta}}, \alpha, \psi\right]}{\int
{\mathrm{k}}^{{\mathbf{x}}}({\mathbf{x}};\theta^{{\mathbf{x}}})
d{\mathds{E}}\left[G(\theta ) \mid {\boldsymbol{\theta}}, \alpha, \psi\right]}
{\mathrm{p}}({\boldsymbol{\theta}}, \alpha,\psi \mid {\mathrm{data}})
d{\boldsymbol{\theta}} d\alpha d\psi,$$ which is the ratio of Pólya urn joint and marginal density point estimates given ${\boldsymbol{\theta}}$ and DP prior parameters $\alpha$, $\psi$, averaged over MCMC draws for these parameters. Unfortunately, (\[muller\]) is [*not*]{} $\mathbb{E}\left[ h({\mathbf{y}} \mid {\mathbf{x}};G) \mid \text{data} \right]$, the posterior expectation for random conditional density $h({\mathbf{y}} \mid {\mathbf{x}};G)=$ $f({\mathbf{x}},{\mathbf{y}};G)/f({\mathbf{x}};G)$, which would be the natural estimate for the conditional mark density at any specified combination of values $({\mathbf{x}},{\mathbf{y}})$. Hence, the regression estimate in @MullErkaWest1996 as well as that proposed in the more recent work of @RodrDunsGelf2009, based on ${\mathrm{p}}({\mathbf{x}},{\mathbf{y}} \mid \text{data})/{\mathrm{p}}({\mathbf{x}} \mid \text{data})$, provide only approximations to $\mathbb{E}\left[ h({\mathbf{y}} \mid {\mathbf{x}};G) \mid \text{data} \right]$; in particular, the latter estimate is approximating the expectation of a ratio with the ratio of expectations. Such approximations become particularly difficult to justify if one seeks inference for non-linear functionals of $h({\mathbf{y}} \mid {\mathbf{x}};G)$.
Hence, to obtain the exact point estimate $\mathbb{E}\left[ h({\mathbf{y}} \mid {\mathbf{x}};G) \mid \text{data} \right]$, and, most importantly, to quantify full posterior uncertainty about general functionals of the NHPP intensity, it is necessary to obtain posterior samples for the mixing distribution, $G$. Note that ${\mathrm{p}}(G \mid \text{data})=$ $\int {\mathrm{p}}(G \mid {\boldsymbol{\theta}}^{\star},{\mathbf{s}},\alpha,\psi)
{\mathrm{p}}({\boldsymbol{\theta}}^{\star},{\mathbf{s}},\alpha,\psi \mid \text{data})
d{\boldsymbol{\theta}}^{\star} d{\mathbf{s}} d\alpha d\psi$, where ${\mathrm{p}}(G \mid {\boldsymbol{\theta}}^{\star},{\mathbf{s}},\alpha,\psi)$ follows a DP distribution with precision parameter $\alpha + N$ and base distribution given by (\[polya\]) (see Appendix A.2). As discussed in [@IshwZare2002], using results from [@Pitman1996], a draw for $G \mid {\boldsymbol{\theta}}^{\star},{\mathbf{s}},\alpha,\psi$ can be represented as $q_{0} G^{*}(\cdot)$ + $\sum_{j=1}^{m} q_{j} \delta_{\theta^{\star}_{j}}(\cdot)$, where $G^{*} \mid \alpha,\psi \sim$ DP$(\alpha,G_{0}(\psi))$, and, independently of $G^{*}$, $(q_{0},q_1,...,q_{m}) \mid \alpha,{\mathbf{s}} \sim$ ${\mathrm{Dir}}(q_{0},q_1,...,q_{m};\alpha, n_{1},...,n_{m})$. Therefore, posterior realizations for $G$ can be efficiently generated, by drawing for each posterior sample $\{ {\boldsymbol{\theta}}^{\star},{\mathbf{s}},\alpha,\psi \}$, $$dG_{L} = q_{0} \left\{ \sum_{l=1}^{L} p_{l} \delta_{\vartheta_{l}}(\cdot) \right\}
+ \sum_{j=1}^{m} q_{j} \delta_{\theta^{\star}_{j}}(\cdot),$$ that is, using a truncation approximation to $G^{*}$ based on the DP stick-breaking definition. Specifically, the $\vartheta_{l}$, $l=1,...,L$, are i.i.d. from $G_{0}(\psi)$, and the $p_{l}$ are constructed through i.i.d. Beta$(1,\alpha)$ draws, $\zeta_{s}$, $s=1,...,L-1$, such that $p_{1}=\zeta_{1}$, $p_{l}=$ $\zeta_{l} \prod_{s=1}^{l-1} (1-\zeta_{s})$, for $l=2,...,L-1$, and $p_{L}=$ $1 - \sum_{l=1}^{L-1} p_{l}$. The truncation level $L$ can be chosen using standard distributional properties for the weights in the DP representation for $G^{*}=$ $\sum_{l=1}^{\infty} \omega_{l} \delta_{\vartheta_{l}}(\cdot)$. For instance, ${\mathds{E}}(\sum\nolimits_{l=1}^{L} \omega_{l} \mid \alpha)=$ $1 - \{ \alpha/(\alpha + 1) \}^{L}$, which can be averaged over the prior for $\alpha$ to estimate ${\mathds{E}}(\sum\nolimits_{l=1}^{L} \omega_{l})$. Given any specified tolerance level for the approximation, this expression yields the corresponding value $L$. Note that even for dispersed priors for $\alpha$, relatively small values for $L$ (i.e., around 50) will generally provide very accurate truncation approximations.
Now, the posterior distribution for any functional (linear or non-linear) of the NHPP density, and thus of the intensity function, can be sampled by evaluating the functional using the posterior realizations $G_{L}$. For example, suppose that ${\mathbf{z}}=$ $[t,y]$, such that we have a temporal process with a single mark, where the mixture kernel factors as ${\mathrm{k}}({\mathbf{z}};\theta)=$ ${\mathrm{k}}^{t}(t;\theta^t){\mathrm{k}}^y(y;\theta^y)$. Given a posterior realization for $G_L$ and a posterior draw for $\Lambda_{{\mathcal{R}}}$, a posterior realization for the marginal process intensity at time $t$ is available as $$\lambda(t;G_L) =
\Lambda_{{\mathcal{R}}} \left[ q_0 \sum\nolimits_{l=1}^L p_l {\mathrm{k}}^t(t;\vartheta^t_l)
+ \sum\nolimits_{j=1}^m q_j {\mathrm{k}}^t(t;\theta^{\star t}_j)\right]$$ where $\vartheta_{l}=$ $(\vartheta_{l}^{t},\vartheta_{l}^{y})$ and $\theta_{j}^{\star}=$ $(\theta_{j}^{\star t},\theta_{j}^{\star y})$, and a realization for the conditional density of mark value $y$ at time $t$ arises through $$\label{approxcnd}
h(y \mid t;G_L) = \frac{ q_0 \sum_{l=1}^L p_l
{\mathrm{k}}^t(t;\vartheta^t_l) {\mathrm{k}}^y(y;\vartheta^y_l) +
\sum_{j=1}^m q_j {\mathrm{k}}^t(t;\theta^{\star t}_j) {\mathrm{k}}^y(y;\theta^{\star y}_j)}
{q_0 \sum_{l=1}^L p_l
{\mathrm{k}}^t(t;\vartheta^t_l) + \sum_{j=1}^m q_j {\mathrm{k}}^t(t;\theta^{\star t}_j)}.$$ Similarly, realized conditional expectation is available as $$\label{cond-mean}
{\mathds{E}}[y\mid t; G_L] = (f(t ; G_L))^{-1}
\left\{
q_0 \sum\nolimits_{l=1}^L p_l {\mathrm{k}}^t(t;\vartheta^t_l) {\mathds{E}}(y \mid \vartheta^y_l)
+ \sum\nolimits_{j=1}^m q_j {\mathrm{k}}^t(t;\theta^{\star t}_j)
{\mathds{E}}(y \mid \theta^{\star y}_j)
\right\}$$ a weighted average of kernel means with time-dependent weights. For multivariate Gaussian kernels, as in (\[mvnmarks\]), one would use conditional kernel means (available through standard multivariate normal theory; see Section \[coal-mining\]). The approach applies similarly to multivariate marks and/or to marked spatial NHPP, and we can thus obtain flexible inference for general functionals of marked NHPPs with full uncertainty quantification.
Model checking
--------------
A basic assumption implied by the Poisson process model is that the number of events within any subregion of the observation window are Poisson distributed, with mean equal to the integrated intensity over that subregion. Hence, a standard approach to assessing model validity is to compare observed counts to integrated intensity within a set of (possibly overlapping) subregions [e.g., @Digg2003; @BTMH2005].
An alternative approach to model checking is to look at goodness-of-fit for simplifying transformations of the observations. In particular, we propose transforming each margin of the point event data (i.e., each spatial coordinate and each mark) into quantities that are assumed, conditional on the intensity model, distributed as i.i.d. uniform random variables. Posterior samples of these (assumed) i.i.d. uniform sets can be compared, either graphically or formally, to the uniform distribution to provide a measure of model validity.
Consider first temporal point processes, and assume that the point pattern $\{ t_{i}: i=1,...,N \}$, with ordered time points $0 = t_0 < t_{1} \leq t_{2} \leq ... \leq t_{N} <
1$, is a realization from a NHPP with intensity function $\lambda(t)$ and cumulative intensity function $\Lambda(t)=$ $\int_{0}^{t} \lambda(s) ds$. Then, based on the time-rescaling theorem [e.g., @DaleVere2003], the transformed point pattern $\{ \Lambda(t_{i}): i=1,...,N \}$ is a realization from a homogeneous Poisson process with unit rate. Let $\Lambda(t;G_{L})$ be the posterior draws for the cumulative intensity, obtained following the approach of Section \[inference\]. Then, with $\Lambda(0;G_L) = 0$ by definition, the rescaled times $\Lambda(t_{i};G_L) - \Lambda(t_{i-1};G_L)$, $i=1,...,N$, are independent exponential random variables with mean one. Thus, the sampled $u_{i}=$ $1 - \exp \{ - ( \Lambda(t_{i};G_L) -
\Lambda(t_{i-1};G_L) ) \}$, $i=1,...,N$, are independent uniform random variables on $(0,1)$.
This approach can be extended to spatial processes by applying the rescaling to each margin of the observation window [e.g., @Cres1993]. If we have data corresponding to a NHPP on ${\mathcal{R}}=(0,1)\times (0,1)$ with intensity $\lambda({\mathbf{x}})$, then point event locations along (say) the first margin of the window are the realization of a one-dimensional NHPP with intensity $\lambda_{1}(x_{1})=$ $\int_{0}^{1} \lambda({\mathbf{x}})dx_2$, and analogously for $\lambda_{2}(x_2)$. Since the kernels in (\[bivarbeta\]) and (\[bivarnormal\]) are easily marginalized, cumulative intensities $\Lambda_1(\cdot)$ and $\Lambda_2(\cdot)$ are straightforward to calculate as sums of marginal kernel distribution functions, based on the sampled $G_L$ as described in Section 4.2. For each dimension $j$, these are then applied to ordered marginals $\{x_{j,1},\ldots,x_{j,N}\}$ to obtain i.i.d. uniform random variables, $u_{ij}=$ $1 - \exp \{ - (\Lambda_j(x_{j,i};G_L) -
\Lambda_j(x_{j,i-1};G_L)) \}$, $i=1,...,N$, where by definition $\Lambda_j(x_{j,0};G_L) = 0$ for $j=1,2$.
Finally, there are a variety of ways that the marks can be transformed into uniform random variables (for instance, the marginal process for continuous marks is also Poisson, such that the time-rescaling theorem applies), but, arguably, the most informative approach is to look at the conditional mark distribution of (\[markcnd\]). Full inference is available for the conditional cumulative distribution function $H(y
\mid {\mathbf{x}};G_{L})=$ $\int_{-\infty}^{y} h(s \mid {\mathbf{x}}; G_L)ds$, through a summation similar to that in (\[approxcnd\]), at any desired points $({\mathbf{x}},y)$. We thus obtain sets of $u_i$ that are assumed to be i.i.d. uniform by taking, for each sampled $G_L$, the distribution function evaluated at the data such that $u_i=$ $H(y_i
\mid {\mathbf{x}}_i;G_L)$, for $i=1,\ldots,N$.
Goodness-of-fit is evaluated through comparison of the $u_{i}$ samples with the uniform distribution, using either graphical or distance-based techniques. For instance, in the context of neuronal data analysis, @BrowBarbVentKassFran2001 used standard tests and quantile-quantile (Q-Q) plots to measure agreement of the estimated $u_{i}$ with the uniform distribution on $(0,1)$. In the examples of Section \[data-examples\], we focus on Q-Q plots for graphical model assessment, and find that these provide an intuitive picture of the marginal fit. In particular, under our Bayesian modeling approach, inference about model validity can be based on samples from the full posterior for each set of $u_{i}$, with each realization corresponding to a single draw for $G_L$, through plots of posterior means and uncertainty bounds for the Q-Q graphs.
The rescaling diagnostics involve a checking of the fit provided by the DP mixture model as well as of the Poisson process model assumption, and thus characterize a general nonparametric model assessment technique. Note that, in evaluating the model for event-location intensity, it is not, in general, feasible under this approach to distinguish the role of the Poisson assumption from the form of the nonparametric model for the NHPP density. The flexibility of the DP mixture modeling framework is useful in this regard, since by allowing general intensity shapes to be uncovered by the data, it enables focusing the goodness-of-fit evaluation on the NHPP assumption for the point process. Furthermore, all of these goodness-of-fit assessments are focused on model validity with respect to marginal processes (although, of course, these are implied marginals from a multidimensional fit). It is possible to extend the rescaling approach to higher dimensions, by defining a distance metric in the higher dimensional space and evaluating cumulative intensity functions with respect to this metric [e.g., @Diggle1990]. However, such procedures are considerably more difficult to implement and will need to be designed specifically for the application of interest.
Examples {#data-examples}
========
We include three data examples to illustrate the methodology. Specifically, Section \[simulation\] involves a simulated data set from a one-dimensional Poisson process with both categorical and continuous marks. In Sections \[coal-mining\] and \[pine-trees\], we consider real data on coal mining disaster events occurring in time with count marks, and on spatial tree locations data with trunk-diameter marks, respectively.
Simulated events with continuous and binary marks {#simulation}
-------------------------------------------------
We first consider a simulated data set from a temporal Poisson process with observation window ${\mathcal{R}}=(0,1)$ and intensity $\lambda(t)=$ $250 \left( b(t ;1/11, 11) + b(t;4/7,7) \right)$, such that $\Lambda_{{\mathcal{R}}} = 500$. The simulated point pattern comprises $N=481$ point events, which are accompanied by binary marks $z$ and continuous marks $y$ generated from a joint conditional density $h(y,z \mid t)=$ $h(y \mid z,t){\mathrm{Pr}}(z \mid t)$. Here, ${\mathrm{Pr}}(z=1 \mid t) = t^2$ and the conditional distribution for $y$, given $z$ and $t$, is built from $y=$ $-10(1-t)^4 + \varepsilon$, with $\varepsilon \sim {\mathrm{N}}(0,1)$ if $z=0$, and $\varepsilon \sim {\mathrm{ga}}(4,1)$ if $z=1$. Hence, the marginal regression function for $y$ given $t$ is non-linear with non-constant error variance, and ${\mathrm{Pr}}(z=1 \mid t)$ increases from 0 to 1 over ${\mathcal{R}}$.
We consider a fully nonparametric DP mixture model consisting of the beta kernel in (\[beta-model\]) for point locations combined with a normal kernel for $y$ and a Bernoulli kernel for $z$. Hence, the full model for the NHPP density is given by $$f(t,y,z;G) = \int b(t;\mu,\tau) {\mathrm{N}}(y;\eta,\phi) q^z(1-q)^{1-z}
dG(\mu, \tau,\eta, \phi, q), \,\,\, G \sim {\mathrm{DP}}(\alpha,G_0)$$ where $g_{0}(\mu,\tau,\eta,\phi,q)=$ ${\mathds{1}}_{\mu \in (0,1)}
{\mathrm{ga}}(\tau^{-1};2,\beta_{\tau}) {\mathrm{N}}(\eta;0,20\phi)
{\mathrm{ga}}(\phi^{-1};2,\beta_{\phi}) b(q;0.5,1)$. We use the reference prior for $\Lambda_{{\mathcal{R}}}$, and for the DP hyperpriors take $\alpha
\sim {\mathrm{ga}}(2,1)$, $\beta_{\tau} \sim {\mathrm{ga}}(1, 1/20)$ and $\beta_{\phi} \sim {\mathrm{ga}}(1,1)$; note that $\beta_{\tau}$ and $\beta_{\phi}$ are the means for $\tau$ and $\phi$, respectively, under $G_0$. The hyperpriors are specified following the guidelines of Appendix A.1, and posterior simulation proceeds as outlined in Appendix A.2. Since the beta kernel specification is non-conjugate, we jointly sample parameters and allocation variables with Metropolis-Hasting draws for each $(\mu_i, \tau_i)$ and $s_i$ given ${\mathbf{s}}^{(-i)}$ and $({\boldsymbol{\mu}}^\star,{\boldsymbol{\tau}}^\star)^{(-i)}$, as in algorithm 5 of @Neal2000.
![Simulation study results. On top, from left to right, we have posterior mean and 90% interval for the marginal intensity $\lambda(t; G)$ (with the true intensity denoted by the grey line), the data (dark grey for $z=1$), and posterior 90% predictive intervals based on both $h(y \mid t;G)$ (solid lines) and GP regression (dotted lines), and posterior mean and 90% intervals for ${\mathrm{Pr}}(z=1 \mid t; G)$ (with the true function denoted by the grey line). The middle row has mean and 90% intervals for conditional densities for $y$ at $t=1/2$, marginalized over $z$ (left panel) and conditional on $z$ (middle and right panels), with true densities plotted in grey. Lastly, the bottom row shows posterior samples for $\beta_{\tau}$ and $\beta_{\phi}$ (dark grey, with priors in the background) and for the number of latent mixture components.[]{data-label="simstudy"}](simstudyBet){width="6.25in"}
Results are shown in Figure \[simstudy\]. In the top row, we see that our methods are able to capture the marginal point intensity and general conditional behavior for $y$ and $z$; note that the uncertainty bounds are based on a full assessment of posterior uncertainty that is made possible through use of the truncated $G_L$ approximations to random mixing measure $G$ (as developed in Section \[inference\]). We also fit a Gaussian process (GP) regression model to the $(t,y)$ data pairs (using the `tgp` package for [R]{} under default parametrization) and, in contrast to our approach based on draws from $h(y \mid t;G_L)$ as in (\[approxcnd\]), the top middle panel shows the GP model’s global variance as unable to adapt to a wider skewed error distribution for larger $t$ values.
The middle row of Figure \[simstudy\] illustrates behavior for a slice of the conditional mark density for $y$, at $t=1/2$, both marginally and given $z=0$ or $1$. The marginal (left-most) plot shows that our model is able to reproduce the skewed response distribution, while the other two plots capture conditional response behavior given each value for $z$. As one would expect, posterior uncertainty around the conditional mark density estimates is highest at the transition from normal to gamma errors. Finally, posterior inference for model characteristics is illustrated in the bottom row of Figure \[simstudy\]. Peaked posteriors for $\beta_{\tau}$ and $\beta_{\phi}$ show that it is possible to learn about hyperparameters of the DP base distribution for both $t$ and $y$ kernel parameters, despite the flexibility of a DP mixture. Moreover, based on the posterior distribution for $m$, we note that the near to 500 observations have been shrunk to (on average) 12 distinct mixture components.
Temporal Poisson process with count marks {#coal-mining}
-----------------------------------------
Our second example involves a standard data set from the literature, the “coal-mining disasters” data [e.g., @AndrewsHerzberg1985 p. 53-56]. The point pattern is defined by the times (in days) of 191 explosions of fire-damp or coal-dust in mines leading to accidents, involving 10 or more men killed, over a total time period of 40,550 days, from 15 March 1851 to 22 March 1962. The data marks $y$ are the number of deaths associated with each accident.
This example will compare two different mixture models for marginal location intensity: a “direct” model with beta-Poisson kernels, and a “transformed” model with data mapped to ${\mathds{R}}^2$ and fit via multivariate normal kernels. The first scheme models data directly on its original scale, but requires Metropolis-Hastings augmented MCMC for the beta kernel parameters, and dependence between $t$ and $y$ is induced only through $G$. The second model affords the convenience of the collapsed Gibbs sampler and correlated kernels, but on a transformed scale.
Following our general modeling approach, both models use the reference prior for $\Lambda_{{\mathcal{R}}}$ and assume NHPP density form $f(t,y;G)=$ $\int {\mathrm{k}}(t,y;\theta)dG(\theta)$ with $G \sim
{\mathrm{DP}}(\alpha, G_0)$ and $\pi(\alpha)= {\mathrm{ga}}(2, 1)$. The distinction between the two models is thus limited to choice of kernel and base distribution. For the direct model, $$\begin{aligned}
\label{truncpois}
{\mathrm{k}}(t,y;\mu,\tau,\phi) &=& b(t;\mu,\tau) {\mathrm{Po}}_{\geq 10}(y;\phi),\\
g_{0}(\mu,\tau,\phi) &=& {\mathds{1}}_{\mu \in (0,1)}
{\mathrm{ga}}(\tau^{-1}; 2, \beta_{\tau}){\mathrm{ga}}(\phi ; 1, 1/60), \notag\end{aligned}$$ where ${\mathrm{Po}}_{\geq 10}(y;\phi)$ is a Poisson density truncated at $y=10$, and with $\pi(\beta_{\tau}) = {\mathrm{ga}}(1, 1/63)$. This leads to prior expectations ${\mathds{E}}[\phi] = 60$ and ${\mathds{E}}[\tau]=$ ${\mathds{E}}[\beta_{\tau}] = 63$ for mean location kernel precision $(1+\tau)/(\mu(1-\mu))$ $\approx$ $4(1+63)$, which translates to a standard deviation of $1/16$. For the transformed model, we take $\tilde{y}=$ $y-9.5$ and $$\begin{aligned}
{\mathrm{k}}(t,y;{\boldsymbol{\mu}},{\boldsymbol{\Sigma}}) &=&
\frac{{\mathrm{N}}\left([{\mathrm{logit}}(t), \log(\tilde{y})]';
{\boldsymbol{\mu}},{\boldsymbol{\Sigma}}\right)}{\tilde{y}t(1-t)}\\
g_{0}({\boldsymbol{\mu}},{\boldsymbol{\Sigma}})
&=& {\mathrm{N}}({\boldsymbol{\mu}}; (0,2.5)',
10{\boldsymbol{\Sigma}})W({\boldsymbol{\Sigma}}^{-1};3,{\boldsymbol{\Omega}}), \notag\end{aligned}$$ with $\pi({\boldsymbol{\Omega}})=$ ${\mathrm{W}}(3, {\mathrm{diag}}[10,20])$ for ${\mathds{E}}(\Sigma)=$ $2/3 {\mathds{E}}(\Omega) = {\mathrm{diag}}[1/5,1/10]$ (${\mathrm{logit}}(t)$ and $\log(\tilde{y})$ range in (-5,5) and (-1,6), respectively). Both models were found to be robust to changes in this parametrization (e.g., ${\mathds{E}}[\phi] \in
[10,100]$ and diagonal elements of ${\mathds{E}}[\Sigma]$ in $[0.1,1]$).
![ Coal-mining disasters. Mean and 90% intervals for (clockwise from top-left): marginal density $f(t;G_{L})$ (with data histogram); conditional expected count ${\mathds{E}}(y \mid t;G_{L})$ (data counts in grey); and posterior Q-Q plots for ${\mathrm{Pr}}(y < y_i \mid t_i; G_{L})$ and $\Lambda(t_i;G_{L})$, respectively.[]{data-label="coal"}](coal){width="6.25in"}
Results under both models are shown in Figure \[coal\]. In the top left panel, we see that marginal process density estimates derived from each model are generally similar, with the normal model perhaps more sensitive to data peaks and troughs. There is no noticeable edge effect for either model. The Q-Q plot in the bottom left panel shows roughly similar fit with the normal model performing slightly better. The top and bottom right panels report inference for the count mark conditional mean and distribution Q-Q plot. For the beta-Poisson model, posterior realizations for ${\mathds{E}}(y \mid t;G_L)$ are obtained using (\[cond-mean\]). The conditional mean calculation for the normal model must account for the correlated kernels (and the transformation to $\tilde{y}$), such that ${\mathds{E}}(y \mid t;G_L) $ is $$\frac{9}{2} +
\left(q_0\sum_{l=1}^L p_l {\mathrm{N}}(t; \mu_{lt}, \sigma^2_{lt})
{\mathds{E}}[y \mid t;\vartheta_l] + \sum_{j=1}^mq_j{\mathrm{N}}(t; \mu^\star_{jt}, \sigma^{\star 2}_{jt})
{\mathds{E}}[y \mid t;\theta^\star_j]\right)/{f(t;G_L)}$$ where ${\mathds{E}}[y \mid t,\theta]=$ $\exp\left[\mu_{y} + \rho
\sigma_{t}^{-2}(t-\mu_{t}) + 0.5(\sigma^{2}_{y} - \rho^{2}
\sigma_{t}^{-2}) \right]$ with ${\boldsymbol{\mu}}=$ $(\mu_{t},\mu_{y})$ and ${\boldsymbol{\Sigma}}$ partitioned into variances $(\sigma^2_{t},\sigma^{2}_{y})$ and correlation $\rho$. Similarly, uniform quantiles for the conditional mark distribution under the beta-Poisson model are available as weighted sums of Poisson distribution functions, while the normal model calculation for ${\mathrm{Pr}}(y < y_i \mid t_i; G_L)$ is as above for ${\mathds{E}}(y \mid
t;G_L)$, but with ${\mathds{E}}[y \mid t,\theta]$ replaced by ${\mathrm{Pr}}(y <
y_i \mid t_i; \theta)=$ $\Phi\left(\left[\tilde{y}_i -
\mu_{y} + \rho \sigma_{t}^{-2} (t_i-\mu_{t})\right]
(\sigma^{2}_{y} - \rho^{2} \sigma_{t}^{-2})^{-1/2} \right)$. The estimated conditional mean functions are qualitatively different, with the poisson model missing the peak at WW1. Indeed, the corresponding QQ plot shows that the normal model provides a better fit to this data; we hypothesize that this is due to the equality of mean and variance assumed in Poisson kernels, and may be fixed by using instead, say, truncated negative binomials.
Spatial Poisson process with continuous marks {#pine-trees}
---------------------------------------------
Our final example considers the locations and diameters of 584 Longleaf pine trees in a $200\times200$ meter patch of forest in Thomas County, GA. The trees were surveyed in 1979 and the measured mark is diameter at breast height (1.5 $m$), or $dbh$, recorded only for trees with greater than 2 [*cm dbh*]{}. The data, available as part of the `spatstat` package for `R`, were analyzed by [@RathCres1994] as part of a space-time survival point process. Poisson processes are generally viewed as an inadequate model for forest patterns, due to the dependent birth process by which trees occur. However, the NHPP should be flexible enough to account for variability in tree counts at a single time point and, in this example, we will concentrate primarily on inference for the conditional $dbh$ mark distribution.
![Longleaf pines. The left panel has data (point size proportional to tree diameter) and a Q-Q plot (mean and 90% interval) for $\int^{y} h(s \mid {\mathbf{x}};G_{L})ds$ evaluated at data. The right panel plots posterior mean and 90% intervals for $h(y \mid {\mathbf{x}};G_{L})$ at four specific ${\mathbf{x}}$ values.[]{data-label="pine"}](pinesQQ "fig:"){width="2.25in"} ![Longleaf pines. The left panel has data (point size proportional to tree diameter) and a Q-Q plot (mean and 90% interval) for $\int^{y} h(s \mid {\mathbf{x}};G_{L})ds$ evaluated at data. The right panel plots posterior mean and 90% intervals for $h(y \mid {\mathbf{x}};G_{L})$ at four specific ${\mathbf{x}}$ values.[]{data-label="pine"}](pineCND "fig:"){width="4.1in"}
To analyse this data set, we employ a spatial version of the model in (\[mvnmarks\]), with tree marks log-transformed to lie on the real line. Thus, our three-dimensional normal kernel model is $$\phi({\mathbf{x}},y;G) = \Lambda_{{\mathcal{R}}} \int \frac{{\mathrm{N}}\left(
[{\mathrm{logit}}({\mathbf{x}}),\log(y-2)]'; {\boldsymbol{\mu}},{\boldsymbol{\Sigma}}
\right) } {(y-2) \prod_{i=1}^2 x_i (1-x_i )} d
G({\boldsymbol{\mu}},{\boldsymbol{\Sigma}}),
\hspace{0.5cm} G \sim {\mathrm{DP}}(\alpha,G_{0}).$$ The base distribution is taken to be $g_{0}({\boldsymbol{\mu}},{\boldsymbol{\Sigma}})=$ ${\mathrm{N}}({\boldsymbol{\mu}}; (0,0,1)',100{\boldsymbol{\Sigma}}){\mathrm{W}}({\boldsymbol{\Sigma}}^{-1};
4,{\boldsymbol{\Omega}})$, with $\pi({\boldsymbol{\Omega}})=$ ${\mathrm{W}}(4,{\mathrm{diag}}[0.1,0.1,0.1,0.1])$. A ${\mathrm{ga}}(2,1)$ prior is placed on $\alpha$. Posterior sampling follows the fully collapsed Gibbs algorithm of Appendix A.2.
In this data set, high density clusters of juveniles trees ([*dbh*]{} $<5$[*cm*]{}) combine with the more even dispersal of larger trees to form conditional mark densities with non-standard shapes and non-homogeneous variability. This behavior is clearly exhibited in the posterior estimates of the conditional density for $dbh$, shown on the right side of Figure \[pine\], at four different locations in the observations window. Although conditional densities vary in shape over the different locations, each appears to show the mixture of a diffuse component for mature trees combined with a sharp increase in density at low $dbh$ values, corresponding to collections of juvenile trees (only some of whom make it to maturity). It is notable that we are able to infer this structure nonparametrically, in contrast to existing approaches where the effect of a tree-age threshold is assumed [*a priori*]{} [as in @RathCres1994]. Finally, the conditional mark distribution Q-Q plot on the bottom right panel of Figure \[pine\] (based on calculations similar to those in Section \[coal-mining\]) shows a generally decent mean-fit with wide uncertainty bands corresponding to the 95% and 5% density percentile Q-Q plots.
Discussion {#summary}
==========
We have presented a nonparametric Bayesian modeling framework for marked non-homogeneous Poisson processes. The key feature of the approach is that it develops the modeling from the Poisson process density. We have considered various forms of Dirichlet process mixture models for this density which, when extended to the joint mark-location process, result in highly flexible nonparametric inference for the location intensity as well as for the conditional mark distribution. The approach enables modeling and inference for multivariate mark distributions comprising both categorical and continuous marks, and is especially appealing with regard to the relative simplicity with which it can accommodate spatially correlated marks. We have discussed methods for prior specification, posterior simulation and inference, and model checking. Finally, three data examples were used to illustrate the proposed methodology.
The Poisson assumption for marked point processes is what enables us to separate modeling for the process density from the integrated intensity. This simplification is particularly useful for applications involving several related intensity functions and mark distributions, and is less restrictive than it may at first appear. For instance, [@Taddy2010] presents an estimation of weekly violent crime intensity surfaces, using autoregressive modeling for marked spatial NHPPs, and [@KBMPO2011] compares neuronal firing intensities recorded under multiple experimental conditions, using hierarchically dependent modeling for temporal NHPPs.
Among the possible ways to relax the restrictions of the Poisson assumption, while retaining the appealing structure of the NHPP likelihood, we note the class of multiplicative intensity models studied, for instance, in @IshwJame2004. These models for marked point processes are under the NHPP setting and, indeed, follow the simpler strategy of separate modeling for the process intensity and mark density as in the semiparametric framework of Section 3.1. More generally, one could envision relaxing the Poisson assumption for the number of marks through a joint intensity function such that the location intensity is not the marginal of the joint intensity over marks. Such extensions would however sacrifice the main feature of our proposed framework – flexible modeling for multivariate mark distributions under a practical posterior simulation inference scheme. As a more basic extension, our factorization in (1) could be combined with alternative specifications for integrated intensity; for example, hierarchical models may be useful to connect intensity across observation windows.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors wish to thank an Associate Editor and a referee for helpful comments. The work of the first author was supported in part by the IBM Corporation Faculty Research Fund at the University of Chicago. The work of the second author was supported in part by the National Science Foundation under awards DEB 0727543 and SES 1024484.
Appendix: Implementation Details for Dirichlet Process Mixture Models {#appendix-implementation-details-for-dirichlet-process-mixture-models .unnumbered}
=====================================================================
A.1 Prior Specification {#a.1-prior-specification .unnumbered}
-------------------------
Prior specification for the DP precision parameter is facilitated by the role of $\alpha$ in controlling the number, $m \leq N$, of distinct mixture components. For instance, for moderately large $N$, ${\mathds{E}}[m \mid \alpha] \approx$ $\alpha \log \left( (\alpha +
N)/\alpha \right)$. Furthermore, it is common to assume a gamma prior for $\alpha$, such that $\pi(\alpha)=$ ${\mathrm{ga}}(\alpha;a_{\alpha}, b_{\alpha})$, and use prior intuition about $m$ combined with ${\mathds{E}}[m \mid \alpha]$ to guide the choice of $a_\alpha$ and $b_\alpha$.
Specification of the base distribution parameters will clearly depend on kernel choice and application details, and DP mixture models are typically robust to reasonable changes in this specification. First, the base distribution for kernel [*location*]{} (usually the mean, but possibly median) can be specified through a prior guess for the data center; for example, this value can be used to fix the mean parameter ${\boldsymbol{\delta}}$ in (\[bivarnormal\]) or the mean of a normal hyperprior for ${\boldsymbol{\delta}}$. In choosing dispersion parameters, note that the DP prior will place most mass on a small number of mixture components, with the remaining components assigned very little weight and, hence, very few observations. At the same time, this behavior can be overcome in the posterior and it is important to not restrict the mixture to overly-dispersed kernels. Thus, the expectation of the kernel variance (or scale, or shape) parameters should be specified with a small number of mixture components in mind, but with low precision. For example, again in the context of (\[bivarnormal\]), the square-root of the hyperprior expectation for diagonal elements of ${\boldsymbol{\Omega}}$ can be set at $1/8$ to $1/16$ of a prior guess at data range, and the precision $\nu$ will be as small as is practical (usually the dimension of the kernel plus 2). The factor $\kappa$ is then chosen to scale the mixture to expected dispersion in ${\boldsymbol{\mu}}$.
Moreover, except when specific prior information about co-dependence is available, it is best to center $G_0$ on kernel parametrization that implies independence between variables, such that the mixture is centered on a model with dependence induced nonparametrically by $G$. For example, in the model of (\[bivarnormal\]), we assume zeros in the off-diagonal elements for the prior expectation of ${\boldsymbol{\Omega}}$, and this is combined with a small $\nu$ to allow for within-kernel dependence where appropriate. A prior expectation of independence also fits with our general approach of building kernels for mixed-type data as the product of multiple independent densities.
Note that we have chosen to introduce prior information into the base measure based on the intuition arising from a small number of large mixture components and $\alpha$ near zero. Recent work in [@BushLeeMacE2010] provides a rigorous treatment of non-informative prior specification, and they advocate a hierarchical scheme for $\alpha | G_0$ that maintains desirable properties at all scales of precision. As the main work here – use of mixtures for modeling joint location-mark Poisson process densities – is independent of prior and base measure choice, these innovations, as well as application-specific prior schemes, could potentially be integrated into our framework.
A.2 Posterior Simulation {#a.2-posterior-simulation .unnumbered}
--------------------------
Using results from @Anto1974, the posterior distribution for the DP mixture model is partitioned as ${\mathrm{p}}(G,{\boldsymbol{\theta}}^{\star},{\mathbf{s}},\alpha,\psi \mid \text{data})=$ ${\mathrm{p}}(G \mid {\boldsymbol{\theta}}^{\star},{\mathbf{s}},\alpha,\psi)
{\mathrm{p}}({\boldsymbol{\theta}}^{\star},{\mathbf{s}},\alpha,\psi \mid \text{data})$, where $G$, given ${\boldsymbol{\theta}}^{\star},{\mathbf{s}},\alpha,\psi$, is distributed as a DP with precision parameter $\alpha + N$ and base distribution given by (\[polya\]). Hence, full posterior inference involves sampling for the finite dimensional portion of the parameter vector, which is next supplemented with draws from the conditional posterior distribution for $G$ (obtained as discussed in Section \[inference\]). A generic Gibbs sampler for posterior simulation from ${\mathrm{p}}({\boldsymbol{\theta}}^{\star},{\mathbf{s}},\alpha,\psi \mid
\text{data})$, derived by combining MCMC methods from @MacEachern1994 and @EscoWest1995, proceeds iteratively as follows:
- For $i=1,\ldots,N$, denote by ${\mathbf{s}}^{(-i)}$ the allocation vector with component $s_{i}$ removed, and by $N_{s}^{(-i)}$ the number of elements of ${\mathbf{s}}^{(-i)}$ that are equal to $s$. Then, if $s=s_{r}$ for some $r \neq i$, the $i$-th allocation variable is updated according to $$\text{Pr}(s_{i} = s \mid {\mathbf{s}}^{(-i)},\alpha,\psi,\text{data}) \propto
\frac{N_{s}^{(-i)}}{N - 1 + \alpha}
\int {\mathrm{k}}({\mathbf{z}}_i;\theta^{\star}) p(\theta^{\star} \mid {\mathbf{s}}^{(-i)},\psi,\text{data})
d\theta^{\star},$$ where $p(\theta^{\star} \mid {\mathbf{s}}^{(-i)},\psi,\text{data})$ is the density proportional to $g_{0}(\theta^{\star};\psi) \prod\nolimits_{
\{ r \neq i: s_{r}=s \} } {\mathrm{k}}({\mathbf{z}}_{r};\theta^{\star})$. Moreover, the probability of generating a new component, that is, $\text{Pr}(s_{i} \neq s_{r} \, \text{for all} \, r \neq i \mid
{\mathbf{s}}^{(-i)},\alpha,\psi,\text{data})$, is proportional to $\alpha (N
- 1 + \alpha)^{-1} \int {\mathrm{k}}({\mathbf{z}}_{i};\theta^{\star})
g_{0}(\theta^{\star};\psi) d\theta^{\star}$.
- For $j=1,...,m$, draw $\theta^\star_j$ from $p(\theta^{\star}_{j} \mid {\mathbf{s}},\psi,\text{data}) \propto$ $g_{0}(\theta^{\star}_{j};\psi) \prod\nolimits_{\{ i: s_{i}=j \}}
{\mathrm{k}}({\mathbf{z}}_{i};\theta^{\star}_{j})$.
- Draw the base distribution hyperparameters from $\pi(\psi) \prod_{j=1}^{m} g_{0}(\theta^{\star}_{j};\psi)$, where $\pi(\psi)$ is the prior for $\psi$. Finally, if $\alpha$ is assigned a gamma hyperprior, it can be updated conditional on only $m$ and $N$ using the auxiliary variable method from @EscoWest1995.
The integrals that are needed to update the components of ${\mathbf{s}}$ can be evaluated analytically for models where $G_0$ is conjugate for ${\mathrm{k}}(\cdot;\theta)$. It is for this reason that conditionally conjugate mixture models can lead to substantially more efficient posterior sampling, especially when $\theta$ is high-dimensional. When this is not true (as for, e.g., beta kernel models or the truncated Poisson of Equation \[truncpois\]), the draw for ${\mathbf{s}}$ requires use of the auxiliary parameters, ${\boldsymbol{\theta}}^{\star}$, sampled as in the second step of our algorithm, in conjunction with a joint Metropolis-Hastings draw for each $\theta_i$ and $s_i$ given ${\boldsymbol{\theta}}^{(-i)}$ and ${\mathbf{s}}^{(-i)}$. In particular, we can make use of algorithms from @Neal2000 for non-conjugate models.
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|
---
abstract: 'We show that the discrete-time evolution of an open quantum system generated by a single quantum channel $T$ can be embedded in the discrete-time evolution of an enlarged closed quantum system, i.e. we construct a unitary dilation of the discrete-time quantum-dynamical semigroup $(T^n)_{n \in \mathbb N_0}$. In the case of a cyclic channel $T$, the auxiliary space may be chosen (partially) finite-dimensional. We further investigate discrete-time quantum control systems generated by finitely many commuting quantum channels and prove a similar unitary dilation result as in the case of a single channel.'
author:
- Frederik
- Gunther
bibliography:
- 'article\_vomende\_dirr\_unitary\_dilations\_update\_2019\_03\_11.bib'
title: 'Unitary Dilations of Discrete-Time Quantum-Dynamical Semigroups'
---
[^1]
Introduction
============
Stimulated by the seminal work of Arveson[@arveson1972], Lindblad[@Lindblad76], Gorini, Kossakowski and Sudarshan[@GKS76] in the mid 1970s many efforts have been made to obtain dilation results of various degrees of generality for semigroups of completely positive operators.
For instance, Davies[@Davies Ch. 9, Thm. 4.3] proved that for any continuous semigroup $(T_t)_{t\in\mathbb R^+_0}$ of completely positive, unital operators acting on a finite-dimensional Hilbert spaces $\mathcal H$ (or, more precisely, on the corresponding $W^*$-algebra $\mathcal B(\mathcal H)$ of all bounded linear operators) there exists a Hilbert space $\mathcal K$, a pure state $\omega$ in $\mathcal B(\mathcal K)$, and a strongly continuous one-parameter group $(U_t)_{t\in\mathbb R}$ of unitaries on $\mathcal H\otimes\mathcal K$ such that $$\begin{aligned}
\label{eq:0}
T_t(A)=\operatorname{tr}_{\omega}(U_t^\dagger(A\otimes\operatorname{id}_{\mathcal K})U_t)\end{aligned}$$ holds for all $A \in \mathcal B(\mathcal H)$ and $t\in\mathbb R^+_0$. For infinite-dimensional $\mathcal H$, there is a whole zoo of similar results. While Davies [@davies78; @Davies79], Evans[@Evans1976groups], and Evans & Lewis[@EvansLewis1976; @EvansLewisOverview77] focused primarily on one-parameter semigroups $(T_t)_{t \in\mathbb R^+_0}$ of different continuity type, Kümmerer[@Kuemmerer83] discussed at great length the discrete-time case $(T^n)_{n \in \mathbb N}$. However, to the best of our knowledge, for arbitrary Hilbert spaces a dilation result of the above form is still not available.
In the following, we give a short chronological overview on those contributions which are relevant and closely related to our work. Further results and a brief survey over the latest developments can be found in [@Bhat1996; @bhat_skeide_2000; @muhly_solel_2002; @Gaebler2015] and [@sawada2018]. For the readers’ convenience we collected some standard terminology and basic results on dilations and (completely) positive maps, which are well known to experts in this area, in the glossary of Appendix \[sec:ditalions\].
In [@Evans1976groups Thm.1 and Thm.2], Evans shows that every family $(T_g)_{g \in G}$ of completely positive, unital operators acting on a unital $C^*$-algebra $\mathcal A$ and indexed by an arbitrary group $G$ admits a unitary dilation, i.e. $$\begin{aligned}
T_g(A) = E(U_g^\dagger J(A)U_g)\end{aligned}$$ for all $A \in \mathcal A$ and $g \in G$, where $(U_g)_{g \in G}$ is a unitary representation of $G$ on some Hilbert space $\mathcal K$ and $E$ a conditional expectation with corresponding injection $J$ into $\mathcal B(\mathcal K)$. Remarkably, he need not assume that $g \mapsto T_g$ is a group homomorphism. His result can be regarded as $C^*$-counterpart to Sz.-Nagy’s [@nagy1953; @nagy1970] and Stroescu’s [@stroescu1973] work on isometric dilations on Hilbert and Banach spaces, respectively. While possible generalizations to $W^*$-algebras are addressed by Evans, continuity issues of the map $g \mapsto U_g$ are disregarded completely. His proof is based on Stinespring’s representation $T_g(x) = V^*_g \pi_g(x) V_g$ which of course exists for all $g \in G$. However, he did not exploit the fact that one can choose a common Hilbert space for all $\pi_g$ which leads to a substantial simplification in our approach. In [@EvansLewis1976 Thm.2] Evans & Lewis focus on norm-continuous semigroups $(T_t)_{t\in\mathbb R^+_0}$ of ultraweakly continuous, completely positive and unital operators acting on a separable Hilbert space $\mathcal H$. They obtain a unitary dilation $$\begin{aligned}
T_t(A) = E(U_t^\dagger J(A)U_t)\,,\end{aligned}$$ for all $A \in \mathcal B(\mathcal H)$ and $t \in \mathbb R^+_0$, where $(U_t)_{t\in\mathbb R}$ is a strongly continuous group of unitary operators acting on some extended Hilbert space $\mathcal K$ (and $E, J$ as above). Their proof exploits the fact that the explicit form of the infinitesimal generator of $(T_t)_{t\in\mathbb R^+_0}$ is well-known due to the work of Lindblad[@Lindblad76]. In [@EvansLewisOverview77] Evans & Lewis provide an overview on dilation results known at that time including some minor generalisations of their previous work [@EvansLewis1976]. For locally compact groups $G$, Davies [@davies78 Thm.2.1 and Thm.3.1] obtains the following rather general result: Let $(T_g)_{g\in G}$ be a strongly continuous family of ultraweakly continuous, completely positive and unital operators on $\mathcal B(\mathcal H)$. Then there exists a Hilbert space $\mathcal K$, a strongly continuous unitary representation $U$ of $G$ on $\mathcal H \otimes \mathcal K$ and conditional expectations $E_n:\mathcal B(\mathcal H\otimes\mathcal K)\to\mathcal B(\mathcal H)$ (for all $n \in \mathbb N$) such that $$\begin{aligned}
\label{eq:0_1}
T_g(A)=\lim_{n\to\infty} E_n(U_g^\dagger(A\otimes\operatorname{id}_{\mathcal K})U_g)\end{aligned}$$ holds for all $A \in \mathcal B(\mathcal H)$ and all $g\in G$ in the weak operator topology. Here, $E_n$ is of the form $E_n(A):=V_n^\dagger AV_n$ where $V_n:\mathcal H\to\mathcal H\otimes\mathcal K$ are isometric embeddings. This seems to be the result which is closest to (\[eq:0\]) in infinite dimensions, but it is not known whether the limit in (\[eq:0\_1\]) is necessary or not[@davies78 cf. p. 335]. For discrete-time systems $T_n := T^n$, $n \in \mathbb N$ or, more accurately, for an appropriate extension to $G = \mathbb Z$ Davies’ approach and ours are quite similar—in particular due to the fact that in this case, the limit in can be avoided as $G$ is discrete. More precisely, Davies first extends the state space from $\mathcal H$ to $L^2(\mathbb Z, \mathcal H) \cong \ell_2(\mathbb Z) \otimes \mathcal H$ such that $(T_n)_{n \in \mathbb Z}$ can be regarded as one completely positive, unital operator from $\mathcal B(\mathcal H)$ to $\mathcal B(L^2(\mathbb Z, \mathcal H))$. He then applies Stinespring’s representation theorem to obtain an dilation of $(T_n)_{n \in \mathbb Z}$ on a larger state space $L^2(\mathbb Z, \mathcal H) \otimes \mathcal{K}$. We, however, exploit Stinespring’s result first to guarantee for all $n \in \mathbb N$ a dilation of the form $$\begin{aligned}
T^n(A)=\operatorname{tr}_{\omega}((U^\dagger)^n(A\otimes\operatorname{id}_{\mathcal K})U^n)\,,\end{aligned}$$ where $\omega$ and $\mathcal K$ are independent of $n \in \mathbb N$, to then enlarge the state space to $\ell_2(\mathbb Z) \otimes \mathcal H \otimes \mathcal K \cong L^2(\mathbb Z, \mathcal H \otimes \mathcal K)$. Although both approaches differ only in the order of the construction steps the resulting dilations behave quite differently: While Davies’ construction is more “flexible” as one can see, e.g., in Section \[section\_control\], Remark \[ch\_4\_bem\_7\].2, ours yields the desired partial trace structure of which is in general not satisfied for even if the limit can be avoided[@Note1] .
Kümmerer[@Kuemmerer83] discussed the discrete-time case $(T^n)_{n \in \mathbb N}$ in detail. However, his setting significantly differs from ours. In his sense, a discrete-time quantum dynamical system consists of a triple $(\mathcal A,\varphi,T)$, where $\mathcal A \subset \mathcal B(\mathcal H)$ is a $W^*$-algebra, $T$ an ultraweakly continuous, completely positive and unital operator which acts on $\mathcal A$ and leaves a faithful normal state $\varphi \in \mathcal A^*$ invariant, i.e. $\varphi\circ T = \varphi$. The latter condition can be thermodynamically motivated as $\varphi$ can be interpreted as an equilibrium state which is preserved under composition with $T$ and every power of it. This constraint on the quantum channel $T$ obviously narrows down the possible choices of $T$. Even more restrictive is Kümmerer’s definition of a first order dilation of $(\mathcal A,\varphi,T)$. Here, he requires the existence of a reversible quantum dynamical system $(\mathfrak A,\hat{\varphi},\hat{T})$, i.e. $\hat{T}$ is a $*$-automorphism on $\mathfrak A$ and $E$ is a conditional expectation with corresponding injection $J$ such that $$\begin{aligned}
T(A) = E\big(\hat{T}(J(A))\big)\qquad \text{and} \qquad \varphi\circ E = \hat{\varphi}\,.\end{aligned}$$ for all $A \in \mathcal A$. In doing so, the condition $\varphi\circ E = \hat{\varphi}$ is the delicate part. For instance, the standard Kraus/Stinspring representation which constitutes a (first order) dilation does in general not satisfy this condition—note that, by definition, $\hat{\varphi}$ has to be a faithful normal state—and therefore even first order dilations in Kümmerer’s sense need not exist as the existence of a $\varphi$-adjoint is not guaranteed, cf. [@Kuemmerer83 Prop. 2.1.8 ff.]. Within his setting, Kümmerer proved (cf. Thm 4.2.1, Cor 4.2.3) that a quantum dynamical system $(\mathcal B(\mathcal H),\varphi,T)$ has a dilation of first order if and only if it admits a Markovian one of first order which in turn implies that $(\mathcal B(\mathcal H),\varphi,T)$ also allows a Markovian dilation of arbitrary order. His definition of Markovianity can be regarded as a $W^*$-algebra counterpart of a well-known subspace condition[@Note2] which guarantees for contractions on Hilbert spaces that a first order unitary dilation $T = P_{\mathcal H}U|_{\mathcal H}$ is already a dilation (of arbitrary order), i.e. $T^n = P_{\mathcal H}U^n|_{\mathcal H}$ holds for all $n \in \mathbb N$. To achieve a Markovian dilation he imbedded the given $W^*$-algebra $\mathcal A = \mathcal B(\mathcal H)$ in an infinite product/sum of $W^*$-algebras. Our approach considerably deviates from his construction since we use first order Stinespring/Kraus dilations for $T^n$ which of course exist for all $n \in \mathbb N$ but in general do not satisfy Kümmerer’s faithful state condition. Probably one of the strongest semigroup dilation results so far was presented by Gaebler[@Gaebler2015 Thm. 5.10]. Using Sauvageot’s theory he showed that for a norm-continuous semigroup $( T_t)_{t\in\mathbb R^+}$ of ultraweakly continuous, completely positive and unital operators acting on a $W^*$-algebra $\mathcal A$ with separable pre-dual, there exists a unital dilation $(\mathfrak A,(\sigma_t)_{t\in\mathbb R^+_0},J,E)$ of $( T_t)_{t\in\mathbb R^+_0}$ (cf. Def. \[def:dilation-von-Neumann-algebras\]) where $\mathfrak A$ has separable pre-dual and $((\sigma_t)_{t\in\mathbb R^+_0},J,E)$ satisfies the strong dilation property, i.e. $T_t\circ E=E\circ\sigma_t$ for all $t\in\mathbb R^+_0$. The strength of this result, however, comes at the cost of lacking any partial trace structure of the form (\[eq:0\]).
The paper is organized as follows: After some preliminaries on trace-class operators and quantum channels, we present our main results in Section \[sec:main\_results\]: (i) For discrete-time quantum-dynamical semigroups on separable Hilbert spaces, a unitary dilation of the form (\[eq:0\]) is proved. (ii) If the semigroup in question is generated by a cyclic quantum channel, then the auxiliary Hilbert space can be chosen partially finite-dimensional. (iii) Finally, for discrete-time quantum control systems, the control of which can be switched between a finite number of commuting channels, a unitary dilation of the form (\[eq:0\]) is derived.
Preliminaries and Notation {#sec:perlim}
==========================
In this section, we fix our notation and recall some basic material on Schrödinger and Heisenberg quantum channels. These results should be known to experts in this area.
Henceforth, let $\mathcal G, \mathcal H$ be infinite-dimensional separable complex Hilbert spaces and $\mathcal X, \mathcal Y$ real or complex Banach spaces. By convention, all scalar products on complex Hilbert spaces are assumed to be conjugate linear in the first argument and linear in the second. Moreover, let $\mathcal B(\mathcal G), \mathcal B(\mathcal H)$ denote the set of all bounded operators acting on $\mathcal G, \mathcal H$ and let $\mathcal B(\mathcal X), \mathcal B(\mathcal Y)$ be defined respectively.
Recall that an operator $A\in\mathcal B(\mathcal H)$ on a complex Hilbert space is said to be positive semi-definite, denoted by $A\geq 0$, iff $\langle x,Ax\rangle\geq 0$ for all $x\in\mathcal H$. Because we consider complex Hilbert spaces, $A\geq 0$ directly implies that $A$ is self-adjoint via the polarization identity, cf. [@kadisonringrose1983 Prop. 2.4.6]—else, self-adjointness would have to be required in the definition of $A\geq 0$.
Quantum Channels {#sec:qu-channels}
----------------
Let $\mathcal B^1(\mathcal H) \subset \mathcal B(\mathcal H)$ be the subset of all *trace-class operators*, i.e. $\mathcal B^1(\mathcal H)$ is the largest subspace of $\mathcal B(\mathcal H)$ which allows to define the *trace* of an operator $A$ via $$\begin{aligned}
\label{def_trace_eq}
\operatorname{tr}(A):=\sum\nolimits_{i\in I}\langle e_i,Ae_i\rangle\end{aligned}$$ such that the right-hand side of is finite and independent of the choice of the orthonormal basis $(e_i)_{i \in I}$. More precisely, $\mathcal B^1(\mathcal H)$ can be defined either as the set of all compact operators $A \in \mathcal B(\mathcal H)$ whose singular values $\sigma_n(A)$ are summable, i.e. $$\label{def1_trace_eq}
\nu_1(A):=\sum\nolimits_{n\in \mathbb{N}} \sigma_n(A) < \infty$$ or, equivalently[@ReedSimon1 Thm.VI.21], as the set of all $A \in \mathcal B(\mathcal H)$ such that $$\begin{aligned}
\label{def2_trace_eq}
\sum\nolimits_{i\in I}\langle e_i, \sqrt{A^\dagger A} e_i\rangle < \infty\end{aligned}$$ is summable for some orthonormal basis $(e_i)_{i \in I}$ of $\mathcal H$.
Because of $\sqrt{A^\dagger A}\geq 0$, all summands in are non-negative and therefore[@ReedSimon1 Thm.VI.18], the value of the left-hand side of is independent of the chosen orthonormal basis. Moreover, one has $\operatorname{tr}(\sqrt{A^\dagger A})=\nu_1(A)$ for all $A\in\mathcal B^1(\mathcal H)$ which readily implies $\operatorname{tr}(A)=\nu_1(A)$ if $A\geq 0$. Finally, we note that for finite-dimensional Hilbert spaces the sets $\mathcal B(\mathcal H)$ and $\mathcal B^1(\mathcal H)$ coincide with the set of all linear operators acting on $\mathcal H$ and that for arbitrary Hilbert spaces, $\mathcal B^1(\mathcal H)$ constitutes a Banach space with respect to the *trace norm* $\nu_1$ given by . For more on these topics we refer to [@ReedSimon1 Ch.VI.6] and [@MeiseVogt Ch.16].
An operator $\rho\in\mathcal B^1(\mathcal H)$ which is positive semi-definite and fulfills $\operatorname{tr}(\rho)=1$ is called a *state* and the set of all states is denoted by $$\begin{aligned}
\mathbb D(\mathcal H):=\lbrace \rho\in\mathcal B^1(\mathcal H)\,|\,\rho\text{ is state}\rbrace\,.\end{aligned}$$ A state $\rho$ is said to be *pure* if it has rank one. Certainly, the corresponding definitions apply to $\mathcal B^1(\mathcal G)$ and $\mathbb D(\mathcal G)$. After these preliminaries, we can introduce the key concepts.
\[def1\]
- A linear map $T:\mathcal B^1(\mathcal H)\to\mathcal B^1(\mathcal G)$ is said to be positive if $T(A)\geq 0$ for all positive semi-definite $A\in \mathcal B^1(\mathcal H)$.
- A linear map $T:\mathcal B^1(\mathcal H)\to\mathcal B^1(\mathcal G)$ is said to be completely positive if for all $n\in\mathbb N$ the maps $T\otimes\operatorname{id}_n: \mathcal B^1(\mathcal H \otimes \mathbb{C}^n)\to\mathcal B^1(\mathcal G \otimes \mathbb{C}^n)$ are positive.
- A Schrödinger quantum channel is a linear, completely positive and trace-preserving map $T:\mathcal B^1(\mathcal H)\to\mathcal B^1(\mathcal G)$. Furthermore, we define $$\begin{aligned}
Q_S (\mathcal H,\mathcal G):=
\lbrace T:\mathcal B^1(\mathcal H)\to\mathcal B^1(\mathcal G)\, |\, T\text{ is Schr\"odinger quantum channel}\,\rbrace\end{aligned}$$ and $Q_S (\mathcal H):= Q_S (\mathcal H,\mathcal H)$.
Note that Definition \[def1\] (a) and (b) also make sense for maps from $\mathcal B(\mathcal H) \to \mathcal B(\mathcal G)$ instead of $\mathcal B^1(\mathcal H)\to\mathcal B^1(\mathcal G)$.
Clearly, every positive, trace-preserving map and thus every Schrödinger quantum channel maps states to states. Further algebraic and topological properties of $Q_S (\mathcal H)$ which are crucial in the following are summarized in the following theorem, the proof of which can be found in Appendix \[app:monoid\].
\[thm\_monoid\] The set $Q_S (\mathcal H)$ is a convex subsemigroup of $\mathcal B(\mathcal B^1(\mathcal H))$ with unity element $\operatorname{id}_{\mathcal B^1(\mathcal H)}$. Moreover, $Q_S(\mathcal H)$ is closed in $\mathcal B(\mathcal B^1(\mathcal H))$ with respect to the weak operator, strong operator and uniform operator topology.
\[rem:boundedness\]
1. Here one should emphasize that it is not necessary to require boundedness of Schrödinger quantum channels. In fact, one can easily prove that any positive linear map is bounded automatically [@Davies Ch. 2, Lemma 2.1].
2. In Proposition \[thm\_q\_norm\_1\] we will see that $Q_S (\mathcal H)$ is actually a closed convex subset of the unit sphere of $\mathcal B(\mathcal B^1(\mathcal H))$. Note that the existence of non-trivial convex subsets on the unit sphere of $\mathcal B(\mathcal B^1(\mathcal H))$ is a consequence of the non-strict convexity of the operator norm.
The following beautiful and well-known representation result for Schrödinger quantum channels which can be traced back to Kraus is the starting of our work.
\[thm1\] For every $T\in Q_S(\mathcal H)$ there exists a separable Hilbert space $\mathcal K$, a pure $\omega\in\mathbb D(\mathcal K)$ and a unitary $U\in\mathcal B(\mathcal H\otimes\mathcal K)$ such that $$\begin{aligned}
\label{eq:T_stinespring}
T(A)=\operatorname{tr}_{\mathcal K}(U(A\otimes \omega)U^\dagger)\end{aligned}$$ for all $A\in\mathcal B^1(\mathcal H)$.
Here $\operatorname{tr}_{\mathcal K}:\mathcal B^1(\mathcal H\otimes\mathcal K)\to\mathcal B^1(\mathcal H)$ is the partial trace *with respect to the Hilbert space* $\mathcal K$ which is defined via $$\begin{aligned}
\label{eq:partial_trace_1}
\operatorname{tr}(B\operatorname{tr}_{\mathcal K}(A))=\operatorname{tr}((B\otimes\operatorname{id}_{\mathcal K})A)\end{aligned}$$ for all $B\in\mathcal B(\mathcal H)$ and all $A\in\mathcal B^1(\mathcal H\otimes\mathcal K)$.
For a complete proof of Theorem \[thm1\], see [@Kraus second part of Thm.2]. Here, we only emphasize that the separable auxiliary space $\mathcal K$ can be chosen independently of $T$; for instance, $\mathcal K := \ell_2(\mathbb N)$ constitutes such a universal auxiliary space. Moreover, once $\mathcal K$ is fixed, $\omega\in\mathbb D(\mathcal K)$ can be chosen as any orthogonal rank-$1$ projection. Thus $\omega$ is pure and independent of $T$, too.
\[coro\_gen\_stinespring\] For every $T\in Q_S(\mathcal H,\mathcal G)$ there exists a separable Hilbert space $\mathcal K$, pure $\omega_G\in\mathbb D(\mathcal G)$, $\omega_K\in\mathbb D(\mathcal K)$ and a unitary $U\in\mathcal B(\mathcal H\otimes\mathcal G\otimes\mathcal K)$ such that $$\begin{aligned}
T(A)=(\operatorname{tr}_{\mathcal H}\circ\operatorname{tr}_{\mathcal K})(U(A\otimes \omega_G\otimes\omega_K)U^\dagger)\end{aligned}$$ for all $A\in\mathcal B^1(\mathcal H)$.
Consider arbitrary $\omega_G\in\mathbb D(\mathcal G)$ and $\omega_H\in\mathbb D(\mathcal H)$ of rank one. Applying Theorem \[thm1\] to $X(\cdot):=\omega_H\otimes T(\operatorname{tr}_{\mathcal G}(\cdot))\in Q_S(\mathcal H\otimes\mathcal G)$ which is obviously a composition of Schrödinger quantum channels yields a separable Hilbert space $\mathcal K$, a pure $\omega_K\in\mathbb D(\mathcal K)$ and a unitary $U\in\mathcal B(\mathcal H\otimes\mathcal G\otimes\mathcal K)$ such that $X$ is of form . For any $A\in\mathcal B^1(\mathcal H)$ one gets $$\begin{aligned}
T(A)=\operatorname{tr}_{\mathcal H}(X(A\otimes\omega_G))=(\operatorname{tr}_{\mathcal H}\circ\operatorname{tr}_{\mathcal K})(U(A\otimes \omega_G\otimes\omega_K)U^\dagger)\end{aligned}$$ with $\omega_G\otimes\omega_K\in\mathbb D(\mathcal G\otimes\mathcal K)$ rank one.
The last result of this subsection provides a characterization of invertible quantum channels which leads to a nice simplification later on (cf. Remark \[ch\_4\_bem\_3\]). For finite dimensions, this was essentially shown in [@wolf08_1 Coro. 3].
\[ch\_3\_Theorem\_14\] Let $T\in Q_S(\mathcal H)$ be bijective. Then the following statements are equivalent.
- $T^{-1}$ is positive.
- There exists unitary $U\in\mathcal B(\mathcal H)$ such that $T(A)=U AU^\dagger$ for all $A\in\mathcal B^1(\mathcal H)$.
In particular if one (and thus both) conditions are fulfilled, then $T\in Q_S(\mathcal H)$ is invertible as a channel, i.e. $T$ is bijective and $T^{-1}\in Q_S(\mathcal H)$.
(b)$\,\Rightarrow\,$(a): $\checkmark$ (a)$\,\Rightarrow\,$(b): The proof idea is the same as in [@Heinosaari Prop. 4.31]. Consider the restricted channel $T|_{\mathbb D}:\mathbb D(\mathcal H)\to\mathcal B^1(\mathcal H)$ which by assumption is convex-linear and injective. As $T$ and $T^{-1}$ are linear, trace-preserving and, by assumption, positive, the restricted channel satisfies $$T(\mathbb D(\mathcal H))\subseteq \mathbb D(\mathcal H)
\qquad
T^{-1}(\mathbb D(\mathcal H))\subseteq \mathbb D(\mathcal H)\,,$$ so $T|_{\mathbb D}:\mathbb D(\mathcal H)\to \mathbb D(\mathcal H)$ is surjective and thus a state automorphism, i.e. convex-linear and bijective. Then Corollary 3.2 in [@Davies] or, more explicitely, Theorem 2.63 in [@Heinosaari] imply the existence of unitary or anti-unitary $U$ such that $T|_{\mathbb D}(\cdot)=U(\cdot)U^\dagger$. If $U$ were anti-unitary, then $T$ would not be completely positive[@Heinosaari Prop. 4.14] hence $U$ has to be unitary. Due to $\operatorname{span}_{\mathbb C}(\mathbb D(\mathcal H))=\mathcal B^1(\mathcal H)$, this representation extends linearily to all of $\mathcal B^1(\mathcal H)$ which concludes the proof.
Dual Channels {#sec:dualchannel}
-------------
It is well known[@MeiseVogt Prop.16.26] that the dual space of $\mathcal B^1(\mathcal H)$ is isometrically isomorphic to $\mathcal B(\mathcal H)$ by means of the map $\psi_{\mathcal H}:\mathcal B(\mathcal H)\to (\mathcal B^1(\mathcal H))'$, $B\mapsto\psi_{\mathcal H}(B)$ with $$\begin{aligned}
(\psi_{\mathcal H}(B))(A):=\operatorname{tr}(BA)\end{aligned}$$ for all $A\in\mathcal B^1(\mathcal H)$. Note that the weak-$*$-topology and the ultraweak topology on $\mathcal B(\mathcal H)$ coincide under the above identification $(\mathcal B^1(\mathcal H))' \cong \mathcal B(\mathcal H)$, cf. [@Davies Section 1.6].
Now, since every positive linear map $T:\mathcal B^1(\mathcal H)\to\mathcal B^1(\mathcal G)$ is bounded (cf. Remark \[rem:boundedness\].1) the dual map $$\begin{aligned}
T':(\mathcal B^1(\mathcal G))'\to(\mathcal B^1(\mathcal H))'\qquad X\mapsto T'(X):=X\circ T\end{aligned}$$ is well defined and this allows us to construct the so called *dual channel* of $T$ $$\begin{aligned}
T^*:\mathcal B(\mathcal G)\to\mathcal B(\mathcal H)\qquad B\mapsto T^*(B):=(\psi_{\mathcal H}^{-1}\circ T' \circ\psi_{\mathcal G})(B)\end{aligned}$$ which then satisfies $$\begin{aligned}
\label{eq:dual_channel}
\operatorname{tr}(BT(A))=\operatorname{tr}(T^*(B)A)\end{aligned}$$ for all $B\in\mathcal B(\mathcal G)$ and $A\in\mathcal B^1(\mathcal H)$. Alternatively, onc can use (\[eq:dual\_channel\]) as defining equation for $T^*$. Furthermore, one has $\Vert T\Vert=\Vert T^*\Vert$ by definition of $T^*$, because $T$ and $T'$ have the same operator norm and $\psi_{\mathcal G}$ and $\psi_{\mathcal H}$ are isometric isomorphisms. Some basic properties of $T^*$ are:
- $T^*$ is positive and ultraweakly continuous.
- $T^*$ is completely positive if and only if $T$ is completely positive.
- $T^*$ is unital (i.e. $T^*(\operatorname{id}_{\mathcal G})=\operatorname{id}_{\mathcal H}$) if and only if $T$ is trace-preserving.
For more details and proofs we refer to [@Kraus p. 35] or [@Heinosaari Ch.4.1.2].
\[defi\_heisenberg\_qc\] A Heisenberg quantum channel is a linear, ultraweakly continuous[@Note2a], completely positive and unital map $S:\mathcal B(\mathcal G)\to\mathcal B(\mathcal H)$. Furthermore, we define $$\begin{aligned}
Q_H(\mathcal G,\mathcal H):=\lbrace S:\mathcal B(\mathcal G)\to\mathcal B(\mathcal H)\, |\, S\text{ is Heisenberg quantum channel}\,\rbrace\end{aligned}$$ and $Q_H (\mathcal H):= Q_H (\mathcal H,\mathcal H)$.
By the properties listed above, it is evident that the map $*:Q_S(\mathcal H,\mathcal G)\to Q_H(\mathcal G,\mathcal H)$ which to any quantum channel assigns its dual channel is well-defined. Furthermore, it is—as we will see next—bijective.
\[thm\_dual\]
- For every $S:\mathcal B(\mathcal G)\to\mathcal B(\mathcal H)$ linear, ultraweakly continuous and positive there exists unique $T:\mathcal B^1(\mathcal H)\to\mathcal B^1(\mathcal G)$ linear and positive such that $T^*=S$.
- For every $S\in Q_H(\mathcal G,\mathcal H)$ there exists unique $T\in Q_S(\mathcal H,\mathcal G)$ such that $T^*=S$.
\(a) By the above construction of the dual channel it is obvious that the map $*$ is one-to-one. Therefore, it suffices to show its surjectivity.
First one shows, similar to [@Davies Ch. 1, Lemma 6.1], that for every positive, linear and ultraweakly continuous functional $\lambda:\mathcal B(\mathcal G)\to\mathbb C$ there exists a unique positive semi-definite $\rho\in\mathcal B^1(\mathcal G)$ such that $\lambda(\cdot)=\operatorname{tr}(\rho(\cdot))$. Next choose arbitrary positive semi-definite $A \in \mathcal B^1(\mathcal H)$ and consider the linear functional $$B \mapsto \operatorname{tr}(S(B)A) % =\operatorname{tr}(B\sigma_A)$$ which by assumption on $S$ is ultraweakly continuous. Our preliminary consideration yields a unique positive semi-definite $\rho_A\in\mathcal B^1(\mathcal G)$ such that $\operatorname{tr}(S(B)A)=\operatorname{tr}(B\rho_A)$ for all $B\in\mathcal B(\mathcal G)$. This allows to define an $\mathbb R^+$-linear map $\hat T$ on the positive semi-definite elements of $\mathcal B^1(\mathcal H)$ via $\hat T(A):=\rho_A$. Finally, $\hat T(A)$ can be uniquely extended to a positive, linear map $T:\mathcal B^1(\mathcal H)\to\mathcal B^1(\mathcal G)$ satisfying $T^*=S$.
Now (b) follows from (a) together with the above connections between properties of a positive, linear map and its dual channel.
1. Note that, again, boundedness is not required in the definition of a Heisenberg quantum channel because, similar to Schrödunger quantum channels, they are automatically bounded, see Proposition \[thm\_q\_norm\_1\] below.
2. In finite dimensions, ultraweak continuity is of course always satisfied and the $*$-map is an involution as the sets of trace-class operators and bounded operators coincide.
\[thm\_q\_norm\_1\] Let $T\in Q_S(\mathcal H,\mathcal G)$ and $S\in Q_H(\mathcal G,\mathcal H)$. Then $\|T\|=1$ and $\|S\|=1$.
As each $S\in Q_H(\mathcal G,\mathcal H) $ in particular is linear, positive and unital it has operator norm $\|S\|=1$ as a consequence of the Russo-Dye Theorem, cf. [@russo1966 Cor. 1] or Rem. \[rem\_19\].1. This directly implies $\|T\|=\|T^*\|=1$.
Alternatively, one can prove Proposition \[thm\_q\_norm\_1\] via the general Stinespring dilation (Corollary \[coro\_gen\_stinespring\]) because all maps involved in the Stinespring representation have operator norm one. Either way, with this one readily verifies that $Q_H(\mathcal H)$ forms a convex subsemigroup of the Banach space $\mathcal B(\mathcal B(\mathcal H))$ with unity element $\operatorname{id}_{\mathcal B(\mathcal H)}$.
The partial trace $\operatorname{tr}_{\omega}:\mathcal B(\mathcal H\otimes\mathcal K)\to\mathcal B(\mathcal H)$ *with respect to a state* $\omega\in\mathbb D(\mathcal K)$ is defined via $$\begin{aligned}
\label{eq:partial_trace_2}
\operatorname{tr}(\operatorname{tr}_{\omega}(B)A)=\operatorname{tr}(B(A\otimes\omega))\end{aligned}$$ for all $B\in\mathcal B(\mathcal H\otimes\mathcal K)$, $A\in\mathcal B^1(\mathcal H)$, cf. [@Davies Ch. 9, Lemma 1.1]. Be aware that the map $\operatorname{tr}_{\mathcal K}$ from (\[eq:partial\_trace\_1\]) and the extension $$i_\omega:\mathcal B^1(\mathcal H)\to \mathcal B^1(\mathcal H\otimes\mathcal K)\qquad A\mapsto A\otimes\omega$$ with some state $\omega\in\mathbb D(\mathcal K)$ are Schrödinger quantum channels so we immediatly get their dual channels $i_{\omega}^*=\operatorname{tr}_{\omega}$ and $\operatorname{tr}_{\mathcal K}^*=i_{\mathcal K}$ with $$\begin{aligned}
i_{\mathcal K}:\mathcal B(\mathcal H)\to\mathcal B(\mathcal H\otimes\mathcal K)
\qquad B\mapsto B\otimes\operatorname{id}_{\mathcal K}\,.\end{aligned}$$
This leads to the following result.
\[coro\_1\] For every $S\in Q_H(\mathcal G,\mathcal H)$ there exists a separable Hilbert space $\mathcal K$, pure states $\omega_G\in\mathbb D(\mathcal G)$ and $\omega_K\in\mathbb D(\mathcal K)$ and a unitary $U\in\mathcal B(\mathcal H\otimes\mathcal G\otimes\mathcal K)$ such that $$\begin{aligned}
S(B)=(\operatorname{tr}_{\omega_G}\circ\operatorname{tr}_{\omega_K})(U^\dagger(\operatorname{id}_{\mathcal H}\otimes B\otimes \operatorname{id}_{\mathcal K})U)\end{aligned}$$ for all $B\in\mathcal B(\mathcal G)$. For $\mathcal G=\mathcal H$ this reduces to $$\begin{aligned}
\label{eq:stinespring_q_h}
S(B)=\operatorname{tr}_{\omega_K}(U^\dagger(B\otimes \operatorname{id}_{\mathcal K})U)\end{aligned}$$ for all $B\in\mathcal B(\mathcal H)$ where the unitary operator $U$ now acts on $\mathcal H\otimes\mathcal K$.
Note that (\[eq:dual\_channel\]) implies $(T_1\circ T_2)^*=T_2^*\circ T_1^*$ for arbitrary positive, linear maps $T_1$ and $T_2$. Hence this is a simple consequence of Theorem \[thm1\], Theorem \[thm\_dual\] (b) and Corollary \[coro\_gen\_stinespring\].
The result in Corollary \[coro\_1\] is a more structured version of Stinespring’s theorem [@Stinespring] for Heisenberg quantum channels due to the following: Let $S\in Q_H(\mathcal H)$ (the same argument works for $S\in Q_H(\mathcal G,\mathcal H)$) and $\omega_K\in\mathbb D(\mathcal K)$ be the state from of rank one, i.e. $\omega_K=\langle y,\cdot\rangle y$ for some $y\in\mathcal K$ with $\|y\|=1$. Defining the isometric embedding $V_y:\mathcal H\to\mathcal H\otimes\mathcal K$, $x\mapsto x\otimes y$, one readily verifies via that $\operatorname{tr}_{\omega_K}(B)=V_y^\dagger BV_y$ for all $B\in\mathcal B(\mathcal H\otimes\mathcal K)$. Now becomes $$S(\cdot)=V^\dagger\pi(\cdot) V$$ with the auxiliary Hilbert space $\mathcal H\otimes\mathcal K$ being of tensor form, the Stinespring isometry $V = UV_y$ and the unital $*$-homomorphism $\pi:\mathcal B(\mathcal H)\to\mathcal B(\mathcal H\otimes\mathcal K) $ being $\pi(B):= B\otimes \operatorname{id}_{\mathcal K}$. To the best of our knowledge, the above representation so far only appeared in an unpublished (as of now) book by S. Attal[@Attal6 Thm. 6.15].
The above concept of dual channels will be useful to transfer dilation results from the Schrödinger to the Heisenberg picture and vice versa so one is independent of the used quantum-mechanical framework.
Main Results {#sec:main_results}
============
Unitary Dilation of Discrete-Time Quantum-Dynamical Systems {#sect:unit_dil_A}
-----------------------------------------------------------
Consider a discrete-time quantum-dynamical system, the evolution of which is described by $$\begin{aligned}
\label{eq:DQDS}
\rho_{n+1}= T(\rho_n), \quad \rho_0\in \mathbb D(\mathcal H)\end{aligned}$$ for arbitrary but fix $T\in Q_S(\mathcal H)$. Obviously, the explicit solution of is given by $$\begin{aligned}
\rho_{n}= T^n(\rho_0)\end{aligned}$$ for all $n \in \mathbb N_0$ By Theorem \[thm\_monoid\], one has $T^n\in Q_S(\mathcal H)$ and thus Theorem \[thm1\] yields separable Hilbert spaces $\mathcal K_n$, pure states $\omega_n \in\mathbb D(\mathcal K_n)$ and unitaries $U_n\in \mathcal B(\mathcal H\otimes\mathcal K_n )$ such that $$\begin{aligned}
\label{eq:3_0}
T^n(A)=\operatorname{tr}_{\mathcal K_n}\left( U_n(A\otimes\omega_n )U_n^\dagger \right)\end{aligned}$$ for all $A\in \mathcal B^1(\mathcal H)$ and all $n\in\mathbb N_0$. Now our goal is to simplify the right-hand side of (\[eq:3\_0\]) in the following sense: We want to embed the evolution of $\rho_0$ into an evolution of a closed discrete-time quantum-dynamical system, i.e. we want to replace the r.h.s. of (\[eq:3\_0\]) by $$\begin{aligned}
\operatorname{tr}_{\tilde{\mathcal K}}\left( V^n(A\otimes\tilde\omega)(V^\dagger)^n \right)\end{aligned}$$ where $V$ is an appropriate unitary operator and the separable Hilbert space $\tilde{\mathcal K}$ as well as the pure state $\tilde\omega$ does no longer depend on $n \in \mathbb N_0$. Our established result reads as follows
\[ch\_4\_satz\_1\] For every $ T\in Q_S(\mathcal H)$ there exists a separable Hilbert space $\mathcal K$, a pure state $\omega\in\mathbb D(\mathcal K)$ and a unitary $V\in\mathcal B(\mathcal H\otimes\mathcal K)$ such that $(\mathcal H\otimes\mathcal{K},(V^n)_{n \in \mathbb Z}, i_\omega, \operatorname{tr}_{\mathcal K})$ is a unitary dilation of $(T^n)_{n\in\mathbb N_0}$ (in the sense of Definition \[def:dilation-Schroedinger-channles\].2). In particular, for all $A\in\mathcal B^1(\mathcal H)$ and $n\in\mathbb N_0$, one has $$\begin{aligned}
\label{eq:3-00}
T^n(A)=\operatorname{tr}_{\mathcal K}\left( V^n(A\otimes\omega)(V^\dagger)^n \right)\,.\end{aligned}$$
First we consider the $n$-dependence of ${\mathcal K}_n$ and $\omega_n$. By construction, cf. Theorem \[thm1\], ${\mathcal K}_n$ does not depend on $T^n$ anymore, thus we can choose $\tilde{\mathcal K}$ with a countably infinite basis, for example $\tilde{\mathcal K}=\ell_2(\mathbb N)$, and replace every ${\mathcal K}_n$ with $\tilde{\mathcal K}$. Moreover, also by construction, the pure state $\omega_n$ is determined via ${\mathcal K}_n$ and thus can be chosen independently of $n$, too. Hence we obtain a joint Hilbert space $\tilde{\mathcal K}$ and a pure state $\tilde\omega$ such that $$\begin{aligned}
T^n( A)=\operatorname{tr}_{\tilde{\mathcal K}}\left( U_n( A\otimes\tilde\omega )U_n^\dagger \right).\end{aligned}$$ for all $n\in\mathbb N_0$. Finally, in order to remove the $n$-dependence of the unitary operators $U_n$ we define $\mathcal K:=\tilde{\mathcal K}\otimes\ell_2(\mathbb Z)$ and $U_n=\operatorname{id}_{\mathcal H\otimes\tilde{\mathcal K} }$ for all $n\leq 0$. Furthermore, let $( e_n)_{n\in\mathbb Z}$ denote the standard basis of $\ell_2(\mathbb Z)$ so $\sigma: \ell_2(\mathbb Z) \to \ell_2(\mathbb Z)$ given by $\sigma=\sum_{i\in\mathbb Z}e_ie_{i-1}^\dagger$ yields the right shift on $\ell_2(\mathbb Z)$. With this $U,W:\mathcal B(\mathcal H\otimes\mathcal K)\to\mathcal B(\mathcal H\otimes\mathcal K)$ are defined by $$\begin{aligned}
{U}:=\sum\nolimits_{n\in\mathbb Z} U_nU_{n-1}^\dagger\otimes e_ne_n^\dagger
\qquad\text{and}\qquad
{W}:= \operatorname{id}_{\mathcal H\otimes\tilde{\mathcal K} }\otimes\,\sigma\,.\end{aligned}$$ Thus $U$ can be visualised as follows: $$\begin{aligned}
\underset{\hspace*{-65pt}\begin{matrix}\uparrow & &\end{matrix}}
{\begin{pmatrix}
\ddots &&&&& \\&\operatorname{id}_{\mathcal H\otimes\tilde{\mathcal K} }&&&& \\&&U_1&&& \\&&&U_2U_1^\dagger&& \\&&&&U_3U_2^\dagger& \\&&&&&\ddots
\end{pmatrix}}
\begin{matrix}
\\\leftarrow\\\\\\\\\\
\end{matrix}\,,\end{aligned}$$ where the arrows indicate the zero-zero entry of this both-sided “infinite matrix”. A simple calculation shows that $U$, $W$ and therefore also $V:=UW$ are unitary. Next, using the results from Section \[sec:dualchannel\], one readily verifies that the maps $E:=\operatorname{tr}_{\mathcal K}$ and $J:=i_\omega$ (where $\omega:=\tilde\omega \otimes e_0e_0^\dagger\in\mathbb D(\mathcal K)$ is obviously pure) satisfy the conditions from Definition \[def:dilation-Schroedinger-channles\].1. Then, by induction, one shows $$V^n (A\otimes\omega) (V^\dagger)^n=U_n( A\otimes\tilde\omega ) U_n^\dagger\otimes e_ne_n^\dagger$$ for all $A \in \mathcal B^1(\mathcal H)$ and $n\in\mathbb N_0$. Finally, $\operatorname{id}_{\tilde{\mathcal K}\otimes\ell_2}=\operatorname{id}_{\tilde{\mathcal K}}\otimes\operatorname{id}_{\ell_2}$ implies $\operatorname{tr}_{\tilde{\mathcal K}\otimes\ell_2}=\operatorname{tr}_{\tilde{\mathcal K}}\circ\operatorname{tr}_{\ell_2}$ so $$\begin{aligned}
\operatorname{tr}_{\mathcal K}\left( V^n(A\otimes\omega)(V^\dagger)^n \right)
= \operatorname{tr}_{\tilde{\mathcal K}}(\operatorname{tr}_{\ell_2}(U_n( A\otimes\tilde\omega ) U_n^\dagger\otimes e_ne_n^\dagger))
= \operatorname{tr}_{\tilde{\mathcal K}}( U_n( A\otimes\tilde\omega )U_n^\dagger ) = T^n( A)\end{aligned}$$ for all $A \in \mathcal B^1(\mathcal H)$ and $n\in\mathbb N_0$. Hence we constructed a unitary dilation of $(T^n)_{n\in\mathbb N_0}$ of the form which concludes the proof.
Note that $i_\omega$ is trace-preserving because $\omega\in\mathbb D(\mathcal K)$, so the above (tensor type) dilation is trace-preserving.
Now we can easily extend this result to Heisenberg quantum channels.
\[coro\_Q\_H\_1\] For every $S\in Q_H (\mathcal H)$ there exists a separable Hilbert space $\mathcal K$, a pure state $\omega\in\mathbb D(\mathcal K)$ and a unitary $V\in\mathcal B(\mathcal H\otimes\mathcal K)$ such that $(\mathcal H\otimes\mathcal{K},((V^\dagger)^n)_{n \in \mathbb Z}, i_{\mathcal K}, \operatorname{tr}_{\omega})$ is a unitary dilation of $(S^n)_{n\in\mathbb N_0}$ (in the sense of Definition \[def:dilation-Heisenberg-channles\].2). In particular, for all $B\in\mathcal B(\mathcal H)$ and $n\in\mathbb N_0$, one has $$\begin{aligned}
S^n(B)= \operatorname{tr}_\omega\left( (V^\dagger)^n(B\otimes\operatorname{id}_{\mathcal K})V^n \right).\end{aligned}$$
By Theorem \[thm\_dual\] (b) there exists a unique $T\in Q_S(\mathcal H)$ such that $S=T^*$ and therefore $S^n=(T^*)^n=(T^n)^*$. Now Theorem \[ch\_4\_satz\_1\] yields a separable Hilbert space $\mathcal K$, a pure state $\omega\in\mathbb D(\mathcal K)$ and a unitary $V$ such that (\[eq:3-00\]) holds. By duality we obtain $$\begin{aligned}
S^n(B) = \Big(\operatorname{tr}_{\mathcal K}\left( V^n i_\omega(\cdot) (V^\dagger)^n \right)\Big)^*(B)
=\operatorname{tr}_\omega\left( (V^\dagger)^n(B\otimes\operatorname{id}_{\mathcal K})V^n \right)\end{aligned}$$ for all $B \in \mathcal B(\mathcal H)$ and for all $n\in\mathbb N_0$.
\[rem\_subspace\]
1. Due to $\operatorname{id}_{\mathcal H}\otimes \operatorname{id}_{\mathcal K} = \operatorname{id}_{\mathcal H\otimes\mathcal K}$ we even constructed a unital dilation (of tensor type).
2. Recall that a “classical” unitary dilation $T^n = P_{\mathcal H} \circ U^n \circ \operatorname{inc}_{\mathcal H}$ of some Hilbert space contraction $T: \mathcal H \to \mathcal H$ (cf. Rem. \[rem:classical-dilations\].3), where $P_{\mathcal H}$ denotes the orthogonal projection onto $\mathcal H$ and $\operatorname{inc}_{\mathcal H}$ the inclusion map, is called *minimal* if the domain of $U\in\mathcal B( \mathcal K )$ is minimal in the sense of $$\label{eq1:classical_dilation}
\mathcal K = \bigvee_{n \in \mathbb Z} U^n \mathcal H\,.$$ Here the right-hand side of denotes the smallest closed subspace of $\mathcal K$ which contains all images $U^n \mathcal H$, $n \in \mathbb Z$, cf. [@NagyFoias1970; @FoiasFrazho1990]. Kümmerer[@Kuemmerer83] captures this idea and defines a dilation $T^n(A) = E\big(\hat{T}^n(J(A))\big)$ of an ultraweakly continuous, completely positive and unital map $T:\mathcal A \to \mathcal A$ on a $W^*$-algebra $\mathcal A$ to be minimal if $$\label{eq2:classical_dilation}
\mathcal A = \bigvee_{n \in \mathbb Z} \hat{T}^n (i(\mathcal A))$$ holds, where the right-hand side of now denotes the smallest closed $W^*$-algebra which contains all images $\hat{T}^n (i(\mathcal A))$, $n \in \mathbb Z$, cf. [@Kuemmerer83 Def. 2.1.5]. It is easy to see that our constructions in Theorem \[ch\_4\_satz\_1\] / Corollary \[coro\_Q\_H\_1\] do in general not lead to a minimal dilation in the above sense. However, one can always restrict a given dilation to the right-hand side of to obtain a minimal one.
3. As seen above in the space $\mathcal H^{\infty}_{-\infty} := \bigvee_{n \in \mathbb Z} U^{-n} \mathcal H$ and its forward and backward invariant counterparts $$% \mathcal H^{\infty}_{-\infty} := \bigvee_{n \in \mathbb Z} U^{-n} \mathcal H\,,
% \quad\text{and}\quad
\mathcal H^{\infty} := \bigvee_{n \in \mathbb N_0} U^n \mathcal H\,,
\quad\text{and}\quad
\mathcal H_{-\infty} := \bigvee_{n \in \mathbb N_0} U^{-n} \mathcal H\,,$$ play an essential role in the theory of “classical” unitary dilations. In particular, they admit orthogonal decompositions $$\label{eq4:classical_dilation}
\mathcal H^{\infty} = \mathcal H \oplus \hat{\mathcal H}^{\infty}\,,
\quad \mathcal H_{\infty} = \mathcal H \oplus \hat{\mathcal H}_{-\infty}
\quad\text{and}\quad
\mathcal H^{\infty}_{-\infty} = \hat{\mathcal H}^{\infty} \oplus \mathcal H \oplus \hat{\mathcal H}_{-\infty}$$ such that $\hat{\mathcal H}^{\infty}$ and $\hat{\mathcal H}_{-\infty}$ are invariant under $U$ and $U^{-1}$, respectively, cf. [@FoiasFrazho1990 Lemma VI.3.1] and [@Note3]. Eventually, establishes the relation to Kümmerer’s notion of Markovianity, cf. [@Kuemmerer83 Prop. 2.2.3 (b)].
Next we want to improve Theorem \[ch\_4\_satz\_1\] for cyclic $T$, i.e. in the case of $T^m= T$ for some $m\in\mathbb N\setminus\lbrace 1\rbrace$.
\[ch\_4\_def\_1\] In doing so, we define a modified modulo function $$\begin{aligned}
\nu:\mathbb N\setminus\lbrace 1\rbrace\times\mathbb N&\to\mathbb N\\
(m,n)&\mapsto (n-1)\operatorname{mod}(m-1)+1\end{aligned}$$ as well as $$\begin{aligned}
\mu:\mathbb N\setminus\lbrace 1\rbrace\times\mathbb N&\to\mathbb N_0\\
(m,n)&\mapsto \frac{n-\nu(m,n)}{m-1}\,.\end{aligned}$$
To connect $\nu(m,n)$ to the above cyclicity condition of $T$ we represent $n-1$ as $$\begin{aligned}
\label{eq:ch_4_bem_1_0}
n-1=j(m-1)+r\end{aligned}$$ with unique $j\in\mathbb N_0$ and $r\in\lbrace 0,\ldots,m-2\rbrace$. This yields $\nu(m,n)=r+1$ as well as $\mu(m,n)=j\in\mathbb N_0$ and we obtain the following result.
\[ch\_4\_lemma\_2\] Let $ T\in Q_S(\mathcal H)$ be cyclic so $ T^m= T$ for $m\in\mathbb N\setminus\lbrace 1\rbrace$. Then $$\begin{aligned}
T^n= T^{\nu(m,n)}\end{aligned}$$ for all $n\in\mathbb N$.
Via (\[eq:ch\_4\_bem\_1\_0\]) we get $T^n=T^{j(m-1)+r+1}=T^{r+1-j}(T^m)^j=T^{r+1-j}T^j=T^{r+1}=T^{\nu(m,n)}$.
Thus $\mu(m,n)$ indicates how often the cyclicity condition of $T$ can be applied to reduce the exponent $n$ to its remaining non-cyclic portion $\nu(m,n)$. With this we obtain the following simplification of Theorem \[ch\_4\_satz\_1\].
\[ch\_4\_koro\_1\] Let $ T\in Q_S(\mathcal H)$ be cyclic, i.e. $ T^m= T$ for some $m\in\mathbb N\setminus\lbrace 1\rbrace$. Then for the unitary dilation $(\mathcal H\otimes\mathcal{K},(V^n)_{n \in \mathbb Z}, i_\omega, \operatorname{tr}_{\mathcal K})$ of $(T^n)_{n\in\mathbb N_0}$ from Theorem \[ch\_4\_satz\_1\], one can choose $\mathcal K=\tilde{\mathcal K}\otimes\mathbb C^m$ such that (after modifying $V$ and $\omega$ accordingly) $$\begin{aligned}
T^n( A)=\operatorname{tr}_{\mathcal K}\left( V^{n+\mu(m,n)}(A\otimes\omega)(V^\dagger)^{n+\mu(m,n)} \right)\end{aligned}$$ for all $A\in\mathcal B^1(\mathcal H)$ and all $n\in\mathbb N_0$. Note that $\omega\in\mathbb D(\mathcal K)$ still is a pure state.
Choose $\tilde{\mathcal K}$ and $\tilde\omega\in\mathbb D(\tilde{\mathcal K})$ as in the proof of Theorem \[ch\_4\_satz\_1\]. For every $ T,\ldots, T^{m-1}$ there again exist unitary $U_1,\ldots,U_{m-1} \in \mathcal B(\mathcal H\otimes\tilde{\mathcal K})$ satisfying Theorem \[thm1\]. This allows to define $$\begin{aligned}
{U}:=\sum\nolimits_{i=1}^m U_iU_{i-1}^\dagger\otimes e_ie_i^\dagger
\quad\text{and}\quad
{W}:=\operatorname{id}_{\mathcal H\otimes\tilde{\mathcal K}}\otimes \sum\nolimits_{i=1}^{m} e_{i+1}e_{i}^\dagger\end{aligned}$$ where $e_{m+1}:=e_1$ and $U_0:=\operatorname{id}_{\mathcal H\otimes\tilde{\mathcal K}}=:U_m$. Then ${W}$ represents a cyclic shift acting on $\mathbb C^m$ and $U$ is of the following form. $$\begin{aligned}
{U}=\begin{pmatrix} U_1&&&& \\ &U_2U_1^\dagger&&& \\&&\ddots&&\\&&&U_{m-1}U_{m-2}^\dagger&\\&&&&U_{m-1}^\dagger \end{pmatrix}\end{aligned}$$ Obviously, $U$, ${W}$ and thus $V:=UW$ are unitary. Again choosing $E:=\operatorname{tr}_{\mathcal K}$ and $J:=i_\omega$ with pure state $\omega:=\tilde\omega\otimes e_me_m^\dagger\in\mathbb D(\mathcal K)$, one readily verifies via indiction $$\begin{aligned}
V^{n+\mu(m,n)} (A\otimes\omega)(V^\dagger)^{n+\mu(m,n)}
=U_{\nu(m,n)}( A\otimes\omega ) U_{\nu(m,n)}^\dagger\otimes e_{\nu(m,n)}e_{\nu(m,n)}^\dagger\,.\end{aligned}$$ for all $ A\in\mathcal B^1(\mathcal H)$ and $n\in\mathbb N$. Together with Lemma \[ch\_4\_lemma\_2\] one gets $$\begin{aligned}
\operatorname{tr}_{\mathcal K}\left( V^{n+\mu(m,n)}(A\otimes\omega)(V^\dagger)^{n+\mu(m,n)} \right)
&=\operatorname{tr}_{\tilde{\mathcal K}}\circ\operatorname{tr}_{\mathbb C^m}\left( U_{\nu(m,n)}( A\otimes\tilde\omega ) U_{\nu(m,n)}^\dagger\otimes e_{\nu(m,n)}e_{\nu(m,n)}^\dagger \right)\\
&=\operatorname{tr}_{\tilde{\mathcal K}}\left( U_{\nu(m,n)}( A\otimes\tilde\omega ) U_{\nu(m,n)}^\dagger\right)
= T^{\nu(m,n)}( A)= T^n( A)\end{aligned}$$ for all $A\in\mathcal B^1(\mathcal H)$ and $n\in\mathbb N$.
\[ch\_4\_bem\_3\] Note that quantum channels which have an inverse channel (or are “just” bijective with positive inverse) can be written as a unitary conjugation $\operatorname{Ad}_U$, cf. Prop. \[ch\_3\_Theorem\_14\]. For such channels, Theorem \[ch\_4\_satz\_1\] is trivially fulfilled by choosing $\mathcal K=\mathbb C$, $E=J=\operatorname{id}_{\mathcal B^1(\mathcal H)}$ and $V=U$. The same holds for cyclic quantum channels which are bijective because cyclicity implies $T^{-1}=T^{m-2}\in Q_S(\mathcal H)$.
Unitary Dilation of Discrete-Time Quantum-Control Systems {#section_control}
---------------------------------------------------------
Here, we investigate discrete-time quantum-mechanical control systems of the form $$\begin{aligned}
\label{eq:chap:3_2_1}
\rho_{n+1}= T_n(\rho_n), \quad \rho_0\in \mathbb D(\mathcal H)\end{aligned}$$ where $T_n$, $n\in\mathbb N_0$ is regarded as control input which can be chosen freely from some subset $\mathcal C \subset Q_S(\mathcal H)$. We define $\rho(\cdot,(T_n)_{n\in\mathbb N_0},\rho_0)$ to be the unique solution of (\[eq:chap:3\_2\_1\]) generated by the control sequence $(T_n)_{n\in\mathbb N_0}$ and the initial value $\rho_0$. In the sequel, we are interested in whether the dynamics of can be embedded in the dynamics of a unitary discrete-time quantum control system of the same form.
\[ch\_4\_defi\_1\] Let $R_N(\rho_0)$ denote the set of all states which can be reached from $\rho_0$ in $N$ time steps via (\[eq:chap:3\_2\_1\]), i.e. $$\begin{aligned}
R_N(\rho_0):=\lbrace \rho(N,(T_n)_{n\in\mathbb N_0},\rho_0)\,|\,(T_n)_{n\in\mathbb N_0}\text{ arbitrary control sequence}\rbrace\,.\end{aligned}$$ Moreover, the overall reachable set of $\rho_0$ is defined by $$\begin{aligned}
R(\rho_0):=\bigcup\nolimits_{N\in\mathbb N_0}R_N(\rho_0).\end{aligned}$$
For the remaining section, we assume $\mathcal C:=\lbrace T,S\rbrace$ where $T$ and $S$ are commuting but otherwise arbitrary quantum channels over $\mathcal H$. Then the following result is a direct consequence of the fact that $T$ and $S$ commute.
\[ch\_4\_lemma\_3\] For all $N\in\mathbb N_0$ one has $R_N(\rho_0):=\lbrace T^kS^{N-k}\rho_0\,|\,k = 0,\ldots,N \rbrace$.
Based on this we are interested in dilations of quantum channels of the form $T^kS^{N-k}$.
\[ch\_4\_satz\_2\] Let $T,S\in Q_S(\mathcal H)$ be commuting. Then there exists a separable Hilbert space $\mathcal K$, a pure state $\omega\in\mathbb D(\mathcal K)$ and unitary $U,V\in\mathcal B(\mathcal H\otimes\mathcal K)$ such that $$\begin{aligned}
T^kS^{N-k}( A)=\operatorname{tr}_{\mathcal K}\left( {U}^{k}{V}^{{N-k}}(A\otimes\omega)({V}^\dagger)^{{N-k}}({U}^{\dagger})^{k} \right)\,.\end{aligned}$$ for all $ A\in\mathcal B^1(\mathcal H)$, $N\in\mathbb N_0$ and $k = 0,\ldots,N$.
For fixed $N\in\mathbb N$ and $k = 0,\ldots,N$, one has ${T}^{k} {S}^{{N-k}}\in Q_S(\mathcal H)$ by Theorem \[thm\_monoid\] and thus Theorem \[thm1\] yields a separable Hilbert space $\mathcal K_{N,k} $, a pure state $\omega_{N,k} \in\mathbb D(\mathcal K_{N,k} )$ and unitary $U_{N,k}\in \mathcal B(\mathcal H\otimes\mathcal K )$ such that $$\begin{aligned}
{T}^{k} {S}^{{N-k}} (A) = \operatorname{tr}_{\mathcal K_{N,k}}\left( U_{N,k}( A\otimes\omega_{N,k} )U_{N,k}^\dagger \right).\end{aligned}$$ for all $ A\in \mathcal B^1(\mathcal H)$. The same line of arguments as in the proof of Theorem \[ch\_4\_satz\_1\] show that $\mathcal K_{N,k}$ and $\omega_{N,k}$ can be chosen independently of $N$ and $k$, so there exists some mutual auxiliary space $\tilde{\mathcal K}$ as well as a mutual pure state $\tilde\omega\in\mathbb D(\tilde{\mathcal K})$ such that $$\begin{aligned}
\label{eq:4-5}
{T}^{k} {S}^{{N-k}} (A) =\operatorname{tr}_{\tilde{\mathcal K}}\left( U_{N,k}( A\otimes\tilde\omega )U_{N,k}^\dagger \right)\end{aligned}$$ for all $ A\in \mathcal B^1(\mathcal H)$, $N\in\mathbb N$ and $k = 0,\ldots,N$. In particular, to every $ {T}^{k}{S}^{N-k}$ we can assign some unitary $U_{N,k}\in\mathcal B(\mathcal H\otimes\tilde{\mathcal K})$ such that (\[eq:4-5\]) holds. Now, choose $\mathcal K:=\tilde{\mathcal K}\otimes\ell_2(\mathbb Z)\otimes \ell_2(\mathbb Z)$ and again $E:=\operatorname{tr}_{\mathcal K}$ and $J:=i_\omega$ with pure state $\omega:=\tilde\omega \otimes e_0e_0^\dagger\otimes e_0e_0^\dagger\in\mathbb D(\mathcal K)$. Moreover, by means of the right shift $\sigma$ from the proof of Theorem \[ch\_4\_satz\_1\] one defines $$\begin{aligned}
{W}_1:=\operatorname{id}_{\mathcal H\otimes\tilde{\mathcal K}}\otimes \,\sigma\otimes \,\sigma\,,\qquad
{U}_1&:=\sum_{m,n\in\mathbb Z} U_{m,n}U_{m-1,n-1}^\dagger \otimes e_me_m^\dagger\otimes e_ne_n^\dagger\,,\\
{W}_2:=\operatorname{id}_{\mathcal H\otimes\tilde{\mathcal K}}\otimes\,\sigma\otimes\operatorname{id}_{\ell_2}\,,\qquad
{U}_2&:=\sum_{n\in\mathbb Z}U_{n,0}U_{n-1,0}^\dagger \otimes e_ne_n^\dagger \otimes \operatorname{id}_{\ell_2}\,,\end{aligned}$$ where $U_{m,n}:=\operatorname{id}_{\mathcal H\otimes\tilde{\mathcal K}}$ if $m<1$ or $n\notin\lbrace 0,\ldots,m\rbrace$. Obviously, $W_1$ and $W_2$ are unitary. The unitarity of $U_1,$ and $U_2$ is readily verified via the unitarity of $U_{N,k}$ so $U:=U_1W_1$ and $V:=U_2W_2$ are unitary, too. As before, by induction one shows $$\begin{aligned}
\label{eq:commuting_lemma_1}
{V}^{j}(A\otimes\omega) ({V}^\dagger)^{j}=U_{{j},0} ( A\otimes\tilde\omega) U_{{j},0}^\dagger\otimes e_{j}e_{j}^\dagger\otimes e_0e_0^\dagger\end{aligned}$$ for all $ A\in\mathcal B^1(\mathcal H)$ and $j \in \mathbb N_0$ and based on this $$\begin{aligned}
\label{eq:4-7}
{U}^{k}{V}^{{N-k}}(A\otimes\omega)({V}^\dagger)^{{N-k}}({U}^{\dagger})^{k} =U_{N,k} ( A\otimes\tilde\omega) U_{N,k}^\dagger\otimes e_{N}e_{N}^\dagger\otimes e_ke_k^\dagger\end{aligned}$$ for all $ A\in\mathcal B^1(\mathcal H)$, $N\in\mathbb N_0$ and $k = 0,\ldots,N$. Note that the case $k=0$ reproduces (\[eq:commuting\_lemma\_1\]) and thus can be omitted. Finally, and imply $$\begin{aligned}
\operatorname{tr}_{\mathcal K}\left( {U}^{k}{V}^{{N-k}}(A\otimes\omega)({V}^\dagger)^{{N-k}}({U}^{\dagger})^{k} \right)
&=\operatorname{tr}_{\tilde{\mathcal K}}(\operatorname{tr}_{\ell_2\otimes \ell_2}( U_{N,k} ( A\otimes\tilde\omega) U_{N,k}^\dagger\otimes e_{N}e_{N}^\dagger\otimes e_{k}e_{k}^\dagger ))\\
&=\operatorname{tr}_{\tilde{\mathcal K}}( U_{N,k}( A\otimes\tilde\omega )U_{N,k}^\dagger )= {T}^{k} {S}^{{N-k}} (A)\end{aligned}$$ for all $ A\in\mathcal B^1(\mathcal H)$, $N\in\mathbb N$ and $k = 0,\ldots,N$ which concludes this proof.
The statement of Theorem \[ch\_4\_satz\_2\] can be extended to finitely many commuting channels $ T_1,\ldots, T_m\in Q_S(\mathcal H)$. Obviously, it is natural to choose $$\begin{aligned}
\mathcal K=\tilde{\mathcal K}\otimes\underbrace{\ell_2(\mathbb Z)\otimes\ldots\otimes \ell_2(\mathbb Z)}_{m\text{-times}}\end{aligned}$$ as common auxiliary space. The rest of the proof is completely analogous.
We can now transfer the above result to obtain a characterization of the reachable set of the control system (\[eq:chap:3\_2\_1\]).
\[ch\_4\_koro\_3\] There exists a separable Hilbert space $\mathcal K$, a pure state $\omega\in\mathbb D(\mathcal K)$ and unitary $U,V\in\mathcal B(\mathcal H\otimes\mathcal K)$ such that $$\begin{aligned}
\rho(N,(T_n)_{n\in\mathbb N_0},\rho_0)=\operatorname{tr}_{\mathcal K}\left( {U}^{k}{V}^{{N-k}}(\rho_0\otimes\omega)({V}^\dagger)^{{N-k}}({U}^{\dagger})^{k} \right)\end{aligned}$$ for all controls $(T_n)_{n\in\mathbb N_0}$, initial states $\rho_0\in\mathbb D(\mathcal H)$ and $N\in\mathbb N_0$, where $k=k(N,(T_n)_{n \in\mathbb N_0}) \in \lbrace 0,\ldots,N\rbrace$ counts how often $T$ occurs in the control sequence $(T_n)_{n\in\mathbb N_0}$ during the first $N$ time steps.
By Definition \[ch\_4\_defi\_1\], $\rho(N,(T_n)_{n\in\mathbb N_0},\rho_0)\in R_N(\rho_0)$ and hence by Lemma \[ch\_4\_lemma\_3\] there exists $k\in\lbrace 0,\ldots,N\rbrace$ such that $\rho(N,(T_n)_{n\in\mathbb N_0},\rho_0)=T^kS^{N-k}(\rho_0)$. Thus the result follows immeditely from Theorem \[ch\_4\_satz\_2\].
\[ch\_4\_koro\_2\] Let $\mathcal K$, $\omega\in\mathbb D(\mathcal K)$ and $U,V\in\mathcal B(\mathcal H\otimes\mathcal K)$ be as in Corollary \[ch\_4\_koro\_3\]. Then, for all $N\in\mathbb N_0$ and $\rho_0\in\mathbb D(\mathcal H)$ one has $$\begin{aligned}
\label{eq:ch_4_koro_2_1}
R_N(\rho_0)\subseteq \operatorname{tr}_{\mathcal K}(\tilde R_N(\rho_0\otimes\omega))\end{aligned}$$ and thus $R(\rho_0)\subseteq \operatorname{tr}_{\mathcal K}(\tilde R(\rho_0\otimes\omega))$. Here, $\tilde R(\tilde\rho_0)$ and $\tilde R_N(\tilde\rho_0)$ denote the reachable sets of the discrete-time closed quantum control system $$\begin{aligned}
\tilde \rho_{n+1}=U_n\tilde \rho_n U_n^\dagger\,, \quad \tilde\rho_0\in\mathbb D(\mathcal H\otimes\mathcal K)\end{aligned}$$ with $U_n\in\lbrace U,V\rbrace$ for all $n\in\mathbb N_0$.
By Lemma \[ch\_4\_lemma\_3\] and Theorem \[ch\_4\_satz\_2\], one has $$\begin{aligned}
R_N(\rho_0)&=\lbrace T^kS^{N-k}(\rho_0)\,|\,k = 0,\ldots,N \rbrace
\\&=\lbrace \operatorname{tr}_{\mathcal K}\left( {U}^{k}{V}^{{N-k}}(\rho_0\otimes\omega)({V}^\dagger)^{{N-k}}({U}^{\dagger})^{k} \right)\,|\,k = 0,\ldots,N\rbrace
\\&=\operatorname{tr}_{\mathcal K}(\lbrace {U}^{k}{V}^{{N-k}}(\rho_0\otimes\omega)({V}^\dagger)^{{N-k}}({U}^{\dagger})^{k} \,|\,k = 0,\ldots,N \rbrace)
\subseteq \operatorname{tr}_{\mathcal K}(\tilde R_N(\rho_0\otimes\omega))\,.\qedhere\end{aligned}$$
\[ch\_4\_bem\_7\]
1. Note that the unitary channels $U$ and $V$ of Corollary \[ch\_4\_koro\_2\] do in general not commute, so (\[eq:ch\_4\_koro\_2\_1\]) states a proper inclusion rather than an equality for $N>1$.
2. Consider the dual problem of (\[eq:chap:3\_2\_1\]), i.e. let $T,S \in Q_H(\mathcal H)$ be two commuting Heisenberg channels. Of course, one can translate the above results—which we will omit here—into the Heisenberg picture via Corollary \[coro\_Q\_H\_1\]. However, we want to comment on the result of Davies[@davies78] which was already mentioned in the introduction and yields a unitary dilation with *commuting* unitary channels, at the cost of our desired partical trace structure.
Let $G=(\mathbb Z\times\mathbb Z,+)$ with subgroup $\mathbb S:=\lbrace (N,k)\in G\,|\, N\in\mathbb N_0\text{ and } 0\leq k\leq N\rbrace$ and define the family $(T_g)_{g\in G}$ of Heisenberg channels via $T_g:= T^k S^{N-k}$ for $g = (N,k)\in \mathbb S$ and $T_g:=\operatorname{id}_{\mathcal B(\mathcal H)}$ otherwise. Adjusting the proof of[@davies78 Thm. 3.1] to discrete groups and using Corollary \[coro\_1\], one gets a Hilbert space $\mathcal K$, a unitary representation $U$ of $G$ on $\mathcal H\otimes\mathcal K$ and a conditional expectation $E$ such that $T_g(B)=E(U_g(B\otimes\operatorname{id}_{\mathcal K})U_g^\dagger)$ for all $B\in\mathcal B(\mathcal H)$. As $(U_g)_{g\in G}$ is a representation of $G$ we may consider the commuting unitary operators $U_{(1,1)}=:U$, $U_{(1,0)}=:V$ resulting in $$T^k S^{N-k}(B)= T_g(B) = E({U}^{k}{V}^{{N-k}}(B\otimes\operatorname{id}_{\mathcal K})({V}^\dagger)^{{N-k}}({U}^{\dagger})^{k})$$ for all $B\in\mathcal B(\mathcal H)$, $N\in\mathbb N_0$ and $k = 0,\ldots,N$. Observe that we did not use the fact that $T,S$ commute so this result even holds for arbitrary channels $T$ and $S$ with the drawback of $E$ lacking any partial trace structure, see also [@Note1].
This article is based on a master thesis[@vomEnde] which was written at the Institute of Mathematics of the University of Würzburg. The authors are grateful to Michael M. Wolf for drawing their attention to several more recent publications on dilations of completely positive maps.
Topological Properties of $Q_S (\mathcal H)$ {#app:monoid}
============================================
For the following definition, we refer to [@dunford1963linear Ch.VI.1].
\[topologies\] Let $\mathcal X$ and $\mathcal Y$ be arbitrary Banach spaces.
- The strong operator topology (s.o.t.) on $\mathcal B(\mathcal X,\mathcal Y)$ is the locally convex topology induced by the family of seminorms of the form $T\to\Vert Tx\Vert$ with $x\in \mathcal X$.
- The weak operator topology (w.o.t.) on $\mathcal B(\mathcal X,\mathcal Y)$ is the locally convex topology induced by the family of seminorms of the form $T\to|y(Tx)|$ with $(x,y)\in \mathcal X\times \mathcal Y'$.
Note that both topologies, the s.o.t. as well as the w.o.t., are Hausdorff so limits are unique.
By the natural isomorphism $(\mathcal B^1(\mathcal H))' \cong \mathcal B(\mathcal H)$, see Section \[sec:dualchannel\], one has the following equivalence: A net $(T_\alpha)_{\alpha\in I}$ in $\mathcal B(\mathcal B^1(\mathcal H))$ converges to $T\in\mathcal B(\mathcal B^1(\mathcal H))$ in w.o.t. if and only if $$\begin{aligned}
\label{eq:AppA_1}
\lim_{\alpha\in I}|\operatorname{tr}(BT_\alpha(A))-\operatorname{tr}(BT(A))|=0\end{aligned}$$ for all $A\in\mathcal B^1(\mathcal H)$ and $B\in\mathcal B(\mathcal H)$.
\[rem\_metrizable\] At this point one might ask whether the strong or weak operator topology is metrizable. If this is the case, closed and sequentially closed sets do coincide which, of course, is of interest for further investigations. The following is well known in the literature, cf. [@Kim Thm. 1.2 and 1.13]: If $\mathcal X$ is separable, then the s.o.t. is metrizable on bounded subsets of $\mathcal B(\mathcal X)$. If $\mathcal X'$ is also separable, then the w.o.t. is metrizable on bounded subsets of $\mathcal B(\mathcal X)$.
Now, recall that $\mathcal H$ is assumed to be separable. Therefore it is evident that the subspace of finite-rank operators $\mathcal F(\mathcal H)$ and hence $\mathcal B^1(\mathcal H)$ itself, which is the $\nu_1$-closure of $\mathcal F(\mathcal H)$ (cf. [@dunford1963linear Lemma XI.9.11]), is separable. Moreover, we already know from Proposition \[thm\_q\_norm\_1\] that $Q_S(\mathcal H)$ is a subset of the unit ball in $\mathcal B(\mathcal B^1(\mathcal H))$. This implies that the s.o.t. on $Q_S(\mathcal H)$ is metrizable and thus convergence, closedness, continuity, etc. can be fully characterized by sequences. On the other hand, it is also well known that $\mathcal B(\mathcal H)$ is not separable with respect to the operator norm topology as the non-separable space $\ell^\infty$ can be isometrically embedded into $\mathcal B(\mathcal H)$. Hence $(\mathcal B^1(\mathcal H))'$ is not separable and the above metrizability result does not apply to the w.o.t. on $Q_S(\mathcal H)$.
However, one could make use of the result that for *convex sets* in $\mathcal B(\mathcal B^1(\mathcal H))$, the closures with respect to the w.o.t. and the s.o.t coincide, cf. [@dunford1963linear Coro. VI.1.6]. Therefore, in the proof of Theorem \[thm\_monoid\] one could focus on the s.o.t. On the other hand, Lemma \[lemma\_2\] ff. show that a direct approach via the w.o.t. is just as simple.
For clarity of the proof of Theorem \[thm\_monoid\], we first state some auxiliary results.
\[lemma\_2\_b\] For every linear map $S:\mathcal B^1(\mathcal H)\to\mathcal B^1(\mathcal G)$ the following statements are equivalent.
- $S$ is positive.
- For all $ A\in\mathcal B^1(\mathcal H)$ and $B\in\mathcal B(\mathcal G)$ with $A,B\geq 0$, one has $\operatorname{tr}(BS(A))\geq 0$.
(a)$\,\Rightarrow\,$(b): For $A,B\geq 0$ and $S$ positive, we obtain $S(A) \geq 0$ and thus $$\operatorname{tr}(BS(A)) = \operatorname{tr}(\sqrt B S(A)\sqrt B) \geq 0\,,
% = \sum_{i\in\mathbb N}\langle (\sqrt B g_i),S(A)(\sqrt B g_i)\rangle \geq 0$$ where $\sqrt B\geq 0$ denotes the unique square root of $B$.
(b)$\,\Rightarrow\,$(a): Choosing $B :=\langle x,\cdot\rangle x$ for arbitrary $x\in\mathcal G$ yields $B \geq 0$ and $$\begin{aligned}
\langle x,S(A)x\rangle=\operatorname{tr}(BS(A))\geq 0\,,\end{aligned}$$ for all $A \geq 0$. Hence it follows $S(A)\geq 0$ so $S$ is positive.
\[lemma\_2\] Let $(T_\alpha)_{\alpha\in I}$ be a net in $\mathcal B(\mathcal B^1(\mathcal H))$ which converges to $T\in\mathcal B(\mathcal B^1(\mathcal H))$ in w.o.t. Then the following statements hold.
- If $T_\alpha$ is trace-preserving for all $\alpha\in I$ then $T$ is trace-preserving.
- If $T_\alpha$ is positive for all $\alpha\in I$ then $T$ is positive.
Both statements follow from (\[eq:AppA\_1\]): (a) by choosing $B=\operatorname{id}_{\mathcal H}$ and (b) by applying Lemma \[lemma\_2\_b\] and taking into account that $[0,\infty)$ is a closed subset of $\mathbb R$.
For the proof of our next result we recall that $\mathcal B^1(\mathcal H\otimes\mathbb C^m)$ and $\mathcal B^1(\mathcal H)\otimes\mathbb C^{m\times m}$ can be identified as follows. Any $A\in\mathcal B(\mathcal H\otimes\mathbb C^m)$ can be represented as $A=\sum_{i,j=1}^m A_{ij}\otimes E_{ij}$ with the standard basis $(E_{ij})_{i,j=1}^m$ of $\mathbb C^{m\times m}$ and appropriate $A_{ij}\in\mathcal B(\mathcal H)$. Then, the following statements are equivalent [@Kraus p. 33-34].
- $A\in\mathcal B^1(\mathcal H\otimes\mathbb C^m)$
- $A_{ij}\in\mathcal B^1(\mathcal H)$ for all $i,j\in\lbrace 1,\ldots,m\rbrace$
\[lemma\_3\] Let $(T_\alpha)_{\alpha\in I}$ be a net in $\mathcal B(\mathcal B^1(\mathcal H))$ converging to $T\in\mathcal B(\mathcal B^1(\mathcal H))$ in w.o.t. Then, for all $m\in\mathbb N$, the net $(T_\alpha\otimes\operatorname{id}_m)_{\alpha\in I}$ converges to $T\otimes\operatorname{id}_m\in\mathcal B(\mathcal B^1(\mathcal H\otimes\mathbb C^m))$ in w.o.t.
According to (\[eq:AppA\_1\]) we have to show $$\begin{aligned}
\label{eq:App1_2}
\lim_{\alpha\in I}|\operatorname{tr}(B(T_\alpha\otimes\operatorname{id}_m-\,T\otimes\operatorname{id}_m)A)|= 0\end{aligned}$$ for all $A\in\mathcal B^1(\mathcal H\otimes\mathbb C^m)$ and $B\in\mathcal B(\mathcal H\otimes\mathbb C^m)$. As seen above every $A\in\mathcal B^{1}(\mathcal H\otimes\mathbb C^m)$ and $B\in\mathcal B(\mathcal H\otimes\mathbb C^m)$ can be represented as finite linear combinations of elements $A_{ij}\otimes E_{ij}\in \mathcal B^{1}(\mathcal H)\otimes \mathbb C^{m\times m}$ and $B_{ij}\otimes E_{ij}\in\mathcal B(\mathcal H)\otimes \mathbb C^{m\times m}$, respectively, with $i,j = 1,\ldots,m$. Hence $$\begin{aligned}
\operatorname{tr}(B(T_\alpha\otimes\operatorname{id}_m-\,T\otimes\operatorname{id}_m)A) =
\operatorname{tr}(B((T_\alpha-T)\otimes\operatorname{id}_m)A)=\sum\nolimits_{i,j=1}^m \operatorname{tr}(B_{ij}(T_\alpha-T)(A_{ji}))\end{aligned}$$ so convergence of $(T_\alpha\otimes\operatorname{id}_m)_{\alpha\in I}$ can easily be related to the convergence of $(T_\alpha)_{\alpha\in I}$.
Now for the main proof of this section.
Since every linear and positive operator on $\mathcal B^1(\mathcal H)$ is naturally norm bounded as a simple consequence of [@Davies Ch. 2, Lemma 2.1], the set $Q_S(\mathcal H)$ of all Schrödinger channels is a bounded subset of $\mathcal B(\mathcal B^1(\mathcal H))$. Now it is readily verified that $Q_S(\mathcal H)$ is a convex subsemigroup of $\mathcal B(\mathcal B^1(\mathcal H))$, cf.[@Heinosaari Ch.4.3]. Next consider a net $(T_\alpha)_{\alpha\in I}$ in $Q_S(\mathcal H)$ converging to some $T\in \mathcal B(\mathcal B^1(\mathcal H))$ in w.o.t. By Lemma \[lemma\_2\] (a), the map $T$ is trace-preserving and by Lemma \[lemma\_3\] $(T_\alpha\otimes\operatorname{id}_m)_{\alpha\in I}$ converges to $T\otimes\operatorname{id}_m$ with respect to the w.o.t. Then applying Lemma \[lemma\_2\] (b) to the net $(T_\alpha\otimes\operatorname{id}_m)_{\alpha\in I}$ yields that $T$ is also $m$-positive for all $m\in\mathbb N$. Hence $T$ is a Schrödinger quantum channel and $Q_S(\mathcal H)$ is closed in $\mathcal B(\mathcal B^1(\mathcal H))$ with respect to the w.o.t. The well-known fact that the w.o.t. is weaker than the s.o.t. and the uniform operator topology concludes the proof.
Note that in the above proof we did not explicitely use the fact that domain and range of the operator $T$ coincides. Therefore, the convexity and closedness results trivially extend to $Q_S(\mathcal H,\mathcal G)$.
Glossary on Dilations {#sec:ditalions}
=====================
For the sake of self-containedness, we recall some basic terminology concerning different types of dilations of linear contractions. Let us start with the Banach space case.
\[def:dilation-Banach-space\] Let $\mathcal X$ be an arbitrary Banach space.
1. Let $T:\mathcal X \to \mathcal X$ be a linear contraction, i.e. $\|T\| \leq 1$. A dilation $(\mathcal Y,\hat T,J,E)$ of $T$ consists of a Banach space $\mathcal Y$ and a triple of maps $(\hat{T}, J, E)$ with $$\label{eq:dilation-general}
T = E \circ \hat{T} \circ J \quad \text{and} \quad E \circ J = \operatorname{id}_{\mathcal X}\,,$$ where the linear maps $\hat{T}$, $J$ and $E$ satisfy:
1. $\hat{T}: \mathcal Y \to \mathcal Y$ is a bi-isometry (i.e. $\hat T$ is bijective and $\hat T,\hat T^{-1}$ are isometries)
2. $J: \mathcal X \to \mathcal Y$ is an isometric embedding of $\mathcal X$ in $\mathcal Y$.
3. $E: \mathcal Y \to \mathcal X$ has operator norm $\|E\| = 1$.
2. Let $S \subset G$ be a semigroup of a group $G$ and $(T_g)_{g \in S}$ be a representation of $S$ with values in the contraction semigroup of $\mathcal X$. A dilation $(\mathcal Y,(\hat{T}_g)_{g \in G_0}, J, E)$ of $(T_g)_{g \in S}$ consists of a Banach space $\mathcal Y$, a subgroup $G_0 \subset G$ and a triple $((\hat{T}_g)_{g \in G_0}, J, E)$ with $$T_g = E \circ \hat{T}_g \circ J \quad \text{and} \quad E \circ J = \operatorname{id}_{\mathcal X}$$ for all $g \in S\subset G_0$, where $(\hat{T}_g)_{g \in G_0}$ is a linear representation of $G_0$ with values in the isometry group of $\mathcal Y$ and $J$, $E$ as before.
\[rem:classical-dilations\]
1. Note that $E \circ J = \operatorname{id}_{\mathcal X}$ implies that $E$ is onto and $J$ is injective. Furthermore, $J \circ E:\mathcal Y \to \mathcal Y$ is a projection of norm $1$ from $\mathcal Y$ onto the range of $J$.
2. If $S$ is assumed to be abelian and there exists a “dilation” of $(T_g)_{g \in S}$ such that $\hat{T}_g$ is well-defined for all $g \in S$ then $(\hat{T}_g)_{g \in S}$ obviously extends to a proper dilation in the above sense, where $G_0$ can be chosen to be the subgroup generated by $S$. For non-abelian $S$, however, this extension property is not obvious.
3. Choosing $S := \mathbb{N}_0$ and $T_n := T^n$ in Def. \[def:dilation-Banach-space\].2, where $T:\mathcal H \to \mathcal H$ is a linear contraction, we recover the “classical” concept of a linear diliation (see also Rem. \[rem\_subspace\].2). To distinguish such a dilation of $T$—which in our sense is actually a dilation of $(T^n)_{n\in\mathbb N_0}$—from a dilation of $T$ in the sense of Def. \[def:dilation-Banach-space\].1 one sometimes calls the latter a “dilation of first order”, cf. [@Kuemmerer83].
4. Once continuity comes into play, things become more sublte as one can either require that the continuity properties of $g \mapsto T_g$ are preserved by $g \mapsto \hat{T}_g$ (which in some cases is unfeasible) or allow that the continuity is relaxed, cf., e.g., [@EvansLewisOverview77 Rem. 17.5]. For our applications, however, this is not an issue as we are only concerned with the case $S = \mathbb{N}_0$.
In the context of quantum channels (or, more generally, completely positive maps) various specializations of the above definitions to
- abstract $C^*$- or $W^*$-algebras
- Heisenberg quantum channels
- and Schrödinger quantum channels
are available in the literature. For more details we refer to [@EvansLewisOverview77; @Gaebler2015].
\[def:dilation-von-Neumann-algebras\] Let $\mathcal A$ be a unital $C^*$-algebra.
1. Let $T: \mathcal A \to \mathcal A$ be linear, completely positive and unital (i.e. identity preserving). A dilation $(\mathfrak A,\hat T,J,E)$ of $T$ consists of a unital $C^*$-algebra $\mathfrak A$ and a triple of maps $(\hat T,J,E)$ with $$\label{eq:cp-dilation}
T = E \circ \hat{T} \circ J \quad \text{and} \quad E \circ J = \operatorname{id}_{\mathcal A}\,,$$ where $\hat{T}$, $J$ and $E$ satisfy:
1. $\hat T:\mathfrak A\to\mathfrak A$ is a $*$-automorphism.
2. $J: \mathcal A \to \mathfrak A$ is a $*$-homomorphism of $\mathcal A$ into $\mathfrak A$.
3. $E:\mathfrak A\to\mathcal A$ is linear and completely positive with operator norm $\|E\| = 1$.
2. Let $S \subset G$ be a semigroup of a group $G$ and let $(T_g)_{g\in S}$ be a semigroup representation of $S$ with values in the set of completely positive, unital maps on $\mathcal A$. A dilation $(\mathfrak A,(\hat{T}_g)_{g \in G_0}, J, E)$ of $(T_g)_{g \in S}$ consists of a unital $C^*$-algebra $\mathfrak A$, a subgroup $G_0 \subset G$ and a triple $((\hat{T}_g)_{g \in G_0}, J, E)$ with $$T_g = E \circ \hat{T}_g \circ J \quad \text{and} \quad E \circ J = \operatorname{id}_{\mathcal A}\,,$$ for all $g \in S\subset G_0$, where $(\hat{T}_g)_{g \in G_0}$ is a representation of $G_0$ with values in the $*$-automorphism group of $\mathfrak A$ and $J$, $E$ as before.
If, in addition, $J(\operatorname{id}_{\mathcal A})=\operatorname{id}_{\mathfrak A}$ then the dilation is said to be unital. On the other hand, if $\mathcal A$ is even a $W^*$-algebra, then all involved maps are in general assumed to be ultraweakly continuous.
\[rem\_19\]
1. Let $\mathcal A$, $\mathcal B$ be unital $C^*$-algebras and let $T: \mathcal A \to \mathcal B$ be unital. Then positivity of $T$ is equivalent to the norm condition $\|T\| = 1$, cf. [@russo1966; @Blackadar06]. In particular, one has $\|T\| = \|T(\operatorname{id_{\mathcal A}})\|$ for every positive map $T: \mathcal A \to \mathcal B$ so unitality of $T$ implies that $T$ is a contraction.
2. As every $*$-homomorphism is trivially completely positive and every injective $*$-homomorphism is always isometric, yields a dilation in the sense of . Moreover, if a dilation is unital, then $E$ is unital as well because implies $\operatorname{id}_{\mathcal A}=(E\circ J)(\operatorname{id}_{\mathcal A})=E(\operatorname{id}_{\mathfrak A})$.
3. Every $*$-automorphism $\hat T:\mathfrak A\to \mathfrak A$ on a unital $C^*$-algebra $\mathfrak A$ is unital itself because of $$\hat T(\operatorname{id}_{\mathfrak A})=\hat T(\operatorname{id}_{\mathfrak A})\operatorname{id}_{\mathfrak A}=\hat T(\operatorname{id}_{\mathfrak A})\hat T(\hat T^{-1}(\operatorname{id}_{\mathfrak A}))=\hat T(\operatorname{id}_{\mathfrak A} T^{-1}(\operatorname{id}_{\mathfrak A}))=\operatorname{id}_{\mathfrak A}\,.$$
4. Let $A$ be a $C^*$-subalgebra of a unital $C^*$-algebra $\mathfrak A$, i.e. $\operatorname{id}_{\mathfrak A} \in \mathcal A \subset \mathfrak A$. Then a linear map $E:\mathfrak A\to\mathcal A$ is said to be a *conditional expectation* (of $\mathfrak A$ onto $\mathcal A$) if it is completely positive with norm $\|E\| =1$ and satisfies $$\label{eq:cond_exp-1}
E(AB)=AE(B) \quad \text{for all $A \in\mathcal A$ and $B\in\mathfrak A$}\,.$$ Obviously, implies that $E$ is a unital (cf. Rem. \[rem\_19\].1) projection onto $\mathcal A$, that is $E(A) = A$ for all $A\in\mathcal A$. The converse is also true, i.e. every projection $E:\mathfrak A\to\mathcal A$ of norm $\|E\| =1$ is a conditional expectation, cf. [@Blackadar06 Thm. II.6.10.2] and [@tomiyama1957]. Moreover, exploiting that $E(B^*) = E(B)^*$ for all $B\in\mathfrak A$, which results from the (complete) positivity of $E$, one can easily show that is equivalent to $$%\label{eq:cond_exp-2}
E(BA) = E(B)A \quad \text{for all $A \in\mathcal A$ and $B\in\mathfrak A$}$$ and, since $\mathcal A$ is unital, also to $$\label{eq:cond_exp}
E(A_1BA_2)=A_1E(B)A_2 \quad \text{for all $A_1,A_2\in\mathcal A$ and $B\in\mathfrak A$}\,.$$ In the literature, is often replaced by the “more symmetric” condition . Now if $\mathcal A\not\subset\mathfrak A$, but $\mathcal A$ can be embedded into $\mathfrak A$ via some unital, injective $*$-homomorphism $J:\mathcal A\to\mathfrak A$, then $E:\mathfrak A \to \mathcal A$ is said to be a *conditional expectation with corresponding injection* $J$, if $E$ is completely positive and $E \circ J =\operatorname{id}_{\mathcal A}$. Note that in this case $E$ is also unital (because $J$ is unital) and thus of norm one. Hence the composed map $J\,\circ\, E:\mathfrak A\to J(\mathcal A)\subset\mathfrak A$ is a projection of norm one and thus a conditional expectation in the above sense. Thus every unital dilation gives rise to a conditional expectation $E$ with corresponding injection $J$.
Now Definition \[def:dilation-von-Neumann-algebras\] directly applies to Heisenberg channels. Taking into account that the only invertible channels are the unitary ones (cf. Prop. \[ch\_3\_Theorem\_14\]) we obtain the following concept.
\[def:dilation-Heisenberg-channles\]
1. Let $T\in Q_H(\mathcal H)$ be a Heisenberg quantum channel, i.e. $T:\mathcal B(\mathcal H)\to\mathcal B(\mathcal H)$ is linear, ultraweakly continuous, completely positive and unital. A unitary dilation $(\mathcal{K},U,J,E)$ of $T$ consists of a Hilbert space $\mathcal{K}$ and a triple of maps $(U,J,E)$ with $$T = E \circ {\operatorname{Ad}_U} \circ J \quad \text{and} \quad E \circ J = \operatorname{id}_{\mathcal B(\mathcal H)}\,,$$ where $U$, $J$ and $E$ satisfy
1. $U\in\mathcal B(\mathcal{K})$ is unitary.
2. $J:\mathcal B(\mathcal{H}) \to \mathcal B(\mathcal{K})$ is an ultraweakly continuous $*$-homomorphism of $\mathcal B(\mathcal{H})$ into $\mathcal B(\mathcal{K})$.
3. $E:\mathcal B(\mathcal{K})\to\mathcal B(\mathcal H)$ is linear, ultraweakly continuous and completely positive with operator norm $\|E\|=1$.
2. Let $S \subset G$ be a semigroup of a group $G$ and let $(T_g)_{g\in S}$ be a semigroup representation of $S$ with values in the set of Heisenberg quantum channels $Q_H(\mathcal H)$. A unitary dilation $(\mathcal{K},(U_g)_{g \in G_0}, J, E)$ of $(T_g)_{g \in S}$ consists of a Hilbert space $\mathcal{K}$, a subgroup $G_0 \subset G$ and a triple $((U_g)_{g \in G_0}, J, E)$ with $$T_g = E \circ \operatorname{Ad}_{U_g} \circ J \quad \text{and} \quad E \circ J = \operatorname{id}_{\mathcal B(\mathcal H)}\,,$$ for all $g \in S\subset G_0$, where $(U_g)_{g \in G_0}$ is a representation of $G_0$ with values in the unitary group on $\mathcal K$ and $J,E$ as before.
If, in addition, $J(\operatorname{id}_{\mathcal H})=\operatorname{id}_{\mathcal K}$ then the dilation is said to be unital.
If the dilation is unital, then $J\in Q_H(\mathcal H,\mathcal K)$ and $E\in Q_H(\mathcal K,\mathcal H)$ are Heisenberg channels (cf. Rem. \[rem\_19\].2). Finally, this concept can be transferred to the Schrödinger quantum channels via duality (cf. Section \[sec:dualchannel\]).
\[def:dilation-Schroedinger-channles\]
1. Let $T\in Q_S(\mathcal H)$ be a Schrödinger quantum channel, i.e. $T:\mathcal B^1(\mathcal H)\to\mathcal B^1(\mathcal H)$ is linear, completely positive and trace-preserving. A unitary dilation $(\mathcal{K},U,J,E)$ of $T$ consists of a Hilbert space $\mathcal{K}$ and a triple of maps $(U,J,E)$ with $$T = E \circ {\operatorname{Ad}_U} \circ J \quad \text{and} \quad E \circ J = \operatorname{id}_{\mathcal B^1(\mathcal H)}\,,$$ where $U$, $J$ and $E$ satisfy
1. $U\in\mathcal B(\mathcal{K})$ is unitary.
2. $J:\mathcal B^1(\mathcal{H})\to\mathcal B^1(\mathcal K)$ is linear and completely positive with operator norm $\|J\|=1$.
3. $E:\mathcal B^1(\mathcal{K}) \to \mathcal B^1(\mathcal{H})$ is linear, completely positive and satisfies $$\label{eq:dual_star_hom}
E(E^*(B)A)=BE(A) \quad \text{for all $B\in\mathcal B(\mathcal H)$ and $A\in\mathcal B^1(\mathcal K)$,}$$ where $E^*$ is the dual channel of $E$.
2. Let $S \subset G$ be a semigroup of a group $G$ and let $(T_g)_{g\in S}$ be a semigroup representation of $S$ with values in the set of Schrödinger quantum channels $Q_S(\mathcal H)$. A unitary dilation $(\mathcal{K},(U_g)_{g \in G}, J, E)$ of $(T_g)_{g \in S}$ consists of a Hilbert space $\mathcal{K}$, a subgroup $G_0 \subset G$ and a triple $((U_g)_{g \in G_0}, J, E)$ with $$T_g = E \circ \operatorname{Ad}_{U_g} \circ J \quad \text{and} \quad E \circ J = \operatorname{id}_{\mathcal B^1(\mathcal H)}\,,$$ for all $g \in S\subset G_0$, where $(U_g)_{g \in G_0}$ is a representation of $G_0$ with values in the unitary group on $\mathcal K$ and $J,E$ as before.
If, in addition, $E$ is trace-preserving then the dilation is said to be trace-preserving.
1. Property which looks quite similar to implies (by direct computation) that the dual channel $E^*$ is a $*$-homomorphism. Moreover, $E^*$ is ultraweakly continuous as this holds for every dual channel. Conversely, for any ultraweakly continuous $*$-homomorphism $J$ from Definition \[def:dilation-Heisenberg-channles\] one can show that together with its pre-dual channel, it satisfies . In this sense, the dilation definitions \[def:dilation-Heisenberg-channles\] and \[def:dilation-Schroedinger-channles\] are dual to each other. Similar as for , one can conclude that is equivalent to $$E(AE^*(B))=E(A)B \quad \text{for all $B\in\mathcal B(\mathcal H)$ and $A\in\mathcal B^1(\mathcal K)$.}$$
2. If a dilation is trace-preserving, then $J$ is trace-preserving as well (cf. Remark \[rem\_19\].2.) so in particular, $J\in Q_S(\mathcal H,\mathcal K)$ and $E\in Q_S(\mathcal K,\mathcal H)$ are Schrödinger channels.
3. Corollary \[coro\_1\] shows that for every Heisenberg channel $T\in Q_H(\mathcal H)$ there exists a unitary (and even unital) dilation of $T$ of the following type $(\mathcal H\otimes\mathcal K,\operatorname{Ad}_U,i_{\mathcal K},\operatorname{tr}_\omega)$, where $\mathcal K$ is a separable Hilbert space, $\omega\in\mathbb D(\mathcal K)$ a pure state and $U\in\mathcal B(\mathcal H\otimes\mathcal K)$ a unitary operator. Such a dilation is also said to be of tensor type. This result holds analogously for every $T\in Q_S(\mathcal H)$ by Theorem \[thm1\].
[^1]: (corresponding author)
|
---
abstract: 'We study the boundary-driven asymmetric simple exclusion process (ASEP) in a one-dimensional chain with long-range links. Shortcuts are added to a chain by connecting $pL$ different pairs of sites selected randomly where $L$ and $p$ denote the chain length and the shortcut density, respectively. Particles flow into a chain at one boundary at rate $\alpha$ and out of a chain at the other boundary at rate $\beta$, while they hop inside a chain via nearest-neighbor bonds and long-range shortcuts. Without shortcuts, the model reduces to the boundary-driven ASEP in a one-dimensional chain which displays the low density, high density, and maximal current phases. Shortcuts lead to a drastic change. Numerical simulation studies suggest that there emerge three phases; an empty phase with $ \rho = 0 $, a jammed phase with $ \rho = 1 $, and a shock phase with $ 0<\rho<1$ where $\rho$ is the mean particle density. The shock phase is characterized with a phase separation between an empty region and a jammed region with a localized shock between them. The mechanism for the shock formation and the non-equilibrium phase transition are explained by an analytic theory based on a mean-field approximation and an annealed approximation.'
address:
- '$^1$ Department of Physics, University of Seoul, Seoul 130-743, Korea'
- '$^2$ Theoretische Physik, Universität des Saarlandes, 66041 Saarbrücken, Germany'
- '$^3$ School of Physics, Korea Institute for Advanced Study, Seoul 130-722, Korea'
author:
- 'Mina Kim$^1$, Ludger Santen$^2$ and Jae Dong Noh$^{1,3}$'
title: 'Asymmetric simple exclusion process in one-dimensional chains with long-range links'
---
Introduction {#sec1}
============
The asymmetric simple exclusion process (ASEP) has been widely studied in the past decades [@Derrida98]. It is a nonequilibrium driven diffusive system of particles subject to the exclusion interaction. Despite its simplicity, the ASEP describes various nonequilibrium processes such as bio-polymerization [@MacDonald68], surface growth [@Spohn06], traffic flow [@Nagel92; @Chowdhury00], for example. Furthermore, the ASEP in one dimension is exactly solvable via the Bethe ansatz [@Gwa92] and the matrix product ansatz [@Derrida93] or direct solution of recursion relations [@Schutz93]. The exact solution contributes to deeper understanding of fluctuation phenomena [@Derrida98_1; @Lee99] and nonequilibrium phase transitions [@Schutz93].
Most studies on the ASEP have been performed on one-dimensional (1D) chains. On the other hand, there are many quasi-1D systems which involve long-range links. If a polymer chain folds randomly, there arise contacts between different polymer segments which are far apart along a backbone [@deGennes]. Those contacts can be regarded as long-range links for a transport process on a network of polymer chains. A gene regulatory protein diffuses along the DNA chain to search for its target gene [@Berg81]. It can make a long-range jump by dissociation from and reassociation with the DNA. Recent studies show that real world traffic networks have a complex structure with long-range links [@Watts98; @BA]. In this respect, interests are growing in the study of the ASEP on complex networks [@Szavits-Nossan; @Ha07; @Otwinowski09].
In this work, we address the question: What is the transport capacity of a complex network for particles interacting via mutual exclusion? In order to contribute to this issue we consider the ASEP on a 1D chain with long-range links, called shortcuts, for open boundary conditions. It will turn out that the shortcuts cause a drastic change in the phase diagram of the ASEP.
We start with a brief review of the 1D ASEP. The original model is defined on a 1D lattice. The lattice sites are either occupied by at most one particle or empty. Multiple occupancy on a site is prohibited \[exclusion interaction\]. Particles may hop to the left and right with a bias to one direction \[ASEP\]. A closely related equilibrium process is the so-called symmetric simple exclusion process (SSEP) where particles hop in both directions with equal rates. For periodic boundary conditions, the model has a trivial steady state where every microscopic configuration is equally likely irrespective of the hopping bias [@Gwa92]. However, the bias is relevant for the dynamic scaling behavior. In the context of growing interfaces, the SSEP belongs to the Edward-Wilkinson (EW) universality class [@Edwards82], and the ASEP to the Kardar-Parisi-Zhang (KPZ) universality class [@Kardar86]. Both classes are characterized by the power-law scaling $\tau \sim \xi^z$ between characteristic time and length scales. The scaling exponent is given by $z=2$ for the EW class and $z=3/2$ for the KPZ class [@Gwa92; @Kim95].
For open boundary conditions the chain is coupled to particle reservoirs at both ends. One acts as a particle source emitting particles at a rate $\alpha$, and the other as a sink absorbing particles at a rate $\beta$. Interestingly, the system driven by the open boundaries displays nonequilibrium phase transitions between low-density (LD), high-density (HD), and maximal-current (MC) phases [@Derrida93; @Schutz93]. The system belongs to the LD phase if the capacity of the particle source $\alpha$ controls the particle flux, i.e. if $\alpha < \beta$ and $\alpha < p_h/2$, where $p_h$ denotes the hopping rate of the particles. The system belongs to the HD phase when the outgoing rate $\beta$ is smaller than $\alpha$ and $p_h/2$. The overall particle density is determined by the outgoing rate $\beta$ and there is a macroscopic congestion of particles. When both $\alpha$ and $\beta$ are larger than $p_h/2$, the capacity of the system is limited by the capacity of the bulk. Such a phase is called the MC phase, where the overall particle density is independent of $\alpha$ and $\beta$.
In the 1D ASEP, particle hopping is a short-ranged process between neighboring sites. There are a few attempts to study the effect of a long-range hopping on the phase diagram of the ASEP for open boundary conditions. Szavits-Nossan and Uzelac considered the ASEP with a probabilistic long-range hopping [@Szavits-Nossan]. A particle can hop to any site at a distance $l$ with probability $p_{l} \sim 1/l^{\sigma+1}$. They obtained that the phase diagram remains the same as that of the 1D ASEP for $\sigma>1$. For $\sigma \leq 1$, however, the system does not display any phase transition. Ha [*et al.*]{} considered a boundary-driven 1D ASEP model where particles may perform a short-range hopping between nearest-neighbor sites or a long-range hopping [@Ha07]. Upon a long-range hopping, a particle jumps to an empty site directly behind a next particle in front of it. It was found that the long-range hopping introduces an instability towards a so-called empty-road phase. Otwinowski and Boettcher considered the ASEP on a one-dimensional chain decorated with hierarchically organized long-range links [@Otwinowski09]. This model displays the LD and HD phases. Besides, depending on the way the long-range hopping is implemented, an intermediate phase may also be realized.
In this work, we investigate the role of long-range hoppings in a driven system in a generic setting. For that purpose, we study the ASEP on a 1D chain with open boundary conditions and additional long range links connecting randomly-selected pairs of sites. The result is a graph similar to a small-world network [@Watts98]. Additionally, we couple this network to two particle reservoirs, a particle source connected to one boundary site and sink to another. In contrast to the 1D ASEP, particle source and sink now are connected via the small-world network. This setup describes a generic scenario for directed transport through a complex network with limited capacity. It is generally believed that long-range links make a system defined on a small-world network homogeneous [@Dorogovtsev08]. In contrast, our study shows that the system develops a localized shock which separates a 1D backbone into an empty region near the entrance and a fully-occupied region near the exit. Such an inhomogeneity is caused by the interplay between the boundary driving and the long-range hopping. The steady-state position of the shock depends on the particle input and output rates, which results in an interesting phase diagram.
This paper is organized as follows: In Sec. \[sec2\], we introduce a boundary-driven ASEP in a 1D chain with long-range links. In Sec. \[sec3\], we present numerical results obtained from Monte Carlo simulations. Measuring the overall particle density $\rho$, we obtain a numerical phase diagram which consists of an empty (E) phase, a jammed (J) phase, and a shock (S) phase. The overall particle density takes the value of $\rho=0~(1)$ in the E (J) phase, while it varies continuously in the S phase. The shock phase is characterized by a localized shock which separates a 1D backbone into empty and jammed domains. In order to understand the mechanism for the shock formation and the phase transition, we develop an approximate analytic theory. This is presented in Sec. \[sec4\]. We summarize and conclude the paper in Sec. \[sec5\].
Boundary-driven ASEP with long-range links {#sec2}
==========================================
![A graph consisting of $L$ sites, short-range links, and long-range links. The site $i=1$ and $L$ are attached to a particle source and sink, respectively.[]{data-label="fig1"}](fig1.eps){width="\columnwidth"}
Consider a graph $\mathcal{G}$ consisting of $L$ sites which are labeled as $i=1,2,\cdots,L$. Every pair $(i,i+1)$ with $i=1,\cdots,L-1$ is connected with a short-range link to form a 1D backbone. In addition, we select $pL$ pairs of sites at random and add long-range links between them (see Figure \[fig1\]). The shortcut density, the total number of long-range links divided by $L$, is denoted by $p$. Every link has an orientation: A link between $i$ and $j$ is directed from $i$ to $j$ if $i<j$ and vice versa. Such a direction will be referred to as a forward direction. It is assumed that the boundary site $i=1$, called an entrance, is attached to a particle source where particles are fed into the system at constant rate. The other boundary site $i=L$, called an exit, is attached to a particle sink which absorbs particles at a constant rate.
We consider the ASEP process on such a graph. More precisely the particle dynamics is defined as follows: A particle enters the system at a rate denoted as $\alpha$. Particles which have entered the system are then allowed to hop along forward links at rate one. If a particle is located at a site which is connected to others via multiple forward links one of these is randomly chosen. For each forward link the selection probability is given by $1/k_i$ where $k_i$ denotes the number of forward links at a given site $i$. A particle can leave the system at site $L$ (the exit) at a rate denoted as $\beta$. Throughout the process, particles are subject to the exclusion principle which forbids multiple occupancy. Therefore any trial move violating the exclusion principle is rejected.
One can represent the structure of a graph $\mathcal{G}$ with an adjacency matrix $\bm{A}$ whose elements $A_{ij}$ take the value of $1$ only if there is a forward link from site $j$ to $i$. Due to the link directionality, $A_{ij}=0$ for all $i<j$. Then, the hopping probability of a particle from site $i$ to $j$ is given by $$\label{p_hop}
u_{ji} = \frac{A_{ji}}{k_i} \ ,$$ where $k_i\equiv \sum_{l>i} A_{li}$ denotes the number of forward links out of site $i$. It should be noticed that the hopping probability is a quenched random variable. It varies from one realization of a graph to another. Consequently, the quenched average over different realizations of the graph is necessary.
Here, we are interested in the particle density distribution and the current. Let $n_i~(=0,1)$ be the occupation number at site $i$. It is a stochastic variable evolving according to the ASEP dynamics. For a given realization of the graph, its average value is governed by the time evolution equation $$\label{dni_dt}
\frac{d}{dt} \langle n_{i} \rangle =
\sum_{j<i} u_{ij} \langle n_j (1-n_i)\rangle -
\sum_{j>i} u_{ji} \langle n_{i} ( 1-n_j )\rangle$$ for $1<i<L$ and $$\begin{aligned}
\frac{d}{dt}\langle n_1\rangle &=& \alpha(1-\langle n_1\rangle) -
\sum_{j=2}^L u_{j1} \langle n_1 ( 1-n_j )\rangle \label{dn1_dt} \\
\frac{d}{dt}\langle n_L \rangle &=&
\sum_{j=1}^{L-1} u_{Lj} \langle n_j (1-n_L)\rangle - \beta \langle
n_L\rangle . \label{dnL_dt} \end{aligned}$$ Here the angle bracket $\langle(\cdot) \rangle$ represents the average of a quantity $(\cdot)$ for different realizations of the stochastic noise.
In contrast to the 1D-ASEP particle conservation in the bulk does not imply that the current is the same for every link. Therefore, the current can be defined in several ways. An obvious choice is to count the number of particles entering or leaving the system since the particle reservoirs are each connected via a single link. The incoming current $J_{in}$ through the entrance and the outgoing current $J_{out}$ from the exit are given by $$\begin{aligned}
J_{in} &=& \alpha ( 1 - \langle n_1\rangle) \ , \label{J_in}\\
J_{out} &=& \beta \langle n_L \rangle \ . \label{J_out}\end{aligned}$$ In the bulk site $i$ ($=1,\cdots,L-1$), the current $J_i$ is defined as the total current of particles departing from sites $j=1,\cdots,i$ and arriving at sites $l=i+1,\cdots,L$. It is given by $$J_i = \sum_{j=1}^i \sum_{l=i+1}^L u_{lj} \langle n_j (1-n_l)\rangle \ .$$ With this definition of the current, the time evolution of the density is given by $d\langle n_i\rangle/dt = J_{i-1}-J_i$ for $1<i<L$ and $d\langle n_1 \rangle/dt = J_{in} - J_1$ and $d\langle n_L\rangle/dt = J_{L-1} - J_{out}$. Hence, all values of the local currents $J_i$, $J_{in}$ and $J_{out}$ should be the same in the steady state, where $d\langle n_i\rangle /dt = 0$.
The time-evolution equations (\[dni\_dt\]), (\[dn1\_dt\]), and (\[dnL\_dt\]) already illustrate the difficulty in calculating the quantities of interest. In order to determine the particle density two-point correlations have to be calculated, which in turn require higher order correlation functions. Furthermore, the quenched average over the random variables $\{u_{ij}\}$ is necessary. An exact solution of this process is not available. So, we will investigate the model using a numerical simulation method in the following section.
Simulation Results {#sec3}
==================
In this section we present numerical results obtained from Monte Carlo simulations. For a simulation, one generates a graph $\mathcal{G}$ with an adjacency matrix $\bf{A}$ consisting of a 1D chain of $L$ sites and $pL$ shortcuts. Particles on $\mathcal{G}$ move in the following way: First, we select a random variable $l\in \{0,1,\cdots,L\}$ with equal probability. (i) If $l=0$, we add a particle to the entrance site $i=1$ with probability $\alpha$ if it is empty. (ii) If $0<l<L$, we try to move a particle (if any) at site $l$ to a target site $j$ selected among $\{l+1,\cdots,L\}$ with probability $u_{jl}$ given in (\[p\_hop\]). The particle move can be carried out if the target site is empty. (iii) If $l=L$, we remove a particle (if any) at the exit site $i=L$ with probability $\beta$. The time is incremented by unity after $(L+1)$ trials.
We are interested in the particle distribution and the current in the steady state. After a transient period of time interval $T_t$, we average the occupation number for the time interval $T_s$ to obtain the steady-state occupation number distribution $\langle n_i\rangle$. Note that the mean occupation number varies from one realization of a graph to another. In a second step a quenched average $[(\cdots)]_{\mathcal{G}}$ over graph realizations is necessary. The quenched averaged quantities will be denoted as $$\rho_i = [\langle n_i\rangle]_{\mathcal{G}} \ .$$ The overall particle density is given by $$\rho = \frac{1}{L}\sum_{i=1}^L \rho_i \ .$$ The steady-state current is given by $$J = \alpha (1-\rho_1) = \beta \rho_L \ .$$
![(Color online) Particle density $\rho$ (a) and current $J$ (b) in the $\alpha\beta$ plane. The system size is $L=400$ and the shortcut density $p=0.2$.[]{data-label="fig2"}](fig2.eps){width="\columnwidth"}
The overall behavior of the particle density $\rho$ and the current $J$ is presented in Figure \[fig2\]. Those data were averaged over $N_s = 1000$ graph realizations over a time interval $T_s = 500000$ after a transient interval $T_t = 10000$. Figure \[fig2\] suggests that there exist three different regimes. When $\alpha$ is small, the overall density $\rho$ is close to zero. On the other hand, $\rho$ is close to 1 when $\beta$ is small. A finite-size-scaling (FSS) analysis reveals that the system indeed undergoes nonequilibrium phase transitions between three different phases, which will be discussed below.
![(Color online) (a) Plot of the density of empty sites vs. $\beta$ at fixed $\alpha=0.8$. (b) Plot of the particle density vs. $\alpha$ at fixed $\beta=0.8$.[]{data-label="fig3"}](fig3.eps){width="\columnwidth"}
Figure \[fig3\](a) shows the plot of $(1-\rho)$ along a line $\alpha=0.8$ at several values of $L=100,\cdots,3200$. Those data are well fitted to a FSS form [@Privman] $$(1-\rho) = L^{-x_v} \mathcal{F}_v((\beta-\beta_c)L^{1/\nu_v})$$ with $\beta_c \simeq 0.15$, $x_v \simeq 0.4$, and $\nu_v \simeq 2.5$. The scaling function has a limiting behavior $\mathcal{F}_v(y \gg 1) \sim y^{\beta_v'}$ with $\beta_v' = x_v \nu_v \simeq 1.0$. The FSS indicates that system undergoes a continuous phase transition at a critical point $\beta = \beta_c$. For $\beta<\beta_c$, the density of vacant sites is zero. That is to say, the system is fully occupied by particles. Such a macroscopic state will be called a [*jammed phase*]{}. Near the critical point with $\epsilon = \beta-\beta_c$, the density of empty sites scales as $$(1-\rho) \sim \epsilon^{\beta_v'} \ .$$
Figure \[fig3\](b) shows the plot of $\rho$ along a line $\beta=0.8$. The data are well fitted to a FSS form [@Privman] $$\rho = L^{-x_p} \mathcal{F}_p((\alpha-\alpha_c)L^{1/\nu_p})$$ with $\alpha_c \simeq 0.36$, $x_p \simeq 0.5$, and $\nu_p \simeq 2$. The scaling function has a limiting behavior $\mathcal{F}_p(y\gg 1)\sim
y^{\beta_p'}$ with $\beta_p' = x_p \nu_p \simeq 1.0$. Hence we conclude that the system undergoes a continuous phase transition at a critical point $\alpha=\alpha_c$. The particle density vanishes for $\alpha<\alpha_c$. Such a macroscopic state will be called an [*empty phase*]{}. Near the critical point with $\epsilon =
\alpha-\alpha_c$, the particle density scales as $$\rho \sim \epsilon^{\beta_p'} \ .$$
![Phase diagram at $p=0.2$. The location of symbols ($\Box$ between the E and S phases, and $\circ$ between the S and J phases) is determined from the FSS analysis. The solid lines are the phase boundary obtained from the mean-field and annealed approximation developed in Sec. \[sec4\].[]{data-label="fig4"}](fig4.eps){width="0.6\columnwidth"}
Repeating the FSS analysis, we obtain the numerical phase diagram as shown in Figure \[fig4\]. The phase diagram consists of the empty (E) phase with $\rho=0$ and the jammed (J) phase with $\rho=1$. The other phase with $0<\rho<1$ will be called a shock (S) phase. Although the phase diagram looks similar to that of the ASEP on 1D chains [@Derrida93; @Schutz93], the nature of the phases is different.
The particle distribution is intriguing in the S phase. Taking $\alpha=1.0$ and $\beta=0.5$, we have measured the particle density distribution $\{\langle n_i\rangle\}$ in the steady state at a given realization of $\mathcal{G}$. Figure \[fig5\] shows typical distributions. There is a phase separation between a region with $\langle
n_i \rangle \simeq 0$ and a region with $\langle n_i
\rangle \simeq 1.0$. The domain boundary between the two regions is called a shock. The localized shock gives a hint why there are phase transitions into the E and J phases. The system can be in the E (J) phase when the shock is absorbed at the exit (entrance). Hence it is crucial to understand the mechanism for the shock formation. This will be discussed in the following section.
![(Color online) Particle density profile in the shock phase. Each panel contains a data set obtained from a different realization of a graph. Monte Carlo simulation data are drawn with a red line. Also shown with a blue line are numerical data from the mean field approximation as explained in Sec. \[sec4\]. Parameter values are $L=1000$, $p=0.2$, $\alpha=1.0$, and $\beta=0.5$.[]{data-label="fig5"}](fig5.eps){width="\columnwidth"}
Before proceeding, we present a physical argument for the phase diagram. It is known that the diameter $l_D$ of the small-world network with nonzero shortcut density $p$ scales as $l_D \sim \frac{1}{p} \ln (pL)$ [@BA]. This means that any site can be reached from a given site within $l_D$ steps [@Watts98; @BA]. Then, a particle injected at the entrance can move toward the exit with a diverging speed $v \simeq L/l_D \sim L/\ln L$ along the backbone in the $L\to\infty$ limit when $\alpha$ is so small that the exclusion is irrelevant near the entrance. Consequently, particles can escape instantly from the system and the E phase can be realized. Using the same argument we can understand the origin of the J phase. A hopping of a particle from one site to another is equivalent to a hopping of a hole in the opposite direction. Hence, when $\beta$ is small enough, a hole injected at the exit can travel toward the entrance with the diverging speed in $\mathcal{O}(\ln L)$ steps and the J phase can be realized. The S phase is a result of competition between the empty domain (which is stabilized by the long-range hopping of particles) and the jammed domain (which is stabilized by the long-range hopping of holes). This argument clearly shows that the small-world property is crucial in the formation of the E, J, and S phases. The small-world property emerges at any nonzero value of $p$ [@BA]. Hence we expect that those phases replace the LD, HD, and MC phases of the conventional ASEP in 1D chains immediately as one turns on the long-range links.
We have also performed the numerical simulations with $p=0.1$ and $p=1.0$. In both cases we observe qualitatively the same phase diagram. As $p$ increases, the E phase expands while the J phase shrinks. For example, when $\beta=0.8$, we found that $\alpha_c = 0.35$, $0.36$, and $0.39$ at $p=0.1$, $0.2$, and $1.0$, respectively. When $\alpha = 0.8$, we found that $\beta_c = 0.20$, $0.15$, and $0.07$ at $p=0.1$, $0.2$, and $1.0$, respectively. We do not find a simple explanation for the observed $p$ dependence of the phase boundary.
Analytic Results {#sec4}
================
Mean field approximation
------------------------
We adopt a mean field approximation by assuming that $$\langle n_i n_j \rangle = \langle n_i \rangle \langle n_j \rangle$$ for all $i$ and $j$. Applying the mean field approximation to (\[dni\_dt\]), (\[dn1\_dt\]), and (\[dnL\_dt\]) yields that $$\label{dndt}
\frac{d}{dt}\langle n_{i}\rangle = -
\langle n_i \rangle R_i + (1-\langle n_i \rangle) Q_i \ ,$$ where the auxiliary quantities are defined as $$R_i = \beta \delta_{i,L} + \left(\sum_{j>i}u_{ji}
\left(1-\langle n_j \rangle\right)\right)(1-\delta_{i,L})$$ and $$Q_i \equiv \alpha \delta_{i,1} + \left(
\sum_{j<i} u_{ij} \langle n_j \rangle \right)(1-\delta_{i,1})$$ with the Kronecker $\delta$ symbol. The quantities $R_i$ and $Q_i$ can be interpreted as particle evaporation and deposition rates at site $i$, respectively. They depend on the structure of an underlying graph through $\{u_{ij}\}$ and the whole particle density distribution $\{\langle n_i \rangle \}$. This interpretation of the rates $R_i$ and $Q_i$ allows to relate our model to the 1D ASEP with constant evaporation and deposition rates which was studied in [@Parmeggiani03; @Evans03; @Juhasz04]. In contrast to that model the rates $R_i$ and $Q_i$ in our model depend on the particle distribution as well as on the realization of the adjacency matrix.
The steady-state density satisfying $d\langle n_i\rangle/dt=0$ is given by $$\label{SCE}
\langle n_i \rangle = \frac{Q_i}{R_i + Q_i} \ .$$ Since $R_i$ and $Q_i$ depends on the particle distribution, (\[SCE\]) should be solved self-consistently. The self-consistent equation can be solved numerically via iteration. Starting from any trial distribution, one updates it by evaluating the right-hand side of (\[SCE\]). A particle distribution converges to a steady-state distribution without difficulty.
In order to test the mean field approximation, we compare the particle density profile obtained from the Monte Carlo method and the mean field approximation. For a given graph $\mathcal{G}$, we have performed the Monte Carlo simulation and solved the self-consistent equation for the steady-state density profile. They are compared in Figure \[fig5\]. The mean field approximation reproduces the phase separation and the shock in the density profile. In many cases, a mean field result is remarkably close to a Monte Carlo result (see the top and middle panels in Figure \[fig5\]). On the other hand, in some cases, there is a noticeable quantitative discrepancy between them (see the bottom panel in Figure \[fig5\]). Nevertheless, the mean field result still indicates the presence of the shock clearly. Therefore, we conclude that the mean field theory is suitable for the description of the phase transitions.
Annealed network approximation
------------------------------
The density profile given by the solution of (\[SCE\]) depends on a graph realization $\mathcal{G}$. We have to perform the quenched average over graph realizations to obtain the disorder-averaged density profile $\rho_i = [\langle n_i\rangle]_{\mathcal{G}}$. The quenched average is analytically intractable. Hence we further make an approximation by replacing an adjacency matrix element $A_{ij}$, which is a quenched random variable, with its disorder-averaged value $[A_{ij}]_{\mathcal{G}}$. Such an approximation is called an annealed approximation. The annealed approximation is useful in studying physical systems on graphs or networks [@Boguna09; @Noh09; @Lee09].
On a graph of $L$ sites, there are $pL$ long-range links. Hence the probability to find a long-range links between two sites $i$ and $j\neq i\pm 1$ is given by $p_1 = 2pL/((L-1)(L-2)) \simeq 2p/L$. Short-range links are connecting sites $i$ and $i+1$ for $i=1,\cdots,L-1$. Taking account of the short- and long-range links, we obtain that $$\label{A_av}
[A_{ij}]_{\mathcal{G}} = \delta_{i,j+1} + ( 1 - \delta_{i,j+1}) \frac{2p}{L}$$ for $i>j$ and $[A_{ij}]_{\mathcal{G}}=0$ for $i\le j$. The parameter $u_{ij}$ is replaced by $u_{ij} = [A_{ij}]_{\mathcal G} / [k_j]_{\mathcal
G}$ with $[k_j]_{\mathcal G} = (1+2p(L-j-1)/L)$.
In the annealed approximation, the self-consistent equation for the disorder-averaged density $\rho_i$ becomes $$\label{rho_i}
\rho_i = \frac{Q_i}{R_i+Q_i}$$ where $R_i$ and $Q_i$ are given by $$\begin{aligned}
R_i &=& 1-\frac{1}{1+\frac{2p}{L}(L-i-1)} \left( \rho_{i+1} + \frac{2p}{L}
\sum_{j=i+2}^L \rho_j\right) \label{R_i_av} \\
Q_i &=& \frac{\rho_{i-1}}{1+\frac{2p}{L}(L-i)} + \sum_{j=1}^{i-2}
\frac{ \frac{2p}{L} \rho_j }{1 + \frac{2p}{L} (L-j-1)} \label{Q_i_av}\end{aligned}$$ with the boundary terms $R_L = \beta$ and $Q_1 = \alpha$. Equations (\[rho\_i\]), (\[R\_i\_av\]), and (\[Q\_i\_av\]) are the starting point for further analysis.
Shock state
-----------
We first consider the sites with $i=\mathcal{O}(1)$ near the entrance. Ignoring $\mathcal{O}(L^{-1})$ corrections, one can approximate $R_i$ and $Q_i$ as $$\begin{aligned}
R_i^{in} & = & \frac{1}{1+2p} ( 1-\rho_{i+1}) + \frac{2p}{1+2p} (1-\rho)
\label{R_in} \\
Q_i^{in} & = & \frac{1}{1+2p} \rho_{i-1} \label{Q_in} \end{aligned}$$ with the boundary term $Q_1^{in}=\alpha$ and the overall particle density $\rho$. These expressions allow for an interpretation for an effective dynamics near the entrance: A particle at site $i$ performs a short-range jump to site $i+1$ with probability $$W^{in}_{h} = \frac{1}{1+2p}$$ or annihilates spontaneously with the probability $$W^{in}_{a}(\rho) = \frac{2p}{1+2p}(1-\rho) \ .$$
Due to the effective annihilation, the particle density should decay to zero exponentially with the distance from the entrance along the backbone with a characteristic length scale $\xi_{in} = 1/W_{a}^{in}(\rho)$ unless $\rho=1$. This feature is consistent with the density profile shown in Figure \[fig5\].
We next consider the sites $i=L-l$ with $l=\mathcal{O}(1)$ near the exit. Ignoring again $\mathcal{O}(L^{-1})$ corrections, one can approximate $R_i$ and $Q_i$ by $$\begin{aligned}
R_i^{out} & = & 1-\rho_{i+1} \label{R_out}\\
Q_i^{out} & = & \rho_{i-1} + \frac{2p}{L}
\sum_{j=1}^{i-2} \frac{\rho_j}{1+2p(L-j-1)/L}
\label{Q_out}\end{aligned}$$ with the boundary term $R_L^{out} = \beta$. These expressions suggest that particles near the exit have a following effective dynamics: A particle at site $i$ performs a short-range hopping to site $i+1$ with the probability $$W^{out}_{h} = 1 \ ,$$ and particles are created spontaneously at each site with the probability given by the second term in (\[Q\_out\]). Due to the creation, the particle density should saturate to unity as one departs from the exit along the backbone.
The density profiles stemming from the both boundaries converge to different values of 0 and 1. So, there must emerge a shock as a domain boundary. The position of the shock along the backbone is denoted by $i_S$, which is related to the overall particle density as $$\rho = 1 - \frac{i_S}{L} \ .$$ Making use of the shock structure, the quantity $Q^{out}_i$ in (\[Q\_out\]) is given by $$\begin{aligned}
Q^{out}_i &=& \rho_{i-1} + \frac{2p}{L} \sum_{j=i_S}^{i-2}
\frac{1}{1+2p(L-j-1)/L} \nonumber \\
&=& \rho_{i-1} + \ln(1+2p\rho)\end{aligned}$$ with $\mathcal{O}(L^{-1})$ corrections being ignored. Correspondingly, the particle creation probability near the exit is given by $$W_c^{out}(\rho) = \ln(1+2p\rho) \ .$$
So far we have established the shock state. Effectively, particles near the entrance hop to the right with the probability $W_h^{in}$ or are annihilated with the probability $W_a^{in}(\rho)$. This dynamics results in a density profile $\{\rho_i^{in}\}$ which decays to zero as $i$ increases. Near the exit, particles are created effectively with the probability $W_{c}^{out}(\rho)$ and hop to the right with the probability $W_h^{out}$. This results in a density profile $\{\rho_i^{out}\}$ which converges to unity as $i$ decreases from $L$. Both profiles should be matched at a position $i_S=(1-\rho)L$ to yield a shock.
This situation is similar to the driven 1D ASEP with particle creation and annihilation [@Parmeggiani03; @Evans03; @Juhasz04] or boundary driven multi-lane systems (see e.g. [@reichenbach_f_f06; @Schiffmann_10]). When the creation and annihilation rates are spatially uniform and inversely proportional to the lattice size, the system also develops a shock in the stead-state density profile [@Parmeggiani03; @Evans03; @Juhasz04]. In comparison with the model studied in [@Parmeggiani03; @Evans03; @Juhasz04; @reichenbach_f_f06; @Schiffmann_10], the creation and annihilation rates are not uniform in space: Particles are annihilated near the entrance and created near the exit. The difference results in the feature that the shock separates the empty and the fully jammed domains.
The parameters $W_a^{in}(\rho)$ and $W_c^{out}(\rho)$ depend on the the overall particle density $\rho = 1-i_S/L$, which should be determined self-consistently. The overall density can be obtained from the current conservation, which will be explained in the following subsection.
Phase diagram
-------------
Particles are injected at the entrance ($i=1$), move to the right, and are removed at the exit ($i=L$). So, the system can carry a nonzero current. The incoming current at the entrance and outgoing current at the exit are given by $$\begin{aligned}
J_{in}(\rho) &=& \alpha (1-\rho_1^{in}) \label{Jin} \\
J_{out}(\rho) &=& \beta \rho_L^{out} \ . \label{Jout}\end{aligned}$$ Because $\rho_1^{in}$ and $\rho_L^{out}$ are governed by the $\rho$-dependent effective dynamics, the incoming and the outgoing currents are given as a function of $\rho$. Particle number conservation requires that the incoming and outgoing currents should be the same in the steady state. The equality $J^{in}(\rho) = J^{out}(\rho)$ determines the overall particle density $\rho$, hence the phase diagram.
The effective dynamics is still too complex and does not allow for the closed-form solution for $\rho_i^{in,out}(\rho)$. Therefore, the exact phase diagram will be obtained from numerical solutions of the self-consistent equations and the current-balance condition. Before doing so, we apply an approximate scheme to the self-consistent equations in order to gain a physical insight.
In terms of the effective dynamics, all particles introduced at the entrance have to be transferred toward the exit and therefore annihilated from the entry area. So, the incoming current can also be written as $$\label{Jin_sum}
J_{in}(\rho) = W_a^{in} \sum_{i\geq 1} \rho_i^{in} \ .$$ Using the explicit forms given in (\[R\_in\]) and (\[Q\_in\]), the self-consistent equation for the density becomes as $$\rho_i^{in} = \frac{\rho_{i-1}^{in}}{ 1+2p(1-\rho) + (\rho_{i-1}^{in} -
\rho_{i+1}^{in})}$$ for $i\geq 2$. Because of the continuous annihilation of particles, we expect that $\rho_{i}^{in}$ decays monotonically and rather fast to zero. In order to gain a qualitative understanding we can ignore $(\rho_{i-1}^{in} - \rho_{i+1}^{in})$ in the denominator to obtain that $$\rho_{i}^{in} \simeq \frac{\rho_1^{in}}{ (1+2p(1-\rho))^{i-1} } \ .$$ Inserting these approximate solutions into (\[Jin\_sum\]), one obtains that $$\label{Jin_trial}
J_{in} \simeq \frac{1+2p(1-\rho)}{1+2p}\rho_1^{in}.$$ Comparing the two expressions for $J_{in}$ given in (\[Jin\]) and (\[Jin\_trial\]), one finds a solution for $\rho_1^{in}(\rho)$, which yields that $$\label{Jin_app}
J_{in}(\rho) \simeq \frac{\alpha(1+2p -2p \rho)}
{(1+\alpha)(1+2p) - 2p \rho} \ .$$ It is a decreasing function of $\rho$ with $J_{in}(0) = \frac{\alpha}{1+\alpha}$ and $J_{in}(1) =
\frac{\alpha}{1+\alpha(1+2p)}$.
One can carry out a similar analysis to obtain an approximate expression for $J_{out}(\rho)$. First, the outgoing current given in (\[Jout\]) should be equal to the total particle creation rate, that is to say, $$J_{out} = W_c^{out} \sum_{i\leq L} (1 -\rho_{i}^{out}) \ .$$ It is easy to show that the void density $(1-\rho_i^{out})$ satisfies self-consistent equations $$1-\rho_{i}^{out} = \frac{1-\rho_{i+1}^{out}}{1+\ln(1+2p\rho) +
(\rho_{i-1}^{out}-\rho_{i+1}^{out})}$$ for $i<L$. Again the transport of particles via long ranged links can be understood as spontaneous creation of particles at sites close to the exit. Therefore, we expect that $(1-\rho_i^{out})$ decays to zero as $i$ decreases from $L$. So we can ignore $(\rho_{i-1}^{out}-\rho_{i+1}^{out})$ in the denominator as in the previous case. A similar calculation then yields $$\label{Jout_app}
J_{out}(\rho) \simeq
\frac{\beta(1+\ln(1+2p\rho))}{1+\beta+\ln(1+2p\rho)} \ .$$ This is an increasing function of $\rho$ with $J_{out}(0) = \frac{\beta}{1+\beta}$ and $J_{out}(1) =
\frac{\beta(1+\ln(1+2p))}{1+\beta+\ln(1+2p)}$.
![Schematic plots of $J_{in}$ (solid lines) and $J_{out}$ (dashed lines) against $\rho$.[]{data-label="fig6"}](fig6.eps){width="\columnwidth"}
The particle number conservation requires that the incoming current and the outgoing current should be the same in the steady state. Figure \[fig6\] shows schematic plots of $J^{in}(\rho)$ and $J^{out}(\rho)$ in three different situations. In the first case, the current curves may intersect with each other at $\rho=\rho_0$ with $0<\rho_0<1$ as shown in Figure \[fig6\](a). This case corresponds to the shock phase. The intersection determines the shock position $i_S = (1-\rho_0)L$. If $\rho$ becomes greater (smaller) than $\rho_0$ due a stochastic fluctuation, then the incoming current becomes smaller (greater) than the outgoing current. Consequently, the particle density is attracted toward the steady-state value and the shock is driven toward the steady-state position. This explains the reason why there is a localized sharp shock [@Parmeggiani03; @Evans03; @Juhasz04].
The second case with $J_{in}(\rho) > J_{out}(\rho)$ for all values of $\rho$ is shown in Figure \[fig6\](b). Then, the shock is localized at the entrance and the steady-state density is equal to one. In this case the system belongs to the jammed phase.
The third case is sketched in Figure \[fig6\](c). If $J_{in}(\rho) <
J_{out}(\rho)$ for all values of $\rho$, the shock is localized at the exit and the steady-state density is equal to zero. Then the system belongs to the empty phase.
We have solved numerically exactly the self-consistent equations for $\rho^{in}_i$ and $\rho^{out}_i$ to obtain the incoming and outgoing currents as a function of $\rho$ at each set of values of $\alpha$, $\beta$, and $p$. The current balance condition allows us to evaluate the steady-state particle density and the current, hence the phase diagram. The resulting numerical phase diagram in the $\alpha\beta$ plane is presented in Figure \[fig4\]. The phase diagram at $p=0.2$ consists of the empty phase, jammed phase, and the shock phase, which is consistent with the Monte Carlo result. There is a quantitative discrepancy in the location of the phase boundaries, which is caused by the approximations.
The annealed approximation requires the self-averaging property [@Wiseman98; @Roy06]. The self-averaging property in the small-world network was tested for the equilibrium Ising model [@Roy06]. To test the self-averaging property in our nonequilibrium model, we have measured the relative sample-to-sample fluctuation of the particle density $X\equiv \sqrt{ \left(
[\langle \frac{1}{L}\sum_i n_i\rangle^2]_{\mathcal G}
- [\langle \frac{1}{L} \sum_i n_i \rangle]_{\mathcal G}^2 \right)} /
[\langle \frac{1}{L} \sum_i n_i \rangle]_{\mathcal G}$. In the E and J phases, it decays to zero as $X \sim L^{-1/2}$ suggesting a strong self-averaging [@Wiseman98]. On the other hand, it converges to a finite value in the S phase, an indication of non-self-averaging. This tells us that the annealed approximation has a limitation. It explains successfully the mechanism of shock formation, but not its average position due to the strong fluctuations. A refined approach beyond the annealed approximation is necessary to study the sample-to-sample fluctuation phenomena in the S phase.
Summary and Conclusion {#sec5}
======================
We have studied the boundary-driven ASEP in the 1D chain with long-range links. This setup represents a generic situation for directed transport in a complex network, e.g., the exchange of data between two sites of a computer network.
The backbone of nearest neighbor links ensures the existence of a path between start and destination. The long ranged links add shortcuts to the transport network that are in principle able to enhance the capacity of the system.
Considering the typical results for transport problems on complex networks one would also expect that long-range shortcuts, added randomly on to a lattice, are believed to suppress fluctuations and make a system homogeneous. This is, however, not the case when there is a boundary driving. Compared to the pure one-dimensional system it turns out that the long-range links play an essential role. They generate a localized shock which separates the 1D chain backbone into an empty and jammed regions.
Adopting the mean field and the annealed approximations, we have derived effective dynamics near the entrance and the exit, which are similar to those of the ASEP with spontaneous particle annihilation and creation, respectively. The effective theory reveals the mechanism for the shock formation and for the phase transition. The phase diagram consists of the empty, shock, and jammed phases. The shock phase is characterized by presence of a localized shock and separating a low and a high density domain. The shock position and the overall particle density $\rho$ vary continuously with the model-parameters. In the empty (jammed) phase, the shock is anchored to the exit (entrance) to yield $\rho=0~(1)$.
In conclusion, our study shows that a driven system on a spatially disordered structure displays an inhomogeneous pattern. The appearance of a localized shock is reminiscent of one-dimensional systems without particle-conservation in the bulk. In contrast to these systems the mechanism driving the localization of the shock is neither the competition between bulk and boundary reservoirs nor an optimal partitioning between multiple lanes. Here, the long-ranged links enhance the mass transfer between entry and exit and thereby stabilize the position of the shock.
Considering more generally the transport capacity of a complex network between arbitrary sites our results have important consequences. They indicate that, as far as the capacity of the feeding particle reservoir does not exceed the capacity of the exit reservoir, the sites of the backbone are only rarely occupied and can be used in parallel for transport issues between other sites. Contrary, overfeeding the backbone leads to a complete blockage of the sites in question which may spread over the whole network.
This work was supported by Mid-career Researcher Program through NRF grant (No. 2010-0013903) funded by the MEST.
References {#references .unnumbered}
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|
---
abstract: 'We have studied the impact of chemical composition on the spectrum of stellar population formed in the starburst. The range of metallicities used - $Z=0.0001\div 0.05-0.1$ covers all the compositions observed; depending on age, wavelength region and IMF used the UV-fluxes differ between extreme compositions [*at least*]{} tenfold, usually more. Independently of age, UV-fluxes shortwards of Balmer jump monotonically decline with the growth of metallicity. Optical and especially IR-region are also influenced, at $> 1$ $\mu$m fluxes typically differ $10 \div 50$ times. In IR the actually brightest emitting composition gets reversed with age.'
author:
- Peeter Traat
title: Metallicity Dependence of Starburst Spectra
---
Starbursts express the ununiformity and complexity of the real global star formation process in galaxies. They are marked by highly enhanced star formation activity and presence of a numerous very young massive star generation. When having an usual, solar-vicinity-like bottom-heavy IMF they cannot be sustained by the gas resources for long. Starbursting galaxies are of diverse sizes and range in composition from about solar to very metal-poor small objects; given patchiness of star formation process and the role of separate starbursts seems to grow towards smaller, lower-mass galaxies. Some very blue lowest mass nearby dwarfs have yet kept their matter nearly pristine (extreme cases SBS 0335-052 and IZw18 are 41 and 50 times undersolar in metals, Thuan [*et al.*]{} 1997) and are probably undergoing the very first substantial star formation event in their history.
Starburst properties get quantified with the use of spectral models. We have here estimated effects of chemical composition on the composite spectrum of formng stellar component on the basis of youngest models in our multicomposite (8 compositions $Z=0.0001\div$0.1, age range 4 Myr$\div$20 Gyr, in $Z$=0.1 models 10 Myr$\div$13 Gyr, 6 IMF slopes, 21 SFR combinations, no gas-dust absorption-reemission) grid of spectrophotometric evolutionary models (Traat 1996). The resulting spectral flux distributions for the standard IMF are given in Fig., heavy line in all panels is the reference $Z$=0.02 curve. The general trend emerging is the progressive growth of absorption and decline of flux towards shorter UV-wavelengths with increasing metallicity, with differences in Balmer continuum slowly growing from $\sim$2 right below Balmer limit to $\sim$5 near Lyman jump at 912Å, progressing also somewhat with the age of stellar generation itself. In the Lyman continuum, fluxes differ more, about 10-20 times at the age 4 Myr, its drop-off with time is much more rapid for compositions richer in metals. At the age 10 Myr, flux output of $Z$=0.1 population is lower than that of $Z$=0.0001 stars by $\sim$100 at the base of Lyman jump and $\sim$6000 at the edge of He 504Åbreak. Flux emitted in He is weakly present in very young populations with higher metallicity, as in the illustrated 4 Myr case, but very rapidly fades away. Somewhat surprisingly metal-weak populations are at very early ages, due to the earlier appearance of red supergiants also in the IR brighter than near-solar metallicity stars. This situation gets quickly reversed with aging. The possibility suspected in some starbursts - the preferential formation of massive stars what lowers the IMF slope $n$ - has also its own but easily accountable impact on the resulting spectrum, namely, it reduces the output in IR and rises UV the more the smaller the $n$ value is.
Thuan, T.X., Izotov, Y.I., Lipovetsky, V.A. 1997, , 477, 661 Traat, P. 1996, [*ASP Conf. Series*]{}, 98, 36
|
---
abstract: |
Thompson’s group $F$ is the group of all increasing dyadic PL homeomorphisms of the closed unit interval. We compute $\Sigma^m(F)$ and $\Sigma^m(F;{\mathbb{Z}})$, the homotopical and homological Bieri-Neumann-Strebel-Renz invariants of $F$, and show that $\Sigma^m(F)
= \Sigma^m(F;{\mathbb{Z}})$. As an application, we show that, for every $m$, $F$ has subgroups of type $F_{m-1}$ which are not of type $FP_{m}$ (thus certainly not of type $F_m$).
address: 'Department of Mathematics, Johann Wolfgang Goethe-Universität Frankfurt, D-60054 Frankfurt am Main, Germany Department of Mathematical Sciences, Binghamton University (SUNY), Binghamton, NY 13902-6000, USA Department of Mathematics, State University of Campinas, Cx. P. 6065, 13083-970 Campinas, SP, Brazil'
author:
- 'Robert Bieri, Ross Geoghegan, Dessislava H. Kochloukova'
date: 'February, 2008'
title: 'The Sigma invariants of Thompson’s Group $F$'
---
[^1]
Introduction
============
The group $F$
-------------
Let $F$ denote the group of all increasing piecewise linear (PL) homeomorphisms[^2] $$x: [0,1] \to [0,1]$$ whose points of non-differentiability $\in [0,1]$ are dyadic rational numbers, and whose derivatives are integer powers of 2. This is known as Thompson’s Group $F$; it first appeared in [@McKenzie].
The group $F$ has an infinite presentation $$\label{*11}
\langle x_0, x_1, x_2, \cdots \mid x_i^{-1} x_n x_i= x_{n+1}\hbox{
for } 0 \leq i< n\rangle$$
Let $F(i)$ denote the subgroup $\langle x_i, x_{i+1}, \ldots \rangle$. The presentation (\[\*11\]) displays $F$ as an HNN extension with base group $F(1)$, associated subgroups $F(1)$ and $F(2)$, and stable letter $x_0$ ; see [@Ross Prop. 9.2.5] or [@RossKen] for a proof. Thus $F$ is an ascending[^3] HNN-extension whose base and associated subgroups are isomorphic to $F$.
The correspondence between the generators $x_i$ in the presentation (\[\*11\]) and PL homeomorphisms is as in [@diagram]. For example, the generator $x_0$ corresponds to the PL homeomorphism with slope $\frac{1}{2}$ on $[0,\frac{1}{2}]$, slope $1$ on $[\frac{1}{2},\frac{3}{4}]$, and slope $2$ on $[\frac{3}{4},1]$.
The group $F$ has type $F_{\infty}$ i.e. there is a $K(F,1)$-complex with a finite number of cells in each dimension [@RossKen]. Therefore $F$ is finitely presented and has type $FP_{\infty}$. Furthermore, $F$ has infinite cohomological dimension [@RossKen], $H^{*}(F,
{\mathbb Z}F)$ is trivial [@RossKen2], $F$ does not contain a free subgroup of rank 2 [@B-S], and the commutator subgroup $F'$ is simple [@Brown2], [@diagram]. It is known that $F$ has quadratic Dehn function [@Guba]. The group of automorphisms of $F$ was calculated in [@Brin].
The Sigma invariants of a group {#review}
-------------------------------
By a [*(real) character*]{} on $G$ we mean a homomorphism $\chi : G \to {\mathbb{R}}$ to the additive group of real numbers. For a finitely generated group $G$ the [*character sphere*]{} $S(G)$ of $G$ is the set of equivalence classes of non-zero characters modulo positive multiplication. This is best thought of as the “sphere at infinity" of the real vector space $Hom(G,{\mathbb{R}})$. The dimension $d$ of that vector space is the torsion-free rank of $G/G'$, and the sphere at infinity has dimension $d-1$. We denote by $[\chi ]$ the point of $S(G)$ corresponding to $\chi $.
We recall the Bieri-Neumann-Strebel-Renz (or Sigma) invariants of a group $G$. Let $R$ denote a commutative ring[^4] with $1\neq 0$, and let $m\geq 0$ be an integer. When $G$ is of type $F_m$ (resp. $FP_{m}(R)$) the homotopical invariant $\Sigma^m(G)$ (resp. the homological invariant $\Sigma^m(G;R)$), is a subset of $S(G)$. In both cases we have $\Sigma^{m+1}\subseteq \Sigma^{m}$. We refer the reader to [@Renz] for the precise definition, confining ourselves here to a brief recollection:
### $m=0$
All groups have type $F_0$ and type $FP_{0}(R)$. By definition $\Sigma^{0}(G)=\Sigma^{0}(G;R)=S(G)$. This will only be of interest when we consider subgroups of $F$ in Section \[Subgroups\].
### $m=1$
Let $X$ be a finite set of generators of $G$ and let $\Gamma ^1$ be the corresponding Cayley graph, with $G$ acting freely on $\Gamma ^1$ on the left. The vertices of $\Gamma ^1$ are the elements of $G$ and there is an edge joining the vertex $g$ to the vertex $gx$ or each $x\in X$.
For any non-zero character $\chi : G \to {\mathbb{R}}$, and for any real number $i$ define $\Gamma ^{1}_{\chi \geq i}$ to be the subgraph of $\Gamma$ spanned by the vertices $$G_{\chi \geq i } = \{ g \in G \mid \chi(g) \geq i \}.$$ By definition, $[\chi] \in \Sigma^1(G)$ if and only if $\Gamma ^{1}_{\chi
\geq 0}$ is connected. For a detailed treatment of $\Sigma ^1$ from a topological point of view, see [@Ross Sec. 16.3].
### $m=2$
Let $\langle X \mid T \rangle $ be a finite presentation of $G$. Choose a $G$-invariant orientation for each edge of $\Gamma ^1$ and then form the corresponding Cayley complex $\Gamma ^2$ by attaching 2-cells equivariantly to $\Gamma ^1$ using attaching maps indicated by the relations in $T$. Define $\Gamma ^{2}_{\chi \geq i }$ to be the subcomplex of $\Gamma ^2$ consisting of $\Gamma ^{1}_{\chi
\geq i}$ together with all the 2-cells which are attached to it.
By definition, $[\chi] \in \Sigma^2(G)$ if and only if $[\chi ]\in \Sigma^{1}(G)$ and there is a nonpositive $d$ such that the map $$\label{desi100}
\pi_1(\Gamma ^{2}_{\chi \geq 0}) \to \pi_1(\Gamma ^{2}_{\chi \geq d }),$$ induced by the inclusion of spaces $\Gamma ^{2}_{\chi \geq
0} \subseteq \Gamma ^{2}_{\chi \geq d}$ is zero (and $\Gamma ^{1}_{\chi
\geq 0}$ is connected). See, for example, [@Renz3]. Note that $\Gamma ^2$ is the $2$-skeleton of the universal cover of a $K(G,1)$-complex which has finite $2$-skeleton.
### $m>2$
The higher $\Sigma^m(G)$ are defined similarly, for groups of type $F_m$, using the $m$-skeleton, $\Gamma ^{m}$, of the universal cover of a $K(G,1)$-complex having finite $m$-skeleton. See [@Renz].
### The homological case
For a commutative ring $R$, the homological Sigma invariants $\Sigma^{m}(G;R)$ are defined similarly when the group $G$ is of type $FP_{m}(R)$, using a free resolution of the trivial (left) $RG$-module $R$ which is finitely generated in dimensions $\leq m$; see [@Renz] for details. Among the basic facts to be used below, which hold for all rings $R$, are: $\Sigma^{1}(G)=\Sigma^{1}(G;R)$; and $\Sigma^{m}(G)\subseteq \Sigma^{m}(G;R)$ when both are defined (i.e. when $G$ has type $F_m$.) If $G$ is finitely presented then “type $F_m$" and “type $FP_m({\mathbb{Z}})$" coincide. In that case, $\Sigma^m(G;{\mathbb{Z}})$ can also be understood from the above topological definition of $\Sigma^m(G)$, replacing statements about homotopy groups by the analogous statements about reduced ${\mathbb{Z}}$-homology groups; more precisely, one requires $$\label{desi101} \tilde
{H}_{k-1}(\Gamma ^{k}_{\chi \geq 0}) \to \tilde{H}_{k-1}(\Gamma ^{k}_{\chi
\geq d }),$$ to be trivial for all $k\leq m$.
[**Remark:**]{} The definition of $\Sigma^1$ given here agrees with the now-established conventions followed, for example, in [@Renz] and in [@RossBieri]. It differs by a sign from the $\Sigma^1$-invariant defined in [@Bieri]. This arises from our convention that $RG$-modules are left modules, while in [@Bieri] they are right modules.
Some facts about Sigma invariants {#conv}
---------------------------------
It is convenient to write “$[\chi ] \in \Sigma^{\infty}$" as an abbreviation for “$[\chi ] \in \Sigma^{m}$ for all $m$".
Among the principal results of $\Sigma$-theory for a group $G$ of type $F_m$ (resp. type $FP_{m}(R)$) are: (1) $\Sigma^m(G)$ (resp. $\Sigma^m(G;R)$) is an open subset of the character sphere $S(G)$, and (2) $\Sigma^m(G)$ (resp. $\Sigma^m(G;R)$) classifies all normal subgroups $N$ of $G$ containing the commutator subgroup $G'$ by their finiteness properties in the following sense:
\[renzbieri\] [@Renz], [@Renz2], [@Renz3] Let $G$ be a group of type $F_m$ (resp. type $FP_m(R)$) with a normal subgroup $N$ such that $G/N$ is abelian. Then $N$ is of type $F_m$ (resp. $FP_m$) if and only if for every non-zero character $\chi$ of $G$ such that $\chi(N) = 0$ we have $[\chi] \in \Sigma^m(G)$ (resp. $[\chi] \in \Sigma^m(G;R)$).
A non-zero character is [*discrete*]{} if its image in ${\mathbb{R}}$ is an infinite cyclic subgroup. A special case of Theorem \[renzbieri\] (the only one we will use) is:
\[renzbiericor\] If the non-zero character $\chi$ is discrete then its kernel has type $F_m$ (resp. type $FP_{m}(R)$) if and only if $[\chi ]$ and $[-\chi ]$ lie in $\Sigma^m(G)$ (resp. $\Sigma^m(G;R)$).
The invariants $\Sigma^m(G)$ and $\Sigma^m(G;R)$ have been calculated for only a few families of groups $G$, even fewer when $m>1$. For metabelian groups $G$ of type $F_m$ there is the still-open $\Sigma^m$-Conjecture: $\Sigma^m(G)^c =\Sigma^m(G;{\mathbb{Z}})^c =conv_{\leq m}\Sigma^1(G)^c$, where[^5] $conv_{\leq m}$ denotes the union of the (spherical) convex hulls of all $\leq m$-tuples; this is known for $m =
2$ [@Desi1] but only for larger $m$ under strong restrictions on $G$ [@Desi2], [@Meinert]. A complete description of $\Sigma^m(G)$ and $\Sigma^m(G;{\mathbb{Z}})$ for any right angled Artin group $G$ is given in [@Meier]. Recently the homotopical invariant $\Sigma^m(G)$ has been generalized to an invariant of group actions on proper CAT(0) metric spaces [@RossBieri]; the corresponding invariants for the natural action of $SL_{n}({\mathbb{R}})$ on its symmetric space have been calculated: for $n=2$ (action by Möbius transformations on the hyperbolic plane) in [@RossBieri2], and for $n>2$ in [@Rehn]. A similar generalization of the homological case, $\Sigma^m(G;R)$, to the CAT(0) setting will appear in [@RossBieri4].
Sigma invariants of $F$
-----------------------
In this paper we calculate the Sigma invariants $\Sigma^m(F)$ and $\Sigma^m(F;R)$ of the group $F$. For $x\in F$ and $i=0$ or $1$ let $\chi _{i}(x) := log_{2}x'(i)$, i.e. the (right) derivative of the map $x$ at 0 is $2^{\chi _{0}(x)}$ and the (left) derivative of $x$ at 1 is $2^{\chi _{1}(x)}$. In terms of the presentation (\[\*11\]) $\chi_{0}(x_0)=-1$ and $\chi_{0}(x_i)=0$ for $i\geq 1$, while $\chi_{1}(x_i)=1$ for all $i\geq 0$. These two characters are linearly independent. Thus $[\chi _{0}]$ and $[\chi _{1}]$ are not antipodal points of the circle $S(F)$. From (\[\*11\]) we see that the real vector space $Hom(F,{\mathbb{R}})$ has dimension 2, so these two characters span $Hom(F,{\mathbb{R}})$. It follows that the convex sum of $[\chi
_{0}]$ and $[\chi _{1}]$ is a well-defined interval in the circle $S(F)$; it members are the points $\{[a\chi _{0}+b\chi _{1}]\:|\:a,b>0\}$. We call it the “shorter interval". We call $\chi_0$ and $\chi_1$ the “special" characters.
There is a useful automorphism $\nu$ of $F$ which is most easily expressed when $F$ is regarded as a group of PL homeomorphisms as above: it is conjugation by the homeomorphism $t \mapsto (1-t)$; if one draws the graph of the PL homeomorphism $x\in F$ in the square $[0,1]\times [0,1]$ then the graph of $\nu (x)$ is obtained by rotating that square through the angle $\pi$. This $\nu$ induces an automorphism of $Hom(F, {\mathbb{R}})$ and consequently an automorphism of $S(F)$ which permutes the elements of $\Sigma^m(F)$ (resp. $\Sigma^m(F;R)$). In particular, it swaps the points $[\chi_0]$ and $[\chi_1]$. We refer to this as “$\nu$-symmetry" of the Sigma invariants.
The Theorems of this paper can now be stated:
One part of this is not new: $\Sigma ^{1}(F)$ was computed in [@Bieri].
Theorem A is proved in Section \[Proof\], and Theorem B is proved (using [@RossBieri3]) in Section \[Subgroups\].
[**Acknowledgment**]{} We thank Dan Farley who asked about the possibility of embedding powers of $F$ in $F$ to get non-normal subgroups of $F$ with more interesting finiteness properties than can be found among the kernels of characters on $F$ itself. His question led to the writing of the paper [@RossBieri3] and thus to our Theorem B.
Proof of Theorem A {#Proof}
==================
$\Sigma^0$ and $\Sigma^1$ {#0and1}
-------------------------
By an [*ascending*]{} HNN extension we mean a group presented by $\langle
H,t|t^{-1} h t=\phi (h)$ for $h\in H\rangle$ where $\phi:H\to H$ is a monomorphism. Such a group is denoted by $H*_{\phi, t}$.
We begin by citing:
\[HNN\] Let $G$ decompose as an ascending HNN extension $H*_{\phi, t}$. Let $\chi :G \to {\mathbb{R}}$ be the character given by $\chi(H) = 0$ and $\chi(t) = 1$.
1. If $H$ is of type $F_m$ (resp. $FP_{m}(R)$) then $[\chi] \in
\Sigma^{m}(G)$ (resp. $[\chi] \in \Sigma^m(G;R)$).
2. If $H$ is finitely generated and $\phi$ is not onto $H$ then $[-
\chi]\in \Sigma^{1}(G)^c$.
The homological case of (1) for all $m$ is [@Meinert Prop. 4.2] and the homotopical case for $m=2$ is a special case of [@Meinert2 Thm. 4.3]. The homotopical case of (1) for all $m$ then follows.
\(2) is elementary: we recall the argument. Let $N$ be the kernel of $\chi$. By (1) and Corollary \[renzbiericor\], (2) is equivalent to claiming that the group $N$ is not finitely generated. The hypothesis that $\phi$ is not onto implies $t^{-1}Ht$ is a proper subgroup of $H$. Thus $N=\cup _{n\geq 1}t^{n}Ht^{-n}$ is a proper ascending union, so it cannot be finitely generated.
Applying Theorem \[HNN\] together with “$\nu$-symmetry" to the group $F$, i.e. $G = F$, $t = x_0$, $H = F(1)$, and $\chi = - \chi_{0}$, we get part of Theorem A:
\[CorB\] $\{[- \chi_0], [- \chi_1]\}\subseteq
\Sigma^{\infty}(F)$ and $\{[\chi_0], [\chi_1]\}\subseteq
\Sigma^{1}(F)^c$.
Theorem 8.1 of [@Bieri] is the assertion that the complement of the two-point set $\{[\chi_0], [\chi_1]\}$ is precisely[^6] $\Sigma^{1}(F)$.
The “longer" interval
---------------------
The following is proved by combining two theorems of H. Meinert, namely [@Meinert Prop. 4.1] and [@Meinert2 Thm. B]:
Let $G$ decompose as an ascending HNN extension $H*_{\phi,
t}$. Let $\chi:G\to {\mathbb{R}}$ be a character such that $\chi |H\neq 0$. If $H$ is of type $F_{\infty }$ and if $[\chi |H]\in \Sigma^{\infty}(H)$ then $[\chi]\in \Sigma^{\infty}(G)$.
We use this to show that whenever $\chi : F \to {\mathbb{R}}$ is such that $\chi(x_1) < 0$ we always have $[\chi] \in \Sigma^{\infty}(F)$. Recall that $F$ is an HNN extension with base group $F(1) = \langle
x_1, x_2,\ldots \rangle$, associated subgroups $F(1)$ and $F(2)$ and with stable letter $x_0$, where $F(i) = \langle x_i, x_{i+1}, \ldots
\rangle$. As $\{ x_i \}_{i \geq 1}$ are conjugate in $F$ we see that $\chi(x_1) = \chi(x_i)<0 $ for all $i \geq 1$. Let $\widetilde{\chi}$ be the restriction of $\chi$ to $F(1)$. If we identify $F(1)$ with $F$ via the isomorphism that sends $x_i$ to $x_{i-1}$ for $i \geq 1$ , then $\widetilde{\chi}$ gets identified with $- \chi_1$ and, by Corollary \[CorB\], $[- \chi_1] \in \Sigma^{\infty}(F)$. Thus we have:
$$\{ [\chi] \in S(F) \mid \chi(x_1) < 0 \} \subseteq \Sigma^{\infty}(F).$$
This shows that the open interval in the circle $S(F)$ from $[\chi _0]$ to $[-\chi _0]$ which contains $[-\chi _1]$ lies in $\Sigma^{\infty}(F)$. By $\nu$-symmetry its image under $\nu$ has the same property, and this enlarges the interval in question to cover the whole “long" open interval between $[\chi _0]$ and $[\chi _1]$. In summary:
\[Meinertcor\] All of $S(F)$ except possibly the closed convex sum of the points $[\chi _0]$ and $[\chi _1]$ lies in $\Sigma^{\infty}(F)$.
The “shorter" interval
----------------------
For the homotopical version of Theorem A we could simply apply the following:
\[desithm\][@Desi] [*Let $G$ be a finitely presented group which has no free non-abelian subgroup. Then[^7] $conv_{\leq 2}\Sigma^1(G)^c
\subseteq \Sigma^2(G)^c$.*]{}
However, the homological version of Theorem \[desithm\] is only known under restrictive conditions, so we proceed in a manner which handles the homotopical and homological versions at the same time. We begin by citing:
\[bieristrebel\] Let $G$ have no non-abelian free subgroups and have type $FP_{2}(R)$. Let $\tilde \chi:G\to {\mathbb{R}}$ be a non-zero discrete character. Then $G$ decomposes as an ascending HNN extension $H*_{\phi,t}$ where $H$ is a finitely generated subgroup of ker$(\tilde \chi)$, and $\tilde \chi (t)$ generates the image of $\tilde \chi$.
This is an immediate consequence of [@BieriStrebel Thm. A]. That theorem yields an HNN extension, and the hypothesis about free subgroups ensures it is an ascending HNN extension[^8].
We apply Theorem \[bieristrebel\] to understand $\Sigma^2(F;R)$. Consider the non-zero character $a\chi _{0}+b\chi _1$ where $a, b\in {\mathbb{Q}}$. Let $G:=$ker$(a\chi _{0}+b\chi _1)$. Since $F/F'$ is a free abelian group of rank 2, it is not hard to see that $G=\langle
F',t\rangle$ for some $t\in F$. For the same reason, there is a non-zero discrete character $\tilde \chi :G\to {\mathbb{R}}$ whose kernel is $F'$ such that $\tilde \chi (t)$ generates $im(\tilde \chi )$. We assume that $G$ has type $FP_{2}(R)$ and we consider what this implies. By Theorem \[bieristrebel\] the existence of $\tilde \chi$ implies that $G$ decomposes as $H*_{\phi, t}$ where $H$ is a finitely generated subgroup of $F'$. The group $F'$ consists of all PL homeomorphisms whose left and right slopes are 1. Since $H$ is finitely generated, there must exist $\epsilon >0$ such that all elements of $H$ are supported in the interval $[\epsilon , 1-\epsilon ]$. We may assume $\epsilon$ is so small that the PL homeomorphism $t$ is linear on $[0, \epsilon ]$ and on $[1-\epsilon ,1]$.
The character $\tilde \chi $ expresses $G$ as a semidirect product of $F'$ and ${\mathbb{Z}}$. Thus we have $F'=\cup_{n\geq 1}t^{n}Ht^{-n}$. So for each $x\in F'$ there is some $n>0$ such that $t^{-n}xt^{n}\in H$, and hence the support of $t^{-n}xt^{n}$ lies in $[\epsilon, 1-\epsilon ]$.
This implies that the support of $x$ lies in $[t^{n}(\epsilon ), t^{n}(1-\epsilon)]$, and hence these end points have subsequences converging to 0 and 1 respectively as $x$ varies in $F'$. If $t$ has slope $\geq 1$ on $[0,\epsilon ]$ then $t(\epsilon ) \geq \epsilon $ so $t^{n}(\epsilon ) \geq \epsilon $ for all $n>0$. Therefore $t$ must have slope $<1$ near 0. Similarly $t$ must have slope $<1$ near 1. Since $a\chi _{0}(t)+b\chi _{1}(t)=0$ it follows that (still assuming $G$ has type $FP_{2}(R)$) $ab<0$. Expressing the contrapositive, we have
If $ab>0$ then ker$(a\chi _{0}+b\chi _{1})$ does not have type $FP_{2}(R)$. $\square$
Now assume $a$ and $b$ are positive and rational. Write $\chi =a\chi _{0}+b\chi _{1}$; thus $\chi $ is discrete. By Corollary \[renzbiericor\], ker$(\chi )$ has type $FP_{2}(R)$ if and only if both $[\chi ]$ and $[- \chi ]$ lie in $\Sigma^2(F;R)$. But by Proposition \[Meinertcor\] $[- \chi ]\in \Sigma^{2}(F;R)$. So $[\chi ]$ cannot lie in $\Sigma^{2}(F;R)$.
\[hull\] No point in the open convex sum of $[\chi _0]$ and $[\chi _1]$ (i.e. the shorter open interval) lies in $\Sigma^{2}(F;R)$.
We have just shown that a dense subset of the open convex sum lies in $\Sigma^{2}(F;R)^c$, and since $\Sigma^{2}(F;R)$ is open in $S(F)$ this is enough.
The proof of Theorem A is completed by recalling that for any ring $R$
1. $\Sigma^{1}(F;R)=\Sigma^{1}(F)$, and
2. $\Sigma^m(F)\subseteq \Sigma^m(F;R).$
Subgroups of $F$ with different finiteness properties {#Subgroups}
=====================================================
As before, we denote the complement of any subset $A$ of a sphere by $A^c$. The Direct Product Formula for homological Sigma invariants (which is not always true) reads as follows:
$$\Sigma^n(G\times H; R)^c =\bigcup^n_{p=0} \Sigma^p(G;R)^c *\Sigma^{n-p} (H;R)^c$$
Here, $*$ refers to “join" of subsets of the spheres $S(G)$ and $S(H)$ which are considered to be subspheres of the sphere $S(G\times H)$. In particular, when $p=0\text{ or }n$ one of these sets is empty, and then the join is treated in the usual way: e.g., $A*\emptyset =A$.
It has been known for many years that one inclusion of the Direct Product Formula is always true:
(Meinert’s Inequality)\[Meinert\] $$\Sigma^n(G\times H; R)^c \subseteq \bigcup^n_{p=0} \Sigma^p(G;R)^c *
\Sigma^{n-p} (H;R)^c$$ and $$\Sigma^n(G\times H)^c \subseteq \bigcup^n_{p=0} \Sigma^p(G)^c *\Sigma^{n-p} (H)^c$$
Meinert did not publish this, but a proof can be found in [@Gehrke2 Section 9]. The paper [@Bieri2] also contains a proof of the homotopy version.
It is proved in [@RossBieri3] that the Direct Product Formula holds when $R$ is a field. On the other hand, an example in [@Schuetz] shows that the Formula does not always hold when $R={\mathbb{Z}}$. However, it is shown in [@RossBieri3] that when $\Sigma^n(G;{\mathbb{Z}})= \Sigma^n(G;{\mathbb{Q}})$ for all $n$ then the Direct Product Formula does hold when $R={\mathbb{Z}}$. Writing $F^r$ for the $r$-fold direct product of copies of $F$, one concludes (by induction on $r$) that the Formula holds for $F^r$ when $R={\mathbb{Z}}$. More precisely, we have:
\[join\] Let $r\geq 2$. Then for all $n$ $$\Sigma^n(F^{r};{\mathbb{Z}})^c =\bigcup^n_{p=0} \Sigma^p(F;{\mathbb{Z}})^c *\Sigma^{n-p}(F^{r-1};{\mathbb{Z}})^c$$ and $\Sigma^{n}(F^{r})=\Sigma^{n}(F^{r};{\mathbb{Z}})$.
Only the last sentence requires some explanation. It follows from Meinert’s Inequality (Theorem \[Meinert\]) together with the fact that for any group $G$ we have $\Sigma^{m}(G)\subseteq \Sigma^{m}(G;R)$.
Theorem A implies that $\Sigma^{m}(F)^{c}$ is a (spherical) 1-simplex if $m\geq 2$, is the 0-skeleton of that 1-simplex when $m=1$, and is empty (i.e., the (-1)-skeleton of the 1-simplex) when $m=0$. And that 1-simplex has the property that it is disjoint from its negative. It follows from Theorem \[join\] that $\Sigma^{m}(F^{r})^{c}$ is the $(m-1)$-skeleton of a spherical $(2r-1)$-simplex in the $(2r-1)$-sphere $S(F^r)$, a simplex which is disjoint from its negative.
We now prove Theorem B. Consider $[\chi]$ in $S(F^r)$ which lies in the $(m-1)$-skeleton but not in the $(m-2)$-skeleton of the $(2r-1)$-simplex. Since the discrete characters are dense we can always choose $\chi $ discrete. Then $[\chi
]$ lies in $\Sigma^{m}(F^{r})^{c}\cap \Sigma^{m-1}(F^{r})$ while $[-\chi
]$ lies in $\Sigma^{m}(F^{r})$. Thus, by Corollary \[renzbiericor\], the kernel of $\chi $ has type $F_{m-1}$ but not type $FP_{m}({\mathbb{Z}})$ when $m<2r-1$. Now, $F$ contains copies of $F^r$ for all $r$; for example, let $0<t_{1}<\cdots <t_{r-1}<1$ be a subdivison of $[0,1]$ into $r$ segments where the subdivision points are dyadic rationals. The subgroup of $F$ which fixes all the points $t_i$ is a copy of $F^r$. Thus Theorem B is proved.
[**Example:**]{} Here is an explicitly described subgroup $G_r\leq F$ which has type $F_{2r-1}$ but does not type $FP_{2r}({\mathbb{Z}})$. Fix a dyadic subdivision of $[0,1]$ into $r$ subintervals as above. Let $G_r$ denote the subgroup of $F$ consisting of all elements $x$ for which the product of the numbers in the following set $D_r$ equals 1. The members of $D_r$ are: the left and right derivatives of $x$ at the $(r-1)$ subdivision points $t_i$, the right derivative of $x$ at 0, and the left derivative of $x$ at 1. This subgroup of $F$ (we consider $F^r$ embedded in $F$ as above) corresponds to the barycenter of the $(2r-1)$-simplex, and thus has the claimed properties.
[This example is “structurally stable" in the following sense: The interior of the $(2r-1)$-simplex is open in the sphere $S(F^r)$. Thus all the points in that open set which correspond to discrete characters on $F^r$ (they are dense) give rise to groups $\tilde
G_r$ with exactly the finiteness properties possessed by $G_r$. These groups $\tilde G_r$ should be thought of as all the normal subgroups of $F^r$ “near" $G_r$ which have infinite cyclic quotients.]{}
[99]{}
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[^1]: The third author is partially supported by “bolsa de produtividade de pesquisa" from CNPq, Brazil.
[^2]: Here, PL homeomorphisms are understood to act on $[0,1]$ on the left as in [@diagram] rather than on the right as in [@RossKen].
[^3]: See Subsection \[0and1\] for the definition.
[^4]: Only the rings ${\mathbb{Z}}$ and ${\mathbb{Q}}$ will play a role in this paper.
[^5]: It is customary to use the notation $A^c$ for the complement of the set $A$ in a character sphere; e.g. $\Sigma^{m}(G)^c$ or $\Sigma^m(G;R)^c$.
[^6]: But note the change of conventions explained in the Remark at the end of Subsection \[review\].
[^7]: See Sec. \[conv\] for the definition of $conv_{\leq 2}$.
[^8]: The equivalence of “almost finitely presented" with respect to $R$, the term actually used in [@BieriStrebel], and $FP_{2}(R)$ is well-known: see, for example, Exercise 3 of [@Brown3 VIII 5].
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